Error Logic: Paving Pathways for Intelligent Error Identification and Management: Programming and Interfacing 3031008197, 9783031008191

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Studies in Systems, Decision and Control 442

Shiyong Liu Kaizhong Guo

Error Logic: Paving Pathways for Intelligent Error Identification and Management

Studies in Systems, Decision and Control Volume 442

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Shiyong Liu · Kaizhong Guo

Error Logic: Paving Pathways for Intelligent Error Identification and Management

Shiyong Liu Beijing Normal University at Zhuhai Zhuhai, China

Kaizhong Guo Guangzhou Vocational and Technical University of Science and Technology Guangdong University of Technology Guangzhou, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-00819-1 ISBN 978-3-031-00820-7 (eBook) https://doi.org/10.1007/978-3-031-00820-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our families and friends who have been giving generous emotional and material support on the journey of writing this book, we couldn’t have done this without you. To my wife Grace, and two lovely sons Shuren and Shuyi, for showing what love really means.

Contents

1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 7

2 Short Introduction on the Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Major Theories in Modern Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Origin of Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Contents of Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Research History of Mathematical Logic . . . . . . . . . . . . . . . 2.1.4 Development of Mathematical Logic . . . . . . . . . . . . . . . . . . 2.1.5 Research Status of Mathematical Logic . . . . . . . . . . . . . . . . 2.1.6 Formalism of Dialectical Logic in China . . . . . . . . . . . . . . . 2.2 Error Logical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Existential Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 16 17 18 19 25 26 29 30 30

3 Error Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Concepts of Error Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Error Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Operations of Error Sets and Their Laws . . . . . . . . . . . . . . . 3.2 Transformation of Error Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Definition of Transformation of Error Sets . . . . . . . . . . . . . . 3.2.2 Transformation Operations of Error Sets . . . . . . . . . . . . . . . 3.2.3 Types and Operation Rules of Transformation . . . . . . . . . . . 3.2.4 Transformation and Elimination of Errors . . . . . . . . . . . . . . 3.3 Classic Error Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Concepts of Classic Error Set . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Categories of Classic Error Set . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Operations of Classic Error Set and Their Laws . . . . . . . . . 3.4 Fuzzy Error Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Concepts of Fuzzy Error Set . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Operations of Fuzzy Error Set and Their Laws . . . . . . . . . . 3.4.3 Error Set with Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Multivariate Error Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 37 40 42 42 44 44 56 57 57 58 59 59 60 61 62 62 vii

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Contents

3.5.1 3.5.2 3.5.3

Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binary Error Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Transformation on Binary Error Set and Their Laws of Operations . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . 4.1 Similarity Transformation Connectives in Error Logic . . . . . . . . . . . 4.1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Similarity Transformation Connectives in Error Logic . . . . 4.1.4 Characteristics of Domain Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Characteristics of Thing Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Characteristics of Property Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.7 Characteristics of Spatial Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.8 Characteristics of Property (or Attribute) Value Similarity Transformation Connectives in Error Logic . . . . 4.1.9 Characteristics of Error Value Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . 4.1.10 Characteristics of Rule Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.11 Characteristics of Error Function Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . 4.1.12 Characteristics of Temporal Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . 4.1.13 Characteristics of Combination Similarity Transformation Connectives in Error Logic . . . . . . . . . . . . . 4.2 Decomposition Transformation Connectives in Error Logic . . . . . . . 4.2.1 Concepts of Decomposition Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Domain Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Thing Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Spatial Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Property Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Property (or Attribute) Value Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . .

62 63 68 71 73 74 74 75 81 83 90 102 114 128 140 152 166 178 191 192 193 199 207 230 251 271

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4.2.7

Error Value Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.8 Error Function Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.9 Time Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.10 Rule Decomposition Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.11 Overall Decomposition Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Displacement Transformation Connectives in Error Logic . . . . . . . . 4.3.1 Concepts of Displacement Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Domain Displacement Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Thing Displacement Transformation Connective in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Increase Transformation Connectives in Error Logic . . . . . . . . . . . . . 4.4.1 Increase Transformation Connectives in Error Logic . . . . . 4.4.2 Characteristics of Error Value Increase Transformation Connective in Error Logic . . . . . . . . . . . . . . 4.5 Destruction Transformation Connectives in Error Logic . . . . . . . . . . 4.5.1 Concept of Destruction Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Principles for Destruction Transformation in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Approaches of Destruction Transformation in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Hierarchy of Destruction Transformation in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Characteristics of Domain Destruction Transformation Connective Th in Error Logic . . . . . . . . . . . 4.5.6 Thing Destruction Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.7 Property Destruction Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.8 Engenderment Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Mathematical Error Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Concept of Mathematical Error Propositional Logic . . . . . . . . . . . . . 5.2 Error Logical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Existential Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Atomic Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Basic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 313 334 356 377 377 377 380 389 396 396 399 414 414 417 417 417 418 429 440 451 467 467 468 468 470 472

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Contents

5.5 Compound Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Atomic Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Error Logical Compound Proposition . . . . . . . . . . . . . . . . . . 5.6 Basic Rules for Error Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Axiom Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Reasoning Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Forms of Error Logical Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Error Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Forms of Error Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Form Language of Error Predicate Logic . . . . . . . . . . . . . . . 5.9 Semantic Explanation of Error Predicate Logical Expression . . . . . .

492 492 492 496 496 496 499 504 504 508 509

6 Applications of Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Method for Finding Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Modeling Error-elimination for a Urban Traffic Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Flowchart Representing Process of Finding Solutions . . . . 6.1.5 Modeling Building and Analysis . . . . . . . . . . . . . . . . . . . . . . 6.2 Computerized Error Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Error Theory-Based Expert System Structure . . . . . . . . . . . 6.2.2 Application Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Knowledge Representation Model for Ecological Civilization Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Decomposition Transformation Connectives in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Principles and Types for Decomposition Transformation in Error Logic . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Knowledge Representation Model for Ecological Civilization Performance Indicators . . . . . . . . . . . . . . . . . . . 6.4 Error Transmission in a System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Critical Factors for Error Transmission in a System . . . . . . 6.4.2 Expression Form for a System . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Error Transmission Function in a System . . . . . . . . . . . . . . . 6.4.4 Application Example for Concept of Error Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Application of Error Logic in Decision Support System for Nanquan Referees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Description on the Error Object of Nanquan in Error Logical Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Error Logic-based Object System of Nanquan Movement Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

511 511 512 513 515 517 518 527 528 536 540 541 542 543 548 548 549 550 552 553 554 558

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6.5.3 6.5.4 6.5.5

Computer Vision-based Error Identification Model for Optional Nanquan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 Application of Computer Vision-based Error Identification Model for Optional Nanquan . . . . . . . . . . . . . 603 Summary for the Application . . . . . . . . . . . . . . . . . . . . . . . . . 608

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

Acronyms

ABI ABV AI AS BES CCEF CE CEF CEFwCP CES CFEF CNnEF CP DL EC EDTC EF EFwCP EM EME ES ES ESCP ESS ET FEF FES HCM IS JMDC JMSAD

All be invalid All be valid Artificial intelligence Aristotelian syllogism Binary error set Correlational classic error function Comparative experiment Classic error function Correlational error function with critical points Classic error set Correlational classic error function Correlational non-negative error function Cooperative Principle Dynamic logic Ecological civilization Error decomposition transformation connective Error function Error function with critical points Error matrix Error matrix equation Error set Expert system Error set with critical points Evaluation standards Error theory Fuzzy error function Fuzzy error set Highway Capacity Manual Interval semantics Joint method of difference and comparison Joint method of statistics, agreement, and difference xiii

xiv

JMSC JMSD KE KPIs KR LoI LOS LPT MACL MEP MES MoHURD MPS MS MSF NDS NnEF ORB PL PTQ PWM PWS RFID RT SBR SoS TCI TKFJ UD VTEF

Acronyms

Joint method of sampling and comparison Joint method of sampling and difference Knowledge engineering Key performance indicators Knowledge representation Law of identity Level of services Logical Presupposition Theory Multi-agent cognitive logic Ministry of Ecology and Environment of the People’s Republic of China Multivariate error set Ministry of Housing and Urban-Rural Development of the People’s Republic of China Moderately prosperous society Modal syllogism Maximum service flow Numerically definite syllogism Non-negative error function Object-recognition-based Partial logic Proper treatment of quantification in ordinary English Possible world model Possible world semantics Radio frequency identification Reflection theory Sequencing Batch Reactor System of systems Theory of Conversational Implicatures Tengkong Feijiao Universe of discourse Vector type error function

Chapter 1

Preface

Errors permeate every corner of the world and the decision-making process in different entities including state government, for-profit enterprises, not-for-profit organizations, and person is often susceptible to error ([1–48]). The occurrence of error does not differentiate the developed countries or underdeveloped countries, well-tuned organizations or under-performing organizations, those famous or infamous persons in history. Errors were often seen in scientific and technological fields whether they are in developed countries or developing countries. History has witnessed many different errors in either primitive or civilized societies ([49]). The causes for an error might result from some elements or a single element of the system in which the error is embedded. The consequences of the errors in an object, a case, a decision, or a theoretical system may lead to minor loss, catastrophic casualties, disband of organizations, dissolution of countries, or even the termination of human being. For instance, the mid-air collision of 1st July 2002 between Bashkirian Airlines Flight 2937(Tu-154 passenger jet)and DHL Flight 611(Boeing757 cargo jet) was caused by a series of errors including problems of the arrangement errors for both personnel and equipment maintenance of SWISS air traffic control service and procedural errors of using Traffic Collision Avoidance System (TCAS).The disaster killed 69 passengers and crew members on Flight 2937 and two crew members of the Flight 611, which offered a thought-provoking lesson for relevant organizations and decision makers ([50]). There was a story about how an old wise farmer divided his heir to his three sons. This old farmer had 17 camels. This distribution scenario was to give half of his assets to the oldest son, 1/3 of them to the second son, and 1/9 of them to the youngest son. However, it was not possible for him to have the scenario implemented since he only had odd number of camels. His neighbor just passed by as he was contemplating how he should do it. Having understood the farmer’s issue, the neighbor decided to lend the farmer one camel. Right now, the old farmer had 18 camels to be divided. The oldest © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Liu and K. Guo, Error Logic: Paving Pathways for Intelligent Error Identification and Management, Studies in Systems, Decision and Control 442, https://doi.org/10.1007/978-3-031-00820-7_1

1

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1 Preface

son then got 9 of the 18 camels. The second son got 6 which is 1/3 of all the camels. And the youngest son got 2 which is 1/9 of the 18 camels. Therefore, total camels divided to three sons were 17 and the neighbor took his camel back. This story told us that increase transformation can help resolve challenging issues. In engineering project, dismantling shabby buildings and erecting new ones employ the concepts of destruction and inverse destruction(creation or construction). In repairing broken electronic appliances, replacing broken parts with new one adopts the thoughts of displacement transformation. In the process of studying a complicated system, it is necessary to investigate how the error or failure of one element or multiple elements (or subsystems) of the system can influence its reliability and functionality ([51–59]). In order to identify the errors or failures and consequently eliminate or remove them, we need to find the ways or laws that error is transmitted and transformed in the system ([60–97]). In order to identify and remove error occurred in an element ei of system S, the first step is to decompose system A into a subsystem set containing the subsystem with ei as one part of it; then a non-erroneous subsystem is used to replace ei . For example, a company A experiencing crisis wants to learn the management system of a successful company B, it is not wise to mechanically implant A’s procedures into B’s system without making adaptive changes. Taking another example, regarding the expansion strategy of chain stores, decision makers have to consider the differences in distinct cultures, religions, social norms, law and regulations, and economic and political contexts and make necessary transformation and changes accordingly. Therefore, it is necessary to examine decomposition transformation, displacement transformation, increase transformation, similarity transformation, destruction transformation, unit transformation, and their corresponding inverse transformations (if necessary) of those successful management systems and the management systems experiencing crises. Error logic is to explore thinking forms, thinking methods and laws, and valid reasoning of related propositions for static and dynamic transformation relationships of intra- and inter-errors in a given system. According to the range of truth values in error logic, error logic can be categorized into: error binary-value logic, fuzzy error logic ([98–112]), and error logic with critical points ([113]). In scientific research and social practices, people have tried to use various transformation (change) methods to resolve actual problems or eliminate errors in a system. Therefore, in order to investigate ways and laws of error transformation such as T (μ) = μ0 , we solve for μ0 given that T and μ are known or for μ given that T and μ0 are known, or for T given that μ0 and μ are known, where T is the transformation mechanism, μ0 and μ are the object of research. Here it is necessary to study the laws and mechanisms on how errors are transformed under different conditions. Definition of error: Suppose that U is the universe of discourse, G is a set of rules for judging error defined within U , if ∃ G  a (including the cases that a can not be completely or partially obtained by exercising G; or the case that a has nothing to do with G.), a is erroneous defined within U under the rule of judging errors G. Based on the

1 Preface

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definition, we know that the error is relative. Its existence pertains to certain universe of discourse, a group of rules, and the implications manifested in the definition. Note: G is a set of predetermined and qualified rules for judging if an object is correct or not. Otherwise, G ⇒ a holds if G is the rule for judging whether an object is erroneous or not. Relevant definitions are presented as below. 1. Rules for judging errors: Research contents in this area contain: (1) conditions for justifying the appropriateness of those chosen rules and their applicable factors such as field and time; (2) relations between chosen rules; (3) relations between chosen rules and the object being investigated; and (4) methods for screening and assessing rules 2. Error set theory: On the basis of classic and fuzzy set theories, the established error set is composed of classic error set, fuzzy error set, and error set with critical points ([114, 115]). In error set theory, we investigate both static and dynamic relations and interactions of things ([116–129]). We especially focus on the study of relation between change and transformation of things. Six basic transformations, namely similarity or equivalence transformation, displacement transformation, decomposition transformation, addition transformation, destruction transformation, and unit transformation are introduced for the purpose of studying change and transformation of things. Dynamic parameters are assigned to elements of error set to capture the change and transformation of things. Thereafter, by using 6 basic transformations, operations can be qualitatively and quantitatively conducted on elements of error set. 3. Error logic: Built on the foundation of classic mathematical logic, fuzzy mathematical logic, and dialectic logic, the concept of error logic is proposed in order to identify the pattern and laws for transition and transformation of errors. In our error logic system, in addition to denotation connective, connotation connective, and individual connective, 6 more transformation connectives are created, namely similarity transformation connective, displacement transformation connective, decomposition transformation connective, addition transformation connective, destruction transformation connective, and unit transformation connective as well as their inverse transformation connectives. More importantly, research in error logic covers many aspects including but not limited to (1) principles and laws of operation of each newly created connective, operation between new connectives, and operation between each of the new connective and existing denotation connective, connotation connective, or individual connective; (2) truth value table, normal forms, and valid reasoning methods in error logic; (3) concept and parameter of error predicate logic; and (4) semantic structure and semantic interpretation of subject term (the variable), quantifier, and predicate, etc.

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Regarding the differences between error logic and correctness logic, five aspects are presented as follows: 1. For parallel system (a) For correctness logic, A ∨ B ∨ · · · ∨ G (disjunction of logical propositions), the system can normally function as long as any of the subsystems A, or B, · · · , or G functions well. (b) For error logic, A ∧ B ∧ · · · ∧ G (conjunction of logical propositions), the system can not normally function if all the subsystems A, B, · · · , and G do not function. 2. For series system (a) For correctness logic, A ∧ B ∧ · · · ∧ G (conjunction of logical propositions), the system can normally function if all the subsystems A, B, · · · , and G can function. (b) For error logic, A ∨ B ∨ · · · ∨ G (disjunction of logical propositions), the system can not normally function if one of the subsystems A, or B, · · · , or G does not function. 3. For whole system (a) For correctness logic, enlarging the scale of a system will not increase the correctness of the system if the fundamental structure of the system does not change. (b) For error logic, enlarging the scale of a system will increase the occurrence of error if the fundamental structure of the system does not change. 4. For logic system Error logical transformation connectives are created based on conventional mathematical logic, fuzzy logic, and mathematical dialectical logic. They are listed as follows: (1) Similarity transformation connectives, Tx ⊆ { Txly , Txsw , Txk j , Txt z , Txlz , Txcz , Txhs , Txs j , Txgz , Tx zh } where similarity includes: {domain similarity, thing similarity, spatial similarity, property similarity, property (attribute) value similarity, error value similarity, error function similarity, temporal similarity, rule similarity, and similarity in comprehensive aspects (different combinations of them)} ); Tx−1 : inverse similarity transformation connectives, Tx−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 ⊆ { Txly , Txsw , Txk j , Txt z , Txlz , Txcz , Txhs , Txs j , Txgz , Tx zh } where inverse similarity includes: {domain inverse similarity, thing inverse similarity, spatial inverse similarity, property inverse similarity, property (attribute) value inverse similarity, error value inverse similarity, error function inverse similarity, temporal inverse similarity, rule inverse similarity, and inverse similarity in comprehensive aspects (different combinations of them)}. (2) Displacement transformation connectives, Tz ⊆ { Tzly , Tzsw , Tzk j , Tzt z , Tzlz , Tzcz , Tzhs , Tzs j , Tzgz , Tzzh } where displacement includes: {domain displacement, thing displacement, spatial displacement, property displacement, property (attribute) value displacement, error value displacement, error function

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displacement, temporal displacement, rule displacement, and displacement in comprehensive aspects (different combinations of them) } ); Tz−1 : inverse −1 −1 displacement transformation connectives, Tz−1 ⊆ { Tzly , Tzsw , Tzk−1j , Tzt−1z , −1 −1 −1 −1 −1 , Tzcz , Tzhs , Tzs−1j , Tzgz , Tzzh } where inverse displacement includes: Tzlz { domain inverse displacement, thing inverse displacement, spatial inverse displacement, property inverse displacement, property (attribute) value inverse displacement, error value inverse displacement, error function inverse displacement, temporal inverse displacement, rule inverse displacement, and inverse displacement in comprehensive aspects (different combinations of them)}. (3) Increase transformation connectives, Tz n ⊆ { Tznly , Tznsw , Tznk j , Tznt z , Tznlz , Tzncz , Tznhs , Tzns j , Tzngz , Tznzh } where increase includes: {domain increase, thing increase, spatial increase, property increase, property (attribute) value increase, error value increase, error function increase, temporal increase, rule increase, and increase in comprehensive aspects (different combinations of them) } ); Tz n −1 : inverse increase transformation connectives, Tz n −1 ⊆ −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 , Tznsw , Tznk { Tznly j , Tznt z , Tznlz , Tzncz , Tznhs , Tzns j , Tzngz , Tznzh } where inverse increase (decrease) includes :{domain inverse increase, thing inverse increase, spatial inverse increase, property inverse increase, property (attribute) value inverse increase, error value inverse increase, error function inverse increase, temporal inverse increase, rule inverse increase, and inverse increase in comprehensive aspects (different combinations of them)}. (4) Decomposition transformation connectives, T f ⊆ { T f ly , T f sw , T f k j , T f t z , T f lz , T f cz , T f hs , T f s j , T f gz , T f zh } where decomposition includes: {domain decomposition, thing decomposition, spatial decomposition, property decomposition, property (attribute) value decomposition, error value decomposition, error function decomposition, temporal decomposition, rule decomposition, and decomposition in comprehensive aspects (different combinations of them) } ); T f−1 : inverse decomposition transformation connectives; T f−1 ⊆ −1 −1 −1 −1 −1 −1 −1 −1 −1 { T f−1 ly , T f sw , T f k j , T f t z , T f lz , T f cz , T f hs , T f s j , T f gz , T f zh } where inverse decomposition includes:{domain inverse decomposition, thing inverse decomposition, spatial inverse decomposition, property inverse decomposition, property (attribute) value inverse decomposition, error value inverse decomposition, error function inverse decomposition, temporal inverse decomposition, rule inverse decomposition, and inverse decomposition in comprehensive aspects (different combinations of them)}. (5) Destruction transformation connectives, Th ⊆ { Thly , Thsw , Thk j , Tht z , Thlz , Thcz , Thhs , Ths j , Thgz , Thzh } where destruction includes: {domain destruction, thing destruction, spatial destruction, property destruction, property (attribute) value destruction, error value destruction, error function destruction, temporal destruction, rule destruction, and destruction in comprehensive aspects (different combinations of them) } ), Th−1 : inverse destruction (combi−1 −1 −1 −1 −1 −1 , Thsw , Thk nation) transformation connectives; Th−1 ⊆ { Thly j , Tht z , Thlz , Thcz , −1 −1 −1 , Ths−1j , Thgz , Thzh }, where inverse destruction (creation) includes:{domain Thhs

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inverse destruction, thing inverse destruction, spatial inverse destruction, property inverse destruction, property (attribute) value inverse destruction, error value inverse destruction, error function inverse destruction, temporal inverse destruction, rule inverse destruction, and inverse destruction in comprehensive aspects (different combinations of them) }; (6) Unit transformation connectives, Td (unit),Td−1 : inverse unit transformation connective. 5. For quantifier system in error logic (1) Existential quantifiers (a) ∃LY , there exists certain domain; (b) ∃SW , there exists certain thing; (c) ∃S J , there exists certain time; (d) ∃K J , there exists certain spatial location; (e) ∃T J , there exists certain constraint; (f) ∃T Z , there exists certain property; (g) ∃L Z , there exists certain value of the property or attribute; (h) ∃C Z , there exists certain error value; (i) ∃H S, there exists certain function; (j) ∃G Z , there exists certain group of rules; (k) ∃F S, there exists certain decomposition method; (l)  there exists certain universe of discourse. (2) Universal quantifiers (a) ∀LY , for each domain; (b) ∀SW , for each thing; (c) ∀S J , for the whole time; (d) ∀K J , for each spatial location; (e) ∀T J , for each constraint; (f) ∀T Z , for each property; (g) ∀L Z , for each value of the property or attribute; (h) ∀C Z , for each error value; (i) ∀H S, for each function; (j) ∀G Z , for each group of rules; (k) ∀F S, for each decomposition method; (l)  for the whole universe of discourse. Readers should be cautious about several points showed as below while reading or referring to this book: (1) error logic was proposed based on classic mathematical logic, fuzzy logic, and mathematical dialectical logic with the purpose of gaining better understand on erring causes and mechanisms, laws for error transmissions and transformations, and methodologies for avoiding or eliminating errors; (2) classic mathematical logic, fuzzy logic, and mathematical dialectical logic are to get the truth values of propositions based on the degree of “correctness”. In the error logic −−→ definition, A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ( μ(t), G(t))), x(t) ∈ { {0, 1}, [0, 1], (−∞, +∞) } is the error logical variable, where U (t) is the

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−−→ −−→ domain of μ(t) = (U (t), S(t), p(t), T (t), L(t)), S(t) is the thing or subject, p(t) is −−→ the spatial location and direction of μ(t) = (U (t), S(t), p(t), T (t) is the property or −−→ predicate of μ(t) = (U (t), S(t), p(t), L(t) is the value of the property or attribute −−→ of μ(t) = (U (t), S(t), p(t), x(t) = f ( μ(t), G(t)) is the truth value or truth value −−→ function of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ( μ(t), G(t))), G(t) is the rule for judging error defined in domain U (t); (3) all relevant definitions in Chap. 4 conform to definitions and notations defined in classic mathematical logic, fuzzy logic, and mathematical dialectical logic except for those related to error transformation connectives; (4) a logic system with error values as the values of error logical propositions is used to define logic connectives; (5) considering the four points mentioned above, special care is required when discussing and applying necessary logical propositions in actual application. Things needing to keep in mind are that: (1) “correctness” logic’s definitions and logical calculus should be used when logical propositions involving classic mathematical logic, fuzzy logic, and mathematical dialectical logic; (2) definitions and logical calculus related to error logic should be employed, otherwise. The book is arranged as follows. Chapter 2 offers the reviews on the history, development, and trend projection of logic as well as the logic-related thoughts of well-known logician and philosophers. In Chap. 3, error sets and relevant operations are presented. As the core part of this book, Chap. 4 presents main transformation connectives in error logic and how they are used in error identification and elimination. In Chap. 5, concepts, definitions, and operations of mathematical error propositional logic are detailed. Chapter 6 provides the detailed application of error logic. The ultimate objective of studying error is to avoid or eliminate errors, and consequently to reduce the loss or even disaster caused by ignored errors. In order to avoid or eliminate errors, it is necessary to understand erring causes and mechanisms, laws for error transmissions and transformations, and methodologies for avoiding or eliminating errors. This is why we have been devoting our efforts in studying error logic. However, due to limitations in experiences and level of research, the causes and mechanisms of erring are far from clear, the laws of error transmissions and transformations (currently found) are far from being enough, and method and tool system for avoiding or eliminating is far from being fully established. Therefore, there are bound to be mistakes of one kind and another in the book and we wholeheartedly invite scholars and readers to enlighten us by providing feedback and critiques.

References 1. Bell, B.S., Kozlowski, S.W.J.: Collective failure: the emergence, consequences, and management of errors in teams. In: Hofmann, D.A., Frese, M. (eds.) Errors in Organizations, pp. 1–44. Routledge, New York (2011) 2. Bogner, M.S.: Human Error in Medicine (Human Error And Safety). CRC Press (1994) 3. Boniface, D.E., Bea, R.G.: Assessing the risk of and countermeasures for human and organizational error. SHAME Trans. 104(1996), 157–177 (1996)

8

1 Preface

4. Bostock, L., Bairstow, S., Fish, S., Macleod, F.: Managing Risk and Minimizing Mistakes in Services to Children and Families. SCIE, London (2005) 5. Brodbeck, F.C., Zapf, D., Prüomper, J., Frese, M.: Error handling in office work with computers: a field study. J. Occup. Organ. Psychol. 66, 303–317 (1993) 6. Burge, T.: Reference and proper names. J. Philos. 70(14), 425–439 (1973) 7. Carroll, J., Christianson, K.M., Frese, M., Lei, Z., Naveh, E., Vogus, T.: Errors in organizations: exploring new frontiers, and developing new opportunities for theory, research and impact. In: Paper presented at AMD Errors in Organization session at the 2018 Academy of Management Conference, Chicago (2018). Accessed from 13 August 2018 8. Cooper, J., Newbower, R., Kitz, R.: An analysis of major errors and equipment failures in anesthesia management: conditions for prevention and detection. Anasthesiology 60, 42–43 (1984) 9. Dahlin, B.K., Chuang, Y., Roulet, T.J.: Opportunity, motivation and ability to learn from failures and errors: review, synthesis and ways to move forward. Acad. Manag. Ann. 12, 252–277 (2017) 10. Dimitrova, N.G., Van Dyck, C., van Hooft, E.A.J., Groenewegen, P.: Error prevention, error management, or both? In: Annual Meeting Proceedings of Academy of Management. Philadelphia, Pennsylvania United States, 1–5 August 2014, 2014(1):15702–15702 (2014) 11. Dismukes, R.K.: Human Error in Aviation. CRC Press, Taylor & Francis Group (2009) 12. Dormann, T., Frese, M.: Error training: Replication and the function of exploratory behavior. Int. J. Hum.-Comput. Interact. 6(4), 365–372 (1994) 13. Dorner, D.: The Logic of Failure: Recognizing and Avoiding Error in Complex Situations, Revised Ed.. Basic Books (1997) 14. Grout, J.R.: Mistake proofing: changing designs to reduce error. Qual. Saf. Health Care 15(S1), i44–i49 (2006) 15. Gully, S.M., Payne, S.C., Koles, K.L.K., Whiteman, J.A.K.: The impact of error training and individual differences on training outcomes: an attribute-treatment interaction perspective. J. Appl. Psychol. 87, 143–155 (2002) 16. Guo, K.: Preliminary analysis on error systems. J. Guangdong Univ. Technol. (1986) 17. Haselton, M.G., Buss, D.M.: Error management theory: a new perspective on biases in crosssex mind reading. J. Person. Soc. Psychol. 78(1), 81–91 (2000) 18. Haselton, M..G., Galperin, A.: Error management in relationships. In: Simpson, J., Campbell, L. (eds.) The Oxford Handbook of Close Relationships (2013) 19. Hasegawa, T., Kemeda, A.: Analysis and evaluation of human error events in nuclear power plants. Presented at the meeting of IAEA?s CRP on Collection and Classification of Human Reliability Data for Use in Probabilistic Safety Assessments: Available from the Institute of Human Factors, Nuclear Power Engineering Corporation 3–17-1 Toranomon. Minato-Ku, Tokyo (1998) (May 1998) 20. Keith, N., Frese, M.: Self-regulation in error management training: emotion control and metacognition as mediators of performance effects. J. Appl. Psychol. 90(4), 677–691 (2005) 21. Lazonder, A.W., Van der Meij, H.: Effect of error information in tutorial documentation. Interact. with Comput. 6(1), 23–40 (1994) 22. Love, P.E.D., Edwards, D.J., Han, S.: Bad apple theory of human error and building information modelling: a systemic model for BIM implementation. In: 2011 Proceedings of the 28th ISARC, Seoul, Korea, pp. 349–354 (2011). Accessed from June 29–July 2 2011 23. Marx, D.A., Graeber, R.C.: Human error in maintenance. In: Johnston, N., McDonald, N., Fuller, R. (eds.) Aviation Psychology in Practice, pp. 87–104. Ashgate Publishing, London (1994) 24. McKay, R., Efferson, C.: The subtleties of error management. Evol. Hum. Behav. 31(5), 309–319 (2010) 25. Nordstrom, C.R., Wendland, D., Williams, K.B.: To err is human: an examination of the effectiveness of error management training. J. Bus. Psychol. 12, 269–282 (1998) 26. Oostinga, M.S.D.: Breaking (the) Ice: Communication error management in law enforcement interactions. Enschede: University of Twente (2018)

References

9

27. Palmer, D.: Normal organizational wrongdoing: A critical analysis of theories of misconduct in and by organizations. Oxford Scholarship Online (2012) 28. Rasmussen, J., Gunter, S.K., Leplat, J. (eds.): New Technology and Human Error (New Technologies and Work: A Wiley Series), 1st edn. Wiley (1987) 29. Rasmussen, J.: Human error and the problem of causality in analysis of accidents. Philos. Trans. R. Soc. Lond. 327, 449–460 (1990) 30. Reason, P.: Human Error. Cambridge University Press, Cambridge (1990) 31. Reason, J., Hobbs, A.: Managing Maintenance Error: A Practical Guide. CRC Press (2003) 32. Reinach, S., Viale, A.: Application of a human error framework to conduction train accident/incident investigation. Accident Anal. Prevent. 38, 396–406 (2006). https://www. sciencedaily.com/releases/2001/01/010109083707.htm. Accessed from 12 January 2019 33. Rybowiak, V., Garst, H., Frese, M., Batinic, B.: Error orientation questionnaire (EOQ): reliability, validity, and different language equivalence. J. Org. Behav. 20, 527–547 (1999) 34. Senders, J..W., Moray, N..P.: Human Error: Cause, Prediction, and Reduction (Applied Psychology Series), 1st edn. CRC Press, Boca Raton (1991) 35. Sheridan, T.B.: Risk, human error, and system resilience: fundamental ideas. Hum. Factors 50(3), 418–426 (2008) 36. Shi, W., Jiang, F., Zheng, Q., Cui, J.: Analysis and control of human error. Proc. Eng. 26(2011), 2126–2132 (2011) 37. Smith, T.A.: Why originalism won’t die-common mistakes in competing theories of judicial interpretation. Duke J. Constit. Law & Public Policy 2(1), 159–216 (2017) 38. Van der Linden, D., Sonnentag, S., Frese, M., Van Dyck, C.: Exploration strategies, performance, and error consequences when learning a complex computer task. Behav. Inf. Techno. 20, 189–198 (2001) 39. Van der Schaaf, T.W., Frese, M., Heimbeck, D.: Human recovery and error management. In: Proceedings of the XV. European Annual Conference on Human Decision Making and Manual Control. Soesterberg, pp. 5.2-1–5.2-10 (1996) 40. Van Dyck, C., Frese, M., Baer, M., Sonnentag, S.: Organizational error management culture and its impact on performance: a two-study replication. J. Appl. Psychol. 90(6), 1228–1240 (2005) 41. Veldkamp, C.: The human fallibility of scientists: Dealing with error and bias in academic research. Doctoral Thesis. Department of Methodology and Statistics, Tilburg University (2017). https://research.tilburguniversity.edu/en/publications/the-human-fallibilityof-scientists-dealing-with-error-and-bias-i. Accessed from 10 April 2019 42. Wallace, B., Ross, A.: Beyond Human Error: Taxonomies and Safety Science. Boca Raton, Taylor & Francis (2006) 43. Wen, Q.: Discussion on Error. Press of Liaoning University, Shenyang (1995) 44. Whittingham, R.: The Blame Machine: Why Human Error Causes Accidents. Oxford, Elsevier Butterworth-Heinemann (2004) 45. Wilson, B.A., Baddeley, A., Evans, J.: Errorless learning in the rehabilitation of memory impaired people. Neuropsychol. Rehabil. 4, 307–326 (1994) 46. Woods, D., Cook, R.: Nine steps to move forward from error. Cognit. Technol. Work 4, 137–144 (2002) 47. Woods, D., Johannesen, L., Cook, L., Sarter, N.: Behind human Error: Cognitive systems, computers and hindsight. Wright-Patterson Air Force Base, Ohio, CSERIAC (1994) 48. Xie, B., Guo, K., Wu, W.: Destruction transformation on the structure of error system. In: China Intelligence and Automation Conference (CIAC). Nanjing, Jiangsu, China (2009). Accessed from 27–29 September 2009 49. Harrald, J.R., Mazzuchi, T.A., Spahna, J., Van Dorp, R., Merrick, J., Shrestha, S., Grabowski, M.: Using system simulation to model the impact of human error in a maritime system. Saf. Sci. 30 (1998), 235–247 (1998) 50. Johns Hopkins Medical Institutions(JHMI),Fewer airline crashes linked to “pilot error”; Inclement weather still major factor (2001). https://www.sciencedaily.com/releases/2001/01/ 010109083707.htm. Accessed from 15 November 2018

10

1 Preface

51. Bregman, O.C., White, C.M.: Bringing Systems Thinking to Life: Expanding the Horizons for Bowen Family Systems Theory. Routledge, New York (2011) 52. Cacciabue, P.C.: Human error risk management for engineering systems: A methodology for design, safety assessment, accident investigation and training. Reliab. Eng. Syst. Saf. 83(2), 229–240 (2004) 53. Campbell, D., Draper, D., Huffington, C.: (Eds.) Teaching Systemic Thinking. Routledge, London (1988) 54. Checkland, P.: Systems Thinking, Systems Practice: Includes a 30-Year Retrospective, 1st edn. Wiley, New York (1999) 55. Charles, M.C.: Dynamic Systems 3e. Wiley, New York (2001) 56. Clarke, B., Hansen, M.B.N., Smith, B.H., Weintraub, E.R. (eds.): Emergence and Embodiment: New Essays on Second-Order Systems Theory (Science and Cultural Theory). Duke University Press (2009) 57. Fischer, S., Frese, M., Mertins, J.C., HardtGawron, J.V.: The role of error management culture for firm and individual innovativeness. Appl. Psychol. 67(3), 428–453 (2018) 58. Fitts, P., Jones, R.: Analysis of factors contributing to 460 ’pilot error’ experiences in operating aircraft controls. Memorandum Rep. Ohio, Aero Medical Laboratory (1947) 59. Fraser, J., Smith, P., Smith, J.: A catalog of errors. Int. J. Man Mach. Stud. 37, 265–393 (1992) 60. Dekker, S.: The Field Guide to Human Error Investigations. Ashgate Publishing (2002) 61. Dekker, S.: Ten Questions About Human Error: A New View of Human Factors and System Safety (Human Factors in Transportation), 1st edn. CRC Press, Boca Raton (2004) 62. Dhillon, B.S.: Human Reliability and Error in Transportation Systems (Springer Series in Reliability Engineering). Springer, London (2007) 63. Dhillon, B.S.: Human Reliability, Error, and Human Factors in Engineering Maintenance: With Reference to Aviation and Power Generation, 1st edn. CRC Press, Boca Raton (2009) 64. Dhillon, B.S.: Safety and Human Error in Engineering Systems, 1st edn. CRC Press (2012) 65. Dhillon, B.S.: Human Reliability, Error, and Human Factors in Power Generation (Springer Series in Reliability Engineering). Springer International Publishing (2014) 66. Elzer, P.F., Kluwe, R.H., Boussoffara, B.:. Human Error and System Design and Management (Lecture Notes in Control and Information Sciences). Springer, London (2000) 67. Frese, M.: To err is human. Review of Human Factors in Hazardous situations by D.E. Broadbent, A. Baddeley & J.T. Reason (Eds.). The Psychologist, 14, 8, 341 (1991) 68. Frese, M.: Error management or error prevention: two strategies to deal with errors in software design. In: Bullinger, H.-J. (Ed.), Human Aspects in Computing: Design and Use of Interactive Systems and Work with Terminals. Elsevier Science Publisher, pp. 776–782 (1991) 69. Frese, M.: Error management in training: Conceptual and empirical results. In: Zucchermaglio, C., Bagnara, S., Stucky, S.U. (eds.) Organizational Learning and Technological Change, pp. 112–124. Springer, Berlin, Heidelberg, New York (1995) 70. Frese, M.: Error management: An alternative concept to error prevention in organizations and in technical system design. In: Sheridan, T.B. (ed.), Proceedings of Man Machine Systems (MMS’95). Cambridge, Massachusetts, June 1995. Elsevier Science Publisher, Amsterdam (1995) 71. Frese, M., Altmann, A.: The treatment of errors in learning and training. In: Bainbridge, L., Quintanilla, S.A.R. (eds.) Developing Skills with New Technology, pp. 65–86. Wiley, Chichester (1989) 72. Frese, M., Brodbeck, F.C., Zapf, D., Prümper, J.: Users’ errors and error handling: its relationships with task structure and social support. SIGCHI Bull. 23(2), 59–62 (1991) 73. Frese, M., Brodbeck, F.C., Heinbokel, T., Mooser, C., Schleiffenbaum, E., Thiemann, P.: Errors in training computer skills: on the positive function of errors. Hum. Comput. Interact. 6, 77–93 (1991) 74. Frese, M., Keith, N.: Action errors, error management, and learning in organizations. Annu. Rev. Psychol. 66, 661–687 (2014) 75. Frese, M., van Dyck, C.: Error management: concept to error prevention in technical system design. Int. J. Psychol. Abstracts of the XXVI. International Congress of Psychology, Montreal, ´ Canada, 16-21 August 1996 31, 147 (1996)

References

11

76. Goodman, P.S., Ramanujam, R., Carroll, J.S., Edmondson, A.C., Hofmann, D.A., Sutcliffe, K.M.: Organizational errors: directions for future research. Res. Organ. Behav. 31, 151–176 (2011) 77. Hagen, J.U.: Confronting Mistakes: Lessons from the Aviation Industry when Dealing with Errors. Palgrave Macmillan (2013) 78. Hagen, J.U.: Error management: not just a wing and a prayer. EFMD Global Focus 8(2), 52–55 (2014) 79. Hagen, J.U.: How Could This Happen?-Managing Errors in Organizations. Palgrave Macmillan (2018) 80. Heimbeck, D., Sonnentag, S., Frese, M., Keith, N.: Integrating errors into the training process: the function of error management instructions and the role of goal orientation. Person. Psychol. 56(2), 333–361 (2003) 81. Helmreich, R.L.: On error management: lessons from aviation. BMJ 320(7237), 781–785 (2000) 82. Henningsen, D.D., Henningsen, M.L.M.: Testing error management theory: exploring the commitment skepticism bias and the sexual overperception bias. Hum. Commun. Res. 36(4), 618–634 (2010) 83. Hofmann, D.A., Frese, M. (eds.): Error in Organizations (SIOP Organizational Frontiers Series), 1st edn. Routledge (2011) 84. Ivancic, K., Hesketh, B.: Learning from errors in a driving simulation: effects on driving skill and self-confidence. Ergonomics 43, 1966–1984 (2000) 85. International Atomic Energy Agency (IAEA): Human Error Classification and Data Collection. Report of a technical committee meeting organized by the IAEA and held in Vienna, 20–24 February 1989 (1989) 86. Johnson, C.W.: Human error, interaction and the development of safety-critical systems. In: Boy, G.A. (ed.) The Handbook of Human-Machine Interaction: A Human-Centered Design approach, pp. 91–106. Ashgate, Farnham (2011) 87. Lei, Z., Naveh, E., Novikov, Z.: Errors in organizations: An integrative review via level of analysis, temporal dynamism, and priority lenses. J. Manag. 42(5), 1315–1343 (2016) 88. Lin, L., Guo, K.: Studies on the similarity transformation connectives of error domain: investigating errors in management system. Theoretical Investigation, pp. 213–216 (2008) 89. Liu, Y.: Theory and method for understanding conflicts and errors in large-scale complicated systems. Press of South China University of Technology, Guangzhou (2000) 90. Munro, E.: Common errors of reasoning in child protection. Child Abuse & Neglect 23, 745–758 (1999) 91. Prümper, J., Zapf, D., Brodbeck, F., Frese, M.: Errors in computerized office work: differences between novice and expert users. SIGCHI Bull. 23(2), 63–66 (1991) 92. Prümper, J., Zapf, D., Brodbeck, F.C., Frese, M.: Errors of novices and experts: some surprising differences in computerized office work. Behav. Inf. Technol. 11, 319–328 (1992) 93. Rabøl, L.I., Andersen, M.L., Østergaard, D., Bjørn, B., Lilja, B., Mogensen, T.: Republished error management: Descriptions of verbal communication errors between staff. An analysis of 84 root cause analysis-reports from Danish hospitals. Postgraduate Med. J. 87(1033), 783–789 (2011) 94. Rzepnicki, T., Johnson, P.: Examining decision errors in child protection: a new application of root cause analysis. Child. Youth Serv. Rev. 27, 393–407 (2005) 95. Sasou, K., Reason, J.: Team errors: definition and taxonomy. Reliab. Eng. Syst. Saf. 65, 1–9 (1999) 96. Zapf, D., Brodbeck, F., Frese, M., Peters, H., Prümper, J.: Errors in working with office computers. A first validation of a taxonomy for observed errors in a field setting. Int. J. Hum. Comput. Interact. 4, 311–339 (1992) 97. Zhang, M., Sun, X.: 1% errors can lead to 100% of the failure: a case from Falklands War between Argentina and UK. Liberation Army Daily. 16 March 2005 (2005) 98. Guo, K., Huang, J., Xiong, H.: Feature additivity-based system optimization. J. Guangdong Univ. Technol. 25(4), 1–4 (2008)

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99. Guo, K., Shi, J.: Equality-type fuzzy error-eliminating programming-based system optimization. In: Presented at 3rd Workshop on Management Studies in China, Lanzhou, China, 13-15 August 2010 (2010) 100. Guo, Q., Wang, Q., Guo, K.: Exploration of fuzzy system in decision making - Relationship of fuzzy error logic decomposition word T f and connotative antithesis +nhd . In: Proceedings of International Conference on Industrial Engineering and Engineering Management. IEEM, Singapore, 8-11 December 2008, pp. 450–454 (2008) 101. Guo, K., Xiong, H.: Research on thing transformation connectives in fuzzy error logic. Fuzzy Syst. Math. 20(2), 34–39 (2006) 102. Guo, K., Zhang, S.: Fuzzy error set. Fuzzy Syst. Math. 5(2), 67–75 (1991) 103. Huang, J., Xiong, H., Guo, K.: The transformation of fuzzy error set in a system intelligent decision making. In: Proceedings of International Conference on Advanced Computer Control(ICACC). Singapore, 22-24 January 2009, pp. 518–522 (2009) 104. Li, M., Guo, K.: Research on decomposition of fuzzy error set. Adv. Model. Anal. A: General Math. Comput. Tools 43(26), 15–26 (2006) 105. Liu, S., Guo, K.: Extreme value of fuzzy error system with change of time and space-zero faults trend of significant decision. Adv. Model. Anal.: B. 45(3), 39–49 (2002) 106. Liu, S., Guo, K.: Fuzzy error system with change of time and space-the effect of change of time and space on decision making. Adv. Model. Anal.: B. 45(3), 49–61 (2002) 107. Liu, S., Guo, K.: Decision making under condition of uncertainty: entropy change of fuzzy error system. Adv. Model. Anal.: B 39(2), 53–62 (2002) 108. Liu, S., Guo, K.: Exploration and application of redundancy system in decision makingrelation between fuzzy error logic increase transformation word and connotative model implication word. Adv. Model. Anal. A: Gen. Math. Comput. Tools 39(4), 17–29 (2002) 109. Liu, S., Guo, K.: Substantial change of decision-making environment-mutation of fuzzy error system. Adv. Model. Anal. A: Gen. Math. Comput. Tools 39(4), 29–39 (2002) 110. Liu, H., Guo, K.: Discussion about risk-control of investment: research on transformation system of fuzzy error set. Advances in Modeling and Analysis D (2006) 111. Xiong, H., Guo, K.: Research on domain transformation connectives in fuzzy error logic. Fuzzy Syst. Math. 20(1), 24–29 (2006) 112. Xiong, H., Huang, J., Guo, K.: Research on the application of fuzzy error logical in system optimization. In: 2008 IEEE International Symposium on Knowledge Acquisition and Modeling Workshop Proceedings. Wuhan, China,21-22 December 2008, pp. 147–149 (2008) 113. Guo, K., Liu, S.: Research on laws of security risk-Error logic system with critical point. Model. Meas. Control D. 22(1–2), 1–10 (2001) 114. Guo, K., Liu, S.: Fundamentals of error theory (Studies in Systems, Decision and Control 267). Springer (2019) 115. Guo, K., Liu, S.: Error Systems: Concepts, Theory and Applications (Studies in Systems, Decision and Control 275). Springer (2020) 116. Banerjee, S., Rondoni, L.: Applications of Chaos and Nonlinear Dynamics in Science and Engineering, vol. 4 (Understanding Complex Systems). Springer (2015) 117. Gall, J., Blechman, R.O.:Systemantics: How Systems Work and Especially How They Fail, 1st edn. Quadrangle (1977) 118. Gell-Mann, M.: The Quark and the Jaguar: Adventures in the Simple and the Complex. W.H. Freeman, San Francisco (1994) 119. Georgiou, I.: Thinking Through Systems Thinking. Routledge, London (2007) 120. Gharajedaghi, J.: Systems Thinking: Managing Chaos and Complexity: A Platform for Designing Business Architecture, 3rd edn. Morgan Kaufmann (2011) 121. Ghosh, A.: Dynamic Systems for Everyone: Understanding How Our World Works. Springer (2015) 122. Gleik, J.: Chaos: Making a New Science. Penguin Books (2008) 123. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmosp. Sci. 20(2), 130–141 (1963a) 124. Lorenz, E.N.: The predictability of hydrodynamic flow. Trans. N. Y. Acad. Sci. 25(4), 409–432 (1963)

References

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125. Lorenz, E.N.: The Essence of Chaos (Jessie and John Danz Lectures), 1 edn. University of Washington Press (1995) 126. Luhmann, N., Gilgen, P.: Introduction to Systems Theory, 1 edn. Polity (2012) 127. Miller, J.H., Page, S.: Complex Adaptive Systems: An Introduction to Computational Models of Social Life (Princeton Studies in Complexity). Princeton University Press (2007) 128. Sterman, J.: Business Dynamics: Systems Thinking and Modeling for a Complex World. McGraw-Hill Education (2000) 129. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Studies In Nonlinearity), 1 edn. CRC Press (2000)

Chapter 2

Short Introduction on the Logic

Logic deals with the systematic study of the form of valid inference. Formal logic was developed in ancient civilized countries such as India, China, and Greece. Aristotelian logic was widely recognized and accepted for the systematic study of logic in western science and mathematics for millennia, which was reflected by Aristotle’s (384BC– 322BC) six logic works (collectively called Organon-Tool), i.e., categories, De interpretatione (On Interpretation), prior analytics, posterior analytics, topics, and sophistic refutations. Stoics scholars represented by Chrysippus (c.279BC–c.206BC) proposed and developed predicate logic. Philosophers such as Anicius Manlius Severinus Boethius (c.477–524 AD), Abu Ali Sina(980–1037 AD), William of Ockha (1285– 1347 AD), and Jean Buridan (c.1301–c.1359 AD) advanced Aristotelian logic and thereafter research on logic declined and did not receive attention until mid-nineteen century.1 The logic discussed here is modern logic that is based on mathematical logic and analytic philosophy developed by George Boole (1815–1864), Friedrich Ludwig Gottlob Frege (1848–1925), Bertrand Arthur William Russell (1872–1970), and Giuseppe Peano (1858–1932). Mathematical logic’s fast progress was attributed to the work of Kurt Gödel (1906–1978) and Alfred Tarski (1901–1983). Modern logic has witnessed rapid development, which has become a primary subject including multiple branches. Modern logic has widely extended its application to mathematics, computer science, artificial intelligence, philosophy, law studies, linguistic studies, economics, and psychology, etc. In 1974, UNESCO (United Nations Educational, Scientific, and Cultural Organization), in the subject category, listed logic as one of the seven fundamental scientific disciplines, which are mathematics, astronomy, earth and space science, astrophysics, physics, chemistry, and life science.

1

Historyoflogic: https://en.wikipedia.org/wiki/History_of_logic.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Liu and K. Guo, Error Logic: Paving Pathways for Intelligent Error Identification and Management, Studies in Systems, Decision and Control 442, https://doi.org/10.1007/978-3-031-00820-7_2

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2.1 Major Theories in Modern Logic Modern logic consists of six fields which are mathematical logic, philosophic logic, logic of natural language, interdisciplinary research in logic and computer science, inductive logic, and philosophy of logic. They are briefly introduced respectively. 1. Mathematical Logic Mathematical logic is the foundation for modern logic which includes model theory, proof theory, recursion theory (also known as computability theory), and set theory. As the foundations of mathematics and theoretical computer science, it explores the application of formal logic to mathematics. It attempts to investigate the expressive power of formal systems and the deductive power of formal proof systems. 2. Philosophic Logic Philosophic logic is a subcategory of logic developed in 1950s. Founded on mathematical logic, it connects the concepts, scope, and problems in traditional philosophy. Philosophical logic has two subcategories. One subcategory is to address extensions to classical logic. For instance, a modal expression such as “necessarily” or “possibly” is used to qualify the truth of a judgment; tense logic (temporal logic) has been broadly used to include all approaches to representation and reasoning regarding time and temporal information within a logical framework in which inferences are studied using formal logical language with tense operators (“earlier than”, “before”, “points in time”); epistemic logic is the logic of knowledge and belief, which uses expressions like “knows that” or “believes that”; deontic logic is a branch of symbolic logic that is concerned with contribution that the following notions make to what follows from what: must, supererogatory (beyond the call of duty), indifferent/significant, the least one can do, better than /best/good/bad, and claim /liberty/power/immunity.2 The other subcategory is to addresses alternatives to “classical” logic known as “nonclassical” logic. For example, intuitionistic logic is as a part of mathematics rather than as the foundation of mathematics, which provides constructive explanation for logic connectives and logic quantifiers; relevance logic was developed to avoid the paradoxes of material and strict implication, in which relevance logicians thought that the paradoxes reside in that the premises and conclusions are on completely different topics. 3. Logic of Natural Language Logic of natural language is an important branch of logic founded on modern logic, modern linguistics, and semiotics. It concerns with not only the linguistic expression of cognitive contents but also linguistic expression of emotional contents. Moreover, it pays attention to functionality of linguistic expression as well as the capability of communicating information.

2

https://plato.stanford.edu/entries/logic-deontic/.

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4. Interdisciplinary Research in Logic and Computer Science This area focuses on the studies and applications in programming logic and procedural coding, logic programming language, design of conversion program, program verification and validation, program integration, formal semantics of programming language, and computational complexity theory, etc. 5. Inductive Logic An inductive logic is a system that provides evidence to extend deductive logic to lessthan-certain inferences. It attempts to quantify, formalize, and axiomatize inductive reasoning through employing mathematical logic, philosophical logic, and theory of probability and statistics. 6. Philosophy of Logic Philosophy of logic was derived from the interaction of modern logic and modern philosophy, which represents the intellectual reflection on issues arising in logic. It includes but not limited to the scope and nature of logic, the nature of logic truth, relationship between logic and other subjects, explanation of fundamental logic concepts, theory of meaning, logical paradox, and the nature of formalism, etc.

2.1.1 Origin of Mathematical Logic In as early as 17 century, scholars attempted to use quantitative method to map the logical reasoning in human beings’ thinking. Gottfried Wilhelm Leibniz (1646–1716), known as the “universal genius” during his era, ever thought to create a “universal scientific language” that quantifies all the reasoning processes using equations in mathematics. Given the constraints of that time, his thoughts were not practiced. Nevertheless, his thoughts sowed the seed of modern mathematical logic and he can be regarded the trailblazer for founding modern mathematical logic. In 1847, Britain mathematician George Boole published the “The Mathematical Analysis of Logic” his first work on symbolic logic. In this book, boolean algebra and associated symbolic system were created to represent all kinds of concepts in logic. Boole established series of laws of operations and algebraic approaches were employed to study logic issues, which laid the cornerstone for mathematical logic. From the end of 19 century to the beginning of 20 century, mathematical logic witnessed rapid development. In 1884, in Gottlob Frege’s book “The Foundations of Arithmetic”, the German philosopher, logician, and mathematician introduced the symbol for quantifier, which improved the symbolic system of mathematical logic. Other scholars including Charles Sanders Peirce (1839–1914) had also made significant contributions to consummate the symbolic system of mathematical logic. With contributions of so many dedicated scholars, mathematical logic finally becomes an independent subject.

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2.1.2 Contents of Mathematical Logic Mathematical logic includes first-order logic, high-order logic, axiomatic set theory, recursion theory, model theory, proof theory, mathematical fuzzy logic, and mathematical dialectical logic. Hereby, two important aspects-i.e. propositional calculus and predicate calculus are briefly introduced. Proposition is the sentence for being the primary bearers of truth and falsity. Propositional calculus is the branch of logic for studying more advanced and complicated propositions (whether they are true or false) through the use of logical connectives. If proposition is looked as the object of an operation such as the numbers, variables, and algebraic expression while logical connectives are treated as operator, e.g. addition, subtraction, multiplication, and division. And the process of forming compound proposition from simple propositions is called the propositional calculus. Proposition calculus, same as algebraic operation, follows certain principles and laws of operations. Those laws of operations include not only commutative law, associative law, and distributive law widely used in algebraic operation but also the laws of identity, the absorption laws, the laws of double negation, the De Morgan’s laws, and the laws of syllogism, etc. in logical operations. By using laws mentioned above, logical reasoning and compound proposition simplification can be conducted. Furthermore, equivalence of two compound propositions can be postulated, which is to verify that their logical truth tables are completely the same. One specific model of proposition calculus is the logical algebra, i.e., Boolean algebra. Boolean algebra is also called switching algebra and its basic operations are logical sum (OR), logical product (AND), and logical negation (NOT). The objects of operation only have two numbers 0 and 1, which correspond to the “True” and “False” in propositional calculus. The features of operation for propositional algebra are similar to “ON/OFF” and “High/Low potentials” for electronic circuits. Therefore, it has been widely applied to the analysis of electronic circuits. Electronic parts can be combined through logic sum (OR), logic product (AND), and logic negation (NOT) operations to form logic gate circuit. Logical circuit can be further constructed into more complicated logical network. Many complex logical relationships can be realized through certain combination of logical circuits, which makes logical circuits possess the capability of logical judging. By this means,logical circuits play critical roles in automatic control. Predicate logic is also called propositional term calculus. In predicate calculus, the internal structure of proposition is divided into the form of “subject” + “predicate”. Proposition is composed of term, logical connectives, and quantifiers. Logical reasoning relationship among propositions is hereby explored. Propositional term is the logical formulation that contains both constants and variables. Constants refer to the objects, quantifier, and relationship that have certainty or precise description. Variables are thought to be creatures but not to any certain ones, which refer to any value within certain range. Propositional terms, different from propositional calculus, do not have “True” or “False”. Propositional term becomes “True” or “False” proposition if variable is replaced with constant. Propositional term becomes universal or existential proposition if universal or existential quantifier is added.

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2.1.3 Research History of Mathematical Logic 2.1.3.1

Research History of Mathematical Logic in the World

Although Aristotle and his successors pioneered the systematic research on logic, it was mainly realized by terms in language instead of symbols widely used in the modern mathematical logic. Mathematician, logicians, and researchers in computer found it impossible for such terms to be used in computers. Leibniz was recognized as the very first logician that indicated the necessity for considering symbolic logic, which was reflected in his essay (1666) named “De arte Combinatoria”. In the essay mentioned above, he expressed his belief in the possibility of developing a common scientific language that can economically and effectively dictate reasoning procedures in logic research. Leibniz’s discussions and statements made significant contributions to the creation of symbolic logic and some concepts proposed by him are still playing important roles in contemporary studies in logic. In 1848, George Boole’s book titled “The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning” reignited logicians’ interests in symbolic logic. As the second book of Boole’s discussion on algebraic logic, “An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities” laid the foundation for the discipline of algebraic logic. Augustus De Morgan’s (1806–1871) book of “Formal Logic, or the Calculus of Inference, Necessary and Probable” intended to presents the quantification of the predicate or a concept of “numerically definite syllogism (NDS)”, which significantly advanced the development of symbolic logic. In his later works, Morgan did extensive studies on the relevance logic that was neglected in previous literature. As “the most original and versatile of American philosophers and America’s greatest logician”,3 Charles Sanders Peirce’s (1839–1914) works had made significant contributions to the development of logic into the fields of epistemology and the philosophy of science. His wisdom in seeing the possible connection between logical operations and electrical switching circuit made his prospective thoughts far exceeded what most scholars can imagine in 1880s. Ernst Schröder’s (1841–1904) work of “Vorlesungen Über die Algebra der Logik” provided a complete picture for Boolean algebraic logic. In reality, modern logicians tend to use Boole-Schröder Algebra to name traditional Boolean symbolic logic. Another path for studying symbolic logic was paved by the works of Gottlob Frege and Giuseppe Peano. Peano made pivotal contributions to the foundations of mathematical logic and set theory,4 which was motivated by his aspiration of representing all known formulas and and theorems of mathematical science using Peano notations invented by himself. In Peano et al.’s “Formulaire de mathematiques -Mathematics Form”, working with his collaborators, they aimed to use symbolic language invented by Peano to express then-known main mathematical formulas and theorems.5 In the 3

Weiss (1934). “Peirce, Charles Sanders”. Dictionary of American Biography. Arisbe. Giuseppe Peano: https://en.wikipedia.org/wiki/Giuseppe_Peano. 5 Mathematics form: https://fr.wikipedia.org/wiki/Formulaire_de_mathématiques. 4

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book of “Begriffsschrift-Concept-script”6 published in 1879, Frege presented the formal system. In “Grundgesetze der Arithmetik-Basic Laws of Arithmetic”7 the book with historical importance in logic research, Frege intended to employ basic logical concepts and symbols to completely represent arithmetic. The three-volume work “Principia Mathematica”8 written by Alfred North Whitehead and Russell in 1910, 1912, and 1913, with historical significance in the field of mathematics and philosophy, had popularized the symbolic logic, which was partially attributed to both Peano’s and Frege’s works. Working with Pual Bernays (1888–1977), David Hilbert (1862–1943)9 wrote the book of “Grundlagen der Mathematik-Foundations of Mathematics” with the intention of finding a new way of using symbolic logic to commensurate the subject of mathematics, which later has affected many mathematicians and logicians. The Journal of Symbolic Logic, founded in 1935, is a channel that publishes relevant studies in this area.

2.1.3.2

Research History of Logic and Mathematical Logic in China

Compared to research level in European countries and USA, there still exists gaps in studies of logic in China. In 1950 and 1960s, the main directions of research in logic focused on traditional formal logic, dialectical logic, and history of logic research in China and the size of scholars in modern logic research was very small. In 1978, the 1st National Symposium on Logic Research blew the trumpet for the logicians in China to march towards modernization of logic studies. From 1978 to 1990, research in logic witnessed rapid development and yielded copious achievements. In 1990, an awarding event sponsored by Jin Yuelin Academic Foundation conferred awards to 12 scholars that had made great contributions to logic research and majority of their studies touched various branches in modern logic and kept abreast of global high level of research in relevant areas. From 1990s onward, books and high-tier journal papers in modern logic continued to appear, which included modal logic, tense (temporal) logic, paraconsistent logic, modern inductive logic, logic of language, modern logic history, modern explanation of classic logic theory, and philosophy of logic. The main contributions are reflected in the following areas: 1. Modal logic and temporal logic: A method of developing a model by grafting one model on another one was used to resolve the integrity of system S1. Temporal logic was enriched and advanced by constructing minimal systems based on tense operators of U (until) and S (since) and tense operators of G (“It always will be the case until...”)and H (“It always was the case since...”), respectively.

6

Begriffsschrift: https://en.wikipedia.org/wiki/Begriffsschrift. Grundgesetze derArithmetik: https://de.wikipedia.org/wiki/Grundgesetze_der_Arithmetik. 8 Principia Mathematica: https://en.wikipedia.org/wiki/Principia_Mathematica. 9 Grundlagen der Mathematik: https://de.wikipedia.org/wiki/Grundlagen_der_Mathematik. 7

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2. Paraconsistent logic: Paraconsistent logic coined by Peruvian philosopher Francisco Miró Quesada Cantuarias deals with inconsistent-tolerant systems of logic that rejects the principle of explosion, whose initial discussion could be arguably dated back to 1910. Chinese scholars in the fields of logic and philosophy have developed some new paraconsistent logic such as paraconsistent conditional statement logical system, paraconsistent modal logical system, and paraconsistent temporal logical system. 3. Philosophic logic: Logical formal systems regarding “OR, OR Else” and “Which” were proposed. Neighborhood semantics that can be applied to generic propositional logic was developed, where portrayal framework and integrity of relevance logic were discussed. All existing theories regarding the logic of infinity were investigated for their connections and differences, the proofs for some theorems were provided, and theoretical significance of important results in the logic of infinity was analyzed, which made the logic of infinity become a comprehensive and systematic theory and significantly drove the developments of this theory. 4. Logic of natural language: By employing Mongtague grammar and generalized quantifier method in modern logic, partial statement system for Chinese quantitative phrases was established. Theory of logic concerning correct thinking and successful communication was proposed based modern logic. Liquan Zhou was recognized as the pioneer in the research of logic of natural language in China. In his book titled “Logic-Theory of Correct Thinking and Successful Communication”, Zhou put forward a system of logic of natural language-namely theory of successful communication, which is a two-triple-theory considering logic, grammar, rhetoric, syntax, semantics, and pragmatics. He improved the Paul Herbert Grice’s “Cooperative Principle (CP)” and “Theory of Conversational Implicatures (TCI)” and stated that implicature obtained through “CP” is also pragmatic. Zhou indicated that both grammatical implication and semantic entailment are necessary. In Zhou’s opinion, it is not necessary to find implicatures of statements in context although statements’ or proposition’s attitude is necessary. He further developed Frege’s and Strawson’s logical presupposition theory (LPT)10 by defining presupposition based on pre-defined rules for presupposition. He deemed that presupposition is pragmatic because it it related to ‘CP”. Zhou also presented a language communication schema. He contended that successful communication depends on successful conveyance and receiver’s accurate understanding on the intention of sender. The system proposed by Zhou, deeply affected by Georg Wilhelm Friedrich Hegel’s (1770–1831) logic, has apparent characteristics of reflection theory (RT). Zhou attempted to organize his theory into a scope system of scientific concepts. Zou Chongli’s main works in logic of natural language are in two books titled “Logic, Language, and Montague Grammar” and “Study on the Logic of Natural Language”. In the first book, Zou investigated challenging problems related to Montague grammar. For example, he simplified the meaning postulates for the 10

Presupposition: https://plato.stanford.edu/entries/presupposition/.

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Montague P T Q (The Proper Treatment of Quantification in Ordinary English) English sentence system and unified the two meaning postulates that involve common names, intransitive verbs, and individual concepts and proved several relevant inferences as well. He also simplified the description for the meaning postulate involving transitive verb and prepositions. In the works for logic of Chinese language , Zou established partial statement system for capturing quantified meanings of Chinese noun phrases and, especially differentiated the semantic features of specific reference and general reference. He examined the semantic features for the signified reference, generic use reference, indefinite reference, definite reference, and determinate reference in Chinese noun phrases and proposed the schemes for formalizing them. In the second book, Zou explored the semantic characteristics of Chinese tense aspects and employed seven set relationships about time interval in interval semantics (IS) to discuss seven tense types in Chinese-i.e., present, imperfect, future, aorist, perfect, pluperfect, and future perfect. This description can help differentiate the time phase and time system (present, past, and future), which created a exquisite Chinese system. In the semantic model, Zou defined temporal operators such as Pr og, Per f , and Gen and provided new ideas in studying temporal logic. 5. Dialectical logic: In Shunfu Jin’s works represented by two books: “Study on Dialectical Thinking” and “Dialectical Logic”, he thought dialectical logic is a subject about thinking by taking dialectic thinking as its research object. In his view, dialectical logic, being different from materialist dialectics and logic in the strict sense, is a logic with nature of philosophy. The relationship between dialectical logic and dialectics is the one between specific and general scopes while dialectical logic focuses on the dialectics of thinking that is unique to itself. As for issues of dialectical thinking and its models, Jin contended that dialectic thinking is the thinking where people get hold of the wholeness of objects in motion through a series of categories of unity of opposites in the thinking process. This is a thinking model for gaining better understanding on the complicated dynamic systems. Jin presented three basic features of dialectic thinking-i.e., concrete wholeness, complementary opposites, and spiraling upward by negating contradictions and operating mechanisms and their general models. As far as the methods of dialectical logic is concerned, Jin thought that relevant methods could have characteristics from both philosophical epistemology and logic methodology and conducted detailed analysis on those features. Yintong Zhuge argued that the so-called dialectical logic (i.e., logic unified with epistemology and dialectics or the unity of logic and history) is a kind of logic not in studying forms of reasoning but in exploring theory of truth, ideology, and philosophy. Zhuge thought that Hegel’s law of identity (LoI) is nothing but sophistry, which violates common senses and formal logic.

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6. Inductive logic: Xiaowu Li proposed a inductive cognitive logic and constructed a inductive logic system with cognitive operators and characterized general connections between probabilistic reasoning. Li introduced measurement function into the relational semantics of model logic. Based on such concept, Li constructed a kind of possible world model and proved that the formal system mentioned above is reliable. Jialong Zhang popularized John Stuart Mill’s (1806–1873) five methods of induction11 and put forward five new inductive reasoning methods-i.e., comparative experiment (CE), joint method of difference and comparison (JMDC), joint method of statistics, agreement, and difference (JMSAD), joint method of sampling and difference (JMSD), and joint method of sampling and comparison (JMSC). 7. History of logic: Yunzhi Zhou systematically studied the development histories of mathematical logic and modal logic. He also conducted a comprehensive review on history of logic in China and published a five-volume masterpiece titled “History of Logic in China”. The book “Ayer” written by Jialong Zhang is a monograph that systematically studies the analytical philosophy of Alfred Jules Ayer (1910–1989). In the book, Zhang discussed the development process of Ayer’s philosophical thought and described his philosophical features during the period of different philosophical movements. He also compared Ayer’s works with the empiricist tradition in UK and phenomenological traditions in Vienna Circle and indicated that Ayer’s philosophical thoughts were the synthesis of the two traditions, which also put an end for both traditions. Ayer’ s works bestowed a new form on empiricism. In the other book “Bradley”, Jialong Zhang systematically studied the philosophical thoughts of Francis Herbert Bradley (1846–1924) who was representative figure of Neo-Hegelianism. Zhang argued that Bradley’s absolute idealism system was composed of internal-relation-based ontology and epistemology, his theory of logic was constructed by dialectical judgment theory and inference theory, and his ethics was characterized by self-actualization. In the last section of his book, Zhang compared Bradley’s philosophy with that of Hegel’s and clarified the differences and similarities between the two. Lu Wang, in his book “Into Analytical Philosophy”, reviewed relevant thoughts and references for analytical philosophy (AP) and discussed how AP was derived from modern logical ideas and methods. He summarized the significant role of logic in the development of philosophy. He stated that, since Western philosophy originated from ancient Greek philosoph´ιa (ϕιλoσ oϕ´ια, the love of wisdom), the discussions on Western philosophy, especially metaphysics have been focusing on the combination of “nature or essence” and “truth” while translation and understanding of “existence” and “truth” in China severed their organic connections and consequently generated biased and even erroneous thoughts on essential features of Western metaphysics. Yuqing Zhang examined and expanded Youding Shen’s (1909–1989) paradox and proposed non-Z paradox. Zhang also reviewed 11

Mill’s Methods: https://en.wikipedia.org/wiki/Mill%27s_Methods.

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and summarized several famous paradoxes such as Russell’s paradox,12 Youding Shen’s paradox, and Curry’s paradox.13 Xiaowu Li discussed the definition of logic and pointed out that a reliable and complete formal system relative to a model class is called logic. Li also revealed the differences between specific logic and logic as a subject (discipline). Li investigated several popular definitions for logic and showed their limitations. Jialong Zhang discussed the scientificalness of Saul Aaron Kripke’s (1940–) and Hilary Whitehall Putnam’s (1926–2016) essentialism and provided new explanation on essentialism by using possible world semantics (PWS). Shangshui Zhang advanced the research on the reference theory of proper names and indicated that proper names are not the abbreviations of an array of descriptivists but the referents. The development of logic in China has witnessed great success. Philosophers and logicians have introduced principles of logic, logic of philosophy, history of logic, temporal logic, deontic logic, conditional logic, paraconsistent logic, modern deductive logic, modern inductive logic, linguistic logic, and informal logic to China and have conducted independent research accordingly and so far have made great progress in each branch mentioned above. In the study of inductive logic and artificial intelligence, scholars, by combining cognitive science and modern inductive logic, pioneered the new field of applying inductive logic. In the area of linguistic logic, researchers reviewed and evaluated the works and contributions in the area of speech acts theory and pragmatics of world well-known linguistic logicians such as John Langshaw Austin (1911–1960),14 John Rogers Searle (1932–),15 and Daniel Vanderveken (1949–2019).16 Based on the above research, they constructed relevant logic system and carried out in-depth research in meta logic. Some scholars attempted to conduct systematic research on Chinese forms, semantic, and pragmatics by using methods in modern logic and obtained some preliminary results. Philosophical issues related to logic have been explored, which include object of logical study, logical truth, evaluation of formal methods, implication, meaning, and paradox. An in-depth analysis of “possible worlds” was also carried out. In the history of logic, important thoughts in logic from well-known logicians and philosophers such as Frege,17 Georg Henrik von Wright (1916–2003),18 and Willard Van Orman Quine (1908–2000)19 were reviewed and evaluated. As for the future of logic, modern deductive logic has made new progress around the world, where relevant research is being integrated with philosophy of science and epistemology of science. The exploration of modal inductive logic and epistemological inductive logic marks the transition of inductive logic from being mathematics12

Russell’s paradox: https://en.wikipedia.org/wiki/Russell%27s_paradox. Curry’s paradox: https://en.wikipedia.org/wiki/Curry%27s_paradox. 14 John Langshaw Austin: https://iep.utm.edu/austin/. 15 John Rogers Searle: https://en.wikipedia.org/wiki/John_Searle. 16 Daniel Vanderveken: https://fr.wikipedia.org/wiki/Daniel_Vanderveken. 17 Friedrich Ludwig Gottlob Frege: https://en.wikipedia.org/wiki/Gottlob_Frege. 18 Georg Henrik von Wright: https://en.wikipedia.org/wiki/Georg_Henrik_von_Wright. 19 Willard Van Orman Quine: https://en.wikipedia.org/wiki/Willard_Van_Orman_Quine. 13

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oriented to philosophy-oriented. Significant results have been made in philosophical logic and study in cognitive logic has become a hot spot. Different branches in non-monotonic logic have also attracted attentions from scholars. In the research of linguistic logic, semantics theory is experiencing accelerated development and new trend is seen in the combination of semantics and pragmatics. Logic, by combining with modern technology, has become a pivotal role in driving the development of computer science and artificial intelligence, among others. Interdisciplinary research between logic and other subjects such as philosophy, linguistics, cognitive science, artificial intelligence, computer science, law, ethics, economics, and management science is becoming more popular. The relationship between theoretical research of logic and applied research is getting more closer than ever. The research on applied logic theory and application techniques and methods of logic will become a new driving force for the development of logic. Two key research areas of logic in China are as follows: 1. Modern logic Scholars should focus on non-classical logic and explorations on the frontier theories of logic while continuing to emphasizing basic research in mathematical logic. More attentions should be paid to branches of logic that have not yet been given in-depth studies such as dynamic logic (DL), partial logic (PL), deontic logic, conditional logic, and multi-agent cognitive logic (MACL). 2. History of logical thought and comparative logic Main studies can be categorized into following directions: (1) (2) (3) (4) (5) (6)

The history of emergence and development of logic in China; Types of reasoning in ancient China; History of the spread and development of western logical thoughts in China; The interaction between logic and cultural construction in China; Thoughts of contemporary world well-known logicians and philosophers; Comparing and summarizing the results of modern and contemporary research in logic; (7) Carrying out research on various branches of logic and promoting the enrichment and perfection of the logic system.

2.1.4 Development of Mathematical Logic Mathematical logic has experienced accelerated development since its inception. Of course, multiple factors have contributed to its growth. For example, the establishment of non-Euclidean geometry makes researchers investigate the consistency of mathematics for both non-Euclidean geometry and Euclidean geometry, which, as a results, drove the development of mathematical logic. The emergence of set theory was one of the significant events over the course of modern mathematics development. However, in the research of set theory, the paradox of set theory created the so-called third crisis in the history of mathematics. In 1903, the Britain idealism philosopher, logician, and mathematician Russell

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raised the “Russell paradox of set theory” which almost shook the cornerstone of mathematics. There are many examples in “Russell paradox”. One of the common examples is the “barber paradox”. One day, a barber living in a village claimed that he only shaves beard for the person who does not shave beard by himself. Then, an issue appeared here whether he will shave beard for himself. He is the person who shaves himself if he shaved his beard. Per his principle, he should not shave himself. However, he becomes the person who does shave himself if he did shave himself. According to his claim, he should shave beard for himself. This generated the paradox, which encouraged many mathematicians to study the consistency of set theory. This consequently led to the generation of axiomatic set theory an important branch in mathematical logic. The birth of non-Euclidean geometry and the finding of paradox of set theory indicated that there exist many issues in the theoretic system of mathematics. In order to examine the consistency of mathematics, the concepts, propositions, and proofs in the theoretic system of mathematics are treated as objects of research. The logical structure and laws of proof in the theoretic system of mathematics are explored, which gave rise to the proof theory another important branch in propositional logic. Recursion theory and model theory are two latest branches in propositional logic. Recursion theory focuses on the study of computability, which has close relationship with the development of computer science. Model theory mainly explores the relationship between formalism system and the mathematical model. The accelerated development of mathematical model benefited from the fact that this subject has made significant impacts on the development of other mathematical branches such as set theory, number theory, algebra, and topology. It has especially driven the rapid development of computer science. Of course, the development of other subjects also pushed the growth of mathematical logic.

2.1.5 Research Status of Mathematical Logic In 20 century, the development of mathematical logic attained great achievements exemplified in the following cases. In 1931, Alfred Gödel put forward the “Gödel’s incompleteness theorems” which include two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency. In 1933, Tarski published a very long paper named “The Concept of Truth in Languages of Deductive Sciences”, which laid foundation for the development of logic semantics. In 1937, Alan Turing proposed his work “On computable numbers, with an applica-

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tion to the Entscheidungsproblem-decision problem”, which can be regarded as the foundation of computer science and of the artificial intelligence program. In 1920s, Jan Łukasiewicz (1878–1956) invented the Polish parenthesis-free notation for the logical connectives. Logicians in China proposed connective-free propositional logic system where parenthesis can act as connectives. Axiomatic set theory is the research using modern axiomatized method to reestablish Cantor set theory. Recursion theory is also called computability theory, which was derived from the research on algorithm. The major purpose of computability theory is to study computing complexity of computable objects and the structure of uncomputable objects. It is to explore the essence of computability and relative computability. It also serves as the theoretic foundation for computer science. Several precise definitions on essence of computing algorithms were proposed in 1930s, which include λ-definable function, recursive function, Turning computable function, and Post’s recursively enumerable sets, etc. Due to the failure of searching for judging process for system of predicate calculus, people turned to the research on uncomputability and found many uncomputable examples. Model theory is a emerging subject dedicated to studying models of theories in formal language and relationship between models. It is also a theory that examines the relationship between semantical elements (meaning and truth) and syntactical elements (formulas and proofs) of formal language. Modern proof theory was thought to be established by David Hilbert in 1920s. Hilbert proposed the so-called Hilbert’s program for proving that if there is a way to find the finitary proofs of consistency for all the complex formal theories needed by mathematicians. Since 1970s, the major characteristics of research on mathematical logic were to focus on the application of newly-derived approaches and theories to mathematics and computer science. It involves analytics, algebra, topology, and combinatorics. For example, first-order logic has very extensive application in computer science. In China, logicians also made remarkable contributions to the development of mathematical logic. Youding Shen (1908–1989) who was famous for finding “ShenYou-Ding Paradox” in 1950s was the pioneer researcher in mathematical logic. Shen developed a system for logical calculus in 1981, which is a narrow predicate calculus with equivalents. Qingyu Zhang accomplished two main achievements in classic logic: (1) created a first-order logic systems without using connectives and quantifiers, in which unique arrangements for parentheses were used to possess the functions of both connectives and quantifiers, representing new logical symbols and notation systems after Łukasiewicz; (2) Zhang built an axiomatic system Z using general disjunction as initial symbol, which is similar to system AB proposed by Alan Ross Anderson (1925–1973) and Nuel Belnap (1930–). Zhang also opened up a new direction in the field of paraconsistent logic and he created conditional logic PIW and CnW, modal logic CnG  , and temporal logic CnG  H  and CnU S, etc. Jialong Zhang, based on modern logic, constructed a tree-shaped natural deduction system for Aristotelian syllogism (AS) and modal syllogism (MS) and proved “all be valid (ABV)” and excluded “all be invalid (ABI)” using formal exclusion method,

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which resolved the decision problem of categorical syllogism and syllogisms with necessary premises. Zhang also constructed the possible world model (PWM) for syllogisms with necessary premises. Xiaowu Li made the following achievements in logical calculus: (1) constructed several axiomatic systems of infinite logic and proved that these systems have important properties of meta theorem; (2) explored the connections and difference among all known indefinite logic and supplemented the proofs for some theorems in his book “all be valid (ABV)”; (3) reviewed and summarized the relations among the neighborhood semantics, relational semantics, and class-selective semantics for conditional logic and the inclusion relations of the above identified systems; (4) investigated the appropriateness of the subjunctive conditional logic and proposed a conditional logic being capable of capturing subjunctive implication association. Xuegang Wang created the minimal temporal logic G  and H  . In the application of logic, great improvements have been made in the interdisciplinary research between logic and computer science. For instance, in program specification, program function can be described by logical formulas; for semantic characterization of programs, axiomatic semantics of programs can be illustrated by logical system; in program verification and validation, logic theorem proving can be used to replace the proof of program correctness; in automatic theorem proving, principles of logic can be employed to determine the provability; in software development, logic-based formalism can be used to aid the process of development; in the hardware/software (HW/SW) co-design, logical tools can be adopted to conduct feasible HW/SW functional partitioning, etc. Logic is indispensable theoretic foundation and practical tool. The well-known computer scientist John McCarthy ever foretold that the relationship between logic and computer science in 21 century is equivalent to the relationship between mathematical analysis and physics. Nevertheless, logic is not omnipotent. Due to the intrinsic limitations and difficulties of logic and computer science, many problems in computer science can be resolved using logic. For example, logic method can not still resolve the efficiency issue that computer science must consider; logic-based formalism has not yet become the mainstream technology; the proof of correctness of large-scale software is still facing the difficulties of combinatorial explosion. In particular, logic-based artificial intelligence has not gained breakthrough in its development and is even experiencing certain stagnation. “Artificial intelligence and the limits of artificial intelligence” were listed as the prominent issues facing mathematics in 21 century. Therefore, studying on machines that can understand human knowledge and intelligence is the most difficult and complicated issues of human being. We believe that logic will gain more extensive and profound applications than ever in computer science with the development of mathematical logic and continuous driving impacts of computer science on logic. In the early stage of 21 century, the prominent features for the development of mathematical logic are: (1) theories in mathematical logic such as axiomatic set theory, recursive theory, and model theory will gain further improvement and development; (2) the newly developed logic that fits the need of social development will emerge. Mathematical logic will continue to investigate the following issues: (1) studies on independence issues and large cardinal theory are still the two primary themes

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in axiomatic set theory; (2) computational complexity theory and local theory will be the major fields on which research of computability theory focus, Lachlan’s use of trees (even in 0 -priority arguments) is still the very necessary tool and approach even though it is very complicated in the early stage of 21 century; (3) model theory and it’s associated methodologies might be used to resolve more difficult mathematical problems and frontier issues in modern mathematics; (4) tense logic and deontic logic maybe need to be established and improved; (5) formalism of dialectical logic; (6) logical laws for changes and transformations of things; (7) logical laws for errors. Moreover, different theories in mathematical logic will be integrated, fused, and/or unified to drive the development of mathematical logic. For instance, when studying set theory, recursive theory, decidability of logical reasoning, and computational complexity, new theories and methodologies derived from the above four aspects can be applied to investigate issues and problems in mathematics and computer science. The derived theories involve not only analytics, algebra, extensionics, and combinatorics but also more branches in modern mathematics, which help resolve many difficult mathematical problems and frontier issues in those subjects. In 21 century, the starting point for research in mathematical logic is to use it to investigate and develop the concepts, structures, and frontiers issues in mathematics as well as to provide more practical mathematical models, deduction tools, and computational methods. Theories and methods in mathematical logic are also used to advance the development of different branches in philosophical logic and consequently the development of philosophical logic itself.

2.1.6 Formalism of Dialectical Logic in China Regarding the formalism of dialectical logic, Yizhong Luo made mathematical analysis on “I Ching” and formalized it mathematically. What Luo has done is to integrate dialectical logic into formal calculus of the classic mathematical logic to form a normal/reverse mathematical logic which unifies classic formal logic and dialectical logic and contains the major elements of epistemic logic and dialectical philosophical methodology. Qiguan Gui, for the purpose of realizing formalization of dialectical logic, proposed ten propositions including feasibility of reasonable reconstruction, limitations of law of contradiction, correspondence principle, and non-classic negation. Shunfu Jin pointed out that formalization of dialectical logic is determined by both characteristics of this subject and readiness of the conditions. So far, dialectical logic is still an immature subject with many theoretic issues needing further study and solution. For example, the structure, mechanism, and method for the thinking process of ascending from the abstract to the concrete. Under this circumstance, what kind of formal languages are used to realize formalization. Some scholars have examined the reasoning of dialectical logic and obtained the relationship between research of dialectical logic and creative thinking. Some other scholars have investigated the characteristics and operating mechanisms of

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dialectical logic and made significant contributions in the formalization of dialectical logic. This type of research has both theoretical and practical significance. Of course, doubts on the formalization of dialectical logic has been raised by some scholars, too. Zongkuan Zhao, in his book “Introduction of Mathematical Dialectical Logic”, has done pioneering work in the formalization of dialectical logic based on the research of attribute set theory

2.2 Error Logical System 2.2.1 Existential Quantifiers (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

∃LY , there exists certain domain; ∃SW , there exists certain thing; ∃S J , there exists certain time; ∃K J , there exists certain spatial location; ∃T J , there exists certain constraint; ∃T Z , there exists certain property; ∃L Z , there exists certain value of the property or attribute; ∃C Z , there exists certain error value; ∃H S, there exists certain function; ∃G Z , there exists certain group of rules; ∃F S, there exists certain decomposition method; Θ there exists certain universe of discourse.

2.2.1.1 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

Universal Quantifiers

∀LY , for each domain; ∀SW , for each thing; ∀S J , for the whole time; ∀K J , for each spatial location; ∀T J , for each constraint; ∀T Z , for each property; ∀L Z , for each value of the property or attribute; ∀C Z , for each error value; ∀H S, for each function; ∀G Z , for each group of rules; ∀F S, for each decomposition method; Ψ for the whole universe of discourse.

2.2 Error Logical System

2.2.1.2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

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Denotative Connectives

¬ negation, ∧ conjunction, ∨ disjunction, ∨bxr exclusive disjunction: or → material implication: if · · · then, ← inverse material implication: only · · · if, ↔ if and only if-biconditional logical connective, = equal, ˙ small AND operation, ∧ ¨ equal AND operation, ∧ ∧ large AND operation, ˙ small OR operation, ∨ ¨ equal OR operation, ∨ ∨ large OR operation.

2.2.1.3

Transformation Connectives

(1) Similarity transformation connectives, Tx ⊆ { Txly , Txsw , Txk j , Txt z , Txlz , Txcz , Txhs , Txs j , Txgz , Tx zh } (similarity), Tx−1 : inverse similarity transformation con−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 , Txsw , Txk nectives, Tx−1 ⊆ { Txly j , Txt z , Txlz , Txcz , Txhs , Txs j , Txgz , Tx zh }; (2) Displacement transformation connectives, Tz ⊆ { Tzly , Tzsw , Tzk j , Tzt z , Tzlz , Tzcz , Tzhs , Tzs j , Tzgz , Tzzh } (displacement), Tz−1 : inverse displacement transformation −1 −1 −1 −1 −1 −1 −1 connectives, Tz−1 ⊆ { Tzly , Tzsw , Tzk−1j , Tzt−1z , Tzlz , Tzcz , Tzhs , Tzs−1j , Tzgz , Tzzh }; (3) Increase transformation connectives, Tz n ⊆ { Tznly , Tznsw , Tznk j , Tznt z , Tznlz , Tzncz , Tznhs , Tzns j , Tzngz , Tznzh } (increase), Tz n −1 : inverse increase transformation −1 −1 −1 −1 −1 −1 −1 −1 −1 , Tznsw , Tznk connectives, Tz n −1 ⊆ { Tznly j , Tznt z , Tznlz , Tzncz , Tznhs , Tzns j , Tzngz , −1 }; Tznzh (4) Decomposition transformation connectives, T f ⊆ { T f ly , T f sw , T f k j , T f t z , T f lz , T f cz , T f hs , T f s j , T f gz , T f zh } (decomposition), T f−1 : inverse decomposition trans−1 −1 −1 −1 −1 −1 −1 formation connectives, T f−1 ⊆ { T f−1 ly , T f sw , T f k j , T f t z , T f lz , T f cz , T f hs , T f s j , −1 T f−1 gz , T f zh }; (5) Destruction transformation connectives, Th ⊆ { Thly , Thsw , Thk j , Tht z , Thlz , Thcz , Thhs , Ths j , Thgz , Thzh } (destruction), Th−1 : inverse destruction transformation −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 , Thsw , Thk connectives, Th−1 ⊆ { Thly j , Tht z , Thlz , Thcz , Thhs , Ths j , Thgz , Thzh }; −1 (6) Unit transformation connective, Td (unit),Td : inverse unit transformation connective; (7) Quantifier system of error logic.

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2.2.1.4

Connotative Connectives

(1) ¬bz , not only negation: there exists error that can be negated before being decomposed; (2) ¬bj , unfinished negation: there exists error that can be negated after being decomposed; (3) ¬bx , unconstrained negation: for the error being negated, there exists its opposite side before being decomposed; (4) ¬bd , uninterrupted negation: for the error being negated, there exists its opposite side after being decomposed; (5) ∧n , connotative conjunction: different errors are compatible; (6) ∨n , connotative disjunction: different errors have mutual infiltration; (7) −n , connotative difference: errors were removed or reduced; (8) |n f l , connotative separation: different errors coexist; (9) |n f h , connotative differentiation: errors with critical points do not exist; (10) nhb , connotative complement: correctness, error with critical points, and error coexist; (11) nhdl , connotative antithesis: correctness and error coexist; (12) →nhy , connotative possibility implication: if . . . , then it is possible; (13) →nby , connotative necessity implication, if . . . , then it must be or it could not be otherwise; (14) →nsy , connotative isness implication: if . . . , then it is . . . ; (15) ↔nhdz , connotative equivalence: if...then it is equivalent; (16) =nhdt , connotative same connective: properties are the same or two errors are the same.

2.2.1.5 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

Concept of Predicate Logic

D (C) , rare case; D (T ) , special case; D (Y B Q) , generic case; D (Y BG) , generic concept; D (C W Q) , erroneous case; D (FC W Q) , non-erroneous case; D (Z Q Q) , correct case; D (C W L Q) , case of error with critical points; D (CC W Q) , pure error case (without critical points); D (RY C W J ) , set of arbitrary error logical variables; D (QC W J ) , set of all error logical variables; D (B FC W J ) , set of partial error logical variables; D (C W J ) , set of error logical variables; D (FC W J ) , set of non-erroneous logical variables; D (L J C W J ) , set of logical variables for error with critical points ; D (Z Q J ) , set of correct logical variables;

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(17) D (W QC W J ) , set of complete error logical variables; (18) D (C Z L J C W J ) , set of pure positive critical error logical variables; (19) D (C Z Q J ) , set of absolute correct (without critical points)logical variables.

2.2.1.6

Error Set

(1) Concepts; (2) Operation of error sets; (3) N-nary error set;

2.2.1.7 (1) (2) (3) (4) (5) (6)

Mathematical Error Propositional Logic

Concepts of error logic; System of error logic; Atomic proposition; Compound proposition; Principles of reasoning in error logic; Propositional paradigm in error logic;

2.2.1.8

Relationship Between Transformation and Denotative Connectives in Error Logic

Relationship between transformation connective and ¬ negation; Relationship between transformation connective and ∧ conjunction; Relationship between transformation connective and ∨ disjunction; Relationship between transformation connective and ∨bxr exclusive disjunction; Relationship between transformation connective and → material implication: if · · · then; (6) Relationship between transformation connective and ← inverse material implication: only · · · if; (7) Relationship between transformation connective and ↔ if and only if-biconditional logical connective; (1) (2) (3) (4) (5)

2.2.1.9

Relationship Between Transformation and Connotative Connectives in Error Logic

(1) Relationship between transformation connective and ¬bz , not only negation; (2) Relationship between transformation connective and ¬bj , unfinished negation; (3) Relationship between transformation connective and ¬bx , unconstrained negation;

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(4) Relationship between transformation connective and ¬bd , uninterrupted negation; (5) Relationship between transformation connective and ∧n , connotative conjunction; (6) Relationship between transformation connective and ∨n , connotative disjunction; (7) Relationship between transformation connective and −n , connotative difference; (8) Relationship between transformation connective and |n f l , connotative separation; (9) Relationship between transformation connective and |n f h , connotative differentiation; (10) Relationship between transformation connective and nhb , connotative complement; (11) Relationship between transformation connective and nhdl , connotative antithesis; (12) Relationship between transformation connective and →nhy , connotative possibility implication: if . . . , then it is possible; (13) Relationship between transformation connective and →nby , connotative necessity implication, if . . . , then it must be or it could not be otherwise; (14) Relationship between transformation connective and →nsy , connotative isness implication: if . . . , then it is . . . ; (15) Relationship between transformation connective and ↔nhdz , connotative equivalence: if...then it is equivalent; (16) Relationship between transformation connective and =nhdt , connotative same connective: properties are the same or two errors are the same.

2.2.1.10

Error Predicate Logic

(1) (2) (3) (4)

Form and representation of predicate logic; Semantic interpretation for representation of predicate logic; Truth value for quantified subject-predicate form of predicate logic; Reasoning rules and theorems for quantified subject-predicate form of predicate logic; (5) Paradigm decision method for quantified subject-predicate form of predicate logic; (6) Truth tree decision method for quantified subject-predicate form of predicate logic.

2.2.1.11

Applications of Error Logic

(1) Applied theory; (2) Practical examples

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In theory, the construction for the system of mathematical error logic will consummate the system of error theory and enlarge the research domain of generic logic system. In practice, error logic has been successfully applied to stock and security investment, decision support system, computer science, and artificial intelligence, etc. The development of error logic connectives will become propelling forces for error theory to address more issues and complicated systems.

Chapter 3

Error Sets

In order to quantitatively investigate and capture errors, it is necessary to have a set of mathematical methods to describe errors and conduct corresponding operations on errors. In this chapter, we introduce the definition of error sets, operations on error sets, and the laws that those operations must satisfy.

3.1 Concepts of Error Sets This section discusses the concept of error sets, categorization of error sets, and operations on error sets and their corresponding laws.

3.1.1 Error Sets 3.1.1.1

Definitions

Definition 3.1 Suppose that U (t) is an object set, G(t) is a set of rules for judging # » error, if C = {((U (t), S(t), p(t), T (t), L(t)), x(t) = f (G  u(t))) | (U (t), S(t), # » p(t), T (t), L(t)) = u(t) ∈ U (t), f ⊆ ×R, x(t) = f (G  u(t))} then C is called an error set defined on U (t) under the rule of judging errors G(t), and UC = {u(t) | (u(t), x(t)) ∈ C, x(t) > 0} U Z = {u(t) | (u(t), x(t)) ∈ C, x(t) < 0} U L = {u(t) | (u(t), x(t)) ∈ C, x(t) = 0}

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Liu and K. Guo, Error Logic: Paving Pathways for Intelligent Error Identification and Management, Studies in Systems, Decision and Control 442, https://doi.org/10.1007/978-3-031-00820-7_3

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is called a domain with errors (erroneous domain),1 a domain without errors (correct domain),2 and a domain with critical points (critical domain)3 of error set C, respectively, where U K = {u(t) | (u(t), x(t)) ∈ C, x(t) ≥ 0, T ( f (G  u(t))) > 0} U K H = {u(t) | (u(t), x(t)) ∈ C, x(t) ≤ 0, T ( f (G  u(t))) > 0} U K L = {u(t) | (u(t), x(t)) ∈ C, T ( f (G  u(t)) = 0} UH = UZ − UK H U S = UC − U K is called, with regard to transformation T, a domain tending to generate no errors,4 a domain tending to generate errors,5 a domain tending to generate error with critical points,6 an absolute benign domain,7 and a vicious domain, respectively.8 And R represents the universe of discourse of all real numbers. ⎧ (1) there is contradiction between u(t) and G(t) ⎪ ⎪ ⎨ (2) none of u(t) can be deduced from G(t) G(t)  u(t) includes (3)u(t) can only be partially deduced from G(t) ⎪ ⎪ ⎩ (4) it is not sure if u(t) can be deduced from G(t) In this definition, f (G  u(t) is generally represented by f (G  u(t)). # » Definition 3.2 If ∀((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) ∈ C, and x(t) ∈ {0, 1}, then C is a typical error set.9 # » Definition 3.3 If ∀((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) ∈ C, and x(t) ∈ [0, 1], then C is a fuzzy error set. # » Definition 3.4 If ∀((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) ∈ C, and x(t) ∈ (-∞, +∞), then C is a error set with critical points. # » Definition 3.5 If ∀((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) ∈ C, and x(t) > 0, then C is a complete error set noted by Cq . # » Definition 3.6 If ∀((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) ∈ C, and x(t) ≤ 0, then C is a set without errors noted by Cw . The value of error is x(t) > 0 before making any transformation. Where value of error is x(t) ≤ 0 before making any transformation. 3 Where value of error function approaches zero. 4 Where transformation enables error to be changed to be nonerroneous. 5 Where transformation engenders erroneous results fromnonerroneous case. 6 Where transformation produces either erroneous or non-erroneous results. 7 Where value of error x(t) ≤ 0, even with transformation exerted, value of error x(t) ≤ 0 still holds. 8 Where the value of error x(t) > 0, even with transformation exerted, value of error x(t) > 0 still holds. » 9 If time is not emphasized, U(t), G(t) and #p(t) are represented by U, G, and #» p respectively. 1 2

3.1 Concepts of Error Sets

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# » Definition 3.7 Suppose that C1 = {((U, S(t), p(t), T (t), L(t)), x(t) = f 1 (G 1  # » u(t))) | U, S(t), p(t), T (t), L(t))=u(t) ∈ U, f 1 ⊆ U × R, x(t)= f 1 (G 1  u(t))} # » # » and C2 = {((U, S(t), p(t), T (t), L(t)), y(t) = f 2 (G 2  u(t))) | (U, S(t), p(t), T (t), L(t)) = u(t) ∈ U, f 2 ⊆ U × R, y(t) = f 2 (G 2  u(t))}. As ∀u(t) ∈ U , and ∀(u(t), x(t)) ∈ C1 , ∀(u(t), y(t)) ∈ C2 , if both x(t)=y(t) and G 1 = G 2 hold, then C1 and C2 are equal under the rules of judging errors G 1 or G 2 , i.e., C1 = C2 . # » Proposition 3.1 In domain U , if G 1 = G 2 , f 1 = f 2 , C1 = {((U, S(t), p(t), T (t), # » L(t)), x(t) = f 1 (G 1  u(t))) | (U, S(t), p(t), T (t), L(t)) = u(t) ∈ U , f 1 ⊆ U × # » R, x(t) = f 1 (G 1  u(t))} and C2 = {((U, S(t), p(t), T (t), L(t)), y(t)= f 2 (G 2  # » u(t))) | (U, S(t), p(t), T (t), L(t))=u(t) ∈ U, f 2 ⊆ U × R, y(t)= f 2 (G 2  u(t))}, and then C1 = C2 , vice versa. Proof With both G 1 = G 2 and f 1 = f 2 . For ∀ u(t) ∈ U, x(t) = f 1 (G 1  u(t))= f 2 (G 2  u(t))=y(t). So, for ∀ u(t) ∈ U , x(t)=y(t) holds when (u(t), x(t)) ∈ C1 and (u(t), y(t)) ∈ C2 . Then C1 = C2 . On the other side, G 1 = G 2 holds if C1 = C2 . And with U if f 1 = f 2 , for ∃ u(t)∈ U, f 1 (G 1  u(t)) = f 2 (G 2  u(t)) holds. So, we have x(t)= y(t) holds when (u(t), x(t)) ∈ C1 and (u(t), y(t)) ∈ C2 . This contradicts the fact that C1 = C2 . Therefore, f 1 = f 2 hold here when C1 = C2 . We conclude that {G 1 = G 2 and f 1 = f 2 }⇔ C1 = C2 . Definition 3.8 If C1 ⊆ C, C1 is called an error subset of C. For simplicity, C1 is called a subset of C. Definition 3.9 If ∀(u(t), x(t)) ∈ C, ∃Tu , Tu (u(t), x(t)) = (u(t), y(t)), y(t) = x(t) does not always hold in U , then C is called a transformable error set. Definition 3.10 If ∀(u(t), x(t)) ∈ C, ∃Tu , y(t)  x(t) holds in U as Tu (u(t), x(t)) = (u(t), y(t)) and y(t) = x(t) does not always hold in U , then C is called an extentionable error set. Definition 3.11 If ∃(u(t), x(t)) ∈ C, there does not exist Tu that makes y(t) < 0 hold when Tu (u(t), x(t)) = (u(t), y(t)), then C is called a dead error set. Definition 3.12 If ∀(u(t), x(t)) ∈ C, there does not exist Tu that makes y(t)  0 hold when Tu (u(t), x(t)) = (u(t), y(t)), then C is called a completely dead error set.

3.1.1.2

Categorization of Error Set

1. Categorization based on properties of elements: # » (1) Classical error set C = {((U, S(t), p(t), T (t), L(t)), x(t)= f (u(t), G(t)) | # » (U, S(t), p(t), T (t), L(t)) = u(t) ∈ U , f ⊆ U × {0, 1}, x(t) = f (G  u(t))}

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# » (2) Fuzzy error set C = {((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) | # » (U, S(t), p(t), T (t), L(t)) = u(t) ∈ U , f ⊆ U × [0, 1], x(t) = f (G  u(t))} # » (3) Error set with critical points C={((U, S(t), p(t), T (t), L(t)), x(t)= f (u(t), # » G(t)) | (U, S(t), p(t), T (t), L(t)) = u(t) ∈ U , f ⊆ U × (−∞, +∞), x(t) = f (G  u(t))} 2. Categorization based on properties of error: (1) Transformable error set (2) Modifiable error set (3) Extentionable error set (4) Dead error set (5) Completely dead error set Note: the categorization here does not represent all categories.

3.1.2 Operations of Error Sets and Their Laws 3.1.2.1

Union of Error Sets

Definition 3.13 Suppose that C1 , C2 are two subsets of error set defined on judg# » ing rule G in domain U , if C3 ={((U, S(t), p(t), T (t), L(t)), z(t) = f (u(t), G(t))) # » # » | ((U, S(t), p(t), T (t), L(t)), z(t) = f (u(t), G(t)))∈ C1 or ((U, S(t), p(t), T (t), L(t)), z(t) = f (u(t), G(t)))∈ C2 }, C3 is called the union operation of C1 and C2 , denoted by C3 = C1 ∪ C2 . Proposition 3.2 C1 ∪ C2 = C2 ∪ C1 . Proposition 3.3 C1 ∪ C1 = C1 .

3.1.2.2

Intersection of Error Sets

Definition 3.14 Suppose that C1 , C2 are two subsets of error set defined on judging # » rule G in U , if C3 ={((U, S(t), p(t), T (t), L(t)), z(t) = f (u(t), G(t))) | ((U, S(t), # » # » p(t), T (t), L(t)), z(t) = f (u(t), G(t)))∈ C1 and ((U, S(t), p(t), T (t), L(t)), z(t)= f (u(t), G(t)))∈ C2 }, C3 is called the intersection operation of C1 and C2 , denoted by C3 = C1 ∩ C2 . Proposition 3.4 C1 ∩ C2 = C2 ∩ C1 . Proposition 3.5 C1 ∩ C1 = C1 . Proposition 3.6 C1 ∩ (C2 ∩ C3 )=(C1 ∩ C2 ) ∩C3 . Proposition 3.7 If C1 ⊆ C2 , then C1 ∩ C2 = C1 .

3.1 Concepts of Error Sets

3.1.2.3

41

Combined Operations of Union and Intersection of Error Sets

Proposition 3.8 Suppose C1 , C2 , and C3 are three subsets of error set defined on judging rule G in domain U , then (1) C1 ∩ (C2 ∪ C3)=(C1 ∩ C2 ) ∪(C1 ∩ C3 ) (2) C1 ∪ (C2 ∩ C3)=(C1 ∪ C2 ) ∩(C1 ∪ C3 ) Proposition 3.9 Suppose C1 and C2 are two subsets of error set defined on judging rule G in domain U , the sufficient and necessary conditions for C1 ∪ C2 = C2 to hold is C1 ∩ C2 = C1 Proposition 3.10 Suppose C1 , C2 and C3 are three subsets of error set defined on judging rule G in domain U , then (1) When C1 ⊂ C2 , we have C1 ∪ C3 ⊂ C2 ∪ C3 ; (2) When C1 ⊂ C2 , we have C1 ∩ C3 ⊂ C2 ∪ C3 .

3.1.2.4

Combined Operations of Union and Intersection of Error Sets under Different Rules

Suppose that G 1 and G 2 are two rules for judging errors defined in domain U . Definition 3.15 Suppose that, in U, C1 and C2 are two subsets of error set defined on # » judging rules G 1 and G 2 respectively, and C3 ={((U, S(t), p(t), T (t), L(t)), z(t) = # » f (u(t), G(t))) | (U, S(t), p(t), T (t), L(t))=u(t) ∈ U, z(t) = max(x(t), y(t))# » f 1 (x(t), y(t), G 1 , G 2 ), ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G 1 (t))) ∈ C1 , # » ((U, S(t), p(t), T (t), L(t)), y(t) = f (u(t), G 2 (t))) ∈ C2 }, then C3 is called the ¯ 2. union of C1 and C2 for rules G 1 and G 2 in U , denoted by C3 = C1 ∪C Definition 3.16 Suppose that, in U, C1 and C2 are two subsets of error set defined on # » judging rules G 1 and G 2 respectively, and C3 ={((U, S(t), p(t), T (t), L(t)), z(t) = # » f (u(t), G(t))) | (U, S(t), p(t), T (t), L(t))=u(t) ∈ U, z(t) = min(x(t), y(t))+ # » f 2 (x(t), y(t), G 1 , G 2 ), ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G 1 (t))) ∈ C1 , # » ((U, S(t), p(t), T (t), L(t)), y(t) = f (u(t), G 2 (t))) ∈ C2 }, then C3 is called the ¯ 2. intersection of C1 and C2 for rules G 1 and G 2 in domain U , denoted by C3 = C1 ∩C Among which, f (x(t), y(t), G 1 , G 2 ) represents coefficient between x(t), y(t) and G 1 , G 2 , and 0 ≤ f (x(t), y(t), G 1 , G 2 ) | x-y |, we have: |x − y| [1 − R(G 1 , G 2 )], 2 |y − x| R(G 1 , G 2 ) (2) f 2 (x(t), y(t), G 1 , G 2 ) = 2 R(G 1 , G 2 ) represents coefficient of rules G 1 and G 2 , it satisfies the condition of 0 ≤ R(G 1 , G 2 )  1. (1) f 1 (x(t), y(t), G 1 , G 2 ) =

42

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3 Error Sets

Complement of Error Sets

Definition 3.17 Suppose that C1 is a subset of error set defined on judging rule G in U, C is the error set defined on judging rule G in U , if C2 = C − C1 , then C2 is called the complement set of C1 with respect to C, simply speaking, C2 is called the complement set of C1 , noted as C2 = C¯1 . Proposition 3.11 (1) C1 = C1 Proposition 3.12 (1) C = ∅ (2) ∅ = C Proposition 3.13 (1) C1 ∩ C2 = C1 ∪ C2 (2) C1 ∪ C2 = C1 ∩ C2

3.2 Transformation of Error Sets As indicated by one of the three laws of dialectics, “the law of the unity and conflict of opposites”, error and correctness always exist in the same context. The conflicting opposites, under certain circumstances, transform to the other side. With lapse of time and change in space, something correct may become erroneous if investigated from different disciplinary perspective or research objective, and vice versa. Therefore, it is necessary to ask the question what the mechanisms or rules for those transformation are. In this chapter, through examining the transformation of error set, we investigate the mechanisms, methodologies, and rules. In our research, we found that the transformation of error sets not only has relationship with element of error set but also has relationship with domain U , G(t) judging rules for errors, and the binary relationship between U and G(t) thereof. # » In the following discussion, ((U, S(t)), p(t), T (t), L(t)), z(t) = f (u(t), G(t))) is expressed by (u(t), x(t)), even simpler as (u, x), G(t) is expressed as G.

3.2.1 Definition of Transformation of Error Sets 1. Definition Definition 3.18 Suppose that C is an error set defined on judging rule G in domain U , if T (C) = {g | c ∈ C, g = T (c)}, then the process of finding solutions for T(C) is called a type of transformation of C.

3.2 Transformation of Error Sets

43

2. Analysis on Definition According to the definition of error set, element c = (u, x) in error set C is a tuple (two elements) composed of element u in object set and element x in R. As u ∈ U, x = f (G  u), therefore in tuple c, u has relationship with domain and x has relationship with both f and G. That is to say the change in c = (u, x) directly affects U , the relationship f , and G. On the other hand, c = (u, x) will change accordingly when any transformation has be made to U , the relationship f , and G. In the definition of transformation, element g is obtained with transformation made on element c in C i.e., g = (v, y) = T (u, x). With the transformation operation on element c = (u, x), we have u → v and x → y. And the element c in C becomes g. Hereby, the following changes will be considered: (1) The change of element in error set i.e., c ∈ C and g ∈ C; (2) The change of element in error set causes the change on U ; (3) The change of element in error set causes the change on rules of jugding error G; (4) The change of element in error set causes the change on f ; (5) The change of element in error set causes the simultaneous change on U and f ; (6) The change of element in error set causes the simultaneous change on f and G; (7) The change of element in error set causes the simultaneous change on U and G; (8) The change of element in error set causes the simultaneous change on U, G, and f. Here we mainly discuss the methods and laws for transforming c = (u, x) to g = (v, y). And we also investigate the transformation methods and laws of inverse change of the above 8 cases i.e.: (1) (2) (3) (4) (5) (6) (7) (8)

The direct transformation of elements in error set C; The change of U ; The change of G; The change of f ; The simultaneous change of U and f ; The simultaneous change of f and G; The simultaneous change of U and G; The simultaneous change of U, G, and f . # » As element c in C, (u, x) = ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))), # » and u = (U, S(t), p(t), T (t), L(t)), x = x(t) = f (u(t), G(t)). Therefore, the transformation on element of error set can be classified into transformation regarding # » domain U , time t(temporal transformation), entity/thing S(t), space p(t) (spatial transformation), properties T (t), property (attribute) value L(t), error value x(t), error function f , and judging rules G(t), and their combination thereof.

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3.2.2 Transformation Operations of Error Sets 1. Conjunction of transformations Definition 3.19 Suppose that two transformations T2 and T1 act on an error set C sequentially, then the relationship between T2 and T1 is conjunction noted by T1 )(C) i.e., (T2 ∧ T1 )(C) = T1 (T2 (C)). (T2 ∧ ∼ ∼ 2. Disjunction of transformations Definition 3.20 Suppose that at least one of two transformations T2 and T1 acts on an error set C, then the relationship between T2 and T1 is disjunction noted by T1 )(C) i.e., (T2 ∨ T1 )(C) = T1 (C) ∨ T2 (C). (T2 ∨ ∼ ∼ ∼ 3. Inverse transformation Definition 3.21 If T1 (c1 ) = c2 , T2 (c2 ) = c1 , then we call the relationship between T2 and T1 reciprocal transformation noted by T1 = T2−1 or T2 = T1−1 4. Rules of transformation operations Proposition 3.14 If the action of T1 and T2 on error set C has no sequential order, T1 = T1 ∧ T2 . then T2 ∧ ∼ ∼ Proposition 3.15 If the action of T1 and T2 on error set C has no sequential order, T1 = T1 ∨ T2 . then T2 ∨ ∼ ∼ Proposition 3.16 (T1−1 )−1 = T1 Proof Suppose T1−1 = T2 , then T2−1 = T1 , therefore (T1−1 )−1 (c) = T2−1 (c) = T1 (c).

3.2.3 Types and Operation Rules of Transformation 1. Unit transformation Definition 3.22 If ∀c ∈ C, there is T (c) = c, then T is called the unit transformation of error set C defined on judging rule G in U noted by Td . 2. Displacement transformation Definition 3.23 If ∀c ∈ C, there is T (c) = b, then T is called the displacement transformation of error set C defined on judging rule G in U noted by Tz In the definition, if (1) If ∀c ∈ C, c = b holds, then Tz is degenerated into unit transformation. (2) If ∃c ∈ C, c = b holds, a. If b ∈ / C, then Tz has possibly made transformation on U, f , and G;

3.2 Transformation of Error Sets

45

i. If c = (u 1 , x1 ), b = (u 2 , x2 ), and u 2 ∈ / U , then Tz must have made transformation on domain U ii. If c = (u 1 , x1 ), b = (u 2 , x2 ), and u 2 ∈ U , the Tz may not have transformation on domain U b. If b ∈ C, then Tz has possibly made transformation on elements of error set C, f , and G;

(1) Element displacement transformation Definition 3.24 If Tz makes no transformation on domain U , judging rule G, and relationship f , then Tz has made element transformation on C noted by Tzy . Therefore in C, ∀c1 ∈ C and Tz (c1 ) = c2 , then c2 ∈ C i.e., Tz (u 1 , x1 ) = (u 2 , x2 ) = c2 and c2 ∈ C. Definition 3.25 ∀c ∈ C, there has T (u 1 , x1 ) = (u 2 , −x1 ), then T is called the negative displacement transformation noted by Tz f . Proposition 3.17 Tz f (Cq ) = Cw Proof ∀c1 ∈ Cq , c1 = (u 1 , x1 ), x1 > 0 and Tz f (c1 ) = Tz f (u 1 , x1 ) = (u 2 , −x1 ) = (u 2 , x2 ), x2 = −x1 < 0, i.e., (u 2 , x2 ) ∈ Cw . Therefore Tz f (Cq ) = Cw Proposition 3.18 Suppose (u 1 , x1 ), (u 2 , x2 ) ∈ C, G is a group of scientific rules for judging error set C in domain U, Tzy represents the unit transformation and Tzy (u 1 , x1 ) = (u 2 , x2 ), if x1 = x2 holds, then u 1 = u 2 holds. Proof According to definition 3.22, Tzy does not change domain U , relationship f , and judging rules G of error set C; in U under G when u 1 = u 2 , x1 = x2 , i.e., for (u 1 , x1 ), (u 2 , x2 ), x1 = x2 holds under judging rule G in U , this contradicts the facts that G is scientific in U and relationship f does not change (G has intrisic contradiction or relationship f has been changed), therefore, the above conclusion holds. (2) Domain displacement transformation Definition 3.26 If Tz changes the domain U of C, then Tz is called the domain displacement transformation of error set C defined under judging rules G in domain U noted by Tzl . From definition, Tzl changes domain (regarding G) of error set C. When domain U of C changes, it may, on one side, render G in the new domain invalid or unscientific. On the other side, it is possible to find the (u, x) under judging rule G. (3) Displacement transformation of judging rule G Definition 3.27 If Tz changes the judging rule G of C, then Tz is called the displacement transformation of judging rule G of error set C in domain U noted by Tzg . Based on definition, we know that Tzg will cause the changes of error set C.

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(4) Displacement transformation of relationship f Definition 3.28 If Tz changes the relationship f of C, then Tz is called the displacement transformation of relationship f of error set C in domain U noted by Tze .

(5) Comprehensive displacement transformation Definition 3.29 If Tz changes any two or more factors among domain U , judging rule G, the relationship f , and element of error set C, then Tz is called the comprehensive displacement transformation of error set C noted by Tzh . Proposition 3.19 Suppose that C J is a classical error set defined under G in domain U, ∀c ∈ C J , Tzh (c) = Tzh (u, x) = (v, y), and the following relationship 

(1) y < x when x > 0 (2) y = x when x = 0

holds, then Tzh (C J ) = C W . (Proof ommited). 3. Decomposition transformation Definition 3.30 If ∀c ∈ C, T (ci ) = {ci1 , ci2 , . . . , cini , }, n i ≥ 2, then T is called the decomposition transformation on error set C noted by T f . In the definition, T f (ci ) = {ci1 , ci2 , . . . , cini , }, n i ≥ 2. If c = (u, x), cik = (vik , xik ), k = 1, 2, . . . , n i , then T f (u i , xi ) = {(vi1 , xi1 ), (vi2 , xi2 ), . . . , (vini , xini )}, ∀ci ∈ C, vik ∈ U holds, k = 1, / U , and 2, . . . , n i , the domain U of error set C does not change; if ∃ci ∈ C, vik ∈ then the domain U of error set C has been changed, at the same time, it may cause the change of judging rule G and relationship f of error set C. There are several types of decomposition as follows:

1. Element decomposition transformation Definition 3.31 If T f does not change the domain U , judging rule G, and relationship f , then T f is called element decomposition transformation noted by T f y . And the # » element in error set C has the formulation of (u, x) = ((U, S(t), p(t), T (t), L(t)), # » x(t) = f (u(t), G(t))), i.e., u = (U, S(t), p(t), T (t), L(t)), x = x(t) = f (u(t), G(t)). Therefore, element decomposition transformation can be classified into: domain transformation U , entity/thing transformation S(t), spatial transformation # » p(t), property transformation T (t), property value transformation L(t), error value transformation x(t), error function transformation f , judging rule transformation G(t), and the transformation of their combination thereof (two or more of them combined).

3.2 Transformation of Error Sets

47

Proposition 3.20 Suppose that C is an error set defined under G in domain U, C1 and C2 are two subsets of C, ∀ci ∈ C, i=1, 2.T f y (ci )=T f y (u i , xi )={(vi1 , xi1 ), (vi2 , xi2 ), . . . , (vin , xin )} = {ci1 , ci2 , . . . , cin }. ∀ci , c j ∈ C, when i = j and ci = c j , cik1 = cik2 k1 = 1, 2, . . . , n j holds. T f y (C1 ∩ C2 ) = T f y (C1 ) ∩ T f y (C2 ). Proof ∵ C1 ∩ C2 = {(u, x) | (u, x) ∈ C1 ∧ (u, x) ∈ C2 } ∴T f y (C1 ∩ C2 ) = T {(u i , xi ) | (u i , xi ) ∈ C1 ∧ (u i , xi ) ∈ C2 } = {(vik , xik ) | (vik , xik ) ∈ T f y (Ci ), ci ∈ C1 ∧ ci ∈ C2 } T f y (C1 ) ∩ T f y (C2 ) = {(vik , xik ) | (vik , xik ) ∈ T f y (C1 ) ∩ (vik , xik ) ∈ T f y (C2 )} ∵ ci ∈ C1 ∧ ci ∈ C2 ∴T f y (Ci ) ⊆ T f y (C1 ) ∧ T f y (Ci ) ⊆ T f y (C2 ) ∴T f y (C1 ∩ C2 ) ⊆ T f y (C1 ) ∩ T f y (C2 ) ∵ ∀ci , c j ∈ C, i  = j, ci  = c j T f y (Ci ) = {Ci1 , Ci2 , . . . , Cin i } T f y (C j ) = {C j1 , C j2 , . . . , C jn j } Cik1  = C jk2 , k1 = 1, 2, . . . , n i , k2 = 1, 2, . . . , n j ∴T f y (C1 ) ∩ T f y (C2 ) ⊆ T f y (C1 ∩ C2 ) ∴T f y (C1 ∩ C2 ) = T f y (C1 ) ∩ T f y (C2 )

Proposition 3.21 Suppose that C is an error set defined under G in domain U, C1 and C2 are two subsets of C, T f y (C1 ∪ C2 ) = T f y (C1 ) ∪ T f y (C2 ). Proof ∵ C1 ∪ C2 = {(u, x) | (u, x) ∈ C1 ∩ (u, x) ∈ C2 } I f ci ∈ C1 ∧ ci ∈ C2 T henT f y (Ci ) ⊆ T f y (C1 ) ∨ T f y (C2 ) I f ci ∈ C1 , ci ∈ C2 T henT f y (Ci ) ⊆ T f y (C1 ) ∴T f y (Ci ) ⊆ T f y (C1 ) ∨ T f y (C2 ) Similarly, i f ci ∈ C2 , ci ∈ / C1 T henT f y (Ci ) ⊆ T f y (C1 ) ∴T f y (C1 ∪ C2 ) ⊆ T f y (C1 ) ∪ T f y (C2 ). I f ci ∈ C1 , thenci ∈ (C1 ∪ C2 ) ∴T f y (C1 ) ⊆ T f y (C1 ∪ C2 ). Similarly, T f y (C2 ) ⊆ T f y (C1 ∪ C2 ) T f y (C1 ∪ C2 ) = T f y (C1 ) ∪ T f y (C2 )

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2. Domain decomposition transformation Definition 3.32 If T f changes the domain U of error set C, i.e., T f (U ) = U1 ∪ U2 ∪, . . . , ∪Un , then T f is called domain decomposition transformation noted by T f l . For T f l , it decomposes element c into c1 , c2 , . . . , cn and it may sometimes does not change the particular elment of C but the domain. For example, if the number of element of C is infinite, the derived elements from decomposition transformation are still elements of error set C. 3. Decomposition transformation of relationship f Definition 3.33 If T f changes the relationship f of error set C definfed under G in domain U , i.e., T f ( f ) = f 1 ∪ f 2 ∪, . . . , ∪ f 2 , then T f is called the decomposition transformation of relationship f noted by T f g .

4. Comprehensive decomposition transformation Definition 3.34 If T f simultaneously changes two or more of the factors among element of error set C, domain U , judging rule G, the relationship f , then T f is called the comprehensive decomposition transformation of error set C noted by T f z . 4. Combination transformation Definition 3.35 Suppose that ci1 , ci2 , . . . , cini (n i ≤ 2) are elements of error set C, if T (cg1 , cg2 , . . . , cgn g ) = ci , then T is called the combination transformation of c noted by Tzu . In the definition, for all ci ∈ C, i ∈ {1, 2, . . . , n, . . .}, then Tzu does not make combination transformation on C; for all i ∈ {1, 2, . . . , n, . . .}, at least one element does not belong to C, then it is possible that Tzu has made combination transformation on domain U , jugding rule G, and relationship f . Combination transformation Tzu is actually inverse operation of decomposition transformation T f . Similarly, we can define 1) element combination transformation Tzuy = T f−1 y , 2) domain combination transformation Tzul = T f−1 , 3) combination transformation on relationship T f−1 ly gx , and −1 4) comprehensive combination transformation T f zh . Proposition 3.22 Suppose that C is an error set defined under G in domain U, C1 and C2 are two subsets of C, if ci1 , ci2 , . . . , cini ∈ C1 , cg1 , cg2 , . . . , cgn g ∈ C2 , ∃cik1 = cgk2 , k1 ∈ {1, 2, . . . , n j }, k2 ∈ {1, 2, . . . , n g }, T Tzuy (ci1 , ci2 , . . . , cini ) = ci = Tzuy (cg1 , cg2 , . . . , cgn g ) = cg , then Tzuy (C1 ∩ C2 ) = Tzuy (C1 ) ∩ Tzuy (C2 ).

3.2 Transformation of Error Sets

49

Proof I f ci1 , ci2 , . . . , cini ∈ C1 ∩ C2 , then ci1 , ci2 , . . . , cini ∈ C1 , cg1 , cg2 , . . . , cgn g ∈ C2 , ∴Tzuy (C1 ∩ C2 ) ⊆ Tzuy (C1 ) ∩ Tzuy (C2 ). I f ci ∈ Tzuy (C1 ) ∩ Tzuy (C2 , ) T henci ∈ Tzuy (C1 ), ci ∈ Tzuy (C2 ). Supposethat{ci1 , ci2 , . . . , cini } ⊆ C1 , {cg1 , cg2 , . . . , cgn g } ⊆ C2 , T henTzuy (ci1 , ci2 , . . . , cini ) = ci , Tzuy (cg1 , cg2 , . . . , cgn g ) = c j , ∴n i = n g , cik = cgk , k1 = 1, 2, . . . , n j , I f not, ∃cik1 = cgk2 , cik1 ∈ {ci1 , ci2 , . . . , cini } ⊆ C1 , cik2 ∈ {cg1 , cg2 , . . . , cgn g } ⊆ C2 , T hus, Tzuy (ci1 , ci2 , . . . , cini ) = Tzuy (cg1 , cg2 , . . . , cgn g ) ∴{ci1 , ci2 , . . . , cini } = {cg1 , cg2 , . . . , cgn g } ⊆ (C1 ∩ C2 ) ∴Tzuy (C1 ) ∩ Tzuy (C2 ) = Tzuy (C1 ∩ C2 ) ∴Tzuy (C1 ∩ C2 ) = Tzuy (C1 ) ∩ Tzuy (C2 ) Proposition 3.23 Suppose that C is an error set defined under G in domain U, C1 and C2 are two subsets of C, if ci1 , ci2 , . . . , cini ∈ C1 , cg1 , cg2 , . . . , cgn g ∈ C2 , then Tzuy (C1 ∪ C2 ) ⊇ Tzuy (C1 ) ∪ Tzuy (C2 ). Proof I f {ci1 , ci2 , . . . , cini } ⊆ C1 , then {ci1 , ci2 , . . . , cini } ⊆ C1 ∪ C2 , I f {cg1 , cg2 , . . . , cgn g } ⊆ C2 , . T hen{cg1 , cg2 , . . . , cgn g } ⊆ C1 ∪ C2 , ∴Tzuy (C1 ∪ C2 ) ⊇ Tzuy (C1 ) ∩ Tzuy (C2 ).

Proposition 3.24 Tzuy (Φ) = Φ. 5. Destruction transformation # » Definition 3.36 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((Φ, Φ), Φ, Φ, Φ)), Φ = Φ(Φ, Φ))), then # » T is called the destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Th . The meaning of destruction is: Th (destruction transformation) −→ {kill, eradicate, annihilate, disappear, fire, sell out, discard, move away, leave, . . . }.

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# » Definition 3.37 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((Φ, S(t), p(t), T (t), L(t)), x(t)= f (u(t), # » G(t))), then T is called the domain destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thly . The meaning of domain destruction is: Thly (domain destruction) −→ domain does not exist −→ this domain does not apply in this case. # » Definition 3.38 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))= ((U, Φ, p(t), T (t), L(t)), x(t)= f (u(t), G(t))), then T is called the thing (entity) destruction transformation on ((U, S(t)), # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thsw . The meaning of thing(entity) destruction is: Thsw (thing/entity destruction) −→ thing/entity does not exist −→ it is not necessary to discuss this thing/entity in this case. # » Definition 3.39 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((U, S(t), Φ, T (t), L(t)), x(t)= f (u(t), # » G(t))), then T is called the spatial destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thk j . The meaning of spatial destruction is: Thk j (spatial destruction) −→ space does not exist −→ this space is not needed in this case. # » Definition 3.40 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((U, S(t), p(t),Φ, L(t)), x(t)= f (u(t), # » G(t))), then T is called the property destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Tht x . The meaning of property destruction is: Tht x (property destruction) −→ properties do not exist −→ those properties do not apply in this case. # » Definition 3.41 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((U, S(t), p(t), T (t), Φ),x(t)= f (u(t), G(t))), then T is called the property value destruction transformation on ((U, S(t)), # » p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Thlz .

3.2 Transformation of Error Sets

51

The meaning of property (attribute) value destruction is: Thlz (property (attribute) value destruction) −→ this property (attribute) value does not exist −→ property (attribute) value does not apply in this case. # » Definition 3.42 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((U, S(t), p(t), T (t), L(t)), Φ = f (u(t), # » G(t))), then T is called the error value destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thcz . The meaning of error value destruction is: Thcz (error value destruction) −→ error value does not exist −→ this error value does not apply in this case. # » Definition 3.43 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((U, S(t), p(t), T (t), L(t)), x(t)=Φ(u(t), G(t))), then T is called the error function destruction transformation on ((U, S(t)), # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thhs . The meaning of error function destruction is: Thgs (error function destruction) −→ error function does not exist −→ this error function does not hold in this case. # » Definition 3.44 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((U, S(t), p(t), T (t), L(t)), x(t)= f (u(t), # » Φ)), then T is called the judging rule destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thgz . The meaning of judging rule destruction is: Thgz (judging rule destruction) −→ judging rule does not exist −→ this judging rule does not hold in this case. # » Definition 3.45 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T # » # » ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))=((U, S(Φ), p(t), T (Φ), L(Φ)), x(Φ) = f (u(Φ), G(Φ))), then T is called the temporal destruction transformation # » on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Ths j . The meaning of temporal destruction is: Ths j (Temporal destruction) −→ time factor does not apply −→ this time range does not hold in this case. # » Definition 3.46 Suppose that ((U, S(t), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)), x(t) = f (u(t), G(t)))=((Φ, Φ, p(t), T (t), L(t)), x(t) = f (u(t),

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G(t))) or =((Φ, Φ, Φ, T (t), L(t)), x(t) = f (u(t), G(t))) or =((U, Φ, Φ, T (t), L(t)), x(t) = f (u(t), G(t))), or · · · · · · =((Φ, Φ, Φ, Φ, Φ), Φ = Φ(Φ), Φ)), then T is # » called the multi-factor destruction transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t)) defined under judging rule G in domain U , it is noted by Thqb . The meaning of multi-factor destruction is: Thqb (multi-factor destruction) −→ two or more factors do not exist −→ two or more factors do not hold in this case. Proposition 3.25 Th (C) = Φ. The creation transformation is actually the inverse operation of destruction transformation noted by Th−1 . Discussion on this aspect is omitted. 6. Increase transformation # » Definition 3.47 Suppose that ((U , S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U , S(t)), # » # » p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U , S(t), p(t), T (t), L(t)), x(t) = f (u(t), # » G(t))), ((U1 , S1 (t), p(t) p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), # » # » p(t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t), G 2 (t))), . . ., ((Un , Sn (t), p(t), Tn (t), L n (t)), xn (t) = f n (u n (t), G n (t)))}, then T is called the increase transformation on ((U , S(t)), # » p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz j . # » In the transformation of T ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))= # » {((U, S(t), p(t), T (t), L(t)), x(t)= f (u(t), G(t))), ((U1 , S1 (t), #p»1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), #p»2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 (u 2 (t), G 2 (t))), . . ., ((U , S (t), #p»(t), T (t), L (t)), x (t) = f (u (t), G (t)))}, if n

n

n

n

n

n

n

n

n

(1) Domain increase transformation ((U1 , S1 (t), #p»1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 (u 1 (t), G 1 (t))) ∈ U1 (t), ((U2 , S2 (t), #p»2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 (u 2 (t), G 2 (t))) ∈ U2 (t), . . ., ((Un , Sn (t), #p»n (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))) · · · Un (t), and U (t) → U (t) ∪ U1 (t) ∪ U2 (t), . . . , ∪Un (t), in U1 (t), U2 (t), . . . , Un (t), at least there is Ui (t) = Φ, Tz j is called the domain # » increase transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jly . Under this circumstance, U (t) → U (t) ∪ U1 (t) ∪ U2 (t), . . . , ∪Un (t) is to make increase transformation on the domain U of object u(t) in order to achieve the expected objective. For instance, while studying the regional economy of Shanghai, we have to extend our research domain from Shanghai herself to the pan-Yangtze River Delta (YRD) region considering the reciprocal impacts between Shanghai and the pan-YRD. It may even be necessary to extend domain to whole China and the world. (2) Thing/entity increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))), here S(t) →

3.2 Transformation of Error Sets

53

S(t)+S1 (t)+S2 (t), + · · · , + Sn (t), Tz j has made the thing/entity increase transforma# » tion on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jsw . For example, it is assumed that the domain under consideration is university A and Si (t) is i school. As university A is going to diversify and strengthen its research by adding a new school, i.e., School of Innovation ([1]). (3) Spatial increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  # » G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))), here p(t) →       # »  p(t)+ p1 (t)+ p2 (t), + · · · , + pn (t), pi (t) ∈ { p1 (t), p2 (t), . . . , pn (t)}, Tz j has made # » the spatial increase transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jk j . For example, company A has planned to promote its product in southeast China. However, having conducted market research, company A determines to add southwest China in the plan. Thus, this involves adding new location (or expanding space) to current object of interest. (4) Property increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n (u n (t), G n (t))), here T → T ∪ T1 ∪ T2 ∪, . . . , ∪Tn , Ti ∈ {T1 , T2 , . . . , Tn }, Tz j has made the property increase # » transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jt x . Considering a critical part for a machine, aside from previously defined property T , two more properties “surface roughness T1 ” and “insulation performance T2 ” are added to the specifications. (5) Property (attribute) value increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))), here L(t) → L(t)+ L 1 (t) + L 2 (t) +, . . ., + L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, Tz j has made # » property (attribute) value increase transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jlz . For example, two more color choices (purple L 1 (t)and blue L 2 (t) are added to the original design (Red, Gray, Silver, and Chocolate represented by L(t) ) of a product, e.g., purse. (6) Error value increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))), here x(t) →

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x(t) + x1 (t) + x2 (t) +, . . ., + xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, Tz j has # » made error value increase transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jcz . For example, it is assumed that x(t) stands for the risk value of using $20,000 to buy stock A. The risk value increases to be x(t) + x1 (t) ( and x(t) = x(t) +x1 (t) ) if $10,000 more are spent to buy stock A. (7) Judging rule increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))), here G(t) → G(t) ∪ G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G i (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, Tz j has # » made judging rule increase transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jgz . For example, in order to strengthen antitrust enforcement, more rules and articles (G 1 (t), G 2 (t), . . . G n (t)) are designed and added to the current legal system. (8) Error function increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n (u n (t), G n (t))), here f (t) → f (t) + f 1 (t) + f 2 (t) + · · · , + f n (t), f i (t) ∈ { f 1 (t), f 2 (t), . . . , f n (t)}, Tz j has # » made error function increase transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz j hs . For ´ example, new error function has been changed from f ïijLt)= trig function to be f (t)=trig function + logarithm function. (9) Temporal increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n (u n (t), G n (t))), here t → t + t1 + t2 + . . ., + tn , ti ∈ {t1 , t2 , . . . , tn }, Tz j has made error function increase transfor# » mation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz js j . In order to increase the effectiveness of dieting, the health consultant encourages those obese children to increase exercise time from 7 hours per week to 12 hours per week. (10) Comprehensive increase transformation  # » Tz j ((U, S(t)), p(t), T (t), L(t)),x(t) = f (u(t), G(t)))={((U1 , S1 (t), p1 (t), T1 (t), 

L 1 (t)), x1 (t) = f 1 (u 1 (t), G 1 (t))), ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 (u 2 (t),  G 2 (t))), . . ., ((Un , Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n (u n (t), G n (t))), here Tz j ω has made comprehensive increase transformation on ((U, S(t)), p, T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tz jq . The comprehensive increase transformation includes simultaneous changes made on

3.2 Transformation of Error Sets

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universe of discourse (UD), thing/entity, property, property (attribute) value, error function, time, space, and judging rules for error. Proposition 3.26 Suppose that C is an error set defined under G in domain U, C1 and C2 are two subsets of C, if ci , cg ∈ C and i = g, ci ∈ C1 , cg ∈ C2 , Tz j (ci )= {ci , vi1 , vi2 , . . . , vini }, Tz j (cg ) ={cg , vg1 , vg2 , . . . , vgn g }, vik1 = vik2 , k1 ∈ {1,2, . . . , n j }, k2 ∈ {1,2, . . . , n g }, then Tz j (C1 ∪ C2 ) = Tz j (C1 ) ∪Tz j ( C2 ). The discussion on inverse operation of Tz j (i.e.,Tz−1 j ) is omitted here. 7. Similarity transformation

# » Definition 3.48 Suppose that ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) is an element of error set C defined under judging rule G in domain U , if T ((U, S(t)), # » # » p(t), T (t), L(t)),x(t)= f (u(t), G(t)))=((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), # » G  (t))), then T is called the similarity transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Tx . (1) Domain similarity transformation # » # » In ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))), if Tx ((U, S(t), p(t), T (t), # » L(t)) , x(t) = f (u  (t), G  (t)))=((U  , S(t), p(t), T (t), L(t)), x(t) = f (u  (t), ω G  (t))), then Tx is called the domain similarity transformation on ((U, S(t)), p, T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txly . In this case, if U1 (t) = kU2 (t), k > 0, U2 (t) is used to replace U1 (t), or U1 (t) is used to replace U2 (t). For example, when talking about labor resources in China, the domain of Guangdong province is U2 (t) and the domain of whole China is U1 (t), there exists a factor k between U1 (t) and U2 (t). (2) Thing/entity similarity transformation # » # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S  (t), p(t), T (t), L(t)), x(t) = f (u  (t), G  (t))), then Tx is called the thing/entity similarity transfor# » mation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txsw . In most cases, Txsw generally makes geometric similarity transformation. (3) Spatial similarity transformation  # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S(t), p (t), T (t), 





L(t)), x(t) = f (u  (t), G  (t))), p (t) ∈ [ p (t) - ε, p (t)+ε], then Tx is called the spa# » tial similarity transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txk j . (4) Property similarity transformation # » # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S(t), p(t), T  (t), L(t)), x(t) = f (u  (t), G  (t))), then Tx is called property similarity transformation # » on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txt x . Considering advantages of aerodynamic design of rocket and airplane, the design can provide some reference for high-speed train, which is an example of property (shap) similarity transformation.

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(5) Property (attribute) value similarity transformation # » # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S(t), p(t), T (t),     L (t)), x(t) = f (u (t), G (t))), then Tx is called property (attribute) value similarity # » transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txlz . For example, product A has silver color and similarity transformation can be made to have more variations, e.g., Silver Chalice, Silver Sand, and Silver Tree, etc. (6) Error value similarity transformation # » # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u(t), G(t))), x  (t) ∈ [x - ε, x+ε], then Tx is called the error value # » similarity transformation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txcz . (7) Judging rule similarity transformation # » # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u(t), G  (t))), then Tx is called the judging rule similarity transfor# » mation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txgz . For instance, G 1 (t) represents the regulations in university A in 2003 and G 2 (t) stands for regulations in same university in 2010. G 1 (t) was replaced with G 2 (t). (8) Error function f similarity transformation # » # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U, S(t), p(t), T (t), L(t)) , x(t) = f  (u(t), G(t))), then Tx is called the error function f similarity trans# » formation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txhs . (9) Comprehensive similarity transformation  # » If Tx ((U, S(t), p(t), T (t), L(t)) , x(t) = f (u  (t), G  (t))) = ((U  , S  (t), p (t), T  (t), L  (t)), x(t) = f  (u  (t), G  (t))), then Tx is called the comprehensive similarity trans# » formation on ((U, S(t)), p(t), T (t), L(t)), x(t) = f (u(t), G(t))) defined under judging rule G in domain U , it is noted by Txq . This transformation Txq has simultaneously made similarity changes in domain, thing/entity, property, property (attribute) value, error function, error value, and judging rules or their certain combination thereof. (10) Inverse similarity transformation All the following similarity transformations including Tx ⊆ { Txly , Txsw , Txk j , Txt x , Txlz , Txgz , Txhs , Txq } have inverse similarity transformation Tx−1 .

3.2.4 Transformation and Elimination of Errors Based on the definition of transformation, we know that the study on transformation is to examine the methodology and principles used to make changes on element (u, x), domain U , judging rule G, and the relationship f of error set C. The objective

3.3 Classic Error Set

57

is to identify the methodology and laws that change error set C into non-erroneous set. Among the element (u, x), domain U , judging rule G, and the relationship f of error set C, it is possible to make transformation on some of them combined or all the factors together. As error set C is a mathematical tool in portraying and modeling errors in reality, the research of transformation on error set is essentially to study the method and laws used to eliminate actual errors ([2–14]). Therefore, we care about the changes of error set when they are individually or jointly affected by transformation(s). Three cases are considered here. First, actual error and transformation method have been known, the objective is to know the laws and results of transformation. Mathematically, in T (C1 )=C2 , T and C1 are given, we ask for C2 . Secondly, actual error and the derived transformation results (or expected results) are given, it attempts to figure out the method and mechanism of transformation. In T (C1 )=C2 , C1 and C2 are given, we ask for T . In the third case, transformation method and transformation results are given, we ask for original actual error. In T (C1 )=C2 , T and C2 are given, we ask for C1 .

3.3 Classic Error Set 3.3.1 Concepts of Classic Error Set From Definition 3.1 we know that: U is an object set, G is a set of rules for judging error, if C J = {(u, x) | u ∈ U, f ⊆ U × {0, 1}, x = f (G  u), then C J is called a “classic error set” defined under judging rule G in domain U . Suppose that erroneous domain of C J is noted by U J C , U J C ={u | (u, x) ∈ C J , x = 1}. And correct domain of C J is noted by U J Z , U J Z ={u | (u, x) ∈ C J , x = 0}. Definition 3.49 Suppose that C J 1 and C J 2 are two classic error sets defined under judging rule G 1 and G 2 in domain U , if ∀(u, x1 ) ∈ C J 1 and (u, x2 ) ∈ C J 2 , x1 = x2 holds, and when G 1 = G 2 , then C J 1 = C J 2 holds. Definition 3.50 Suppose that C J 1 ⊆ C J , then C J 1 is called a classic error subset of C J . In the case of not causing confustion, C J 1 is called a subset of C J . Definition 3.51 Suppose that C J 1 and C J 2 are two classic error subsets defined under judging rule G 1 and G 2 in domain U , if ∀(u, x1 ) ∈ C J 1 , there exists (u, x1 ) ∈ C J 2 , then C J 1 is included in C J 2 , which is noted by C J 1 ⊆ C J 2 or C J 2 ⊇ C J 1 . Suppose that C J 1 , C J 2 , and C J 3 are three classic error subsets defined under judging rule G in domain U and C J 1 ⊆ C J 1 , if C J 1 ⊆ C J 2 and C J 2 ⊆ C J 3 , then C J 1 ⊆ C J 3 holds. Definition 3.52 If a set does not contain any element that belongs to U , it is called a classic error empty set noted by Φ. For any subset C J i , i ∈ {1,2,. . ., n } in C J , there exists Φ ⊆ C J i .

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Definition 3.53 Suppose that C J w is an error set defined under judging rule G in domain U , if C J w ={(u, x) |∈ U, x = 0 }, then C J w is called a classic nonerroneous set. Definition 3.54 Suppose that C J q is an error set defined under judging rule G in domain U , if C J 1 ={(u, x) |∈ U, x = 1 }, then C J q is called a classic complete error set. In the following definitions, C J is a classic error set defined under judging rule G in domain U . Definition 3.55 If ∀(u, x) ∈ C J , ∃ transformation Tu , Tu (u, x) = (u, y) and u = u, if y = x does not always hold in U , then C J is called a transformable classic error set. Definition 3.56 If ∀(u, x) ∈ C J , ∃ transformation Tu , Tu (u, x) = (u, y), if y = 0, then C J is called a modifiable classic error set. Definition 3.57 If ∀(u, x) ∈ C J ,  transformation Tu that makes y = 0 hold as Tu (u, x) = (u, y), then C J is called a dead classic error set. Definition 3.58 Suppose that C J 1 and C J 2 are two classic error subsets defined under judging rule G 1 and G 2 in domain U, C J 1 ={(u, x) | u ∈ U, f 1 ⊆ U × {0, 1}, x = f 1 (G 1  u) }, C J 2 ={(u, y) | u ∈ U, f 2 ⊆ U × {0, 1}, y = f 2 (G 2  u) }. If ∀u ∈ U , the following (u, x) ∈ C J 1 , (u, y) ∈ C J 2 , and x ≤ y hold, then there exists C J 1 ≤ C J 2 under judging rules G 1 and G 2 in domain U . For example, C J w is a classic nonerroneous set defined under G 1 in domain U and C J q is a classic complete error set defined under G 2 in domain U , then C J w ≤ C J q . Suppose that C J 1 , C J 2 , and C J 3 are three classic error sets defined under judging rules G 1 , G 2 , and G 3 , then I) C J 1 ≤ C J 1 ; II) If C J 1 ≤ C J 2 and C J 2 ≤ C J 3 , then C J 1 ≤ C J 3 holds.

3.3.2 Categories of Classic Error Set Based on the characteristics of error, classic error set can be classified into: (1) (2) (3) (4)

Transformable classic error set; Extensionable classic error set; Modifiable classic error set; Dead classic error set.

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3.3.3 Operations of Classic Error Set and Their Laws 1. Union of classic error sets Definition 3.59 Suppose that C J 1 and C J 2 are two classic error subsets defined under judging rule G 1 and G 2 respectively in domain U, C J 3 ={(u, z) | (u, x) ∈ C J 1 or (u, y) ∈ C J 2 , z = max[x, y] }, then C J 3 is called the union of C J 1 and C J 2 defined under judging rules G 1 and G 2 in domain U , it is noted by C J 3 = C J 1 ∨ C J 2 . Proposition 3.27 Suppose that C J 1 and C J 2 are two classic error subsets defined under judging rule G 1 and G 2 in domain U , and C J 1 ≤ C J 2 , then: C J 1 ≤ C J 1 ∨ C J 2 = C J 2. Proof

∵ C J 1 ≤ C J 2, ∴∀u ∈ U, (u, x1 ) ∈ C J 1 , (u, x2 ) ∈ C J 2 , the x1 ≤ x2 holds. Suppose that ∀u ∈ U, (u, y) ∈ C J 1 ∨ C J 2 , Then there exists (u, y) ∈ C J 1 or (u, y) ∈ C J 2 , And: (1) as (u, y) ∈ C J 1 , x1 ≤ y = x2 , (2) as (u, y) ∈ C J 2 , x1 ≤ y = x2 , Combining I) and II), we have C J 1 ≤ C J 1 ∨ C J 2 = C J 2 . Proof is over.

2. Intersection of classic error sets In Definition 3.16, based on characteristics of classic error set, if f (x, y, G 1 , G 2 )=0, this resorts to the following definition of the intersection of C J 1 and C J 2 defined under judging rules G 1 and G 2 , respectively. Definition 3.60 Suppose that C J 1 and C J 2 are two classic error subsets defined under judging rule G 1 and G 2 in domain U, C J 3 ={(u, z) | (u, x) ∈ C J 1 or (u, y) ∈ C J 2 , z = min[x, y] }, then C J 3 is called the intersection of C J 1 and C J 2 defined under judging rules G 1 and G 2 in domain U , it is noted by C J 3 = C J 1 ∧ C J 2 .

3.4 Fuzzy Error Set From the definition of classic error set, given (u, x) ∈ C J , when x = 0, element (u, x) has no error; if x = 1, element u has error. The establishment of classic error set tells quantitative description on whether an element is erroneous or not. In this case, an element must be wrong or right and there is nothing in between. However, for element u having certain degree of membership that belongs to right or wrong, it cannot be addressed in classic error set as judging rules, in this case, has separated things into dichotomous pairs, i.e., right and wrong. Under certain circumstances, people perceive the object of interest in a vague or fuzzy manner. Therefore, the error may also possess the property of vagueness or fuzziness. In this session, we mainly

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discuss the fuzzy error set in which an element could be both right and wrong with certain degree of membership.

3.4.1 Concepts of Fuzzy Error Set 1. Definition on fuzzy error set Definition 3.61 Suppose that U is an object set and G is a set of judging rules defined ˜ in U, C={(u, x) | u ∈ U, x = f (G  u), f ⊆ U × [0, 1] }, then C˜ is called a fuzzy error set defined under judging rule G in domain U . 2. Relationship of Fuzzy Error Sets (1) Equal Definition 3.62 Suppose that U is an object set and G is a set of judging rules defined in U, C˜1 ={(u, x) | u ∈ U, x = f 1 (G 1  u), f 1 ⊆ U × [0, 1] }, C˜2 ={(u, y) | u ∈ U, y = f 2 (G 2  u), f 2 ⊆ U × [0, 1] }, if ∀u ∈ U , there exists (u, x) ∈ C1 and (u, y) ∈ C2 that makes x = y and G 1 = G 2 hold. Then C˜1 and C˜2 are equal under rules G 1 or G 2 , noted by C˜1 = C˜2 (2) Subset Definition 3.63 Suppose that U1 and U2 are two fuzzy error subsets of U, C˜1 ={(u, x) | u ∈ U, x = f (G  u), f ⊆ U1 × [0, 1] }, C˜2 ={(u, x) | u ∈ U, x = f (G  u), f ⊆ U2 × [0, 1] }, if U1 ⊆ U2 , then C˜1 is a subset of C˜2 under rules G, noted by C˜1 ⊆ C˜2 or C˜2 ⊇ C˜1 . Proposition 3.28 Suppose that C˜1 , C˜2 , and C˜3 are three fuzzy error subsets of C˜ defined under G in U , the following items hold 1)C˜1 ⊆ C˜2 ; 2) if C˜1 ⊆ C˜2 and C˜2 ⊆ C˜3 , then C˜1 ⊆ C˜3 . (3) Less Definition 3.64 Suppose that C˜1 and C˜2 are two fuzzy error sets defined under judging rules G 1 and G 2 respectively in domains U, C˜1 ={(u, x) | u ∈ U, x = f 1 (G 1  u), f 1 ⊆ U × [0, 1] }, C˜2 ={(u, y) | u ∈ U, y = f (G 2  u), f 2 ⊆ U × [0, 1] }, if ∀u ∈ U , there exists (u, x) ∈ C˜1 and (u, y) ∈ C˜2 that make x ≤ y hold. Then under judging rules G 1 and G 2 in domain U, C˜1 is less than or equal to C˜2 , note by C˜1 ≤ C˜2 or C˜2 ≥ C˜1 . If ∀u ∈ U , there exists (u, x) ∈ C˜1 and (u, y) ∈ C˜2 that make x < y hold. Then under judging rules G 1 and G 2 in domain U, C˜1 is less than C˜2 , note by C˜1 < C˜2 or C˜2 > C˜1 . Proposition 3.29 Suppose that C˜1 , C˜2 , and C˜3 are three fuzzy sets defined under three judging rules G 1 , G 2 , and G 3 respectively in domain U , the following items hold 1) C˜1 ⊆ C˜1 ; 2) if C˜1 ⊆ C˜2 and C˜2 ⊆ C˜3 , then C˜1 ⊆ C˜3 .

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3.4.2 Operations of Fuzzy Error Set and Their Laws In Definition 3.15, if f (x, y, G 1 , G 2 ) ≡ 0, the definition on the union of fuzzy error sets C1 and C2 under two rules G 1 and G 2 respectively. 1. Union of fuzzy error sets Definition 3.65 Suppose that C˜1 and C˜2 are two fuzzy sets defined under judging rules G 1 and G 2 respectively in domain U, C˜3 ={(u, z) | (u, x) ∈ C1 , (u, y) ∈ C2 , z = max(x, y)}, then C˜3 is called the union of C˜1 and C˜2 under judging rules G 1 and G 2 respectively, noted by C˜3 = C˜1 ∨ C˜2 Proposition 3.30 Suppose that C˜1 and C˜2 are two fuzzy sets defined under judging rules G 1 and G 2 respectively in domain U , then there exists 1) C˜1 ∨ C˜1 = C˜1 ; 2) C˜1 ∨ C˜2 = C˜2 ∨ C˜1 ; 3) if C˜1 ≤ C˜2 , then C˜1 ≤ C˜1 ∨ C˜2 = C˜2 . (1) ∀u ∈ U , suppose that (u, y) ∈ C˜1 ∨ C˜2 According to Definition 3.65, y = max(x, x) = x, ∴C˜1 ∨ C˜1 = C˜1 . (2) For ∀u ∈ U , suppose that (u, y) ∈ C˜1 ∨ C˜2 and (u, z) ∈ C˜2 ∨ C˜1 , Based on the definition for union operation: y = max(x1 , x2 ), z = max(x1 , x2 ), ∵ max(x1 , x2 ) = max(x1 , x2 ), ∴y = z, ∴C˜1 ∨ C˜2 = C˜2 ∨ C˜1 .

Proof

2. Intersection of fuzzy error sets Definition 3.66 Suppose that C˜1 and C˜2 are two fuzzy sets defined under judging rules G 1 and G 2 respectively in domain U, C˜3 ={(u, z) | (u, x) ∈ C1 , (u, y) ∈ C2 , z = min(x, y)}, then C˜3 is called the intersection of C˜1 and C˜2 under judging rules G 1 and G 2 respectively, noted by C˜3 = C˜1 ∧ C˜2 Proposition 3.31 Suppose that C˜1 , C˜2 , and C˜3 are three random fuzzy sets defined under judging rules G 1 , G 2 , and G 2 respectively in domain U , then there exists (1) C˜1 ∧ C˜1 = C˜1 ; (2) C˜1 ∧ C˜2 = C˜2 ∧ C˜1 ; (3) C˜1 ∧ (C˜2 ∧ C˜3 )= (C˜1 ∧ C˜2 ) ∧C˜3 ). Proof is omitted here. 3. Complement of fuzzy error sets Definition 3.67 Suppose that C˜1 a fuzzy set defined under judging rules G in domain U, C˜2 ={(u, y) | (u, x) ∈ C1 , y = 1 − x }, then C˜2 is called the complement of C˜1 under judging rules G, noted by C˜2 = C˜1 . Proposition 3.32 Suppose that C˜1 a fuzzy set defined under judging rules G in domain U , there exists C˜1 = C˜1

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Proof From definition, we know that C˜1 = {(u, y) | (u, x) ∈ C1 , y = 1 − x} C˜1 = {(u, z) | (u, y) ∈ C1 , z = 1 − y}, ∵ z = 1 − y = 1 − (1 − x) = x. ∴C˜1 = C˜1 ,

3.4.3 Error Set with Critical Points The discussion of fuzzy error set with critical points is omitted because its properties are similar to those of general error set.

3.5 Multivariate Error Set 3.5.1 Concepts In reality, the solutions for many questions are determined by multiple interacting and sometimes mutual constricting factors. For example, in Cartesian coordinate system, (4, 2) and (2, 4) are two distinct points where the x and y axles profiling those points in the two-dimension space have interacting and constricting relationship. In general, it is not possible to investigate the properties of two-dimension space by studying two one-dimension spaces. This can be exemplified by characteristics and relationship between one-variable function and two-variable function, one-variable calculus and two-variable calculus, ordinary differential equation and partial differential equation. Taking another example in daily life, suppose that one wants to pour water from one kettle k1 to another k2 and the precondition is that one can not pour the water into any other container, this case involves multi-factors such as spatial coordinates of two kettles (x1 , y1 , z 1 ) and (x2 , y2 , z 2 ), states of water in two kettles (q-quantity and c-temperature, mineral content), direction of pouring water f , and volume flow rate a. Those factors interact and constrict each other and simultaneously influence the error value of ch(x) (water not being poured from k1 into k2 or from k2 into k1 ) of this matter(or process). The relationship between n-variable error set and (n + 1)-ary relation: the nvariable error set is a special type of (n + 1)-ary relation. Why don’t we replace the research of n-variable error set with the research of (n + 1)-ary relation? This is just like n-variable function is a special (n + 1)-ary relation and we don’t substitute the research of (n + 1)-ary relation for the research of n-variable error set. This is because that studying a special type of relationship is simpler and the expected research findings and implications must be more appropriate and applicable to the particular case or domain.

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The relationship between n-variable error set and n-ary function: in n-ary function, we focus on the research on the features of f : An → 1 . While for n-variable error set, we mainly concern the relationships of n-variable sets, method/mechanism and laws of change and transformation of those relationships. In summary, in order to achieve the objective of this research, i.e., avoiding or eliminating errors, it is necessary to study n-variable error set. There are fundamentally different characteristics between one-variable error set and two-variable error set. As there is no much difference in features between two-variable error set and multi-variable error set, we only focus on the introduction of two-variable error set.

3.5.2 Binary Error Set 1. Concept of binary error set Definition 3.68 Suppose that U ⊆ U1 × U2 , C={((u 1 (t), u 2 (t)), x(t)) |, (u 1 (t),  u 2 (t)) ⊆ U, f ⊆ U × R, x(t) = f (u 1 (t), u 2 (t), G) }={(((U1 , S1 (t), p1 , T1 (t),  L 1 (t))=u 1 (t)), (U2 , S2 (t), p2 , T2 (t), L 2 (t))=u 2 (t)), x(t)) | (u 1 (t), u 2 (t))∈ U, f ⊆ U × R, x(t) = f (u 1 (t), u 2 (t), G)}, then C is a binary error set defined under judging rule G in domain U . And Ubc = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C, x(t) > 0} Ubz = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C, x(t) < 0} Ubl = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C, x(t) = 0} is respectively called a domain with errors (erroneous domain),10 a domain without errors (correct domain),11 a domain with critical points (critical domain)12 of binary Error Set C. R represents the set of all real numbers. • Ubk ={((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C, x(t)≥0, T ( f (u 1 (t), u 2 (t), G)) 0} • Ubkl ={((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C, x(t) ∈ (−∞, +∞), T ( f (u 1 (t), u 2 (t), G))=0 }

the value of error value x(t) > 0 before making any transformation. where error value x(t) ≤ 0 before making any transformation. 12 U : where value of error function approaches zero. bl 10 U

bc :

11 U

bz :

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are called, with respect to transformation T , domain tending to generated no errors,13 domain tending to generate errors,14 domain tending to generate critical points15 of binary Error Set C. Ubh = Ubq − Ubkh Ubs = Ubc − Ubk are called absolute benign domain16 and vicious domain17 of binary Error Set C. Definition 3.69 Suppose that Ua ⊆ U1 × U2 and Ub ⊆ Ua × U2 , C1 ={((u 1 (t), u 2 (t)), x(t)) |, (u 1 (t), u 2 (t)) ⊆ U1 , f 1 ⊆ Ua × R, x(t) = f 1 (u 1 (t), u 2 (t), G 1 ))}, C2 ={((u 1 (t), u 2 (t)), y(t)) |, (u 1 (t), u 2 (t)) ⊆ Ub , f 2 ⊆ Ub × R, y(t) = f 2 (u 1 (t), u 2 (t), G 2 ))}, if Ua = Ub , f 1 = f 2 , G 1 = G 2 , then C1 = C2 holds. Proposition 3.33 Suppose that C1 = C2 , then ∀((u 1 (t), u 2 (t)), x(t)) ∈ C1 and ((u 1 (t), u 2 (t)), y(t)) ∈ C2 , if ((u 1 (t), u 2 (t)), x(t))=((u 1 (t), u 2 (t)), y(t)). Suppose that U = Ua = Ub , G = G 1 = G 2 , ∀((u 1 (t), u 2 (t)) ∈ U, ∵ G 1 = G 2 , f1 = f2 , ∴x(t) = f 1 ((u 1 (t), u 2 (t)), G 1 ) = f 2 ((u 1 (t), u 2 (t)), G 1 ) = f 2 ((u 1 (t), u 2 (t)), G 2 ) = y(t) ∴x(t) = y(t), ∴((u 1 (t), u 2 (t)), x(t)) = ((u 1 (t), u 2 (t)), y(t)).

Proof

Proposition 3.34 Suppose that C1 and C2 are two binary error sets defined in the same U, G 1 = G 2 , if ∀ ((u 1 (t), u 2 (t)), x(t)) ∈ C1 and ((u 1 (t), u 2 (t)), y(t)) ∈ C2 , then C1 = C2 . Proof

∵ Ua = U b , G 1 = G 2 , ∵ ∀((u 1 (t), u 2 (t)) ∈ U , ∴((u 1 (t), u 2 (t)), x(t)) = ((u 1 (t), u 2 (t)), y(t)), ∴ f1 = f2 . Proof is completed.

Proposition 3.35 Suppose that C1 and C2 are two binary error sets defined in Ua and Ub respectively, G 1 = G 2 , and i) ∀(u 1 (t), u 2 (t)) ∈ Ua , ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , 13 U

Where transformation enables error to be changed to be nonerroneous. : bkh where transformation engenders erroneous results from nonerroneous case. 15 U : where transformation produces either erroneous or nonerroneous results of binary Error bkl Set C. 16 U : where error value x(t) ≤ 0, even with transformation exerted, error value x(t) ≤ 0 still bh holds. 17 U : where the error value x(t) > 0, even with transformation exerted, error value x(t) > 0 still bs holds. 14 U

bk :

3.5 Multivariate Error Set

65

∃(u 1 (t), u 2 (t)) ∈ Ub , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 , then ((u 1 (t), u 2 (t)), x(t)) = ((u 1 (t), u 2 (t)), y(t)) holds. ii) ∀(u 1 (t), u 2 (t)) ∈ Ub , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 , ∃(u 1 (t), u 2 (t)) ∈ Ua , ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , then ((u 1 (t), u 2 (t)), x(t)) = ((u 1 (t), u 2 (t)), y(t)) holds. Then C1 = C2 holds. Proof

Based on Proposition 2.1, we only need to prove Ua = Ub , ∵ i) ∀(u 1 (t), u 2 (t)) ∈ Ua ⇒ (u 1 (t), u 2 (t)) ∈ Ub , ∵ ii) ∀(u 1 (t), u 2 (t)) ∈ Ub ⇒ (u 1 (t), u 2 (t)) ∈ Ua , ∴Ua = Ub , Proof is completed.

Definition 3.70 Suppose that ∀((u 1 (t), u 2 (t)), x(t)) ∈ C, we have (1) x(t) ∈ {0, 1}; (2) x(t) ∈ [0, 1]; (3) x(t) ∈ (−∞, +∞); (4) x(t) > 0; and (5) x(t) ≤ 0. Error set C under above-mentioned case is respectively called a binary classic error set, a binary fuzzy error set, a binary error set with critical points, a binary complete error set, and a binary set without errors. Definition 3.71 Suppose that C1 and C are two binary error sets if 1) C1 ⊂ C and 2) C1 ⊆ C, then C1 is called a subset of C, and when 1) holds, C1 is called a proper subset of C. 2. Operations on binary error sets (1) Definition of operations on binary error subsets Definition 3.72 Suppose that C is a binary error set, C1 and C2 are two subsets of C, then (a) C1 ∪ C2 = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C1 or ((u 1 (t), u 2 (t)), x(t)) ∈ C2 } is called the union of C1 and C2 . (b) C1 ∩ C2 = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C1 and ((u 1 (t), u 2 (t)), x(t)) ∈ C2 } is called the intersection of C1 and C2 . (c) C1 = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C and ((u 1 (t), u 2 (t)), x(t))∈C1 } then C1 is called the complement of C. (d) C1 − C2 = {((u 1 (t), u 2 (t)), x(t)) | ((u 1 (t), u 2 (t)), x(t)) ∈ C1 and ((u 1 (t), u 2 (t)), x(t))∈C2 } is called the difference set of C1 and C2 . (2) Operation laws of binary error subsets Suppose that C1 , C2 , and C3 are three binary error subsets of error set C, then (a) C1 ∪ C1 = C1 ,

66

3 Error Sets

(b) C1 ∩ C1 = C1 , (c) C1 ∪ C2 = C2 ∪ C1 , (d) C1 ∩ C2 = C2 ∩ C1 , (e) C1 ∪ (C2 ∪ C3 ) = (C1 ∪ C2 ) ∪ C3 , (f) C1 ∩ (C2 ∩ C3 ) = (C1 ∩ C2 ) ∩ C3 , (g) C1 ∪ (C1 ∩ C2 ) = C1 , (h) C1 ∩ (C1 ∪ C2 ) = C1 , (i) C1 ∪ (C2 ∩ C3 ) = (C1 ∪ C2 ) ∩ (C1 ∪ C3 ), (j) C1 ∩ (C2 ∪ C3 ) = (C1 ∩ C2 ) ∪ (C1 ∩ C3 ). Suppose that C1 , C2 , and C3 are three binary error subsets of error set C, if C1 ⊆ C2 , then (a) C1 ∪ C3 ⊆ C2 ∪ C3 , C1 ∩ C3 ⊆ C2 ∩ C3 , (b) C1 ∪ C1 = C, (c) C1 ∩ C1 = Φ, (d) C = C, (e) C1 ∪ C2 = C1 ∩ C2 , (f) C1 ∩ C2 ) = C1 ∪ C2 , (3) Concepts of operations on binary error sets Definition 3.73 Suppose that C1 and C2 are two binary error sets defined in U ⊆ U1 × U2 , if ∀(u 1 (t), u 2 (t)) ∈ U , ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , and ((u 1 (t), u 2 (t)), y(t)) ∈ C2 , which makes x(t) ≤ y(t) hold, then C1 is less than and equal to C2 noted by C1 ≤ C2 . If x(t) < y(t) hold, then C1 is less than C2 noted by C1 < C2 . Here we need to differentiate the distinction among error subset and error set and value of error. Definition 3.74 Suppose that C1 and C2 are two binary error sets defined under judging rules G 1 and G 2 in U ⊆ U1 × U2 , if C3 = {((u 1 (t), u 2 (t)), z(t)) | (u 1 (t), u 2 (t)) ∈ U, z(t)=max(x(t), y(t))- f (x(t), y(t), G 1 , G 2 ), ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 }, then C3 is called the union of C1 and C2 defined under G 1 and G 2 noted by C3 = C1 ∪ C2 . Particularly, as f (x(t), y(t), G 1 , G 2 ) =(x(t)-y(t))(1R(G 1 , G 2 ))/2, C1 ∪ C2 = C1 ∨C2 , ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , ((u 1 (t), u 2 (t)), x(t)) ∈ C2 . When f (x(t), y(t), G 1 , G 2 ) ≡ 0, C1 ∪ C2 ={((u 1 (t), u 2 (t)), z(t)) | (u 1 (t), u 2 (t)) ∈ U, z(t) = min(x(t), y(t)), ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 }, R(G 1 , G 2 ) is coefficient of G 1 and G 2 . Definition 3.75 Suppose that C1 and C2 are two binary error sets defined under judging rules G 1 and G 2 in U ⊆ U1 × U2 , if C3 = {((u 1 (t), u 2 (t)), z(t)) | (u 1 (t), u 2 (t)) ∈ U, z(t)=min(x(t), y(t))+ f (x(t), y(t), G 1 , G 2 ), ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 }, then C3 is called the intersection of C1 and C2 defined under G 1 and G 2 noted by C3 = C1 ∩ C2 . As f (x(t), y(t), G 1 , G 2 ) =| x(t)-y(t) | R(G 1 , G 2 )/2, C1 ∩ C2 ={((u 1 (t), u 2 (t)), z(t)) | (u 1 (t), u 2 (t)) ∈ U, z(t) = min(x(t), y(t))+ | x(t)-y(t) | R(G 1 , G 2 )/2, ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 }. When f (x(t), y(t), G 1 , G 2 ) ≡ 0, C1 ∩ C2 ={((u 1 (t), u 2 (t)), z(t))|(u 1 (t), u 2 (t)) ∈ U, z(t)=min(x(t), y(t)), ((u 1 (t), u 2 (t)), x(t)) ∈ C1 , ((u 1 (t), u 2 (t)), y(t)) ∈ C2 }.

3.5 Multivariate Error Set

67

Definition 3.76 Suppose that C is a binary error set defined in U ⊆ U1 × U2 , then C={((u 1 (t), u 2 (t)), −x(t))|(u 1 (t), u 2 (t)) ∈ U , ((u 1 (t), u 2 (t)), x(t)) ∈ C} is called the complement set of C. When C is classic error set, C= {((u 1 (t), u 2 (t)), z(t)) | (u 1 (t), u 2 (t))∈ U , ((u 1 (t), u 2 (t)), x(t)) ∈ C; when x(t) = 1, x(t)=1; when x(t) = 0, z(t) = 1} Definition 3.77 Suppose that C is a binary error set defined in U ⊆ U1 × U2 , if T (C)={g | c ∈ C, g = T (c)}, then the process finding solutions for T (C) is called the transformation on C. This is similar to the transformation on unary error set discussed previously. One can define the conjunction, disjunction, and inverse transformation. (4) Laws of operations on binary error sets Suppose that C1 , C2 , and C3 are three binary error sets defined under judging rules G 1 , G 1 , and G 2 in U ⊆ U1 × U2 , when f (x, y, G 1 , G 2 ) ≡, then (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n)

C1 ∪ C2 = C2 ∪ C1 , C1 ∩ C2 = C2 ∩ C1 , C1 ∪ (C2 ∪ C3 ) = (C1 ∪ C2 ) ∪ C3 , C1 ∩ (C2 ∩ C3 ) = (C1 ∩ C2 ) ∩ C3 , C1 ∪ (C1 ∩ C2 ) = C1 , C1 ∩ (C1 ∪ C2 ) = C1 , C1 ∪ (C2 ∩ C3 ) = (C1 ∪ C2 ) ∩ (C1 ∪ C3 ), C1 ∩ (C2 ∪ C3 ) = (C1 ∩ C2 ) ∪ (C1 ∩ C3 ). C1 ∪ C1 = C1 , C1 ∩ C1 = C1 , C = C, C1 ∪ C2 = C1 ∩ C2 , C1 ∩ C2 ) = C1 ∪ C2 , i f C1 ≤ C2 , then, (a) i f C1 ∪ C3 ≤ C2 ∪ C3 , (b) i f C1 ∩ C2 ≤ C2 ∩ C3 ,

when f (x(t), y(t), G 1 , G 2 )=| x(t)-y(t) | (1-R(G 1 , G 2 ))/2, for union ∪, and when f (x(t), y(t), G 1 , G 2 )=| x(t)-y(t) | R(G 1 , G 2 )/2, for intersection ∩, we have (a) (b) (c) (d)

C1 ∪ C2 C1 ∩ C2 C1 ∪ C1 C1 ∩ C1

= C2 ∪ C1 , = C2 ∩ C1 , = C1 , = C1 ,

(5) Transformation on binary error sets Based on the definition of binary error set, as the domain of binary error set is U ⊆ U1 × U2 , therefore it is a binary set (in discrete mathematics, it is called binary

68

3 Error Sets

relationship). For the element ((u 1 (t), u 2 (t)), x(t)) ∈ C of binary error set18 is composed of two parts, namely, the element of binary set (u 1 (t), u 2 (t)) and error value x(t) = f ((u 1 (t), u 2 (t)), G) of this element defined under judging rule G in domain U . Therefore, the difference that it has with unary error set resides: 1) the domain is a binary set while the domain of unary error set is a unary set; 2) the element is composed of the element of a binary set and error value of this element defined under judging rule G in domain U , while the element of unary error set is comprised of element of a unary set and error value of this element defined under judging rule G in domain U . As there are essential differences between u ∈ U , and (u 1 (t), u 2 (t)) ∈ U ⊆ U1 × U2 , it is impossible and inappropriate to treat (u 1 (t), u 2 (t)) as an integral part when being investigated. For example, when enlarging domain of error set, the scaling factor for different dimension might be different and the scaling mechanism for different dimension might be absolutely different. The transformations on binary error set are more flexible and more complicated than that of unary error set. Therefore, we are going to address the particular difference of transformation on binary error set.

3.5.3 Types of Transformation on Binary Error Set and Their Laws of Operations 1. Displacement transformation of binary error set Definition 3.78 Suppose that ∀c ∈ C, there exists T (c)=b, then T is called the displacement transformation on C noted by Tz . (1) Domain displacement transformation Definition 3.79 Suppose that Tz changes the domain U ⊆ U1 × U2 of error set C, then Tz is called the domain displacement transformation on C defined under judging rule G in U ⊆ U1 × U2 noted by Tzly . From definition, domain displacement Tzly will change the domain U ⊆ U1 × U2 of error set defined under judging rule G. Because U ⊆ U1 × U2 is a binary set, so the displacement transformations of C have following: 



• Tzly (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = 



f (u(t), G(t))) = (((V1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t)));

18

From the definition on discrete mathematical relationship, it is a ternary relationship. However, for the sake of convenience and necessity of investigating the problems while we are quantitatively examining impact of multiple factors on the object of interest and studying the transformation and transfer of error, we call it binary error set.

3.5 Multivariate Error Set

69





• Tzly (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = 







f (u(t), G(t))) = (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (V2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t)));   • Tzly (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t))) = (((V1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (V2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t))). (2) Entity/thing displacement transformation 

Definition 3.80 Suppose that Tz changes the thing/entity of (((U1 , S1 (t), p1 (t),  T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t))), then Tz is called the thing/entity displacement transformation on C defined under judging rule G in U ⊆ U1 × U2 noted by Tzsw . The thing/entity displacement transformations have: 



Tzsw (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) =   f (u(t), G(t))) = (((V1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t)));   Tzsw (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) =   f (u(t), G(t))) = (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (V2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t)));   Tzsw (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) =   f (u(t), G(t))) = (((V1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (V2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = f (u(t), G(t))); 2. Decomposition transformation of binary error set (1) Domain decomposition transformation 



• If T f ly (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = 



f (u(t), G(t)))={(((U11 , S11 , p11 (t), T11 (t), L 11 (t)), (U2 , S2 (t), p2 (t), T2 (t),  L 2 (t))), x1 (t) = f 1 (u 1 (t), G 1 (t)), (((U12 , S12 , p12 (t), T12 (t), L 12 (t)), (U2 ,   S2 (t), p2 (t), T2 (t), L 2 (t))), x2 (t) = f 2 (u 2 (t), G 2 (t)), . . ., (((U1n , S1n , p1n (t),  T1n (t), L 1n (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), xn (t)= f n (u n (t), G n (t)))}, and U1 (t) = U11 (t) ∪ U12 (t)∪, . . . , ∪U1n (t), it has made decomposition trasformation on the domain U1 (t) of object u(t) in order to achieve our expected objective.   • If T f ly (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p12 (t), T2 (t), L 2 (t))), x(t)= 



f (u(t), G(t)))={(((U1 , S1 , p1 (t), T1 (t), L 1 (t)), (U21 , S21 (t), p21 (t), T21 (t),  L 21 (t))), x1 (t) = f 1 (u 1 (t), G 1 (t)), (((U1 , S1 , p1 (t), T1 (t), L 1 (t)), (U22 , S22 (t),   p22 (t), T22 (t), L 22 (t))), x2 (t) = f 2 (u 2 (t), G 2 (t)), . . ., (((U1 , S1 , p1 (t), T1 (t),  L 1 (t)), (U2n , S2n (t), p2n (t), T2n (t), L 2n (t))), xn (t) = f n (u n (t), G n (t)))}, and U2 (t) = U21 (t) ∪ U22 (t)∪, . . . , ∪U2n (t), it has made decomposition transformation on the domain U2 (t) of object u(t) in order to achieve our expected objective.

70

3 Error Sets 



• If T f ly (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t) = 



f (u(t), G(t)))={(((U11 , S11 , p11 (t),T11 (t), L 11 (t)), (U21 , S21 (t), p21 (t), T21 (t),  L 21 (t))), x1 (t) = f 1 (u 1 (t), G 1 (t)), (((U12 , S12 , p12 (t), T12 (t), L 12 (t)), (U22 ,   S22 (t), p22 (t), T22 (t), L 22 (t))), x2 (t)= f 2 (u 2 (t), G 2 (t)), . . ., (((U1n , S1n , p1n (t),  T1n (t), L 1n (t)), (U2n , S2n (t), p2n (t), T2n (t), L 2n (t))), xn (t)= f n (u n (t), G n (t)))}, and U1 (t) × U2 (t)=U11 (t) × U21 (t) ∪ U12 (t) × U22 (t)∪, . . ., ∪U1n (t) × U1n (t), it has made decomposition trasformation on the domain U1 (t) of object u(t) in order to achieve our expected objective. (2) Thing/entity decomposition transformation 



• If T f sw (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t)=  f (u(t), G(t)))={(((U11 , S11 , p11 (t), T11 (t),



L 11 (t)), (U2 , S2 (t), p2 (t), T2 (t),  L 2 (t))), x1 (t) = f 1 (u 1 (t), G 1 (t)), (((U12 , S12 , p12 (t), T12 (t), L 12 (t)), (U2 , S2 (t),   p2 (t), T2 (t), L 2 (t))), x2 (t) = f 2 (u 2 (t), G 2 (t)), . . ., (((U1n , S1n , p1n (t), T1n (t),  L 1n (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), xn (t) = f n (u n (t), G n (t)))}, and S1 (t) = S11 (t) + S12 (t)+, . . . , +S1n (t), it has made decomposition transformation on the thing/entity S1 (t) of object u(t) in order to achieve our expected objective.   • If T f sw (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t)= 



f (u(t), G(t)))={(((U1 , S1 , p1 (t), T1 (t), L 1 (t)), (U21 , S21 (t), p21 (t), T21 (t),  L 21 (t))), x1 (t) = f 1 (u 1 (t), G 1 (t)), (((U1 , S1 , p1 (t), T1 (t), L 1 (t)), (U22 , S22 (t),   p22 (t), T22 (t), L 22 (t))), x2 (t) = f 2 (u 2 (t), G 2 (t)), . . ., (((U1 , S1 , p1 (t), T1 (t),  L 1 (t)), (U2n , S2n (t), p2n (t), T2n (t), L 2n (t))), xn (t) = f n (u n (t), G n (t)))}, and S2 (t) = S21 (t) + S22 (t)+, . . . , +S2n (t), it has made decomposition transformation on the thing/entity S2 (t) of object u(t) in order to achieve our expected objective.   • If T f sw (((U1 , S1 (t), p1 (t), T1 (t), L 1 (t)), (U2 , S2 (t), p2 (t), T2 (t), L 2 (t))), x(t)= 



f (u(t), G(t)))={(((U11 , S11 , p11 (t), T11 (t), L 11 (t)), (U21 , S21 (t), p21 (t), T21 (t),  L 21 (t))), x1 (t)= f 1 (u 1 (t), G 1 (t)), (((U12 , S1 , p12 (t), T12 (t), L 12 (t)), (U22 , S22 (t),   p22 (t), T22 (t), L 22 (t))), x2 (t)= f 2 (u 2 (t), G 2 (t)), . . ., (((U1n , S1n , p1n (t), T1n (t),  L 1n (t)), (U2n , S2n (t), p2n (t), T2n (t), L 2n (t))), xn (t) = f n (u n (t), G n (t)))}, and S1 (t) = S11 (t) + S12 (t)+, . . . , +S1n (t), S2 (t)=S21 (t) + S22 (t)+, . . . , +S2n (t), it has made decomposition transformation on the things/entities S1 (t) and S2 (t) of object u(t) in order to achieve our expected objective. Similarly, we can discuss the destruction transformation, increase transformation, similarity transformation, and the reciprocal transformation of them

References

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References 1. Bian Y, Guo K (2011). Optimization of the non-additive feature systems. Mathematics in Practice and Theory 41(9):90–96 2. Guo K (1992). Research of eliminating error of system transformation. Advances in Modeling and Analysis: D. 19(4):9–14 3. Guo K (1994). Research of eliminating error in system. In: International Conference of AMSE System, Control Information Methodologies and Applications, Wuhan, China, Oct 1994 4. Guo K (1997). Logic of error elimination. Journal of Guangdong University of Technology 5. Guo K (2015). How to avoid and correct errors. China Science Press, Beijing 6. Guo Q, Guo K (2009). Error-eliminating theory of intelligent system: The optimization of systems, subsystems and elements. In: 2009 International Workshop on Intelligent Systems and Applications, ISA, Wuhan, China, 23-24 May 2009 7. Guo K, Huang J (2008). 1% error leading to 100% system failure: Investigation on the mechanism of error generation and methods for eliminating and avoiding errors. Journal of Guangdong University of Technology. 25(2):1–5 8. Liu H, Guo K (2010). Fuzzy multi-element error-matrix and its operation. Advances in Modeling and Analysis A: General Mathematical and Computer Tools 47(1-2):21–32 9. Liu JJ, Zhang Y, Zhao F (2006). Robust distributed node localization with error management. In: 7th ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 2006), Florence; Italy 22-25 2006 May, p 250–261 10. Liu S, Guo K, Sun D (2010). Progress and prospect of research on theory of error-elimination: A new thrust in management science. Chinese Journal of Management 7(12):1749–1758 11. Min X, Huang J, Qi J, Guo K (2012). Error matrix equation based error-elimination and erroravoidance expert system. Journal of Guangdong University of Technology. 29(2):21–27 12. Ye Q, Guo Q, Guo K (2010). Methods for the error-eliminating in system decision-making. In: Proceedings of The 2010 International Conference on Measuring Technology and Mechatronics Automation (ICMTMA). Changsha, China, 13-14 March 2010, 2:652–655 13. Guo K, Zhang S (1995a). Introduction to error elimination. Press of South China University of Technology, Guangzhou 14. Guo K, Zhang S (1995b). Theory and method for judging decision-making errors in enterprise capital asset investment. Press of South China University of Technology, Guangzhou

Chapter 4

Transformation Connectives in Error Logic

In the process of studying a complicated system, it is necessary to investigate how the error or failure of one element or multiple elements (or subsystems) of the system can influence its reliability and functionality. In order to identify the errors or failures and consequently eliminate or remove them, we need to find the ways or laws that error is transmitted and transformed in the system. In order to identify and remove error occurred in an element ei of system S, the first step is to decompose system A into a subsystem set containing the subsystem with ei as one part of it; then a non-erroneous subsystem is used to replace ei . For example, a company A experiencing crisis wants to learn the management system of a successful company B, it is not wise to mechanically implant A’s procedures into B’s system without making adaptive changes. Taking another example, regarding the expansion strategy of chain stores, decision makers have to consider the the differences in distinct cultures, religions, social norms, law and regulations, and economic and political contexts and make necessary transformation and changes accordingly. Therefore, it is necessary to examine decomposition transformation, displacement transformation, increase transformation, similarity transformation, destruction transformation, unit transformation, and their corresponding inverse transformations (if necessary) of those successful management systems and the management systems experiencing crises. Error logic is to explore thinking forms, thinking methods and laws, and valid reasoning of related propositions for static and dynamic transformation relationships of intra- and inter-errors in a given system. According to the range of truth values in error logic, error logic can be categorized into: error binary-value logic, fuzzy error logic, and error logic with critical points.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Liu and K. Guo, Error Logic: Paving Pathways for Intelligent Error Identification and Management, Studies in Systems, Decision and Control 442, https://doi.org/10.1007/978-3-031-00820-7_4

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4.1 Similarity Transformation Connectives in Error Logic 4.1.1 Basic Concepts # » Definition 4.1 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, where U (t) is # » the domain of μ(t) = (U (t), S(t), p(t), T (t), L(t)), S(t) is the thing (corresponding # » to subject) or subject (corresponding to the structure of a proposition) of μ(t), p(t) is the spatial location and direction of μ(t), T (t) is the property (corresponding to subject) or predicate (correponding to the structure of a proposition) of μ(t), L(t) is the property (or attribute) value (corresponding to subject) or predicative (corre# » ponding to the structure of a proposition) of μ(t), x(t) = f ((μ(t), p(t)), G A (t)) is the truth value or truth function of μ(t), G(t) is the rule for judging error defined # » # » in domain U (t), A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is the error logical variable defined in U (t) under the rule of judging errors G(t). # » Definition 4.2 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}(G(t) is the rule for judging error, U (t) is the domain) is the error logical variable defined in U (t) under G(t), then the set C composed of all error logical variables is called error logical variable set defined in U (t) under G(t). # » Definition 4.3 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((V (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domains U (t) and V (t) under # » judging rule GA (t) and GB (t) , if ω(t) in C(ω(t), z(t)) = C((W (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is composed of equivalent parts or certain parts of μ(t) and ν(t) in A(μ(t), x(t)) and B(ν(t), y(t)) respectively, where W (t) is the domain of ω(t) and GC (t) is the judging rule for errors defined in W (t). Then C(ω(t), z(t)) is called the mediator variable of A(μ(t), x(t)) and B(ν(t), y(t)) # » defined in different domains noted by C Azy B ((W (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))). # » Definition 4.4 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule # » GA (t) and GB (t), if ω(t) in C(ω(t), z(t)) = C((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is composed of equivalent parts or certain parts of μ(t) and ν(t) in A(μ(t), x(t)) and B(ν(t), y(t)), respectively, where GC (t) is the judging rule for errors defined in W (t). Then C(ω(t), z(t)) is called the mediator variable of A(μ(t), x(t)) and B(ν(t), y(t)) defined in the same domain noted by # » C Azt B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))).

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# » Definition 4.5 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule GA (t) and GB (t), if the properties TA (t) and TB (t) in A(μ(t), x(t)) and B(ν(t), y(t)) respectively have the relationship of connotative inclusion, i.e., TA (t) ⊃nhb TB (t), # » then C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) = A(μ(t), x(t)) ⊃n B(ν(t), y(t)) is called the connotative inclusion variable of A(μ(t), x(t)) # » and B(ν(t), y(t)) denoted by C AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))). # » Definition 4.6 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule GA (t) and GB (t), if the properties of A(μ(t), x(t)) and B(ν(t), y(t)) are # » the same , i.e., TA (t) =nhdt TB (t), then C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = n f (ω(t), G C (t))) = A(μ(t), x(t)) = B(ν(t), y(t)) is called the connotative same # » variable for A(μ(t), x(t)) and B(ν(t), y(t)) denoted by C Andt B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))). # » Definition 4.7 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule GA (t) and GB (t), if the properties of A(μ(t), x(t)) and B(ν(t), y(t)) are equiv# » alent, i.e., TA (t) ⇐⇒nhd j TB (t), then C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = n f (ω(t), G C (t))) = A(μ(t), x(t)) = B(ν(t), y(t)) is called the connotative equivalence variable for A(μ(t), x(t)) and B(ν(t), y(t)) denoted by C And j B ((U (t), SC (t), # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))).

4.1.2 Basic Operations # » Definition 4.8 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule GA (t) and GB (t), if  A(μ(t), x(t)) ∨ B(ν(t), y(t)) =

A(μ(t), x(t)), if x(t) ≥ y(t) B(ν(t), y(t)), if x(t) ≤ y(t)

Then ∨ is called the denotation disjunction of A(μ(t), x(t)) and B(ν(t), y(t)).

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# » Definition 4.9 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule GA (t) and GB (t), if  A(μ(t), x(t)) ∧ B(ν(t), y(t)) =

A(μ(t), x(t)), if x(t) ≤ y(t) B(ν(t), y(t)), if x(t) ≥ y(t)

Then ∧ is called the denotation conjunction of A(μ(t), x(t)) and B(ν(t), y(t)). # » Definition 4.10 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging rule GA (t) and GB (t), if ¬A(μ(t), x(t)) = B(ν(t), y(t)),where ⎧ y(t) = 0, if x(t) = 1, x(t) ∈ {0, 1}; ⎪ ⎪ ⎨ y(t) = 1, if x(t) = 0, x(t) ∈ {0, 1}; = y(t) = 1 − x(t), if x(t) ∈ [0, 1]; ⎪ ⎪ ⎩ y(t) = −x(t), if x(t) ∈ (−∞, +∞). Then ¬ is called the negation operation on A(μ(t), x(t)). # » Definition 4.11 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging # » rule GA (t) and GB (t), C AnhbB ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the connotative inclusion variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) →nsy B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C AnhbB # » ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (¬A(μ(t), x(t)) # » ∧ B(ν(t), y(t)) ∧ C AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C AnhbB ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then →nsy is called the connotative isness implication connective of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “if . . . , then it is . . . ”. # » Definition 4.12 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging # » rule GA (t) and GB (t), C Anhbhd B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), # » GC (t))) = C AnhbB ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) ∨ # » C And j B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the connotative inclusion variable or connotative equivalence variable for A(μ(t), x(t)) and

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B(ν(t), y(t)), if A(μ(t), x(t)) →nhy B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), # » y(t)) ∧ C Anhbhd B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ # » (A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C Anhbhd B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C Anhbhd B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then →nhy is called the connotative possibility implication connective of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “if . . . , then it is possible”. # » Definition 4.13 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging # » rule GA (t) and GB (t), C Anhbhd B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), # » GC (t))) = C AnhbB ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) ∨ # » C And j B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the connotative inclusion variable or connotative equivalence variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) →nby B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), # » y(t)) ∧ C Anhbhd B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ # » (¬A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Anhbhd B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C Anhbhd B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then →nby is called the connotative necessity implication connective of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “if . . . , then it must be or it could not be otherwise”. # » Definition 4.14 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judging # » rule GA (t) and GB (t), C Andt B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) = A(μ(t), x(t)) =n B(ν(t), y(t)) is the connotative same variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) =nhdt B(ν(t), y(t)) = (A(μ(t), # » x(t)) ∧ B(ν(t), y(t)) ∧ C Andt B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), # » GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C Andt B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then =nhdt is called the connotative same connective of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “properties are the same or two errors are the same”. # » Definition 4.15 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), # » # » pB (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), # » TB (t), LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under # » judging rule GA (t) and GB (t), C Andt B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = # » f (ω(t), G C (t))) = C Anhdt B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t))) ∨ C Anhdthd j B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative same or equivalence variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) ←→nhdz B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧

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# » C Anhdthd j B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (¬ # » A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C Anhdthd j B ((U (t), SC (t), pC (t), TC (t), LC (t)), nhdz is called the connotative equivalence conz(t) = f (ω(t), GC (t)))), then ←→ nective of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “two properties coexist or two errors coexist”. # » Definition 4.16 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), # » # » L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), # » TB (t), LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judg# » ing rule GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) −n B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ ¬B(ν(t), # » y(t)) ∧ C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then −n is called the connotative difference connective of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “errors were removed or reduced”. # » Definition 4.17 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judg# » ing rule GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) ∨n B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (A(μ(t), x(t)) ∧ B(ν(t), # » y(t)) ∧ ¬C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ # » (¬A(μ(t), x(t))∧ B(ν(t), y(t)) ∧ C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = # » f (ω(t), GC (t)))) ∨ (A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ ¬(ν(t), y(t)) ∧ C Az B # » ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t)= f (ω(t), GC (t)))) ∨ (¬A(μ(t), x(t)) ∧ # » B(ν(t), y(t)) ∧ ¬C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t)= f (ω(t), GC (t)))) # » LC (t)), ∨ (A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ ¬C Az B ((U (t), SC (t), pC (t), TC (t), z(t) = f (ω(t), GC (t)))), then ∨n is called the connotative disjunction of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “different errors are compatible”. # » Definition 4.18 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judg# » ing rule GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) ∧n B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (A(μ(t), x(t)) ∧ ¬ B(ν(t), # » y(t)) ∧ C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) # » ∨ (¬A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)),

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z(t) = f (ω(t), GC (t)))) ∨(¬A(μ(t), x(t)) ∧ ¬ B(ν(t), y(t)) ∧ C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then ∧n is called the connotative conjunction of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “different errors have mutual infiltration”. # » Definition 4.19 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judg# » ing rule GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) nhb B(ν(t), y(t)) =A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))), then nhb is called the connotative complement of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “correctness, error with critical points, and error coexist”. # » Definition 4.20 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judg# » ing rule GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) nhdl B(ν(t), y(t)) = A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ ¬C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))), then nhdl is called the connotative antithesis of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “correctness and error coexist”. # » Definition 4.21 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are two error logical variables respectively defined in domain U (t) under judg# » ing rule GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) |n f l B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ C Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (A(μ(t), x(t)) ∧ B(ν(t), # » y(t)) ∧ ¬C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), then |n f l is called the connotative separation of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “different errors coexist”. # » Definition 4.22 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » # » x(t) = f ((μ(t), pA (t)), GA (t))), B(ν(t), y(t)) = B((U (t), SB (t), pB (t), TB (t), # » LB (t)), y(t) = f ((ν(t), pB (t)), GB (t))), x(t), y(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, are the error logical variables respectively defined in domain U (t) under judging rule # » GA (t) and GB (t), ω(t) in C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(μ(t), x(t)) and B(ν(t), y(t)), if A(μ(t), x(t)) |n f h B(ν(t), y(t)) = (A(μ(t), x(t)) ∧ B(ν(t), y(t)) ∧ ¬C Az B ((U (t), SC (t),

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# » pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))) ∨ (A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ # » ¬C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t)))), (¬A(μ(t), # » x(t)) ∧ B(ν(t), y(t)) ∧ ¬C Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), # » GC (t)))), (¬A(μ(t), x(t)) ∧ ¬B(ν(t), y(t)) ∧ ¬C Az B ((U (t), SC (t), pC (t), TC (t), nf h is called the connotative differentiation LC (t)), z(t) = f (ω(t), GC (t)))), then | of A(μ(t), x(t)) and B(ν(t), y(t)), which means that “errors with critical points do not exist”. # » Definition 4.23 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » x(t) = f ((μ(t), pA (t)), GA (t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, is an error logical variable defined in domain U (t) under judging rule GA (t), A(n) (μ(t), x(t)) = # » # » A(n) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) and A(n+1) # » # » (μ(t), x(t)) = A(n+1) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) are respectively the n th and (n + 1)th layer error logical variables of # » A(μ(t), x(t)), it is assumed that ω(t) in C A(n)z A(n+1) ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(n) (μ(t), x(t)) and A(n+1) (μ(t), x(t)), if ¬bx A(n) (μ(t), x(t)) = A(n) (μ(t), x(t)) ∧ A(n+1) (μ(t), x(t)) # » ∧ C A(n)z A(n+1) ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))), then ¬bx is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the error being negated, there exists its opposite side before being decomposed”. # » Definition 4.24 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » x(t) = f ((μ(t), pA (t)), GA (t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, is an error logical variable defined in domain U (t) under judging rule GA (t), A(n) (μ(t), x(t)) = # » # » A(n) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) and A(n−1) # » # » (μ(t), x(t)) = A(n−1) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) are respectively the n th and (n − 1)th layer error logical variables of # » A(μ(t), x(t)), it is assumed that ω(t) in C (A(n−1)z A(n) ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the mediator variable for A(n−1) (μ(t), x(t)) and A(n) (μ(t), x(t)), if ¬bd A(n) (μ(t), x(t)) = A(n−1) (μ(t), x(t)) ∧ A(n) (μ(t) ∧ # » C A(n−1)z A(n) ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))), then ¬bd is called the connotative uninterrupted negation on A(n) (μ(t), x(t)), which means that “for the error being negated, there exists its opposite side after being decomposed”. # » Definition 4.25 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » x(t) = f ((μ(t), pA (t)), GA (t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, is an error logical variable defined in domain U (t) under judging rule GA (t), A(n) (μ(t), x(t)) = # » # » A(n) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) and A(n+1) # » # » (μ(t), x(t)) = A(n+1) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) are respectively the n th and (n + 1)th layer error logical variables of A(μ(t), x(t)), and ¬A(n+1) (μ(t), x(t)) is the (n + 1)th layer error logical complementary variable of A(n+1) (μ(t), x(t)), it is assumed that ω(t) in C (n+1)Az B ((U (t), # » SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))) is the (n + 1)th mediator variable of A(μ(t), x(t)), if ¬bz A(n) (μ(t), x(t)) = A(n+1) (μ(t), x(t)) ∧ ¬A(n+1) (μ(t), # » x(t)) ∧ C (n+1)Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t), GC (t))), then

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¬bz is called the connotative “not-only” negation on A(n) (μ(t), x(t)), which means that “there exists error that can be negated before being decomposed”. # » Definition 4.26 Suppose that A(μ(t), x(t)) = A((U (t), SA (t), pA (t), TA (t), L A (t)), # » x(t) = f ((μ(t), pA (t)), GA (t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, is an error logical variable defined in domain U (t) under judging rule GA (t), A(n) (μ(t), x(t)) = # » # » A(n) ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), GA (t))) and A(n−1) # » # » (n−1) (μ(t), x(t)) = A ((U (t), SA (t), pA (t), TA (t), L A (t)), x(t) = f ((μ(t), pA (t)), th th GA (t))) are the (n − 1) and n layer error logical variables of A(μ(t), x(t)), and ¬b A(n−1) (μ(t), x(t)) is the (n − 1)th layer error logical complementary variable for # » A(n−1) (μ(t), x(t)), it is assumed that ω(t) in C (n−1)Az B ((U (t), SC (t), pC (t), TC (t), th LC (t)), z(t) = f (ω(t), GC (t))) is the (n − 1) layer mediator variable for A(n−1) (μ(t), x(t)) and ¬A(n−1) (μ(t), x(t)), if ¬bj A(n) (μ(t), x(t)) = A(n−1) (μ(t), x(t)) ∧ # » ¬A(n−1) (μ(t), x(t)) ∧ C (n−1)Az B ((U (t), SC (t), pC (t), TC (t), LC (t)), z(t) = f (ω(t),  GC (t))),then ¬bj is called the connotation unfinished negation on A(n) ((U, S A (t), p A , T A (t), L A (t)), x(t) = f (μ(t), G A (t))), which means that “there exists error that can be negated after being decomposed”.

4.1.3 Similarity Transformation Connectives in Error Logic # » Definition 4.27 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, is an error logical variable defined in domain U (t) under judging rule G(t), if T(A(μ(t), x(t))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))), then T is called similarity transformation connective with respect to G(t) and A(μ(t), x(t)) defined in U (t), which is denoted by Tx . # » # » (1) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U  (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), then Tx is called the domain similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txly . In this case, U2 (t) can be replaced by U1 (t) or U1 (t) can be replaced by U1 (t) if U1 (t) = kU2 (t) and k > 0. The domain in the object μ(t) is transformed in order to achieve the expected objective. U (t) = kU  (t) indicates the fact that the scale or power of U (t) is k times as much as U  (t) in that respect. For example, when talking about issue regarding human resources, the domain Guangdong province U  (t) and the domain China U (t) are two similar domains. The domain Guangdong province U  (t), if necessary, can be replaced by the domain China U (t), and vice versa. # » # » (2) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U (t), # » # » S  (t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), then Tx is called the thing similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txsw .

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# » # » (3) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U (t), # » # » S  (t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), then Tx is called the spatial similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txk j . In this case, the spatial position of thing in the object μ is transformed to attain the expected objective. # » # » (4) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U (t), # » # »  S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), then Tx is called the property similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txt x . In this case, T1 (t) and T2 (t) are the same type of properties. T1 (t) is replaced by T2 (t) or T2 (t) is replaced by T1 (t) is to # » conduct property similarity transformation on the object (μ, p(t)) to achieve desired objective. For example, the length T1 (t) and width T2 (t) of a machine represent the volume property. The length T1 (t) and width T2 (t) can be replaced with each other depending on how you define them. # » # » (5) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U (t), # » # » S(t), p(t), T (t), L  (t)), x  (t) = f ((μ (t), p(t)), G(t))), then Tx is called the property (or attribute) value similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txlz . In this case, L 1 (t) can be replaced by L 2 (t), and vice versa if L 1 (t) = k L 2 (t) and k > 0. For example, suppose that L 1 (t) = k L 2 (t) representing travel time between two cities. The travel time L 1 (t) can be replaced with T2 (t) and vice versa given that different transport modes are used. (6) x  (t) ∈ [x − , x + ], then Tx is called is called the error value similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txcz . In this case, transformation is conducted on x = # » f (μ(t), p(t)), G(t)). # » # » (7) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t), G  (t))), then Tx is called the rule similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txgz . In such a case, G 1 (t) can be replaced by G 2 (t), and vice versa if G 1 (t) and G 2 (t) are same or similar type of rules. For example, the modified constitution G 1 in 2004 in China is a modification for constitution G 2 (t) in 2003, which was a typical case of rule similarity transformation for serving the emerging needs of social and economic development. # » # » (8) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t), G  (t))), then Tx is called the error function similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txhs . In this case, transformation is carried on the error function f . # » # » (9) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f (u  (t), p  (t), G  (t))) = A((U (t  ), # » # » then Tx is S(t  ), p(t  ), T (t  ), L(t  )), x  (t  ) = f (μ(t  ), p(t  ), G(t  ))), called the temporal similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txs j . In this case, transformation is conducted on time t.

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# » # » (10) A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f (u  (t), p  (t), G  (t))) = A((U  (t  ), # » # » S  (t  ), p  (t  ), T  (t  ), L  (t  )), x  (t  ) = f  (μ (t  ), p  (t  ), G  (t  ))), then Tx is called the comprehensive similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Txq . In this situation, in order to attain one’s objective, comprehensive transformation is conducted on domain, thing, property, property (or attribute) value, error function, time, space, error value, and rules for judging errors in the object μ(t). (11) The corresponding inverse transformation connectives of similarity transformation connectives Tx ⊆ {Txly , Txsw , Txk j , Txt x , Txlz , Txcz , Txgz , Txhs , Txs j , Txq } −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 , Txsw , Txk are Tx−1 ⊆ {Txly j , Txt x , Txlz , Txcz , Txgz , Txhs , Txs j , Txq }.

4.1.4 Characteristics of Domain Similarity Transformation Connectives in Error Logic # » # » Txly A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t), G  (t))) = A((U  (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), Txly is called the similarity transformation connective with respect to G(t) and A(μ(t), x(t)) defined in U (t). In this case, U (t) = kU  (t) and k > 0 is the scale of change, U (t) can be replaced by U  (t), and vice versa. In the following section, we discuss the characteristics of domain similarity transformation connective in error logic. # » Proposition 4.1 Suppose that two error logical variables A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) and A((U  (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) and U  (t) respectively under G(t) # » # » the rules for judging errors. If ∀(μ(t), p(t)) = (U (t), S(t), p(t), T (t), L(t)) ∈ U (t) # » # » are corresponding to A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), where x0 (t) and x1 (t) are the lower- and upper bounds respectively for x(t) = # » # » # » f ((μ(t), p(t)), G(t))); ∀(μ (t), p(t)) = (U (t) , S  (t), p  (t), T  (t), L  (t)) ∈ U  (t) # » # » are corresponding to A((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), where x0 (t) and x1 (t) are the lower- and upper bounds respectively for # » x  (t) = f ((μ (t), p(t)), G(t))), then: (1) if k1, then x0 (t)  x0 (t) and x1 (t)  x1 (t); (2) if k1, then x0 (t)  x0 (t) and x1 (t)  x1 (t). Proof (1) ∵ k1, ∴ U  (t) ⊆ U (t), suppose that V (t) = U (t) − U  (t), if ∀(μ(t), # » x(t)) ∈ V (t) corresponding to A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); c0 (t) and c1 (t) are assumed to be the lower- and upper # » bounds respectively for x(t) = f ((μ(t), p(t)), G(t)), if c0 (t)  x0 (t), then x0 (t) = x0 (t), and if c0 (t)  x0 (t), then x0 (t)  x0 (t); if c1 (t)  x1 (t), then x1 (t)  x1 (t), and if c1 (t)  x1 (t), then x1 (t) = x1 (t). (2) Similarly, ∵ k1, ∴ U (t) ⊆ U  (t), suppose that V (t) = U  (t) − U (t), if ∀(μ(t), # » x(t)) ∈ V (t) corresponding to A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); c0 (t) and c1 (t) are assumed to be the lower- and upper

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# » bounds respectively for x(t) = f ((μ(t), p(t)), G(t)), if c0 (t)  x0 (t), then   x0 (t) = x0 (t), and if c0 (t)  x0 (t), then x0 (t)  x0 (t); if c1 (t)  x1 (t), then x1 (t)  x1 (t), and if c1 (t)  x1 (t), then x1 (t) = x1 (t). Proof is completed. # » Proposition 4.2 Suppose that two error logical variables A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) and B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for judging # » errors; suppose that another two error logical variables A((U  (t), S(t), p(t), T (t), # » # »  L(t)), x(t) = f ((μ(t), p(t)), G(t))) and B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors; then: # » # » (1) Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), # » # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = A((U  (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t))); # » # » (2) Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G(t))) ∧ B((U (t), # » # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = A((U  (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U  (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t))); # » # » (3) Txly (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » ¬A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))). Proof (1) Txly does not change the error function f and G(t) rules for judging # » errors but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t))) and B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t) and x  (t) will not be changed in the transformation. # » If x(t)  x  (t), then the left side Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # »  p(t)), G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # »  G(t))); and right side is A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # »  G(t))) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), left side = right side; # » (2) If x(t)  x  (t), then the left side Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t)))) = B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))), and the right side is A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t))) = B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), left side = right side. Proof is completed. Similarly, (2) and (3) can be proved.

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# » Proposition 4.3 Suppose that three error logical variables A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # »  f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » ing errors; suppose that another three error logical variables A((U  (t), S(t), p(t), # » # »   T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), x (t) = # » # » f ((μ (t), p(t)), G(t))), and C Az B ((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » p(t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors; then: # » # » (1) Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txly # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∨n Txly # » # » (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t), G(t)))); # » # » (2) Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∧n # » # » Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof (a) Txly does not change the error function f and G(t) rules for judging # » # » errors but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # »   G(t))), B((U (t), S(t), p(t), T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t))), and # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which   means that x(t), x (t), and x (t) will not be changed in the transformation. If # » x(t)  x  (t)  x  (t), then the left side is Txly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t)))) = (A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t))), and the right side is (A((U  (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t))) ∨n B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t))) ∨n C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t)))) = (A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), left side = right side. (b) Other cases can be proved in the similar way. Proof is completed. Similarly,(2) can be proved.

# » Proposition 4.4 Suppose that three error logical variables A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t))), and C AnhbB ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » ing errors; suppose that another three error logical variables A((U  (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t))), and C AnhbB ((U  (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U  (t) under G(t) the rules for # » # » judging errors, then:Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)),

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# » # » G(t))) −n B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) −n Txly # » # » (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof Since Txly does not change the error function f and G(t) rules for judging # » # » errors but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))), B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C AnhbB # » # » ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t), x  (t), and x  (t) will not be changed in the transformation, then:Txly (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) −n B((U (t), S(t), p(t), # » # » T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # »  AnhbB ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), f ((μ (t), p(t)), G(t))) ∧ C # » # » # » p(t)), G(t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) # » # »   ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t))) ∧ C AnhbB # » # » ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = (A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ ¬B((U  (t), S(t), p(t), # » # »   AnhbB  ((U (t), S(t), p(t), T (t), T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t))) ∧ C # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) ∨ (¬A((U  (t), S(t), p(t), T (t), L(t)), # » # »  x(t) = f ((μ(t), p(t)), G(t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t))) ∧ C AnhbB ((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » p(t)), G(t)))). Proof is completed. Proposition 4.5 Suppose that three error logical variables A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » ing errors; suppose that another three error logical variables A((U  (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t))), and C Az B ((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » p(t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, # » # » then:Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) |n f l Txly (B((U (t), S(t), # » # » p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof Since Txly does not change the error function f and G(t) rules for judging errors # » # » but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t), x  (t), and x  (t) will not be changed in the transformation, then:Txly (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) |n f l B((U (t), S(t), p(t), T (t), # » # »   L(t)), x (t) = f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t),

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# » # » # » p(t)), G(t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t)))) ∨ (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) ∧ ¬C Az B ((U (t), # » # »   S(t), p(t), T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) ∨ (C Az B ((U (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t)))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∧ # » # » Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U  (t), S(t), p(t), T (t), # » L(t)), x  (t) = f ((μ (t), p(t)), G(t))). Proof is completed. Proposition 4.6 Suppose that three error logical variables A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for # » # » judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U  (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U  (t), S(t), # » # » p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) are defined in domain U  (t) under # » G(t) the rules for judging errors, then:Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t))) |n f h B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) |n f h Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof Since Txly does not change the error function f and G(t) rules for judging errors # » # » but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t), x  (t), and x  (t) will not be changed in the transformation, then:Txly (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) |n f h B((U (t), S(t), p(t), T (t), # » # »   Az B L(t)), x (t) = f ((μ (t), p(t)), G(t)))) = Txly (¬C ((U (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = ¬C Az B ((U  (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t))) |n f h B((U  (t), S(t), p(t)), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))) |n f h Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof is completed.

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Proposition 4.7 Suppose that three error logical variables A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), # » # »   Az B x (t) = f ((μ (t), p(t)), G(t))), and C ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U  (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U  (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t))) are defined in domain U  (t) under G(t) the rules for # » # » judging errors, then:Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))) nhb B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) nhb Txly # » # » (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof Since Txly does not change the error function f and G(t) rules for judging errors # » # » but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), # » # »   B((U (t), S(t), p(t), T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t),   x (t), and x (t) will not be changed in the transformation, then:Txly (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) nhb B((U (t), S(t), p(t), # » # » T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) ∧ Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) ∧ # » # » Txly (C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U  (t), S(t), # » # » # » p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) ∧ C Az B ((U  (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) nhb B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # »  f ((μ (t), p(t)), G(t)))). Proof is completed. Proposition 4.8 Suppose that three error logical variables A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), B((U  (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U  (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t))) are defined in domain U  (t) under G(t) the rules for

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# » # » judging errors, then:Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))) nhdl B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) nhdl Txly # » # » (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof Since Txly does not change the error function f and G(t) rules for judging errors # » # » but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), and C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t), x  (t), and x  (t) will not be changed in the transformation, then:Txly (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) nhdl B((U (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t)))) = Txly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∧ # » # » Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) ∧ Txly # » # » (¬C Az B ((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∧ B((U  (t), S(t), # » # » # » p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) ∧ ¬C Az B ((U  (t), S(t), p(t), # » # » T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) = Txly (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) nhdl Txly (B((U (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t)))). Proof is completed. # » Proposition 4.9 Suppose that two error logical variables A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) and B((U (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); suppose that another two error logical variables # » # » A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) and B((U  (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) are defined in domain # » U  (t) under G(t) the rules for judging errors, then:Txly (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) → B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t)))) = Txly (¬A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) ∨ Txly (B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))). Proof Since Txly does not change the error function f and G(t) rules for judging errors # » # » but the domain of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) # » # » and B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), which means that x(t) and x  (t) will not be changed in the transformation, then:Txly (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) → B((U (t), S(t), p(t), # » # »   T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t)))) = Txly (¬A((U (t), S(t), p(t), T (t),

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# » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # »  f ((μ (t), p(t)), G(t)))) = Txly (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) ∨ Txly (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t)))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) = A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) → B((U  (t), S(t), p(t), # » # » T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) = Txly (A((U  (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t)))) → Txly (B((U  (t), S(t), p(t), T (t), L(t)), # » x  (t) = f ((μ (t), p(t)), G(t)))). Proof is completed. Because →nsy connotative isness implication connective, →nhy connotative possibility implication connective, →nby connotative necessity implication connective, =nhdt connotative same connective, and ←→nhdz connotative equivalence connective can be expressed by denotation connective →, we will not discuss each of them here.

4.1.5 Characteristics of Thing Similarity Transformation Connectives in Error Logic # » # » For Txsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t), # » # » S  (t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), Txsw is the thing geographic similarity transformation connective with respect to G(t) and A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in U (t), as S  (t) = Txsw S(t), S  (t) and S(t) have the relationship of geographic similarity. Generally, the property T (t), property (or attribute) value L(t), error value x(t) of both S  (t) and S(t) also have relationship of geographic similarity. In some rare occasion, the property T (t), property (or attribute) value L(t), error value x(t) of both S  (t) and S(t) might be absolutely different (or sometimes equivalent). For example, if # » # » Txsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = A ((U  (t), # » # » S  (t), p  (t), T  (t), L  (t)), x  (t) = f ((μ (t), p  (t)), G(t))), Txsw has conducted thing # » geographic similarity transformation on thing of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))) and it also simultaneously changed domain (U (t) to U  (t)), property (T (t) to T  (t)), property (or attribute) value(L(t) to L  (t)), and error value (x(t) to x  (t)) of A(μ(t), x(t)). # » Proposition 4.10 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txsw

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91

# » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (1) Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ # » # » Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ # » # » Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txsw (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) # » # » = ¬Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))); And if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (4) Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ # » # » Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (5) Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ # » # » Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (6) Txsw (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) # » # » = ¬Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). Proof As indicated in the above description as S  (t) = Txly S(t), S  (t) and S(t) have the relationship of geographic similarity. Generally, the property T (t), property (or attribute) value L(t), error value x(t) of both S  (t) and S(t) also have relationship of geographic similarity. In some rare occasion, the property T (t), property (or attribute) value L(t), error value x(t) of both S  (t) and S(t) might be absolutely different (or sometimes equivalent). Because the operators ∨, ∧, and ¬ can only act on x(t), if # » x1 (t)  x2 (t) then the left side: Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p2 (t)), G(t)))) = Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # » p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » G(t))). if x1 (t)  x2 (t) then the right side Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∨ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t),

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# » # » # » p2 (t)), G(t))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (2) and (3) can be proved; under the conditions of “if x1 (t)  x2 (t) then x1 (t)  x2 (t) or if x1 (t)  x2 (t) then x1 (t)  x2 (t)”, cases (4) to (6) can also be proved in similar way. # » Proposition 4.11 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txsw # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # »        B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txsw # » # » (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  x2 (t)  x3 (t), then: Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), S1 (t), # » #2 » 2 p (t), T (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txsw (B((U (t), S2 (t), # » #1 » 1 p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » Proof Based on the definition on ∨n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »2 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » Az B x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B

4.1 Similarity Transformation Connectives in Error Logic

93

# » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the conditions of 0  # » x1 (t)  x2 (t)  x3 (t), because the left side = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), # » # » n L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # »2 # » # » p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » # » p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((− x1 (t)) ∧ x2 (t) ∧ (−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) =

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4 Transformation Connectives in Error Logic

# » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » #  »1 p (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # 2 » p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.12 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txsw # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txsw # » # » (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  x2 (t)  x3 (t), then:Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), S1 (t), # » #2 » 2 p (t), T (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txsw (B((U (t), S2 (t), # » #1 » 1 p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » Proof Based on the definition on ∧n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »2 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t),

4.1 Similarity Transformation Connectives in Error Logic

95

# » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » Az B x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 # » # » (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # » p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). For 0  x1 (t)  x2 (t)  x3 (t), because the left side = Txsw (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw ((A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » #2 » p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A # » # » ((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) = x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) =

96

4 Transformation Connectives in Error Logic

# » # » f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. Proposition 4.13 Suppose that three error logical variables A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txsw (B # » # » ((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txsw (C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x3 (t)  (−x2 (t)), x3 (t)  x1 (t) and x3 (t)  (−x2 (t)), x3 (t)  x1 (t), then:Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p (t), # » #2 » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txsw (B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » Proof Based on the definition on −n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »2 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the left side = # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw ((A # » # » ((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txsw (C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))).  And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = Txsw # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txsw

4.1 Similarity Transformation Connectives in Error Logic

97

# » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  # » # » (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. Proposition 4.14 Suppose that three error logical variables A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), # » # » Az B L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C ((U (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the # » rules for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » p(t)), G(t))) is the complement error logical variable of A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logi# » # » cal variables Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), # » # » Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and # » # » Txsw (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | # » and x3 (t) have the same sign, then:Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). Proof Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | and x3 (t) have the same sign, the left side = # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw # » # » ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t),

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# » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txsw ((A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) ∧ (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » Az B x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txsw (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » Az B x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign,the right side = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 # » # » (t), p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # » p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  # » # » (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.15 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # »  # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txsw (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under

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G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) # » and 0  x1 (t)  x2 (t)  (−x3 (t)), then:Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f h Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). Proof Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the left side = Txsw # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw ((A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)),the right side = Txsw # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txsw # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #  »3 p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A # » # » ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.16 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)),

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# » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # »  # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txsw (C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x1 (t)  x2 (t)  x3 (t) and # » x1 (t)  x2 (t)  x3 (t), then:Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » p (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » #2 » p (t)), G(t)))) nhb Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))). Proof Under the conditions of x1 (t)  x2 (t)  x3 (t), the left side = Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) # » # » = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f # » ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t),the right side = Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.17 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules

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101

# » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical variables # » # » Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txsw (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x1 (t)  x2 (t)  (−x3 (t)) and # » x1 (t)  x2 (t)  (−x3 (t)), then:Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » #2 » p (t)), G(t)))) nhdl Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))). Proof Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the left side = Txsw # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U # » # » (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)),the right side = Txsw (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txsw (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.18 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txsw # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  # » # » (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txsw (B((U (t),

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# » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t) and 0  # » x1 (t)  x2 (t), then: Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) → Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the conditions of 0  x1 (t)  x2 (t), the left side = Txsw (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txsw (¬A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # » f ((μ (t), p(t)), G(t)))) = Txsw (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) ∨ Txsw (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t)))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) Under the condi# » tions of 0  x1 (t)  x2 (t),the right side = Txsw (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txsw (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) → B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 # » # » # » (t), p2 (t)), G(t))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

4.1.6 Characteristics of Property Similarity Transformation Connectives in Error Logic # » # » For Txt x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t), # » # » S(t), p(t), T  (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))), Txt x is the property simi# » larity transformation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in U (t), as T  (t) = Txt x T (t), T  (t) and T (t) have the relationship of similarity. Generally, the error values x(t) and x  (t) have # » some intrinsic relationship. In some case such that Txt x (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t)))) == A ((U  (t), S  (t), p  (t), T  (t), L  (t)), # » x  (t) = f ((μ (t), p  (t)), G  (t))), it is said that property similarity transformation connective Txt x has caused the simultaneous changes in domain, thing, spatial status, property, property (or attribute) value, and error value of error logic variable # » # » A((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ(t), p(t)), G(t))).

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# » Proposition 4.19 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  p (t), T (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txt x (B((U (t), S2 (t), # » # » #1 » 1 p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (1) Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txt x # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txt x # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txt x (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » ¬Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))));   And if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (4) Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txt x # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (5) Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txt x # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (6) Txt x (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » ¬Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). Proof As indicated in the above description as T  (t) = Txt x T (t), T  (t) and T (t) have the relationship of property similarity. Generally, the error values x(t) and x  (t) have some intrinsic relationship. If x1 (t)  x2 (t) then the left side: Txt x (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). # » If x1 (t)  x2 (t) then the right side= Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) =

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# » # » f ((μ1 (t), p1 (t)), G(t))) ∨ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), #  »2 p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (2) and (3) can be proved; under the conditions of “if x1 (t)  x2 (t) then x1 (t)  x2 (t) or if x1 (t)  x2 (t) then x1 (t)  x2 (t)”, cases (4) to (6) can also be proved in similar way. # » Proposition 4.20 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txt x (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txt x (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  x2 (t)  # » # » x3 (t), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n # » # » Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » Proof Based on the definition on ∨n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »2 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » Az B G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t),

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# » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » Az B f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t)))). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the # » # » left side = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txt x ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t))))) # » # » = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ (−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » p3 (t), T3 (t), L 3 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), # » # » x3 (t) = f ((μ3 (t), p  (t)), G(t)))) ∨ (¬A ((U  (t), S  (t), p  (t), T1 (t), L 1 (t)), #  »3 #  »1  1      x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B ((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 # » # » (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)),

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# » # » G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  # » # » (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), #  »3 p1 (t)), G(t))). Left side = right side. Proof is completed. Proposition 4.21 Suppose that three error logical variables A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables # » # » Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U # » # » (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txt x (C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  # » x3 (t) and 0  x1 (t)  x2 (t)  x3 (t), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) ∧n Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). # » Proof Based on the definition on ∧n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »2 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t),

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# » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 # » # » (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). For 0  x1 (t)  x2 (t)  x3 (t), because the left side = Txt x (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x ((A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) = x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txt x (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) =

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# » # » f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.22 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txt x (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t)= f ((μ2 (t), p2 (t)), G(t))), and Txt x (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x3 (t)  (−x2 (t)), x3 (t)  x1 (t) and x3 (t)  # » (−x2 (t)), x3 (t)  x1 (t), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))))=Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), # » # » G(t)))) −n Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » Proof Based on the definition on −n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »2 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the left side = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))))=Txt x ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) # » # » = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))))) =Txt x (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txt x

4.1 Similarity Transformation Connectives in Error Logic

109

# » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), # » #  »1 p2 (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » p (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # 3 » p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.23 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txt x (C Az B ((U (t), S3 (t), # » #2 » 2 # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have the same sign, then:Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l Txt x # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | and x3 (t) have the same sign, the left side = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) |n f l B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x ((A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))))) = Txt x ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) =

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# » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t)))) ∧ (C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » Az B G(t))) ∨ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t))))) # » # » = Txt x (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ # » # » Az B ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign, the right side = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » nf l f ((μ1 (t), p1 (t)), G(t)))) | Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.24 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for # » # » judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txt x (C Az B ((U (t), S3 (t), # » #2 » 2 # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) # » and 0  x1 (t)  x2 (t)  (−x3 (t)), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)),

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# » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f h Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). Proof Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the left side = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x ((A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » Az B G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txt x # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #  »3 p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.25 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)

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# » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for # » # » judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # »        A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txt x (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x1 (t)  x2 (t)  x3 (t) and # » x1 (t)  x2 (t)  x3 (t), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), #2 » # » p (t)), G(t)))) nhb Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))). Proof Under the conditions of x1 (t)  x2 (t)  x3 (t), the left side = Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t)= f ((μ3 (t), p3 (t)), G(t))))=Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.26 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)),

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# » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical variables # » # » Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txt x (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x1 (t)  x2 (t)  (−x3 (t)) and # » x1 (t)  x2 (t)  (−x3 (t)), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » #2 » p (t)), G(t)))) nhdl Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))). Proof Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the left side = Txt x # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) # » # » = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txt x (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » # » = f ((μ3 (t), p3 (t)), G(t)))) = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.27 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txt x (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t),

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# » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t) and 0  x1 (t)  # » # » x2 (t), then:Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txt x (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) # » # » → Txt x (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the conditions of 0  x1 (t)  x2 (t), the left side = Txt x (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txt x (¬A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), # » # » x  (t) = f ((μ (t), p(t)), G(t)))) = Txt x (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) ∨ Txt x (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t)))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) Under the con# » ditions of 0  x1 (t)  x2 (t), the right side = Txt x (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txt x (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # »2 p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

4.1.7 Characteristics of Spatial Similarity Transformation Connectives in Error Logic # » Suppose that an error logical variable A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))) is defined in domain U (t) under G(t) the rules for judging # » # » errors. For Txk j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t), S(t), # » # » p (t), T (t), L(t)), x(t) = f ((μ(t), p  (t)), G(t))), Txk j is the spatial similarity trans# » formation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) # » # » # » = f ((μ(t), p(t)), G(t))) defined in U (t), as p(t) and p (t) are spatial locations of # » # » # » # » # » (U (t), S(t), p(t), T (t), L(t)), p(t) is replaced by p  (t) or p  (t) is replaced by p(t), which is to conduct transformation on the spatial location of subject μ(t). Generally, # » # » # » # » # » p(t) and p  (t) have some intrinsic relationship-i.e., Txk j ( p(t)) = p  (t) ∈ [ p(t) − # » # » #»  , p(t) + #»  ]. In some case such that Txk j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) == A ((U  (t), S  (t), p  (t), T  (t), L  (t)), x  (t) = f ((μ (t), # » p (t)), G(t))), it is said spatial similarity transformation connective Txk j has caused simultaneous changes in domain, thing, spatial status, property, property (or attribute)

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# » value, and error value of error logic variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))). # » Proposition 4.28 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  p (t), T (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txk j (B((U (t), S2 (t), # » # » #1 » 1 p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under # » # » G(t) the rules for judging errors; if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds # » # » when p (t) = p(t), which means that the error value did not change because the spatial location was not changed; the relationship between x(t)) and x  (t) is represented in the following forms: 

x (t) = x  (t) = x  (t) = x  (t) =

# » # » # » | p  (t) | # » cos( p (t), p(t))x(t) | p(t) | # » # » # » | p(t) | #  » cos( p (t), p(t))x(t) | p (t) | # » | p  (t) | # » # » # » | p(t) | cos( p  (t), p(t))x(t) # » | p(t) | # » # » # » | p  (t) | cos( p  (t), p(t))x(t)

Here we have:

# » # » (1) Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txk j (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) # » # » =¬Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). # » # » # » (t)| Proof In (1), from the assumption x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very # » # » # » # » # » small positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = | p (t)| k > 0, therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), without loss of generality, it

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# » is assumed that x1 (t)  x2 (t), then the left side = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))))=Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) # » # » = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). As x1 (t)  x2 (t) holds when x1 (t)  x2 (t) then the right side= Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (2) and (3) can be proved; under the conditions of “if x1 (t)  x2 (t) then x1 (t)  x2 (t) or if x1 (t)  x2 (t) then x1 (t)  x2 (t)”. # » Proposition 4.29 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txk j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txk j (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for # » # » # » # » judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when p  (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) and x  (t) is represented in # » # » # » # » (t)| the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t) Txk j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txk j (B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very # » # » # » # » # » small positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = | p (t)| k > 0, therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t) 

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x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t). Based on the definition on ∨n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 # » # » (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the # » conditions of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j ((A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)),

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# » x1 (t)= f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ (−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » #  »2 p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), # » # » # »  p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » #  »2 p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), # »  # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.30 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t)= f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging # » errors; suppose that another three error logical variables Txk j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) =

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119

# » # » f ((μ2 (t), p2 (t)), G(t))), and Txk j (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging # » # » # » # » errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when p  (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) and x  (t) is represented in # » # » # » (t)| the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), then:Txk j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txk j (B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very # » # » # » # » # » small positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| cos( p  (t), p(t)) =  » | p (t)| k > 0, therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t). Based on the definition on ∧n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » # »2 p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t)))). Under condition of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t),

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# » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t))=x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). Under the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » #  »2 p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), # » # » # »  p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.31 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txk j (C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under # » # » G(t) the rules for judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds # » # » when p (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between # » # » (t)| x(t)) and x  (t) is represented in the following forms: x  (t) = ||pp(t)| # » cos( p (t),

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# » # » # » p(t))x(t), then:Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n # » # » Txk j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very # » # » # » # » # » small positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = | p (t)| k > 0, therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t). Based on the definition on −n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the left side = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j ((A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ C # » # » f ((μ3 (t), p3 (t)), G(t))))) =Txk j (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » #  »1 p (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # 2 » p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » p3 (t)), G(t))). Left side = right side. Proof is completed.

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# » Proposition 4.32 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txk j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the # » # » rules for judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when # » # » p (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) and # » # » # » (t)| x  (t) is represented in the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), then: # » # » Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very small # » # » # » # » # » positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = k > 0, | p (t)| therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j ((A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t))))) = Txk j ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) ∧ # » # » Az B (C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txk j (C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))).

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Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign, the right side = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txk j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.33 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txk j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) # » # » the rules for judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when # » # » p (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) # » # » # » (t)| and x  (t) is represented in the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), # » # » then:Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))).

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# » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very # » # » # » # » # » small positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = | p (t)| k > 0, therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  # » # » (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #  »3 p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.34 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)

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# » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txk j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) # » # » the rules for judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when # » # » p (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) and # » # » # » (t)| x  (t) is represented in the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), then: # » # » Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very small # » # » # » # » # » positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = k > 0, | p (t)| therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) # » # » = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))))=(A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txk j (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))))=(A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))).

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Left side = right side. Proof is completed. # » Proposition 4.35 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txk j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txk j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) # » # » the rules for judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when # » # » p (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) # » # » # » (t)| and x  (t) is represented in the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), # » # » then:Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very small # » # » # » # » # » positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = k > 0, | p (t)| therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) # » # » Az B = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » # » = f ((μ3 (t), p3 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))))=(A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txk j (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t),

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# » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » # » = f ((μ3 (t), p3 (t)), G(t)))) = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.36 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txk j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txk j (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) # » # » the rules for judging errors, if x  (t) = f ( p  (t), p(t), x(t)), x  (t) = x(t) holds when # » # » p (t) = p(t), which means that the error value did not change because the spatial location was not changed; it is assumed that the relationship between x(t)) and # » # » # » (t)| x  (t) is represented in the following forms: x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), then: # » # » Txk j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txk j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » # » (t)| Proof From the assumption of x  (t) = ||pp(t)| # » cos( p (t), p(t))x(t), as  is a very small # » # » # » # » # » positive number, so 0 < cos( p  (t), p(t))  1, and |#p(t)| » cos( p (t), p(t)) = k > 0, | p (t)| therefore x1 (t)  x2 (t) holds when x1 (t)  x2 (t), ∃(t)  0, 0  x1 (t)  x2 (t)  x3 (t) holds when 0  x1 (t)  x2 (t)  x3 (t), the left side = Txk j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txk j (¬A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = # » # »  f ((μ (t), p(t)), G(t)))) = Txk j (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) ∨ Txk j (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » # » G(t)))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) Under the con# » ditions of 0  x1 (t)  x2 (t), the right side = Txk j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txk j (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) =

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# » # » f ((μ2 (t), p2 (t)), G(t))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

4.1.8 Characteristics of Property (or Attribute) Value Similarity Transformation Connectives in Error Logic # » # » For Txlz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))))=A((U (t), # » # » # » S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t), S(t), p(t), # »    T (t), L (t)), x (t) = f ((μ (t), p(t)), G(t))), Txlz is the property (or attribute) value # » similarity transformation connective with respect to G(t) and A((U (t), S(t), p(t), # »  T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in U (t), as L (t) = Txlz (or L  (t) = k L(t)L(t), L  (t) and L(t) have the relationship of similarity. Generally, the error values x(t) and x  (t) corresponding to L  (t) and L(t) respectively have some intrin# » sic relationship. In some case such that Txlz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) == A ((U  (t), S  (t), p  (t), T  (t), L  (t)), x  (t) = f ((μ (t), # » p (t)), G  (t))), it is said property (or attribute) value similarity transformation connective Txlz has caused the simultaneous changes in domain, thing, spatial status, property, property (or attribute) value, and error value of error logic variable # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))). # » Proposition 4.37 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txlz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) and Txlz (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (1) Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txlz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txlz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txlz (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) # » # » =¬Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))));

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And if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (4) Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txlz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (5) Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txlz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (6) Txlz (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) # » # » =¬Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). Proof As indicated in the above description as L  (t) = Txlz L(t), L  (t) and L(t) have the relationship of property similarity. Generally, the error values x(t) and x  (t) corresponding to L  (t) and L(t) respectively have some intrinsic relationship and they can # » change according to some laws. However, in cases, U (t), S(t), p1 (t), and G(t) will not be changed. If x1 (t)  x2 (t) then x1 (t)  x2 (t), the left side: Txlz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). # » If x1 (t)  x2 (t) then the right side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t)= f ((μ1 (t), p1 (t)), G(t)))) ∨ Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))))=A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (2) and (3) can be proved; under the conditions of “if x1 (t)  x2 (t) then x1 (t)  x2 (t) or if x1 (t)  x2 (t) then x1 (t)  x2 (t)”, cases (4) to (6) can also be proved in similar way. # » Proposition 4.38 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txlz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) =

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# » f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  # » x2 (t)  x3 (t), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))))=Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t)))) # » # » ∨n Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most cases, # » U (t), S(t), p1 (t), and G(t) will not be changed. Based on the definition on ∨n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # »3 # » # » p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under condi# » tions of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) # » # » = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # » p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), # »1 # » # » p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)),

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# » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » Az B L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ (−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txlz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txlz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), #  »2 # » # » p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # »  # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » #  »1 p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » # » p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed.

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# » Proposition 4.39 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txlz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  # » x2 (t)  x3 (t), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » n G(t)))) ∧ Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most cases, # » U (t), S(t), p1 (t), and G(t) will not be changed. Based on the definition on ∧n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » # »2 p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t)))). # » For 0  x1 (t)  x2 (t)  x3 (t), because the left side = Txlz (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txlz ((A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) # » # » = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t),

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# » # » # » p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » Az B G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) = x1 (t) ∨ (− x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txlz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txlz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.40 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txlz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules

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for judging errors, it is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then # » x3 (t)  (−x2 (t)) and x3 (t)  x1 (t), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) −n Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most cases, # » U (t), S(t), p1 (t), and G(t) will not be changed. Based on the definition on −n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), # » x3 (t)  x1 (t), the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » n f ((μ (t), p1 (t)), G(t))) − B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p2 (t)), G(t)))) = Txlz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) G(t))) ∧ C # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) =Txlz (C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))).    And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = # » # » Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n # » # » Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed.

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# » Proposition 4.41 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have the same sign, # » # » then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l # » # » Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most # » cases, U (t), S(t), p1 (t), and G(t) will not be changed. Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | # » and x3 (t) have the same sign, the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txlz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t))))) = Txlz # » # » ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) ∧ (C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), # » # » Az B T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txlz (C ((U (t), S3 (t), p3 (t), # » # » Az B T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign, the right side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) =

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# » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.42 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # »  # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) # » then 0  x1 (t)  x2 (t)  (−x3 (t)), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f h Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most # » cases, U (t), S(t), p1 (t), and G(t) will not be changed. Under the condition of 0  # » x1 (t)  x2 (t)  (−x3 (t)), the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)),

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# » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txlz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » # » Az B p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = # » # » Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h # » # » Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » p (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » # 2 » p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.43 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t),

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# » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C Az B ((U (t), S3 (t), # » #2 » 2 # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if x1 (t)  x2 (t)  x3 (t) and # » x1 (t)  x2 (t)  x3 (t), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) nhb Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most cases, # » U (t), S(t), p1 (t), and G(t) will not be changed. Under the conditions of x1 (t)  # » x2 (t)  x3 (t), the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txlz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txlz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.44 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) =

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# » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txlz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txlz (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if x1 (t)  x2 (t)  (−x3 (t)) and # » x1 (t)  x2 (t)  (−x3 (t)), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » #2 » p (t)), G(t)))) nhdl Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t), x2 (t) and x3 (t) corresponding to L 1 (t), L 2 (t) and L 3 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most cases, # » U (t), S(t), p1 (t), and G(t) will not be changed. Under the conditions of x1 (t)  # » x2 (t)  (−x3 (t)), the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txlz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txlz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.45 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txlz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txlz (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t),

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# » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if 0  x1 (t)  x2 (t) and 0  x1 (t)  # » # » x2 (t), then:Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → # » # » Txlz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof As indicated in the above description, generally, the error values x1 (t) and x2 (t) corresponding to L 1 (t) and L 2 (t) respectively have some intrinsic relationship and they can change according to some laws. However, in most cases, # » U (t), S(t), p1 (t), and G(t) will not be changed. Under the conditions of 0  x1 (t)  # » x2 (t), the left side = Txlz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txlz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txlz (¬A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∨ Txlz (B((U (t), S(t), p(t), # » # »    T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # »  p(t)), G(t))) Under the conditions of 0  x1 (t)  x2 (t), the right side = Txlz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txlz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B  ((U  (t), S2 (t), # »  # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = A((U  (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

4.1.9 Characteristics of Error Value Similarity Transformation Connectives in Error Logic # » # » For Txcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t), # » # »   S(t), p(t), T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t))), Txcz is the error value sim# » ilarity transformation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in U (t), i.e., Txcz (x(t)) = x  (t) ∈ # » [x(t)-(t), x(t)+(t)]. In some case such that Txcz (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t)))) == A ((U  (t), S  (t), p  (t), T  (t), L  (t)), x  (t) = # » f ((μ (t), p  (t)), G  (t))), it is said error value similarity transformation connective Txcz has caused the simultaneous changes in domain, thing, spatial status, property, property (or attribute) value, and error value of error logic variable # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))).

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# » Proposition 4.46 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txcz (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, if x1 (t)  x2 (t), then x1 (t)  x2 (t), or if x1 (t)  x2 (t), then x1 (t)  x2 (t), then: # » # » (1) Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ # » # » Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ # » # » Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txcz (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » ¬Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t) + (t)], then ∃(t)  0,if x1 (t)  x2 (t) then x1 (t)  x2 (t), the left side= Txcz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). # » If x1 (t)  x2 (t) then the right side = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∨ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (2) and (3) can be proved; under the conditions of “if x1 (t)  x2 (t) then x1 (t)  x2 (t) or if x1 (t)  x2 (t) then x1 (t)  x2 (t)”, cases (4) to (6) can also be proved in similar way. # » Proposition 4.47 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for

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judging errors; suppose that another three error logical variables Txcz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txcz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  # » x2 (t)  x3 (t), then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » n G(t)))) ∨ Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t) + (t)], then ∃(t)  0, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t), Based on # » the definition on ∨n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the left side = # » # » Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txcz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t),

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# » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » # » Az B p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ (−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txcz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » #  »1 p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t),

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# » # » # » p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.48 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))),and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txcz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))),and Txcz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if 0  x1 (t)  x2 (t)  x3 (t) and 0  x1 (t)  # » x2 (t)  x3 (t), then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) ∧n Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t) + (t)], then ∃(t)  0, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t), based on # » # » the definition on ∧n : A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))).

4.1 Similarity Transformation Connectives in Error Logic

145

Under the condition of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txcz ((A((U (t), S1 (t), #2 » 2 # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # »3 # » # » p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬ # » # » ((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) = x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txcz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.49 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txcz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t),

146

4 Transformation Connectives in Error Logic

# » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txcz (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  # » (−x2 (t)) and x3 (t)  x1 (t) hold, then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) −n Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t) + (t)], then ∃(t)  0, if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  (−x2 (t)) and x3 (t)  # » x1 (t), based on the definition on −n , we have: A((U (t), S1 (t), p1 (t), T1 (t), # » # » n L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) − B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the left side = Txcz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txcz ((A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t))))) =Txcz (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = # » # » Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txcz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) =

4.1 Similarity Transformation Connectives in Error Logic

147

# » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.50 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # » # »  p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txcz (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors,it is assumed that if | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have the same sign, # » # » then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txcz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l Txcz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t) + (t)], then ∃(t)  0,if | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have the same sign. Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | and x3 (t) have # » the same sign, the left side = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) |n f l B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t)))) = Txcz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #2 » p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), # » # » p (t), T (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txcz ((A((U (t), S1 (t), # » # » #3 » 3 p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) ∧ (C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txcz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) =

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# » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign,the right side = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.51 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txcz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txcz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) then 0  # » x1 (t)  x2 (t)  (−x3 (t)), then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), #2 » # » p (t)), G(t)))) |n f h Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))).

4.1 Similarity Transformation Connectives in Error Logic

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Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t)+(t)], then ∃(t)  0, if 0  x1 (t)  x2 (t)  (−x3 (t)) then 0  x1 (t)  x2 (t)  (−x3 (t)). Under the condition of 0  x1 (t)  x2 (t)  (−x3 (t)), the left side = Txcz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txcz ((A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = Txcz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txcz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #  »3 p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.52 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)),

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# » x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txcz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txcz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x1 (t)  x2 (t)  x3 (t) then x1 (t)  # » x2 (t)  x3 (t), then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) nhb Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t)+(t)], then ∃(t)  0, if x1 (t)  x2 (t)  x3 (t) and x1 (t)  x2 (t)  x3 (t). Under the conditions of # » x1 (t)  x2 (t)  x3 (t), the left side = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txcz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.53 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txcz

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# » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txcz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txcz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judging errors, it is assumed that if x1 (t)  x2 (t)  (−x3 (t)) and x1 (t)  # » x2 (t)  (−x3 (t)), then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) nhdl Txcz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t)+(t)], then ∃(t)  0, if x1 (t)  x2 (t)  (−x3 (t)) and x1 (t)  x2 (t)  (−x3 (t)). Under the conditions # » of x1 (t)  x2 (t)  (−x3 (t)), the left side = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txcz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » # » = f ((μ3 (t), p3 (t)), G(t)))) = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))))=(A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.54 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txcz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txcz (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under G(t) the

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rules for judging errors, it is assumed that if 0  x1 (t)  x2 (t) then 0  x1 (t)  # » # » x2 (t), then:Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txcz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption, error value x  (t) ∈ [x(t)-(t), x(t)+(t)], then ∃(t)  0, if 0  x1 (t)  x2 (t) and 0  x1 (t)  x2 (t). Under the conditions of 0  x1 (t)  # » x2 (t), the left side = Txcz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txcz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txcz (¬A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∨ Txcz (B((U (t), S(t), p(t), # » # »    T (t), L(t)), x (t) = f ((μ (t), p(t)), G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # »  p(t)), G(t))) Under the conditions of 0  x1 (t)  x2 (t), the right side = Txcz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txcz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B  ((U  (t), S2 (t), # » # »  # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = A((U  (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = # » f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

4.1.10 Characteristics of Rule Similarity Transformation Connectives in Error Logic # » # » For Txgz (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G(t))))=A((U (t), # » # » # » S(t), p(t), T (t), L(t)) , x  (t)= f ((μ (t), p  (t)), G  (t)))=A((U (t), S(t), p(t), T (t), # » L  (t)), x  (t) = f ((μ (t), p(t)), G(t))), Txgz is the rule similarity transformation # » connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), # » p(t)), G(t))) defined in U (t), i.e., Txgz (G(t)) = G  (t). Generally, (G(t)) and G  (t) have some intrinsic relationship and they belong to the same type of rules. In # » # » some case such that Txgz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) == A ((U  (t), S  (t), p  (t), T  (t), L  (t)), x  (t) = f ((μ (t), p  (t)), G  (t))), it is said rule similarity transformation connective Txgz has caused the simultaneous changes in domain, thing, spatial status, property, property (or attribute) # » value, and error value of error logic variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))).

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# » Proposition 4.55 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  p (t), T (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txgz (B((U (t), S2 (t), # » # » #1 » 1 p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under # » G(t) the rules for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t)  holds when G(t) = G (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by 1 x1 (t), where R(G  (t), G(t)) x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| = 0, the rule is not changed when R(G  (t), G(t)) = 1. We have: # » # » (1) Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txgz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txgz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txgz (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » ¬Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = 1 x (t), if let k = | R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| 1 1 ,where R(G  (t), G(t)) = 0, then 1  k  ∞, therefore, x1 (t)  x2 (t) |R(G  (t),G(t))| holds if x1 (t)  x2 (t). Without loss of generality, it is assumed that x1 (t)  x2 (t), # » then the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). # » If x1 (t)  x2 (t) then the right side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ Txgz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) # » # » = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))) ∨ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # » p2 (t)), G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (2) and (3) can be proved; under the conditions of “x1 (t)  x2 (t) holds if x1 (t)  x2 (t)”.

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# » Proposition 4.56 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txgz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judg# » ing errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when G(t) = G  (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by x1 (t) =| R(G  (t), G(t)) | 1 x1 (t), where R(G  (t), G(t)) = 0, the rule is not changed x1 (t) or x1 (t) = |R(G  (t),G(t))| when R(G  (t), G(t)) = 1, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  # » x3 (t) holds, we have: Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) ∨n Txgz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = 1 1 x (t), if let k= | R(G  (t), G(t)) |, then 0  k  1 or if let k= |R(G  (t),G(t))| , |R(G  (t),G(t))| 1  where R(G (t), G(t)) = 0, then 1  k  ∞, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t) holds. Without loss of generality, it is assumed that 0  x1 (t)  x2 (t)  x3 (t), then the left side = Based on the definition on ∨n , # » # » we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) =

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155

# » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the # » conditions of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txgz (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz ((A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » x1 (t)= f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ (−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txgz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)),

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# » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » #  »1 p (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » # 2 » p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.57 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txgz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules for judg# » ing errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when G(t) =  G (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by x1 (t) =| R(G  (t), G(t)) | 1 x1 (t), where R(G  (t), G(t)) = 0, the rule is not changed x1 (t) or x1 (t) = |R(G  (t),G(t))| when R(G  (t), G(t)) = 1, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  # » x3 (t) holds, We have: Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) ∧n Txgz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). 1 Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| 1 x1 (t), if let k =| R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| ,where

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R(G  (t), G(t)) = 0, then 1  k  ∞, therefore, 0  x1 (t)  x2 (t)  x3 (t) holds if 0  x1 (t)  x2 (t)  x3 (t). Without loss of generality, it is assumed that 0  # » x1 (t)  x2 (t)  x3 (t), based on the definition on ∧n : A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » Az B T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » # »2 p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the condition of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t))=x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txgz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) =

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# » # » f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # » p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.58 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txgz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the rules # » for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when  G(t) = G (t) meaning that the error value will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by 1 x1 (t), where R(G  (t), G(t)) x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| = 0, the rule is not changed when R(G  (t), G(t)) = 1. It is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) hold, we have: # » # » Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))))−n Txgz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). 1 Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| 1 x1 (t), if let k =| R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| ,where  R(G (t), G(t)) = 0, then 1  k  ∞. It is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) hold,based on the defini# » # » tion on −n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t),

4.1 Similarity Transformation Connectives in Error Logic

159

# » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t)= f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  # » x1 (t), the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) =Txgz (C AnhbB ((U (t), S3 (t), # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), # »  # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txgz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » p (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # 2 » p (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # 3 » p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.59 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the

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# » rules for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when G(t) = G  (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by 1 x1 (t), where R(G  (t), G(t)) x1 (t) = | R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| = 0, the rule is not changed when R(G  (t), G(t)) = 1. It is assumed that if | x3 (t) | x2 (t) and | x3 (t) | x1 (t) then | x3 (t) | x2 (t) then | x3 (t) | x1 (t) hold, | x3 (t) | # » and x3 (t) have the same sign, then:Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) |n f l B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p2 (t)), G(t))))=Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » # » p1 (t)), G(t)))) |n f l Txgz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » p2 (t)), G(t)))). 1 Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| 1 x1 (t), if let k =| R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| ,where  R(G (t), G(t)) = 0, then 1  k  ∞. It is assumed that if | x3 (t) | x2 (t) and | x3 (t) | x1 (t) then | x3 (t) | x2 (t) and | x3 (t) | x1 (t) hold, | x3 (t) | and x3 (t) have the same sign. Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | and x3 (t) have the same sign, # » # » the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) |n f l B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) # » # » = Txgz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txgz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » Az B f ((μ2 (t), p2 (t)), G(t)))) ∧ (C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » Az B f ((μ3 (t), p3 (t)), G(t))))) = Txgz (C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))) Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | # » x3 (t) | and x3 (t) have the same sign, the right side = Txgz (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l Txgz (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » # » p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧

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# » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))=C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.60 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))=C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the # » rules for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when G(t) = G  (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by 1 x1 (t), where R(G  (t), G(t)) x1 (t) = | R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))|  = 0, the rule is not changed when R(G (t), G(t)) = 1. It is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) then 0  x1 (t)  x2 (t)  (−x3 (t)) holds, then:Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz (A((U (t), S1 (t), # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txgz (B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). 1 Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| 1 x1 (t), if let k =| R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| ,where  R(G (t), G(t)) = 0, then 1  k  ∞. Under the condition of 0  x1 (t)  x2 (t)  # » (−x3 (t)), the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧

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# » # » ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))).    Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txgz # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) |n f h B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #  »3 p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.61 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))=C Az B ((U  (t), S3 (t), p3 (t), T3 (t),

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# » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the # » rules for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when G(t) = G  (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by 1 x1 (t), where R(G  (t), G(t)) = x1 (t)= | R(G  (t), G(t)) | x1 (t) or x1 (t)= |R(G  (t),G(t))|  0, the rule is not changed when R(G (t), G(t)) = 1. It is assumed that if x1 (t)  # » x2 (t)  x3 (t) then x1 (t)  x2 (t)  x3 (t), then:Txgz (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) # » # » = f ((μ1 (t), p1 (t)), G(t)))) nhb Txgz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). 1 Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| 1  x1 (t), if let k =| R(G (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| ,where R(G  (t), G(t)) = 0, then 1  k  ∞. Under the conditions of x1 (t)  x2 (t)  # » x3 (t), the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » Az B C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))))=(A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txgz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t))))=(A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.62 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t),

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# » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txgz (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txgz (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))=C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the # » rules for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) = x1 (t) holds when G(t) = G  (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be represented by 1 x1 (t), where R(G  (t), G(t)) x1 (t) = | R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))|  = 0, the rule is not changed when R(G (t), G(t)) = 1. It is assumed that if x1 (t)  x2 (t)  (−x3 (t)) then x1 (t)  x2 (t)  (−x3 (t)) holds , then:Txgz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))))=Txgz (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txgz (B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). 1 Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))| 1 x1 (t), if let k =| R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| ,where  R(G (t), G(t)) = 0, then 1  k  ∞. Under the conditions of x1 (t)  x2 (t)  # » (−x3 (t)), the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » # » # » p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txgz # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))).   Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txgz (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » # » = f ((μ3 (t), p3 (t)), G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.63 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the

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rules for judging errors; suppose that another two error logical variables Txgz (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  p (t), T (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txgz (B((U (t), S2 (t), # » # » #1 » 1 p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t) under # » G(t) the rules for judging errors. If x1 (t) = f ((μ1 (t), p1 (t)), G  (t))), x1 (t) =  x1 (t) holds when G(t) = G (t) meaning that the error will not change if the rule is kept unchanged; here the relationship between x1 (t) and x1 (t) can be rep1 x1 (t), where resented by x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = |R(G  (t),G(t))|   R(G (t), G(t)) = 0, the rule is not changed when R(G (t), G(t)) = 1. It is assumed that if 0  x1 (t)  x2 (t) then 0  x1 (t)  x2 (t) holds, then:Txgz (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txgz (B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof From the assumption x1 (t) =| R(G  (t), G(t)) | x1 (t) or x1 (t) = 1 x (t), if let k =| R(G  (t), G(t)) |, then 0  k  1 or if let k = |R(G  (t),G(t))| 1 1 , where R(G  (t), G(t)) = 0, then 1  k  ∞. Under the conditions of |R(G  (t),G(t))| # » 0  x1 (t)  x2 (t), the left side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » # » p2 (t)), G(t)))) = Txgz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » Txgz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∨ # » # » Txgz (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), # » L(t)), x  (t) = f ((μ (t), p(t)), G(t))) Under the conditions of 0  x1 (t)  x2 (t), # » # » the right side = Txgz (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) → Txgz (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = A((U  (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), # » L(t)), x  (t) = f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

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4.1.11 Characteristics of Error Function Similarity Transformation Connectives in Error Logic # » # » For Txhs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f  ((μ (t), p(t)), G(t))), Txhs is the error function # » similarity transformation connective with respect to G(t) and A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in U (t),i.e., Txhs f ((μ(t), p(t)), # » # » G(t)) = f  ((μ (t), p(t)), G(t)). Generally, f ((μ(t), p(t)), G(t)) and f  ((μ (t), # » p(t)), G(t)) have some intrinsic relationship,e.g. linear relationship represented # » # » by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b. In some case such # » # » that Txhs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) == # » # » A ((U  (t), S  (t), p  (t), T  (t), L  (t)), x  (t) = f ((μ (t), p  (t)), G  (t))), it is said error function similarity transformation connective Txhs has caused the simultaneous changes in domain, thing, spatial status, property, property (or attribute) value, and # » error value of error logic variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G(t))). # » Proposition 4.64 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined # » in domain U  (t) under G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) # » # » and f  ((μ (t), p(t)), G(t)) have linear relationship represented by f  ((μ (t), p(t)), # » G(t)) = a( f ((μ(t), p(t)), G(t))) + b, where a  0, then we have: # » # » (1) Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); When a  0 # » # » (3) Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (4) Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) =

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# » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (5) Txhs (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » ¬Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). # » # » Proof In (1), based on the assumption f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), # »   G(t)) have linear relationship represented by f ((μ (t), p(t)), G(t)) = a( f ((μ(t), # » p(t)), G(t))) + b, when a  0, x1 (t)  x2 (t) holds if x1 (t)  x2 (t). Without loss of generality,it is assumed that x1 (t)  x2 (t), then the left side = Txhs (A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))). # » If x1 (t)  x2 (t) then the right side = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » # » p2 (t)), G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly, cases of (3)–(5) can be proved. # » Proposition 4.65 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txhs (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txhs (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the # » # » rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # »   have linear relationship represented by f ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)),   G(t))) + b, when a  0, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  # » x3 (t) holds,we have: Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) ∨n Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))).

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# » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # »   have linear relationship represented by f ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)),   G(t))) + b, when a  0, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t) holds. Without loss of generality,it is assumed that 0  x1 (t)  x2 (t)  x3 (t), then the left side = Based on the definition on ∨n , we have: A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the # » conditions of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txhs (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txhs ((A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » # » p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)),

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# » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t))) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t)∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧(−x3 (t))) ∨ (x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txhs (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » # » p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.66 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txhs (A((U (t), S1 (t),

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# » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txhs (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under G(t) the # » # » rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  # » x3 (t) holds,We have: Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) ∧n Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, 0  x1 (t)  x2 (t)  x3 (t) holds if 0  x1 (t)  x2 (t)  x3 (t). Without loss of generality,it is assumed that 0  x1 (t)  x2 (t)  x3 (t), based # » on the definition on ∧n : A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) ∧n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B¬((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))). Under the condition of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txhs ((A((U (t), S1 (t), # » # » #2 » 2 p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧

4.1 Similarity Transformation Connectives in Error Logic

171

# » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) = x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txhs (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » # » p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), # »  # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.67 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txhs (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under # » # » G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), # » G(t)) have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), # » p(t)), G(t))) + b, when a  0, it is assumed that if x3 (t)  (−x2 (t)) and x3 (t) 

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x1 (t) then x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) hold,we have: Txhs (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txhs (B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, it is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) hold,based on the definition on −n , we have: # » # » A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), # » # » the left side = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) =Txhs (C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed.

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# » Proposition 4.68 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txhs # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), # » # »  p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) # » # » under G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), # » G(t)) have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), # » p(t)), G(t))) + b, when a  0, it is assumed that if | x3 (t) | x2 (t) and | x3 (t) | x1 (t) then | x3 (t) | x2 (t) then | x3 (t) | x1 (t) hold, | x3 (t) | and x3 (t) have the # » # » same sign, then:Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » nf l G(t))) | B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). # » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # »   have linear relationship represented by f ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, it is assumed that if | x3 (t) | x2 (t) and | x3 (t) | x1 (t) then | x3 (t) | x2 (t) and | x3 (t) | x1 (t) hold, | x3 (t) | and x3 (t) have the same sign. Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | and x3 (t) have the same sign,the left side = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = Txhs ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) ∧ (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))))) = Txhs (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))).

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Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign,the right side = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.69 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), Txhs (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), # »  # » p (t), T (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C Az B ((U (t), S3 (t), # » # » #2 » 2 p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) under # » # » G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, it is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) then # » 0  x1 (t)  x2 (t)  (−x3 (t)) holds, then:Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f h Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))).

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# » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # »   have linear relationship represented by f ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, under the condition of 0  x1 (t)  x2 (t)  (−x3 (t)), the # » # » left side = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » nf h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = G(t))) | # » # » Txhs ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))).    Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h # » # » Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » # » p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.70 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules # » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t),

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# » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txhs # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), # » # »  p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) # » under G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), # » # » p(t)), G(t)) have linear relationship represented by f  ((μ (t), p(t)), G(t)) = # » a( f ((μ(t), p(t)), G(t))) + b, when a  0, it is assumed that if x1 (t)  x2 (t)  x3 (t) # » then x1 (t)  x2 (t)  x3 (t) holds, then:Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) nhb Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). # » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, under the conditions of x1 (t)  x2 (t)  x3 (t), the left side # » # » = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txhs # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txhs (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.71 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules

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# » # » for judging errors,where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), # » G(t))) is the complement error logical variable of A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))); suppose that another three error logical vari# » # » ables Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txhs # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txhs (C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), # » # »  p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t) # » # » under G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), # » G(t)) have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), # » p(t)), G(t))) + b, when a  0, it is assumed that if x1 (t)  x2 (t)  (−x3 (t)) then # » x1 (t)  x2 (t)  (−x3 (t)) holds, then:Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) nhdl Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » f ((μ2 (t), p2 (t)), G(t)))). # » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the # » # » left side = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txhs # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txhs (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.72 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t),

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# » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables # » # » Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txhs # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain # » U  (t) under G(t) the rules for judging errors. If f ((μ(t), p(t)), G(t)) and f  ((μ (t), # » # » p(t)), G(t)) have linear relationship represented by f  ((μ (t), p(t)), G(t)) = # » a( f ((μ(t), p(t)), G(t))) + b, when a  0, it is assumed that if 0  x1 (t)  x2 (t) # » then 0  x1 (t)  x2 (t) holds, then:Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p2 (t)), G(t)))) = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » p (t)), G(t)))) → Txhs (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #1 » p2 (t)), G(t)))). # » # » Proof From the assumption of f ((μ(t), p(t)), G(t)) and f  ((μ (t), p(t)), G(t)) # » # » have linear relationship represented by f  ((μ (t), p(t)), G(t)) = a( f ((μ(t), p(t)), G(t))) + b, when a  0, under the conditions of 0  x1 (t)  x2 (t), the left side # » # » = Txhs (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txhs (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), # » # » S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = Txhs (¬A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) ∨ Txhs (B((U (t), S(t), p(t), T (t), # » # » L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) Under the conditions of 0  x1 (t)  x2 (t), the right side = Txhs (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txhs (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = A((U  (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » p(t)), G(t))). Left side = right side. Proof is completed.

4.1.12 Characteristics of Temporal Similarity Transformation Connectives in Error Logic # » # » For Txs j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)) , x  (t) = f ((μ (t), p  (t)), G  (t))) = A((U (t  ), # » # » S(t  ), p(t  ), T (t  ), L(t  )), x(t  ) = f ((μ (t  ), p(t  )), G(t  ))), Txs j is the temporal # » similarity transformation connective with respect to G(t) and A((U (t), S(t), p(t),

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# » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in U (t) under G(t) rules for judging errors. In the transformation process, t  is used to replace t, when time is continuous t  ∈ {t − Δt, t + Δt} when time is discrete t  ∈ {tn−1 , tn+1 }. In some case # » # » such that Txs j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) == # » # » A ((U  (t  ), S  (t  ), p  (t  ), T  (t  ), L  (t  )), x  (t  ) = f ((μ (t  ), p  (t  )), G  (t  ))), it is said error function similarity transformation connective Txs j has caused the simultaneous changes in domain, thing, spatial status, time, property, property (or attribute) # » value, and error value of error logic variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))). # » Proposition 4.73 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t  ), S1 (t  ), #  »   # » p1 (t ), T1 (t ), L 1 (t  )), x1 (t  ) = f ((μ1 (t  ), p1 (t  )), G(t  ))) and Txs j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t  ), S2 (t  ), p2 (t  ), # » T2 (t  ), L 2 (t  )), x2 (t  ) = f ((μ2 (t  ), p2 (t  )), G(t  ))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, # » # » t  ∈ {t − Δt, t + Δt}, error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), # » # » G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time,then we have: # » # » (1) Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨ # » # » Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (2) Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ # » # » Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))); # » # » (3) Txs j (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = # » # » ¬Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, x1 (t)  x2 (t) holds if x1 (t)  x2 (t). Without loss of generality,it is assumed that # » x1 (t)  x2 (t), then the left side = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∨ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # »1 p2 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))).

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# » If x1 (t)  x2 (t) then the right side = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧ Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » # » p2 (t)), G(t))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = Right side. Proof is completed. Similarly,cases of (2) and (3) can be proved. # » Proposition 4.74 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txs j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, # » # » error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), # » #  » p1 (t )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time,if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t) hold, we have: # » # » Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txs j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t) holds . Without loss of generality, it is assumed that 0  x1 (t)  x2 (t)  x3 (t), then the left side = # » Based on the definition on ∨n , we have: A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t),

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# » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » Az B G(t))) ∧ C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), # » # » # »2 p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), # » # » # » p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ x2 (t) ∧ (−x3 (t)))∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧x2 (t) ∧(−x3 (t))) ∨(x1 (t) ∧ (−x2 (t)) ∧ (−x3 (t))) = x1 (t) ∨ (−x3 (t)) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) ∨ (−x3 (t)) ∨ (−x3 (t)) = x1 (t). Under the conditions of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∨n Txs j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t),

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# » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∨n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » #  »1 p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # » # » p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ # » # » ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = # » # » A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.75 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txs j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error # » # » # » functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), # » G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, if 0  x1 (t)  x2 (t)  x3 (t) then 0  x1 (t)  x2 (t)  x3 (t) holds, We have: Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t),

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# » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txs j (B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, 0  x1 (t)  x2 (t)  x3 (t) holds if 0  x1 (t)  x2 (t)  x3 (t). Without loss of generality, it is assumed that 0  x1 (t)  x2 (t)  x3 (t), based on the definition on ∧n : # » # » A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), # » # » S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # »2 p3 (t)), G(t)))). Under the condition of 0  x1 (t)  x2 (t)  x3 (t), the left side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B¬((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Because (x1 (t) ∧ x2 (t) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ x2 (t) ∧ x3 (t)) ∨ (x1 (t) ∧ (−x2 (t)) ∧ x3 (t)) ∨ ((−x1 (t)) ∧ (−x2 (t)) ∧ x3 (t)) = x1 (t) ∨ (−x1 (t)) ∨ (−x2 (t)) ∨ (−x2 (t)) = x1 (t). And for the condition of 0  x1 (t)  x2 (t)  x3 (t), the right side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) ∧n Txs j (B((U (t), S2 (t),

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# » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧n B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), #  »1 # » # » p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.76 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another three error logical variables Txs j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, # » # » error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), # » #  » p1 (t )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, it is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  (−x2 (t)) # » and x3 (t)  x1 (t) hold, we have: Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), # » # » G(t)))) −n Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0,

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it is assumed that if x3 (t)  (−x2 (t)) and x3 (t)  x1 (t) then x3 (t)  (−x2 (t)) and # » x3 (t)  x1 (t) hold,based on the definition on −n , we have: A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » # » # »3 p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the left side = Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j ((A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C AnhbB ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C # » # » f ((μ3 (t), p3 (t)), G(t))))) =Txs j (C AnhbB ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » f ((μ3 (t), p3 (t)), G(t))). And for the condition of x3 (t)  (−x2 (t)), x3 (t)  x1 (t), the right side = Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) −n Txs j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) −n B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » #  »1 p (t)), G(t))) ∧ C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # » # 2 » p (t)), G(t)))) = C AnhbB ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), # 3 » p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.77 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t),

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# » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, # » # » t + Δt}, error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), # » # » f ((μ1 (t  ), p1 (t  )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, it is assumed that if | x3 (t) | x2 (t) and | x3 (t) | x1 (t) then | x3 (t) | x2 (t) then | x3 (t) | x1 (t) hold, | x3 (t) | and x3 (t) have the same sign, # » # » then:Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) # » # » |n f l B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f l # » # » Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t) and | x3 (t) | x2 (t), | x3 (t) | x1 (t), where | x3 (t) | and x3 (t) have the same sign, the left side = Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f l B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j ((A((U (t), # » # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))))) = Txs j ((A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 Az B p2 (t)), G(t)))) ∧ (C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » Az B G(t))) ∨ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t))))) = Txs j (C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t))) ∨ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) # » # » = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) ∨ ¬ # » # » C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))).     Under the conditions of | x3 (t) | x2 (t), | x3 (t) | x1 (t), | x3 (t) | and x3 (t) have # » the same sign, the right side = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) |n f l Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) |n f l B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B

4.1 Similarity Transformation Connectives in Error Logic

187

# » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ (C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t))) ∨ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.78 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − # » # » Δt, t + Δt}, error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), # » # » f ((μ1 (t  ), p1 (t  )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, it is assumed that if 0  x1 (t)  x2 (t)  (−x3 (t)) then 0  x1 (t)  # » x2 (t)  (−x3 (t)) holds, then:Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # » # » # »1 p2 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t)))) |n f h Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, under the condition of 0  x1 (t)  x2 (t)  (−x3 (t)), the left side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B((U (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j ((A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t),

188

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# » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), # »1 # » # » Az B p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), # » # » G(t)))) ∨ (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ # » # » ¬B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Under the conditions of 0  x1 (t)  x2 (t)  (−x3 (t)), the right side = Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) |n f h Txs j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) |n f h B  ((U  (t), S2 (t), # » # » # »  p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ (t), p3 (t)), G(t)))) ∨ (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » #  »3 p1 (t)), G(t))) ∧ ¬B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t)= f ((μ3 (t), p3 (t)), G(t)))) # » # » ∨ (¬A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ ¬ # » # » B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B # » # » ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))))=(A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.79 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C Az B ((U (t), S3 (t), p3 (t),

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# » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, # » # » t + Δt}, error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), # » # » f ((μ1 (t  ), p1 (t  )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, it is assumed that if x1 (t)  x2 (t)  x3 (t) then x1 (t)  x2 (t)  x3 (t) # » # » holds, then:Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) nhb B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb # » # » Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, under the conditions of x1 (t)  x2 (t)  x3 (t), the left side = Txs j (A((U (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  x3 (t), the right side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhb Txs j (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # »  # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhb B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) = # » # » f ((μ3 (t), p3 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.80 Suppose that three error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and C Az B ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judg# » # » ing errors, where B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) # » is the complement error logical variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); suppose that another three error logical variables Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t),

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# » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))), Txs j (B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))), and Txs j (C Az B ((U (t), S3 (t), p3 (t), # » # » T3 (t), L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = C Az B ((U  (t), S3 (t), p3 (t), T3 (t), # » L 3 (t)), x3 (t) = f ((μ3 (t), p3 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, t + # » # » Δt}, error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), # » #  » p1 (t )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, it is assumed that if x1 (t)  x2 (t)  (−x3 (t)) then x1 (t)  x2 (t)  (−x3 (t)) # » # » holds, then:Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), # » # » G(t))) nhdl B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = # » # » Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txs j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the left side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B((U (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B((U (t), S2 (t), p2 (t), T2 (t), # » # » Az B L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C ((U (t), S3 (t), p3 (t), T3 (t), L 3 (t)), # » # » x3 (t) = f ((μ3 (t), p3 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Under the conditions of x1 (t)  x2 (t)  (−x3 (t)), the right side = Txs j (A((U (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) nhdl Txs j (B((U (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), # » # » # »  p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) nhdl B  ((U  (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) ∧ B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f ((μ2 (t), p2 (t)), G(t))) ∧ ¬C Az B ((U  (t), S3 (t), p3 (t), T3 (t), L 3 (t)), x3 (t) # » # » = f ((μ3 (t), p3 (t)), G(t)))) = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) = (A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), # » p1 (t)), G(t))). Left side = right side. Proof is completed. # » Proposition 4.81 Suppose that two error logical variables A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and B((U (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U (t) under G(t) the rules for judging errors; suppose that another two error logical variables Txs j (A((U (t),

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# » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t), S1 (t), # » # »  # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) and Txs j (B((U (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = B  ((U  (t), S2 (t), p2 (t), T2 (t), # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t))) are defined in domain U  (t  ) under G(t  ) the rules for judging errors. Under the condition of continuous time, t  ∈ {t − Δt, t + # » # » Δt}, error functions of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), # » #  » p1 (t )), G(t  ))), f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, it is assumed that if 0  x1 (t)  x2 (t) then 0  x1 (t)  x2 (t) holds, # » # » then:Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → # » # » B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j # » # » (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) → Txs j # » # » (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))). Proof Under the condition of continuous time, t  ∈ {t − Δt, t + Δt}, error functions # » # » # » of f ((μ1 (t), p1 (t)), G(t))), f ((μ2 (t), p2 (t)), G(t))), f ((μ1 (t  ), p1 (t  )), G(t  ))), # » f ((μ2 (t  ), and p2 (t  )), G(t  ))) are all continuous with respect to time, ∃Δt > 0, # » under the conditions of 0  x1 (t)  x2 (t), the left side = Txs j (A((U (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t))) → B((U (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), G(t)))) = Txs j (¬A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), # » # » # » p(t)), G(t)))) = Txs j (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) ∨ Txs j (B((U (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t)))) = # » # » A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ B((U  (t), S(t), # » # » p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))) Under the conditions of 0  # » x1 (t)  x2 (t), the right side = Txs j (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ1 (t), p1 (t)), G(t)))) → Txs j (B((U (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f ((μ2 (t), p2 (t)), G(t)))) = A ((U  (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ1 (t), # » # » # » p1 (t)), G(t))) → B  ((U  (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t)), # » # » G(t))) = A((U  (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) ∨ # » # » B((U  (t), S(t), p(t), T (t), L(t)), x  (t) = f ((μ (t), p(t)), G(t))). Left side = right side. Proof is completed.

4.1.13 Characteristics of Combination Similarity Transformation Connectives in Error Logic # » Suppose that an error logical variable A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))) is defined in domain U (t) under G(t) the rules for judging # » errors, the other error logical variable Txq (A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) # » # » = f ((μ1 (t), p1 (t)), G(t)))) = A ((U  (t  ), S1 (t  ), p1 (t  ), T1 (t  ), L 1 (t  )),

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# » x1 (t  ) = f  ((μ1 (t  ), p1 (t  )), G  (t  ))) is defined in domain U  (t  ) under G  (t  ) the rules for judging errors, it is said that Txq has conducted combination similarity trans# » formation on the error logic variable of A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » f ((μ1 (t), p1 (t)), G(t))). Generally, Txq is any of the subsets of {Txly , Txsw , Txt x , Txk j , Txlz , Txgz , Txhs , Txs j , Txq }, Txq ⊆ Tx ={Txly , Txsw , Txt x , Txk j , Txlz , Txgz , Txhs , Txs j , Txq }. Multiple elements or subsets of the above-mentioned set Tx can act on the error logical variable at the same time, which can be classified into 5 categories: (1) Conjunction of transformation connectives: For example, connectives Txly , Txk j , and Txlz can simultaneously act on error log# » # » ical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) in # » a way as: (Txly ∧ Txk j ∧ Txlz )(A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G(t)))); (2) Disjunction of transformation connectives: for example, connectives Txhs , Txs j , and Txgz can simultaneously act on an error # » # » logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) # » in a way as: (Txhs ∨ Txs j ∨ Txgz )(A((U (t), S(t), p(t), T (t), L(t)), x(t)= # » f ((μ(t), p(t)), G(t)))); (3) Inverse transformation connectives: # » # » For example, if Txk j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t))))=A((U (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f ((μ1 (t), p1 (t)), G(t))), # » −1 then the inverse transformation connective Txk j (A((U (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f ((μ1 (t), p1 (t)), G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))) ; (4) Conjunction of inverse transformation connectives: −1 −1 For example, two inverse transformation connectives Txk j and Txs j simultane# » ously act on an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » −1 −1 f ((μ(t), p(t)), G(t))) in a way as: (Txk j ∧ Txs j )(A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t)))); (5) Disjunction of inverse transformation connectives: −1 −1 −1 For example,three inverse transformation connectives Txly , Txk j , and Txlz simul# » taneously act on an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) # » # » −1 −1 −1 = f ((μ(t), p(t)), G(t))) in a way as: (Txly ∨ Txk j ∨ Txlz )(A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))));

4.2 Decomposition Transformation Connectives in Error Logic In this section, we mainly talk about the mechanisms of physical decomposition, mathematical decomposition, and comprehensive decomposition for the decomposition transformation connectives in error logic. We also discuss the ways of decom-

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position (e.g. equal division, unequal division, and decomposition according to actual needs or objective feasibility) and their inverse operations. The relationship between decomposition transformation connectives, denotation connectives, connotation connectives, and the particulars.

4.2.1 Concepts of Decomposition Transformation Connectives in Error Logic 4.2.1.1

Definition of Decomposition Transformation Connectives in Error Logic

# » Definition 4.28 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors, if T f (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ1 (t), p1 (t)), # » # » G(t))))={A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), p2 (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, here μ1 (t)hμ2 (t)h . . . hμn (t) = μ(t), then T f is called the error decomposition # » transformation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))). (1) In {xi (t), i = 1, 2, . . . , m}, if xi (t)  x(t), i = 1, 2, . . . , n, then T f is called the error-quantity-increasing decomposition transformation noted by T f z ; (2) In {xi (t), i = 1, 2, . . . , m}, if xi (t)  0, i = 1, 2, . . . , n, then T f is called the error-elimination decomposition transformation noted by T f x ; (3) In {xi (t), i = 1, 2, . . . , m}, if xi (t) = kx(t), i = 1, 2, . . . , n, then T f is called the error-value-amplifying decomposition transformation noted by T f k . (a) If k  1, then T f k is called the error-value positive amplification decomposition transformation noted by T f zk ; (b) If k  −1, then T f k is called the error-value negative amplification decomposition transformation noted by T f f k ; (c) If 0 < k < 1, then T f k is called the error-value positive reduction decomposition transformation noted by T f zs ; (d) If −1 < k < 0, then T f k is called the error-value negative reduction decomposition transformation noted by T f f s ; (e) If k = 0, then T f k is called the error-elimination decomposition transformation noted by T f hl .

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4.2.1.2

Principles and Types of Decomposition Transformation in Error Logic

1. Principles of decomposition transformation: (1) Actual needs; (2) Feasibility of actual conditions and resources; (3) Minimum cost. 2. Ways of decomposition transformation: (1) Physical decomposition: suppose that μ is a diesel-powered vehicle, μ can be decomposed into cooling system, fuel supply system, electronic control system, and transmission system, etc. In general, the object of interest μ(t) not only has vertical and horizontal structure but also contains many hierarchies and complicated relationships. Please refer to Fig. 4.1 (2) Mathematical decomposition: if the object of interests is a given mathematical equation such as differential equation, difference equation, common algebra equation or other equation: . μ(t) : x = f (t, x) + g(t, x); . μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); We can decompose the object according to Lyapunov approach accordingly. (3) Decomposing based on actual needs and requirements (4) Comprehensive decomposition: base on the definition for T f and the ele# » # » ments of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), decomposition transformation can be conducted on the universe of discourse U (t), object μ(t), error value x(t), error function f , time t, and # » rule G(t) for judging errors of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))), therefore, T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }. The type of error logical variable of A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) does not change if decomposition transformation is not carried out on error function f .

4.2.1.3

Preparation for Application of Decomposition Transformation Connectives in Error Logic

# » Definition 4.29 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors, if there exists T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) # » # » = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), # » # » #1 » 1 p2 (t), G 2 (t))), . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), xk (t) = f k ((μk (t), pk (t),

4.2 Decomposition Transformation Connectives in Error Logic

195

Fig. 4.1 Physical decomposition for object in research μ(t)

G k (t)))} (simplified as {A1 , A2 , . . . , Ak }) where L(t) = L 1 (t) + L 2 (t)+, . . . , # » +L k (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L k (t)}, and T f lz (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G(t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = # » # » f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , Bk ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), yk (t)= # » f k ((νk (t), pk (t), G k (t)))} (simplified as {B1 , B2 , . . . , Bk }), where L(t) = L 1 (t) + L 2 (t)+, . . . , +L k (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L k (t)}. If {A1 , A2 , . . . , Ak } ∧ {B1 , # » # » B2 , . . . , Bk } = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G 1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G 2 (t))), # » # » . . . , Ck ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), z k (t) = f k ((ωk (t), pk (t), G k (t)))} (simplified as {C1 , C2 , . . . , Ck }), where  Ci =

Ai , xi (t)  yi (t); Bi , xi (t)  yi (t).

Then {C1 , C2 , . . . , Ck } is the result for the conjunction of {A1 , A2 , . . . , Ak } and {B1 , B2 , . . . , Bk }. For example, if two same-type parallel connection systems are decomposed in the same way and the obtained constituent subsystems are then reconfigured to form two new parallel connection systems, then the error value of each new parallel system is the smallest error value of the corresponding constituent subsystems.

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4 Transformation Connectives in Error Logic

# » Definition 4.30 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors, if there exists T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), xk (t) = f k ((μk (t), pk (t), G k (t)))} (simplified as {A1 , A2 , . . . , Ak }) where L(t) = L 1 (t) + L 2 (t)+, . . . , # » +L k (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L k (t)}, and T f lz (B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G(t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = # » # » f ((ν (t), p1 (t)), G 1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), # » # » # 1 »1 p2 (t), G 2 (t))), . . . , Bk ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), yk (t) = f k ((νk (t), pk (t), G k (t)))} (simplified as {B1 , B2 , . . . , Bk }), where L(t) = L 1 (t) + L 2 (t)+, . . . , +L k (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L k (t)}. If {A1 , A2 , . . . , Ak } ∨ {B1 , B2 , . . . , # » # » Bk } = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G 1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G 2 (t))), . . . , # » # » Ck ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), z k (t) = f k ((ωk (t), pk (t), G k (t)))} (simplified as {C1 , C2 , . . . , Ck }), where  Ci =

Ai , xi (t)  yi (t); Bi , xi (t)  yi (t).

Then {C1 , C2 , . . . , Ck } is the result for the disjunction of {A1 , A2 , . . . , Ak } and {B1 , B2 , . . . , Bk }. For example, if two same-type series connection systems are decomposed in the same way and the obtained constituent subsystems are then reconfigured to form two new series connection systems, then the error value of each series connection system is the largest error value of the corresponding constituent subsystems. # » Definition 4.31 Suppose that error logical variables {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), # » xk (t) = f k ((μk (t), pk (t), G k (t)))}, i ∈ {1, 2, . . . , k}(simplified as {A1 , A2 , . . . , # » # » Ak }) and {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , # » # » Bm ((Um (t), Sm (t), pm (t), Tm (t), L m (t)), ym (t) = f m ((νm (t), pm (t), G m (t)))}, i ∈ {1, 2, . . . , m} (simplified as {B1 , B2 , . . . , Bm }) are defined in domain U (t) under G i (t) ∈ {1, 2, . . . , k} ∪ {1, 2, . . . , m} the rules for judging errors, it is assumed that a1 = min {x1 (t), x2 (t), . . . , xk (t)}, b1 = min {y1 (t), y2 (t), . . . , yk (t)}, a2 = x1 (t) + x2 (t)+, . . . , +xk (t), b2 = y1 (t) + y2 (t)+, . . . , +yk (t), a3 = max {x1 (t), x2 (t), . . . , xk (t)}, b3 = max {y1 (t), y2 (t), . . . , yk (t)}, {A1 , A2 , . . . , Ak } ∧ {B1 , B2 , . . . , Bk }  =

{A1 , A2 , . . . , Ak }, a1 < b1 or a1 = b1 a3 < b3 or a2  b2 , a1 = b1 , a3 = b3 ; {B1 , B2 , . . . , Bk }, otherwise.

Then ∧ is called the small AND operation.

4.2 Decomposition Transformation Connectives in Error Logic

197

For example, if two same-type parallel connection systems are decomposed in the different ways and the obtained constituent subsystems are then reconfigured to form two new parallel connection systems, then the one with the smallest total error value is the chosen new system. # » Definition 4.32 Suppose that error logical variables {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), # » xk (t) = f k ((μk (t), pk (t), G k (t)))}, i ∈ {1, 2, . . . , k}(simplified as {A1 , A2 , . . . , # » # » Ak }) and {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , # » # » Bm ((Um (t), Sm (t), pm (t), Tm (t), L m (t)), ym (t) = f m ((νm (t), pm (t), G m (t)))}, i ∈ {1, 2, . . . , m} (simplified as {B1 , B2 , . . . , Bm }) are defined in domain U (t) under G i (t) ∈ {1, 2, . . . , k} ∪ {1, 2, . . . , m} the rules for judging errors, it is assumed that a1 = min {x1 (t), x2 (t), . . . , xk (t)}, b1 = min {y1 (t), y2 (t), . . . , yk (t)}, a2 = x1 (t) + x2 (t)+, . . . , +xk (t), b2 = y1 (t) + y2 (t)+, . . . , +yk (t), a3 = max {x1 (t), x2 (t), . . . , xk (t)}, b3 = max {y1 (t), y2 (t), . . . , yk (t)}, {A1 , A2 , . . . , Ak } ∧ {B1 , B2 , . . . , Bk }  =

{A1 , A2 , . . . , Ak }, a3  b3 , a1 < b1 or a3 = b3 a2  b2 , a1 = b1 ; {B1 , B2 , . . . , Bk }, otherwise.

.

then ∧ is called the large AND operation.

# » Definition 4.33 Suppose that error logical variables {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), # » xk (t) = f k ((μk (t), pk (t), G k (t)))}, i ∈ {1, 2, . . . , k}(simplified as {A1 , A2 , . . . , # » # » Ak }) and {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , # » # » Bm ((Um (t), Sm (t), pm (t), Tm (t), L m (t)), ym (t) = f m ((νm (t), pm (t), G m (t)))}, i ∈ {1, 2, . . . , m} (simplified as {B1 , B2 , . . . , Bm }) are defined in domain U (t) under G i (t) ∈ {1, 2, . . . , k} ∪ {1, 2, . . . , m} the rules for judging errors, it is assumed that a1 = min {x1 (t), x2 (t), . . . , xk (t)}, b1 = min {y1 (t), y2 (t), . . . , yk (t)}, a2 = {x1 (t), x1 (t) + x2 (t)+, . . . , +xk (t), b2 = y1 (t) + y2 (t)+, . . . , +yk (t), a3 = max .. x2 (t), . . . , xk (t)}, b3 = max {y1 (t), y2 (t), . . . , yk (t)}, {A1 , A2 , . . . , Ak } ∧ {B1 , B2 , . . . , Bk }  =

{A1 , A2 , . . . , Ak }, a2 < b2 or a2 = b2 a1 < b1 or a2 = b2 , a1 = b1 , a3  b3 ; {B1 , B2 , . . . , Bk }, otherwise. ..

then ∧ is called the sum-of-logic-value AND operation.

# » Definition 4.34 Suppose that error logical variables {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), # » xk (t) = f k ((μk (t), pk (t), G k (t)))}, i ∈ {1, 2, . . . , k}(simplified as {A1 , A2 , . . . ,

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4 Transformation Connectives in Error Logic

# » # » Ak }) and {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , # » # » Bm ((Um (t), Sm (t), pm (t), Tm (t), L m (t)), ym (t) = f m ((νm (t), pm (t), G m (t)))}, i ∈ {1, 2, . . . , m} (simplified as {B1 , B2 , . . . , Bm }) are defined in domain U (t) under G i (t) ∈ {1, 2, . . . , k} ∪ {1, 2, . . . , m} the rules for judging errors, it is assumed that a1 = min {x1 (t), x2 (t), . . . , xk (t)}, b1 = min {y1 (t), y2 (t), . . . , yk (t)}, a2 = x1 (t) + x2 (t)+, . . . , +xk (t), b2 = y1 (t) + y2 (t)+, . . . , +yk (t), a3 = max {x1 (t), x2 (t), . . . , xk (t)}, b3 = max {y1 (t), y2 (t), . . . , yk (t)}, {A1 , A2 , . . . , Ak } ∧ {B1 , B2 , . . . , Bk }  =

{A1 , A2 , . . . , Ak }, a1 > b1 or a1 = b1 a3 > b3 or a2  b2 , a1 = b1 , a3 = b3 ; {B1 , B2 , . . . , Bk }, otherwise.

then ∨ is called the small OR operation. # » Definition 4.35 Suppose that error logical variables {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), # » xk (t) = f k ((μk (t), pk (t), G k (t)))}, i ∈ {1, 2, . . . , k}(simplified as {A1 , A2 , . . . , # » # » Ak }) and {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , # » # » Bm ((Um (t), Sm (t), pm (t), Tm (t), L m (t)), ym (t) = f m ((νm (t), pm (t), G m (t)))}, i ∈ {1, 2, . . . , m} (simplified as {B1 , B2 , . . . , Bm }) are defined in domain U (t) under G i (t) ∈ {1, 2, . . . , k} ∪ {1, 2, . . . , m} the rules for judging errors, it is assumed that a1 = min {x1 (t), x2 (t), . . . , xk (t)}, b1 = min {y1 (t), y2 (t), . . . , yk (t)}, a2 = {x1 (t), x1 (t) + x2 (t)+, . . . , +xk (t), b2 = y1 (t) + y2 (t)+, . . . , +yk (t), a3 = max . x2 (t), . . . , xk (t)}, b3 = max {y1 (t), y2 (t), . . . , yk (t)}, {A1 , A2 , . . . , Ak } ∨ {B1 , B2 , . . . , Bk }  =

{A1 , A2 , . . . , Ak }, a3 > b3 or a3 = b3 a1 > b1 or a3  b3 , a1 = b1 , a2  b2 ; {B1 , B2 , . . . , Bk }, otherwise. .

then ∨ is called the large OR operation. # » Definition 4.36 Suppose that error logical variables {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , Ak ((Uk (t), Sk (t), pk (t), Tk (t), L k (t)), # » xk (t) = f k ((μk (t), pk (t), G k (t)))}, i ∈ {1, 2, . . . , k}(simplified as {A1 , A2 , . . . , # » # » Ak }) and {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G 1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G 2 (t))), . . . , # » # » Bm ((Um (t), Sm (t), pm (t), Tm (t), L m (t)), ym (t) = f m ((νm (t), pm (t), G m (t)))}, i ∈ {1, 2, . . . , m} (simplified as {B1 , B2 , . . . , Bm }) are defined in domain U (t) under G i (t) ∈ {1, 2, . . . , k} ∪ {1, 2, . . . , m} the rules for judging errors, it is assumed that a1 = min {x1 (t), x2 (t), . . . , xk (t)}, b1 = min {y1 (t), y2 (t), . . . , yk (t)}, a2 = x1 (t) + x2 (t)+, . . . , +xk (t), b2 = y1 (t) + y2 (t)+, . . . , +yk (t), a3 = max {x1 (t),

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199 ..

x2 (t), . . . , xk (t)}, b3 = max {y1 (t), y2 (t), . . . , yk (t)}, {A1 , A2 , . . . , Ak } ∨ {B1 , B2 , . . . , Bk }  =

{A1 , A2 , . . . , Ak }, a2 > b2 or a2 = b2 a1 > b1 or a2  b2 , a1 = b1 , a3  b3 ; {B1 , B2 , . . . , Bk }, otherwise. ..

then ∨ is called the sum-of-logic-value OR operation. In addition, since denotation connectives such as ¬, ⇒, and ⇐, etc. and connotation connectives such as ∨n , ∧n , and ¬bj , etc. have been discussed in the section of “Mathematical Logic”, they will not be repeated here.

4.2.2 Domain Decomposition Transformation Connective in Error Logic 4.2.2.1

Conditions and Ways of Domain Decomposition in Error Logic

# » # » Suppose that T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if μ1 (t) ∈ U1 (t), μ2 (t) ∈ U2 (t), . . . , μn (t) ∈ Un (t) and U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t), then T f ly is called the domain decomposition transformation connective. For example, in the process of clinical and pathological research, the whole body is investigate by considering different subsystems such as digestive system, respiratory system, . . . , neural system, etc. 1. The conditions for domain decomposition are: fl

(1) Legal conditions T Ju ; kg (2) Actual conditions T Ju ; (3) Objective conditions (target) T Jumd . For example, in the discussion of climate zones in Canada, the intention to kg investigate tropical climate zone is constricted by the T Ju . 2. Ways of domain decomposition: (1) Equal division: # » in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), G n (t)))}, Ui (t) = U j (t)i, j ∈ {1, 2, . . . , n}; (2) Unequal division: # » in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) =

# » f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » f 2 ((μ2 (t), p2 (t), G 2 (t))), # » xn (t) = f n ((μn (t), pn (t), # » f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » f 2 ((μ2 (t), p2 (t), G 2 (t))),

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4 Transformation Connectives in Error Logic

# » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, Ui (t) = U j (t)i, j ∈ { 1, 2, . . . , n }. (3) Decomposing based on special needs and requirements.

4.2.2.2

Characteristics of Domain Decomposition in Error Logic

# » # » We have T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))},if μ1 (t) ∈ U1 (t), μ2 (t) ∈ U2 (t), . . . , μn (t) ∈ Un (t) and U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) Ways of domain decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors,if there exists T f ly (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if both Ai ((Ui (t), # » # » / U (t), i ∈ {1, 2, Si (t), pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t), G i (t))) ∈ # » # » . . . , n} and A j ((U j (t), S j (t), p j (t), T j (t), L j (t)), x j (t) = f j ((μ j (t), p j (t), G j (t))) ∈ U (t), j ∈ {1, 2, . . . , n} exist,then it is said that T f ly has forced # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain increase transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors,if there exists T f ly (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists Ai ((Ui (t), Si (t), # » # » / U (t), i ∈ {1, 2, . . . , n}, pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t), G i (t))) ∈ # » then it is said that T f ly has forced A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))) to carry out transformations on domain,rules for judging errors,time,object,or error function; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors,if there exists T f ly (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, ∀Ai ((Ui (t), Si (t), pi (t), Ti (t), # » / U (t), i ∈ {1, 2, . . . , n}, then it is said L i (t)), xi (t) = f i ((μi (t), pi (t), G i (t))) ∈ that T f ly has carried out domain displacement transformation on A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))).

4.2 Decomposition Transformation Connectives in Error Logic

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# » Proposition 4.82 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors, T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G(t) the rules for judging errors. If G(t) = G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G i (t) is error-judging rules defined in domain Ui (t), i ∈ {1, 2, . . . , n} if Ai ((U (t), # » # » Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((μi (t), pi (t)), G(t))) ∈ U (t) and Ai ((Ui (t), # » # » Si (t), pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G i (t))) ∈ Ui (t)i ∈ {1, 2, . . . , n}, then yi (t) = xi (t). # » # » Proof In Ai ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G i (t))) ∈ # » # » Ui (t), xi (t) = f i ((μi (t), pi (t)), while for G i (t)), Ai ((U (t), Si (t), pi (t), Ti (t), # » L i (t)), yi (t) = f i ((μi (t), pi (t)), G(t))), as G i (t) ⊆ G(t), therefore yi (t) = # » # » f i ((μi (t), pi (t)), G(t)) = f i ((μi (t), pi (t)), G i (t)) = xi (t). Left side = right side. Proof is completed. For example, suppose that Corporate X = { Company A, Company B, . . . , Company n } and employee P belongs to both Company B and Corporate X, if employee P had an error in his decision-making process, the error value is the same with respect to Corporate X and Company B because the same rules are applied. # » Proposition 4.83 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors,if G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), Ai ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ U and Ai ((Ui (t), Si (t), # » #i » i pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ Ui i ∈ {1, 2, . . . , n}; sup# » pose that we have another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)),

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# » # » G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, # » # » where B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ # » # » U1 (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ # » # » U2 (t), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) is error-judging rule defined in domain Ui (t), Bi ((Ui (t), # » # » Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ U and Bi ((Ui (t), # » # » Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ Ui i ∈ {1, 2, . . . , n}; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, we have . # » # » (1) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f ly . # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f ly # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); .. # » # » (2) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f ly .. # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f ly # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f ly (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » ¬T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof From assumption T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} and # » # » T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » #2 » 2 x(t)  y(t), ∀i, pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, when . i ∈ {1, 2, . . . , n}, xi (t)  yi (t) and from the definition of ∨: a1 = min {x1 (t), x2 (t), . . . , xn (t)}, b1 = min {y1 (t), y2 (t), . . . , yn (t)}, a2 = x1 (t) + x2 (t)+, . . . , +xn (t), b2 = y1 (t) + y2 (t)+, . . . , +yn (t), a3 = max {x1 (t), x2 (t), . . . , xn (t)}, b3 = # » max {y1 (t), y2 (t), . . . , yn (t)}, if {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), # » #1 » 1 p2 (t), G A2 (t))), .. . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » # » # » pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}

4.2 Decomposition Transformation Connectives in Error Logic

 =

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{A1 , A2 , . . . , An }, a3 > b3 or a3 = b3 a1 > b1 or a1 = b1 , a3 = b3 , a2  b2 ; {B1 , B2 , . . . , Bn }, otherwise.

# » As x(t)  y(t), the left side = T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = . # » # » # » f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; As # » a3 > b3 , the right ride = T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2)and (3) can also be proved. # » Proposition 4.84 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), Ai ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ U and Ai ((Ui (t), Si (t), # » #i » i pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ Ui i ∈ {1, 2, . . . , n}; sup# » pose that we have another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, # » # » where B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ # » # » U1 (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ # » # » U2 (t), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) is error-judging rule defined in domain Ui (t), Bi ((Ui (t), Si (t), # » # » pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ U and Bi ((Ui (t), Si (t),

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# » # » pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ Ui i ∈ {1, 2, . . . , n}. C Az B # » # » ((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((ω(t), p(t)), G A (t))) is the mediator # » # » variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f ly (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) is error-judging rule defined in domain Ui (t), Ci ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), z i (t) = f i ((ωi (t), pi (t)), G Ci (t))) ∈ U and Ci ((Ui (t), Si (t), #i » i # » pi (t), Ti (t), L i (t)), z i (t) = f i ((ωi (t), pi (t)), G Ci (t))) ∈ Ui i ∈ {1, 2, . . . , n}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, we have: # » # » (1) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l T f ly # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f ly # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h T f ly # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (5) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f ly # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb T f ly # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (6) T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof According to the assumptions for A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))), B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))), and C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), # » as x(t)  y(t)  z(t)  0, then the left side = T f ly (A((U (t), S(t), p(t), T (t), L(t)),

4.2 Decomposition Transformation Connectives in Error Logic

205

# » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » Az B p(t)), G B (t)))) = T f ly (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t)))); And the right side = T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t)))) ∨n T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), # » # » G An (t)))} ∨n {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), # » # » G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), # » # » G Bn (t)))} = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), # » # » G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), # » # » G Cn (t)))} = T f ly (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))). Left side = right side. Proof is completed. Similarly, (2)–(6) can also be proved. # » Proposition 4.85 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), Ai ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ U and Ai ((Ui (t), Si (t), # » #i » i pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ Ui i ∈ {1, 2, . . . , n}; sup# » pose that we have another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, # » # » where B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ # » # » U1 (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ # » # » U2 (t), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪,

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. . . , ∪G Bn (t), G Bi (t) is error-judging rule defined in domain Ui (t), Bi ((Ui (t), # » # » Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ U and Bi ((Ui (t), # » # » Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ Ui i ∈ {1, 2, . . . , n}. # » # » C AnhbB ((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((ω(t), p(t)), G A (t))) is the con# » notative inclusion variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » T f ly (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) is error-judging rule defined in domain # » # » Ui (t), Ci ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), z i (t) = f i ((ωi (t), pi (t)), G Ci (t))) ∈ U # » # » and Ci ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), z i (t) = f i ((ωi (t), pi (t)), G Ci (t))) ∈ Ui i ∈ {1, 2, . . . , n}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  # » # » z i (t)  0 holds, then T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n # » # » T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.86 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f ly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors,iG A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), Ai ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ U and Ai ((Ui (t), Si (t), # » #i » i pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ Ui i ∈ {1, 2, . . . , n}; sup# » pose that we have another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), # » # » G Bn (t)))}, where B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))) ∈ U1 (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), # » G B2 (t))) ∈ U2 (t), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), # » pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined

4.2 Decomposition Transformation Connectives in Error Logic

207

in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) is error-judging rule defined in domain Ui (t), # » # » Bi ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ U and # » # » Bi ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ Ui i ∈ # » # » Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((ω(t), p(t)), {1, 2, . . . , n}. C G A (t))) is the connotative same or connotative equivalence variable for A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f ly (C Anhdthd j B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # » # » f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = # » f n ((ωn (t), pn (t), G Cn (t)))}, G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) is # » error-judging rule defined in domain Ui (t), Ci ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), # » # » z i (t) = f i ((ωi (t), pi (t)), G Ci (t))) ∈ U and Ci ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), # » z i (t) = f i ((ωi (t), pi (t)), G Ci (t))) ∈ Ui i ∈ {1, 2, . . . , n}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds,then T f ly (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f ly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f ly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted.

4.2.3 Thing Decomposition Transformation Connective in Error Logic # » # » Suppose that T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where S(t) = S1 (t)hS2 (t)h, . . . , hSn (t). This transformation is carried out on the thing of the object of interest. For example,in order to move a giant machine into a plant with small door,the machine may need to be disassembled.

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4.2.3.1

Conditions for Thing Decomposition in Error Logic

1. The conditions for thing decomposition are: (1) (2) (3) (4) (5) (6)

fl

Legal conditions T Jsw ; kg Actual conditions T Jsw ; md Objective conditions (target) T Jsw ; sm ; Conditions for sustaining life T Jsw gj Technical conditions T Jsw ; nl . Energy conditions T Jsw

4.2.3.2

Principles for Thing Decomposition in Error Logic

The principles for thing decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.3.3

Ways of Thing Decomposition in Error Logic

2. Ways of thing decomposition: (1) Physical decomposition: For example, if the thing of interest is a drone, it can be disassembled into drone propeller, flight controller, GPS module, obstacle avoidance sensor, power port module, drone camera,and drone battery, etc. (2) Mathematical decomposition: For example, .

.

μ(t) : x = f (t, x) + g(t, x); μ(t) : x(k + 1) = Ax(k); . . . . . . . . . . . . . . . . . . . . . ; μ(t) : x = f (x1 , x2 , . . . , xn );

Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs; (4) Equal division: # » # » in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, Si (t) = S j (t)i, j ∈ {1, 2, . . . , n}; (5) Unequal division: # » # » in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, Si (t) = S j (t)i, j ∈ {1, 2, . . . , n};

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(6) Decomposing based on special needs and requirements.

4.2.3.4

Characteristics of Thing Decomposition in Error Logic

# » # » Suppose that T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, T f sw is the thing decomposition transformation connective, Ways of domain decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors,if there exists T f sw (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if both Ai ((Ui (t), Si (t), pi (t), # » / U (t), i ∈ {1, 2, . . . , n} and Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t), G i (t))) ∈ # » # » A j ((U j (t), S j (t), p j (t), T j (t), L j (t)), x j (t) = f j ((μ j (t), p j (t), G j (t))) ∈ U (t), j ∈ {1, 2, . . . , n} exist, then it is said that T f sw has enabled A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors,if there exists T f sw (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists Ai ((Ui (t), Si (t), # » # » / U (t), i ∈ {1, 2, . . . , n}, pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t), G i (t))) ∈ # » then it is said that T f ly has enabled A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))) to carry out transformations on domain,rules for judging errors,time,object,or error function,etc. T f sw ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors,if there exists T f sw (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, ∀Ai ((Ui (t), Si (t), pi (t), Ti (t), # » / U (t), i ∈ {1, 2, . . . , n}, then it is said L i (t)), xi (t) = f i ((μi (t), pi (t), G i (t))) ∈ that T f sw has carry out domain displacement transformation on A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))). # » Proposition 4.87 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the

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# » rules for judging errors, T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), Ai ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ U and Ai ((Ui (t), Si (t), #i » i # » pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ Ui i ∈ {1, 2, . . . , n}, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, T f sw (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ U1 (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ U2 (t), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) is error-judging rule defined # » # » in domain Ui (t), Bi ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), # » # » G Bi (t))) ∈ U and Bi ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ Ui i ∈ {1, 2, . . . , n}, G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t). For # » # » both A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), there is S(t) = S1 (t)hS2 (t)h, . . . , hSn (t); if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds,then the following relationships hold: # » # » (1) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f sw (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f sw (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t),

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# » # » # » p(t)), G B (t)))) = T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.88 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), Ai ((Ui (t), Si (t), # » # » p (t), T (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ U and Ai ((Ui (t), Si (t), # » #i » i pi (t), Ti (t), L i (t)), xi (t) = f i ((μi (t), pi (t)), G Ai (t))) ∈ Ui i ∈ {1, 2, . . . , n}, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, T f sw (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ U1 (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ U2 (t), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . ,

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∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) is error-judging rule defined # » # » in domain Ui (t), Bi ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), # » # » G Bi (t))) ∈ U and Bi ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), yi (t) = f i ((νi (t), pi (t)), G Bi (t))) ∈ Ui i ∈ {1, 2, . . . , n}, G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t). # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the media# » # » tor variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f sw (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) and G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t). For T f sw (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))), T f sw (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))), and T f sw (C Az B ((U (t), S(t), p(t), T (t), # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))), there is S(t) = S1 (t)hS2 (t)h, . . . , hSn (t). If x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds,then the following relationships hold: # » # » (1) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f sw (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = T f sw (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # » # » f ((ω (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), # » # » #1 » 1 p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  # » # » 0, the right side = T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) ∨n T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} ∨ # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))),

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# » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))} = # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.89 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors,iG A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t); suppose that another # » # » error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where # » # » B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ U1 (t), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ U2 (t), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), # » G Bi (t) is error-judging rule defined in domain Ui (t); C AnhbB ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative inclusion variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f sw (C AnhbB ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) # » and G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t). For T f sw (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t)))), T f sw (B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t)))), and T f sw (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), # » z(t) = f ((ω(t), p(t)), G C (t)))), there is S(t) = S1 (t)hS2 (t)h, . . . , hSn (t). If ∀i,

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i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t). then the following relationship holds: T f sw (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) −n B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f sw (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n T f sw (B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.90 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors,iG A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t); suppose that another # » # » error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where # » # » B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ U1 (t), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ U2 (t), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), # » G Bi (t) is error-judging rule defined in domain Ui (t); C Az B ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable of A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f sw (C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # » # » f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = # » f n ((ωn (t), pn (t), G Cn (t)))}, where U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) and # » G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t). For T f sw (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t)))), T f sw (B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t)))), and T f sw (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), # » z(t) = f ((ω(t), p(t)), G C (t)))), there is S(t) = S1 (t)hS2 (t)h, . . . , hSn (t). If ∀i,

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i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.91 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) the rules for judging errors,iG A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t); suppose that another # » # » error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » T f sw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where # » # » B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))) ∈ U1 (t), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))) ∈ U2 (t), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t))) ∈

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Un (t), U (t) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G B (t) the rules for judging errors. If G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), # » G Bi (t) is error-judging rule defined in domain Ui (t); C Anhdthd j B ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative same or equivalence # » # » variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f sw (C Anhdthd j B # » # » ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where U (t) = U1 (t) ∪ U2 (t) ∪, . . . , ∪Un (t) and G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t). For T f sw (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))), T f sw (B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))), and T f sw (C AnhbB ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))), there is S(t) = S1 (t)hS2 (t)h, . . . , hSn (t). If ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following # » # » relationship holds: T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) →nby B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f sw # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.92 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » T f sw (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # (n) » (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where A1 ((U1 (t), S1 (t), # » # (n) » (n) (n) (n) (n) (n) (n) p1 (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ U1 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) G (n) A2 (t))) ∈ U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » (n) (n) pn(n) (t), G (n) (t) = U1(n) (t) ∪ U2(n) (t)∪, . . . , f n(n) ((μ(n) n (t), An (t))) ∈ Un (t), U (n) (n) ∪Un (t) are also defined in domain U (t) under G (n) A (t) the rules for judg(n) (n) (n) (n) ing errors, G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G (n) Ai (t) is error-judging rule defined in domain Ui(n) (t), where S (n) (t) = S1(n) (t)hS2(n) (t)h, . . . , hSn(n) (t); sup# » pose that A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) is the (n + 1)th order error logical variA (n+1) (t) under G (n+1) (t) the rules for judging errors, able defined in domain U A

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# » T f sw (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), 1 A # (n+1) » # » p1 (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), 1 1 # » (n+1) G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = 2 A1 (t))), A2 # » # » (n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), f 2(n+1) ((μ(n+1) 2 A2 (t))), . . . , An # » (n+1) Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n n An (t)))}, A1 # » ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), 1 # (n+1) » # 1(n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t)), G A1 (t))) ∈ U1 (t), A2 ((U2 (t), S2 (t), p2 (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ T2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L (n+1) (t)), xn(n+1) U2 n # » (n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t))) ∈ Un(n+1) (t), U (n+1) (t) = U1(n+1) (t) ∪ n U2(n+1) (t)∪, . . . , ∪Un(n+1) (t) are also defined in domain U (n+1) (t) under G (n+1) (t) A (n+1) (n+1) (n+1) (t) = G (t) ∪ G (t)∪, . . . , ∪G (t), the rules for judging errors, G (n+1) A A1 A2 An (n+1) (n+1) G (n+1) (t) is error-judging rule defined in domain U (t), where S (t) = Ai i S1(n+1) (t)hS2(n+1) (t)h, . . . , hSn(n+1) (t); suppose that error logical variable # » B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = # » (t))) is the complement error logical varif (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » (n+1) (n+1) (n+1) ((U (t), S (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = able for A # » (t))) which is defined in domain U (n+1) (t) under f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » (n+1) G B (t) the rules for judging errors, T f sw (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) {B1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 B1 (t))), B2 # » (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) L (n+1) 2 2 B2 (t))), . . . , Bn # » ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), n n # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) pn (t), G Bn (t)))}, where B1 ((U1 (t), S (t), p1 (t), T1 (t), # (n+1) 1» (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G B1 (t))) ∈ U1 (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 B2 # (n+1) »2 (n+1) (n+1) (n+1) (n+1) (n+1) ((μ2 (t), p2 (t), G B2 (t))) ∈ U2 (t), . . . , Bn ((Un (t), Sn(n+1) (t), # (n+1) » (n+1) # » pn (t), Tn (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n n Bn (t))) ∈ Un(n+1) (t), U (n+1) (t) = U1(n+1) (t) ∪ U2(n+1) (t)∪, . . . , ∪Un(n+1) (t) are also defined in domain U (n+1) (t) under G (n+1) (t) the rules for judging errors, G (n+1) (t) = B B (n+1) (n+1) (n+1) (t) ∪ G (t)∪, . . . , ∪G (t), G (t) is error-judging rule defined in G (n+1) B1 B2 Bn Bi domain Ui(n+1) (t), where S (n+1) (t) = S1(n+1) (t)hS2(n+1) (t)h, . . . , hSn(n+1) (t); # » C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the mediator variable for

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# » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » (t))), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) B # » T f sw (C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = {C1(n+1) ((U1(n+1) (t), S1(n+1) (t), # (n+1) » # » p1 (t), T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), 1 # » (n+1) G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), z 2(n+1) (t) = 2 # » # » (n+1) f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), # » (n+1) Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn (t)))}, where n (n+1) U (n+1) (t) = U1(n+1) (t) ∪ U2(n+1) (t)∪, . . . , ∪Un(n+1) (t) and G C(n+1) (t) = G C1 (t) ∪ (n+1) (n+1) (n+1) (n+1) (n+1) G C2 (t)∪, . . . , ∪G Cn (t), where S (t) = S1 (t)hS2 (t)h, . . . , hSn(n+1) (t); suppose that T f sw has carried out thing decomposition transformation # » on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of (n) (n+1) (t), and z i(n+1) (t) is the same as that of x (n) (t), x (n+1) (t), size for xi (t), xi and z (n+1) (t), then the following relationship holds: T f sw (¬bz A(n) ((U (n) (t), S (n) (t), # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bz (n) (n) (n) (n) (n) (n) ¬ T f sw (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f sw (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f sw (A # (n+1) » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t))) ∧ B (n+1) A # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), B # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f sw (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f sw (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f sw (C (n+1)Az B B # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # » # (n+1) » p (t)), G C(n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) G (n+1) n A2 (t))) . . . An # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((μn (t), pn (t), G An (t)))} ∧ {B1 ((U1 (t), S1(n+1) (t), = fn # (n+1) » (n+1) # » p1 (t), T1 (t), L (n+1) (t)), x1(n+1) (t)= f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))),

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# » B2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = f 2(n+1) 2 # » # » (n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 B2 (t))) . . . Bn # » (n+1) (t)), xn(n+1) (t)= f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n Bn (t)))} ∧ {C 1 # » ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), 1 # (n+1) » # » (n+1) p1 (t)), G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) z2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))) . . . Cn ((U (t), Sn (t), # (n+1) » (n+1) # (n+1)n » (n+1) (n+1) (n+1) (n+1) pn (t), Tn (t), L (n+1) (t)), z (t)= f ((ω (t), p (t), G (t)))}; n n n n n Cn # » And the right side of the equation = ¬bz T f sw A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bz T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) 1 An (t)))} = {A1 # » # » (n+1) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), 1 A1 (t))), A2 # » (t)), x2(n+1) (t)= f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n+1) An ((Un # » # » (n+1) (n+1) (n+1) (n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n+1) (t), n # » (n+1) T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), B2 # » ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t)= f 2(n+1) ((μ(n+1) (t), 2 2 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), Tn (t), # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) L n (t)), xn (t)= f n ((μn (t), pn (t), G Bn (t)))} ∧ {C1 ((U1 (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G C1 (t))), C2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n+1) » 2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) f ((ω (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), # 2(n+1) » 2 (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) pn (t), Tn (t), L (n+1) (t)), z (t)= f ((ω (t), p (t), G (t)))}. n n n n n Cn Left side = right side. Proof is completed. # » Proposition 4.93 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f sw (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t)))) # (n) » (n) #A(n) » (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p2 (t), G A2 (t))), . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) =

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# (n) » # (n) » (n) (n) (n) (n) (n) f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ U1 (t), A2 ((U2 (t), S2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) p2 (t), T2 (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ U2 (t), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), (n) (n) G (n) (t) = U1(n) (t) ∪ U2(n) (t)∪, . . . , ∪Un(n) (t) are also defined An (t))) ∈ Un (t), U (n) (n) (n) in domain U (t) under G (n) A (t) the rules for judging errors, G A (t) = G A1 (t) ∪ (n) (n) (n) G (n) A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), (n) (n) (n) (n) where S (t) = S1 (t)hS2 (t)h, . . . , hSn (t); # » suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) # » (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) is the (n + 1)th order error logical A (n+1) (t) under G (n+1) (t) the rules for judging errors, variable defined in domain U A # (n+1) » (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), x (n+1) (t)= f (n+1) T f sw (A # » # » (n+1) (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = {A1 ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), # » (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) T1(n+1) (t), L (n+1) 1 1 A1 (t))), A2 # » ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = 2 # » # » (n+1) f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n An (t)))}, where # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n+1) (t) = f 1(n+1) # » (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t), A(n+1) ((U2(n+1) (t), S2(n+1) (t), 1 2 A1 (t))) ∈ U1 # (n+1) » (n+1)1 # » p2 (t), T2 (t), L (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) 2 2 A2 (t))) # (n+1) » (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), ∈ U2(n+1) (t), . . . , A(n+1) n n n n n # (n+1) n » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f n ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) xn = U1(n+1) (t) ∪ U2(n+1) (t)∪, . . . , ∪Un(n+1) (t) are defined in domain U (n+1) (t) under (n+1) G (n+1) (t) the rules for judging errors, G (n+1) (t) = G (n+1) A A A1 (t) ∪ G A2 (t)∪, . . . , (n+1) (n+1) (n+1) (t), where ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (n+1) (n+1) (n+1) (n+1) (n)Az B(n+1) (n)(n+1) S (t)=S (t)hS2 (t)h, . . . , hSn (t); C ((U (t), S (n)(n+1) # (n)(n+1)1 » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t), p (t), T (t), L (t)), x (t) = f ((μ(n)(n+1) (t), # (n)(n+1) » p (t)), G (n)(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1)C # » p (t), T (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) A # » # » and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), G (n) A (t))), T f sw (C # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » {C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) 1 # » (n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), # » S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), L (n)(n+1) (t)), z 2(n)(n+1) (t)= f 2(n)(n+1) ((ω2(n)(n+1) 2

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# » (n)(n+1) # » (t), p2(n)(n+1) (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), # » Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), n (n)(n+1) G Cn (t)))}, where U (n)(n+1) (t)=U1(n)(n+1) (t) ∪ U2(n)(n+1) (t)∪, . . . , ∪Un(n)(n+1) (t) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) and G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), S (n)(n+1) (t) = (n)(n+1) (n)(n+1) S1 (t)hS2 (t)h, . . . , hSn(n)(n+1) (t); suppose that T f sw has carried out thing decomposition transformation on (U (n) (t), # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the # » following relationship holds: T f sw (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » bx (n) (n) x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ T f sw (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f sw (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f sw (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # (n)(n+1) » (n)Az B(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t), S (t), p (t), T (t), L (t)), ∧C # » x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = T f sw (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f sw (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A # » (n)Az B(n+1) x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) ((U (n)(n+1) (t), A (t)))) ∧ T f sw (C # » S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) # » # » (t), p ((n)n+1) (t)), G C(n)(n+1) (t))))={A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) G (n+1) n A2 (t))), . . . , An # (n) » # » (n) (n) (n) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n An (t)))} ∧ {A1 ((U1 (t), S1 (t), p1 (t), # (n) » (n) (n) (n) (n) (n) (n) T1(n) (t), L (n) (t)), x (t) = f ((μ (t), p1 (t)), G (n) 1 1 1 A1 (t))), A2 ((U2 (t), S2 (t), # (n) » 1(n) # » (n) (n) (n) (n) (n) (n) p2 (t), T2 (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An # » # » (n) (n) (n) (n) (n) ((Un(n) (t), Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) n (t)), x n (t)= f n ((μn (t), pn (t), G An (t)))} # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L (n)(n+1) (t)), ∧ {C1 1 # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) z1 (t)= f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2(n)(n+1) # » (n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t),

222

# (n)(n+1) » p (t), Tn(n)(n+1) (t), # n(n)(n+1) » (n)(n+1) pn (t), G Cn (t)))}.

4 Transformation Connectives in Error Logic

L (n)(n+1) (t)), n

z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t),

# » And the right side of the equation = ¬bx T f sw (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) 1 An (t)))} = {A1 # » # » (n+1) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f n(n+1) A(n+1) n n n n n n n # » # » (n) (n) (n) (n) (n) (n) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n An (t)))} ∧ {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # (n) » (n) (n)(n+1) (n) (n) (n) ((U1(n)(n+1) (t), Tn(n) (t), L (n) n (t)), x n (t)= f n ((μn (t), pn (t), G An (t)))} ∧ {C 1 # » S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) 1 # (n)(n+1) » # » (n)(n+1) (t), p1 (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), # » T2(n)(n+1) (t), L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), 2 # » (n)(n+1) (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), G C2 # » (n)(n+1) L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn (t)))}. n Left side = right side. Proof is completed. # » Proposition 4.94 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f sw (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t)))) # (n) » (n) #A(n) » (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) p2 (t), G A2 (t))), . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » # » (n) (n) (n) (n) (n) (n) (n) f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ U1 (t), A2 ((U2 (t), S2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) p2 (t), T2 (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ U2 (t), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), (n) (n) G (n) (t) = U1(n) (t) ∪ U2(n) (t)∪, . . . , ∪Un(n) (t) are also defined An (t))) ∈ Un (t), U (n) (n) in domain U (n) (t) under G (n) A (t) the rules for judging errors, G A (t) = G A1 (t) ∪

4.2 Decomposition Transformation Connectives in Error Logic

223

(n) (n) (n) G (n) A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), (n) (n) where S (n) (t) = S1 (t)hS2 (t)h, . . . , hSn(n) (t); # » suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error A (n−1) (t) under G (n−1) (t) the rules for judging logical variable defined in domain U # (n−1) » A(n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) errors, T f sw (A # » (n−1) (n−1) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t)))) = {A1 ((U1(n−1) (t), S1(n−1) (t), # (n−1) » (n−1) # » p1 (t), T1 (t), L (n−1) (t)), x1(n−1) (t)= f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n−1) A2 ((U2 # » # » (n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), 2 A2 (t))), . . . , An # » Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) n n An (t)))}, where # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = f 1(n−1) # » (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t), A(n−1) ((U2(n−1) (t), S2(n−1) (t), 1 2 A1 (t))) ∈ U1 # (n−1) » (n−1)1 # » p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) 2 2 A2 (t))) # (n−1) » (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), xn(n−1) ∈ U2(n−1) (t), . . . , A(n−1) n n n n n n # » (n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) (t), U (n−1) (t) = U1(n−1) (t) ∪ n An (t))) ∈ Un (n−1) U2 (t)∪, . . . , ∪Un(n−1) (t) are also defined in domain U (n−1) (t) under G (n−1) (t) A (n−1) (n−1) (n−1) (t) = G (t) ∪ G (t)∪, . . . , ∪G (t), the rules for judging errors, G (n−1) A A1 A2 An G (n−1) (t) is error-judging rule defined in domain Ui(n−1) (t), where S (n−1) (t) = Ai S1(n−1) (t)hS2(n−1) (t)h, . . . , hSn(n−1) (t); # » suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » logical complementary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (n−1) (t) under G B (t) the rules for judging errors, T f sw (B (n−1) ((U (n−1) (t), S (n−1) U # (n−1) » (n−1) # » (t), p (t), T (t), L (n−1) (t)), x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) G (n−1) 1 B # » (n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), B1 (t))), B2 # (n−1) » (n−1)1 # » p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) 2 2 B2 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n−1) . . . , Bn # » # » (n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), n Bn (t)))}, where B1 # » (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) T1(n−1) (t), L (n−1) 1 1 B1 (t))) ∈ # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2(n−1) (t) U1 # » (n−1) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) (t), . . . , Bn(n−1) ((Un(n−1) (t), 2 B2 (t))) ∈ U2 # » # » Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), n n (n−1) G (n−1) (t), U (n−1) (t) = U1(n−1) (t) ∪ U2(n−1) (t)∪, . . . , ∪Un(n−1) (t) are Bn (t))) ∈ Un

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also defined in domain U (n−1) (t) under G (n−1) (t) the rules for judging errors, B (n−1) (n−1) (n−1) (n−1) (t) is error-judging rule G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G (n−1) Bi defined in domain Ui(n−1) (t), where S (n−1) (t)=S1(n−1) (t)hS2(n−1) (t)h, . . . , hSn(n−1) (t); # » suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))), T f sw T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) (C ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {C1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), # » (n−1) T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), C2(n−1) 1 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), z 2(n−1) (t) = f 2(n−1) ((ω2(n−1) (t), 2 # (n−1) » (n−1) # » p2 (t), G C2 (t))) . . . Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), n # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) zn (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where U (t)=U1 (t) ∪ (n−1) (n−1) (n−1) U2(n−1) (t)∪, . . . , ∪Un(n−1) (t) and G C(n−1) (t)=G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), (n−1) (n−1) (n−1) (n−1) S (t) = S1 (t)hS2 (t)h, . . . , hSn (t); suppose that T f sw has carried out # » thing decomposition transformation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following rela# » tionship holds: T f sw (¬bj A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » bj (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), A (t)))) = ¬ T f sw (A ((U # » L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation=T f sw ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f sw (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » p (t). # » And the right side of the equation =¬bj T f sw (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n−1) G (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) 1 An (t)))} = {A1

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# » # » (n−1) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 A1 (t))), A2 # (n−1) » (n−1) (n−1) (n−1) (t)), x (t) = f ((μ (t), p2 (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 2 2 A2 (t))), . . . , # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n−1) # » # » (n−1) (n−1) (n−1) (n−1) ((μ(n−1) (t), pn(n−1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n−1) (t), n # » (n−1) T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 B1 (t))), B2 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), Tn (t), # (n−1) » n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t)), x (t)= f ((μ (t), p (t), G (t)))} ∧ {C ((U L (n−1) n n n n 1 1 Bn # (n−1) n» (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t)= f 1 ((ω1 (t), # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n−1) » 2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) z2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , Cn ((Un(n−1) (t), # (n−1) » 2(n−1) # » (n−1) (n−1) (n−1) (n−1) (n−1) Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn(n−1) (t), (n−1) G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.95 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f sw (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t)))) # » #A(n) » (n) (n) (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) p2 (t), G A2 (t))), . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » # » (n) (n) (n) (n) (n) (n) (n) f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))) ∈ U1 (t), A2 ((U2 (t), S2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) p2 (t), T2 (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) ∈ U2 (t), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), (n) (n) G (n) (t) = U1(n) (t) ∪ U2(n) (t)∪, . . . , ∪Un(n) (t) are also defined An (t))) ∈ Un (t), U (n) (n) in domain U (n) (t) under G (n) A (t) the rules for judging errors, G A (t) = G A1 (t) ∪ (n) (n) (n) G (n) A2 (t)∪, . . . , ∪G An (t), G Ai (t) is error-judging rule defined in domain Ui (t), (n) (n) (n) (n) where S (t) = S1 (t)hS2 (t)h, . . . , hSn (t); # » suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) # » (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logical A (t) the rules for judging errors, variable defined in domain U (n−1) (t) under G (n−1) A # (n−1) » (n−1) (n−1) (n−1) (n−1) T f sw (A ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t)= f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), 1 A

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# » (n−1) T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), A2 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), Tn (t), # (n−1) »n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f n ((μn (t), pn (t), G An (t)))}, where A1 L n (t)), xn # (n−1) » (n−1) (n−1) (n−1) (n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))) ∈ U1 (t), A2 ((U2 (t), S2 (t), p2 (t), # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 A2 (t))) ∈ # (n−1) » (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), xn(n−1) (t) U2(n−1) (t), . . . , A(n−1) n n n n n n # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((μn (t), pn (t), G An (t))) ∈ Un (t), U (t) = U1 (t) ∪ = fn (n−1) (n−1) (n−1) (n−1) (t)∪, . . . , ∪Un (t) are also defined in domain U (t) under G A (t) U2 (n−1) (n−1) (t) = G (t) ∪ G (t)∪, . . . , ∪G (n−1) the rules for judging errors, G (n−1) A A1 A2 An (t), (n−1) (n−1) G Ai (t) is error-judging rule defined in domain Ui (t), where S (n−1) (t) = S1(n−1) (t)hS2(n−1) (t)h, . . . , hSn(n−1) (t); suppose that C (n)Az B(n−1) ((U (n)(n−1) (t), # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) # » (t), p (n)(n−1) (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) G (n−1) A # (n) » (n)Az B(n−1) (t), p (t)), G (n) ((U (n)(n−1) (t), A (t))), there exists the relationship of T f sw (C # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t)= f (n)(n−1) ((μ(n)(n−1) # » # » (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), # » T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), 1 # » (n)(n−1) G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), # » (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), G C2 (t))), . . . , L (n)(n−1) 2 # » Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), n # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where U (t) zn (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) = U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t) and G C (t) = G C1 (t) ∪ (n)(n−1) (n)(n−1) G C2 (t)∪, . . . , ∪G Cn (t), S (n)(n−1) (t) = S1(n)(n−1) (t)hS2(n)(n−1) (t)h, . . . , hSn(n)(n−1) (t); suppose that T f sw has carried out thing decomposition transformation # » on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of (n) (n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and size for xi (t), xi z (n)(n−1) (t), then the following relationship holds: T f sw (¬bd A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bd p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f sw # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side

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# » of the equation= T f sw ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f sw (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), G (n) A (t))) ∧ C # » L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » # » T f sw (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), # » (n−1) G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) A (t)))) ∧ T f sw (A # » (n−1) (n−1) (t)), x (t)= f ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f sw C (n)Az B(n−1) A # » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = # (n) » # » (n) (n) f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))))={A(n) 1 ((U1 (t), S1 (t), p1 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) T1(n) (t), L (n) 1 (t)), x 1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p2 (t), T2 (t), L (n) 2 (t)), x 2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), # » # » (n) (n−1) (n) (n) (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) n (t)), x n (t)= f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))), A2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n−1) » 2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un(n−1) (t), x2 # » # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n n # » (n)(n−1) G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), An (t)))} ∧ {C 1 # » (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), L (n)(n−1) 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C2 ((U2 (t), S2 (t), p (t), T (t), L 2 (t)), z 2 # (n)(n−1) 2 » (n)(n−1) 2 (n)(n−1) (n)(n−1) (n)(n−1) (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un(n)(n−1) (t), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) Sn (t), p (t), T (t), L n (t)), z n (t) = f n ((ωn(n)(n−1) # (n)(n−1) n» (n)(n−1) n (t), pn (t), G Cn (t)))}. # » And the right side of the equation = ¬bd T f sw (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t)= f 1 ((μ1 (t), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) p1 (t)), G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 # » # » (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n−1) » # » (n) (n−1) (n−1) (n−1) (n) ((U1 (t), S (t), p1 (t), xn(n) (t)= f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {A1 # (n−1) » 1 (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2

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# » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G A2 (t))) . . . An ((Un (t), Sn (t), pn (t), Tn (t), # (n−1) » (n−1) (n)(n−1) (n−1) (n−1) (n−1) (n−1) (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {C1 L n (t)), xn # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = 1 # » (n)(n−1) f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), # » S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) 2 # (n)(n−1) » (n)(n−1) # » (t), p2 (t), G C2 (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), # » Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), n (n)(n−1) G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.96 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » bx G A1 (t))), ¬ A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bx G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bx if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 sw {¬ # » # » A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), ¬bx A2 # » # » ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , ¬bx An # » # » ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the thing being negated, there exists its opposite side before being decom# » # » posed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the thing becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.97 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f sw (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)),

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229

# » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 sw # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An # » # » ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) is called the connotative uninterrupted negation on A(n) (μ(t), x(t)), which means that “for the thing being negated, there exists its opposite side after being decomposed”; if # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the thing becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.98 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f sw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bz if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 sw {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), ¬bz A2 # » # » ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , ¬bz An # » # » ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) (n) is called the connotative “not-only” negation on A (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decomposed”; and the # » # » logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the

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state of being erroneous, or correctness) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.99 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f sw (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 sw {A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.187. Proof Proof is omitted.

4.2.4 Spatial Decomposition Transformation Connective in Error Logic # » # » Suppose that T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t)))) # » # » = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where # » # » # » # » p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), it is said that T f k j has conducted spatial trans# » formation on the object of interest μ(t, p(t)). For instance, if one wants to go to Beijing from Shanghai, he/she can choose to fly to Shanghai and makes transfer to Beijing from there if the direct flight is canceled due to some reason.

4.2 Decomposition Transformation Connectives in Error Logic

4.2.4.1

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Conditions for Spatial Decomposition in Error Logic

1. The conditions for spatial decomposition are: fl

(1) Legal conditions T Jk j ; kg

(2) Actual conditions T Jk j ; (3) Objective conditions (target) T Jkmd j ; (4) Conditions for sustaining life T Jksm j ; gj

(5) Technical conditions T Jk j ; (6) Energy conditions T Jknlj . 4.2.4.2

Principles for Spatial Decomposition in Error Logic

The principles for spatial decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.4.3

Ways of Spatial Decomposition in Error Logic

2. Ways of spatial decomposition: (1) Physical decomposition: For example, if the things of interest are a group of warehouses, they can be spatially located in different regions according to principles of minimizing logistic costs. (2) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs; (4) Equal division: # » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))),

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# » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), # » # » G n (t)))}, pi (t) = p j (t)i, j ∈ {1, 2, . . . , n}. (5) Unequal division:

# » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), # » # » G n (t)))}, pi (t) = p j (t)i, j ∈ {1, 2, . . . , n};

(6) Decomposing based on special needs and requirements.

4.2.4.4

Characteristics of Spatial Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f k j and the elements of the error log# » # » ical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f k j can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f k j ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be changed if T f k j does not change its error func# » # » tion f ; for T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), # » # » # » # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where p(t) = p1 (t)+ # » # » # » p2 (t)+, . . . , + pn (t), pi (t) # » # » # » ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}, then T f k j is the spatial decomposition transfor# » mation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))) defined in domain U (t), in this case, T f k j has conducted trans# » # » formation on the element of p(t) in the object (μ(t), p(t)). Ways of domain decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where p(t) = p1 (t)+ p2 (t)+, # » # » # » # » # » . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}, if both (Ui (t), Si (t), # » # » / U (t), i ∈ {1, 2, . . . , n} and (U j (t), S j (t), p j (t), T j (t), pi (t), Ti (t), L i (t)) ∈ L j (t)) ∈ U (t), j ∈ {1, 2, . . . , n} exist, then it is said that T f k j has enabled # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation;

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# » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where p(t) = p1 (t)+ p2 (t)+, # » # » # » # » # » . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}, if there exists (Ui (t), # » / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f k j has Si (t), pi (t), Ti (t), L i (t)) ∈ # » # » enabled A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain, rules for judging errors, time, object, or error function, etc. T f k j ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where p(t) = p1 (t)+ p2 (t)+, # » # » # » # » # » # » . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}, if ∀(Ui (t), Si (t), pi (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f k j has carry out Ti (t), L i (t)) ∈ # » domain displacement transformation on A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where p(t) = p1 (t)+ p2 (t)+, # » # » # » # » # » # » . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}, if ∀(Ui (t), Si (t), pi (t), # » Ti (t), L i (t)) ∈ U (t), i ∈ {1, 2, . . . , n}, and if (U (t), S(t), p(t), T (t), L(t)) and # » (Ui (t), Si (t), pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said that T f k j has conducted decomposition transformation on (U (t), S(t), # » p(t), T (t), L(t)), and T f k j did not carry out decomposition transformation on # » (U (t), S(t), p(t), T (t), L(t)), otherwise. # » Proposition 4.100 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » # » # » # » # » # » # » where p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) =

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# » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where p(t) = # » # » # » # » # » # » # » p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) ∨ B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f k j (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f k j (B((U (t), S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) ∧ B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f k j (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ T f k j (B((U (t), S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f k j (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f k j (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side # » # » = T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f k j # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= # » # » f ((μ (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), # » #1 » 1 p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.101 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)),

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# » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » # » # » # » # » # » # » where p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where p(t) = # » # » # » # » # » # » # » p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator # » # » variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f k j (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » # » # » # » # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where p(t)= p1 (t)+ p2 (t)+, . . . , + pn (t), # » # » # » # » pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold: # » # » (1) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f k j # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n T f k j # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f k j # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n T f k j # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f k j (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))))=T f k j (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t)= f ((ω(t), # » # » # » p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  0, the right # » # » side = T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨n {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}={A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) =

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# » # » f ((μ (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), # » #1 » 1 p (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), #2 » # » # » pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} ∨ # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t)= f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), # » # » # » pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))} = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.102 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » # » # » # » # » # » # » where p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where p(t) = # » # » # » # » # » # » # » p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that # » # » C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the con# » notative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » T f k j (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t)= f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), # » # » # » Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where p(t) = # » # » # » # » # » # » # » p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), # » and z(t). then the following relationship holds: T f k j (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) −n T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted.

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# » Proposition 4.103 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » # » # » # » # » # » # » where p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where p(t) = # » # » # » # » # » # » # » p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator # » # » variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f k j (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » # » # » # » # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where p(t)= p1 (t)+ p2 (t)+, . . . , + pn (t), # » # » # » # » pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and-z(t), then the following relationships hold: # » # » (1) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l T f k j (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h T f k j (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb T f k j (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl T f k j (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted.

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# » Proposition 4.104 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, # » # » # » # » # » # » # » # » where p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f k j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where p(t) = # » # » # » # » # » # » # » p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that # » # » C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the con# » notative same or equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » p(t)), G B (t))), T f k j (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, # » # » # » # » # » # » # » # » where p(t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship # » # » holds: T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f k j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f k j (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.105 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, # » T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » # » # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where p (t) = p1 (t)+ » # » # » # (n) » # » # # » p2 (t)+, . . . , + pn(n) (t), pi(n) (t) ∈ { p1(n) (t)(t), p2(n) (t)(t), . . . , pn(n) (t)(t)}; suppose that # » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » (t))) is the (n + 1)th order error logical varif (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A

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able defined in domain U (n+1) (t) under G (n+1) (t) the rules for judging errors, # (n+1) »A (n+1) (n+1) (n+1) (n+1) T f k j (A ((U (t), S (t), p (t), T (t), L (n+1) (t)), x (n+1) (t) = # » (n+1) (n+1) f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = {A1 ((U1(n+1) (t), S1(n+1) (t), # » # (n+1) » (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), p1 1 1 # » (n+1) G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = 2 A1 (t))), A2 # » # » (n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), f 2(n+1) ((μ(n+1) 2 A2 (t))), . . . , An # » Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n n An (t)))}, # » # » # (n+1) » # (n+1) » # (n+1) » # (n+1) » (n+1) (n+1) where p (t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), # (n+1) » # » p2 (t)(t), . . . , pn(n+1) (t)(t)}; suppose that error logical variable B (n+1) ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), (t))) is the complement error logical variable for A(n+1) ((U (n+1) (t), S (n+1) (t), G (n+1) B # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), (t))) which is defined in domain U (n+1) (t) under G (n+1) (t) the rules for G (n+1) A # (n+1) » B (n+1) (n+1) (n+1) (n+1) judging errors, T f k j (B ((U (t), S (t), p (t), T (t), L (n+1) (t)), # » (n+1) (n+1) (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G B (t))))={B1 ((U1 (t), S1(n+1) (t), # (n+1) » # » p1 (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), 1 1 # » (n+1) G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = 2 B1 (t))), B2 # » # » (n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), f 2(n+1) ((μ(n+1) 2 B2 (t))), . . . , Bn # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n Bn (t)))}, # (n+1) » # (n+1) » # (n+1) » # (n+1) » # (n+1) » # (n+1) » where p (t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), # (n+1) » # » (t)(t), . . . , pn(n+1) (t)(t)}; suppose that C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p2 # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » G C(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) and B (n+1) A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n+1) (t), # (n+1) » # » (n+1) p (t)), G B (t))), T f k j (C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t))))={C1(n+1) ((U1(n+1) (t), # » # » (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) 1 # » (n+1) G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), z 2(n+1) (t) = 2 # » (n+1) f (n+1) ((ω(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), # 2(n+1) » 2 # » (n+1) (n+1) pn (t), Tn (t), L n (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), » # » » # » # # » # (n+1) G Cn (t)))}, where p (n+1) (t) = p1(n+1) (t)+ p2(n+1) (t)+, . . . , + pn(n+1) (t), pi(n+1) (t) ∈ # » # » # » { p1(n+1) (t)(t), p2(n+1) (t)(t), . . . , pn(n+1) (t)(t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n+1) (t), y (n+1) (t), and z (n+1) (t), then the following relationship holds: T f k j (¬bz A(n) ((U (n) (t), S (n) (t),

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# (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bz (n) (n) (n) (n) (n) (n) ¬ T f k j (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f k j (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f k j (A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) A # » (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), B # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f k j (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f k j (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f k j (C (n+1)Az B B # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # » # (n+1) » p (t)), G C(n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), L (n+1) 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), G (n+1) n A2 (t))), . . . , An # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {B1 ((U1 (t), xn # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), S1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), f2 # (n+1) »n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L n (t)), xn (t) = f n ((μ (t), pn (t), G Bn (t)))} ∧ Tn # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1(n+1) (t) = # » # » (n+1) f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), # » (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , T2(n+1) (t), L (n+1) 2 # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) Cn ((Un (t), Sn (t), p (t), Tn (t), L n (t)), z n (t) = # » (n+1)n (t)))}; f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn # » And the right side of the equation = ¬bz T f k j A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), 1 An (t)))} = {A1

4.2 Decomposition Transformation Connectives in Error Logic

241

# » (n+1) x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), 1 A1 (t))), A2 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (t), T2 (t), L (n+1) (t)), x (t) = f ((μ (t), p2 (t), G A2 (t))), p2 2 2 2 # (n+1) »2 (n+1) (n+1) (n+1) (n+1) (n+1) . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n+1) (t) = # » # » (n+1) (n+1) (n+1) (n+1) f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n+1) (t), n # » T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = B2 # (n+1) » (n+1) 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n+1) (t), L (n+1) n n n n Bn # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), # (n+1) »2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , # » 2 Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = n # » (n+1) (t)))}. f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn Left side = right side. Proof is completed. # » Proposition 4.106 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # (n) » # » # » (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where p (t) = p1 (t)+ # (n) » # (n) » # (n) » # (n) » # (n) » # (n) » p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that # » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » (t))) is the (n + 1)th order error logical varif (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) (t) under G (n+1) (t) the rules for judging errors, able defined in domain U # (n+1) »A (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), x (n+1) (t) = T f k j (A # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), 1 A # (n+1) » # » p1 (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), 1 1 # (n+1) » (n+1) (n+1) (n+1) (n+1) G (n+1) (t))), A ((U (t), S (t), p2 (t), T2 (t), L (n+1) (t)), x2(n+1) (t) = 2 2 2 2 A1 # » # » (n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), f 2(n+1) ((μ(n+1) 2 A2 (t))), . . . , An # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) (t)))}, where Tn(n+1) (t), L (n+1) n n # » # » # »An # (n+1) » # (n+1) » # (n+1) » (n+1) (n+1) (n+1) p (t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), # (n+1) » # (n+1) » (n)Az B(n+1) (n)(n+1) (n)(n+1) p2 (t)(t), . . . , pn (t)(t)}; assuming that C ((U (t), S (t),

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# (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), # (n)(n+1) » p (t)), G (n)(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1)C # » p (t), T (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) A # » # » and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), G (n) A (t))), T f k j (C # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » {C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) (t)), 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) z1 (t) = f 1 ((ω1 (t), p (t)), G C1 (t))), C2 # (n)(n+1) » 1(n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U2 , S2 (t), p2 (t), T2 (t), L 2 (t)), z 2(n)(n+1) (t) = # » (n)(n+1) f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), # » Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) n # (n)(n+1) » (n)(n+1) # (n)(n+1) » # (n)(n+1) » # (n)(n+1) » (t), pn (t), G Cn (t)))}, where p (t) = p1 (t)+ p2 (t)+, . . . , # (n)(n+1) » # (n)(n+1) » # (n)(n+1) » # (n)(n+1) » # (n)(n+1) » + pn (t), pi (t) ∈ { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that T f k j has carried out spatial decomposition transformation on (U (n) (t), # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then # » the following relationship holds: T f k j (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bx (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ T f k j (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f k j (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f k j (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), G (n) A (t))) ∧ C # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » T f k j (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), A # » (n)Az B(n+1) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) A (t)))) ∧ T f k j (C # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = # ((n)n+1) » (n)(n+1) (n+1) (n+1) (n+1) (n)(n+1) (n)(n+1) ((μ (t), p (t)), G C (t)))) = {A1 ((U1 (t), S1 (t), f # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ (t), p1 (t)), G A1 (t))), # (n+1) » 1 (n+1)1 (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2(n+1) (t) = A2 ((U2 # » # » (n+1) f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An

4.2 Decomposition Transformation Connectives in Error Logic

243

# » Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n n An (t)))} ∧ # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1(n) (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # (n)(n+1) » # » (n) (n)(n+1) (n)(n+1) (n)(n+1) (n) ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T (t), # (n)(n+1) » (n)(n+1)2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t)), z 2 (t) = f 2 ((ω2 (t), p (t), G (t))), . . . , L2 # (n)(n+1) » 2 (n)(n+1) C2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n)(n+1) (t)))}. z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn # » And the right side of the equation = ¬bx T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # (n+1) » # » (n) (n+1) (n+1) (n+1) pn(n) (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), # » T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n+1) » (n+1)2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n+1) (t), L (n+1) n n n An # (n) » (n) n (n) n # (n) » (n) (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # (n)(n+1) » # » (n) (n)(n+1) (n)(n+1) (n)(n+1) (n) ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) L2 (t)), z 2 (t) = f 2 ((ω2 (t), p (t), G (t))), . . . , # (n)(n+1) » 2 (n)(n+1) C2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), T (t), L n (t)), # » (n)(n+1)n (t)))}. z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn Left side = right side. Proof is completed. Proposition 4.107 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t),

244

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# (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » # » # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where p (t) = p1 (t)+ # (n) » » # » # » # » # # » p2 (t)+, . . . , + pn(n) (t), pi(n) (t) ∈ { p1(n) (t)(t), p2(n) (t)(t), . . . , pn(n) (t)(t)}; # » suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logA (t) the rules for judging errors, ical variable defined in domain U (n−1) (t) under G (n−1) # » A T f k j (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 A # (n−1) » # » p1 (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) (t)))}, where Tn(n−1) (t), L (n−1) n n # (n−1) An » # » # (n−1) » # (n−1) » # (n−1) » # (n−1) » p (t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1(n−1) (t)(t), # (n−1) » # » p2 (t)(t), . . . , pn(n−1) (t)(t)}; # » suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » logical complementary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (t) the rules for judging errors, T ((U (n−1) (t), U (n−1) (t) under G (n−1) f k j (B B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), G (n−1) 1 B # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), x1 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G B2 (t))), # » (t)), xn(n−1) (t) = . . . , Bn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n # » # » # » (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G Bn (t)))}, where p (n−1) (t) = p1(n−1) (t) n # » # (n−1) » # » # (n−1) » # » + p2(n−1) (t)+, . . . , + pn(n−1) (t), pi (t) ∈ { p1(n−1) (t)(t), p2 (t)(t), . . . , # (n−1) » pn (t)(t)}; # » suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (t))), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # » T f k j (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) =

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# » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {C1(n−1) ((U1(n−1) (t), S1(n−1) (t), # (n−1) » # » p1 (t), T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), 1 # » (n−1) G C1 (t))), C2(n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), z 2(n−1) (t) = 2 # » # » (n−1) (t))) . . . Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((ω2(n−1) (t), p2(n−1) (t), G C2 # » (n−1) Tn(n−1) (t), L (n−1) (t)), z n(n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}, where n # (n−1) » # » # (n−1) » # (n−1) » # (n−1) » # (n−1) » p (t) = p1 (t)+ p2 (t)+ . . . + pn (t), pi (t) ∈ { p1(n−1) (t)(t), # (n−1) » # » p2 (t)(t), . . . , pn(n−1) (t)(t)}; suppose that T f k j has carried out spatial decomposition transformation on (U (n) (t), # » (n) S (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following relationship holds: T f k j (¬bj A(n) ((U (n) (t), S (n) (t), # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bj (n) (n) (n) (n) (n) (n) ¬ T f k j (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f k j ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f k j (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = T f k j (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f k j (B (n−1) ((U (n−1) (t), A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) ∧ T f k j (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # (n−1) » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (t)), G C(n−1) (t)))) = {A(n−1) ((U1(n−1) (t), 1 # » # » (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))} ∧ # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 B2 (t))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) =

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# » # » (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), n Bn (t)))} ∧ {C 1 # » (n−1) T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. # » And the right side of the equation = ¬bj T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # » # » (n−1) pn(n) (t), G (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), f n(n) ((μ(n) n (t), An (t)))} = {A1 # » T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n−1) » (n−1)2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n−1) »n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ Tn # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 B2 (t))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = Bn # » # » (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), n Bn (t)))} ∧ {C 1 # » (n−1) T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.108 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # (n) » (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # (n) » # (n) » # (n) » (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where p (t) = p1 (t)+

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# (n) » » # » # » # » # # » p2 (t)+, . . . , + pn(n) (t), pi(n) (t) ∈ { p1(n) (t)(t), p2(n) (t)(t), . . . , pn(n) (t)(t)}; suppose that # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » (t))) is the (n − 1)th order error logical varif (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A (t) the rules for judging errors, able defined in domain U (n−1) (t) under G (n−1) # (n−1) »A (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = T f k j (A # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 A # (n−1) » # » p1 (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))}, # » # » # » # (n−1) » # (n−1) » # (n−1) » (n−1) (n−1) (n−1) where p (t) = p1 (t)+ p2 (t)+, . . . , + pn (t), pi (t) ∈ { p1 (t)(t), # (n−1) » # » p2 (t)(t), . . . , pn(n−1) (t)(t)}; suppose that C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), # (n)(n−1) » p (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), # (n)(n−1) » p (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) ((U (n−1) (t), S (n−1) (t), # (n−1) » # » p (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = G (n−1) A # » (n)Az B(n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), A (t))), and T f k j (C # (n)(n−1) » p (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), # » # (n)(n−1) » p (t)), G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), # » T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), 1 # » (n)(n−1) G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t)), z (t) = f ((ω (t), p2 (t), G C2 (t))), L (n)(n−1) 2 2 2 2 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n)(n−1) » (n)(n−1) pn (t), G Cn (t)))}, where z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), # (n)(n−1) » # (n)(n−1) » # (n)(n−1) » # (n)(n−1) » # » p (t) = p1 (t)+ p2 (t)+, . . . , + pn(n)(n−1) (t), pi (t) ∈ # (n)(n−1) » # (n)(n−1) » # (n)(n−1) » { p1 (t)(t), p2 (t)(t), . . . , pn (t)(t)}; suppose that T f k j has carried out spatial decomposition transformation on (U (n) (t), # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), # » then the following relationship holds: T f k j (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bd (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ T f k j (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side # » of the equation= T f k j ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) =

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# » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f k j (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), G (n) A (t))) ∧ C # » L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n−1) p (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), A (t)))) ∧ T f k j (A # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f k j C (n)Az B(n−1) A # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U (t), S (t), p (t), T (t), L (t)), x (n)(n−1) (t) = # » (n)(n−1) (n) p (n)(n−1) (t)), G C (t)))) = {A1 ((U1(n) (t), S1(n) (t), f (n)(n−1) ((μ(n)(n−1) (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n−1) G (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), 1 An (t)))} ∧ {A1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), # (n−1) »2 (n−1) (n−1) (n−1) (n−1) . . . , A(n−1) ((U (t), S (t), p (t), T (t), L (t)), xn(n−1) (t) = n n n n n n # » (n)(n−1) (t), pn(n−1) (t), G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), f (n−1) ((μ(n−1) n An (t)))} ∧ {C 1 # n(n)(n−1) » p (t), T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), 1 # 1(n)(n−1) » # » (n)(n−1) p1 (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), # » T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), 2 # » (n)(n−1) (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), G C2 # » (n)(n−1) L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn (t)))}. n # » And the right side of the equation = ¬bd T f k j (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) f n(n) ((μ(n) n (t), pn (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # (n) » # » (n) (n−1) (n) (n) (n) (n) pn (t), Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)),

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249

# (n−1) » (n−1) x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2 (t), G (n−1) ((Un(n−1) (t), 2 A2 (t))) . . . An # » # » Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), n n # » (n)(n−1) G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), An (t)))} ∧ {C 1 # » (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), L (n)(n−1) 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) z2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn # » ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = n # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) ((ωn (t), pn (t), G Cn (t)))}. fn Left side = right side. Proof is completed. # » Proposition 4.109 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , ¬bx An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 kj # » # » {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » bx ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the space being negated, there exists its opposite side before being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the space becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.110 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f k j (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t),

250

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# » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » T f−1 k j {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative uninterrupted negation on A(n) (μ(t), x(t)), which means that “for the space being negated, there exists its opposite side after being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the space becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.111 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f k j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bz if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 k j {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), (n) G A (t))) is called the connotative “not-only” negation on A (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decom# » posed”; and the logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the

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251

erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ {1, 2, . . . , n}. Left side = right side. Proof is completed. # » Proposition 4.112 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) # » the rules for judging errors; if T f k j (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » T f−1 k j {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.30. Proof Proof is omitted.

4.2.5 Property Decomposition Transformation Connective in Error Logic # » # » Suppose that T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), it is said that T f t x has conducted property # » transformation on the object of interest μ(t, p(t)). For instance,the volume property of an object can be decomposed/divided into properties of width, length, and height.

4.2.5.1

Conditions for Property Decomposition in Error Logic

1. The conditions for property decomposition are: fl

(1) Legal conditions T Jt x ;

252

(2) (3) (4) (5) (6)

4 Transformation Connectives in Error Logic kg

Actual conditions T Jt x ; Objective conditions (target) T Jtmd x ; Conditions for sustaining life T Jtsm x ; gj Technical conditions T Jt x ; Energy conditions T Jtnlx .

4.2.5.2

Principles for Property Decomposition in Error Logic

The principles for property decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.5.3

Ways of Property Decomposition in Error Logic

2. Ways of property decomposition: (1) Physical decomposition; (2) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs; (4) Equal division: # » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, Ti (t) = T j (t)i, j ∈ {1, 2, . . . , n}; (5) Unequal division:

# » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » pn (t), G n (t)))}, Ti (t) = T j (t) or L i (t) = L j (t)i, j ∈ {1, 2, . . . , n};

(6) Decomposing based on special needs and requirements.

4.2 Decomposition Transformation Connectives in Error Logic

4.2.5.4

253

Characteristics of Property Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f and the elements of the error logical vari# » # » able A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be changed if T f does not change its error function f ; for # » # » T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), #2 » 2 # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}, then T f t x is the property decomposition transformation connective with respect to G(t) and A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in domain U (t), in this case, T f t x has conducted transformation on the element of T (t) in the object # » (μ(t), p(t)). Ways of property decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, # » . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}, if both (Ui (t), Si (t), pi (t), Ti (t), # » / U (t), i ∈ {1, 2, . . . , n} and (U j (t), S j (t), p j (t), T j (t), L j (t)) ∈ U (t), L i (t)) ∈ # » j ∈ {1, 2, . . . , n} exist, then it is said that T f t x has enabled A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, # » . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}, if there exists (Ui (t), Si (t), pi (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f t x has enabled Ti (t), L i (t)) ∈ # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain, rules for judging errors, time, object, or error function, etc. T f t x ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q };

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4 Transformation Connectives in Error Logic

# » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, # » . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}, if ∀(Ui (t), Si (t), pi (t), Ti (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f t x has carried out L i (t)) ∈ # » domain displacement transformation on A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, # » . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}, if ∀(Ui (t), Si (t), pi (t), Ti (t), # » L i (t)) ∈ U (t), i ∈ {1, 2, . . . , n}, and if (U (t), S(t), p(t), T (t), L(t)) and (Ui (t), # » Si (t), pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said # » that T f t x has conducted decomposition transformation on (U (t), S(t), p(t), T (t), L(t)), and T f t x did not carry out decomposition transformation on (U (t), S(t), # » p(t), T (t), L(t)), otherwise. # » Proposition 4.113 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f t x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) ∨ B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f t x (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f t x (B((U (t), S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) ∧ B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f t x (A((U (t), S(t),

4.2 Decomposition Transformation Connectives in Error Logic

255

# » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ T f t x (B((U (t), S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f t x (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f t x (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side # » # » = T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f t x # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), # » # » L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}={A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = # » f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.114 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f t x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose that C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t),

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# » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f t x (C Az B ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold: # » # » (1) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n T f t x (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n T f t x (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f t x (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))))=T f t x (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  0, the right # » # » side = T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f t x (B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨n {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t)= f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), # » #1 » 1 p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » # » # » pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} ∨ # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t)= f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), # » # » # » pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))} = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side.

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Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.115 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f t x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose that C AnhbB ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative inclusion # » # » variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))), T f t x (C AnhbB ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t). then the follow# » # » ing relationship holds: T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n T f t x # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.116 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f t x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t),

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# » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose that C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f t x (C Az B ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l T f t x (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h T f t x (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb T f t x (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f t x (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl T f t x (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.117 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f t x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t),

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# » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose that C Anhdthd j B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative same or # » # » equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » T f t x (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t)= f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship holds: T f t x (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f t x (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f t x (B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.118 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, T f t x (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t)))) # » #A(n) » (n) (n) (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) G (n) (t))), A ((U (t), S (t), p2 (t), T2 (t), L (n) 2 2 2 (t)), x 2 (t) = f 2 ((μ2 (t), A1 # (n) » (n)2 # » (n) (n) (n) (n) (n) (n) p2 (t), G A2 (t))), . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » (n) (n) (n) (n) (n) (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn(n) (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where T (n) (n) (n) (n) Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) A is the (n + 1)th order error logical variable defined in domain U (n+1) (t) under # » G (n+1) (t) the rules for judging errors, T f t x (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) {A1 ((U1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 A1 (t))), A2 # » (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) L (n+1) 2 2 A2 (t))), . . . , An # » # » (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t)= f n(n+1) ((μ(n+1) (t), pn(n+1) (t), n n (n+1) G (n+1) (t)=T1(n+1) (t) ∪ T2(n+1) (t)∪, . . . , ∪Tn(n+1) (t), Ti(n+1) (t) ∈ An (t)))}, where T (n+1) (n+1) (t), T2 (t), . . . , Tn(n+1) (t)}; suppose that error logical variable B (n+1) {T1 # » (n+1) (n+1) ((U (t), S (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t),

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# (n+1) » p (t)), G (n+1) (t))) is the complement error logical variable for A(n+1) ((U (n+1) (t), B # (n+1) » # » (n+1) S (t), p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), (t))) which is defined in domain U (n+1) (t) under G (n+1) (t) the rules for G (n+1) A # (n+1) » B (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), judging errors, T f t x (B # » (n+1) (n+1) (n+1) x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G B (t))))={B1 ((U1 (t), S1(n+1) (t), # (n+1) » (n+1) # » p1 (t), T1 (t), L (n+1) (t)), x1(n+1) (t)= f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n+1) B2 # » # » (n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 B2 (t))), . . . , Bn # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n Bn (t)))}, where (n+1) (n+1) (n+1) (n+1) (n+1) T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1(n+1) (t), # » (n+1) T2 (t) . . . Tn(n+1) (t)}; suppose that C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # (n+1) » (n+1) # » (n+1) (n+1) (n+1) p (t), T (t), L (t)), x (t)= f ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))), B # (n+1) » (n+1) (n+1)Az B (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = T f t x (C # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((μ (t), p (t)), G C (t))))={C1 ((U1 (t), S (t), p1 (t), f # (n+1) » 1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2(n+1) ((ω2(n+1) (t), # (n+1) » # » (n+1) p2 (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), # » (n+1) L (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn (t)))}, where T (n+1) (t) = n (n+1) (n+1) (n+1) T1 (t) ∪ T2 (t)∪, . . . , ∪Tn(n+1) (t), Ti (t) ∈ {T1(n+1) (t), T2(n+1) (t), . . . , Tn(n+1) (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n+1) (t), y (n+1) (t), and z (n+1) (t), then the following # » relationship holds: T f t x (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » bz (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), A (t)))) = ¬ T f t x (A ((U # » L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f t x (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f t x (A # » L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) B # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f t x (A(n+1) ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)),

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# » G (n+1) (t)))) ∧ T f t x (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), A # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f t x (C (n+1)Az B ((U (n+1) (t), B # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) G C(n+1) (t))))={A(n+1) 1 1 # » # » (n+1) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n+1) An ((Un # » # » (n+1) (n+1) (n+1) (n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n+1) (t), n # » (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) T1(n+1) (t), L (n+1) 1 1 B1 (t))), B2 # » ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t)= f 2(n+1) ((μ(n+1) (t), 2 2 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), Tn (t), # (n+1) » n(n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t)= f n ((μn (t), pn (t), G Bn (t)))} ∧ {C1 ((U1 (t), L n (t)), xn # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), S1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) Tn(n+1) (t), L (n+1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn # » And the right side of the equation = ¬bz T f t x A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) 1 An (t)))} = {A1 # » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), p (t)), G (t))), A ((U (t), S (t), p (t), = f 1(n+1) ((μ(n+1) 1 1 2 2 A1 # (n+1) » 2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , T2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n+1) (t) = f n(n+1) # » # » (n+1) (n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {B1 ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) n # » (n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) 1 1 B1 (t))), B2 # » # » (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t)= f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) G (n+1) n B2 (t))), . . . , Bn # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f n ((μ (t), pn (t), G Bn (t)))} ∧ {C1 ((U1 (t), S1(n+1) (t), # (n+1) » (n+1)n # » (n+1) p1 (t), T1 (t), L (n+1) (t)), z 1(n+1) (t)= f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), 1 # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2(n+1) C2 # » # » (n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), # » (n+1) Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn (t)))}. n

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Left side = right side. Proof is completed. # » Proposition 4.119 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, T f t x (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t)))) # (n) » (n) #A(n) » (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) p2 (t), G A2 (t))), . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » (n) (n) (n) f n(n) ((μ(n) (t) = T1(n) (t) ∪ T2(n) (t)∪, . . . , ∪Tn(n) (t), n (t), pn (t), G An (t)))}, where T (n) (n) (n) Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn(n) (t)}; suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) A is the (n + 1)th order error logical variable defined in domain U (n+1) (t) under # » G (n+1) (t) the rules for judging errors, T f t x (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) {A1 ((U1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 A1 (t))), A2 # » (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) L (n+1) 2 2 A2 (t))), . . . , An # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) pn (t), G An (t)))}, where T (t) = T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti(n+1) (t) ∈ {T1(n+1) (t), T2(n+1) (t), . . . , Tn(n+1) (t)}; assuming that C (n)Az B(n+1) # » ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = # » f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the mediator variable for A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = A # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), T f t x (C # (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), # » # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = {C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), # » T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), 1 # » (n)(n+1) G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) , S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), L (n)(n+1) 2 # » (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , Cn(n)(n+1) # » ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = n # » (n)(n+1) (t)))}, where T (n)(n+1) (t) = T1(n)(n+1) (t) f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn (n)(n+1) (n)(n+1) (n)(n+1) ∪ T2 (t)∪, . . . , ∪Tn (t), Ti (t) ∈ {T1(n)(n+1) (t), T2(n)(n+1) (t), . . . , (n)(n+1) (t)}; suppose that T f t x has carried out property decomposition transforTn # » mation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order

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of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the following relationship holds: T f t x (¬bx A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bx p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f t x # » # » (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), (n) G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f t x (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f t x (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # » ∧ C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), # » x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = T f t x (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f t x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) A # » (n)Az B(n+1) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) ((U (n)(n+1) (t), S (n)(n+1) A (t)))) ∧ T f t x (C # (n)(n+1) » (t), p (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), # » # ((n)n+1) » p (t)), G C(n)(n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), G (n+1) n A2 (t))), . . . , An # (n+1) » (n+1) (n) (n) (n) (n+1) (n+1) (n+1) (t) = f ((μn (t), pn (t), G An (t)))} ∧ {A1 ((U1 (t), S1 (t), xn # (n) » (n) n # (n) » (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n)(n+1) G (n) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) 1 An (t)))} ∧ {C 1 # » (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), C2(n)(n+1) # » ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), L (n)(n+1) (t)), z 2(n)(n+1) (t) = 2 # » (n)(n+1) f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), # » Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) n # (n)(n+1) » (n)(n+1) (t), pn (t), G Cn (t)))}. # » And the right side of the equation = ¬bx T f t x (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))),

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# (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) 1 An (t)))} = {A1 # » # » (n+1) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f n(n+1) A(n+1) n n n n n n n # » # » (n) (n) (n) (n) (n) (n) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n An (t)))} ∧ {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # (n) » (n) (n)(n+1) (n) (n) (n) ((U1(n)(n+1) Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {C 1 # » (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L 1(n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) # » (n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), # (n)(n+1) » p (t), T2(n)(n+1) (t), L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), 2 # 2(n)(n+1) » # » (n)(n+1) p2 (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), # » (n)(n+1) Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn n (t)))}. Left side = right side. Proof is completed. Proposition 4.120 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, T f t x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # (n) » (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (t)=T1(n) (t) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where T (n) (n) (n) (n) ∪ T2 (t)∪, . . . , ∪Tn(n) (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn(n) (t)}; # » Suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t)= # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logical variA (n−1) (t) under G (n−1) (t) the rules for judging errors, able defined in domain U # (n−1) » A(n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) T f t x (A # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))))={A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 A # (n−1) » (n−1) # » p1 (t), T1 (t), L (n−1) (t)), x1(n−1) (t)= f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n−1) A2 ((U2 # » # » (n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))}, where

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T (n−1) (t) = T1(n−1) (t) ∪ T2(n−1) (t)∪, . . . , ∪Tn(n−1) (t), Ti(n−1) (t) ∈ {T1(n−1) (t), T2(n−1) (t), . . . , Tn(n−1) (t)}. # » Suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » logical complementary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (n−1) (t) under G B (t) the rules for judging errors, T f t x (B (n−1) ((U (n−1) (t), U # » # » (n−1) S (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) G (n−1) 1 B # » (n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), B1 (t))), B2 # (n−1) » (n−1)1 # » p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) 2 2 B2 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n−1) . . . , Bn # » (n−1) (n−1) (n−1) ((μ(n−1) (t), pn(n−1) (t), G Bn (t)))}, where T (n−1) (t) = T1 (t) ∪ T2 (t)∪, . . . , n (n−1) ∪Tn (t), Ti(n−1) (t) ∈ {T1(n−1) (t), T2(n−1) (t), . . . , Tn(n−1) (t)}; # » suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))), T f t x T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) (C ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))))={C1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) # » (n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), C2(n−1) ((U2(n−1) 1 # » # » (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), z 2(n−1) (t)= f 2(n−1) ((ω2(n−1) (t), p2(n−1) (t), 2 # » (n−1) (t))) . . . Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), z n(n−1) G C2 n # » (n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}, where T (n−1) (t) = T1(n−1) (t) ∪ T2(n−1) (t)∪, . . . , ∪Tn(n−1) (t), Ti(n−1) (t) ∈ {T1(n−1) (t), T2(n−1) (t), . . . , Tn(n−1) (t)}. Suppose that T f t x has carried out property decomposition transformation on (U (n) (t), # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), # » then the following relationship holds: T f t x (¬bj A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bj (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t))))=¬ T f t x (A ((U # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f t x ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f t x (A

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# » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) # » = T f t x (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f t x (B (n−1) ((U (n−1) (t), S (n−1) (t), A # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) B # » ∧ T f t x (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) # » = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 # (n−1) » (n−1) # » p1 (t), T1 (t), L (n−1) (t)), x1(n−1) (t)= f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n−1) A2 ((U2 # » # » (n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), 2 A2 (t))), . . . , An # » (n−1) (t)), xn(n−1) (t)= f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))} ∧ {B1 # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G B1 (t))), B2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n−1) » 2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) (t) = f 2 ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un(n−1) (t), x2 # » # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n n # » (n−1) G (n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), 1 Bn (t)))} ∧ {C 1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) z1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), z 2 (t) = f ((ω (t), p (t), G (t))), # (n−1) » 2(n−1) 2 (n−1) 2 (n−1) C2 (n−1) (n−1) (n−1) (n−1) . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n # » (n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}. # » And the right side of the equation = ¬bj T f t x (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), # » # (n) » (n) pn (t), G An (t)))} = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), 1 1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), x1 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), # (n−1) » 2(n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f (n−1) . . . , An # (n−1) » n(n−1) # » (n−1) (n−1) (n−1) (n−1) ((μ(n−1) (t), pn(n−1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1 (t), T1 n # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2(n−1) # » (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2

4.2 Decomposition Transformation Connectives in Error Logic

267

# (n−1) » (n−1) # » p2 (t), G B2 (t))), . . . , Bn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t)), xn (t) = f ((μ (t), pn (t), G Bn (t)))} ∧ {C1 ((U1 (t), # (n−1) » n (n−1) n # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), S1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) Tn(n−1) (t), L (n−1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn Left side = right side. Proof is completed. # » Proposition 4.121 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, T f t x (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A # (n) » (n) (n) (n) (n) (n) (n) (n) (t)))) = {A(n) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), 1 1 1 1 1 1 1 1 # (n) » # (n) » 1 (n) (n) (n) (n) (n) (n) p1 (t)), G (n) (t))), A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f 2(n) 2 2 2 2 2 2 2 A1 # » # » (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n) (n) (n) (t) = T1(n) (t) ∪ T2(n) (t)∪, . . . , xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where T (n) (n) (n) (n) (n) ∪Tn (t), Ti (t) ∈ {T1 (t), T2 (t), . . . , Tn (t)}; suppose that A(n−1) ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), (t))) is the (n − 1)th order error logical variable defined in domain U (n−1) (t) G (n−1) A (t) the rules for judging errors, T f t x (A(n−1) ((U (n−1) (t), S (n−1) (t), under G (n−1) # (n−1) » A (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n−1) = {A1 ((U1 # » # » (n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), 1 A1 (t))), A2 # » (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) L (n−1) 2 2 A2 (t))), . . . , An # » # » (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t)= f n(n−1) ((μ(n−1) (t), pn(n−1) (t), n n (n−1) G (n−1) (t)=T1(n−1) (t) ∪ T2(n−1) (t)∪, . . . , ∪Tn(n−1) (t), Ti(n−1) (t) ∈ An (t)))}, where T {T1(n−1) (t), T2(n−1) (t), . . . , Tn(n−1) (t)}; suppose that C (n)Az B(n−1) ((U (n)(n−1) (t), # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) # » (t), p (n)(n−1) (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) ((U (n−1) (t), # (n−1) » S (n−1) (t), p (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) # (n−1) » # » (n−1) (n) (n) (n) (t), p (t)), G A (t))) and A ((U (t), S (t), p (n) (t), T (n) (t), L (n) (t)), # » (n)Az B(n−1) ((U (n)(n−1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), and T f t x (C # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) # » # » (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), # » T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), 1 # » (n)(n−1) G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), L (n)(n−1) 2

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4 Transformation Connectives in Error Logic

# » (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), G C2 (t))), . . . , Cn(n)(n−1) # » ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = n # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((ωn (t), pn (t), G Cn (t)))}, where T (t) = T1(n)(n−1) (t) fn (n)(n−1) (n)(n−1) (n)(n−1) ∪ T2 (t) ∪ · · · ∪ Tn(n)(n−1) (t), Ti (t) ∈ {T1 (t), T2(n)(n−1) (t), . . . , Tn(n)(n−1) (t)}; suppose that T f t x has carried out property decomposition transforma# » tion on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2,. . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), then the following relationship holds: T f t x (¬bd A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bd (n) p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f t x (A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side of # » the equation = T f t x ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f t x (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # » ∧ C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), # » x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = T f t x (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) # » ∧ T f t x (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # (n−1) » f (n−1) ((μ(n−1) (t), p (t)), G (n−1) (t)))) ∧ T f t x C (n)Az B(n−1) ((U (n)(n−1) (t), A # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) # (n) » (n) # » (n) (n) (n) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = {A(n) 1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # (n) » (n) (n−1) (n) (n) (n) ((U1(n−1) (t), Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # » # » S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) n n An (t)))} ∧ # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1(n)(n−1) {C1 # » (n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), # » S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) 2 # (n)(n−1) » (n)(n−1) # » (t), p2 (t), G C2 (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), # » Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), n (n)(n−1) G Cn (t)))}.

4.2 Decomposition Transformation Connectives in Error Logic

269

# » And the right side of the equation = ¬bd T f t x (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) An (t)))}={A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), # (n) » # » (n) (n) (n) (n) (n) (n) (n) (n) p1 (t)), G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 # » # » (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » (n) (n−1) (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {A1 # » (n−1) T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), A2 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G A2 (t))) . . . An ((Un (t), Sn (t), pn (t), Tn (t), L n # (n−1) » (n−1) (n)(n−1) (n−1) (n−1) (n−1) (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {C1 ((U1(n)(n−1) (t), # » S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) 1 # (n)(n−1) » # » (n)(n−1) (t), p1 (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), # » T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), 2 # » (n)(n−1) (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), G C2 # » (n)(n−1) L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn (t)))}. Left n side = right side. Proof is completed. # » Proposition 4.122 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » bx G A1 (t))), ¬ A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bx G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bx if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 t x {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), ¬bx A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , ¬bx An ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ;

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# » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) (n) is called the connotative unconstrained negation on A (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side before being decom# » # » posed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.123 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f t x (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 t x {A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) is called the connotative uninterrupted negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side after being decom# » # » posed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.124 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f t x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » bz p1 (t)), G A1 (t))), ¬ A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bz G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bz if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 t x {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), ¬bz A2 ((U2 (t),

4.2 Decomposition Transformation Connectives in Error Logic

# » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t),

271

# » p (t), G (t))), . . . , ¬bz An ((Un (t), # 2 » A2 pn (t), G An (t)))}, then

(1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) is called the connotative “not-only” negation on A(n) (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decomposed”; and the # » # » logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.125 Suppose that an error logical variable A((U (t), S(t), p(t), T (t) # » , L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f t x (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f t x T f−1 tx # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.43. Proof Proof is omitted.

4.2.6 Property (or Attribute) Value Decomposition Transformation Connective in Error Logic # » # » Suppose that T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))))= # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where L(t) =

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4 Transformation Connectives in Error Logic

L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, it is said that T f lz has conducted property (or attribute) value transformation on the object of interest # » μ(t, p(t)).

4.2.6.1

Conditions for Property (or Attribute) Value Decomposition in Error Logic

1. The conditions for property (or attribute) value decomposition are: (1) (2) (3) (4) (5) (6)

fl

Legal conditions T Jlz ; kg Actual conditions T Jlz ; Objective conditions (target) T Jlzmd ; Conditions for sustaining life T Jlzsm ; gj Technical conditions T Jlz ; Energy conditions T Jlznl .

4.2.6.2

Principles for Property (or Attribute) Value Decomposition in Error Logic

The principles for property (or attribute) value decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.6.3

Ways of Property (or Attribute) Value Decomposition in Error Logic

2. Ways of property (or attribute) value decomposition: (1) Physical decomposition; (2) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs;

4.2 Decomposition Transformation Connectives in Error Logic

273

(4) Equal division:

# » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, L i (t) = L j (t)i, j ∈ {1, 2, . . . , n};

(5) Unequal division:

# » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, L i (t) = L j (t) or L i (t) = L j (t)i, j ∈ {1, 2, . . . , n};

(6) Decomposing based on special needs and requirements.

4.2.6.4

Characteristics of Property (or Attribute) Value Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f and the elements of the error logical vari# » # » able A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be changed if T f does not change its error function # » # » f ; for T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = # » {A1 ((U1 (t), S (t), p1 (t), T1 (t), L 1 (t)), # 1» # » x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = # » f n ((μn (t), pn (t), G n (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, then T f lz is the property (or attribute) value decomposition # » transformation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))) defined in domain U (t), in this case, T f lz has con# » ducted transformation on the element of T (t) in the object (μ(t), p(t)). Ways of property (or attribute) value decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f lz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where L(t) = L 1 (t) +

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4 Transformation Connectives in Error Logic

L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, if both (Ui (t), Si (t), # » # » / U (t), i ∈ {1, 2, . . . , n} and (U j (t), S j (t), p j (t), T j (t), pi (t), Ti (t), L i (t)) ∈ L j (t)) ∈ U (t), j ∈ {1, 2, . . . , n} exist, then it is said that T f lz has enabled # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f lz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, if there exists (Ui (t), # » / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f lz has Si (t), pi (t), Ti (t), L i (t)) ∈ # » # » enabled A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain, rules for judging errors, time, object, or error function, etc. T f lz ⊆ {T f ly , T f sw , T f k j , T f lz , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f lz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where L(t) = L 1 (t) + # » L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, if ∀(Ui (t), Si (t), pi (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f lz has carry out Ti (t), L i (t)) ∈ # » domain displacement transformation on A((U (t), S(t), p(t), T (t), L(t)), x(t) # » = f ((μ(t), p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f lz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where L(t) = L 1 (t) + # » L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, if ∀(Ui (t), Si (t), pi (t), # » Ti (t), L i (t)) ∈ U (t), i ∈ {1, 2, . . . , n}, and if (U (t), S(t), p(t), T (t), L(t)) # » and (Ui (t), Si (t), pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said that T f lz has conducted decomposition transformation on (U (t), # » S(t), p(t), T (t), L(t)), and T f lz did not carry out decomposition transfor# » mation on (U (t), S(t), p(t), T (t), L(t)), otherwise. According to error theory, # » # » (Ui (t), Si (t), pi (t), Ti (t), L i (t)) can not be equal to (U j (t), S j (t), p j (t), T j (t), L j (t)) if xi (t) = x j (t). Therefore,T f lz has possibly conducted decomposition transformation on U (t), T (t), f , or G(t) in error logical variable A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))); if (Ui (t), Si (t), pi (t), Ti (t), # » L i (t)) and (U (t), S(t), p(t), T (t), L(t)) belong to different order (layer), then it

4.2 Decomposition Transformation Connectives in Error Logic

275

# » is said T f lz has conducted decomposition transformation on A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))). In general, as L(t) = L 1 (t) + L 2 (t) +, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}, T f lz has possibly carried out # » transformation on S(t) or t in (U (t), S(t), p(t), T (t), L(t)). # » For example, (U (t), trip, p(t), distance, 1500km), the distance can be decom# » # » posed to be (U (t), trip, p(t), distance, 1500km) = {(U (t), trip, p(t), # » # » distance, 500 km), (U (t), trip, p(t), distance, 600km), (U (t), trip, p(t), # » distance, 400km)}; taking another example, (U (t), GDP(Year 2004-2020) p(t), # » growth, $3.5 trillion)={(U (t), GDP(Year 2004-2010) p(t), growth, $1.2 trillion), # » (U (t), GDP(Year 2011-2015), p(t), growth, $0.8 trillion), (U (t), GDP(Year # » 2016-2020) p(t), growth, $1.5trillion)}. # » Proposition 4.126 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; sup# » pose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f lz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)),

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4 Transformation Connectives in Error Logic

# » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side = T f lz # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f lz (B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » # » L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = # » # » f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = # » # » f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), # » #1 » 1 p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.127 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; sup# » pose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose # » # » that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the medi# » # » ator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f lz (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , + L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold:

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# » # » (1) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f lz (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = T f lz (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # » # » f ((ω (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), # » #1 » 1 p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), # » pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  # » z i (t)  0, the right side = T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) ∨n T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨n # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} ∨ # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))} = # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.128 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t),

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# » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; sup# » pose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose # » # » that C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » connotative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = T f lz (C # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), # » −y(t), and z(t). then the following relationship holds: T f lz (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t)))) = T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t)))) −n T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.129 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; sup# » pose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose # » # » that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the medi# » # » ator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f lz (C Az B ((U (t),

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# » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , + L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.130 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; sup# » pose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judg# » # » ing errors, T f lz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose # » # » that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » connotative same or equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t),

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# » # » p(t)), G B (t))), T f lz (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where L(t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship # » # » holds: T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f lz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f lz (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.131 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » # » T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) x1(n) (t) = G (n) 1 (t)), A (t)))) = {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where L (t) = L 1 (t) + (n) (n) (n) (n) (n) (n) L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose that # » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » (t))) is the (n + 1)th order error logical varif (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (t) the rules for judging errors, able defined in domain U (n+1) (t) under G (n+1) # »A T f lz (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), 1 A # (n+1) » # » p1 (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), 1 1 # (n+1) » G (n+1) A(n+1) ((U2(n+1) (t), S2(n+1) (t), p2 (t), T2(n+1) (t), L (n+1) 2 2 A1 (t))), # » (n+1) (n+1) (n+1) (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p (t), G (t))), . . . , A ((U (t), n n 2 A2 # (n+1) » (n+1) 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), Sn G (n+1) where L (n+1) (t) = L (n+1) (t) + L (n+1) (t)+, . . . , +L (n+1) (t), n 1 2 An (t)))}, (n+1) (n+1) (n+1) (n+1) Li (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose that error logical variable # » B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = # » (t))) is the complement error logical varif (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » (n+1) (n+1) (n+1) ((U (t), S (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = able for A # » (t))) which is defined in domain U (n+1) (t) under f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » (t) the rules for judging errors, T f lz (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), G (n+1) B

4.2 Decomposition Transformation Connectives in Error Logic

281

# » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = {B1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S (t), p (t), # (n+1) »2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) (t), L 2 (t)), x2 (t) = f 2 ((μ (t), p2 (t), G B2 (t))), . . . , T2 # » 2 (t)), xn(n+1) (t) = Bn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) n # » (t), pn(n+1) (t), G (n+1) where T (n+1) (t) = T1(n+1) (t) ∪ f n(n+1) ((μ(n+1) n Bn (t)))}, (n+1) (n+1) (n+1) T2 (t)∪, . . . , ∪Tn(n+1) (t), Ti (t) ∈ {T1 (t), T2(n+1) (t) . . . Tn(n+1) (t)}; sup# » pose that C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the mediator variable for # » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » (t))), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) B # » T f lz (C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = {C1(n+1) ((U1(n+1) (t), S1(n+1) (t), # (n+1) » # » p1 (t), T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), 1 # » (n+1) G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) z2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), # (n+1) » 2(n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), (n+1) (n+1) (n+1) G Cn (t)))}, where L (n+1) (t) = L (n+1) (t) + L (t)+, . . . , +L (t), n 1 2 (n+1) (n+1) (n+1) (n+1) Li (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n+1) (t), y (n+1) (t), and z (n+1) (t), then the following relationship holds: T f lz (¬bz A(n) ((U (n) (t), S (n) (t), # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bz (n) (n) (n) (n) (n) (n) ¬ T f lz (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f lz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f lz (A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) A # » (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), B # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f lz (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f lz (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f lz (C (n+1)Az B B

282

4 Transformation Connectives in Error Logic

# » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # » # (n+1) » p (t)), G C(n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), L (n+1) 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), G (n+1) n A2 (t))), . . . , An # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {B1 ((U1 (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), f2 # (n+1) »n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L n (t)), xn (t) = f n ((μ (t), pn (t), G Bn (t)))} ∧ Tn # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1(n+1) (t) = # » # » (n+1) f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), # » (n+1) T2(n+1) (t), L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , 2 # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) Cn ((Un (t), Sn (t), p (t), Tn (t), L n (t)), z n (t) = # » (n+1)n (t)))}. f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn # » And the right side of the equation = ¬bz T f lz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), 1 An (t)))} = {A1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), p2 # (n+1) »2 (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), xn(n+1) (t) = . . . , A(n+1) n n n n n n # » # » (n+1) (n+1) (n+1) (n+1) f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n+1) (t), n # » T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = B2 # (n+1) » (n+1) 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n+1) (t), L (n+1) n n n n n Bn # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = {C1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), # (n+1) »2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , # » 2 Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = n # » (n+1) (t)))}. f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn

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283

Left side = right side. Proof is completed. # » Proposition 4.132 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » # » T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), # (n) » # » (n) (n) (n) (n) (n) (n) (n) p1 (t)), G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n) (n) (n) (n) (n) xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where L (t) = L 1 (t) + L 2 (t)+, . . . , (n) (n) (n) (n) (n) (n+1) +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose that A ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), (t))) is the (n + 1)th order error logical variable defined in domain U (n+1) (t) G (n+1) A (t) the rules for judging errors, T f lz (A(n+1) ((U (n+1) (t), S (n+1) (t), under G (n+1) # (n+1) » A # » p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), G (n+1) 1 1 A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), x1 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), # (n+1) »2 (n+1) (n+1) (n+1) (n+1) (n+1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n+1) (t) = . . . , An # » (n+1) (t), pn(n+1) (t), G An (t)))}, where L (n+1) (t) = L (n+1) (t) + f n(n+1) ((μ(n+1) n 1 (n+1) (n+1) (n+1) (n+1) (n+1) L (n+1) (t)+, . . . , +L (t), L (t) ∈ {L (t), L (t), . . . , L (t)}; n n 2 1 2 i # (n)(n+1) » (n)Az B(n+1) (n)(n+1) (n)(n+1) (n)(n+1) assuming that C ((U (t), S (t), p (t), T (t), # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # (n) » # (n) » (n) (n) (n) (n) p (t), T (t), L (t)), x (t) = f ((μ(n) (t), p (t)), G (n) A (t))), # » (n)Az B(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t), S (t), p (t), T (t), L (t)), T f lz (C # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) x (t) = f ((μ (t), p (t)), G C (t)))) = {C1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1(n)(n+1) (t) = # » (n)(n+1) f (n)(n+1) ((ω(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) , S2(n)(n+1) (t), # 1(n)(n+1) » 1 p (t), T2(n)(n+1) (t), L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), 2 # 2(n)(n+1) » # » (n)(n+1) p2 (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), # » Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), n (n)(n+1) G Cn (t)))}, where L (n)(n+1) (t) = L (n)(n+1) (t) + L (n)(n+1) (t)+, . . . , + 1 2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) L (n)(n+1) (t), L (t) ∈ {L (t), L (t), . . . , L (t)}; suppose that n n 1 2 i T f lz has carried out error property (or attribute) value decomposition transforma-

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# » tion on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the following relationship holds: T f lz (¬bx A(n) ((U (n) (t), S (n) (t), # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bx (n) (n) (n) (n) (n) (n) ¬ T f lz (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f lz (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f lz (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), G (n) A (t))) ∧ C # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » T f lz (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), A # » (n)Az B(n+1) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) A (t)))) ∧ T f lz (C # » ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = # » ((U1(n+1) (t), S1(n+1) (t), f (n)(n+1) ((μ(n)(n+1) (t), p ((n)n+1) (t)), G C(n)(n+1) (t)))) = {A(n+1) 1 # » # (n+1) » (n+1) (t), T1 (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) p1 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n+1) A2 ((U2 # » # » (n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n An (t)))} ∧ # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1(n) (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # » (n) (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » # » (n) (n)(n+1) (n) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # » (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), T1(n)(n+1) (t), L (n)(n+1) 1 # » (n)(n+1) G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), # » (n)(n+1) L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , 2 # (n)(n+1) » Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn (t), Tn(n)(n+1) (t), L (n)(n+1) n # » (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn (t)))}. # » And the right side of the equation = ¬bx T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))),

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# (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # (n+1) » # » (n) (n+1) (n+1) (n+1) pn(n) (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), # » (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) T1(n+1) (t), L (n+1) 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n+1) » (n+1)2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n+1) (t), L (n+1) n n n An # (n) » (n) n (n) n # (n) » (n) (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # (n)(n+1) » # » (n) (n)(n+1) (n)(n+1) (n)(n+1) (n) ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , L2 # » Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), L (n)(n+1) (t)), n # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t) = f n ((ωn (t), pn (t), G Cn (t)))}. zn Left side = right side. Proof is completed. Proposition 4.133 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where L (t) = L 1 (t) + (n) (n) (n) (n) (n) L 2 (t)+, . . . , +L (n) n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose that # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) A ((U (t), S (t), p (t), T (t), L (t)), x (n−1) (t) = # » (t))) is the (n − 1)th order error logical varif (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A (n−1) (t) under G (n−1) (t) the rules for judging errors, able defined in domain U # (n−1) »A (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = T f lz (A # » (n−1) (n−1) f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t)))) = {A1 ((U1(n−1) (t), S1(n−1) (t), # (n−1) » # » p1 (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # (n−1) » G (n−1) A(n−1) ((U2(n−1) (t), S2(n−1) (t), p2 (t), T2(n−1) (t), L (n−1) 2 2 A1 (t))),

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# » (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), 2 A2 (t))), . . . , An # » # » Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), n n G (n−1) where L (n−1) (t) = L (n−1) (t) + L (n−1) (t)+, . . . , +L (n−1) (t), n 1 2 An (t)))}, (n−1) (n−1) (n−1) (n−1) Li (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose that B (n−1) ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), (t))) is the (n − 1)th order error logical complementary variable of G (n−1) B # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » (t))) defined in domain U (n−1) (t) under f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (t) the rules for judging errors, T f lz (B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), G (n−1) B # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = B # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = {B1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) f1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S (t), p (t), # (n−1) »2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) (n−1) (t), L 2 (t)), x2 (t) = f 2 ((μ (t), p2 (t), G B2 (t))), . . . , T2 # (n−1) » 2 (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = Bn # (n−1) » (n−1) (n−1) (n−1) (n−1) ((μn (t), pn (t), G Bn (t)))}, where L (t) = L (n−1) (t) + fn 1 (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; # » suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (t))), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = T f lz (C # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {C1(n−1) ((U1(n−1) (t), S1(n−1) (t), # » # (n−1) » (t), T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), p1 1 # » (n−1) G C1 (t))), C2(n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), z 2(n−1) (t) = 2 # » # » (n−1) f 2(n−1) ((ω2(n−1) (t), p2(n−1) (t), G C2 (t))) . . . Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), # » (n−1) Tn(n−1) (t), L (n−1) (t)), z n(n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}, n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) where L (t) = L 1 (t) + L 2 (t)+, . . . , +L n (t), L i (t) ∈ {L 1 (t), (n−1) L (n−1) (t), . . . , L (t)}. n 2 Suppose that T f lz has carried out error property (or attribute) value decomposition # » transformation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following relationship holds: T f lz (¬bj A(n) ((U (n) (t), # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # » ¬bj T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » p (t)), G (n) A (t)))).

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287

Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f lz ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f lz (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ C (n−1)Az B ((U (n−1) (t), B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = T f lz (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f lz (B (n−1) ((U (n−1) (t), A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) ∧ T f lz (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {A(n−1) ((U1(n−1) (t), 1 # » # » (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))} ∧ # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 B2 (t))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = Bn # (n−1) » # » (n−1) (n−1) (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) (t)))} ∧ {C ((U (t), S (t), p1 (t), n 1 1 Bn # (n−1) 1» (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2(n−1) (t) = # » # » (n−1) f 2(n−1) ((ω2(n−1) (t), p2(n−1) (t), G C2 (t))), . . . , Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), # » (n−1) Tn(n−1) (t), L (n−1) (t)), z n(n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}. n # » And the right side of the equation = ¬bj T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n−1) G (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) 1 An (t)))} = {A1 # » # » (n−1) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 A1 (t))), A2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 A2 (t))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n−1) An ((Un

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# » # » (n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), n An (t)))} ∧ {B1 # » (n−1) T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 B1 (t))), B2 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), Tn (t), # (n−1) »n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) L n (t)), xn (t) = f n ((μn (t), pn (t), G Bn (t)))} ∧ {C1 ((U1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1(n−1) (t), # (n−1) » # » (n−1) p1 (t)), G C1 (t))), C2(n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), z2 # (n−1) » 2(n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), (n−1) G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.134 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, # » # » T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (n) (n) (n) (n) (n) (n) (n) G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), # (n) » # » (n) (n) (n) (n) (n) (n) (n) p1 (t)), G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = # » # (n) » (n) (n) (n) (n) (n) (n) (n) f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n) (n) (n) (n) (n) xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where L (t) = L 1 (t) + L 2 (t)+, . . . , (n) (n) (n) (n) (n) +L n (t), L i (t) ∈ {L 1 (t), L 2 (t), . . . , L n (t)}; suppose that A(n−1) ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), (n−1) G A (t))) is the (n − 1)th order error logical variable defined in domain U (n−1) (t) (t) the rules for judging errors, T f lz (A(n−1) ((U (n−1) (t), S (n−1) (t), under G (n−1) # (n−1) » A # » p (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), G (n−1) 1 1 A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S (t), x1 # (n−1) » # (n−1) » 2 (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ (t), p2 (t), G A2 # (n−1) »2 (n−1)2 (n−1) (n−1) (n−1) (n−1) (t))), . . . , A(n−1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = n n n n n n n # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((μn (t), pn (t), G An (t)))}, where L (t) = L 1 (t) + L 2 (t)+, fn (n−1) (n−1) (n−1) . . . , +L (n−1) (t), L (t) ∈ {L (t), L (t), . . . , L (n−1) (t)}; suppose that n n 1 2 i # (n)(n−1) » (n)Az B(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C ((U (t), S (t), p (t), T (t), L (n)(n−1) (t)), # » (n)(n−1) (t))) is the mediator x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C # » variable for A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (n)Az B(n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), and T f lz (C

4.2 Decomposition Transformation Connectives in Error Logic

289

# » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = # » f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t), # » S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) 1 # (n)(n−1) » # » (n)(n−1) (t), p1 (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), # » T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), 2 # » (n)(n−1) (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), G C2 # » (n)(n−1) L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn (t)))}, n (n)(n−1) (n)(n−1) where L (n)(n−1) (t) = L (n)(n−1) (t) + L (t)+, . . . , +L (t), n 1 2 (n)(n−1) (n)(n−1) L i(n)(n−1) (t) ∈ {L (n)(n−1) (t), L (t), . . . , L (t)}; suppose that T has carf lz n 1 2 ried out error property (or attribute) value decomposition transformation on (U (n) (t), # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2,. . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), # » then the following relationship holds: T f lz (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bd (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ T f lz (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side # » of the equation = T f lz ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f lz (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # » ∧ C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), # » x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = T f lz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n) A (t)))) ∧ T f lz (A # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f lz C (n)Az B(n−1) ((U (n)(n−1) A # » (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) # (n) » (n) # » (n) (n) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = {A(n) 1 ((U1 (t), S1 (t), p1 (t), T1 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) (n) L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # (n) » (n) (n) (n) (n) (n) (n) T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A(n−1) 1 # » (n−1) (n−1) (n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x (t) = f ((μ (t), 1 1 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))), A2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n−1) » 2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un(n−1) (t), x2 # » # » Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), n n

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# (n)(n−1) » p1 (t), T1(n)(n−1) (t), # » (n)(n−1) L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C2 ((U2 (t), S2 (t), p (t), T (t), L 2 (t)), z 2 # (n)(n−1) 2 » (n)(n−1) 2 (n)(n−1) (n)(n−1) (n)(n−1) (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un(n)(n−1) (t), # » Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) n # (n)(n−1) » (n)(n−1) (t), pn (t), G Cn (t)))}. # » And the right side of the equation = ¬bd T f lz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n) (n) G (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) x1(n) (t) = 1 (t)), An (t)))} = {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n−1) (n−1) (n−1) (n) (n) (n) (n) ((U1 (t), S1 (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((μ2 (t), p2 (t), G A2 (t))) . . . An ((Un (t), Sn (t), pn (t), f2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n−1) (t), L (n−1) n n n n n An # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) z1 (t) = f 1 ((ω1 (t), p (t)), G C1 (t))), C2 # (n)(n−1) » 1 (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2(n)(n−1) (t) = # » (n)(n−1) f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), G C2 (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) Sn (t), p (t), T (t), L n (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) # (n)(n−1) n» (n)(n−1) n (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.135 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » bx p1 (t)), G A1 (t))), ¬ A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bx G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (n)(n−1) G (n−1) ((U1(n)(n−1) (t), An (t)))} ∧ {C 1

S1(n)(n−1) (t),

(1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0;

4.2 Decomposition Transformation Connectives in Error Logic

291

# » # » bx if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 lz {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bx An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side before being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.136 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f lz (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 lz {A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), (n) G A (t))) is called the connotative uninterrupted negation on A (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side after being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.137 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f lz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t),

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# » # » p(t)), G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » bz p1 (t)), G A1 (t))), ¬ A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bz G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bz if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 lz {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative “not-only” negation on A(n) (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decom# » posed”; and the logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.138 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f lz (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 lz {A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.56. Proof Proof is omitted.

4.2 Decomposition Transformation Connectives in Error Logic

293

4.2.7 Error Value Decomposition Transformation Connective in Error Logic # » # » Suppose that T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) # » # » = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, it is said that # » T f cz has conducted error value transformation on the object of interest μ(t, p(t)). For example, error value stands for the total security risk which is the sum of a security’s notdiversifiable and diversifiable risk.

4.2.7.1

Conditions for Error Value Decomposition in Error Logic

1. The conditions for error value decomposition are: (1) (2) (3) (4) (5) (6)

fl

Legal conditions T Jcw ; kg Actual conditions T Jcw ; md ; Objective conditions (target) T Jcw sm ; Conditions for sustaining life T Jcw gj Technical conditions T Jcw ; nl . Energy conditions T Jcw

4.2.7.2

Principles for Error Value Decomposition in Error Logic

The principles for error value decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.7.3

Ways of Error Value Decomposition in Error Logic

2. Ways of error value decomposition: (1) Physical decomposition; (2) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k);

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.....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs; (4) Equal division: # » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, xi (t) = x j (t)i, j ∈ {1, 2, . . . , n}; (5) Unequal division:

# » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, xi (t) = x j (t) or L i (t) = L j (t)i, j ∈ {1, 2, . . . , n};

(6) Decomposing based on special needs and requirements.

4.2.7.4

Characteristics of Error Value Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f and the elements of the error logical vari# » # » able A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be changed if T f does not change its error function f ; for # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, then T f cz is the error value decomposition transformation connective with respect to G(t) and A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in domain U (t), in this case, T f cz has conducted transformation on the element of T (t) in the object # » (μ(t), p(t)). Ways of error value decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f cz (A((U (t),

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# » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where x(t) = x1 (t) + x2 (t)+, # » . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, if both (Ui (t), Si (t), pi (t), Ti (t), # » / U (t), i ∈ {1, 2, . . . , n} and (U j (t), S j (t), p j (t), T j (t), L j (t)) ∈ U (t), L i (t)) ∈ # » j ∈ {1, 2, . . . , n} exist, then it is said that T f cz has enabled A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f cz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where x(t) = x1 (t) + x2 (t)+, # » . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, if there exists (Ui (t), Si (t), pi (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f cz has enabled Ti (t), L i (t)) ∈ # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain, rules for judging errors, time, object, or error function, etc. T f cz ⊆ {T f ly , T f sw , T f k j , T f lz , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f cz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where x(t) = x1 (t) + x2 (t)+, # » . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, if ∀(Ui (t), Si (t), pi (t), Ti (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f cz has carry out domain disL i (t)) ∈ # » placement transformation on A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f cz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where x(t) = x1 (t) + x2 (t)+, # » . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, if ∀(Ui (t), Si (t), pi (t), Ti (t), # » L i (t)) ∈ U (t), i ∈ {1, 2, . . . , n}, and if (U (t), S(t), p(t), T (t), L(t)) and (Ui (t), # » Si (t), pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said that # » T f cz has conducted decomposition transformation on (U (t), S(t), p(t), T (t), L(t)), and T f cz did not carry out decomposition transformation on (U (t), S(t), # » # » p(t), T (t), L(t)), otherwise. According to error theory, (Ui (t), Si (t), pi (t), # » Ti (t), L i (t)) can not be equal to (U j (t), S j (t), p j (t), T j (t), L j (t)) if xi (t) =

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x j (t). Therefore,T f cz possibly has conducted decomposition transformation on # » U (t), T (t), f , or G(t) in error logical variable A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))); if (Ui (t), Si (t), pi (t), Ti (t), L i (t)) and (U (t), # » S(t), p(t), T (t), L(t)) belong to different order (layer), then it is said T f cz has # » conducted decomposition transformation on A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))). In general, as where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}, T f cz has possibly carried out transfor# » mation on S(t) or t in (U (t), S(t), p(t), T (t), L(t)). # » Proposition 4.139 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where y(t) = y1 (t) + y2 (t)+, . . . , +yn (t), yi (t) ∈ {y1 (t), y2 (t), . . . , yn (t)}; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f cz # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f cz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f cz # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ T f cz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f cz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f cz (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))};

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297

And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f cz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), # » # » L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » xn (t) = f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.140 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where y(t) = y1 (t) + y2 (t)+, . . . , +yn (t), yi (t) ∈ {y1 (t), y2 (t), . . . , yn (t)}; suppose that C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f cz (C Az B ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where z(t) = z 1 (t) + z 2 (t)+, . . . , +z n (t), z i (t) ∈ {z 1 (t), z 2 (t), . . . , z n (t)}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold: # » # » (1) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))));

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# » # » (2) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f cz (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = T f cz (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # » # » f ((ω (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), #1 » 1 # » p (t), G (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), # 2 » C2 pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  # » yi (t)  z i (t)  0, the right side = T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t)))) ∨n T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨n # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = # » # » {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 # » # » (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} ∨ {C1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))} = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # » # » f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = # » f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2)can also be proved. # » Proposition 4.141 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t),

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# » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where y(t) = y1 (t) + y2 (t)+, # » . . . , +yn (t), yi (t) ∈ {y1 (t), y2 (t), . . . , yn (t)}; suppose that C AnhbB ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative inclusion variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f cz (C AnhbB ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where z(t) = z 1 (t) + z 2 (t)+, . . . , +z n (t), z i (t) ∈ {z 1 (t), z 2 (t), . . . , z n (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t). then the following # » # » relationship holds: T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n T f cz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.142 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where y(t) = y1 (t) + y2 (t)+, . . . , +yn (t), yi (t) ∈ {y1 (t), y2 (t), . . . , yn (t)}; suppose that C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f cz (C Az B ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where z(t) = z 1 (t) + z 2 (t)+, . . . , +z n (t), z i (t) ∈ {z 1 (t), z 2 (t), . . . , z n (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t),

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−xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.143 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) = x1 (t) + x2 (t)+, . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, T f cz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), # » # » B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # » Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where y(t) = y1 (t) + y2 (t)+, . . . , +yn (t), yi (t) ∈ {y1 (t), y2 (t), . . . , yn (t)}; suppose that # » # » C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the con# » notative same or equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » p(t)), G B (t))), T f cz (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where z(t) = z 1 (t) + z 2 (t)+, . . . , +z n (t), z i (t) ∈ {z 1 (t), z 2 (t), . . . , z n (t)}; if ∀i,

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i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship # » # » holds: T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f cz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f cz (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.144 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f cz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p2 (t), G A2 (t))), . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » (n) (n) (n) (n) (n) (n) (n) f n ((μn (t), pn (t), G An (t)))}, where x (t) = x1 (t) + x2 (t)+, . . . , +xn(n) (t), xi(n) (t) ∈ {x1(n) (t), x2(n) (t), . . . , xn(n) (t)}; suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) A is the (n + 1)th order error logical variable defined in domain U (n+1) (t) under # » G (n+1) (t) the rules for judging errors, T f cz (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) {A1 ((U1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 A1 (t))), A2 # » (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) L (n+1) 2 2 A2 (t))), . . . , An # » ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), n n # (n+1) » (n+1) (n+1) (n+1) pn (t), G An (t)))}, where x (t) = x1 (t) + x2(n+1) (t)+, . . . , +xn(n+1) (t), xi(n+1) (t) ∈ {x1(n+1) (t), x2(n+1) (t), . . . , xn(n+1) (t)}; suppose that error logical vari# » able B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) # » ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) is the complement error logical variable for B # » (n+1) (n+1) (n+1) ((U (t), S (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) A # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) which is defined in domain U (n+1) (t) under A # » (t) the rules for judging errors, T f cz (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), G (n+1) B # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = B # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) {B1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 B1 (t))), B2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p2 (t), G (n+1) L (n+1) 2 2 2 2 B2 (t))), . . . , Bn # » ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), n n

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# (n+1) » (n+1) pn (t), G Bn (t)))}, where T (n+1) (t) = T1(n+1) (t) ∪ T2(n+1) (t)∪, . . . , ∪Tn(n+1) (t), Ti(n+1) (t) ∈ {T1(n+1) (t), T2(n+1) (t) . . . Tn(n+1) (t)}; suppose that C (n+1)Az B ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » G C(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) and A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) B ((U (t), S (t), p (t), T (t), L (t)), x (t) = # » (n+1)Az B (n+1) (n+1) (t))), T (C ((U (t), S (t), f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) f cz B # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) z1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2(n+1) (t), # » # » S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), 2 # » (n+1) (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), G C2 n # » (n+1) z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn (t)))}, where x (n+1) (t) = x1(n+1) (t) + x2(n+1) (t)+, . . . , +xn(n+1) (t), xi(n+1) (t) ∈ {x1(n+1) (t), x2(n+1) (t), . . . , xn(n+1) (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n+1) (t), y (n+1) (t), and z (n+1) (t), then the following rela# » tionship holds: T f cz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » bz (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), A (t)))) = ¬ T f cz (A ((U # » L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f cz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f cz (A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) A # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), B # (n+1) » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G C(n+1) (t)))) = T f cz L (n+1) (t)), # » (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f cz (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f cz B # (n+1) » (n+1) (n+1)Az B (n+1) (n+1) (n+1) (n+1) (C ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n+1) # (n+1) » # » (n+1) (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = {A(n+1) ((U (t), S (t), p1 (t), 1 1 # (n+1) » 1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n+1) ((μ(n+1) (t), 2 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), Tn (t), # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {B1 ((U1

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# » (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), 1 1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t)), G B1 (t))), B2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n+1) » 2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (t) = f 2 ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un(n+1) (t), x2 # » # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) n n # » (n+1) G (n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), 1 Bn (t)))} ∧ {C 1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) z1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » (n+1) . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z (t) = f n(n+1) n n # (n+1) » (n+1) (n+1) ((ωn (t), pn (t), G Cn (t)))}. # » And the right side of the equation = ¬bz T f cz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) 1 An (t)))} = {A1 # » # » (n+1) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f n(n+1) A(n+1) n n n n n n n # » # » (n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), n An (t)))} ∧ {B1 # » (n+1) T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), B2 # » ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), 2 2 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), Tn (t), # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ {C ((U L (n+1) n n n n 1 1 Bn # »n (n+1) (n+1) (n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), z (t) = f ((ω (t), 1 1 1 1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n+1) » 2 (n+1) 2 (n+1) (n+1) (n+1) z2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), # (n+1) »2 (n+1) (n+1) Sn (t), pn (t), Tn (t), L (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), n # (n+1) » (n+1) pn (t), G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.145 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f cz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (n) (n) (n) (n) (n) (n) (n) G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1

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# » # (n) » (n) (n) (n) (n) (n) (n) (t), p1(n) (t)), G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n) (n) (n) (n) (n) xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where x (t) = x 1 (t) + x 2 (t)+, . . . , (n) (n) (n) (n) (n) +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose that A(n+1) ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), (n+1) G A (t))) is the (n + 1)th order error logical variable defined in domain U (n+1) (t) (t) the rules for judging errors, T f cz (A(n+1) ((U (n+1) (t), S (n+1) (t), under G (n+1) # (n+1) » A (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) = {A1 ((U1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 A1 (t))), A2 # » (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) L (n+1) 2 2 A2 (t))), . . . , An # » ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), n n # (n+1) » (n+1) (n+1) (n+1) pn (t), G An (t)))}, where x (t) = x1 (t) + x2(n+1) (t)+, . . . , +xn(n+1) (t), xi(n+1) (t) ∈ {x1(n+1) (t), x2(n+1) (t), . . . , xn(n+1) (t)}; assuming that C (n)Az B(n+1) # » ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = # » f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the mediator variable for A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = A # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), T f cz (C # (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), # » # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = {C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), # » T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), 1 # » (n)(n+1) G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) , S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), L (n)(n+1) 2 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((Un (t), Sn (t), p (t), T (t), L n (t)), z n(n)(n+1) (t) = # (n)(n+1)n » (n)(n+1)n (n)(n+1) (n)(n+1) (n)(n+1) ((ωn (t), pn (t), G Cn (t)))}, where x (t) = x1(n)(n+1) (t) fn (n)(n+1) (n)(n+1) (n)(n+1) + x2 (t)+, . . . , +xn(n)(n+1) (t), xi (t) ∈ {x1 (t), x2(n)(n+1) (t), . . . , xn(n)(n+1) (t)}; suppose that T f cz has carried out error value decomposition transfor# » mation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the following relationship holds: T f cz (¬bx A(n) ((U (n) (t), S (n) (t), # (n) » # » bx p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f cz # » # » (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of

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# » the equation = T f cz (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f cz (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # » ∧ C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), # » x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = T f cz (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f cz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A # » (n)Az B(n+1) x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) ((U (n)(n+1) (t), A (t)))) ∧ T f cz (C # » S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) # » # » (t), p ((n)n+1) (t)), G C(n)(n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) G (n+1) n A2 (t))), . . . , An # (n) » # » (n) (n) (n) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n An (t)))} ∧ {A1 ((U1 (t), S1 (t), p1 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p2 (t), T2 (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ # » {C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) (t)), 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) z1 (t) = f 1 ((ω1 (t), p (t)), G C1 (t))), C2 # (n)(n+1) » 1 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2(n)(n+1) (t) = # » (n)(n+1) f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), # » Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) n # (n)(n+1) » (n)(n+1) (t), pn (t), G Cn (t)))}. # » And the right side of the equation = ¬bx T f cz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) 1 An (t)))} = {A1 # » # » (n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), = f 1(n+1) ((μ(n+1) 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n+1) An ((Un # » # » (n) (n) (n) (n) (n) (n) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n An (t)))} ∧ {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)),

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# (n) » # (n) » (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) (t)), x (t) = f ((μ (t), p2 (t), G A2 (t))), . . . , A(n) L (n) n ((Un (t), Sn (t), pn (t), 2 2 2 2 # (n) » (n) (n)(n+1) (n) (n) (n) Tn(n) (t), L (n) ((U1(n)(n+1) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {C 1 # » (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L 1(n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) # » (n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), # (n)(n+1) » p (t), T2(n)(n+1) (t), L 2(n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), # 2(n)(n+1) » # » (n)(n+1) p2 (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), # » Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), n (n)(n+1) G Cn (t)))}. Left side = right side. Proof is completed. Proposition 4.146 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging # » errors, T f cz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) = x 1 (t) + (n) (n) (n) (n) x2 (t)+, . . . , +xn(n) (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn(n) (t)}; # » Suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) # » (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logical A (n−1) (t) under G (n−1) (t) the rules for judging errors, variable defined in domain U A # » (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) T f cz (A # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), 1 A # » (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) T1(n−1) (t), L (n−1) 1 1 A1 (t))), A2 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), Tn (t), # (n−1) » n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) = L n (t)), xn x1(n−1) (t) + x2(n−1) (t)+, . . . , +xn(n−1) (t), xi(n−1) (t) ∈ {x1(n−1) (t), x2(n−1) (t), . . . , xn(n−1) (t)}. # » Suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » (n−1) (n−1) ((U (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), logical complementary variable of A # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A

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U (n−1) (t) under G (n−1) (t) the rules for judging errors, T f cz (B (n−1) ((U (n−1) (t), B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) G (n−1) 1 B # » (n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), B1 (t))), B2 # (n−1) » (n−1)1 # » p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) 2 2 B2 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), S (t), p (t), Tn (t), L n (t)), xn (t) = . . . , Bn # (n−1) »n (n−1) n (n−1) (n−1) (n−1) (t), p (t), G (t)))}, where x (t) = x (t) + x (t)+, f n(n−1) ((μ(n−1) n n 1 2 Bn (n−1) (n−1) (n−1) (n−1) (n−1) . . . , +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; # » suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))), T f cz T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) (C ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n−1) # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) ((μ(n−1) (t), p (n−1) (t)), G C (t))))={C1 ((U1 (t), S1 (t), p1 (t), T1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2(n−1) # » (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), z 2(n−1) (t) = f 2(n−1) ((ω2(n−1) (t), 2 # (n−1) » (n−1) # » p2 (t), G C2 (t))) . . . Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n # » (n−1) (t)), z n(n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}, where x (n−1) (t) = x1(n−1) (t) + x2(n−1) (t)+, . . . , +xn(n−1) (t), xi(n−1) (t) ∈ {x1(n−1) (t), x2(n−1) (t), . . . , xn(n−1) (t)}; suppose that T f cz has carried out error value decomposition transformation on # » (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of (n−1) (n−1) (t), yi (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), size for xi and z (n−1) (t), then the following relationship holds: T f cz (¬bj A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bj (n) p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f cz (A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f cz ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f cz (A # » # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) # » (n−1) (n−1) p (t), T (t), L (t)), x (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) # » = T f cz (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) =

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# » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f cz (B (n−1) ((U (n−1) (t), S (n−1) (t), A # (n−1) » (n−1) # » (n−1) (n−1) p (t), T (t), L (t)), x (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) ∧ T f cz (C ((U (t), S (t), p (t), T (t), L (t)), x (t) = # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((μ (t), p (t)), G C (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), f # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) T1 (t), L 1 (t)), x1 (t) = f ((μ (t), p1 (t)), G A1 (t))), # (n−1) »1 (n−1)1 (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2(n−1) (t) = f 2(n−1) A2 ((U2 # » # » (n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))} ∧ # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n−1) {B1 # » # » (n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), 1 B1 (t))), B2 # » (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) L (n−1) 2 2 B2 (t))), . . . , Bn # » (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), n n # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) pn (t), G Bn (t)))} ∧ {C1 ((U1 (t), S (t), p1 (t), T1 (t), # (n−1) »1 (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f ((ω (t), p (t), S2 # (n−1) » 2 (n−1) 2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n G C2 (t))), . . . , Cn # » (n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}. # » And the right side of the equation = ¬bj T f cz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), # (n−1) » (n−1) # (n) » (n) (n−1) (n−1) (n−1) (n−1) pn (t), G An (t)))} = {A1 ((U1 (t), S (t), p1 (t), T1 (t), L 1 (t)), # (n−1) » 1 (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2(n−1) (t), # (n−1) » (n−1) # » p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) 2 2 A2 (t))), # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n−1) . . . , An # » # » (n−1) (n−1) (n−1) (n−1) ((μ(n−1) (t), pn(n−1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n−1) (t), n # » (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) T1(n−1) (t), L (n−1) 1 1 B1 (t))), B2 # » ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), 2 2 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), Tn (t), # (n−1) » n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ {C ((U L (n−1) n n n n n 1 1 Bn # » (n−1) (n−1) (n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), z (t) = f ((ω (t), 1 1 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), T2 (t), L 2 (t)), # (n−1) » 2 (n−1) 2 (n−1) (n−1) (n−1) z2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn(n−1) ((Un(n−1) (t),

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# » # » Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), z n(n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), n (n−1) G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.147 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f cz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A # (n) » (n) (n) (n) (n) (n) (n) (n) (t)))) = {A(n) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), 1 1 1 1 1 1 1 1 # (n) » # (n) » 1 (n) (n) (n) (n) (n) (n) p1 (t)), G (n) (t))), A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f 2(n) 2 2 2 2 2 2 2 A1 # » # » (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n) (n) (n) (n) (n) xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))}, where x (t) = x 1 (t) + x 2 (t)+, . . . , (n) (n) (n) (n) (n) (n−1) +xn (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn (t)}; suppose that A ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), (t))) is the (n − 1)th order error logical variable defined in domain U (n−1) (t) G (n−1) A (t) the rules for judging errors, T f cz (A(n−1) ((U (n−1) (t), S (n−1) (t), under G (n−1) # (n−1) » A (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n−1) = {A1 ((U1 # » # » (n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), 1 A1 (t))), A2 # » (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) L (n−1) 2 2 A2 (t))), . . . , An # » # » (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), n n (n−1) G (n−1) (t) = x1(n−1) (t) + x2(n−1) (t)+, . . . , +xn(n−1) (t), xi(n−1) (t) An (t)))}, where x (n−1) (n−1) ∈ {x1 (t), x2 (t), . . . , xn(n−1) (t)}; suppose that C (n)Az B(n−1) ((U (n)(n−1) (t), # (n)(n−1) » (n)(n−1) (n)(n−1) (t), p (t), T (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) S # (n)(n−1) » (t), p (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) ((U (n−1) (t), S (n−1) # (n−1) » # » (t), p (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n−1) A # (n) » # » (n)Az B(n−1) p (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), A (t))), and T f cz (C # » T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), # » G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), # » (n)(n−1) L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C2 ((U2 (t), S2 (t), p (t), T (t), L 2 (t)), z 2 # (n)(n−1) 2 » (n)(n−1)2 (n)(n−1) (n)(n−1) (n)(n−1) (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un(n)(n−1) (t), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) Sn (t), p (t), T (t), L n (t)), z n (t) = f n ((ωn(n)(n−1) # (n)(n−1) n» (n)(n−1)n (t), pn (t), G Cn (t)))}, where x (n)(n−1) (t) = x1(n)(n−1) (t) + x2(n)(n−1) (t)+, (n)(n−1) (t) ∈ {x1(n)(n−1) (t), x2(n)(n−1) (t), . . . , xn(n)(n−1) (t)}; sup. . . , +xn(n)(n−1) (t), xi

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pose that T f cz has carried out error value decomposition transformation on (U (n) (t), # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2,. . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), # » then the following relationship holds: T f cz (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bd (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ T f cz (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side of # » the equation = T f cz ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f cz (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), G (n) A (t))) ∧ C # » L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » # » T f cz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n−1) G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), A (t)))) ∧ T f cz (A # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f cz C (n)Az B(n−1) ((U (n)(n−1) A # » (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) # (n) » (n) # » (n) (n) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = {A(n) 1 ((U1 (t), S1 (t), p1 (t), T1 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) (n) L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # (n) » # » (n) (n−1) (n) (n) (n) (n) pn (t), Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn(n−1) x2 # (n−1) »2 # » (n−1) (n−1) (n−1) (n−1) (n−1) (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn(n−1) (t), # » (n)(n−1) G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), An (t)))} ∧ {C 1 # » (n)(n−1) L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), C2 # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) z2 (t) = f 2 ((ω2 (t), p (t), G C2 (t))), . . . , Cn(n)(n−1) # (n)(n−1) » 2 (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n(n)(n−1) (t) = # » (n)(n−1) (t)))}. f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn # » And the right side of the equation = ¬bd T f cz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2

4.2 Decomposition Transformation Connectives in Error Logic

311

# » # (n) » (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) p1 (t)), G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 # » # (n) » (n) (n) (n) (n) (n) (n) (n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n−1) » # » (n) (n−1) (n−1) (n−1) (n) ((U1 (t), S (t), p1 (t), xn(n) (t) = f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {A1 # (n−1) » 1(n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n−1) ((μ(n−1) (t), 2 # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), G A2 (t))) . . . An ((Un (t), Sn (t), pn (t), Tn (t), L n # (n−1) » (n−1) (n)(n−1) (n−1) (n−1) (n−1) (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {C1 ((U1(n)(n−1) (t), # » (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), L (n)(n−1) 1 # (n)(n−1) » # » (n)(n−1) (t), p1 (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), # » (n)(n−1) T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), G C2 2 # » (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), n # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t) = f n ((ωn (t), pn (t), G Cn (t)))}. zn Left side = right side. Proof is completed. # » Proposition 4.148 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , ¬bx An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bx if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 cz {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » bx ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side before being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))

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is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.149 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f cz (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 cz {A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), (n) G A (t))) is called the connotative uninterrupted negation on A (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side after being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.150 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f cz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » bz p1 (t)), G A1 (t))), ¬ A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bz G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » bz if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 cz {¬ A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then

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313

(1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative “not-only” negation on A(n) (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decom# » posed”; and the logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.151 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f cz (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 cz {A1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.69. Proof Proof is omitted.

4.2.8 Error Function Decomposition Transformation Connective in Error Logic # » # » Suppose that T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) # » # » = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An # » # » ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t), it is said that T f hs has conducted error function # » transformation on the object of interest μ(t, p(t)). For example, f (t) = sin(x) = x3 5 7 x x x + − +, . . . . 3! 5! 7!

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4.2.8.1

Conditions for Error Function Decomposition in Error Logic

1. The conditions for error function decomposition are: (1) (2) (3) (4) (5) (6)

fl

Legal conditions T Jhs ; kg Actual conditions T Jhs ; md ; Objective conditions (target) T Jhs sm Conditions for sustaining life T Jhs ; gj Technical conditions T Jhs ; nl . Energy conditions T Jhs

4.2.8.2

Principles for Error Function Decomposition in Error Logic

The principles for error function decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.8.3

Ways of Error Function Decomposition in Error Logic

2. Ways of error function decomposition: (1) Physical decomposition; (2) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs; (4) Decomposition transformation acting on property: # » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, f i (t) acting on Ti (t)i ∈ {1, 2, . . . , n}; (5) Decomposition transformation acting on domain:

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315

# » # » (a) in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), # » # » . . . . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, f i (t) acting on Ui (t)i ∈ {1, 2, . . . , n}; (6) Decomposing based on special needs and requirements.

4.2.8.4

Characteristics of Error Function Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f and the elements of the error logical vari# » # » able A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be changed if T f does not change its error function f ; for # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t), then T f hs is the error value decomposition transformation con# » nective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G(t))) defined in domain U (t), in this case, T f hs has conducted transformation # » on the element of T (t) in the object (μ(t), p(t)). Ways of error value decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f hs (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where f (t) = f 1 (t) + f 2 (t)+, # » / U (t), i ∈ {1, 2, . . . , . . . , + f n (t), if both (Ui (t), Si (t), pi (t), Ti (t), L i (t)) ∈ # » n} and (U j (t), S j (t), p j (t), T j (t), L j (t)) ∈ U (t), j ∈ {1, 2, . . . , n} exist, # » then it is said that T f hs has enabled A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f hs (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where f (t) = f 1 (t) + f 2 (t)+,

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# » . . . , + f n (t), if there exists (Ui (t), Si (t), pi (t), Ti (t), L i (t)) ∈ / U (t), i ∈ {1, 2, # » . . . , n}, then it is said that T f hs has enabled A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain, rules for judging errors, time, object, or error function, etc. T f hs ⊆ {T f ly , T f sw , T f k j , T f lz , T f lz , T f hs , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f hs (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where f (t) = f 1 (t) + f 2 (t)+, # » / U (t), i ∈ {1, 2, . . . , n}, then . . . , + f n (t), if ∀(Ui (t), Si (t), pi (t), Ti (t), L i (t)) ∈ it is said that T f hs has carry out domain displacement transformation on A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f hs (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where f (t) = f 1 (t) + f 2 (t)+, # » . . . , + f n (t), if ∀(Ui (t), Si (t), pi (t), Ti (t), L i (t)) ∈ U (t), i ∈ {1, 2, . . . , n}, and # » # » if (U (t), S(t), p(t), T (t), L(t)) and (Ui (t), Si (t), pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said that T f hs has conducted decom# » position transformation on (U (t), S(t), p(t), T (t), L(t)), and T f hs did not carry # » out decomposition transformation on (U (t), S(t), p(t), T (t), L(t)), otherwise. # » According to error theory, (Ui (t), Si (t), pi (t), Ti (t), L i (t)) can not be equal # » to (U j (t), S j (t), p j (t), T j (t), L j (t)) if xi (t) = x j (t). Therefore,T f hs possibly has conducted decomposition transformation on U (t), T (t), f , or G(t) in error # » # » logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))); # » # » if (Ui (t), Si (t), pi (t), Ti (t), L i (t)) and (U (t), S(t), p(t), T (t), L(t)) belong to different order (layer), then it is said T f hs has conducted decomposition trans# » # » formation on A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))). In general, as where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t), T f hs has possibly carried # » out transformation on S(t) or t in (U (t), S(t), p(t), T (t), L(t)). # » Proposition 4.152 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined

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in domain U (t) under G B (t) the rules for judging errors, T f hs (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f hs # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f hs # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f hs (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f hs (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side # » # » = T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f hs # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = # » f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.153 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the

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# » rules for judging errors, T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, T f hs (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); # » # » suppose that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is # » # » the mediator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » T f hs (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold: # » # » (1) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f hs (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » Az B f ((ν(t), p(t)), G B (t)))) = T f hs (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  0, the right # » # » side = T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » n Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)),

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# » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), #1 » 1 # » p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » # » # » pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} ∨ # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))} = # » # » {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), # » # » C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # » Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.154 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, T f hs (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); # » # » suppose that C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) # » is the connotative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t))), T f hs (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t). then the follow# » # » ing relationship holds: T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n T f hs # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted.

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# » Proposition 4.155 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, T f hs (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); # » # » suppose that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is # » # » the mediator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » T f hs (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 # » # » ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f hs (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted.

# » Proposition 4.156 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the

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321

# » rules for judging errors, T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, T f hs (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); # » # » suppose that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t))) is the connotative same or equivalence variable for A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))), T f hs (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # » # » f ((ω (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), # » # » #1 » 1 p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship holds: T f hs (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f hs (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f hs (B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted.

# » Proposition 4.157 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # (n) » (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (t) = f 1(n) (t) + L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where f # » f 2(n) (t)+, . . . , + f n(n) (t); suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » (t))) is the T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (t) (n + 1)th order error logical variable defined in domain U (n+1) (t) under G (n+1) A # (n+1) » (n+1) (n+1) (n+1) (n+1) the rules for judging errors, T f hs (A ((U (t), S (t), p (t), T (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) L (t)), x (t) = f ((μ (t), p (t)), G A (t)))) = {A1 # (n+1) » (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), 1

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# (n+1) » # » (n+1) p1 (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), 2 A1 (t))), A2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f 2 ((μ (t), p2 (t), G A2 (t))), . . . , An ((Un (t), x2 # (n+1) » 2(n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), (n+1) (n+1) (n+1) (n+1) G (n+1) (t)))}, where f (t) = f (t) + f (t)+, . . . , + f (t); suppose 1 2 An # n » that error logical variable B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) is the complement B # (n+1) » (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), x (n+1) (t) = variable for A # » (n+1) f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t))) which is defined in domain U (n+1) (t) under # » (t) the rules for judging errors, T f hs (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), G (n+1) B # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = {B1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S (t), p2 (t), # (n+1) »2 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G B2 (t))), . . . T2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n+1) (t) = Bn # » (n+1) (t), pn(n+1) (t), G Bn (t)))}, where f (n+1) (t) = f 1(n+1) (t) + f n(n+1) ((μ(n+1) n f 2(n+1) (t)+, . . . , + f n(n+1) (t); suppose that C (n+1)Az B ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » G C(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) and A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = B # » (n+1)Az B (n+1) (n+1) (t))), T (C ((U (t), S (t), f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) f hs B # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S (t), p1 (t), T1 (t), L 1 (t)), G C (t)))) = {C1 # (n+1) »1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) z1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2(n+1) (t), # (n+1) » (n+1) # » (n+1) p2 (t), T2 (t), L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), 2 # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = # » (n+1) (t)))}, where f (n+1) (t) = f 1(n+1) (t) + f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn f 2(n+1) (t)+, . . . , + f n(n+1) (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n+1) (t), y (n+1) (t), and z (n+1) (t), # » then the following relationship holds: T f hs (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bz (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ T f hs (A ((U # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f hs (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f hs (A

4.2 Decomposition Transformation Connectives in Error Logic

323

# (n+1) » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t))) ∧ B (n+1) A # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n+1) (t), # (n+1) » # » (n+1) p (t)), G B (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f hs (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f hs (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f hs (C (n+1)Az B B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n+1) ((μ(n+1) (t), # » # (n+1) » p (t)), G C(n+1) (t)))) = {A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 # » (n+1) L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), G (n+1) n A2 (t))), . . . , An # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {B1 ((U1 (t), xn # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), f2 # (n+1) »n (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n+1) (t), L (n+1) n n n n Bn # (n+1) » n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), # (n+1) »2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , # » 2 Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = n # » (n+1) (t)))}. f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn # » And the right side of the equation = ¬bz T f hs A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), 1 An (t)))} = {A1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), # (n+1) »2 (n+1) (n+1) (n+1) (n+1) (n+1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n+1) (t) = . . . , An # » # » (n+1) f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), n An (t)))} ∧ {B1 # » T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = B2

324

4 Transformation Connectives in Error Logic

# » # » (n+1) f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 B2 (t))), . . . , Bn # » Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n n Bn (t)))} ∧ # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1(n+1) (t) = {C1 # » # » (n+1) f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), # » (n+1) T2(n+1) (t), L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , 2 # » (n+1) Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z (t) = n n # (n+1) » (n+1) (n+1) (n+1) ((ωn (t), pn (t), G Cn (t)))}. fn Left side = right side. Proof is completed. # » Proposition 4.158 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, # » T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (t) = f 1(n) (t) + L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where f # » (n) f 2 (t)+, . . . , + f n(n) (t); suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » (t))) is the T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A th (n+1) (t) under G (n+1) (t) (n + 1) order error logical variable defined in domain U A # » the rules for judging errors, T f hs (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = {A(n+1) 1 A # » (n+1) (n+1) (n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x (t) = f ((μ (t), 1 1 1 1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t)), G A1 (t))), A2 ((U2 (t), S (t), p2 (t), T2 (t), L 2 (t)), # (n+1) » 2 (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un(n+1) (t), x2 # » # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) n n (n+1) G (n+1) (t) = f 1(n+1) (t) + f 2(n+1) (t)+, . . . , + f n(n+1) (t); it is An (t)))}, where f # » assumed that C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and A(n) ((U (n) (t), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), # » T f hs (C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), # » x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = {C1(n)(n+1) # » ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = 1

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325

# » (n)(n+1) f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) , S2(n)(n+1) (t), # (n)(n+1) » p (t), T2(n)(n+1) (t), L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), 2 # 2(n)(n+1) » # » (n)(n+1) p2 (t), G C2 (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), # » Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), n (n)(n+1) G Cn (t)))}, where f (n)(n+1) (t) = f 1(n)(n+1) (t) + f 2(n)(n+1) (t)+, . . . , + (n)(n+1) fn (t); suppose that T f hs has carried out error function decomposition transfor# » mation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order (n) (n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and of size for xi (t), xi z (n)(n+1) (t), then the following relationship holds: T f hs (¬bx A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bx p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f hs # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f hs (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f hs (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), G (n) A (t))) ∧ C # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » T f hs (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), A # » (n)Az B(n+1) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) A (t)))) ∧ T f hs (C # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = # ((n)n+1) » (n)(n+1) (n+1) (n+1) (n+1) (n)(n+1) (n)(n+1) ((μ (t), p (t)), G C (t)))) = {A1 ((U1 (t), S1 (t), f # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t), T1 (t), L 1 (t)), x1 (t) = f ((μ (t), p1 (t)), G A1 (t))), # (n+1) » 1 (n+1)1 (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2(n+1) (t) = A2 ((U2 # » # » (n+1) f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n An (t)))} ∧ # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1(n) (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # » (n) (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » # » (n) (n)(n+1) (n) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # » T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), 1 # » (n)(n+1) G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t),

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# » (n)(n+1) L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , 2 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » (n)(n+1) (t)))}. z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn # » And the right side of the equation = ¬bx T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p2 (t), A(n) 2 2 2 2 2 2 2 # 2(n) » 2 (n) (n) (n) (n) (n) (n) (n) G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = # » # (n) » (n+1) pn (t), G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), f n(n) ((μ(n) n (t), An (t)))} = {A1 # » T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n+1) A2 ((U2 # » # » (n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n An (t)))} ∧ # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1(n) (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # (n)(n+1) » # » (n) (n)(n+1) (n)(n+1) (n)(n+1) (n) ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) L2 (t)), z 2 (t) = f 2 ((ω2 (t), p (t), G (t))), . . . , # (n)(n+1) » 2 (n)(n+1) C2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), T (t), L n (t)), # » (n)(n+1)n (t)))}. z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn Left side = right side. Proof is completed. Proposition 4.159 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging # » errors, T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # (n) » (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (t) = f 1(n) (t) + L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where f f 2(n) (t)+, . . . , + f n(n) (t); # » suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logA

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ical variable defined in domain U (n−1) (t) under G (n−1) (t) the rules for judging errors, # (n−1) » A(n−1) (n−1) (n−1) (n−1) T f hs (A ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = # » (n−1) (n−1) f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t)))) = {A1 ((U1(n−1) (t), S1(n−1) (t), # » # (n−1) » (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), p1 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) n n An (t)))}, where (n−1) (n−1) (n−1) (n−1) f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t). # » (n−1) (n−1) (n−1) ((U (t), S (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), Suppose that B # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » logical complementary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (n−1) (t) under G B (t) the rules for judging errors, T f hs (B (n−1) ((U (n−1) (t), U # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), G (n−1) 1 B # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), x1 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G B2 (t))), # (n−1) »2 (n−1) (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = . . . , Bn # » (t), pn(n−1) (t), G (n−1) where f (n−1) (t) = f 1(n−1) (t) + f n(n−1) ((μ(n−1) n Bn (t)))}, (n−1) (n−1) f2 (t)+, . . . , + f n (t). # » (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), Suppose that C # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (t))), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # » T f hs (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {C1(n−1) ((U1(n−1) (t), S1(n−1) (t), # » # (n−1) » (t), T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), p1 1 # » (n−1) G C1 (t))), C2(n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), z 2(n−1) (t) = 2 # » # » (n−1) f 2(n−1) ((ω2(n−1) (t), p2(n−1) (t), G C2 (t))) . . . Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), # » (n−1) Tn(n−1) (t), L (n−1) (t)), z n(n−1) (t) = f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}, where n f (n−1) (t) = f 1(n−1) (t) + f 2(n−1) (t)+, . . . , + f n(n−1) (t); suppose that T f hs has carried out error function decomposition transformation on # » (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following relationship holds: T f hs (¬bj A(n) ((U (n) (t), S (n) (t),

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# (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bj (n) (n) (n) (n) (n) (n) ¬ T f hs (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f hs ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f hs (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = T f hs (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f hs (B (n−1) ((U (n−1) (t), A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) ∧ T f hs (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # (n−1) » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (t)), G C(n−1) (t)))) = {A(n−1) ((U1(n−1) (t), 1 # » # » S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # (n−1) » (n−1) (n−1) (n−1) (n−1) G (n−1) (t))), A ((U (t), S (t), p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = 2 2 2 2 A1 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) n n An (t)))} ∧ # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = {B1 # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 B2 (t))), . . . , # » (t)), xn(n−1) (t) = Bn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n # » # » (n−1) (n−1) (n−1) (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G Bn (t)))} ∧ {C1 ((U1 (t), S1 (t), p1(n−1) (t), n # » (n−1) T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = C2 # (n−1) » (n−1) 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) Tn(n−1) (t), L (n−1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn # » And the right side of the equation = ¬bj T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # (n−1) » # » (n) (n−1) (n−1) (n−1) pn(n) (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t),

4.2 Decomposition Transformation Connectives in Error Logic

329

# » x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 A1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n−1) » (n−1)2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n−1) »n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ Tn # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 B2 (t))), . . . , # » (t)), xn(n−1) (t) = Bn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n # » # » (n−1) (n−1) (n−1) (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G Bn (t)))} ∧ {C1 ((U1 (t), S1 (t), p1(n−1) (t), n # » (n−1) T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) Tn(n−1) (t), L (n−1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn Left side = right side. Proof is completed. T1(n−1) (t),

L (n−1) (t)), 1

# » Proposition 4.160 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, # » T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) = x 1 (t) + (n) (n) (n) (n) x2 (t)+, . . . , +xn(n) (t), xi (t) ∈ {x1 (t), x2 (t), . . . , xn(n) (t)}; suppose that # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » (t))) is the (n − 1)th order error logical varif (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A (n−1) (t) under G (n−1) (t) the rules for judging errors, able defined in domain U # (n−1) »A (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = T f hs (A # » (n−1) (n−1) f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t)))) = {A1 ((U1(n−1) (t), S1(n−1) (t), # (n−1) » # » p1 (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))}, where (n−1) (n−1) (n−1) (n−1) f (t) = f 1 (t) + f 2 (t)+, . . . , + f n (t); suppose that C (n)Az B(n−1) # » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) =

330

4 Transformation Connectives in Error Logic

# » f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (n)Az B(n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), and T f hs (C # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((μ (t), p (t)), G C (t)))) = {C1 ((U1 (t), f # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) S1 (t), p1 (t), T (t), L 1 (t)), z 1 (t) = f 1 # (n)(n−1) » 1 (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2(n)(n−1) (t), # (n)(n−1) » p (t), T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), 2 # 2(n)(n−1) » # » (n)(n−1) p2 (t), G C2 (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), # » Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), n (n)(n−1) G Cn (t)))}, where f (n)(n−1) (t) = f 1(n)(n−1) (t) + f 2(n)(n−1) (t)+, . . . , + (n)(n−1) (t); suppose that T f hs has carried out error function decomposition transforfn # » mation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), then the following relationship holds: T f hs (¬bd A(n) ((U (n) (t), S (n) (t), # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) bd (n) (n) (n) (n) (n) (n) ¬ T f hs (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side of # » the equation = T f hs ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f hs (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), G (n) A (t))) ∧ C # » L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » T f hs (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n−1) p (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), A (t)))) ∧ T f hs (A # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f hs C (n)Az B(n−1) A # » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = # (n)(n−1) » (n) (n) p (t)), G C(n)(n−1) (t)))) = {A(n) f (n)(n−1) ((μ(n)(n−1) (t), 1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n−1) G (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), 1 An (t)))} ∧ {A1

4.2 Decomposition Transformation Connectives in Error Logic

331

# » (n−1) x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), 1 A1 (t))), A2 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (t), T2 (t), L (n−1) (t)), x (t) = f ((μ (t), p2 (t), G A2 (t))), p2 2 2 2 # (n−1) »2 (n−1) (n−1) (n−1) (n−1) (n−1) . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = # (n−1) » (n−1) (n)(n−1) (n)(n−1) (n−1) (n−1) ((μn (t), pn (t), G An (t)))} ∧ {C1 ((U1 (t), S1(n)(n−1) (t), f # n(n)(n−1) » (t), T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p 1 # 1(n)(n−1) » # » (n)(n−1) p1 (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), # » T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), 2 # » (n)(n−1) (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), G C2 # » (n)(n−1) L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn (t)))}. n # » bd (n) (n) (n) And the right side of the equation = ¬ T f hs (A ((U (t), S (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) f n(n) ((μ(n) n (t), pn (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # (n) » # » (n) (n−1) (n) (n) (n) (n) pn (t), Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))), A2 ((U2 (t), S (t), p2 (t), T2 (t), L 2 (t)), # (n−1) »2 (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))) . . . An ((Un(n−1) (t), x2 # » # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n n # » (n)(n−1) G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), An (t)))} ∧ {C 1 # » (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), L (n)(n−1) 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) z2 (t) = f 2 ((ω2 (t), p (t), G C2 (t))), . . . , Cn # (n)(n−1) » 2 (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n(n)(n−1) (t) = # » (n)(n−1) (t)))}. f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn Left side = right side. Proof is completed. # » Proposition 4.161 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t),

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# » # » G A2 (t))), . . . , ¬bx An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 hs # » # » {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bx An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side before being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.162 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f hs (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » T f−1 hs {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), (n) G A (t))) is called the connotative uninterrupted negation on A (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side after being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))

4.2 Decomposition Transformation Connectives in Error Logic

333

is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.163 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f hs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » bz T f−1 hs {¬ A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) (n) is called the connotative “not-only” negation on A (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decomposed”; and the # » # » logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.164 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) # » the rules for judging errors; if T f hs (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » T f−1 hs {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then

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(1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.82. Proof Proof is omitted.

4.2.9 Time Decomposition Transformation Connective in Error Logic # » # » Suppose that T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn , it is said that T f s j has conducted # » error function transformation on the object of interest μ(t, p(t)). For example,in order to develop strategic economic plan over long course,China uses Five-Year Plan to strategize the development for sectors of industry,agriculture,and service.

4.2.9.1

Principles for Time Decomposition in Error Logic

The principles for time decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2.9.2

Conditions for Time Decomposition in Error Logic

1. The conditions for error function decomposition are: fl

(1) Legal conditions T Js j ; kg

(2) Actual conditions T Js j ; (3) Objective conditions (target) T Jsmd j ; (4) Conditions for sustaining life T Jssm j ; gj

(5) Technical conditions T Js j ; (6) Energy conditions T Jsnlj .

4.2 Decomposition Transformation Connectives in Error Logic

4.2.9.3

335

Ways of Time Decomposition in Error Logic

2. Ways of time decomposition: (1) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (2) Decomposition based on actual needs; (3) Equal division: # » # » (a) in {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), # » # » p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = # » f n ((μn (tn ), pn (tn ), G An (tn )))}, if ti = t j ; i, j ∈ {1, 2, . . . , n}; (4) Unequal division:

# » # » (a) in {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), # » # » p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = # » f n ((μn (tn ), pn (tn ), G An (tn )))}, if ti = t j ; i, j ∈ {1, 2, . . . , n};

(5) Decomposing based on special needs and requirements.

4.2.9.4

Characteristics of Time Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f and the elements of the error logical variable A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be # » changed if T f does not change its error function f ; for T f s j (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), # » # » L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » # » L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ),

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# » Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn , then T f s j is the time decomposition transformation connective with respect to # » # » G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in domain U (t), in this case, T f s j has conducted transformation on the element of t in # » the object (μ(t), p(t)). Ways of time decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f s j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), # » # » p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), # » # » S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , # » # » An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), # » G An (tn )))}, where t = t1 + t2 +, . . . , +tn , if both (Ui (ti ), Si (ti ), pi (ti ), Ti (ti ), # » / U (t), i ∈ {1, 2, . . . , n} and (U j (t j ), S j (t j ), p j (t j ), T j (t j ), L j (t j )) ∈ L i (ti )) ∈ U (t), j ∈ {1, 2, . . . , n} exist, then it is said that T f s j has enabled A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f s j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), # » # » p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), # » # » S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , # » # » An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), # » G An (tn )))}, where t = t1 + t2 +, . . . , +tn , if there exists (Ui (t), Si (t), pi (t), / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f s j has enabled Ti (t), L i (t)) ∈ # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain,rules for judging errors, time, object, or error function, etc. T f s j ⊆ {T f ly , T f sw , T f k j , T f lz , T f lz , T f s j , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f s j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), # » # » p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), # » # » S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , # » # » An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), # » G An (tn )))}, where t = t1 + t2 +, . . . , +tn , if ∀(Ui (t), Si (t), pi (t), Ti (t), L i (t)) ∈ / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f s j has carry out domain dis# » placement transformation on A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f s j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), # » # » p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), # » # » S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . ,

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# » # » An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), # » G An (tn )))}, where t = t1 + t2 +, . . . , +tn , if ∀(Ui (t), Si (t), pi (t), Ti (t), L i (t)) ∈ # » U (t), i ∈ {1, 2, . . . , n}, and if (U (t), S(t), p(t), T (t), L(t)) and (Ui (t), Si (t), # » pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said that T f s j # » has conducted decomposition transformation on (U (t), S(t), p(t), T (t), L(t)), # » and T f s j did not carry out decomposition transformation on (U (t), S(t), p(t), # » T (t), L(t)), otherwise. According to error theory, (Ui (ti ), Si (ti ), pi (ti ), Ti (ti ), # » L i (ti )) can not be equal to (U j (t j ), S j (t j ), p j (t j ), T j (t j ), L j (t j )) if xi (t) = x j (t). Therefore, T f s j possibly has conducted decomposition transformation on U (t), # » T (t), f , or G(t) in error logical variable A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G(t))); if (Ui (t), Si (t), pi (t), Ti (t), L i (t)) and (U (t), # » S(t), p(t), T (t), L(t)) belong to different order (layer), then it is said T f s j has # » conducted decomposition transformation on A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))). In general, as where t = t1 + t2 +, . . . , +tn , T f s j # » has possibly carried out transformation on S(t) or t in (U (t), S(t), p(t), T (t), # » L(t)). For example, ((U (t), grain yield (1979-2019), p(t), growth, 332 million # » metric tons), 100) = {((U (t), grain yield (1979-1989), p(t), growth, 62 million # » metric tons), 18.68), ((U (t), grain yield (1989-1999), p(t), growth, 145 million # » metric tons), 43.67), ((U (t), grain yield (1999-2009), p(t), growth, −31 million # » metric tons), −9.33), ((U (t), grain yield (2009-2019), p(t), growth, 156 million metric tons), 46.98)}. Where [1979, 2019] = {[1979, 1989], [1989, 1999], [1999, 2009], [2009, 2019], 332 = 62 + 145 − 31 + 156, 100 = 18.68 + 43.67 − 9.33 + 156.} # » Proposition 4.165 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that another error logical # » # » variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in # » domain U (t) under G B (t) the rules for judging errors, T f s j (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), # » # » y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # » y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ), G Bn (tn )))}, where t = t1 + t2 +, . . . , +tn ; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds,then the following relationships hold: # » # » (1) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))));

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# » # » (2) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f s j (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f s j (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » pn (tn ), G An (tn )))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  # » yi (t), the right side = T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) ∨ T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » # » p (t ), G (t )))} ∨ {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), y1 (t1 ) = f 1 ((ν1 (t1 ), # n n» An n # » p (t )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), # » # 1 1» p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), yn (tn ) = # » # » f n ((νn (tn ), pn (tn ), G Bn (tn )))} = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = # » # » f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = # » # » f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # » xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.166 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), # » # » A p (t )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), # » # 1 1» p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = # » f n ((μn (tn ), pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that another # » # » error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), # » # » B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , # » # » Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ),

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# » G Bn (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable of A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))), T f s j (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = # » # » f 1 ((ω1 (t1 ), p1 (t1 )), G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = # » # » f 2 ((ω2 (t2 ), p2 (t2 ), G C2 (t2 ))), . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # » z n (tn ) = f n ((ωn (tn ), pn (tn ), G Cn (tn )))}, where t = t1 + t2 +, . . . , +tn ; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds,then the following relationships hold: # » # » (1) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f s j (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = T f s j (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = # » # » f 1 ((ω1 (t1 ), p1 (t1 )), G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = # » # » f 2 ((ω2 (t2 ), p2 (t2 ), G C2 (t2 ))), . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # » z n (tn ) = f n ((ωn (tn ), pn (tn ), G Cn (tn )))}; And from the assumption of ∀i, i ∈ # » {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  0, the right side = T f s j (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n T f s j (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), # » # » x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # » x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » # » L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))} ∨n {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), # » # » T1 (t1 ), L 1 (t1 )), y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), # » T2 (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), # » # » pn (tn ), Tn (tn ), L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ), G Bn (tn )))} = {A1 ((U1 (t1 ), # » # » S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), # » # » S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , # » # » An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))} ∨ # » # » {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), # » # » B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , # » # » Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ), # » # » G Bn (tn )))} ∨ {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = f 1 ((ω1 (t1 ), p1 (t1 )), # » # » G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = f 2 ((ω2 (t2 ), p2 (t2 ),

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# » G C2 (t2 ))), . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), z n (tn ) = f n ((ωn (tn ), # » pn (tn ), G Cn (tn )))}. Left side = right side. Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.167 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that another error logical # » # » variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in # » domain U (t) under G B (t) the rules for judging errors, T f s j (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), # » # » y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # » y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ), G Bn (tn )))}, where t = t1 + t2 +, . . . , +tn ; sup# » # » pose that C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) # » is the connotative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), p(t)), G B (t))), T f s j (C # » # » G C (t)))) = {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = f 1 ((ω1 (t1 ), p1 (t1 )), # » # » G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = f 2 ((ω2 (t2 ), p2 (t2 ), # » G C2 (t2 ))), . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), z n (tn ) = f n ((ωn (tn ), # » pn (tn ), G Cn (tn )))}, where t = t1 + t2 +, . . . , +tn ; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t). # » then the following relationship holds: T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) # » # » −n T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.168 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that another error logical # » # » variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in # » domain U (t) under G B (t) the rules for judging errors, T f s j (B((U (t), S(t), p(t), T (t),

4.2 Decomposition Transformation Connectives in Error Logic

341

# » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), # » # » y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # » y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ), G Bn (tn )))}, where t = t1 + t2 +, . . . , +tn ; sup# » # » pose that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is # » # » the mediator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » T f s j (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = f 1 ((ω1 (t1 ), p1 (t1 )), G C1 (t1 ))), # » # » C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = f 2 ((ω2 (t2 ), p2 (t2 ), G C2 (t2 ))), . . . , # » # » Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), z n (tn ) = f n ((ωn (tn ), pn (tn ), G Cn (tn )))}, where t = t1 + t2 +, . . . , +tn ; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f s j (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.169 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), # » # » G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), # » G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # » pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that another error logical # » # » variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in # » domain U (t) under G B (t) the rules for judging errors, T f s j (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )),

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# » # » y1 (t1 ) = f 1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # » y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » L n (tn )), yn (tn ) = f n ((νn (tn ), pn (tn ), G Bn (tn )))}, where t = t1 + t2 +, . . . , +tn ; sup# » # » pose that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) # » is the connotative same or equivalence variable for A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))), T f s j (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = # » # » f 1 ((ω1 (t1 ), p1 (t1 )), G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = # » # » f 2 ((ω2 (t2 ), p2 (t2 ), G C2 (t2 ))), . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # » z n (tn ) = f n ((ωn (tn ), pn (tn ), G Cn (tn )))}, where t = t1 + t2 +, . . . , +tn ; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship # » # » holds: T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f s j (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f s j (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.170 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » # » T f s j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (n) (n) (n) (n) (n) G (n) A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = # » # » (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t2 )), x 2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # » # (n) » (n) (n) (n) (n) (n) (n) pn (tn ), Tn (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, where t = # » (n+1) (n+1) (n+1) t1 + t2 +, . . . , +tn ; suppose that A ((U (t), S (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) is the (n + 1)th A (n+1) order error logical variable defined in domain U (t) under G (n+1) (t) the rules A # » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), for judging errors, T f s j (A # » (n+1) (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = {A1 ((U1(n+1) (t1 ), # » (t1 )), x1(n+1) (t1 ) = f 1(n+1) ((μ(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) 1 1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An L 2 (t2 )), x2 # » ((Un(n+1) (tn ), Sn(n+1) (tn ), pn(n+1) (tn ), Tn(n+1) (tn ), L (n+1) (tn )), xn(n+1) (tn ) = f n(n+1) n # (n+1) » (n+1) (n+1) ((μn (tn ), pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that # » error logical variable B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) is the complement variable for B

4.2 Decomposition Transformation Connectives in Error Logic

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# » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » (t))), which is defined in domain U (n+1) (t) f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) under G B (t) the rules for judging errors, T f s j (B (n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » G (n+1) (t)))) = {B1(n+1) ((U1(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) (t1 )), 1 B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t1 ) = f 1 ((ν (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), y1 # (n+1) »1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), S2 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (tn ) = f n ((νn (tn ), pn (tn ), G Bn (tn )))}, where L n (tn )), yn # » (n+1)Az B t = t1 + t2 +, . . . , +tn ; suppose that C ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and B (n+1) ((U (n+1) (t), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))), T f s j (C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) B # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = {C1(n+1) ((U1(n+1) (t1 ), # » (t1 )), z 1(n+1) (t1 ) = f 1(n+1) ((ω1(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) 1 # (n+1) » # » (n+1) p1 (t1 )), G C1 (t1 ))), C2(n+1) ((U2(n+1) (t2 ), S2(n+1) (t2 ), p2(n+1) (t2 ), T2(n+1) (t2 ), # (n+1) » (n+1) L (n+1) (t2 )), z 2(n+1) (t2 ) = f 2(n+1) ((ω2(n+1) (t2 ), p2 (t2 ), G C2 (t2 ))), . . . , 2 # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), z n (tn ) = # (n+1) » (n+1) (n+1) (n+1) ((ωn (tn ), pn (tn ), G Cn (tn )))}, where t = t1 + t2 +, . . . , +tn ; if ∀i, fn i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n) (t), y (n) (t), and z (n) (t), then the following relationship holds: # » T f s j (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bz (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ T f s j (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f s j (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f s j (A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) A # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n+1) (t), # (n+1) » # » (n+1) p (t)), G B (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f s j (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f s j (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A

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# » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f s j (C (n+1)Az B B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n+1) ((μ(n+1) (t), # » # (n+1) » p (t)), G C(n+1) (t)))) = {A(n+1) ((U1(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), 1 # » L (n+1) (t1 )), x1(n+1) (t1 ) = f 1(n+1) ((μ(n+1) (t1 ), p1(n+1) (t1 )), G (n+1) A(n+1) 1 1 2 A1 (t1 ))), # » (n+1) (n+1) ((U2(n+1) (t2 ), S2(n+1) (t2 ), p2(n+1) (t2 ), T2(n+1) (t2 ), L (n+1) (t )), x (t ) = f 2 2 2 2 2 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) ((μ(n+1) (t ), p (t ), G (t ))), . . . , A ((U (t ), S (t ), p (tn ), 2 2 2 n n n n 2 2 A2 # (n+1) n » (n+1)n (n+1) (n+1) (n+1) (n+1) (n+1) (tn ), L n (tn )), xn (tn ) = f n ((μ (tn ), pn (tn ), G An (tn )))} ∧ Tn # (n+1) » n(n+1) (n+1) (n+1) (n+1) (n+1) {B1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), y1(n+1) (t1 ) = # » (n+1) f (n+1) ((ν (n+1) (t1 ), p1(n+1) (t1 )), G (n+1) ((U2(n+1) (t2 ), S2(n+1) (t2 ), B1 (t1 ))), B2 # 1(n+1) »1 # » p2 (t2 ), T2(n+1) (t2 ), L (n+1) (t2 )), y2(n+1) (t2 ) = f 2(n+1) ((ν2(n+1) (t2 ), p2(n+1) (t2 ), 2 # » (n+1) ((Un(n+1) (tn ), Sn(n+1) (tn ), pn(n+1) (tn ), Tn(n+1) (tn ), L (n+1) (tn )), G (n+1) n B2 (t2 ))), . . . , Bn # » (n+1) (n+1) (t )))} ∧ {C ((U (t1 ), yn(n+1) (tn ) = f n(n+1) ((νn(n+1) (tn ), pn(n+1) (tn ), G (n+1) n 1 1 Bn # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = f 1 ((ω1 (t1 ), # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t1 )), G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) L 2 (t2 )), z 2 (t2 ) = f 2 ((ω2 (t2 ), p2 (t2 ), G C2 (t2 ))), . . . , Cn # » ((Un(n+1) (tn ), Sn(n+1) (tn ), pn(n+1) (tn ), Tn(n+1) (tn ), L (n+1) (tn )), z n(n+1) (tn ) = n # » (n+1) (tn )))}. f n(n+1) ((ωn(n+1) (tn ), pn(n+1) (tn ), G Cn # » And the right side of the equation = ¬bz T f s j A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) (t1 )), 1 An (t)))} = {A1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t1 ) = f 1 ((μ (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), x1 # (n+1) » 1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), S2 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t )), x (t ) = f ((μ (t ), p (t ), G (t )))} ∧ {B L (n+1) n n n n n n n n n 1 An # »n (n+1) (n+1) ((U1(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) (t )), y (t ) = f 1 1 1 1 1 # » # » (n+1) ((ν1(n+1) (t1 ), p1(n+1) (t1 )), G (n+1) ((U2(n+1) (t2 ), S2(n+1) (t2 ), p2(n+1) (t2 ), B1 (t1 ))), B2 # » (t2 )), y2(n+1) (t2 ) = f 2(n+1) ((ν2(n+1) (t2 ), p2(n+1) (t2 ), G (n+1) T2(n+1) (t2 ), L (n+1) 2 B2 (t2 ))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), yn(n+1) (tn ) = Bn # » (n+1) ((U1(n+1) (t1 ), S1(n+1) (t1 ), f (n+1) ((ν (n+1) (tn ), pn(n+1) (tn ), G (n+1) Bn (tn )))} ∧ {C 1 # n(n+1) »n # » p1 (t1 ), T1(n+1) (t1 ), L (n+1) (t1 )), z 1(n+1) (t1 ) = f 1(n+1) ((ω1(n+1) (t1 ), p1(n+1) (t1 )), 1 # » (n+1) G C1 (t1 ))), C2(n+1) ((U2(n+1) (t2 ), S2(n+1) (t2 ), p2(n+1) (t2 ), T2(n+1) (t2 ), L (n+1) (t2 )), 2

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# » (n+1) z 2(n+1) (t2 ) = f 2(n+1) ((ω2(n+1) (t2 ), p2(n+1) (t2 ), G C2 (t2 ))), . . . , Cn(n+1) ((Un(n+1) (tn ), # » Sn(n+1) (tn ), pn(n+1) (tn ), Tn(n+1) (tn ), L (n+1) (tn )), z n(n+1) (tn ) = f n(n+1) ((ωn(n+1) (tn ), n # (n+1) » (n+1) pn (tn ), G Cn (tn )))}. Left side = right side. Proof is completed. # » Proposition 4.171 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f s j (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (n) (n) (n) (n) (n) G (n) A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = # » # » (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t2 )), x 2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # (n) » (n) # » (n) (n) (n) (n) (n) pn (tn ), Tn (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, where t = # » (n+1) (n+1) (n+1) t1 + t2 +, . . . , +tn ; suppose that A ((U (t), S (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) is the (n + 1)th A (t) the rules order error logical variable defined in domain U (n+1) (t) under G (n+1) A # (n+1) » (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), for judging errors, T f s j (A # » (n+1) (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = {A1 ((U1(n+1) (t1 ), # » (t1 )), x1(n+1) (t1 ) = f 1(n+1) ((μ(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) 1 1 # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t2 ) = f 2 ((μ (t2 ), p (t2 ), G A2 (t2 ))), . . . , An L 2 (t2 )), x2 # (n+1) »2 (n+1) 2 (n+1) (n+1) ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L (n+1) (tn )), xn(n+1) (tn ) = f n(n+1) n # (n+1) » (n+1) (n+1) ((μn (tn ), pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; it is assumed # » (n)Az B(n+1) that C ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), # » x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the mediator # » variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » (n)Az B(n+1) ((U (n)(n+1) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), T f s j (C # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) S (t), p (t), T (t), L (t)), x (t) = f ((μ(n)(n+1) # » # (n)(n+1) » (t), p (t)), G C(n)(n+1) (t)))) = {C1(n)(n+1) ((U1(n)(n+1) (t1 ), S1(n)(n+1) (t1 ), p1(n)(n+1) (t1 ), # » T1(n)(n+1) (t1 ), L (n)(n+1) (t1 )), z 1(n)(n+1) (t1 ) = f 1(n)(n+1) ((ω1(n)(n+1) (t1 ), p1(n)(n+1) (t1 )), 1 # » (n)(n+1) G C1 (t1 ))), C2(n)(n+1) ((U2(n)(n+1) (t2 ), S2(n)(n+1) (t2 ), p2(n)(n+1) (t2 ), T2(n)(n+1) (t2 ), # » (n)(n+1) (t2 )), z 2(n)(n+1) (t2 ) = f 2(n)(n+1) ((ω2(n)(n+1) (t2 ), p2(n)(n+1) (t2 ), G C2 (t2 ))), L (n)(n+1) 2 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (tn ) = f n ((ωn (tn ), pn (tn ), G Cn (tn )))}, where t = t1 + zn t2 +, . . . , +tn ; suppose that T f s j has carried out temporal decomposition transforma-

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# » tion on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the following relationship holds: T f s j (¬bx A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bx (n) p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ T f s j (A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f s j (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f s j (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # (n)(n+1) » (n)Az B(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t), S (t), p (t), T (t), L (t)), ∧C # » x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = T f s j (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f s j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), A # » (n)Az B(n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) ((U (n)(n+1) A (t)))) ∧ T f s j (C # » (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) # (n) » # » (n) (n) ((μ(n)(n+1) (t), p ((n)n+1) (t)), G C(n)(n+1) (t)))) = {A(n) 1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), # (n) » (n) (n) (n) (n) (n) (n) T1(n) (t1 ), L (n) 1 (t1 )), x 1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), # » # » (n) (n) (n) (n) (n) S2(n) (t2 ), p2(n) (t2 ), T2(n) (t2 ), L (n) 2 (t2 )), x 2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), # » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), x n (tn ) = f n ((μn (tn ), # (n+1) » # (n) » (n) (n+1) (n+1) (n+1) (n+1) pn (tn ), G An (tn )))} ∧ {A1 ((U1 (t1 ), S1 (t1 ), p (t1 ), T1 (t1 ), # (n+1) » 1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) L 1 (t1 )), x1 (t1 ) = f 1 ((μ (t1 ), p1 (t1 )), G A1 (t1 ))), A2 # (n+1) » 1 (n+1) (n+1) (n+1) (n+1) ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2(n+1) (t2 ) = f 2(n+1) # » # » (n+1) ((μ(n+1) (t2 ), p2(n+1) (t2 ), G (n+1) ((Un(n+1) (tn ), Sn(n+1) (tn ), pn(n+1) (tn ), 2 A2 (t2 ))), . . . , An # » (tn )), xn(n+1) (tn ) = f n(n+1) ((μ(n+1) (tn ), pn(n+1) (tn ), G (n+1) Tn(n+1) (tn ), L (n+1) n n An (tn )))} ∧ # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) {C1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L (n)(n+1) (t1 )), 1 # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) z1 (t1 ) = f 1 ((ω1 (t1 ), p (t1 )), G C1 (t1 ))), C2 # (n)(n+1) » 1 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2(n)(n+1) (t2 ) = # » (n)(n+1) f 2(n)(n+1) ((ω2(n)(n+1) (t2 ), p2(n)(n+1) (t2 ), G C2 (t2 ))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (tn ), # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), z n(n)(n+1) (tn ) = f n(n)(n+1) # » (n)(n+1) ((ωn(n)(n+1) (tn ), pn(n)(n+1) (tn ), G Cn (tn )))}. # » And the right side of the equation = ¬bx T f s j (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t),

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347

# » # (n) » (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p2 (t), A(n) 2 2 2 2 2 2 2 2 # (n) » 2 (n) (n) (n) (n) (n) (n) (n) (n) G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μ(n) n (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) pn (t), G An (t)))} = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = # (n) » # (n) » (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t2 )), x 2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # (n) » (n) # » (n) (n+1) (n) (n) (n) (n) pn (tn ), Tn (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))} ∧ {A1 # » ((U1(n+1) (t1 ), S1(n+1) (t1 ), p1(n+1) (t1 ), T1(n+1) (t1 ), L (n+1) (t1 )), x1(n+1) (t1 ) = f 1(n+1) 1 # » # » (n+1) ((μ(n+1) (t1 ), p1(n+1) (t1 )), G (n+1) ((U2(n+1) (t2 ), S2(n+1) (t2 ), p2(n+1) (t2 ), 1 A1 (t1 ))), A2 # » (t2 )), x2(n+1) (t2 ) = f 2(n+1) ((μ(n+1) (t2 ), p2(n+1) (t2 ), G (n+1) T2(n+1) (t2 ), L (n+1) 2 2 A2 (t2 ))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t ), S (t ), p (t ), T (t ), L (t )), x (tn ) = f n(n+1) A(n+1) n n n n n n n n n n n n # » (n)(n+1) ((μ(n+1) (tn ), pn(n+1) (tn ), G (n+1) ((U1(n)(n+1) (t1 ), S1(n)(n+1) (t1 ), n An (tn )))} ∧ {C 1 # (n)(n+1) » p (t1 ), T1(n)(n+1) (t1 ), L (n)(n+1) (t1 )), z 1(n)(n+1) (t1 ) = f 1(n)(n+1) ((ω1(n)(n+1) (t1 ), 1 # 1(n)(n+1) » # » (n)(n+1) p1 (t1 )), G C1 (t1 ))), C2(n)(n+1) ((U2(n)(n+1) (t2 ), S2(n)(n+1) (t2 ), p2(n)(n+1) (t2 ), # » T2(n)(n+1) (t2 ), L (n)(n+1) (t2 )), z 2(n)(n+1) (t2 ) = f 2(n)(n+1) ((ω2(n)(n+1) (t2 ), p2(n)(n+1) (t2 ), 2 # » (n)(n+1) (t2 ))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (tn ), Sn(n)(n+1) (tn ), pn(n)(n+1) (tn ), Tn(n)(n+1) (tn ), G C2 # » (n)(n+1) L (n)(n+1) (tn )), z n(n)(n+1) (tn ) = f n(n)(n+1) ((ωn(n)(n+1) (tn ), pn(n)(n+1) (tn ), G Cn (tn )))}. n Left side = right side. Proof is completed. Proposition 4.172 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging # » errors, T f s j (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = # » # » (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t2 )), x 2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # (n) » (n) # » (n) (n) (n) (n) (n) pn (tn ), Tn (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, where t = # » t1 + t2 +, . . . , +tn ; suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th A (t) the rules order error logical variable defined in domain U (n−1) (t) under G (n−1) A # (n−1) » (n−1) (n−1) (n−1) (n−1) for judging errors, T f s j (A ((U (t), S (t), p (t), T (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t1 ), 1 A # » (t1 )), x1(n−1) (t1 ) = f 1(n−1) ((μ(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An L 2 (t2 )), x2

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# » ((Un(n−1) (tn ), Sn(n−1) (tn ), pn(n−1) (tn ), Tn(n−1) (tn ), L (n−1) (tn )), xn(n−1) (tn ) = f n(n−1) n # (n−1) » (n−1) (n−1) ((μn (tn ), pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that # » (n−1) (n−1) B ((U (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logical compleB # » (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), mentary variable of A # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain U (n−1) (t) A (n−1) under G B (t) the rules for judging errors, T f s j (B (n−1) ((U (n−1) (t), S (n−1) (t), # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) B # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), y1 (t1 ) = = {B1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) f1 ((ν1 (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S (t2 ), p (t2 ), # (n+1) »2 (n+1) 2 (n−1) (n+1) (n+1) (n+1) (n+1) (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), p2 (t2 ), G B2 (t2 ))), . . . , T2 # (n−1) » (n−1) (n+1) (n−1) (n−1) (n−1) Bn ((Un (tn ), S (tn ), p (tn ), Tn (tn ), L n (tn )), yn(n−1) (tn ) = f n(n−1) # (n−1) n» (n−1) n (n−1) ((νn (tn ), pn (tn ), G Bn (tn )))}, where t = t1 + t2 +, . . . , +tn ; # » (n−1)Az B suppose that C ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))), T f s j T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # » (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {C1(n−1) ((U1(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), # » (n−1) T1(n−1) (t1 ), L (n−1) (t1 )), z 1(n−1) (t1 ) = f 1(n−1) ((ω1(n−1) (t1 ), p1(n−1) (t1 )), G C1 (t1 ))), 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = f 2(n−1) # » # » (n−1) ((ω2(n−1) (t2 ), p2(n−1) (t2 ), G C2 (t2 ))), . . . , Cn(n−1) ((Un(n−1) (tn ), Sn(n−1) (tn ), pn(n−1) (tn ), # » (n−1) Tn(n−1) (tn ), L (n−1) (tn )), z n(n−1) (tn ) = f n(n−1) ((ωn(n−1) (tn ), pn(n−1) (tn ), G Cn (tn )))}, n where t = t1 + t2 +, . . . , +tn ; suppose that T f s j has carried out temporal decom# » position transformation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following relationship holds: # » T f s j (¬bj A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bj (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ T f s j (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f s j ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f s j (A # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) (n−1) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n−1) (t)),

4.2 Decomposition Transformation Connectives in Error Logic

349

# » G (n−1) (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) A # » = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), B # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) # » = T f s j (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f s j (B (n−1) ((U (n−1) (t), S (n−1) (t), A # (n−1) » (n−1) # » (n−1) (n−1) p (t), T (t), L (t)), x (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = ∧ T f s j (C # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) f ((μ (t), p (t)), G C (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f ((μ (t1 ), p1 (t1 )), p1 # (n−1) »1 (n−1) 1 (n−1) (n−1) (n−1) (n−1) (n−1) G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2(n−1) # » (n−1) (t2 ) = f 2(n−1) ((μ(n−1) (t2 ), p2(n−1) (t2 ), G (n−1) ((Un(n−1) (tn ), Sn(n−1) A2 (t2 ))), . . . , An # (n−1) » 2(n−1) # » (tn ), pn (tn ), Tn (tn ), L (n−1) (tn )), xn(n−1) (tn ) = f n(n−1) ((μ(n−1) (tn ), pn(n−1) (tn ), n n # » (n−1) G (n−1) ((U1(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) (t1 )), 1 An (tn )))} ∧ {B1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) y1 (t1 ) = f 1 ((ν (t1 ), p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), # (n−1) »1 (n−1) (n−1) (n+1) (n−1) (n−1) (n−1) (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), y2 (t2 ) = f 2 ((ν2 (t2 ), S2 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) p2 (t2 ), G B2 (t2 ))), . . . , Bn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) L (n−1) (t )), y (t ) = f ((ν (t ), p (t ), G (t )))} ∧ {C n n n n n n n n 1 Bn #n »n (n−1) (n−1) ((U1(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) (t )), z (t ) = f 1 1 1 1 1 # » # » (n−1) ((ω1(n−1) (t1 ), p1(n−1) (t1 )), G C1 (t1 ))), C2(n−1) ((U2(n−1) (t2 ), S2(n−1) (t2 ), p2(n−1) (t2 ), # » (n−1) T2(n−1) (t2 ), L (n−1) (t2 )), z 2(n−1) (t2 ) = f 2(n−1) ((ω2(n−1) (t2 ), p2(n−1) (t2 ), G C2 (t2 ))), . . . , 2 # » (n−1) (n−1) Cn(n−1) ((Un(n−1) (tn ), Sn(n−1) (tn ), pn(n−1) (tn ), Tn(n−1) (tn ), L (n−1) (t )), z (t ) n n = fn n n # » (n−1) ((ωn(n−1) (tn ), pn(n−1) (tn ), G Cn (tn )))}. # » And the right side of the equation = ¬bj T f s j (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n−1) G (n) ((U1(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) (t1 )), x1(n−1) 1 An (t)))} = {A1 # » (n−1) (t1 ) = f 1(n−1) ((μ(n−1) (t1 ), p1(n−1) (t1 )), G (n−1) ((U2(n−1) (t2 ), S2(n−1) (t2 ), 1 A1 (t1 ))), A2 # (n−1) » # » p2 (t2 ), T2(n−1) (t2 ), L (n−1) (t2 )), x2(n−1) (t2 ) = f 2(n−1) ((μ(n−1) (t2 ), p2(n−1) (t2 ), 2 2 # » (n−1) ((Un(n−1) (tn ), Sn(n−1) (tn ), pn(n−1) (tn ), Tn(n−1) (tn ), L (n−1) (tn )), G (n−1) n A2 (t2 ))), . . . , An # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (tn ) = f n ((μ (tn ), pn (tn ), G An (tn )))} ∧ {B1 ((U1 (t1 ), xn # (n−1) » n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), y1 (t1 ) = f ((ν (t1 ), # (n−1) » # (n−1) » 1 (n−1) 1 (n+1) (n−1) (n−1) (n−1) (n−1) p1 (t1 )), G B1 (t1 ))), B2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2

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# » (n−1) (t2 )), y2(n−1) (t2 ) = f 2(n−1) ((ν2(n−1) (t2 ), p2(n−1) (t2 ), G (n−1) ((Un(n−1) B2 (t2 ))), . . . , Bn # » (tn ), Sn(n−1) (tn ), pn(n−1) (tn ), Tn(n−1) (tn ), L (n−1) (tn )), yn(n−1) (tn ) = f n(n−1) ((νn(n−1) (tn ), n # » # (n−1) » (n−1) (n−1) (n−1) pn (tn ), G Bn (tn )))} ∧ {C1 ((U1 (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), # » (n−1) (t1 )), z 1(n−1) (t1 ) = f 1(n−1) ((ω1(n−1) (t1 ), p1(n−1) (t1 )), G C1 (t1 ))), C2(n−1) ((U2(n−1) L (n−1) 1 # » (t2 ), S2(n−1) (t2 ), p2(n−1) (t2 ), T2(n−1) (t2 ), L (n−1) (t2 )), z 2(n−1) (t2 ) = f 2(n−1) ((ω2(n−1) (t2 ), 2 # (n−1) » (n−1) # » p2 (t2 ), G C2 (t2 ))), . . . , Cn(n−1) ((Un(n−1) (tn ), Sn(n−1) (tn ), pn(n−1) (tn ), Tn(n−1) (tn ), # » (n−1) L (n−1) (tn )), z n(n−1) (tn ) = f n(n−1) ((ωn(n−1) (tn ), pn(n−1) (tn ), G Cn (tn )))}. Left side = n right side. Proof is completed. # » Proposition 4.173 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f s j (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) G (n) A (t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = f 1 # » # » (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # » (n) (n) (n) (n) (n) (n) x2(n) (t2 ) = f 2(n) ((μ(n) 2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), # (n) » (n) (n) (n) (n) Tn(n) (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, where t = t1 + # » t2 +, . . . , +tn ; suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) # » (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order A (n−1) (t) under G (n−1) (t) the rules for error logical variable defined in domain U # (n−1) » A (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), judging errors, T f s j (A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) x (t) = f ((μ (t), p (t)), G A (t)))) = {A1 ((U1(n−1) (t1 ), # » (t1 )), x1(n−1) (t1 ) = f 1(n−1) ((μ(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) 1 # (n−1) » # (n−1) » (n−1) 1 (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S (t2 ), p (t2 ), T2 (t2 ), L 2 # (n−1) 2 » (n−1)2 (n−1) (n−1) (n−1) (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , A(n−1) ((Un(n−1) n # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (tn ), Sn (tn ), p (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), # (n−1) » (n−1)n pn (tn ), G An (tn )))}, where t = t1 + t2 +, . . . , +tn ; suppose that C (n)Az B(n−1) # » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = # » f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) # » ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), # (n−1) » # » p (t)), G (n−1) (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = A # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), and T f s j (C # (n)(n−1) » p (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), # » # (n)(n−1) » p (t)), G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t1 ), S1(n)(n−1) (t1 ), p1(n)(n−1) (t1 ), # » T1(n)(n−1) (t1 ), L (n)(n−1) (t1 )), z 1(n)(n−1) (t1 ) = f 1(n)(n−1) ((ω1(n)(n−1) (t1 ), p1(n)(n−1) (t1 )), 1

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# » (n)(n−1) G C1 (t1 ))), C2(n)(n−1) ((U2(n)(n−1) (t2 ), S2(n)(n−1) (t2 ), p2(n)(n−1) (t2 ), T2(n)(n−1) (t2 ), # » (n)(n−1) L (n)(n−1) (t2 )), z 2(n)(n−1) (t2 ) = f 2(n)(n−1) ((ω2(n)(n−1) (t2 ), p2(n)(n−1) (t2 ), G C2 (t2 ))), 2 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # » (n)(n−1) (tn )))}; suppose that z n(n)(n−1) (tn ) = f n(n)(n−1) ((ωn(n)(n−1) (tn ), pn(n)(n−1) (tn ), G Cn T f s j has carried out temporal decomposition transformation on (U (n) (t), S (n) (t), # (n) » (n) p (t), T (t), L (n) (t)), if ∀i, i ∈ {1, 2,. . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), then the fol# » lowing relationship holds: T f s j (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » bd (n) (n) x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t)))) = ¬ T f s j (A ((U # (n) » (n) # » (n) p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side of # » the equation = T f s j ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f s j (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A(n) A # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) # » ∧ C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), # » x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = T f s j (A(n) ((U (n) # » # » (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) ∧ # » T f s j (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f s j C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), A # (n)(n−1) » p (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), # (n) » (n) # (n)(n−1) » (n) (n) (n) p (t)), G C(n)(n−1) (t)))) = {A(n) 1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t1 ) = f 1(n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t2 )), x 2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # (n) » (n) # » (n) (n−1) (n) (n) (n) (n) pn (tn ), Tn (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))} ∧ {A1 # » ((U1(n−1) (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) (t1 )), x1(n−1) (t1 ) = f 1(n−1) 1 # » # » (n−1) ((μ(n−1) (t1 ), p1(n−1) (t1 )), G (n−1) ((U2(n−1) (t2 ), S2(n−1) (t2 ), p2(n−1) (t2 ), 1 A1 (t1 ))), A2 # » (t2 )), x2(n−1) (t2 ) = f 2(n−1) ((μ(n−1) (t2 ), p2(n−1) (t2 ), G (n−1) T2(n−1) (t2 ), L (n−1) 2 2 A2 (t2 ))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t ), S (t ), p (t ), T (t ), L (t )), x (tn ) = f n(n−1) A(n−1) n n n n n n n n n n n n # » (n)(n−1) ((μ(n−1) (tn ), pn(n−1) (tn ), G (n−1) ((U1(n)(n−1) (t1 ), S1(n)(n−1) (t1 ), n An (tn )))} ∧ {C 1 # (n)(n−1) » (t1 ), T1(n)(n−1) (t1 ), L (n)(n−1) (t1 )), z 1(n)(n−1) (t1 ) = f 1(n)(n−1) ((ω1(n)(n−1) (t1 ), p 1 # 1(n)(n−1) » # » (n)(n−1) p1 (t1 )), G C1 (t1 ))), C2(n)(n−1) ((U2(n)(n−1) (t2 ), S2(n)(n−1) (t2 ), p2(n)(n−1) (t2 ), # » T2(n)(n−1) (t2 ), L (n)(n−1) (t2 )), z 2(n)(n−1) (t2 ) = f 2(n)(n−1) ((ω2(n)(n−1) (t2 ), p2(n)(n−1) (t2 ), 2

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# » (n)(n−1) G C2 (t2 ))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (tn ), Sn(n)(n−1) (tn ), pn(n)(n−1) (tn ), Tn(n)(n−1) (tn ), # » (n)(n−1) L (n)(n−1) (tn )), z n(n)(n−1) (tn ) = f n(n)(n−1) ((ωn(n)(n−1) (tn ), pn(n)(n−1) (tn ), G Cn (tn )))}. n # » bd (n) (n) (n) And the right side of the equation = ¬ T f s j (A ((U (t), S (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 # » # » (n) (n) (n) (n) (n) ((U2(n) (t), S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) . . . , A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # (n) » (n) (n) (n) (n) (n) (n) (n) G (n) An (t)))} = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = f 1 # » # » (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 1 (t1 ), p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » # (n) » (n) (n) (n) (n) (n) x2(n) (t2 ) = f 2(n) ((μ(n) 2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), # » (n) (n−1) (n) (n) (n) (n) ((U1(n−1) Tn(n) (tn ), L (n) n (tn )), x n (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))} ∧ {A1 # » (t1 ), S1(n−1) (t1 ), p1(n−1) (t1 ), T1(n−1) (t1 ), L (n−1) (t1 )), x1(n−1) (t1 ) = f 1(n−1) ((μ(n−1) (t1 ), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (t2 ) = f 2 ((μ (t2 ), p (t2 ), G A2 (t2 ))), . . . , A(n−1) L 2 (t2 )), x2 n # (n−1) »2 (n−1) 2 (n−1) (n−1) (n−1) (n−1) ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L (n−1) (t )), x (t ) = f n n n n n # (n−1) » (n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((μ(n−1) (t ), p (t ), G (t )))} ∧ {C ((U (t ), S (t1 ), n n n 1 n n 1 1 1 An # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) p (t1 ), T1 (t1 ), L 1 (t1 )), z 1 (t1 ) = f 1 ((ω1 (t1 ), # 1(n)(n−1) » # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) p1 (t1 )), G C1 (t1 ))), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) T2 (t2 ), L 2 (t2 )), z 2 (t2 ) = f 2 ((ω2 (t2 ), p (t2 ), # (n)(n−1) » 2(n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t2 ))), . . . , Cn ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), G C2 # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) Ln (tn )), z n (tn ) = f n ((ωn (tn ), pn (tn ), G Cn (tn )))}. Left side = right side. Proof is completed. # » Let’s consider an example, if T f s j (A(3) ((U (3) (t), S (3) (t), p (3) (t), T (3) (t), L (3) (t)), # (3) » (3) (3) (3) (3) (3) x (3) (t) = 0.8)) = {A(3) 1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = # » (3) (3) (3) (3) (3) (3) 0.5)), A(3) 2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x 2 (t2 ) = 0.6)), . . . , # » (3) (3) (3) (3) (3) (3) A(3) n ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), x n (tn ) = 0.3))}, where n = 3, time t = t1 + t2 +, . . . , +tn ; # » T f s j (A(2) ((U (2) (t), S (2) (t), p (2) (t), T (2) (t), L (2) (t)), x (2) (t) = 0.75)) = # » (2) (2) (2) (2) (2) (2) (2) (2) {A(2) 1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = 0.55)), A2 ((U2 (t2 ), # » (2) (2) (2) (2) S2(2) (t2 ), p2(2) (t2 ), T2(2) (t2 ), L (2) 2 (t2 )), x 2 (t2 ) = 0.6)), . . . , An ((Un (tn ), Sn (tn ), # (2) » (2) (2) (2) pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = 0.22))}, where n = 3, time t = t1 + t2 +, . . . , +tn . # » T f s j (C (2)Az B(3) ((U (2)(3) (t), S (2)(3) (t), p (2)(3) (t), T (2)(3) (t), L (2)(3) (t)), z (2)(3) (t) = # » (t1 )), z 1(2)(3) 0.5)) = {C1(2)Az B(3) ((U1(2)(3) (t1 ), S1(2)(3) (t1 ), p1(2)(3) (t1 ), T1(2)(3) (t1 ), L (2)(3) 1

4.2 Decomposition Transformation Connectives in Error Logic

353

# » (t1 ) = 0.2), C2(2)(3) ((U2(2)(3) (t2 ), S2(2)(3) (t2 ), p2(2)(3) (t2 ), T2(2)(3) (t2 ), L (2)(3) (t2 )), 2 # » z 2(2)(3) (t2 ) = 0.3), . . . , Cn(2)(3) ((Un(2)(3) (tn ), Sn(2)(3) (tn ), pn(2)(3) (tn ), Tn(2)(3) (tn ), (tn )), z n(2)(3) (tn ) = 0.1)}, where n = 3, time t = t1 + t2 +, . . . , +tn . L (2)(3) n # » The left side = ¬bd T f s j (A(3) ((U (3) (t), S (3) (t), p (3) (t), T (3) (t), L (3) (t)), x (3) (t) = # (3) » (3) (3) (3) (3) (3) 0.8)) = ¬bd {A(3) 1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = 0.5)), # » (3) (3) (3) (3) (3) (3) (3) (3) A(3) 2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x 2 (t2 ) = 0.6)), . . . , An ((Un # » (3) (3) (3) (3) (tn ), Sn(3) (tn ), pn(3) (tn ), Tn(3) (tn ), L (3) n (tn )), x n (tn ) = 0.3))} = {A1 ((U1 (t1 ), S1 (t1 ), # (3) » (3) # » (3) (3) (3) (3) (3) (3) p1 (t1 ), T1 (t1 ), L (3) 1 (t1 )), x 1 (t1 ) = 0.5)), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), # » (3) (3) (3) (3) (3) (3) (3) L (3) 2 (t2 )), x 2 (t2 ) = 0.6)), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), # » (2) (2) (2) (2) (2) (2) xn(3) (tn ) = 0.3))} ∧ {A(2) 1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x 1 (t1 ) = # » (2) (2) (2) (2) (2) (2) 0.55)), A(2) 2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x 2 (t2 ) = 0.6)), . . . , # » (2)Az B(3) (2) (2) (2) (2) (2) (2) A(2) n ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), x n (tn ) = 0.22))} ∧ {C 1 # » ((U1(2)(3) (t1 ), S1(2)(3) (t1 ), p1(2)(3) (t1 ), T1(2)(3) (t1 ), L (2)(3) (t1 )), z 1(2)(3) (t1 ) = 0.2), C2(2)(3) 1 # » ((U2(2)(3) (t2 ), S2(2)(3) (t2 ), p2(2)(3) (t2 ), T2(2)(3) (t2 ), L (2)(3) (t2 )), z 2(2)(3) (t2 ) = 0.3), . . . , 2 # » (tn )), z n(2)(3) (tn ) = 0.1)} = Cn(2)(3) ((Un(2)(3) (tn ), Sn(2)(3) (tn ), pn(2)(3) (tn ), Tn(2)(3) (tn ), L (2)(3) n # » {C1(2)Az B(3) ((U1(2)(3) (t1 ), S1(2)(3) (t1 ), p1(2)(3) (t1 ), T1(2)(3) (t1 ), L (2)(3) (t1 )), z 1(2)(3) (t1 ) = 1 # » 0.2), C2(2)(3) ((U2(2)(3) (t2 ), S2(2)(3) (t2 ), p2(2)(3) (t2 ), T2(2)(3) (t2 ), L (2)(3) (t2 )), z 2(2)(3) (t2 ) = 2 # » 0.3), . . . , Cn(2)(3) ((Un(2)(3) (tn ), Sn(2)(3) (tn ), pn(2)(3) (tn ), Tn(2)(3) (tn ), L n(2)(3) (tn )), z n(2)(3) (tn ) = 0.1)}. # » The right side = T f s j (¬bd (A(3) ((U (3) (t), S (3) (t), p (3) (t), T (3) (t), L (3) (t)), x (3) (t) # » = 0.8)) = T f s j ((A(3) ((U (3) (t), S (3) (t), p (3) (t), T (3) (t), L (3) (t)), x (3) (t) = 0.8)) ∧ # » (A(2) ((U (2) (t), S (2) (t), p (2) (t), T (2) (t), L (2) (t)), x (2) (t) = 0.75)) ∧ (C (2)Az B(3) # » (3) ((U (2)(3) (t), S (2)(3) (t), p (2)(3) (t), T (2)(3) (t), L (2)(3) (t)), z (2)(3) (t) = 0.5))) = {A(3) 1 ((U1 # » (3) (3) (3) (3) (t1 ), S1(3) (t1 ), p1(3) (t1 ), T1(3) (t1 ), L (3) 1 (t1 )), x 1 (t1 ) = 0.5)), A2 ((U2 (t2 ), S2 (t2 ), # (3) » (3) # (3) » (3) (3) (3) (3) p2 (t2 ), T2 (t2 ), L (3) 2 (t2 )), x 2 (t2 ) = 0.6)), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), # » (2) (2) (2) (2) (2) (3) Tn(3) (tn ), L (3) n (tn )), x n (tn ) = 0.3))} ∧ {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), # » (2) (2) (2) (2) (2) (2) (2) L (2) 1 (t1 )), x 1 (t1 ) = 0.55)), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # » (2) (2) (2) (2) (2) (2) x2(2) (t2 ) = 0.6)), . . . , A(2) n ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), x n (tn ) = # » 0.22))} ∧ {C1(2)Az B(3) ((U1(2)(3) (t1 ), S1(2)(3) (t1 ), p1(2)(3) (t1 ), T1(2)(3) (t1 ), L (2)(3) (t1 )), 1 # (2)(3) » (2)(3) (2)(3) (2)(3) (2)(3) (2)(3) (2)(3) z 1 (t1 ) = 0.2), C2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), # (2)(3) » (2)(3) (2)(3) (2)(3) (2)(3) ((Un (tn ), Sn (tn ), pn (tn ), Tn(2)(3) (tn ), z 2 (t2 ) = 0.3), . . . , Cn # » L (2)(3) (tn )), z n(2)(3) (tn ) = 0.1)} = {C1(2)Az B(3) ((U1(2)(3) (t1 ), S1(2)(3) (t1 ), p1(2)(3) (t1 ), n # » T1(2)(3) (t1 ), L (2)(3) (t1 )), z 1(2)(3) (t1 ) = 0.2), C2(2)(3) ((U2(2)(3) (t2 ), S2(2)(3) (t2 ), p2(2)(3) (t2 ), 1 T2(2)(3) (t2 ), L (2)(3) (t2 )), z 2(2)(3) (t2 ) = 0.3), . . . , Cn(2)(3) ((Un(2)(3) (tn ), Sn(2)(3) (tn ), 2 # (2)(3) » (2)(3) pn (tn ), Tn (tn ), L (2)(3) (tn )), z n(2)(3) (tn ) = 0.1)}. n

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# » Proposition 4.174 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » # » for judging errors; if T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » G(t)))) = {¬bx A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), # » # » p1 (t1 )), G A1 (t1 ))), ¬bx A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = # » # » f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , ¬bx An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, (1) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; (2) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; # » # » bx if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 s j {¬ A1 # » # » ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), # » # » ¬bx A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), # » # » . . . , ¬bx An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, then (1) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0 ; (2) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side before being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.175 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f s j (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), # » # » p1 (t1 )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), # » # » p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = # » f n ((μn (tn ), pn (tn ), G An (tn )))}, (1) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0; (2) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 s j {A1 # » # » ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), # » # » A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), # » # » . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, then (1) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; (2) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0;

4.2 Decomposition Transformation Connectives in Error Logic

355

# » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), (n) G A (t))) is called the connotative uninterrupted negation on A (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side after being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.176 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bz A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), # » # » p1 (t1 )), G A1 (t1 ))), ¬bz A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = # » # » f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), . . . , ¬bz An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), # » L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, (1) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; (2) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; # » # » bz if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 s j {¬ A1 # » # » ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), # » # » ¬bz A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), # » # » bz . . . , ¬ An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, then (1) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0; (2) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative “not-only” negation on A(n) (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decom# » posed”; and the logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.177 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules # » for judging errors; if T f s j (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), # » # » p (t )), G A1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), # » # 1 1» p2 (t2 ), G A2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = # » f n ((μn (tn ), pn (tn ), G An (tn )))}, then

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(1) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0 ; (2) if x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = T f−1 s j {A1 # » # » ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G A1 (t1 ))), # » # » A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G A2 (t2 ))), # » # » . . . , An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G An (tn )))}, then (1) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; (2) if x(t)  0, then x(t1 )1 , x(t2 )2 , . . . , x(tn )n  0; Similar to the proof of 3.2.95. Proof Proof is omitted.

4.2.10 Rule Decomposition Transformation Connective in Error Logic # » # » Suppose that T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) # » # » = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where G(t) = G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G i (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, it is said that T f bz has conducted rule decomposition transformation on the object of interest # » μ(t, p(t)). For instance, economic law consists of a set of legal rules, namely labor law, tort law, and investment law, etc.

4.2.10.1

Conditions for Rule Decomposition in Error Logic

1. The conditions for rule decomposition are: (1) (2) (3) (4)

fl

Legal conditions T Jgz ; kg Actual conditions T Jgz ; md ; Objective conditions (target) T Jgz sm ; Conditions for sustaining life T Jgz

4.2.10.2

Principles for Rule Decomposition in Error Logic

The principles for rule decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.2 Decomposition Transformation Connectives in Error Logic

4.2.10.3

357

Ways of Rule Decomposition in Error Logic

2. Ways of rule decomposition: (1) Physical decomposition; (2) Mathematical decomposition: For example, . μ(t) : x = f (t, x) + g(t, x); .

μ(t) : x(k + 1) = Ax(k); .....................; μ(t) : x = f (x1 , x2 , . . . , xn ); Decomposition can be conducted according to Lyapunov approach. (3) Decomposition based on actual needs; (4) Decomposition based on property; # » # » in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if G 2 (t) acts on Ti (t), i ∈ {1, 2, . . . , n}; (5) Decomposition according to domain; # » # » in {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if G 2 (t) acts on Ui (t), i ∈ {1, 2, . . . , n}; (6) Decomposing based on special needs and requirements.

4.2.10.4

Characteristics of Rule Decomposition in Error Logic

# » # » Suppose that A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is an error logical variable defined in domain U (t) under G(t) the rules for judging errors; based on the definition for T f and the elements of the error logical vari# » # » able A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))), T f can conduct transformation on the domain U (t), the object μ(t), the error value x(t), the error function f , the time t, and G(t) the rules of judging errors, therefore T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; the type of error logical variable will not be changed if T f does not change its error function f ; for # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . . . . , An ((Un (t), Sn (t), #2 » 2 # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where G(t) = G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G n (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, then T f gz is the rule decom-

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# » position transformation connective with respect to G(t) and A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) defined in domain U (t), in this case, T f gz # » has conducted transformation on the element of G(t) in the object (μ(t), p(t)). Ways of rule decomposition: # » # » (1) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f gz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where G(t) = G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G i (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, if both (Ui (t), Si (t), # » # » / U (t), i ∈ {1, 2, . . . , n} and (U j (t), S j (t), p j (t), T j (t), pi (t), Ti (t), L i (t)) ∈ L j (t)) ∈ U (t), j ∈ {1, 2, . . . , n} exist, then it is said that T f gz has enabled # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out the domain enlargement transformation; # » # » (2) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f gz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where G(t) = G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G i (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, if there exists (Ui (t), # » / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f gz has Si (t), pi (t), Ti (t), L i (t)) ∈ # » # » enabled A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) to carry out transformations on domain, rules for judging errors, time, object, or error function, etc. T f gz ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }; # » # » (3) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f gz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where G(t) = G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), G i (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, if ∀(Ui (t), Si (t), # » / U (t), i ∈ {1, 2, . . . , n}, then it is said that T f gz has carry pi (t), Ti (t), L i (t)) ∈ # » out domain displacement transformation on A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)), G(t))); # » # » (4) A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the rules for judging errors, if there exists T f gz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, where G(t) = G 1 (t) ∪ # » G 2 (t)∪, . . . , ∪G n (t), G i (t) ∈ {G 1 (t), G 2 (t), . . . , G n (t)}, if ∀(Ui (t), Si (t), pi (t),

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# » Ti (t), L i (t)) ∈ U (t), i ∈ {1, 2, . . . , n}, and if (U (t), S(t), p(t), T (t), L(t)) and # » (Ui (t), Si (t), pi (t), Ti (t), L i (t)) do not belong to the same order (layer), then it is said that T f gz has conducted decomposition transformation on (U (t), S(t), # » p(t), T (t), L(t)), and T f gz did not carry out decomposition transformation on # » (U (t), S(t), p(t), T (t), L(t)), otherwise. For example, suppose that (({Diesel # » engine}, Diesel engine, p(t), functioning, L(t)), x(t), Quality criteria) is used to describe the functioning status of a diesel engine, the quality criteria can be decomposed into the quality standards for fueling system, cooling system, electronic control system, and transmission system, etc. # » Proposition 4.178 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , # » G An (t)}; suppose that another error logical variable B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the # » rules for judging errors, T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) ∈ {G B1 (t), G B2 (t), . . . , G Bn (t)}; if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) T f gz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » Proof As x(t)  y(t), the left side = T f gz (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))),

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# » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side # » # » = T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ T f gz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), # » # » L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » xn (t) = f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.179 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , # » G An (t)}; suppose that another error logical variable B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the # » rules for judging errors, T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) ∈ {G B1 (t), G B2 (t), . . . , # » # » G Bn (t)}; suppose that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t))) is the mediator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » Az B p(t)), G B (t))), T f gz (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ {G C1 (t), G C2 (t), . . . , G Cn (t)}; if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold: # » # » (1) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) =

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# » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = T f gz (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » Az B f ((ν(t), p(t)), G B (t)))) = T f gz (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), # » # » # » p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), # » # » G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}; And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  0, the right side= # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n T f gz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # » # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # » # » Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨n {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = # » # » f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = # » # » f n ((μn (t), pn (t), G An (t)))} ∨ {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = # » # » f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = # » # » f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = # » # » f n ((νn (t), pn (t), G Bn (t)))} ∨ {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 # » # » (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), # » # » p (t), G (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), # » # » # 2 » C2 pn (t), G Cn (t)))} = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), # » # » G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2) can also be proved. # » Proposition 4.180 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))),

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# » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , # » G An (t)}; suppose that another error logical variable B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the # » rules for judging errors, T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) ∈ {G B1 (t), G B2 (t), . . . , # » # » G Bn (t)}; suppose that C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t))) is the connotative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t))), T f gz (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ {G C1 (t), G C2 (t), . . . , G Cn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t). then the following relationship holds: # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) −n B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f gz (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n T f gz (B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted.

# » Proposition 4.181 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , # » G An (t)}; suppose that another error logical variable B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the # » rules for judging errors, T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) ∈ {G B1 (t), G B2 (t), . . . , # » # » G Bn (t)}; suppose that C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t))) is the mediator variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » Az B p(t)), G B (t))), T f gz (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)),

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# » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ {G C1 (t), G C2 (t), . . . , G Cn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), , −yi (t), z i (t), and −z i (t) is the same as that of x(t), −x(t), y(t), , −y(t), z(t), and −z(t), then the following relationships hold: # » # » (1) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.182 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # » # » G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), # » # » . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , # » G An (t)}; suppose that another error logical variable B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the # » rules for judging errors, T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), # » # » G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), # » # » . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where G B (t) = G B1 (t) ∪ G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) ∈ {G B1 (t), G B2 (t), . . . , # » G Bn (t)}; suppose that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » p(t)), G C (t))) is the connotative same or equivalence variable for A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))), T f gz (C Anhdthd j B ((U (t), S(t), p(t), T (t),

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# » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # » # » f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = # » f n ((ωn (t), pn (t), G Cn (t)))}, where G C (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ {G C1 (t), G C2 (t), . . . , G Cn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship holds: T f gz (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f gz (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t)))) →nby T f gz (B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted.

# » Proposition 4.183 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, T f gz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t)))) # (n) » (n) #A(n) » (n) (n) (n) (n) (n) (n) (n) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » (n) # » (n) p2 (t), G A2 (t))), . . . , A(n) ((Un(n) (t), Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) n (t)), x n (t) = # (n) » (n)n (n) (n) (n) (n) (n) f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G (n) An (t), (n) (n) (n) (n) (n+1) (n+1) (n+1) G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , G An (t)}; suppose that A ((U (t), S (t), # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) A is the (n + 1)th order error logical variable defined in domain U (n+1) (t) under # » G (n+1) (t) the rules for judging errors, T f gz (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1(n+1) {A1 ((U1 # » # » (n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), 1 A1 (t))), A2 # » (n+1) (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) L (n+1) 2 2 A2 (t))), . . . , An # » # » (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t)= f n(n+1) ((μ(n+1) (t), pn(n+1) (t), n n (n+1) (n+1) (n+1) (n+1) G (n+1) (t) = G (n+1) (t) An (t)))}, where G A A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (n+1) (n+1) (n+1) ∈ {G A1 (t), G A2 (t), . . . , G An (t)}; suppose that error logical variable B (n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), # (n+1) » p (t)), G (n+1) (t))) is the complement error logical variable for A(n+1) ((U (n+1) (t), B # (n+1) » # » (n+1) (t), p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), S (t))), which is defined in domain U (n+1) (t) under G (n+1) (t) the rules for G (n+1) A # (n+1) » B (n+1) (n+1) (n+1) (n+1) judging errors, T f gz (B ((U (t), S (t), p (t), T (t), L (n+1) (t)), # » (n+1) (n+1) (n+1) x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G B (t))))={B1 ((U1 (t), S1(n+1) (t), # (n+1) » (n+1) # » p1 (t), T1 (t), L (n+1) (t)), x1(n+1) (t)= f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))),

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# » B2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), x2(n+1) (t) = f 2(n+1) 2 # » # » (n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 B2 (t))), . . . , Bn # » (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n Bn (t)))}, where (n+1) (n+1) (n+1) (n+1) G (n+1) (t) = G (t) ∪ G (t)∪, . . . , ∪G (t), G (t) ∈ {G (n+1) B B1 B2 Bn Bi B1 (t), # (n+1) » (n+1) (n+1) (n+1)Az B (n+1) (n+1) G B2 (t), . . . , G Bn (t)}; suppose that C ((U (t), S (t), p (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))), B # (n+1) » (n+1) (n+1)Az B (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = T f gz (C # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((μ (t), p (t)), G C (t))))={C1 ((U1 (t), S (t), p1 (t), f # (n+1) » 1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2(n+1) ((ω2(n+1) (t), # (n+1) » (n+1) # » p2 (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) n # » (n+1) (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn (t)))}, where G C(n+1) (t)=G C1 (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ {G C1 (t), G C2 (t), . . . , G Cn (t)}; if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n+1) (t), y (n+1) (t), and z (n+1) (t), then the following rela# » tionship holds: T f gz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » bz (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), A (t)))) = ¬ T f gz (A ((U # » L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), the left side of the equation # » = T f gz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = T f gz (A # (n+1) » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t))) ∧ B (n+1) A # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), B # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = T f gz (A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) ∧ T f gz (B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # (n+1) » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t)))) ∧ B # » (n+1)Az B (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = T f gz (C # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f ((μ (t), p (t)), G C (t)))) = {A1 ((U1 (t), S1 (t), # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) =

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# » # » (n+1) f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An # » Tn(n+1) (t), L (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) n n An (t)))} ∧ # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n+1) (t) = {B1 # » # » (n+1) f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), 1 B1 (t))), B2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 B2 (t))), . . . , # » (t)), xn(n+1) (t) = Bn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) n # » # » (n+1) f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), n Bn (t)))} ∧ {C 1 # » (n+1) T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), 1 # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) C2 ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n+1) » (n+1) 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) Tn(n+1) (t), L (n+1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn # » And the right side of the equation = ¬bz T f gz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n) (n) (n) bz L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n+1) G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), 1 An (t)))} = {A1 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), # (n+1) »2 (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), xn(n+1) (t) = . . . , A(n+1) n n n n n n # » # » (n+1) (n+1) (n+1) (n+1) f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {B1 ((U1 (t), S1 (t), p1(n+1) (t), n # » T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 B1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = B2 # (n+1) » (n+1) 2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))} ∧ Tn(n+1) (t), L (n+1) n n n n n Bn # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) {C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) f1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S (t), p (t), # (n+1) »2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (n+1) T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω (t), p2 (t), G C2 (t))), . . . , # » 2 Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = n # » (n+1) (t)))}. f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn Left side = right side. Proof is completed. # » Proposition 4.184 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable

4.2 Decomposition Transformation Connectives in Error Logic

367

defined in domain U (n) (t) under G (n) A (t) the rules for judging errors, # (n) » (n) (n) (n) (n) T f gz (A ((U (t), S (t), p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ (n) (n) (n) (n) (n) (n) G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G A1 (t), G A2 (t), . . . , G An (t)}; suppose that # » A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » (t))) is the (n + 1)th order error logical varif (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) (t) under G (n+1) (t) the rules for judging errors, able defined in domain U # (n+1) »A (n+1) (n+1) (n+1) (n+1) T f gz (A ((U (t), S (t), p (t), T (t), L (n+1) (t)), x (n+1) (t) = # » # » (n+1) (n+1) f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t))))={A1 ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), # » (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) T1(n+1) (t), L (n+1) 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n+1) » (n+1)2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p (t), G (t)))}, where Tn(n+1) (t), L (n+1) n n n n n An (n+1) (n+1) (n+1) (n+1) G (n+1) (t) = G (t) ∪ G (t)∪, . . . , ∪G (t), G (t) ∈ {G (n+1) A A1 A2 An Ai A1 (t), (n+1) (n+1) (n)Az B(n+1) (n)(n+1) G A2 (t), . . . , G An (t)}; assuming that C ((U (t), S (n)(n+1) (t), # (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), # (n)(n+1) » p (t)), G C(n)(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = G (n+1) A # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), T f gz (C # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) p (t), T (t), L (t)), x (t) = f (n)(n+1) ((μ(n)(n+1) (t), # » # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = {C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), # (n)(n+1) » T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1 (t)), 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) G C1 (t))), C2 ((U2 , S2 (t), p2 (t), T (t), # (n)(n+1) » (n)(n+1)2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) L2 (t)), z 2 (t) = f 2 ((ω2 (t), p (t), G (t))), . . . , # (n)(n+1) » 2 (n)(n+1) C2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where zn (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ G C(n)(n+1) (t) = G C1 (n)(n+1) (n)(n+1) (n)(n+1) {G C1 (t), G C2 (t), . . . , G Cn (t)}; suppose that T f gz has carried out rule # » decomposition transformation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2 . . . n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the following relationship holds: # » T f gz (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t),

368

4 Transformation Connectives in Error Logic

# (n) » # » bx (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ T f gz (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), the left side of # » the equation = T f gz (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n+1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A (t)))) = T f gz (A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), G (n) A (t))) ∧ C # » L (n)(n+1) (t)), x (n)(n+1) (t) = f (n)(n+1) ((μ(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » T f gz (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) ∧ T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), A # » (n)Az B(n+1) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t) ), G (n) A (t)))) ∧ T f gz (C # » ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), x (n)(n+1) (t) = # » ((U1(n+1) (t), S1(n+1) (t), f (n)(n+1) ((μ(n)(n+1) (t), p ((n)n+1) (t)), G C(n)(n+1) (t)))) = {A(n+1) 1 # (n+1) » (n+1) # » p1 (t), T1 (t), L (n+1) (t)), x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n+1) » (n+1)2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L n (t)), xn (t) = f ((μ (t), pn (t), G An (t)))} ∧ Tn # (n) » (n) n (n) n # (n) » (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ(n) 1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # » (n) (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » # » (n) (n)(n+1) (n) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # » T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), 1 # » (n)(n+1) G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), # » (n)(n+1) L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , 2 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t) = f n ((ωn (t), pn (t), G Cn (t)))}. zn # » And the right side of the equation = ¬bx T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bx T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # » # » (n+1) pn(n) (t), G (n) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), f n(n) ((μ(n) n (t), An (t)))} = {A1

4.2 Decomposition Transformation Connectives in Error Logic

369

# » x1(n+1) (t) = f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n+1) » (n+1)2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n+1) »n (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L n (t)), xn (t) = f ((μ (t), pn (t), G An (t)))} ∧ Tn # (n) » (n) n (n) n # (n) » (n) (n) (n) (n) (n) {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ(n) 1 (t), p1 (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) p2 (t), G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = # » # » (n) (n)(n+1) (n) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), f n(n) ((μ(n) n (t), pn (t), G An (t)))} ∧ {C 1 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (t)), z (t) = f ((ω (t), p1 (t)), T1(n)(n+1) (t), L (n)(n+1) 1 1 1 #1 (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) L2 (t)), z 2 (t) = f 2 ((ω2 (t), p (t), G (t))), . . . , # (n)(n+1) » 2 (n)(n+1) C2 (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), T (t), L n (t)), # » (n)(n+1)n (t)))}. z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn Left side = right side. Proof is completed. T1(n+1) (t),

L (n+1) (t)), 1

Proposition 4.185 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging # » errors, T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) p (t)), G (n) A (t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ (n) (n) (n) (n) (n) G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G (n) A1 (t), G A2 (t), . . . , G An (t)}; # » suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » (n−1) x (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logA (t) the rules for judging errors, ical variable defined in domain U (n−1) (t) under G (n−1) # » A T f gz (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 A # (n−1) » # » p1 (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))}, where (n−1) (n−1) (n−1) (n−1) G (n−1) (t) = G (t) ∪ G (t)∪, . . . , ∪G (t), G (t) ∈ {G (n−1) A A1 A2 An Ai A1 (t), (n−1) (n−1) G A2 (t), . . . , G An (t)};

370

4 Transformation Connectives in Error Logic

# » suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » logical complementary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (n−1) (t) under G B (t) the rules for judging errors, T f gz (B (n−1) ((U (n−1) (t), U # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) = {B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), G (n−1) 1 B # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), x1 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G B2 (t))), # (n−1) »2 (n−1) (n−1) (n−1) (n−1) (n−1) . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = # » (t), pn(n−1) (t), G (n−1) where G (n−1) (t) = G (n−1) f n(n−1) ((μ(n−1) n Bn (t)))}, B B1 (t) ∪ (n−1) (n−1) (n−1) (n−1) (n−1) G B2 (t)∪, . . . , ∪G Bn (t), G Bi (t) ∈ {G B1 (t), G B2 (t), . . . , G (n−1) (t)}; # (n−1) » (n−1) Bn (n−1) (n−1)Az B (n−1) (n−1) suppose that C ((U (t), S (t), p (t), T (t), L (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (t))), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) T f gz (C ((U (t), S (t), p (t), T (t), L (t)), x (t) = # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) f ((μ (t), p (t)), G C (t)))) = {C1 ((U1 (t), S1 (t), # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((ω2 (t), p2 (t), G C2 (t))) . . . Cn ((Un (t), S (t), pn (t), f2 # (n−1) » n(n−1) (n−1) (n−1) (n−1) (n−1) (n−1) Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where (n−1) (n−1) (n−1) (n−1) (n−1) G C(n−1) (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ {G C1 (t), (n−1) (n−1) G C2 (t), . . . , G Cn (t)}; suppose that T f gz has carried out rule decomposition transformation on (U (n) (t), # » (n) S (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following relationship holds: T f gz (¬bj A(n) ((U (n) (t), S (n) (t), # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » bj (n) (n) (n) (n) (n) (n) (n) ¬ T f gz (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ(n) (t), # (n) » p (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t). # » The left side of the equation = T f gz ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = T f gz (A

4.2 Decomposition Transformation Connectives in Error Logic

371

# » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = T f gz (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f gz (B (n−1) ((U (n−1) (t), A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t)))) ∧ T f gz (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) 1 # (n−1) » # » (t), p1 (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), 1 1 # (n−1) » (n−1) (n−1) (n−1) (n−1) G (n−1) (t))), A ((U (t), S (t), p2 (t), T2 (t), L (n−1) (t)), x2(n−1) (t) = 2 2 2 2 A1 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))} ∧ # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) {B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 B2 (t))), . . . , # » (t)), xn(n−1) (t) = Bn(n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) n # » # » (n−1) (n−1) (n−1) (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G Bn (t)))} ∧ {C1 ((U1 (t), S1 (t), p1(n−1) (t), n # » (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), T1(n−1) (t), L (n−1) 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) Tn(n−1) (t), L (n−1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn # » And the right side of the equation = ¬bj T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bj T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) xn(n) (t) = G (n) n (t)), A2 (t))), . . . , An ((Un (t), # (n−1) » # » (n) (n−1) (n−1) (n−1) pn(n) (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), f n(n) ((μ(n) n (t), # » (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) T1(n−1) (t), L (n−1) 1 1 A1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p (t), T2 (t), L 2 (t)), x2 (t) = A2 ((U2 # (n−1) » (n−1)2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), S (t), pn (t), # (n−1) »n (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∧ # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1(n−1) (t) = {B1 # » # » (n−1) f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), 1 B1 (t))), B2

372

4 Transformation Connectives in Error Logic

# » T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) 2 2 B2 (t))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn(n−1) (t) = # » # » (n−1) (n−1) (n−1) f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G Bn (t)))} ∧ {C1 ((U1 (t), S1(n−1) (t), p1(n−1) (t), n # » (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), T1(n−1) (t), L (n−1) 1 # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) C2 ((U2 (t), S2 (t), p (t), T2 (t), L 2 (t)), z 2 (t) = # (n−1) » (n−1) 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) f2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) Tn(n−1) (t), L (n−1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn Left side = right side. Proof is completed. # » Proposition 4.186 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, # » # » T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) x1(n) (t) = G (n) 1 (t)), A (t)))) = {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x2(n) (t) = f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » (n) (n) (n) (n) (n) (n) (n) L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where G A (t) = G A1 (t) ∪ (n) (n) (n) (n) (n) G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ {G (n) A1 (t), G A2 (t), . . . , G An (t)}; suppose that # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » (t))) is the (n − 1)th order error logical varif (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A (t) the rules for judging errors, able defined in domain U (n−1) (t) under G (n−1) # »A T f gz (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 A # » # (n−1) » (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), p1 1 1 # » (n−1) G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), x2(n−1) (t) = 2 A1 (t))), A2 # » # » (n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), f 2(n−1) ((μ(n−1) 2 A2 (t))), . . . , An # (n−1) » Tn(n−1) (t), L (n−1) (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn (t), G (n−1) n n An (n−1) (n−1) (n−1) (n−1) (n−1) (t)))}, where G A (t) = G A1 (t) ∪ G A2 (t)∪, . . . , ∪G An (t), G Ai (t) ∈ (n−1) (n−1) (n)Az B(n−1) ((U (n)(n−1) (t), {G (n−1) A1 (t), G A2 (t), . . . , G An (t)}; suppose that C # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) # » ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) # » ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), # (n−1) » # » p (t)), G (n−1) (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = A # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), and T f gz (C # (n)(n−1) » p (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), # » # (n)(n−1) » p (t)), G C(n)(n−1) (t)))) = {C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t),

4.2 Decomposition Transformation Connectives in Error Logic

373

# » z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), # » (n)(n−1) G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), # » (n)(n−1) L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), G C2 (t))), . . . , 2 # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) Cn ((Un (t), Sn (t), pn (t), T (t), L n (t)), # (n)(n−1) » n (n)(n−1) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn (t), G Cn (t)))}, where z n(n)(n−1) (t) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) G C(n)(n−1) (t) = G C1 (t) ∪ G C2 (t)∪, . . . , ∪G Cn (t), G Ci (t) ∈ (n)(n−1) (n)(n−1) (n)(n−1) {G C1 (t), G C2 (t), . . . , G Cn (t)}; suppose that T f gz has carried out rule # » decomposition transformation on (U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), then the following rela# » tionship holds: T f gz (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » bd (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), A (t)))) = ¬ T f gz (A ((U # » L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). T1(n)(n−1) (t),

L (n)(n−1) (t)), 1

Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), the left side of # » the equation = T f gz ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A (t)))) = T f gz (A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), G (n) A (t))) ∧ C # » L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » # » T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n−1) G (n) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), A (t)))) ∧ T f gz (A # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) ∧ T f gz C (n)Az B(n−1) A # » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = # (n)(n−1) » (n) (n) p (t)), G C(n)(n−1) (t)))) = {A(n) f (n)(n−1) ((μ(n)(n−1) (t), 1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) n ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t) = f n ((μn (t), pn (t), # » (n−1) G (n) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), 1 An (t)))} ∧ {A1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), x1 # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p2 (t), T2 (t), L 2 (t)), x2 (t) = f ((μ2 (t), p2 (t), G A2 (t))), # (n−1) »2 (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), xn(n−1) (t) = . . . , A(n−1) n n n n n n # » (n)(n−1) (t), pn(n−1) (t), G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), f (n−1) ((μ(n−1) n An (t)))} ∧ {C 1 # n(n)(n−1) » p1 (t), T1(n)(n−1) (t), L 1(n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t),

374

4 Transformation Connectives in Error Logic

# (n)(n−1) » (n)(n−1) p1 (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t),

# » S2(n)(n−1) (t), p2(n)(n−1) (t), # (n)(n−1) » T2(n)(n−1) (t), L (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2 (t), 2 # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t))), . . . , Cn ((Un (t), Sn (t), p (t), T (t), G C2 # (n)(n−1)n » (n)(n−1) n (n)(n−1) (n)(n−1) (n)(n−1) L (n)(n−1) (t)), z (t) = f ((ω (t), p (t), G (t)))}. n n n n n Cn # » And the right side of the equation = ¬bd T f gz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n) (n) bd T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ {A1 ((U1 (t), # » # » (n) (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # » (n) (n) Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) (t)), xn(n) (t) = G (n) A2 (t))), . . . , An ((Un (t), # (n) »n (n) # » (n) (n) (n) (n) (n) (n) f n(n) ((μ(n) n (t), pn (t), G An (t)))} = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # (n) » # » (n) (n−1) (n) (n) (n) (n) pn (t), Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A1 # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = f 1(n−1) ((μ(n−1) (t), 1 1 # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) p1 (t)), G A1 (t))), A2 ((U2 (t), S (t), p2 (t), T2 (t), L 2 (t)), # (n−1) » 2(n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f ((μ2 (t), p2 (t), G A2 (t))) . . . An ((Un (t), Sn(n−1) x2 # (n−1) »2 # » (n−1) (n−1) (n−1) (n−1) (n−1) (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn(n−1) (t), # » (n)(n−1) G (n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), An (t)))} ∧ {C 1 # » (n)(n−1) L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) z2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn # » ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = n # » (n)(n−1) (t)))}. f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn Left side = right side. Proof is completed. # » Proposition 4.187 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , ¬bx An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » −1 T f gz {¬bx A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))),

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# » # » ¬bx A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » bx ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bx A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative unconstrained negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side before being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.188 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) # » the rules for judging errors; if T f gz (¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » if ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » −1 T f gz {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » Proof Because ¬bd A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative uninterrupted negation on A(n) (μ(t), x(t)), which means that “for the property being negated, there exists its opposite side after being # » # » decomposed”; if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is erroneous, then the property becomes non-erroneous after negation operation, therefore x(t)i and x(t) have different signs, i ∈ { 1, 2,. . . , n }. Left side = right side. Proof is completed. # » Proposition 4.189 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) the # » rules for judging errors; if T f gz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)), G(t)))) = {¬bz A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t),

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# » # » # » p1 (t)), G A1 (t))), ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » bz G A2 (t))), . . . , ¬ An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; # » # » if A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » bz T f−1 gz {¬ A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » ¬bz A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » ¬bz An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; # » # » Proof Because ¬bz A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is called the connotative “not-only” negation on A(n) (μ(t), x(t)), which means that “there exists characteristics that can be negated before being decom# » posed”; and the logical value of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is denoted by its error value, the characteristics being negated is the erroneity (i.e., the state of being erroneous, or correct) of the error logical variable, therefore x(t)i and x(t) have the same signs, i ∈ {1, 2, . . . , n}. Left side = right side. Proof is completed. # » Proposition 4.190 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) is defined in domain U (t) under G(t) # » the rules for judging errors; if T f gz (¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G(t)))) = {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), # » # » # » p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), # » # » G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. # » # » if ¬bj A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G(t))) = # » # » T f−1 gz {A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, then (1) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; (2) if x(t)  0, then x(t)1 , x(t)2 , . . . , x(t)n  0; Similar to the proof of 3.2.108. Proof Proof is omitted.

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4.2.11 Overall Decomposition Transformation Connectives in Error Logic # » # » Suppose that T f (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » {A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 )), G 1 (t1 ))), # » # » A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G 2 (t2 ))), . . . , # » # » An ((Un (tn ), Sn (tn ), pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G n (tn )))}, # » T f has carried out overall decomposition transformation on (A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))), T f is called the overall decomposition transformation logical connective defined in U (t), where T f ⊆ {T f ly , T f sw , T f k j , T f t x , T f lz , T f cz , T f gz , T f hs , T f s j , T f q }.

4.3 Displacement Transformation Connectives in Error Logic 4.3.1 Concepts of Displacement Transformation Connective in Error Logic Definition 4.37 Suppose that an error logical variable {Ai ((Ui , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} is defined in domain U (t) under G A (t) the rules for judging errors, if Tzcz ({Ai ((Ui , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m}) = {B j ((U j , S jv (t), #p jv», T jv (t), L jv (t)), y j (t) = g j (v j (t), G j B (t))), j = 1, 2, . . . , n}, then Tz is called the displacement transformation connective with respect to G(t) and {Ai ((Ui , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m}. Here, {B j ((U j , S jv (t), #p », T (t), L (t)), y (t) = g (v (t), G (t))), j = 1, 2, . . . , n} is used to replace jv jv jv j j j jB {Ai ((Ui , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m}. In the transformation operation,we hope to replace the error value of {xi (t), i = 1, 2, . . . , m} with the desired error value of {y j (t), j = 1, 2, . . . , n}. In the displacement transformation operation {xi (t), i = 1, 2, . . . , m} → {y j (t), j = 1, 2, . . . , n}, Tz is called the error value opposite displacement transformation if xi (t) = −y j (t)(i = 1, 2, . . . , m, j = 1, 2, . . . , n), which is noted by Tz f cz . There are more displacement transformations listed as below. (1) Suppose that a = max{xi (t), i = 1, 2, . . . , m}; b = min{xi (t), i = 1, 2, . . . , m}; c = max{y j (t), j = 1, 2, . . . , m}; d = min{yi (t), j = 1, 2, . . . , m}. If c < a and d < b, then Tzcz is called the error value increasing displacement transformation noted by Tzycz ; (2) Suppose that a = max{xi (t), i = 1, 2, . . . , m}; b = min{xi (t), i = 1, 2, . . . , m}; c = max{y j (t), j = 1, 2, . . . , m}; d = min{yi (t), j = 1, 2, . . . , m}. If c >

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a and d > b, then Tzcz is called the error value decreasing displacement transformation noted by Tzhcz ; (3) In the displacement transformation {xi (t), i = 1, 2, . . . , m} → {y j (t), j = 1, 2, . . . , m}, if yi (t) = kxi (t)(i, j = 1, 2, . . . , m), then Tzcz is called the error-value amplification displacement transformation noted by Tzkcz . (a) If k ≥ 1, then Tzcz is called the error-value positive amplification displacement transformation noted by Tzkzcz ; (b) If k ≤ −1, then Tzcz is called the error-value negative amplification displacement transformation noted by Tzk f cz ; (c) If 0 < k < 1, then Tzcz is called the error-value positive reduction displacement transformation noted by Tzkzscz ; (d) If −1 < k < 0, then Tzcz is called the error-value negative reduction displacement transformation noted by Tzk f scz ; (e) If k = 0, then Tzcz is called the error-elimination displacement transformation noted by Tzkhlcz .

4.3.1.1

Principles for Displacement Decomposition in Error Logic

The principles for displacement decomposition are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.3.1.2

Types of Displacement Transformation in Error Logic

In general, the object of interest u(t) not only has vertical and horizontal structures but also contains many hierarchies and relationships. Therefore, it is necessary to conduct displacement transformation at different hierarchies. Suppose that in {Ai ((Ui , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} and {B j ((U j , S jv (t), #p jv», T jv (t), L jv (t)), y j (t) = g j (v j (t), G j B (t))), j=1, 2, . . . , m}: (1) {u i (t), i = 1, 2, . . . , m} → {v j (t), j = 1, 2, . . . , m}, Tz is called the domain displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzly . In this case U (t) ∩ V (t) = Φ or U (t) ∩ V (t) = Φ. For example, when discussing issues related to human resource in China, suppose that domain Shanghai U (t) and domain China V (t) are two domains, V (t) can be used to replace U (t). (2) {Siu (t), i = 1, 2, . . . , m} → {S jv (t), j = 1, 2, . . . , m}, Tz is called the thing displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzsw . In this case, the displacement transformation is conducted on thing in (Ui , Siu (t), #piu», Tiu (t),

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L iu (t)) to achieve the expected goal. (3) { #piu», i = 1, 2, . . . , m} → { #p jv», j = 1, 2, . . . , m}, Tz is called the spatial displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzk j . In this case, the displacement transformation is carried out on the location of thing in (Ui , Siu (t), #piu», Tiu (t), L iu (t)) to achieve the expected goal. (4) {Tiu , i = 1, 2, . . . , m} → {T jv , j = 1, 2, . . . , m}, Tz is called the property displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzt z . In this case, the displacement transformation is conducted on the property of thing in (Ui , Siu (t), #piu», Tiu (t), L iu (t)) to achieve the expected goal. For example, the length Tu (t) and width Tv (t) are volume property of a product, sometimes it is appropriate to replace Tu (t) with Tv (t). (5) {L iu , i = 1, 2, . . . , m} → {L jv , j = 1, 2, . . . , m}, Tz is called the property (or attribute) value displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzlz . In this case, the displacement transformation is carried out on the property (attribute) value of property T (t) in (Ui , Siu (t), #piu», Tiu (t), L iu (t)) to achieve the expected goal. For example, the property (attribute) value of length Tu (t) and the property (attribute) value of width Tv (t) can be exchanged. (6) {xi (t), i = 1, 2, . . . , m} → {y j (t), j = 1, 2, . . . , m}, Tz is called the error value displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzcz . In this case, the displacement transformation is carried out on the error value in (Ui , Siu (t), #piu», Tiu (t), L iu (t)) to achieve the expected goal. For example, we hope to use the anticipated error value {y j (t), j = 1, 2, . . . , m} to replace the undesirable error value {xi (t), i = 1, 2, . . . , m}. (7) {G i A (t), i = 1, 2, . . . , m} → {G j B (t), j = 1, 2, . . . , m}, Tz is called the rule displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzgz . In this case, the displacement transformation is carried out on the rule to achieve the expected goal. For example, the modified constitution G 2 in 2004 in China is a modification for constitution G 1 (t) in 2003, which was a typical case of rule displacement transformation for serving the emerging needs of social and economic development. 8) { f i , i = 1, 2, . . . , m} → {g j , j = 1, 2, . . . , m}, Tz is called the error function displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzhs . In this case, the displacement transformation is implemented on error function to achieve the expected goal. 9) t A → t B is called the temporal displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzs j . In this case, the displacement transformation is conducted on time to achieve the expected goal.

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Suppose that that displacement transformation connective Tz has enabled the following transformations-i.e., {u i (t), i = 1, 2, . . . , m} → {v j (t), j = 1, 2, . . . , m}, thing {Siu (t), i = 1, 2, . . . , m} → {S jv (t), j = 1, 2, . . . , m}, space { #piu», i = 1, 2, . . . , m} → { #p jv», j = 1, 2, . . . , m}, property {Tiu , i = 1, 2, . . . , m} → {T jv , j = 1, 2, . . . , m}, property (or attribute) value {L iu , i = 1, 2, . . . , m} → {L jv , j = 1, 2, . . . , m}, error function { f i , i = 1, 2, . . . , m} → {g j , j = 1, 2, . . . , m}, error value {xi (t), i = 1, 2, . . . , m} → {y j (t), j = 1, 2, . . . , m}, rule {G i A (t), i = 1, 2, . . . , m} → {G j B (t), j = 1, 2, . . . , m}, time t A → t B , then Tz is called the overall displacement transformation with respect to G A (t) and {Ai ((Vi , Siu (t), #piu», Tiu (t), L iu (t)), xi (t) = f i (u i (t), G i A (t))), i = 1, 2, . . . , m} noted by Tzq . In this situation, the displacement transformation is conducted on domain, thing, space, property, property (attribute) value, error function, error value, time, and rules for judging errors to achieve the expected goal. Here, the displacement transformation connectives are Tz ⊆ {Tzly , Tzsw , Tzk j , Tzt z , Tzlz , Tzcz , Tzgz , Tzhs , Tzs j , Tzq } (displacement transformation) and Tz−1 (inverse displacement transformation connectives).

4.3.2 Domain Displacement Transformation Connective in Error Logic 4.3.2.1

Characteristics of Domain Displacement Decomposition Connectives in Error Logic

# » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y(t) = g((ν(t), p(t)), G A (t))), Tzly has conducted domain displacement transformation on the object of μ(t). # » Proposition 4.191 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x(t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under G A (t) the rules for judging errors, another error logical variable A((V (t), Sv (t), # » # » pv (t), Tv (t), L v (t)), y(t) = g((ν(t), pv (t)), G A (t))) is defined in domain V (t) under # » # » G B (t) the rules for judging errors; if ∀(μ(t), pu (t)) = (U (t), Su (t), pu (t), Tu (t), # » L u (t)) ∈ U (t) corresponds to A((U (t), Su (t), pu (t), Tu (t), L u (t)), x(t) = f ((μ(t), # » # » # » pu (t)), G A (t))) and if ∀(ν(t), pv (t)) ∈ V (t) = (V (t), Sv (t), pv (t), Tv (t), L v (t)) # » # » corresponds to A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y(t)=g((ν(t), pv (t)), G A (t))),   where a  b and a  b , in general, (1) if U (t) = V (t), then a = a  , b = b ; (2) if x(t)1 , x(t)2 , . . . , x(t)n  0, then x(t)  0 ; hold. Proof Proof is omitted. # » Proposition 4.192 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) =

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# » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tzly (B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = B((V (t), Sv (t), # » # » pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors,if x A (t)  x B (t) then y A (t)  y B (t) or if x A (t)  x B (t) then y A (t)  y B (t), there exist the following relationships: # » # » (1) Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∨ # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨ # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (2) Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (3) Tzly (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) = # » # » ¬Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))); hold. Similarly,(2) and (3) can be proved. Proof From the assumption we know that if x A (t)  x B (t) then y A (t)  y B (t) or if x A (t)  x B (t) then y A (t)  y B (t), it is assumed that x A (t)  x B (t), then the left # » side of the equation is: Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) ∨ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))). # » As y A (t)  y B (t), the right side of the equation is: Tzly (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨ Tzly (B((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∨ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » # » g((ν(t), pv (t)), G B (t))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))), left side = right side; it is assumed that x A (t)  x B (t), then the left # » side of the equation is: Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) ∨ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))); # » As y A (t)  y B (t), the right side of the equation is: Tzly (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨ Tzly (B((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∨ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » # » g((ν(t), pv (t)), G B (t))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), # » pv (t)), G B (t))), left side = right side.

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Proof is completed. # » Proposition 4.193 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tzly (B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = B((V (t), Sv (t), # » # » pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) # » under G B (t) the rules for judging errors; suppose that C Az B ((U (t), Su (t), pu (t), # » Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) is the mediator variable of A((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) and B((U (t), Su (t), # » # » pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))), which is defined in domain # » U (t) under G C (t) the rules for judging errors, Tzly (C Az B ((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = C Az B ((V (t), Sv (t), pv (t), Tv (t), # » L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t) or if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t), there exist the following relationships: # » # » (1) Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∨n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨n # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (2) Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧n # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). hold. Proof From the assumption we know that if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t) or if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t), it is assumed that x A (t)  x B (t)  xC (t), then the left side of the equation is: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∨n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))); As y A (t)  y B (t)  yC (t), the right side of the equation is: Tzly (A((U (t), Su (t), # » # » pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨n Tzly (B((U (t), Su (t), # » # » # » pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), Sv (t), pv (t), # » # » Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∨n B((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∨ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) =

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# » # » g((ν(t), pv (t)), G B (t))) ∨ C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), # » # » # » pv (t)), G C (t))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))), left side = right side. Similarly, other situations can also be proved. Proof is completed. Similarly, (2) can be proved. # » Proposition 4.194 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; # » # » suppose that C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), # » G C (t))) is the connotative inclusion variable of A((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x B (t) = f ((μ(t), pu (t)), G B (t))), is defined in domain U (t) under G C (t) the rules # » for judging errors, Tzly (C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), # » # » p (t)), G C (t)))) = C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), #u » pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), then yi (t)  y j (t) (or yi (t)  −y j (t)), where i = j, i, j ∈ {A, B, C}, there exists the following relationships: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) −n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) −n # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), then yi (t)  y j (t) (or yi (t)  −y j (t)), where i = j, i, j ∈ {A, B, C}, then the # » left side of the equation is: Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t))) −n B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » pu (t)), G B (t)))) = Tzly ((A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) ∧ ¬B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t))) ∧ ¬C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), # » # » G C (t)))) ∨ (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t))) ∧ ¬B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t))) ∧ ¬C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), # » # » G C (t))))) = (A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), # » # » G A (t))) ∧ ¬B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧

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# » # » C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) ∨ # » # » (¬A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ # » # » ¬B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ C AnhbB # » # » ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) = A((V (t), # » # » Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) −n B((V (t), Sv (t), # » # » # » pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = Tzly (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) −n Tzly (B((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. # » Proposition 4.195 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; sup# » # » pose that C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) # » is the mediator variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))), which is defined in domain U (t) under G C (t) the rules for judging errors, # » # » Tzly (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)),where i = j, i, j ∈ {A, B, C}, there exists the following relationship: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) |n f l # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) |n f l # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)),where i = j, i, j ∈ {A, B, C}, then the left # » side of the equation is: Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) |n f l B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzly ((A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t))) ∧ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) ∧ # » # » C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) ∨ # » # » (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))¬C Az B ((U (t), Su (t),

4.3 Displacement Transformation Connectives in Error Logic

385

# » # » pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))))) = (Tzly (A((U (t), Su (t), # » # » # » pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ Tzly (B((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) ∧ Tzly (C Az B ((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))))) ∨ (Tzly (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ Tzly (B((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))))Tzly (¬C Az B ((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))))) = (A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » # » g((ν(t), pv (t)), G B (t))) ∧ C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = # » # » g((ν(t), pv (t)), G C (t)))) ∨ (A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » # » # » pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), # » # » AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G B (t))) ∧ ¬C # » # » G C (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), # » # » nf l G A (t))) | B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) |n f l # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. # » Proposition 4.196 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; sup# » # » pose that C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) # » is the mediator variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))), which is defined in domain U (t) under G C (t) the rules for judging errors, # » # » Tzly (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)),where i = j, i, j ∈ {A, B, C}, there exists the following relationship: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) |n f h # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) |n f h # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))).

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Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), then yi (t)  y j (t) (or yi (t)  −y j (t)), where i = j, i, j ∈ {A, B, C}, then the left # » side of the equation is: Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) |n f h B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzly ((A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t))) ∧ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) ∧ # » # » Az B ¬C ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) ∨ # » # » (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))¬C Az B ((U (t), Su (t), # » # » # » pu (t), Tu (t), L u (t)), xC (t)= f ((μ(t), pu (t)), G C (t)))) ∨ (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ ¬B((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))¬C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), # » # » xC (t) = f ((μ(t), pu (t)), G C (t)))) ∨ (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t))) ∧ ¬B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))¬C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), # » # » G C (t))))) = (A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), # » # » G A (t))) ∧ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ # » # » ¬C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) ∨ # » # » (¬A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), # » # » Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ ¬C AnhbB ((V (t), # » # » Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) ∨ (A((V (t), Sv (t), # » # » # » pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ ¬B((V (t), Sv (t), pv (t), # » # » Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ ¬C Az B ((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) ∨ (¬A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∧ ¬B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » # » g((ν(t), pv (t)), G B (t))) ∧ ¬C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = # » # » g((ν(t), pv (t)), G C (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » # » # » pv (t)), G A (t))) |n f h B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), # » # » G B (t))) = Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t)))) |n f h Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. # » Proposition 4.197 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; sup-

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# » # » pose that C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) # » is the mediator variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))), which is defined in domain U (t) under G C (t) the rules for judging errors, # » # » Tzly (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)),where i = j, i, j ∈ {A, B, C}, there exists the following relationship: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ⊃nhb # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ⊃nhb # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)),where i = j, i, j ∈ {A, B, C}, then the left # » side of the equation is: Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) ⊃nhb B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzly ((A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t))) ∧ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) ∧ # » # » Az B C ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) ∧ # » # » Tzly (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), # » # » Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ C Az B ((V (t), Sv (t), # » # » # » pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) = Tzly (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) |n f h Tzly (B((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. # » Proposition 4.198 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; sup# » # » pose that C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) # » is the mediator variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t),

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# » # » # » pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))), which is defined in domain U (t) under G C (t) the rules for judging errors, # » # » Tzly (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)), where i = j, i, j ∈ {A, B, C}, there exists the following relationship: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) nhdl # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) nhdl # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), then yi (t)  y j (t) (or yi (t)  −y j (t)) holds, where i = j, i, j ∈ {A, B, C}: Tzly (A((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) nhdl B((U (t), Su (t), # » # » pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = Tzly (A((U (t), Su (t), # » # » # » pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ B((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) ∧ ¬C Az B ((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = Tzly (A((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ Tzly (B((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) ∧ Tzly ¬(C Az B ((U (t), Su (t), pu (t), Tu (t), # » # » L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » # » g((ν(t), pv (t)), G B (t))) ∧ C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), # » # » pv (t)), G C (t))) = Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » pu (t)), G A (t)))) nhdl Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » pu (t)), G B (t)))). Proof is completed. # » Proposition 4.199 Suppose that two error logical variables A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) # » the rules for judging errors, Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = # » # » f ((μ(t), pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), # » pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; if −x A (t)  x B (t) then −y A (t)  y B (t), or −x A (t)  x B (t) then −y A (t)  y B (t)),there exists the following relationship: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) → # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) =

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389

# » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) → # » # » Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if −x A (t)  x B (t) then −y A (t)  y B (t) holds,or if −x A (t)  x B (t) then −y A (t)  y B (t) holds: # » # » Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) → # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzly (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))))=¬A((V (t), # » # » # » Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), # » # » Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = (A((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) → B((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y B (t) = g((ν(t), pv (t)), G B (t)))) = Tzly (A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » # » x A (t) = f ((μ(t), pu (t)), G A (t)))) → Tzly (B((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. Because →nhy connotative possibility implication connective (if...then it is possible...), →nsy connotative isness implication connective (if...then it is...), →nby connotative necessity implication connective (if... then it is necessary), =nhdt connotative same connective (if... then it is the same), and ←→nhdz connotative equivalence connective (if...then it is equivalent) can be expressed by denotation connective →, we will not discuss each of them here.

4.3.3 Thing Displacement Transformation Connective in Error Logic 4.3.3.1

Concepts of Thing Displacement Decomposition Connectives in Error Logic

# » # » Suppose that Tz (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x(t) = f ((μ(t), pu (t)), G A (t)))) produces the result of {Siu (t), i = 1, 2, . . . , m} → {S jv (t), j = 1, 2, . . . , m}, then it is said that Tz is the thing transformation connective with regard to rule # » # » G A (t) and {Ai ((Ui (t), Si u(t), pi u(t), Ti u(t), L i u(t)), xi (t) = f i ((μi (t), pi u(t)), G i A (t)))), i = 1, 2, . . . , m}, which is denoted by Tzsw . In this case, displacement transformation is conducted on the thing of the object of interest (Ui (t), Si u(t), # » pi u(t), Ti u(t), L i u(t)).

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4.3.3.2

4 Transformation Connectives in Error Logic

Characteristics of Thing Displacement Decomposition Connectives in Error Logic

# » # » In Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y(t) = g((ν(t), p(t)), G A (t))), Tzsw has con# » ducted thing displacement transformation on the object of (U (t), Su (t), pu (t), Tu (t), L u (t)), where Tzsw is defined within U (t). Here Su (t) → Sv (t) indicates that thing Sv (t) is used to replace thing Su (t) for the purpose of achieving expected results. # » Proposition 4.200 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x(t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under G A (t) the rules for judging errors, another error logical variable A((V (t), Sv (t), # » # » pv (t), Tv (t), L v (t)), y(t) = g((ν(t), pv (t)), G A (t))) is defined in domain V (t) under # » # » G B (t) the rules for judging errors; if ∀(μ(t), pu (t)) = (U (t), Su (t), pu (t), Tu (t), L u (t)) ∈ U (t), Tzsw did not conduct domain displacement transformation on # » # » A((U (t), Su (t), pu (t), Tu (t), L u (t)), x(t) = f ((μ(t), pu (t)), G A (t))); if ∀(μ(t), # » # » / U (t)Tzsw has conducted domain dispu (t)) = (U (t), Su (t), pv u(t), Tu (t), L u (t)) ∈ # » placement transformation on A((U (t), Su (t), pu (t), Tu (t), L u (t)), x(t) = f ((μ(t), # » pu (t)), G A (t))). Proof Proof is omitted. # » Proposition 4.201 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tzsw (B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = B((V (t), Sv (t), # » # » pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors,if x A (t)  x B (t) then y A (t)  y B (t) or if x A (t)  x B (t) then y A (t)  y B (t), there exist the following relationships: # » # » (1) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∨ # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∨ # » # » B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))); # » # » (2) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ # » # » B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))); # » # » (3) Tzsw (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) = # » # » ¬A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))); hold. Proof From the assumption we know that if x A (t)  x B (t) then y A (t)  y B (t) or if x A (t)  x B (t) then y A (t)  y B (t), it is assumed that x A (t)  x B (t), then the left

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391

# » side of the equation is: Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) ∨ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))). # » As y A (t)  y B (t), the right side of the equation is: Tzsw (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨ Tzsw (B((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∨ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y B (t) = g((ν(t), pv (t)), G B (t))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = # » g((ν(t), pv (t)), G A (t))), left side = right side; it is assumed that x A (t)  x B (t), then # » the left side of the equation is: Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t))) ∨ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » pu (t)), G B (t)))) = Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))). # » As y A (t)  y B (t), the right side of the equation is: Tzsw (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨ Tzsw (B((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∨ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y B (t) = g((ν(t), pv (t)), G B (t))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » g((ν(t), pv (t)), G B (t))), left side = right side; Proof is completed. Similarly , (2) and (3) can be proved.

# » Proposition 4.202 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tzsw (B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = B((V (t), Sv (t), # » # » pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) # » under G B (t) the rules for judging errors; suppose that C Az B ((U (t), Su (t), pu (t), # » Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) is the mediator variable of A((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) and B((U (t), Su (t), # » # » pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))), which is defined in domain # » U (t) under G C (t) the rules for judging errors, Tzsw (C Az B ((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = C Az B ((V (t), Sv (t), pv (t), Tv (t), # » L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t) or if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t), there exist the following relationships:

392

4 Transformation Connectives in Error Logic

# » # » (1) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∨n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨n # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (2) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧n # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). hold. Proof From the assumption we know that if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t) or if x A (t)  x B (t)  xC (t), then y A (t)  y B (t)  yC (t), it is assumed that x A (t)  x B (t)  xC (t), then the left side of the equation is: # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∨n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), # » # » Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))); As y A (t)  y B (t)  yC (t), the right side of the equation is: Tzsw (A((U (t), Su (t), # » # » p (t), T (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∨n Tzsw (B((U (t), Su (t), # » # » #u » u pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = A((V (t), Sv (t), pv (t), # » # » Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∨n B((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y A (t) = g((ν(t), pv (t)), G A (t))) ∨ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = # » # » g((ν(t), pv (t)), G B (t))) ∨ C Az B ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), # » # » # » pv (t)), G C (t))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))), left side = right side. Similarly,other situations can also be proved. Proof is completed. Similarly,(2) can be proved. # » Proposition 4.203 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; suppose # » # » that C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) is # » the connotative inclusion variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » pu (t)), G B (t))), is defined in domain U (t) under G C (t) the rules for judging # » # » errors, Tzsw (C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)),

4.3 Displacement Transformation Connectives in Error Logic

393

# » # » G C (t)))) = C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), yi (t)  y j (t) (or yi (t)  −y j (t))hold, where i = j, i, j ∈ {A, B, C}, there exists the following relationship: # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) −n # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) −n # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), yi (t)  y j (t) (or yi (t)  −y j (t)) hold,where i = j, i, j ∈ {A, B, C}, then the # » left side of the equation is: Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t))) −n B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » p (t)), G B (t)))) = Tzsw ((A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » #u » pu (t)), G A (t))) ∧ ¬B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t))) ∧ ¬C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), # » # » G C (t)))) ∨ (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t))) ∧ ¬B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t))) ∧ ¬C AnhbB ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), # » # » G C (t))))) = (A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), # » # » G A (t))) ∧ ¬B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ # » # » C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) ∨ # » # » (¬A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ # » # » ¬B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ C AnhbB # » # » ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t)))) = A((V (t), # » # » Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) −n B((V (t), Sv (t), # » # » # » pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = Tzsw (A((U (t), Su (t), pu (t), # » # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) −n Tzsw (B((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. # » Proposition 4.204 Suppose that an error logical variable A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; sup# » # » pose that C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))) # » is the mediator variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) and B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)),

394

4 Transformation Connectives in Error Logic

G B (t))), which is defined in domain U (t) under G C (t) the rules for judging errors, # » # » Tzsw (C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) = # » # » Az B C ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), pv (t)), G C (t))) is defined in domain V (t) under G C (t) the rules for judging errors; if ∀xi (t)  x j (t) (or xi (t)  −x j (t)),then yi (t)  y j (t) (or yi (t)  −y j (t)),where i = j, i, j ∈ {A, B, C}, there exist the following relationships: # » # » (1) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) |n f l # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t)= f ((μ(t), pu (t)), G A (t)))) |n f l # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (2) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t)= f ((μ(t), pu (t)), G A (t))) |n f h # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t)= f ((μ(t), pu (t)), G A (t)))) |n f h # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (3) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t)= f ((μ(t), pu (t)), G A (t))) ⊃nhb # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t)= f ((μ(t), pu (t)), G A (t)))) ⊃nhb # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))); # » # » (4) Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) nhdl # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t)))) nhdl Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t)= f ((μ(t), pu (t)), G B (t)))); hold. Proof From the assumption we know that if ∀xi (t)  x j (t) (or xi (t)  −x j (t)), then yi (t)  y j (t) (or yi (t)  −y j (t)), where i = j, i, j ∈ {A, B, C}, then the left # » side of the equation is: Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), # » # » # » pu (t)), G A (t))) |n f l B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), # » # » G B (t)))) = Tzsw ((A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), # » # » G A (t))) ∧ B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) ∧ # » # » C Az B ((U (t), Su (t), pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t)))) ∨ # » # » (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) ∧ B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))¬C Az B ((U (t), Su (t), # » # » p (t), T (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))))) = (Tzsw (A((U (t), Su (t), # » #u » u p (t), T (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ Tzsw (B((U (t), Su (t), # » #u » u p (t), T (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) ∧ Tzsw (C Az B ((U (t), Su (t), # » #u » u p (t), T (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))))) ∨ (Tzsw (A((U (t), Su (t), # » #u » u p (t), T (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) ∧ Tzsw (B((U (t), Su (t), # » #u » u p (t), T (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))))Tzsw (¬C Az B ((U (t), Su (t), # » # » #u » u pu (t), Tu (t), L u (t)), xC (t) = f ((μ(t), pu (t)), G C (t))))) = (A((V (t), Sv (t), pv (t), # » # » Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) ∧ C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » yC (t) = g((ν(t), pv (t)), G C (t)))) ∨ (A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = # » # » g((ν(t), pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t),

4.3 Displacement Transformation Connectives in Error Logic

395

# » # » p (t)), G B (t))) ∧ ¬C AnhbB ((V (t), Sv (t), pv (t), Tv (t), L v (t)), yC (t) = g((ν(t), # » # » #v » pv (t)), G C (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), # » # » nf l G A (t))) | B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) |n f l # » # » Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. Similarly, (2), (3), and (4) can be proved. # » Proposition 4.205 Suppose that an error logical variables A((U (t), Su (t), pu (t), # » Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = # » # » f ((μ(t), pu (t)), G A (t)))) = A((V (t), Sv (t), pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), # » pv (t)), G A (t))) is defined in domain V (t) under G A (t) the rules for judging errors; # » # » suppose that B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t))) # » is the complement error logical variable of A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x A (t) = f ((μ(t), pu (t)), G A (t))), which is defined in domain U (t) under G B (t) the # » rules for judging errors, Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), # » # » # » pu (t)), G B (t)))) = B((V (t), Sv (t), pv (t), Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) is defined in domain V (t) under G B (t) the rules for judging errors; if −x A (t)  x B (t) then −y A (t)  y B (t) holds, or if −x A (t)  x B (t) then −y A (t)  y B (t)) holds, there exists the following relationship: # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) → # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = Tzsw # » # » (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t)))) → Tzsw # » # » (B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof From the assumption we know that if −x A (t)  x B (t) then −y A (t)  y B (t), or −x A (t)  x B (t) then −y A (t)  y B (t): # » # » Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t) = f ((μ(t), pu (t)), G A (t))) → # » # » B((U (t), Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = Tzsw # » # » (¬A((U (t), Su (t), pu (t), Tu (t), L u (t)), x A (t)= f ((μ(t), pu (t)), G A (t))) ∧ B((U (t), # » # » Su (t), pu (t), Tu (t), L u (t)), x B (t) = f ((μ(t), pu (t)), G B (t)))) = ¬A((V (t), Sv (t), # » # » # » pv (t), Tv (t), L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) ∧ B((V (t), Sv (t), pv (t), # » # » Tv (t), L v (t)), y B (t) = g((ν(t), pv (t)), G B (t))) = (A((V (t), Sv (t), pv (t), Tv (t), # » # » L v (t)), y A (t) = g((ν(t), pv (t)), G A (t))) → B((V (t), Sv (t), pv (t), Tv (t), L v (t)), # » # » y B (t) = g((ν(t), pv (t)), G B (t)))) = Tzsw (A((U (t), Su (t), pu (t), Tu (t), L u (t)), # » # » x A (t) = f ((μ(t), pu (t)), G A (t)))) → Tzsw (B((U (t), Su (t), pu (t), Tu (t), L u (t)), # » x B (t) = f ((μ(t), pu (t)), G B (t)))). Proof is completed. Because →nhy connotative possibility implication connective (if...then it is possible...), →nsy connotative isness implication connective (if...then it is...), →nby connotative necessity implication connective (if... then it is necessary), =nhdt connotative same connective (if... then it is the same), and ←→nhdz connotative equivalence connective (if...then it is equivalent) can be expressed by denotation connective →, we will not discuss each of them here.

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4.4 Increase Transformation Connectives in Error Logic 4.4.1 Increase Transformation Connectives in Error Logic # » Definition 4.38 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G(t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, is defined # » in domain U (t) under judging rule G(t), if T(A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t), G(t)))) = {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » # » G(t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An # » # » ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, then T is called the increase transformation connective defined in U (t) under G(t) rules for judging errors, which is denoted by Tz j . # » # » In Tz j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t), G(t))))={A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), S1 (t), p1 (t), # » # » T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » xn (t) = f n ((μn (t), pn (t), G n (t)))}, # » # » (1) A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))) ∈ # » # » U1 (t), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t)= f 2 ((μ2 (t), p2 (t), G 2 (t))) ∈ # » # » U2 (t), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t))) ∈ Un (t), and U (t) → U (t) ∪ U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t), then Tz j called the domain increase transformation connective with regard to A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t), G(t))), which is denoted by Tz jly . In this situation, increase transformation is conducted on the domain U (t) of object μ(t). For example, we can change the original domain of containing economy of Guangdong to the augmented domain of considering both economies of Guangdong, Hong Kong, and Macau if one wants to consider the impacts of economy for pan-pearl river delta region on economy of Guangdong province. # » # » (2) In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists μ(t) → μ(t) h μ1 (t) h μ2 (t) h, . . . , h μn (t), then Tz j has conducted thing increase transformation on A(μ(t), x(t)), which is denoted by Tz jsw named thing increase transformation connective. For example, suppose that university A is the domain and Si (t) represents ith college in this university, increase transformation can be carried out by adding the program of business analytics to the college of business. # » # » (3) In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t),

4.4 Increase Transformation Connectives in Error Logic

(4)

(5)

(6)

(7)

(8)

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# » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists the rela# » # » # » # » # » tionship of p(t) → p(t) + p1 (t) + p2 (t), . . . , pn (t), then Tz j is called the spatial increase transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jk j . # » # » In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), #2 » 2 # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists the relationship of T (t) → T (t) ∪ T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), then Tz j is called the property increase transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jt x . For example, in choosing the materials used for propelling nozzle of jet plane, the engineers will add some metals such as rhenium, chromium, carbon, titanium, and nickle to enhance the property of heat resistance besides qualities such as oxidation resistance and corrosion resistance. # » # » In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists the relationship of L(t) → L(t) + L 1 (t) + L 2 (t)+, . . . , +L n (t), then Tz j is called the property (or attribute) value increase transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jlz . For example, suppose that L(t) represents the length of drill pipe for crude oil exploitation, an Tz jlz transformation can increase the length the pipe to reach even deeper layer of the oil well. # » # » In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists the relationship of x(t) → x(t) + x1 (t) + x2 (t)+, . . . , + xn (t), then Tz j is called the error value increase transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jcz . # » # » In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p (t), T (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t), # » #2 » 2 pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists the relationship of G(t) → G(t) ∪ G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), then Tz j is called the error value increase transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jgz . For example, the addition of anti-unfair competition law of the People’s Republic of China to the current law system is a kind of increase transformation. # » # » In {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t), # » # » S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), # » # » p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G 2 (t))), . . . , An ((Un (t), Sn (t),

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# » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G n (t)))}, if there exists the relationship of f (t) → f (t) ∪ f 1 (t) ∪ f 2 (t)∪, . . . , ∪ f n (t), then Tz j is called the error value increase transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz j hs . For example, f (x) (Triangular function)= f (x) (Triangular function) + g(x) (Logarithm function). # » # » (9) Tz j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t), G(t)))) = {A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), # » # » T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 ), G 1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), # » T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G 2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # » # » pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G n (tn )))}, then Tz j is called the temporal similarity transformation connective with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jcz . For example, appropriately increasing storage time can improve the liquor quality. # » # » (10) Tz j (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t), G(t)))) = {A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G(t))), A1 ((U1 (t1 ), S1 (t1 ), p1 (t1 ), # » # » T1 (t1 ), L 1 (t1 )), x1 (t1 ) = f 1 ((μ1 (t1 ), p1 (t1 ), G 1 (t1 ))), A2 ((U2 (t2 ), S2 (t2 ), p2 (t2 ), # » T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 ((μ2 (t2 ), p2 (t2 ), G 2 (t2 ))), . . . , An ((Un (tn ), Sn (tn ), # » # » pn (tn ), Tn (tn ), L n (tn )), xn (tn ) = f n ((μn (tn ), pn (tn ), G n (tn )))}, there exist U (t) → U (t) ∪ U1 (t) ∪ U2 (t)∪, . . . , ∪Un (t), μ(t) → μ(t) h μ1 (t) h μ2 (t) # » # » # » # » # » h, . . . , h μn (t), p(t) → p(t) + p1 (t) + p2 (t), . . . , pn (t), T (t) → T (t) ∪ T1 (t) ∪ T2 (t)∪, . . . , ∪Tn (t), L(t) → L(t) + L 1 (t) + L 2 (t)+, . . . , + L n (t), x(t) → x(t) + x1 (t) + x2 (t)+, . . . , + xn (t), G(t) → G(t) ∪ G 1 (t) ∪ G 2 (t)∪, . . . , ∪G n (t), and f (t) → f (t) ∪ f 1 (t) ∪ f 2 (t)∪, . . . , ∪ f n (t), then Tz j is with respect to G(t) and A(μ(t), x(t)) in U (t), which is denoted by Tz jq . In this case, in order to attain one’s objective, overall transformation has been conducted on domain, thing, property, value of the property or attribute, error function, time, space, error value, and rules for judging errors in the object μ(t). (11) The corresponding inverse transformation connectives of similarity transformation connectives Tz j ⊆ {Tz jly , Tz jsw , Tz jk j , Tz jt x , Tz jlz , Tz jcz , Tz jgz , Tz j hs , Tz js j , −1 −1 −1 −1 −1 −1 −1 −1 −1 Tz jq } are Tx−1 ⊆ {Tz−1 jly , Tz jsw , Tz jk j , Tz jt x , Tz jlz , Tz jcz , Tz jgz , Tz j hs , Tz js j , Tz jq }. 4.4.1.1

Principles for Error Value Increase Transformation in Error Logic

The principles for increase transformation are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

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4.4.2 Characteristics of Error Value Increase Transformation Connective in Error Logic # » Proposition 4.206 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » # » G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) → x(t) + x1 (t) + x2 (t)+, . . . , +xn (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tz jcz (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = # » # » f ((ν (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), # » # » # 1 »1 p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where y(t) → y(t) + y1 (t) + y2 (t)+, . . . , +yn (t); if x(t)  y(t), ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t) holds, then the following relationships hold: # » # » (1) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) ∨ B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Tz jcz (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ Tz jcz (B((U (t), S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t))) ∧ B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Tz jcz (A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ Tz jcz (B((U (t), S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) Tz jcz (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » # » Proof As Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » # » L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} and Tz jcz (B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), # » # » # » p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), # » # » G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), # » # » G Bn (t)))}, if {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), # » # » A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B((U (t), S(t),

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# » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {C((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t))), C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), # » # » # » p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), # » # » G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), # » # » G Cn (t)))}, where Ci ((Ui (t), Si (t), pi (t), Ti (t), L i (t)), z i (t) = f i ((ωi (t), pi (t), G Ci (t))) = 

# » # » Ai ((Ui , Si (t), pi (t), Ti (t), L i (t)), xi (t)= f i ((μi (t), pi (t), G Ai (t))), xi (t)  yi (t) # » # » Bi ((Ui , Si (t), pi (t), Ti (t), L i (t)), yi (t)= f i ((νi (t), pi (t), G Bi (t))), xi (t)  yi (t)

# » As x(t)  y(t), the left side = Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))))=Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t)= f ((μ(t), p(t)), G A (t)))) = # » # » {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), A1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}. And from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t), the right side # » # » = Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ Tz jcz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » # » xn (t) = f n ((μn (t), pn (t), G An (t)))} ∨ {B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = f 1 ((ν1 (t), # » # » # » p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), # » # » G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), # » # » G Bn (t)))} = {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), # » # » A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), A2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}. Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.207 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » # » G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))),

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# » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) → x(t) + x1 (t) + x2 (t)+, . . . , +xn (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tz jcz (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = # » # » f ((ν (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), # 1 »1 # » p (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), #2 » pn (t), G Bn (t)))}, where y(t) → y(t) + y1 (t) + y2 (t)+, . . . , +yn (t); suppose that # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the media# » # » tor variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Tz jcz (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = {C((U (t), S(t), p(t), # » # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = # » # » f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = # » f n ((ωn (t), pn (t), G Cn (t)))}, where z(t) → z(t) + z 1 (t) + z 2 (t)+, . . . , +z n (t); if x(t)  y(t)  z(t)  0, ∀i, i ∈ {1, 2, . . . , n}, then xi (t)  yi (t)  z i (t)  0 holds, then the following relationships hold: # » # » (1) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=Tz jcz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n Tz jcz (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=Tz jcz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n Tz jcz (B((U (t), S(t), # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Tz jcz # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l Tz jcz # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=Tz jcz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h Tz jcz (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (5) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=Tz jcz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb Tz jcz (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (6) Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))))=T f s j (A((U (t),

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4 Transformation Connectives in Error Logic

# » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl Tz jcz (B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » Proof As x(t)  y(t)  z(t)  0, the left side = Tz jcz (A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))))=Tz jcz (C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t)= f ((ω(t), # » # » # » p(t)), G C (t)))) = {C((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), # » # » C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t)= f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), # » # » pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}; and from the assumption of ∀i, i ∈ {1, 2, . . . , n}, xi (t)  yi (t)  z i (t)  0, the right side = Tz jcz (A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n Tz jcz (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t), G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = # » # » f ((μ (t), p1 (t), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), # » #1 » 1 p (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # » # » #2 » pn (t), G An (t)))} ∨n {B((U (t), S(t), p(t), T (t), L(t)), y(t)= f ((ν(t), p(t)), G B (t))), # » # » B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t)= f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((Un (t), # » # » Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))} = {C((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), C1 ((U1 (t), S1 (t), p1 (t), T1 (t), # » # » L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), # » # » z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # » z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}. Left side = right side. Proof is completed. Similarly, (2)–(6) can also be proved. # » Proposition 4.208 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » # » G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x(t) → x(t) + x1 (t) + x2 (t)+, . . . , +xn (t); suppose that another error logical vari# » # » able B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tz jcz (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = # » # » f ((ν (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), # » # 1 »1 p (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), #2 » pn (t), G Bn (t)))}, where y(t) → y(t) + y1 (t) + y2 (t)+, . . . , +yn (t); suppose that

4.4 Increase Transformation Connectives in Error Logic

403

# » # » C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the con# » notative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » AnhbB Tz jcz (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » {C((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), C1 ((U1 (t), S1 (t), # » # » # » p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), # » # » T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), # » L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where z(t) → z(t) + z 1 (t) + z 2 (t)+, . . . , +z n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −yi (t), and z i (t) is the same as that of x(t), −y(t), and z(t), then the following relationship holds: # » # » Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) −n B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Tz jcz (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n Tz jcz (B((U (t), S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted.

# » Proposition 4.209 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tz jcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = {A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » # » G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t) = f 1 ((μ1 (t), p1 (t), G A1 (t))), # » # » A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , # » # » An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), pn (t), G An (t)))}, where error value x(t) → x(t) + x1 (t) + x2 (t)+, . . . , +xn (t); suppose that another error # » # » logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, Tz jcz (B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = {B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), y1 (t) = # » # » f ((ν (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f 2 ((ν2 (t), # » # » # 1 »1 p2 (t), G B2 (t))), . . . , Bn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f n ((νn (t), pn (t), G Bn (t)))}, where error value y(t) → y(t) + y1 (t) + y2 (t)+, . . . , +yn (t); suppose # » # » that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) # » is the connotative same or equivalence variable for A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))), Tz jcz (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) = {C((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t))), C1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), # » # » G C1 (t))), C2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), # » # » . . . , Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t) = f n ((ωn (t), pn (t), G Cn (t)))}, where error value z(t) → z(t) + z 1 (t) + z 2 (t)+, . . . , z n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi (t), −xi (t), yi (t), −yi (t), and z i (t) is the same as that of x(t), −x(t), y(t), −y(t), and z(t), then the following relationship holds: Tz jcz (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Tz jcz (A((U (t), S(t), p(t), T (t),

404

4 Transformation Connectives in Error Logic

# » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby Tz jcz (B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted.

# » Proposition 4.210 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Tz jcz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), G (n) A (t))))={(A ((U # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) p (t)), G (n) A (t)))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t)= # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) → x (t) + x1 (t) + # » x2(n) (t)+, . . . , +xn(n) (t); suppose that A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) order error logical variable defined in domain U (t) under G (n+1) (t) the rules for A # » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), judging errors, Tz jcz (A # » (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = {A(n+1) ((U (n+1) (t), S (n+1) # » # » (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » G (n+1) (t))), A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = 1 1 A # » # » (n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), f 1(n+1) ((μ(n+1) 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n+1) An ((Un # » (n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))}, where error value x (n+1) (t) → x (n+1) (t) + n (n+1) x1 (t) + x2(n+1) (t)+, . . . , +xn(n+1) (t); suppose that error logical variable B (n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))) # » is the complement variable for A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » (n) x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t) under A (t))), which is defined in domain U # » (n) (n) (n) (n) (n) (n) G B (t) the rules for judging errors, Tz jcz (B ((U (t), S (t), p (t), T (t), # » # » (n) (n) L (n) (t)), y (n) (t)= f (n) ((ν (n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), B (t))))={B ((U # » (n) (n) (n) T (n) (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))), B1 ((U1 (t), S1 (t), # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), y1 (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), y2 (t) = f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , # » # (n) » (n) (n) (n) Bn(n) ((Un(n) (t), Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) n (t)), yn (t) = f n ((νn (t), pn (t), (n) (n) (n) (n) (n) G (n) Bn (t)))}, where error value y (t) → y (t) + y1 (t) + y2 (t)+, . . . , +yn (t); # » suppose that error logical variable B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) is the complement B

4.4 Increase Transformation Connectives in Error Logic

405

# » variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » (t))), which is defined in domain U (n+1) (t) f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) under G B (t) the rules for judging errors, Tz jcz (B (n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), y (n+1) (t)= f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t)))) B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), y (t) = f (n+1) = {B # (n+1) » (n+1) # » (n+1) (n+1) (n+1) (n+1) ((ν (n+1) (t), p (n+1) (t)), G B (t))), B1 ((U1 (t), S1 (t), p1 (t), T1 (t), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), L 1 (t)), y1 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (t), p2 (t), T2 (t), L 2 (t)), y2 (t) = f ((ν (t), p (t), S2 # (n+1) » 2 (n+1) 2 (n+1) 2 (n+1) (n+1) (n+1) (n+1) (t))), . . . , B ((U (t), S (t), p (t), T (t), L (t)), y G (n+1) n n n n n n B2 # n » (n+1) (n+1) (t)= f n(n+1) ((νn(n+1) (t), pn(n+1) (t), G (n+1) (t)))}, where error value y (t) → y Bn (n+1) (n+1) (t) + y1 (t) + y2 (t)+, . . . , +yn(n+1) (t); suppose that C (n)Az B ((U (n) (t), S (n) (t), # (n) » (n) # » (n) p (t), T (t), L (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t))) is the mediator # » variable for A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), y (n) (t) = A (t))) and B ((U # » # » (n)Az B f (n) ((ν (n) (t), p (n) (t)), G (n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), B (t))), Tz jcz (C # » # » L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t))))={C (n) ((U (n) (t), S (n) (t), p (n) (t), # » T (n) (t), L (n) (t)), z (n) (t)= f (n) ((ω(n) (t), p (n) (t)), G C(n) (t))), C1(n) ((U1(n) (t), S1(n) (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) (n) p1 (t), T1 (t), L (n) 1 (t)), z 1 (t) = f 1 ((ω1 (t), p1 (t)), G C1 (t))), C 2 ((U2 (t), # » # » (n) (n) (n) (n) (n) S2(n) (t), p2(n) (t), T2(n) (t), L (n) 2 (t)), z 2 (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , # » # (n) » (n) (n) (n) Cn(n) ((Un(n) (t), Sn(n) (t), pn(n) (t), Tn(n) (t), L (n) n (t)), z n (t) = f n ((ωn (t), pn (t), (n) G Cn (t)))}, where z (n) (t) → z (n) (t) + z 1(n) (t) + z 2(n) (t)+, . . . , +z n(n) (t); suppose that # » C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) # » ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) G (n+1) A # » (t)= f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))), Tz jcz (C (n+1)Az B ((U (n+1) (t), S (n+1) (t), B # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), # » G C(n+1) (t))))={C (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) # » # » = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))), C1(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), # » (n+1) T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), C2(n+1) 1 # » ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), 2 # (n+1) » # » (n+1) p2 (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), # » (n+1) L (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), pn(n+1) (t), G Cn (t)))}, where z (n+1) (t) → n (n+1) (n+1) (n+1) (n+1) (t) + z 1 (t) + z 2 (t)+, . . . , +z n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order z of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x (n) (t), y (n) (t), and # » z (n) (t), then the following relationship holds: Tz jcz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t),

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4 Transformation Connectives in Error Logic

# » bz (n) (n) T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t))))=¬ Tz jcz (A ((U # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n+1) (t), yi(n+1) (t), and z i(n+1) (t) is the same as that of x n (t), y n (t), and z n (t), here it is assumed that x n (t)  y n (t)  z n (t), and there exists the relationship of xi(n+1) (t)  yi(n+1) (t)  z i(n+1) (t), based on the definition for ¬bz , the left side of the equation = # » Tz jcz (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = Tz jcz (A # » L (n+1) (t)), x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), A # » # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), p (t), T (t), L (t)), y (t) = f ((ν (n+1) (t), p (n+1) (t)), S # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) B # » z (n+1) (t)= f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = Tz jcz (C (n+1)Az B ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))). # » The right side of the equation = ¬bz Tz jcz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bz (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t)))) = ¬ {A ((U # (n) » (n) # » (n) (n) p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G A (t))), A1 ((U1(n) (t), # » # (n) » (n) (n) (n) (n) S1(n) (t), p1(n) (t), T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t)= f n ((μn (t), # (n) » (n) # » pn (t), G An (t)))} = {A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » x (n+1) (t)= f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))), A(n+1) ((U1(n+1) (t), S1(n+1) (t), 1 A # (n+1) » (n+1) # » p1 (t), T1 (t), L (n+1) (t)), x1(n+1) (t)= f 1(n+1) ((μ(n+1) (t), p1(n+1) (t)), G (n+1) 1 1 A1 (t))), # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n+1) A2 ((U2 # » # » (n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), 2 A2 (t))), . . . , An # » (n+1) (t)), xn(n+1) (t) = f n(n+1) ((μ(n+1) (t), pn(n+1) (t), G (n+1) Tn(n+1) (t), L (n+1) n n An (t)))} ∧ {B # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), # » # (n+1) » p (t)), G (n+1) (t))), B1(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), 1 B # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f 1 ((ν1 (t), p1 (t)), G B1 (t))), B2 ((U2 (t), S2 (t), y1 # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p2 (t), T2 (t), L 2 (t)), y2 (t) = f ((ν (t), p (t), G (t))), # (n+1) » 2(n+1) 2 (n+1) 2 (n+1) B2 (n+1) (n+1) (n+1) (n+1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), yn (t) = f . . . , Bn # (n+1) » (n+1) # (n+1) » n(n+1) (n+1) (n+1) (n+1) (n+1) ((νn (t), pn (t), G Bn (t)))} ∧ {C ((U (t), S (t), p (t), T # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), L (t)), z (t) = f ((ω (t), p (t)), G C (t))), C1 ((U1(n+1) # » (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) ((ω1(n+1) (t), 1 # (n+1) » # » (n+1) p1 (t)), G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t),

4.4 Increase Transformation Connectives in Error Logic

407

# » (n+1) L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , Cn(n+1) 2 # » ((Un(n+1) (t), Sn(n+1) (t), p (n+1) (t), Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) n # (n+1) »n (n+1) (n+1) (n+1) (n+1) (n+1) ((ωn (t), pn (t), G Cn (t)))} = {C ((U (t), S (n+1) (t), = fn # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), z (n+1) (t)= f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))), # » C1(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), z 1(n+1) (t) = f 1(n+1) 1 # » # » (n+1) ((ω1(n+1) (t), p1(n+1) (t)), G C1 (t))), C2(n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), # » (n+1) L (n+1) (t)), z 2(n+1) (t) = f 2(n+1) ((ω2(n+1) (t), p2(n+1) (t), G C2 (t))), . . . , Cn(n+1) ((Un(n+1) 2 # » (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), z n(n+1) (t) = f n(n+1) ((ωn(n+1) (t), n # (n+1) » (n+1) # » pn (t), G Cn (t)))} = Tz jcz (C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))). Left side = right side. Proof is completed. # » Proposition 4.211 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Tz jcz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = G (n) A (t))))={A ((U # (n) » (n) # » (n) (n) (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # (n) » (n) (n) (n) (n) (n) (n) Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) → x (t) (n) (n) (n) (n+1) (n+1) (n+1) + x1 (t) + x2 (t)+, . . . , +xn (t); suppose that A ((U (t), S (t), # (n+1) » (n+1) # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) p (t), T (t), L (t)), x (t)= f ((μ (t), p (t)), G A (t))) is the (n + 1)th order error logical variable defined in domain U (n+1) (t) under # » G (n+1) (t) the rules for judging errors, Tz jcz (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n+1) {A # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))), A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), 1 A # (n+1) » (n+1) (n+1) (n+1) (n+1) (t)), x (t) = f ((μ (t), p1 (t)), G (n+1) ((U2(n+1) (t), L (n+1) 1 1 1 1 A1 (t))), A2 # » # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), S2(n+1) (t), p2(n+1) (t), T2(n+1) (t), L (n+1) 2 2 # » (n+1) ((Un(n+1) (t), Sn(n+1) (t), pn(n+1) (t), Tn(n+1) (t), L (n+1) (t)), xn(n+1) G (n+1) n A2 (t))), . . . , An # » (n+1) (n+1) (n+1) (n+1) (n+1) (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) → x (n+1) (t) + (n+1) (n+1) x1 (t) + x2 (t)+, . . . , +x (n+1) (t); suppose that C (n)Az B(n+1) ((U (n)(n+1) (t), # (n)(n+1) » (n)(n+1)n (n)(n+1) (t), p (t), T (t), L (n)(n+1) (t)), z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) S # (n)(n+1) » (t), p (t)), G C(n)(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)),

408

4 Transformation Connectives in Error Logic

# » G (n+1) (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) A # » ((μ(n) (t), p (n) (t)), G (n) Tz jcz (C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), A (t))), # (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), # (n)(n+1) » # » p (t)), G C(n)(n+1) (t)))) = {C (n)(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), # » T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), # » G C(n)(n+1) (t))), C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), # » (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), G C1 (t))), L (n)(n+1) 1 # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) C2 ((U2 (t), S2 (t), p (t), T (t), L 2 (t)), z 2 # (n)(n+1) 2» (n)(n+1) 2 (n)(n+1) (n)(n+1) (n)(n+1) (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un(n)(n+1) (t), # » Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) n # (n)(n+1) » (n)(n+1) (t), pn (t), G Cn (t)))}, here error value z (n)(n+1) (t) → z (n)(n+1) (n)(n+1) (n)(n+1) (t)+z 1 (t) + z 2 (t)+, . . . , +z n(n)(n+1) (t); if ∀i, i ∈ {1, 2 . . . n}, the order (n) (n+1) of size for xi (t), xi (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), then the following relationship holds: Tz jcz (¬bx A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » bx p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ Tz jcz # » # » (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n+1) (t), and z i(n)(n+1) (t) is the same as that of x n (t), x n+1 (t), and z (n)(n+1) (t), it is assumed that x n (t)  x n+1 (t)  z (n)(n+1) (t), then there exists xi(n) (t)  xi(n+1) (t)  z i(n)(n+1) (t), # » the left side of the equation = Tz jcz (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), A (t)))) = Tz jcz (A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), G (n+1) A # (n) » # » (n)Az B(n+1) p (t)), G (n) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), A (t))) ∧ C # » L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » Tz jcz (C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), # » z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p ((n)n+1) (t)), G C(n)(n+1) (t)))). # » And the right side of the equation = ¬bx Tz jcz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bx (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ {A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t))), A(n) 1 # » # (n) » (n) (n) (n) (n) ((U1(n) (t), S1(n) (t), p1(n) (t), T1(n) (t), L (n) (t)), x (t)= f ((μ (t), p (t)), G (t))), 1 1 1 1 1 A1 # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p2 (t), A(n) 2 2 2 2 2 2 2 2 2 # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n ((μ(n) n (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) (n) pn (t), G An (t)))} = {A ((U (t), S (t), p (t), T (t), L (t)), x (t) = # (n) » (n) # » (n) (n) (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)),

4.4 Increase Transformation Connectives in Error Logic

409

# (n) » # (n) » (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # (n) » (n) # (n) » (n) (n) (n) (n) (n) (t)), x (t)= f ((μ (t), p2 (t), G A2 (t))), . . . , A(n) L (n) n ((Un (t), Sn (t), pn (t), 2 2 2 2 # (n) » (n) (n) (n) (n) (n+1) Tn(n) (t), L (n) ((U (n+1) (t), n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A # » # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n+1) (t)), # » G (n+1) (t))), A(n+1) ((U1(n+1) (t), S1(n+1) (t), p1(n+1) (t), T1(n+1) (t), L (n+1) (t)), x1(n+1) (t) = 1 1 A # » # » (n+1) (t), p1(n+1) (t)), G (n+1) ((U2(n+1) (t), S2(n+1) (t), p2(n+1) (t), f 1(n+1) ((μ(n+1) 1 A1 (t))), A2 # » (t)), x2(n+1) (t) = f 2(n+1) ((μ(n+1) (t), p2(n+1) (t), G (n+1) T2(n+1) (t), L (n+1) 2 2 A2 (t))), . . . , # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n+1) An ((Un # » (n+1) ((μ(n+1) (t), pn(n+1) (t), G An (t)))} ∧ {C (n)(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), n # (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), # » # (n)(n+1) » p (t)), G C(n)(n+1) (t))), C1(n)(n+1) ((U1(n)(n+1) (t), S1(n)(n+1) (t), p1(n)(n+1) (t), # » T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) (t), p1(n)(n+1) (t)), 1 # » (n)(n+1) G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), # » (n)(n+1) L (n)(n+1) (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), G C2 (t))), . . . , 2 # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Cn ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (t) = f n ((ωn (t), p (t), G Cn (t)))} = {C zn # (n)(n+1) » n (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t), S (t), p (t), T (t), L (t)), z (n)(n+1) (t) = # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((ω (t), p (t)), G C (t))), C1 ((U1(n)(n+1) (t), f # » S1(n)(n+1) (t), p1(n)(n+1) (t), T1(n)(n+1) (t), L (n)(n+1) (t)), z 1(n)(n+1) (t) = f 1(n)(n+1) ((ω1(n)(n+1) 1 # (n)(n+1) » # » (n)(n+1) (t), p1 (t)), G C1 (t))), C2(n)(n+1) ((U2(n)(n+1) (t), S2(n)(n+1) (t), p2(n)(n+1) (t), # » (t)), z 2(n)(n+1) (t) = f 2(n)(n+1) ((ω2(n)(n+1) (t), p2(n)(n+1) (t), T2(n)(n+1) (t), L (n)(n+1) 2 # » (n)(n+1) (t))), . . . , Cn(n)(n+1) ((Un(n)(n+1) (t), Sn(n)(n+1) (t), pn(n)(n+1) (t), Tn(n)(n+1) (t), G C2 # » (n)(n+1) L (n)(n+1) (t)), z n(n)(n+1) (t) = f n(n)(n+1) ((ωn(n)(n+1) (t), pn(n)(n+1) (t), G Cn (t)))} = n # » (n)Az B(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) Tz jcz (C ((U (t), S (t), p (t), T (t), L (t)), # » z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p ((n)n+1) (t)), G C(n)(n+1) (t)))). Left side = right side. Proof is completed. Proposition 4.212 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, Tz jcz (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = A (t)))) = {A ((U # (n) » (n) # » (n) (n) (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t),

410

4 Transformation Connectives in Error Logic

# (n) » (n) (n) (n) (n) (n) (n) Tn(n) (t), L (n) n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) → x (t) (n) (n) (n) (n−1) (n−1) (n−1) + x1 (t) + x2 (t)+, . . . , +xn (t); suppose that A ((U (t), S (t), # (n−1) » (n−1) # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p (t), T (t), L (t)), x (t)= f ((μ (t), p (t)), G A (t))) is the (n − 1)th order error logical variable defined in domain U (n−1) (t) under # » G (n−1) (t) the rules for judging errors, Tz jcz (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f (n−1) {A # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t) = f 1 ((μ1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), L 1 (t)), x1 # (n−1) » (n−1) # (n+1) » (n−1) (n+1) (n+1) (n+1) (n+1) (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2 ((μ2 (t), p2 (t), S2 # (n−1) » (n+1) (n−1) (n−1) (n−1) (n−1) (t))), . . . , A ((U (t), S (t), p (t), T (t), L (t)), G (n+1) n n n n n A2 # (n−1) » n(n−1) (n−1) (n−1) (t), p (t), G (t)))}, where x (t) → x (t) + xn(n−1) (t)= f n(n−1) ((μ(n−1) n n An x1(n−1) (t) + x2(n−1) (t)+, . . . , +xn(n−1) (t); suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))) B is the (n − 1)th order error logical complementary variable of A(n−1) ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), (n−1) (n−1) G A (t))) defined in domain U (n−1) (t) under G B (t) the rules for judging errors, # » Tz jcz (B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = # » # » f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t)))) = {B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), B # » (t))), B1(n−1) T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), y1(n−1) (t) = f 1(n−1) ((ν1(n−1) (t), 1 # (n−1) » # » (n−1) p1 (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n+1) (t)), 2 B1 (t))), B2 # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n−1) (n−1) (t)= f 2 ((ν2 (t), p2 (t), G B2 (t))), . . . , Bn ((U (t), Sn (t), y2 # (n−1) » (n−1) # (n−1)n » (n−1) (n−1) (n−1) (n−1) (n−1) pn (t), Tn (t), L n (t)), yn (t)= f n ((νn (t), pn (t), G Bn (t)))}, where y (n−1) (t) → y (n−1) (t) + y1(n−1) (t) + y2(n−1) (t)+, . . . , +yn(n−1) (t). # » Suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))), Tz jcz T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) (C ((U (t), S (t), p (t), T (t), L (t)), z (t) = f (n−1) # » # » (n−1) ((ω(n−1) (t), p (n−1) (t)), G C (t)))) = {C (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), # » T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))), C1(n−1) # » ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) ((ω1(n−1) (t), 1 # (n−1) » # » (n−1) p1 (t)), G C1 (t))), C2(n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n−1) (t)), 2 # (n−1) » (n−1) (n−1) (n−1) (n−1) (t)= f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn(n−1) ((Un(n−1) (t), Sn(n−1) (t), z2

4.4 Increase Transformation Connectives in Error Logic

411

# (n−1) » (n−1) # » (n−1) pn (t), Tn (t), L (n−1) (t)), z n(n−1) (t)= f n(n−1) ((ωn(n−1) (t), pn(n−1) (t), G Cn (t)))}, n (n−1) (n−1) (n−1) (n−1) (n−1) where z (t) → z (t) + z 1 (t) + z 2 (t)+, . . . , +z n (t); if ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), then the following relationship holds: # » Tz jcz (¬bj A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bj (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) A (t)))) = ¬ Tz jcz (A ((U # » (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n−1) (t), yi(n−1) (t), and z i(n−1) (t) is the same as that of x (n−1) (t), y (n−1) (t), and z (n−1) (t), it is assumed that x (n−1) (t)  y (n−1) (t)  z (n−1) (t), there exists xi(n−1) (t)  yi(n−1) (t)  z i(n−1) (t), the # » left side of the equation = Tz jcz ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » (n−1) x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), S (n−1) (t), A (t)))) = Tz jcz (A # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), y (t) = f (n−1) ∧B # (n−1) » # » (n−1) (n−1) (n−1)Az B (n−1) (n−1) ((ν (t), p (t)), G B (t))) ∧ C ((U (t), S (t), p (n−1) (t), # » T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = Tz jcz # » (C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) # » ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))). # » And the right side of the equation = ¬bj Tz jcz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bj (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ {A ((U # » # » (n) S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G A (t))), A(n) 1 # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) (n) ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x1 (t)= f 1 ((μ1 (t), p1 (t)), G A1 (t))), # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) A(n) 2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t) = f 2 ((μ2 (t), p2 (t), # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G (n) A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), x n (t)= f n ((μn (t), # (n) » (n) # » pn (t), G An (t)))} = {A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))), A(n−1) ((U1(n−1) (t), S1(n−1) (t), 1 A # (n−1) » (n−1) # » p1 (t), T1 (t), L (n−1) (t)), x1(n−1) (t)= f 1(n−1) ((μ(n−1) (t), p1(n−1) (t)), G (n−1) 1 1 A1 (t))), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x2 (t) = f 2(n−1) A2 ((U2 # » # » (n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), 2 A2 (t))), . . . , An # » (t)), xn(n−1) (t) = f n(n−1) ((μ(n−1) (t), pn(n−1) (t), G (n−1) Tn(n−1) (t), L (n−1) n n An (t)))} ∧ # » {B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) # » # » ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))), B1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), B # » (n−1) (t)), y1(n−1) (t) = f 1(n−1) ((ν1(n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), L (n−1) 1 B1 (t))), B2 # » # » (t)), y2(n−1) (t) = f 2(n−1) ((ν2(n−1) (t), p2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), L (n+1) 2 # » (n−1) ((Un(n−1) (t), Sn(n−1) (t), pn(n−1) (t), Tn(n−1) (t), L (n−1) (t)), yn(n−1) G (n−1) n B2 (t))), . . . , Bn # » (n−1) (t) = f n(n−1) ((νn(n−1) (t), pn(n−1) (t), G Bn (t)))} ∧ {C (n−1) ((U (n−1) (t), S (n−1) (t),

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4 Transformation Connectives in Error Logic

# (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), z (n−1) (t)= f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))), # » C1(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), z 1(n−1) (t) = f 1(n−1) 1 # » # » (n−1) ((ω1(n−1) (t), p1(n−1) (t)), G C1 (t))), C2(n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), T2(n−1) (t), # » (n−1) (t)), z 2(n−1) (t) = f 2(n−1) ((ω2(n−1) (t), p2(n−1) (t), G C2 (t))), . . . , Cn(n−1) ((Un(n−1) L (n−1) 2 # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) (t), Sn (t), pn (t), Tn (t), L n (t)), z n (t)= f n ((ωn(n−1) (t), pn(n−1) (t), (n−1) G Cn (t)))}. Left side = right side. Proof is completed. # » Proposition 4.213 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Tz jcz (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t))))={A ((U # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) p (t)), G (n) A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), x 1 (t) = f 1 # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), L 2 (t)), x 2 (t)= # » # » (n) (n) (n) (n) (n) (n) (n) (n) f 2(n) ((μ(n) 2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), # (n) » (n) (n) (n) (n) (n) (n) (n) xn (t) = f n ((μn (t), pn (t), G An (t)))}, where x (t) → x (t) + x1 (t) + # » x2(n) (t)+, . . . , +xn(n) (t); suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) # » (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th A (n−1) order error logical variable defined in domain U (t) under G (n−1) (t) the rules for A # » (n−1) (n−1) (n−1) (n−1) (n−1) judging errors, Tz jcz (A ((U (t), S (t), p (t), T (t), L (n−1) (t)), # » (n−1) x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t)))) = {A(n−1) ((U (n−1) (t), # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (t), p (t), T (t), L (t)), x (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), S # » (t))), A(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = G (n−1) 1 1 A # » # » (n−1) (t), p1(n−1) (t)), G (n−1) ((U2(n−1) (t), S2(n−1) (t), p2(n−1) (t), f 1(n−1) ((μ(n−1) 1 A1 (t))), A2 # » (t)), x2(n−1) (t) = f 2(n−1) ((μ(n−1) (t), p2(n−1) (t), G (n−1) T2(n−1) (t), L (n−1) 2 2 A2 (t))), . . . , # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n(n−1) # » (n−1) ((μ(n−1) (t), pn(n−1) (t), G An (t)))}, where x (n−1) (t) → x (n−1) (t)+x1(n−1) n (t) + x2(n−1) (t)+, . . . , +xn(n−1) (t); it is assumed that C (n)Az B(n−1) ((U (n)(n−1) (t), # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) # (n)(n−1) » (t), p (t)), G C(n)(n−1) (t))) is the mediator variable for A(n−1) ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) G (n−1) A # (n) » # » (n)Az B(n−1) (t), p (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), A (t))), and Tz jcz (C # » T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), # » G C(n)(n−1) (t)))) = {C (n)(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t),

4.4 Increase Transformation Connectives in Error Logic

413

# » L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))), # » C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), L (n)(n−1) (t)), z 1(n)(n−1) 1 # » (n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), # » (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), L (n)(n−1) 2 # (n)(n−1) » (n)(n−1) # » (t), p2 (t), G C2 (t))), . . . , Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), # » Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), n (n)(n−1) (n)(n−1) G Cn (t)))} where z (t) → z (n)(n−1) (t) + z 1(n)(n−1) (t) + z 2(n)(n−1) (t)+, . . . , (n)(n−1) +z n (t); if ∀i, i ∈ {1, 2,. . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and (n)(n−1) zi (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), then the fol# » lowing relationship holds: Tz jcz (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » # » bd (n) (n) x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), A (t))))=¬ Tz jcz (A ((U # » T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof Because ∀i, i ∈ {1, 2, . . . , n}, the order of size for xi(n) (t), xi(n−1) (t), and z i(n)(n−1) (t) is the same as that of x (n) (t), x (n−1) (t), and z (n)(n−1) (t), it is assumed that x (n) (t)  x (n−1) (t)  z (n)(n−1) (t), then there exists xi(n) (t)  xi(n−1) (t)  z i(n)(n−1) (t), # » the left side of the equation = T f s j ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Tz jcz (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), G (n−1) A # (n) » # » (n)Az B(n−1) p (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), A (t))) ∧ C # » L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » Tz jcz (C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), # » z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))). # » And the right side of the equation = ¬bd Tz jcz (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bd (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ {A ((U # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t))), A(n) 1 # » # (n) » (n) (n) (n) (n) ((U1(n) (t), S1(n) (t), p1(n) (t), T1(n) (t), L (n) (t)), x (t)= f ((μ (t), p (t)), G (t))), 1 1 1 1 1 A1 # (n) » (n) # (n) » (n) (n) (n) (n) (n) (n) ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p2 (t), A(n) 2 2 2 2 2 2 2 2 2 # (n) » (n) (n) (n) (n) (n) (n) (n) (n) G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t)= f n ((μ(n) n (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) (n) pn (t), G An (t)))} = {A ((U (t), S (t), p (t), T (t), L (t)), x (t) = # (n) » (n) # » (n) (n) (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), A1 ((U1 (t), S1 (t), p1 (t), T1 (t), L 1 (t)), # » # » (n) (n) (n) (n) (n) (n) (n) x1(n) (t) = f 1(n) ((μ(n) 1 (t), p1 (t)), G A1 (t))), A2 ((U2 (t), S2 (t), p2 (t), T2 (t), # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) L (n) 2 (t)), x 2 (t)= f 2 ((μ2 (t), p2 (t), G A2 (t))), . . . , An ((Un (t), Sn (t), pn (t), # (n) » (n) (n) (n) (n) (n−1) Tn(n) (t), L (n) ((U (n−1) (t), n (t)), x n (t) = f n ((μn (t), pn (t), G An (t)))} ∧ {A # » # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n−1) (t)),

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# » G (n−1) (t)))A(n−1) ((U1(n−1) (t), S1(n−1) (t), p1(n−1) (t), T1(n−1) (t), L (n−1) (t)), x1(n−1) (t) = 1 1 A # » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (t), p (t)), G (t))), A ((U (t), S (t), p2 (t), f 1(n−1) ((μ(n−1) 1 1 2 2 2 A1 # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t), L 2 (t)), x2 (t)= f ((μ (t), p (t), G (t))), . . . , An T2 # (n−1) » 2(n−1) 2 (n−1) 2 (n−1) A2 (n−1) (n−1) (n−1) (n−1) ((Un (t), Sn (t), pn (t), Tn (t), L n (t)), xn (t) = f n ((μn (t), # (n−1) » # (n)(n−1) » (n)Az B(n−1) (n)(n−1) (n)(n−1) pn (t), G (n−1) (t)))} ∧ {C ((U (t), S (t), p (t), An # » T (n)(n−1) (t), L (n)(n−1) (t)), x (n)(n−1) (t) = f (n)(n−1) ((μ(n)(n−1) (t), p (n)(n−1) (t)), # » G C(n)(n−1) (t)))), C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), T1(n)(n−1) (t), # » (n)(n−1) L (n)(n−1) (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), G C1 (t))), 1 # » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U2 (t), S2 (t), p (t), T (t), L 2 (t)), z 2 C2 # (n)(n−1) 2 » (n)(n−1)2 (n)(n−1) (n)(n−1) (n)(n−1) (t) = f 2 ((ω2 (t), p2 (t), G C2 (t))), . . . , Cn ((Un(n)(n−1) (t), # » Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), z n(n)(n−1) (t) = f n(n)(n−1) n # » (n)(n−1) ((ωn(n)(n−1) (t), pn(n)(n−1) (t), G Cn (t)))} = {C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), # (n)(n−1) » (n)(n−1) (n)(n−1) p (t), T (t), L (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), # » # (n)(n−1) » p (t)), G C(n)(n−1) (t)))), C1(n)(n−1) ((U1(n)(n−1) (t), S1(n)(n−1) (t), p1(n)(n−1) (t), # » (t)), z 1(n)(n−1) (t) = f 1(n)(n−1) ((ω1(n)(n−1) (t), p1(n)(n−1) (t)), T1(n)(n−1) (t), L (n)(n−1) 1 # » (n)(n−1) G C1 (t))), C2(n)(n−1) ((U2(n)(n−1) (t), S2(n)(n−1) (t), p2(n)(n−1) (t), T2(n)(n−1) (t), L (n)(n−1) 2 # » (n)(n−1) (t)), z 2(n)(n−1) (t) = f 2(n)(n−1) ((ω2(n)(n−1) (t), p2(n)(n−1) (t), G C2 (t))), . . . , # » Cn(n)(n−1) ((Un(n)(n−1) (t), Sn(n)(n−1) (t), pn(n)(n−1) (t), Tn(n)(n−1) (t), L (n)(n−1) (t)), n # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)Az B(n−1) (t) = f n ((ωn (t), pn (t), G Cn (t)))}=Tz jcz (C zn # » ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = # » f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))). Left side = right side. Proof is completed.

4.5 Destruction Transformation Connectives in Error Logic 4.5.1 Concept of Destruction Transformation Connectives in Error Logic # » Definition 4.39 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in universe of discourse U (t) under # » # » judging rule G(t), if Th (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((Φ, Φ, Φ, Φ, Φ), Φ = Φ((Φ, Φ, Φ)), then Th is called the destruction # » transformation connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)G(t))). The meaning of destruction transformation is: Th (object of interest ) → { erroneous objects or errors are eliminated through actions such as

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415

abolish, annihilate, decimate, disappear, discard, destroy, eliminate, eradicate, erase, exterminate, extinguish, extirpate, kill, leave, uproot, do away with, get rid of, move away, sell out, weed out, wipe out}. # » Definition 4.40 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), then Thly is called the domain destruction transformation connective with regard to G(t) and # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of domain destruction is: Thly ( domain ) →( domain does not exist ) → ( there is no domain to discuss or there is no need to discuss the object in current domain). # » Definition 4.41 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), then Thsw is called the thing destruction transformation connective with regard to G(t) and # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of thing destruction is: Thsw ( thing ) → (thing does not exist → (there is no thing to discuss or there is no need to discuss the thing in current domain, or current thing was eliminated, destroyed, discarded, removed, fired, or eradicated). # » Definition 4.42 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thk j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((U (t), S(t), Φ, T (t), L(t)), x(t) = f ((μ(t), ΦG(t))), then Thk j is called the spatial destruction transformation connective with regard to G(t) and # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of spatial destruction is: Thk j (space or location) → (space or location does not exist) → (there is no spatial characteristics or location to discuss or there is no need to discuss the object in current domain). # » Definition 4.43 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)G(t))), then Tht x is called the property destruction transformation connective with regard to G(t) # » # » and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of property destruction is: Tht x (property) → (property does not exist) → (there is no property to discuss or the object in discussion does not have the current property). # » Definition 4.44 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thlz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), S(t), p(t), T (t), Φ), x(t) = f ((μ(t), p(t)G(t))), then Thlz is

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called the property (or attribute) value destruction transformation connective with # » # » regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of property (or attribute) value destruction is: Thlz (property (or attribute) value) → (property (or attribute) value does not exist) → (there is no property (or attribute) value to discuss or the property of the object in discussion does not have property (or attribute) value). # » Definition 4.45 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thcz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), Φ = f ((μ(t), p(t)G(t))), then Thcz is called the error value destruction transformation connective with regard to G(t) and # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of error value destruction is: Thcz (error value) → (error value does not exist) → (there is no error value to discuss or the object in discussion does not have error). # » Definition 4.46 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thhs (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = Φ((μ(t), p(t)G(t))), then Thhs is called the error function destruction transformation connective with regard to G(t) # » # » and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of error value destruction is: Thhs (error function → (error function does not exist) → (there is no error function to discuss or error function has not been established). # » Definition 4.47 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Thgz (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)Φ)), then Thgz is called the rule destruction transformation connective with regard to G(t) and # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of rule destruction is: Thgz (rules for judging errors) → (rules do not exist) → (there are no rules to discuss or rules had been abolish). # » Definition 4.48 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule # » # » for judging errors, if Ths j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (Φ), S(Φ), p(Φ), T (Φ), L(Φ)), x(Φ) = f ((μ(Φ), p(Φ)G(Φ))), then Ths j is called the temporal destruction transformation connective with regard to # » # » G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of temporal destruction is: Ths j ( time factor ) → ( time does not apply ) → ( there are no time factor to discuss or the object of interest should not be studied during the given time horizon). # » Definition 4.49 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule

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# » # » for judging errors, if Thqb (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) ∈ {A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), A((Φ, Φ, # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), A((U (t), S(t), Φ, T (t), Φ), x(t) = # » # » f ((Φ, p(t)G(t))), . . . , A((U (Φ), S(Φ), p(Φ), T (Φ), L(Φ)), x(Φ) = f ((μ(Φ), # » p(Φ)G(Φ))), A((Φ, Φ, Φ, Φ, Φ), Φ = f ((Φ, ΦΦ))}, then Thqb is called the partial or complete destruction transformation connective with regard to G(t) and # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of complete destruction is: Thqb (more than two elements or all elements) → (more than two elements or all element do not exist) → (more than two elements do not need to be discussed; partial or all elements have been destroyed, removed, eliminated, demolished, or discarded).

4.5.2 Principles for Destruction Transformation in Error Logic The principles for destruction transformation are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.5.3 Approaches of Destruction Transformation in Error Logic There are many different ways to conduct destruction transformation, which are: { abolish, annihilate, decimate, disappear, discard, destroy, eliminate, eradicate, erase, exterminate, extinguish, extirpate, kill, leave, uproot, do away with, get rid of, move away, sell out, weed out, wipe out }

4.5.4 Hierarchy of Destruction Transformation in Error Logic In general, the object of interest μ(t) has both vertical and horizontal structures and multiple hierarchical relationships as well. Therefore, destruction transformation can also be carried out in different hierarchies.

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4.5.5 Characteristics of Domain Destruction Transformation Connective Th in Error Logic # » Based on the definition for Th and elements of A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)G(t))), Th can conduct transformation on the domain U (t), # » thing S(t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t), therefore there is Th ⊆ {Thly , Thsw , Thk j , Tht z , Thlz , Thcz , Thgz , Thhs , Ths j , Thq } (destruction transformation connectives). The variable type of A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) does not change if Th does not conduct destruction transformation on error function f . # » Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judg# » # » ing errors, if Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))), then Thly is called the error domain destruction transformation connective with respect to A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) and G(t). As far as an error logical # » # » variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) is concerned, if there is no domain defined, this implies that there are no thing S(t), spatial factor # » (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t) to be discussed. If there is no need to examine the object of interest μ(t) in current defined domain, it # » implies that there is no need to investigate the thing S(t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t), which suggests the error logical variable is Φ in current defined domain, or it may be necessary to refine # » domain to make the discussion of thing S(t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t) for the object of interest meaningful, which means other transformation needs to be carried out on the error logical variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). # » (1) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging # » # » errors, if Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))), then the transformation connective here Thly can be replaced by domain displacement transformation connective Tzly or inverse domain increase transformation connective # » # » −1 Tzly ; if Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))) = Φ, then engen−1 derment transformation Thly (inverse domain destruction transformation connective) can be employed to find feasible domain. # » (2) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging

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# » # » errors, if Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((Φ, Φ, Φ, Φ, Φ), Φ = f ((Φ, Φ, Φ)), it is said that Thly has forced A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) to carry out transformation # » on the elements of thing S(t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t), where Th ⊆ {Thly , Thsw , Thk j , Tht z , Thlz , Thcz , Thgz , Thhs , Ths j , Thq }. # » Proposition 4.214 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))); # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » g((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), the following relationships hold: # » # » (1) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) Thly (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » # » Proof As Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) and Thly (B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((Φ, S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), if x(t)  y(t), then the left side = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((Φ, S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t), G A (t))). # » # » And the right side = Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) ∨ Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ B((Φ, S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t), G A (t))). Left side = right side.

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# » If if x(t)  y(t), then the left side = Thly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))). # » # » And the right side = Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) ∨ Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ B((Φ, S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))). Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.215 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))); suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((Φ, # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose that C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Thly (C Az B ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = C((Φ, S(t), p(t), T (t), L(t)), # » z(t) = f ((ω(t), p(t)), G C (t))), the following relationships hold: # » # » (1) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))).

4.5 Destruction Transformation Connectives in Error Logic

421

# » # » (5) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (6) Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » Proof The left side = Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thly ((A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), # » # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » Az B p(t)), G B (t))) ∧ ¬C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))))) = (A((Φ, # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = # » # » # » f ((ω(t), p(t)), G C (t)))) ∨ (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∧ B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B # » # » ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((Φ, S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) ∨ (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » ¬B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((Φ, S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), L(t)), y(t) = # » # » # » f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((Φ, S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((Φ, S(t), p(t), T (t),

422

4 Transformation Connectives in Error Logic

# » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), L(t)), y(t) = # » # » # » Az B f ((ν(t), p(t)), G B (t))) ∧ ¬C ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = Φ # » # » And the right side = Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) ∨n Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » # » A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((Φ, S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = (A((Φ, S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((Φ, S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((Φ, S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » # » (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((Φ, S(t), p(t), T (t), L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » ¬C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = Φ. Left side = right side. Proof is completed. Similarly, (2)–(6) can also be proved. # » Proposition 4.216 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » G A (t))); suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose # » # » that C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » connotative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » Thly (C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » C AnhbB ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), the follow# » # » ing relationship holds: Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) =

4.5 Destruction Transformation Connectives in Error Logic

423

# » # » Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n # » # » Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.217 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » G A (t))); suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose # » # » that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » connotative same or equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » p(t)), G B (t))), Thly (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = C Anhdthd j B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t))), the following relationship holds: Thly (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = Thly (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) →nby Thly (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.218 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Thly (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (n) (n) (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = (A ((Φ , S (t), p (t), T # (n) » # » (n) (n+1) (n+1) p (t)), G A (t))); suppose that A ((U (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) order error logical variable defined in domain U (t) under G (n+1) (t) the rules A # » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), for judging errors, Thly (A # » (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = A(n+1) ((Φ (n+1) , S (n+1) (t), # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))); # (n) » (n) A (n) (n) (n) (n) suppose that error logical variable B ((U (t), S (t), p (t), T (t), L (t)), # » (n) (n) y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) (t), B (t))) is the complement variable for A ((U # » # » (n) S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G A (t))), which (n) is defined in domain U (n) (t) under G (n) B (t) the rules for judging errors, Thly (B # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)),

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# (n) » (n) (n) (n) (n) G (n) (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), B (t)))) = B ((Φ , S (t), p (t), T # (n) » (n) p (t)), G (t))); suppose that error logical variable B (n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » B(n+1) # » p (t), T (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) # (n+1) » B(n+1) (n+1) (n+1) (n+1) is the complement variable for A ((U (t), S (t), p (t), T (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))), which is defined in A (n+1) (n+1) (n+1) (n+1) (t) under G B (t) the rules for judging errors, Thly (B ((U (t), domain U # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), # » (t)))) = B (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = G (n+1) B # » (t))); suppose that C (n)Az B ((U (n) (t), S (n) (t), f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # (n) » (n) # » (n) (n) (n) p (t), T (t), L (t)), z (t) = f ((ω(n) (t), p (n) (t)), G C(n) (t))) is the mediator # » variable for A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), y (n) (t) = A (t))) and B ((U # » # » (n)Az B ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))), Thly (C # » L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t)))) = C (n)Az B ((Φ (n) (t), S (n) (t), # (n) » (n) # » p (t), T (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t))); suppose that # » C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) # » ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) G (n+1) A # » (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))), Thly (C (n+1)Az B ((U (n+1) (t), B # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), # » G C(n+1) (t)))) = C (n+1)Az B ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))), the following relationship # » holds: Thly (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » bz (n) (n) f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), A (t)))) = ¬ Thly (A ((U # » L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). Proof The left side of the equation = # » Thly (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = Thly (A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) A # » (n+1) (n+1) (n+1) (n+1) (n+1) (t), S (t), p (t), T (t), L (t)), y (t) = f (n+1) ((ν (n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), B # » L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = A(n+1) ((Φ (n+1) , # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ B (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = G (n+1) A

4.5 Destruction Transformation Connectives in Error Logic

425

# » # » f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ C (n+1)Az B ((Φ (n+1) , S (n+1) (t), p (n+1) (t), B # » T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) = Φ. # » The right side of the equation = ¬bz Thly A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bz (n) (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ (A ((Φ , S (t), # (n) » # » (n) p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G A (t)))) = A(n+1) # » ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ B (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), A # » p (n+1) (t)), G (n+1) (t))) ∧ C (n+1)Az B ((Φ (n+1) , y (n+1) (t) = f (n+1) ((ν (n+1) (t), B # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) = Φ. Left side = right side. Proof is completed. # » Proposition 4.219 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Thly (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # (n) » (n) (n) (n) (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = (A ((Φ , S (t), p (t), T # (n) » # » (n) (n+1) (n+1) p (t)), G A (t))); suppose that A ((U (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A order error logical variable defined in domain U (n+1) (t) under G (n+1) (t) the rules A # » for judging errors, Thly (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A(n+1) ((Φ (n+1) , S (n+1) (t), A # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))); A # » it is assumed that C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), # » L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # (n) » (n) # » (n)Az B(n+1) p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), Thly (C # » ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = # » f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = C (n)Az B(n+1) ((Φ (n)(n+1) , # » S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) # (n)(n+1) » (t), p (t)), G C(n)(n+1) (t))), the following relationship holds: # » Thly (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bx (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) A (t)))) = ¬ Thly (A ((U # » = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))).

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4 Transformation Connectives in Error Logic

# » Proof The left side of the equation = Thly (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), A (t)))) = Thly (A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n+1) A # (n) » # » (n)Az B(n+1) p (t)), G (n) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), A (t))) ∧ C # » L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » (A(n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A # » (n)Az B(n+1) ((Φ (n)(n+1) , S (n)(n+1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # (n)(n+1) » (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = Φ. # » And the right side of the equation = ¬bx Thly (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bx (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ (A ((Φ , # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))) = # » (A(n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A # » (n)Az B(n+1) ((Φ (n)(n+1) , S (n)(n+1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # (n)(n+1) » p (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = Φ. Left side = right side. Proof is completed. Proposition 4.220 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, Thly (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # (n) » (n) (n) (n) (n) p (t)), G (n) (t), L (n) (t)), x (n) (t) = f (n) A (t)))) = A ((Φ , S (t), p (t), T # » # » (n) (n) (n−1) (n−1) (n−1) ((μ (t), p (t)); suppose that A ((U (t), S (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th A (n−1) order error logical variable defined in domain U (t) under G (n−1) (t) the rules A # » (n−1) (n−1) (n−1) (n−1) (n−1) for judging errors, Thly (A ((U (t), S (t), p (t), T (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = A(n−1) ((Φ (n−1) , S (n−1) (t), A # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)); # » suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = # » f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error logical compleB # » mentary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain U (n−1) (t) A (n−1) under G B (t) the rules for judging errors, Thly (B (n−1) ((U (n−1) (t), S (n−1) (t),

4.5 Destruction Transformation Connectives in Error Logic

427

# (n−1) » # » p (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t)))) = B (n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) G (n−1) B # » (t))); = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » suppose that C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))), Thly B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) (C ((U (t), S (t), p (t), T (t), L (t)), z (t) = f (n−1) # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((ω (t), p (t)), G C (t)))) = C ((Φ ,S (t), p (t), T (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) L (t)), z (t) = f ((ω (t), p (t)), G C (t))), the following relationship holds: # » Thly (¬bj A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bj (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) A (t)))) = ¬ Thly (A ((U # » = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Thly ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Thly (A # » # » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) # » (n−1) (n−1) p (t), T (t), L (t)), z (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) # » = (A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ B (n−1) ((Φ (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))) ∧ T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) ((Φ , S (t), p (t), T (t), L (t)), z (t) = f (n−1) C # » (n−1) ((ω(n−1) (t), p (n−1) (t)), G C (t)))) = Φ. # » And the right side of the equation = ¬bj Thly (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bj (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ A ((Φ , # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) = # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Φ , S (t), p (t), T (t), L (t)), x (t) = f (n−1) (A # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((μ (t), p (t)), G A (t))) ∧ B ((Φ ,S (t), p (t), T (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1)Az B (t)), y (t) = f ((ν (t), p (t)), G B (t))) ∧ C L # (n−1) » (n−1) (n−1) (n−1) (n−1) ((Φ , S (t), p (t), T (t), L (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), # (n−1) » (n−1) p (t)), G C (t)))) = Φ. Left side = right side. Proof is completed.

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4 Transformation Connectives in Error Logic

# » Proposition 4.221 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Thly (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (n) (n) (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = A ((Φ , S (t), p (t), T # (n) » # » p (t)); suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error A (n−1) (t) under G (n−1) (t) the rules for judging logical variable defined in domain U A # » (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = errors, Thly (A # (n−1) » # » (n−1) (n−1) (n−1) (n−1) (n−1) f ((μ (t), p (t)), G A (t)))) = A ((Φ , S (n−1) (t), p (n−1) (t), # » (t))); suppose T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) # (n)(n−1) » (n)(n−1)A (n)Az B(n−1) (n)(n−1) (n)(n−1) ((U (t), S (t), p (t), T (t), L (n)(n−1) (t)), that C # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t) = f ((ω (t), p (t)), G C (t))) is the mediator variz # (n−1) » (n−1) (n−1) (n−1) (n−1) able for A ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = # » # » (n−1) f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n)Az B(n−1) ((U (n)(n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), and Thly (C # » (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) # » ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = C (n)Az B(n−1) ((Φ (n)(n−1) , S (n)(n−1) (t), # (n)(n−1) » p (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), # (n)(n−1) » p (t)), G C(n)(n−1) (t))), the following relationship holds: # » Thly (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bd (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ Thly (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Thly ¬bd (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Thly (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n−1) A # (n) » # » (n)Az B(n−1) p (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), A (t))) ∧ C # » L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » (A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A # » (n)Az B(n−1) ((Φ (n)(n−1) , S (n)(n−1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) p (t), T (t), L (t)), z (t) = f (n)(n−1) ((ω(n)(n−1) (t), # (n)(n−1) » p (t)), G C(n)(n−1) (t)))) = Φ. # » And the right side of the equation = ¬bd Thly (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bd (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ A ((Φ ,

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429

# » # » (n−1) S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) = (A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Φ ,S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), # (n−1) » # (n) » (n) (n−1) (n) (n) (n) (n) (n) p (t)), G A (t))) ∧ A ((Φ , S (t), p (t), T (t), L (t)), x (t) = # » # » (n)Az B(n−1) f (n) ((μ(n) (t), p (n) (t)), G (n) ((Φ (n)(n−1) , S (n)(n−1) (t), p (n)(n−1) (t), A (t))) ∧ C # » T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = Φ. Left side = right side. Proof is completed.

4.5.6 Thing Destruction Transformation Connectives in Error Logic 4.5.6.1

Concept of Thing Destruction Transformation in Error Logic

# » Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging errors, # » # » if Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((U (t), # » # » Φ, p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))), then Thsw is called the error thing # » destruction transformation connective with respect to A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) and G(t).

4.5.6.2

Characteristics of Thing Destruction Transformation Connective Th in Error Logic

# » Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging errors, # » # » if Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((U (t), # » # » Φ, p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))). As for error logical variable # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), if there is no thing # » to discuss, this implies that there are no domain U (t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t) to be discussed. If there is no need to examine the thing element in current domain, it implies that other thing # » S(t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t) will be discussed in the current domain, which suggests that logical variables # » # » such as A((U (t), S  (t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) other than # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) could be discussed. # » (1) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judg-

430

4 Transformation Connectives in Error Logic

# » # » ing errors, if Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))), then the transformation connective here Thsw can be replaced by thing displacement transformation connective Tzsw or inverse thing increase transformation con# » # » −1 ; if Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), nective Tzsw # » # » G(t)))) = A(( U (t), Φ, p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))) = Φ, −1 then engenderment transformation Thsw (inverse thing destruction transformation connective) can be employed to find desired thing. # » (2) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging # » # » errors, if Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((Φ, Φ, Φ, Φ, Φ), Φ = f ((Φ, Φ, Φ)), it is said that Thsw has forced A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) to carry out transformation # » on the elements of domain U (t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t), where Th ⊆ {Thly , Thsw , Thk j , Tht z ,Thlz , Thcz , Thgz , Thhs , Ths j , Thq }. # » Proposition 4.222 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))); # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » g((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), the following relationships hold: # » # » (1) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thsw # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ Thsw # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thsw # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ Thsw # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) Thsw (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » # » Proof As Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) and Thsw (B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), Φ, p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))), if x(t)  y(t), then the left side = Thsw # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thsw (A((U (t), S(t), p(t),

4.5 Destruction Transformation Connectives in Error Logic

431

# » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((U (t), Φ, p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t), G A (t))). # » And the right side = Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) ∨ Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ # » # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = A((U (t), Φ, p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))). Left side = right side. # » If if x(t)  y(t), then the left side = Thsw (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))). # » And the right side = Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) ∨ Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ # » # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = B((U (t), Φ, p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))). Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.223 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))); # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose that C Az B # » # » ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator # » # » variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Thsw (C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = C((U (t), Φ, p(t), T (t), # » L(t)), z(t) = f ((ω(t), p(t)), G C (t))), the following relationships hold: # » # » (1) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (3) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) =

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4 Transformation Connectives in Error Logic

# » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (5) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (6) Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) # » # »

nhdl B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f s j # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl Thsw # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f (( ν(t), p(t)), G B (t)))). Proof The left side = # » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thsw ((A((U (t), S(t), # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » # » f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t))))) = (A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » Az B C ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), # » # » # » Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), Φ, p(t), T (t), # » # » Az B L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C ((U (t), Φ, p(t), T (t), L(t)), z(t) = # » # » # » f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∧ B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B # » # » ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), Φ, # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), Φ, p(t), T (t), # » # » Az B L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C ((U (t), Φ, p(t), T (t), L(t)), z(t) =

4.5 Destruction Transformation Connectives in Error Logic

433

# » # » f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ ¬B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » Az B C ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), # » # » # » Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), Φ, p(t), T (t), # » # » Az B L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C ((U (t), Φ, p(t), T (t), L(t)), z(t) = # » # » # » f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∧ ¬B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B # » # » ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))). # » And the right side = Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) ∨n Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = (A((U (t), Φ, # » # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), Φ, p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) ∨ (A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), Φ, # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), Φ, p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), Φ, p(t), T (t), L(t)), y(t) = # » # » # » f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬ # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), Φ, # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), Φ, p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), Φ, p(t), T (t), L(t)), y(t) = # » # » # » f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (¬A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), # » # » # » Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), Φ, p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), Φ, p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), # » p(t)), G C (t)))). Left side = right side. Proof is completed. Similarly, (2)–(6) can also be proved. # » Proposition 4.224 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » rules for judging errors, Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))); # » suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging # » # » errors, Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose that C AnhbB # » # » ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connota# » # » tive inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)),

434

4 Transformation Connectives in Error Logic

# » # » G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = Thsw (C # » # » C AnhbB ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), the following # » # » relationship holds: Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » n G A (t))) − B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n Thsw # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.225 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = A((U (t), Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), # » G A (t))); suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((U (t), Φ, p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose # » # » that C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » connotative same or equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » f ((μ(t), p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » p(t)), G B (t))), Thsw (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) = C Anhdthd j B ((U (t), Φ, p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » G C (t))), the following relationship holds: Thsw (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) →nby B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = Thsw (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t)))) →nby Thsw (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.226 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, Thsw (A(n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), Φ (n) , p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = (A ((U # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t))); suppose that A # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A order error logical variable defined in domain U (n+1) (t) under G (n+1) (t) the rules A # (n+1) » (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), for judging errors, Thsw (A # » (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = A(n+1) ((U (n+1) (t), Φ (n+1) , # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))); # (n) » (n) A (n) (n) (n) (n) suppose that error logical variable B ((U (t), S (t), p (t), T (t), L (t)), # » (n) (n) y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) (t), B (t))) is the complement variable for A ((U

4.5 Destruction Transformation Connectives in Error Logic

435

# » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), which (n) (n) is defined in domain U (t) under G B (t) the rules for judging errors, Thsw (B (n) # » # » ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), Φ (n) , p (n) (t), T (n) (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), G (n) B (t)))) = B ((U # (n) » p (t)), G (n) (t))); suppose that error logical variable B (n+1) ((U (n+1) (t), S (n+1) (t), # (n+1) » B(n+1) # » p (t), T (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) # » B is the complement variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))), which is defined A (t) the rules for judging errors, Thsw (B (n+1) in domain U (n+1) (t) under G (n+1) B # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t)))) = B (n+1) ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t), L (n+1) (t)), B # » (t))); suppose that C (n)Az B ((U (n) (t), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t))) is the # » mediator variable for A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = # » # » (n) (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) and B ((U # » # » (n)Az B ((U (n) (t), S (n) (t), p (n) (t), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))), Thsw (C # » T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t)))) = C (n)Az B ((U (n) (t), # » # » Φ (n) , p (n) (t), T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t))); suppose # » that C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = # » f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the mediator variable for A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # » (t))), Thsw (C (n+1)Az B L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), z (t)= f (n+1) ((ω(n+1) (t), # (n+1) » # » p (t)), G C(n+1) (t)))) = C (n+1)Az B ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))), the following relationship holds: # » Thsw (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bz (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) A (t)))) = ¬ Thsw (A ((U # » (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Thsw (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), # » (n+1) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = Thsw (A # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A # » (t))) ∧ C (n+1)Az B L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t),

436

4 Transformation Connectives in Error Logic

# (n+1) » # » p (t)), G C(n+1) (t)))) = A(n+1) ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), Φ (n+1) , A # » # (n+1) » (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), p # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) B # » (n+1) z (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) = Φ. # » The right side of the equation = ¬bz Thsw A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bz (n) (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = ¬ (A ((Φ , S (t), # (n) » # (n) » (n) (n) (n) (n) (n) (n) p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))) = A(n+1) # » ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t), L (n+1) (t)), A # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » # » Φ (n+1) , p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) = Φ. Left side = right side. Proof is completed. # » Proposition 4.227 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » # » Thsw (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), Phi (n) , p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = (A ((U # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t))); suppose that A # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) order error logical variable defined in domain U (t) under G (n+1) (t) the rules A # » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), for judging errors, Thsw (A # » (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = A(n+1) ((U (n+1) (t), Φ (n+1) , # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), (t))); G (n+1) A # » it is assumed that C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), # » L (n)(n+1) (t)),z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the # » mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # (n) » # (n) » (n) (n) (n) (n) p (t), T (t), L (t)), x (t) = f ((μ(n) (t), p (t)), G (n) A (t))), # » (n)Az B(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ((U (t),S (t), p (t),T (t),L (t)), Thsw (C # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)Az B(n+1) (t) = f ((ω (t), p (t)), G C (t)))) = C z # » ((U (n)(n+1) (t), Φ (n)(n+1) , p (n)(n+1) (t), T (n)(n+1) (t),L (n)(n+1) (t)),z (n)(n+1) (t)= # » f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))), the following relationship holds: # » Thsw (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t),

4.5 Destruction Transformation Connectives in Error Logic

437

# (n) » # » bx (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ Thsw (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Thsw (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), A (t)))) = Thsw (A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n+1) A # (n) » # » (n)Az B(n+1) p (t)), G (n) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), A (t))) ∧C # » L (n)(n+1) (t)), z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » (A(n+1) ((U (n+1) (t), Φ (n+1) , p (n+1) (t), T (n+1) (t),L (n+1) (t)), x (n+1) (t) = # (n+1) » # (n) » (n) (n+1) (n+1) (n+1) (n) (n) (n) ((μ (t), p (t)), G A (t))) ∧ A ((U (t), Φ , p (t), T (t), f # » (n) (n) (n) (n)Az B(n+1) ((U (n)(n+1) (t), L (t)), x (t) = f ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧C # » (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) (n)(n+1) ,p (t), T (t), L (t)), z (t)= f ((ω(n)(n+1) (t), Φ # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = Φ. # » And the right side of the equation = ¬bx Thsw (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bx (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) Φ , p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t)))) = # » Φ (n+1) , p (n+1) (t), T (n+1) (t),L (n+1) (t)), x (n+1) (t) = (A(n+1) ((U (n+1) (t), # » # (n) » (n) (n+1) (n+1) (n+1) (n+1) (n) (n) (n) ((μ (t), p (t)), G A (t))) ∧ A ((U (t), Φ , p (t), T (t), f # » (n) (n) (n) (n)Az B(n+1) ((U (n)(n+1) (t), L (t)), x (t) = f ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧C # » Φ (n)(n+1) , p (n)(n+1) (t), T (n)(n+1) (t),L (n)(n+1) (t)),z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = Φ. Left side = right side. Proof is completed. Proposition 4.228 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, Thsw (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), # (n) » # » (n) (n) p (t)), G (n) (t), Φ (n) , p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = A (t)))) = A ((U # » # » f (n) ((μ(n) (t), p (n) (t)); suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th A (n−1) order error logical variable defined in domain U (t) under G (n−1) (t) the rules A # » (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), for judging errors, Thsw (A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) x (t) = f ((μ (t), p (t)), G A (t)))) = A ((U (t), Φ (n−1) , # (n−1) » (n−1) # » p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)); # » suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » logical complementary variable of A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t),

438

4 Transformation Connectives in Error Logic

# » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (n−1) (t) under G B (t) the rules for judging errors, Thsw (B (n−1) ((U (n−1) (t), U # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t)))) = B (n−1) ((U (n−1) (t), Φ (n−1) , p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # » (t))); y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » (n−1)Az B (n−1) (n−1) suppose that C ((U (t), S (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (t))), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # (n−1) » (n−1) (n−1)Az B (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), z (t) = Thsw (C # » # » f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = C (n−1) ((U (n−1) (t), Φ (n−1) , p (n−1) (t), # » T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))), the following relationship holds: # » Thsw (¬bj A(n) ((U (n) (t),S (n) (t), p (n) (t), T (n) (t),L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), # (n) » # » (n) bj (n) (n) p (t)), G A (t)))) = ¬ Thsw (A ((U (t), S (n) (t), p (n) (t),T (n) (t),L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Thsw ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Thsw (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = (A(n−1) ((U (n−1) (t), Φ (n−1) , p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ B (n−1) ((U (n−1) (t), Φ (n−1) , A # (n−1) » # » p (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), Φ (n−1) , p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = Φ. # » And the right side of the equation =¬bj Thsw (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » # » bj (n) (n) L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), p (n) (t)),G (n) (t), Φ (n) , p (n) (t), A (t))))= ¬ A ((U # » (n−1) ((U (n−1) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) = (A # » # » Φ (n−1) , p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), Φ (n−1) , p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = G (n−1) A # » # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), Φ (n−1) , p (n−1) (t), f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = Φ. Left side = right side. Proof is completed.

4.5 Destruction Transformation Connectives in Error Logic

439

# » Proposition 4.229 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # (n) » Thsw (A(n) ((U (n) (t), S (n) (t), p (t), T (n) (t), L (n) (t)), x (n) (t) = # » # (n) » (n) (n) (n) (n) (n) (n) (n) (n) f ((μ (t), p (t)), G A (t)))) = A ((U (t), Φ , p (t), T (t), L (n) (t)), # » # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)); suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the A th (n−1) (t) under G (n−1) (t) (n − 1) order error logical variable defined in domain U A # » the rules for judging errors, Thsw (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = A(n−1) A # » ((U (n−1) (t), Φ (n−1) , p (n−1) (t),T (n−1) (t),L (n−1) (t)),x (n−1) (t) = f (n−1) ((μ(n−1) (t), # (n−1) » (n−1) p (t)),G (t))); suppose that C (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), # (n)(n−1) » A p (t), T (n)(n−1) (t),L (n)(n−1) (t)),z (n)(n−1) (t)= f (n)(n−1) ((ω(n)(n−1) (t), # (n)(n−1) » (n)(n−1) p (t)), G C (t))) is the mediator variable for A(n−1) ((U (n−1) (t), S (n−1) (t), # (n−1) » # » (n−1) p (t), T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = G (n−1) A # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), f (n) ((μ(n) (t), p (n) (t)),G (n) A (t))), and Thsw (C # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) p (t), T (t), L (t)), z (t) = f (n)(n−1) ((ω(n)(n−1) (t), # (n)(n−1) » # » p (t)), G C(n)(n−1) (t)))) = C (n)Az B(n−1) ((U (n)(n−1) (t), Φ (n)(n−1) , p (n)(n−1) (t), # » T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))), the following relationship holds: # » Thsw (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bd (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ Thsw (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Thsw ¬bd (A(n) ((U (n) (t),S (n) (t), p (n) (t),T (n) (t), # » (n−1) L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Thsw (A # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) (n−1) S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n−1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) G (n−1) A # (n) » # » (n)Az B(n−1) (t), p (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t), A (t))) ∧ C # (n)(n−1) » T (n)(n−1) (t), L (n)(n−1) (t)),z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (t)), # (n−1) » (n)(n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (t)))) = (A ((U (t), Φ , p (t), T (t), L (t)), GC # » # (n) » (n) (n) (n) x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A ((U (t), Φ , p (t), A # (n) » (n) (n) (n) (n) (n) (n) (n)Az B(n−1) T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t))) ∧ C # » ((U (n)(n−1) (t), Φ (n)(n−1) , p (n)(n−1) (t),T (n)(n−1) (t), L (n)(n−1) (t)),z (n)(n−1) (t) = # » f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = Φ. # » And the right side of the equation = ¬bd Thsw (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bd (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ A ((U

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# » # » Φ (n) , p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) = # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) (A ((U (t), Φ , p (t), T (t), L (t)), x (t) = # (n−1) » # (n) » (n) (n−1) (n−1) (n−1) (n) (n) (n) ((μ (t), p (t)), G A (t))) ∧ A ((U (t), Φ , p (t), T (t), f # » (n)Az B(n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n)(n−1) (t), A (t))) ∧ C # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ,p (t),T (t), L (t)),z (t) = f ((ω(n)(n−1) (t), Φ # (n)(n−1) » (n)(n−1) p (t)), G C (t)))) = Φ. Left side = right side. Proof is completed.

4.5.7 Property Destruction Transformation Connectives in Error Logic 4.5.7.1

Concept of Property Destruction Transformation in Error Logic

# » Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging errors, # » # » if Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((U (t), # » # » S(t), p(t), Φ, L(t)), x(t) = f ((u(t), p(t), G(t))), then Tht x is called the error thing # » destruction transformation connective with respect to A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) and G(t).

4.5.7.2

Characteristics of Property Destruction Transformation Connective Th in Error Logic

# » Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging errors, # » # » if Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((U (t), # » # » S(t), p(t), Φ, L(t)), x(t) = f ((u(t), p(t), G(t))). As for error logical variable # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), if there is no property # » to discuss, this implies that there are no domain U (t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t) to be discussed. If there is no need to examine the property in current domain, it implies that other thing S(t), # » spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t) will be discussed in the current domain, which suggests that logical variables such as # » # » A((U (t), S(t), p(t), T  (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) other than A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) could be discussed. # » (1) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judg-

4.5 Destruction Transformation Connectives in Error Logic

441

# » # » ing errors, if Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((u(t), p(t), G(t))), then the transformation connective Tht x can be replaced by property displacement transformation connective Tzt x or inverse property increase transformation con# » # » nective Tzt−1x ; if Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), # » # » G(t)))) = A(( U (t), S(t), p(t), Φ, L(t)), x(t) = f ((u(t), p(t), G(t))) = Φ, then engenderment transformation Tht−1x (inverse property destruction transformation connective) can be employed to find desired property. # » (2) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging # » # » errors, if Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = A((Φ, Φ, Φ, Φ, Φ), Φ = f ((Φ, Φ, Φ)), it is said that Tht x has forced A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) to carry out transformation # » on the elements of domain U (t), spatial factor (space) p(t), property T (t), property (or attribute) value L(t)), error value x(t), error function f , temporal factor (time) t, and rules for judging error G(t), where Th ⊆ {Thly , Thsw , Thk j , Tht x ,Thlz , Thcz , Thgz , Thhs , Ths j , Thq }. # » Proposition 4.230 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), G A (t))); suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = g((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), # » # » S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))), the following relationships hold: # » # » (1) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (3) Tht x (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » ¬Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » # » Proof As Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), G A (t))) and Tht x (B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), S(t), p(t), # » Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))), if x(t)  y(t), then the left side = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Tht x (A((U (t), S(t), p(t),

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4 Transformation Connectives in Error Logic

# » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((U (t), S(t), p(t), Φ, L(t)), # » x(t) = f ((μ(t), p(t), G A (t))). And the right side = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = A((U (t), # » # » # » S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ B((U (t), S(t), p(t), Φ, L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), # » p(t), G A (t))). Left side = right side. # » If if x(t)  y(t), then the left side = Tht x (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t)))) = Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))). # » # » And the right side = Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) ∨ Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » # » A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ B((U (t), S(t), p(t), # » # » Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = B((U (t), S(t), p(t), Φ, L(t)), y(t) = # » f ((ν(t), p(t)), G B (t))). Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. # » Proposition 4.231 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), G A (t))); suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), # » # » S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose that C Az B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the mediator variable # » # » of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and B((U (t), # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Tht x (C Az B ((U (t), S(t), # » # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = C((U (t), S(t), p(t), Φ, L(t)), # » z(t) = f ((ω(t), p(t)), G C (t))), the following relationships hold: # » # » (1) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (2) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧n # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))).

4.5 Destruction Transformation Connectives in Error Logic

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# » # » (3) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f l # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (4) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » (5) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » (6) Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » T f s j (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f (( ν(t), p(t)), G B (t)))). # » # » Proof The left side = Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x ((A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), # » # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), T (t), L(t)), # » # » x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » # » # » p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), # » # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » Az B G B (t))) ∧ ¬C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))))) = # » # » (A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), # » # » # » p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), Φ, L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » Az B ¬C ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), # » # » # » S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), Φ, L(t)), # » # » Az B y(t) = f ((ν(t), p(t)), G B (t))) ∧ C ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧

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# » # » ¬B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), # » # » # » S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), S(t), p(t), Φ, # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), Φ, L(t)), y(t) = # » # » # » f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (¬A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), # » # » # » S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), S(t), p(t), Φ, # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), Φ, L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), # » p(t)), G C (t)))); # » # » And the right side = Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) ∨n Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = A((U (t), U (t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n # » # » B((U (t), U (t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = (A((U (t), S(t), # » # » # » p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), Φ, L(t)), # » # » y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), # » # » # » p(t)), G C (t)))) ∨ (A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), # » # » # » S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), S(t), p(t), # » # » Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), Φ, L(t)), y(t) = # » # » # » f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » ¬B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), # » # » # » S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), S(t), p(t), Φ, # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), Φ, L(t)), y(t) = # » # » # » f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), # » # » G C (t)))) ∨ (¬A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ # » # » B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), # » # » # » S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), S(t), p(t), Φ, # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), Φ, L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), # » p(t)), G C (t)))). Left side = right side. Proof is completed. Similarly, (2)–(6) can also be proved. # » Proposition 4.232 Suppose that an error logical variable A((U (t), S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under # » G A (t) the rules for judging errors, Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t)))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), # » G A (t))); suppose that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules # » # » for judging errors, Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t)))) = B((U (t), S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose

4.5 Destruction Transformation Connectives in Error Logic

445

# » # » that C AnhbB ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » connotative inclusion variable of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » AnhbB Tht x (C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » C AnhbB ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t))), the follow# » # » ing relationship holds: Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) −n B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n Tht x # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. # » Proposition 4.233 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) is defined in domain U (t) under G A (t) the # » # » rules for judging errors, Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t)))) = A((U (t), S(t), p(t), Φ, L(t)), x(t) = f ((μ(t), p(t), G A (t))); suppose # » that another error logical variable B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), # » p(t)), G B (t))) is defined in domain U (t) under G B (t) the rules for judging errors, # » # » Tht x (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), # » # » S(t), p(t), Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))); suppose that C Anhdthd j B ((U (t), # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative same or # » # » equivalence variable for A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) and B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), # » # » Tht x (C Anhdthd j B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = # » # » C Anhdthd j B ((U (t), S(t), p(t), Φ, L(t)), z(t) = f ((ω(t), p(t)), G C (t))), the follow# » # » ing relationship holds: Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) →nby B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » Tht x (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby Tht x # » # » (B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.234 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable defined in domain U ( n)(t) under G ( n) A (t) the rules for judging errors, # » # » Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = (A ((U # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t))); suppose that A # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (n+1) order error logical variable defined in domain U (t) under G (n+1) (t) the rules A # » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (n+1) (t)), for judging errors, Tht x (A # » (n+1) x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G A (t)))) = A(n+1) ((U (n+1) (t), # (n+1) » # » (n+1) (n+1) (n+1) (n+1) (t), p (t), Φ , L (t)), x (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), S # » (t))); suppose that error logical variable B (n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), G (n+1) A

446

4 Transformation Connectives in Error Logic

# » L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))) is the complement variable for # (n) » (n) # » (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), (n) (t) under G (n) G (n) A (t))), which is defined in domain U B (t) the rules for judging # » (n) (n) (n) (n) (n) (n) errors, Tht x (B ((U (t), S (t), p (t), T (t), L (t)), y (n) (t) = f (n) ((ν (n) (t), # (n) » # » (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), y (n) (t) = B (t)))) = B ((U # » (n+1) ((U (n+1) (t), f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))); suppose that error logical variable B # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), # » (t))) is the complement variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), G (n+1) B # » (t))), which is T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A (t) the rules for judging errors, defined in domain U (n+1) (t) under G (n+1) # (n+1)B » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), y (t) = Tht x (B # (n+1) » # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) (n+1) ((ν (t), p (t)), G B (t)))) = B ((U (t), S (t), p (t), f # » (t))); suppose Φ (n+1) , L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » that C (n)Az B ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), # (n) » # » p (t)), G C(n) (t))) is the mediator variable for A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » # » (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), A (t))) and B ((U # » (n)Az B ((U (n) (t), T (n) (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))), Tht x (C # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), G C(n) (t)))) = # » # » C (n)Az B ((U (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), # » G C(n) (t))); suppose that C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), # » L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the mediator vari# » able for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » (t))) and B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » (t))), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # (n+1) » (n+1)Az B (n+1) (n+1) (n+1) (n+1) (n+1) Tht x (C ((U (t), S (t), p (t), T (t), L (t)), z (t)= # » f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = C (n+1)Az B ((U (n+1) (t), S (n+1) (t), # (n+1) » (n+1) (n+1) # » p (t), Φ ,L (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))), the following relationship holds: # » Tht x (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bz (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ Tht x (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Tht x (¬bz A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), A (t)))) = Tht x (A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) A # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)),

4.5 Destruction Transformation Connectives in Error Logic

447

# » S (n+1) (t), p (n+1) (t), Φ (n+1) , L (n+1) (t)), # (n+1) » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), A # » # » S (n+1) (t), p (n+1) (t), Φ (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), Φ (n+1) , L (n+1) (t)), G (n+1) B # » (n+1) z (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) = Φ. # » The right side of the equation = ¬bz Tht x A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bz (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t)))) = ¬ (A ((U # (n) » # (n) » (n) (n) (n) (n) (n) (n) p (t), Φ , L (t)), x (t) = f ((μ (t), p (t)), G A (t)))) = A(n+1) # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), Φ (n+1) , L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), # (n+1) » # » p (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), Φ (n+1) (t), A # » (t))) ∧ C (n+1)Az B L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » ((U (n+1) (t), S (n+1) (t), p (n+1) (t), Φ (n+1) , L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), # (n+1) » p (t)), G C(n+1) (t))) = Φ. Left side = right side. Proof is completed. G C(n+1) (t)))) = A(n+1) ((U (n+1) (t),

# » Proposition 4.235 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » # » Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), S (n) (t), p (n) (t), Phi (n) , L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = (A ((U # (n) » # » (n+1) p (t)), G (n) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t))); suppose that A # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A order error logical variable defined in domain U (n+1) (t) under G (n+1) (t) the rules A # » for judging errors, Tht x (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # (n+1) » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t)))) = A(n+1) ((U (n+1) (t), A # » # » S (n+1) (t), p (n+1) (t), Φ (n+1) , L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))); it is assumed that C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), G (n+1) A # (n)(n+1) » T (n)(n+1) (t), L (n)(n+1) (t)),z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), p (t)), # » G C(n)(n+1) (t))) is the mediator variable for A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) and A # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n) (t)), # » Tht x (C (n)Az B(n+1) ((U (n)(n+1) (t),S (n)(n+1) (t), p (n)(n+1) (t),T (n)(n+1) (t), G (n) A (t))), # » (n)(n+1) L (t)),z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » C (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) , p (n)(n+1) (t), Φ (n)(n+1) , L (n)(n+1) (t)), # » z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))), the following relationship holds: # » Tht x (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t),

448

4 Transformation Connectives in Error Logic

# (n) » # » bx (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ Tht x (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Tht x (¬bx A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n+1) (t), A (t)))) = Tht x (A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n+1) A # (n) » # » (n)Az B(n+1) p (t)), G (n) ((U (n)(n+1) (t), S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), A (t))) ∧C # » L (n)(n+1) (t)), z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), Φ (n+1) ,L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), A # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧C # (n)(n+1) » (n)(n+1) (n)(n+1) (n)(n+1) p (t), Φ (t), L (t)), z (t)= f (n)(n+1) ((ω(n)(n+1) (t), # (n)(n+1) » p (t)), G C(n)(n+1) (t)))) = Φ. # » And the right side of the equation = ¬bx Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bx (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ (A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), Φ , L (t)), x (t) = f ((μ (t), p (t)), G A (t)))) = # » (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), Φ (n+1) ,L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), A # » (n)Az B(n+1) ((U (n)(n+1) (t), S (n)(n+1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # (n)(n+1) » p (t), Φ (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), # (n)(n+1) » (n)(n+1) p (t)), G C (t)))) = Φ. Left side = right side. Proof is completed. Proposition 4.236 Suppose that an error logical variable A(n) ((U (n) (t), S (n) (t), # (n) » (n) # » th p (t), T (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order ( ( error logical variable defined in domain U n)(t) under G n) A (t) the rules for judging # » errors, Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), # (n) » # » (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), x (n) (t) = A (t)))) = A ((U # » # » f (n) ((μ(n) (t), p (n) (t)); suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th A (n−1) order error logical variable defined in domain U (t) under G (n−1) (t) the rules A # » (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), for judging errors, Tht x (A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) x (t) = f ((μ (t), p (t)), G A (t)))) = A ((U (n−1) (t), # » # » S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)); # » suppose that B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error B # » (n−1) (n−1) logical complementary variable of A ((U (t), S (n−1) (t), p (n−1) (t), T (n−1) (t),

4.5 Destruction Transformation Connectives in Error Logic

449

# » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined in domain A (n−1) (n−1) (t) under G B (t) the rules for judging errors, Tht x (B (n−1) ((U (n−1) (t), U # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t)))) = B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), G (n−1) B # » (t))); suppose that C (n−1)Az B y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)),G (n−1) B # » ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), # (n−1) » p (t)), G C(n−1) (t))) is the mediator variable for A(n−1) ((U (n−1) (t), S (n−1) (t), # (n−1) » (n−1) # » p (t),T (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) A # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (t)), y (t) = and B # » (n−1)Az B (n−1) (n−1) (t))), T (C ((U (t), S (t), f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) ht x B # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) p (t), T (t), L (t)), z (t) = f ((ω (t), p (t)), # » G C(n−1) (t)))) = C (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))), the following relationship holds: # » T (n) (t),L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), Tht x (¬bj A(n) ((U (n) (t),S (n) (t), p (n) (t), # (n) » # » (n) p (t)), G A (t)))) = ¬bj Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t),T (n) (t),L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Tht x ¬bj (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » (n−1) L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Tht x (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ C (n−1)Az B ((U (n−1) (t), B # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) , A # (n−1) » # » p (t), Φ (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), G (n−1) B # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))) = Φ. # » And the right side of the equation =¬bj Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), # » bj (n) (n) L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), p (n) (t)),G (n) (t), S (n) (t), A (t))))= ¬ A ((U # (n) » # » (n−1) p (t), Φ (n) , L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) = (A # » ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), # (n−1) » # » p (t)), G (n−1) (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) , p (n−1) (t), Φ (n−1) (t), L (n−1) (t)), A # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » # » S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), (n−1) G C (t)))) = Φ.

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4 Transformation Connectives in Error Logic

Left side = right side. Proof is completed. # » Proposition 4.237 Suppose that A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n ( ( defined in domain U n)(t) under G n) A (t) the rules for judging errors, # » # » Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n) A (t)))) = A ((U # (n) » # » p (t)); suppose that A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error A (n−1) (t) under G (n−1) (t) the rules for judging logical variable defined in domain U A # » (n−1) (n−1) (n−1) (n−1) (n−1) ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = errors, Tht x (A # » # » (n−1) f (n−1) ((μ(n−1) (t), p (n−1) (t)), G A (t)))) = A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), # » (t))); suppose Φ (n−1) , L (n−1) (t)),x (n−1) (t) = f (n−1) ((μ(n−1) (t) , p (n−1) (t)), G (n−1) A # (n)(n−1) » (n)(n−1) (n)Az B(n−1) (n)(n−1) (n)(n−1) that C ((U (t), S (t), p (t), T (t),L (n)(n−1) (t)), # » (n)(n−1) (t))) is the mediator variz (n)(n−1) (t)= f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C # (n−1) » (n−1) (n−1) (n−1) (n−1) able for A ((U (t), S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) and A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » and Tht x (C (n)Az B(n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)),G (n) A (t))), # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) ((U (t), S (t), p (t), T (t), L (t)), z (n)(n−1) (t) = # » (n)(n−1) (t)))) = C (n)Az B(n−1) ((U (n)(n−1) (t), f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C # » S (n)(n−1) (t), p (n)(n−1) (t), Φ (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) # » ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))), the following relationship holds: # » Tht x (¬bd A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » bd (n) (n) p (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), A (t)))) = ¬ Tht x (A ((U # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » Proof The left side of the equation = Tht x ¬bd (A(n) ((U (n) (t),S (n) (t), p (n) (t),T (n) (t), # » (n−1) L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((U (n−1) (t), A (t)))) = Tht x (A # (n−1) » (n−1) # » (n−1) (n−1) (n−1) (n−1) (n−1) (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n−1) (t)), S # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n−1) A # (n) » # » (n)Az B(n−1) p (t)), G (n) ((U (n)(n−1) (t), S (n)(n−1) (t), p (n)(n−1) (t),T (n)(n−1) (t), A (t))) ∧ C # » L (n)(n−1) (t)),z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), x (n−1) (t)= # » # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), Φ (n) , f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (n)Az B(n−1) ((U (n)(n−1) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # » S (n)(n−1) (t), p (n)(n−1) (t),Φ (n)(n−1) , L (n)(n−1) (t)),z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), # (n)(n−1) » p (t)), G C(n)(n−1) (t)))) = Φ.

4.5 Destruction Transformation Connectives in Error Logic

451

# » And the right side of the equation = ¬bd Tht x (A(n) ((U (n) (t), S (n) (t), p (n) (t), # » bd (n) (n) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t)))) = ¬ A ((U # » # » (n) (n) (n) (n) (n) (n) (n) (n) (n) S (t), p (t), Φ , L (t)), x (t) = f ((μ (t), p (t)), G A (t))) = # » (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), Φ (n−1) , L (n−1) (t)), x (n−1) (t)= f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), Φ (n) , L (n) (t)), A # » (n)Az B(n−1) ((U (n)(n−1) (t), S (n)(n−1) (t), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # (n)(n−1) » p (t), Φ (n)(n−1) , L (n)(n−1) (t)),z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), # (n)(n−1) » (n)(n−1) p (t)), G C (t)))) = Φ. Left side = right side. Proof is completed.

4.5.8 Engenderment Transformation Connectives in Error Logic 4.5.8.1

Concept of Engenderment Transformation Connectives in Error Logic

# » Definition 4.50 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in universe of discourse U (t) under judging rule G(t), if Th−1 (A((X , X , X , X , X ), X = X ((X , X ), X ))) = A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))), where X represents Φ or element at corresponding position, then Th−1 is called the engenderment transformation # » connective with respect to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)G(t))). The meaning of engenderment transformation is: Th−1 ( object of interest ) → { accept, acquire, add, appear, arrive, beget, buy back, build, build up, construct, come back to life, create, develop, erect, establish, found, foster, implant, import, include, increase, instill, move back, move in, pick up, plant, populate, produce, raise, reappear, reconstruct, recover, replenish, retrieve, restore}. # » Definition 4.51 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule −1 (A((Φ, X , X , X , X ), X = X ((X , X ), X ))) = A((U (t), for judging errors, if Thly −1 is called the domain engenderment X , X , X , X ), X = X ((X , X ), X )), then Thly # » transformation connective with regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » −1 x(t) = f ((μ(t), p(t)G(t))). The meaning of domain engenderment is: Thly (Φ) → ( valid domain is defined or confirmed or there exists appropriate domain ). # » Definition 4.52 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for −1 (A((X , Φ, X , X , X ), X = X ((X , X ), X ))) = A((X , S(t), X , X , judging errors, if Thly −1 is called the thing engenderment transformation X ), X = X ((X , X ), X )), then Thly

452

4 Transformation Connectives in Error Logic

# » connective with regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » −1 (Φ) → ( thing exists or there p(t)G(t))). The meaning of thing engenderment is: Thly is designated thing to discuss ). # » Definition 4.53 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule −1 for judging errors, if if Thk j (A((X , X , Φ, X , X ), X = X ((X , X ), X ))) = A((X , # » −1 X , p(t), X , X ), X = X ((X , X ), X )), then Thk j is called the spatial engenderment # » transformation connective with regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » −1 x(t) = f ((μ(t), p(t)G(t))). The meaning of spatial engenderment is: Thk j (Φ) → ( there exists appropriate space or location for the object of interest ). # » Definition 4.54 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging errors, if Tht−1x (A((X , X , X , Φ, X ), X = X ((X , X ), X ))) = A((X , X , X , T (t), X ), X = X ((X , X ), X )), then Tht−1x is called the property engenderment # » transformation connective with regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)G(t))). The meaning of property engenderment is: Tht−1x (Φ) → ( there exists property or there is necessary property to discuss ). # » Definition 4.55 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule −1 for judging errors, if Thlz (A((X , X , X , X , Φ), X = X ((X , X ), X ))) = A((X , X , −1 is called the property (or attribute) X , X , L(t)), X = X ((X , X ), X )), then Thlz value engenderment transformation connective with regard to G(t) and A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of property (or −1 (Φ) → ( there exists property (or attribute) attribute) value engenderment is: Thlz value or there is appropriate property (or attribute) value ). # » Definition 4.56 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule −1 for judging errors, if Thcz (A((X , X , X , X , X ), Φ = X ((X , X ), X ))) = A((X , −1 is called the error value engenderX , X , X , X ), x = X ((X , X ), X )), then Thcz # » ment transformation connective with regard to G(t) and A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of error value engenderment is: −1 (Φ) → ( there exists error value or there is appropriate error value to discuss). Thcz # » Definition 4.57 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule −1 (A((X , X , X , X , X ), X = Φ((X , X ), X ))) = A((X , X , for judging errors, if Thhs −1 is called the error function engenderX , X , X ), X = f ((X , X ), X )), then Thhs # » ment transformation connective with regard to G(t) and A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of error value engenderment is: −1 Thhs (Φ) → (there exists error function or error function has been established).

4.5 Destruction Transformation Connectives in Error Logic

453

# » Definition 4.58 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for −1 (A((X , X , X , X , X ), X = X ((X , X ), Φ))) = A((X , X , X , X , judging errors, if Thgz −1 X ), X = X ((X , X ), G(t))), then Thgz is called the rule engenderment transformation # » connective with regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » −1 p(t)G(t))). The meaning of rule engenderment is: Thgz (Φ) → (there exist rules or rules have been developed). # » Definition 4.59 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging errors, if Ths−1j (A((X , X , X , X , X ), X = X ((X , X ), Φ))) = A((X (t), X (t), X (t), X (t), X (t)), X (t) = X ((X (t), X (t)), X (t))), then Ths−1j is called the temporal engenderment transformation connective with regard to G(t) and A((U (t), S(t), # » # » p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). The meaning of temporal engenderment is: Ths−1j (Φ) → (there exist time factor or time factor is considered). # » Definition 4.60 Suppose that an error logical variable A((U (t), S(t), p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule −1 (A((Φ, Φ, Φ, Φ, Φ), Φ = Φ((Φ, Φ), Φ))) ∈ {A((U (t), for judging errors, if Thqb # » S(t), X , X , X ), X = X ((X , X X )), A((X , X , p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » p(t)G(t))), A((U (t), S(t), X , X , X ), x(t) = f ((X , p(t)G(t))), . . . , A((Φ, Φ, Φ, −1 is called the partial or complete engenderment Φ, Φ), X = f ((Φ, X X ))}, then Thqb # » transformation connective with regard to G(t) and A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t)G(t))). The meaning of partial or complete engenderment is: −1 (Φ, Φ, . . . , Φ) → (there exist those aforementioned corresponding elements). Thqb 4.5.8.2

Principles for Engenderment Transformation in Error Logic

The principles for engenderment transformation are: (1) Actual needs; (2) Feasibility of actual conditions; (3) The minimum cost.

4.5.8.3

Approaches of Engenderment Transformation in Error Logic

There are many different ways to carry out engenderment transformation, which are: { accept, acquire, add, appear, arrive, beget, buy back, build, build up, construct, come back to life, create, develop, erect, establish, found, foster, implant, import, include, increase, instill, move back, move in, pick up, plant, populate, produce, raise, reappear, reconstruct, recover, replenish, retrieve, restore }

454

4.5.8.4

4 Transformation Connectives in Error Logic

Hierarchy of Engenderment Transformation in Error Logic

In general, the object of interest μ(t) has both vertical and horizontal structures and multiple hierarchical relationships as well. Therefore, engenderment transformation can also be carried out in different hierarchies.

4.5.8.5

Characteristics of Engenderment Transformation Connective Th−1 in Error Logic

# » Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain U (t) under G(t) the rule for judging −1 errors, if Thly (A((Φ, X , X , X , X ), X = X ((X , X ), X ))) = A((U (t), X , X , X , X ), X = X ((X , X ), X )), where X represents Φ or element in corresponding posi−1 is the engenderment transformation connective with respect to G(t) and tion, Thly # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))). As for an error logical # » # » variable (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))), if there does not exist domain or there is no need to investigate the object of interests in cur# » rent domain, that is A((Φ, X , X , X , X ), X = X ((X , X ), X )) = A((Φ, S(t), p(t), # » T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) or A((Φ, X , X , X , X ), X = X ((X , X ), # » # » X )) = A((Φ, Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), . . . , A((Φ, Φ, Φ, Φ, Φ), Φ = Φ((Φ, Φ), Φ)). If A((Φ, X , X , X , X ), X = X ((X , X ), X )) = A((Φ, # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), error can be eliminated through −1 (A((Φ, X , X , X , X ), X = X ((X , X ), domain engenderment transformation-i.e., Thly # » # » X ))) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))); while if A((Φ, # » X , X , X , X ), X = X ((X , X ), X )) = A((Φ, Φ, p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)G(t))), . . . , A((Φ, Φ, Φ, Φ, Φ), Φ = Φ((Φ, Φ), Φ)), then other transformations need to be conducted to eliminate errors in A((Φ, X , X , X , X ), X = X ((X , X ), X )). # » (1) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain Φ under G(t) the rule for judging # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t)))) = errors, if Thly # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((u(t), p(t), G(t))), where x(t) ∈ , # » # » then A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))) a meaningful or feasible error logical variable. # » (2) Suppose that an error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f ((μ(t), p(t)G(t))) is defined in domain Φ under G(t) the rule for judging # » errors, if A((Φ, X , X , X , X ), X = X ((X , X ), X )) = A((Φ, Φ, p(t), T (t), # » L(t)), x(t) = f ((μ(t), p(t)G(t))), . . . , A((Φ, Φ, Φ, Φ, Φ), Φ = Φ((Φ, Φ), # » −1 (A((Φ, X , X , X , X ), X = X ((X , X ), X ))) = A((U (t), S(t), p(t), Φ)) and Thly # » −1 T (t), L(t)), x(t) = f ((μ(t), p(t)G(t))), it is said that Thly has forced (A((Φ, X , X , X , X ), X = X ((X , X ), X ))) to carry out transformation on the elements # » of thing S(t), spatial factor (space) p(t), property T (t), property (or attribute)

4.5 Destruction Transformation Connectives in Error Logic

455

value L(t)), error value x(t), error function f , temporal factor (time)t, and rules −1 −1 −1 −1 −1 −1 −1 −1 , Thsw , Thk for judging error G(t), where Th−1 ⊆ {Thly j , Tht z ,Thlz , Thcz , Thgz , Thhs , −1 }. Ths−1j , Thq Proposition 4.238 Suppose that a meaningful error logical variable A((Φ, X , X , # » X , X ), X = X ((X , X ), X )) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is defined in domain Φ under G A (t) the rules for judging errors, # » # » −1 (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((Φ, Thly # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), where x(t) ∈ ; suppose that another meaningful error logical variable B((Φ, X , X , X , X ), X = X ((X , # » # » X ), X )) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = g((ν(t), p(t)), G B (t))) is defined # » −1 (B((Φ, S(t), p(t), in domain Φ under G B (t) the rules for judging errors, Thly # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))), where y(t) ∈ , the following relationships hold: # » # » −1 (1) Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((Φ, # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), # » # » # » −1 p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨ Thly (B((Φ, S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » −1 (2) Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), # » # » # » −1 p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ Thly (B((Φ, S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » −1 (3) Thly (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) = # » # » −1 ¬Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))). # » # » −1 Proof As Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) and Thly (B((Φ, # » # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), S(t), p(t), T (t), # » −1 (A((Φ, L(t)), y(t) = f ((ν(t), p(t)), G B (t))), if x(t)  y(t), then the left side = Thly # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨ B((Φ, S(t), p(t), T (t), # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly # » # » # » f ((μ(t), p(t)), G A (t)))) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))). # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), And the right side = Thly # » # » −1 G A (t)))) ∨ Thly (B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = A((U (t), S(t), p(t), T (t), L(t)), # » x(t) = f ((μ(t), p(t), G A (t))). Left side = right side. # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = If if x(t)  y(t), then the left side = Thly # » # » # » f ((μ(t), p(t)), G A (t))) ∨ B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)),

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# » # » −1 G B (t)))) = Thly (B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))). # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), And the right side = Thly # » # » −1 G A (t)))) ∨ Thly (B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))) ∨ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))). Left side = right side. Proof is completed. Similarly, (2) and (3) can also be proved. Proposition 4.239 Suppose that a meaningful error logical variable A((Φ, X , X , # » X , X ), X = X ((X , X ), X )) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is defined in domain Φ under G A (t) the rules for judging errors, # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((U (t), Thly # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), where x(t) ∈ ; suppose that another meaningful error logical variable B((Φ, X , X , X , X ), X = X ((X ,e X ), # » # » X )) = B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined # » −1 (B((Φ, S(t), p(t), in domain Φ under G B (t) the rules for judging errors, Thly # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))), where y(t) ∈ ; suppose that a meaningful error log# » # » ical variable C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the # » # » mediator variable of A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) # » # » −1 (C Az B ((Φ, and B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Thly # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = C Az B ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), where z(t) ∈ , the following relationships hold: # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((Φ, (1) Thly # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), # » # » # » −1 (B((Φ, S(t), p(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∨n Thly # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧n B((Φ, (2) Thly # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), # » # » # » n −1 p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) ∧ Thly (B((Φ, S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f l B((Φ, (3) Thly # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), # » # » # » −1 nf l p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) | Thly (B((Φ, S(t), p(t), # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) |n f h B((Φ, (4) Thly # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t),

4.5 Destruction Transformation Connectives in Error Logic

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# » # » # » −1 (B((Φ, S(t), p(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) |n f h Thly # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhb B((Φ, (5) Thly # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), # » # » # » −1 (B((Φ, S(t), p(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhb Thly # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))); # » # » −1 (6) Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) nhdl B((Φ, # » # » S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = T f s j (A((Φ, S(t), # » # » # » −1 (B((Φ, S(t), p(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) nhdl Thly # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). # » # » −1 Proof The left side = Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∨n B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » −1 Thly ((A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((Φ, S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » Az B G B (t))) ∧ ¬C ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » # » (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((Φ, S(t), p(t), # » # » Az B T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C ((Φ, S(t), p(t), T (t), L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » Az B C ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((Φ, # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = # » # » # » f ((ω(t), p(t)), G C (t)))) ∨ (¬A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), # » # » G A (t))) ∧ B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » ¬C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((Φ, # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((Φ, S(t), p(t), T (t), # » # » L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((Φ, S(t), p(t), T (t), L(t)), z(t) = # » # » f ((ω(t), p(t)), G C (t))))) = ((A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨

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4 Transformation Connectives in Error Logic

# » # » (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » Az B G B (t))) ∧ ¬C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))))). # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), And the right side = Thly # » # » −1 G A (t)))) ∨n Thly (B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = # » # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) = ((A((U (t), S(t), p(t), T (t), # » # » L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∨n B((U (t), S(t), p(t), T (t), L(t)), y(t) = # » # » f ((ν(t), p(t)), G B (t)))) = ((A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » # » # » p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ # » # » C Az B ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (A((U (t), # » # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), L(t)), # » # » z(t) = f ((ω(t), p(t)), G C (t)))) ∨ (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » Az B G B (t))) ∧ C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » Az B G B (t))) ∧ C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨ # » # » (¬A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) ∧ B((U (t), S(t), # » # » # » p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) ∧ ¬C Az B ((U (t), S(t), p(t), T (t), # » # » L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) ∨(A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » # » # » f ((μ(t), p(t)), G A (t))) ∧ ¬B((U (t), S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), # » # » Az B G B (t))) ∧ ¬C ((U (t), S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))))). Left side = right side. Proof is completed. Similarly, (2)–(6) can also be proved. Proposition 4.240 Suppose that a meaningful error logical variable A((Φ, X , X , # » X , X ), X = X ((X , X ), X )) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is defined in domain Φ under G A (t) the rules for judging errors, # » # » −1 Thly (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((U (t), # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), where x(t) ∈ ; suppose that another meaningful error logical variable B((Φ, X , X , X , X ), X = X ((X , X ), # » # » X )) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined # » −1 (B((Φ, S(t), p(t), in domain Φ under G B (t) the rules for judging errors, Thly # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))), where y(t) ∈ ; suppose that a meaningful error logical variable C AnhbB ((Φ, X , X , X , X ), X = X ((X , X ), X )) = C AnhbB ((U (t),

4.5 Destruction Transformation Connectives in Error Logic

459

# » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative inclu# » # » sion variable of A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » −1 (C AnhbB ((Φ, B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Thly # » # » # » S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = C AnhbB ((U (t), S(t), p(t), # » T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), where z(t) ∈ ; the following relationship holds: # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) −n B((Φ, S(t), Thly # » # » # » −1 (A((Φ, S(t), p(t), T (t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly # » # » −1 L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) −n Thly (B((Φ, S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t)))); Proof Proof is omitted. Proposition 4.241 Suppose that a meaningful error logical variable A((Φ, X , X , # » X , X ), X = X ((X , X ), X )) = A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), # » p(t)), G A (t))) is defined in domain Φ under G A (t) the rules for judging errors, # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) = A((U (t), Thly # » # » S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t), G A (t))), where x(t) ∈ ; suppose that another meaningful error logical variable B((Φ, X , X , X , X ), X = X ((X , X ), # » # » X )) = B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))) is defined # » −1 (B((Φ, S(t), p(t), in domain Φ under G B (t) the rules for judging errors, Thly # » # » T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = B((U (t), S(t), p(t), T (t), L(t)), # » y(t) = f ((ν(t), p(t)), G B (t))), where y(t) ∈ ; suppose that C Anhdthd j B ((Φ, S(t), # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))) is the connotative same or equiva# » # » lence variable for A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) and # » # » −1 (C Anhdthd j B ((Φ, B((Φ, S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t))), Thly # » # » Anhdthd j B S(t), p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t)))) = C ((U (t), S(t), # » # » p(t), T (t), L(t)), z(t) = f ((ω(t), p(t)), G C (t))), the following relationship holds: # » # » −1 (A((Φ, S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t))) →nby B((Φ, Thly # » # » # » −1 S(t), p(t), T (t), L(t)), y(t) = f ((ν(t), p(t)), G B (t)))) = Thly (A((Φ, S(t), p(t), # » # » −1 T (t), L(t)), x(t) = f ((μ(t), p(t)), G A (t)))) →nby Thly (B((Φ, S(t), p(t), T (t), # » L(t)), y(t) = f ((ν(t), p(t)), G B (t)))). Proof Proof is omitted. # » Proposition 4.242 Suppose that A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the nth order error logical variable −1 (A(n) ((Φ (n) , defined in domain Φ ( n) under G ( n) A (t) the rules for judging errors, Thly # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n) (t)), # » n (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), G (n) A (t))), where x ∈ ; suppose that A # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A

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4 Transformation Connectives in Error Logic

order error logical variable defined in domain Φ (n+1) under G (n+1) (t) the rules A # (n+1) » (n+1) −1 (n+1) (n+1) (n+1) for judging errors, Thly (A ((Φ ,S (t), p (t), T (t), L (n+1) (t)), # » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t)))) = A(n+1) ((U (n+1) (t), A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), (n+1) G A (t))), where x n+1 ∈ ; suppose that error logical variable B (n) ((Φ (n) , S (n) (t), # (n) » (n) # » p (t), T (t), L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))) is the comple# (n) » (n) (n) (n) (n) ment variable for A ((Φ , S (t), p (t), T (t), L (n) (t)), x (n) (t) = # » (n) under G (n) f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))), which is defined in domain Φ B (t) # » −1 (B (n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), the rules for judging errors, Thly # » # » (n) (n) y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) (t), S (n) (t), p (n) (t), T (n) (t), B (t)))) = B ((U # » n L (n) (t)), y (n) (t) = f (n) ((ν (n) (t), p (n) (t)), G (n) B (t))), where y ∈ ; suppose that # » error logical variable B (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))) is the complement variable for # (n+1) » B (n+1) (n+1) (n+1) (n+1) ((Φ , S (t), p (t), T (t), L (n+1) (t)), x (n+1) (t) = A # » (t))), which is defined in domain Φ (n+1) under f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # » (n+1) −1 (B (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), G B (t) the rules for judging errors, Thly # » T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t)))) = B # » B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), y (n+1) (t) = # » (t))), where y n+1 ∈ ; suppose that f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # » # » C (n)Az B ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), # » G C(n) (t))) is the mediator variable for A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » # (n) » (n) (n) (n) (n) x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), A (t))) and B ((Φ , S (t), p (t), T # » (n) −1 (n) (n) (n) (n) (n) (n)Az B (n) (n) ((Φ , S (t), L (t)), y (t) = f ((ν (t), p (t)), G B (t))), Thly (C # (n) » # (n) » (n) (n) (n) (n) (n) (n) p (t), T (t), L (t)), z (t) = f ((ω (t), p (t)), G C (t)))) = C (n)Az B # » # » ((U (n) (t)(t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), z (n) (t) = f (n) ((ω(n) (t), p (n) (t)), # » G C(n) (t))), where z n ∈ ; suppose that C (n+1)Az B ((Φ (n+1) , S (n+1) (t), p (n+1) (t), # » T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))) is the # » mediator variable for A(n+1) ((Φ (n+1) ,S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), # » (t))) and B (n+1) ((Φ (n+1) , S (n+1) (t), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A # (n+1) » (n+1) # » p (t), T (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) (t))), B # » −1 (C (n+1)Az B ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t)= Thly # » f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))) = C (n+1)Az B ((U (n+1) (t), S (n+1) (t), # (n+1) » # » p (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t))), where z n+1 ∈ , the following relationship holds: # » −1 Thly (¬bz A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t),

4.5 Destruction Transformation Connectives in Error Logic

461

# (n) » # (n) » (n) bz −1 (n) (n) (n) p (t)), G (n) (t), L (n) (t)), x (n) (t) = A (t)))) = ¬ Thly (A ((Φ , S (t), p (t), T # » (n) f (n) ((μ(n) (t), p (n) (t)), G A (t)))). Proof The left side of the equation = # » −1 Thly (¬bz A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # » −1 (n+1) p (t)), G (n) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), A (t)))) = Thly = (A # » L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((Φ (n+1) , A # (n+1) » (n+1) # » (n+1) (n+1) (n+1) (n+1) (t), p (t), T (t), L (t)), y (t) = f ((ν (n+1) (t), p (n+1) (t)), S # » (t))) ∧ C (n+1)Az B ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) B # » p (n+1) (t)), G C(n+1) (t)))) = (A(n+1) ((U (n+1) (t), z (n+1) (t) = f (n+1) ((ω(n+1) (t), # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), G (n+1) A # » (t))) ∧ C (n+1)Az B ((U (n+1) (t), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)), G (n+1) B # (n+1) » (n+1) # » (n+1) (n+1) (n+1) (t), p (t), T (t), L (t)), z (t) = f (n+1) ((ω(n+1) (t), p (n+1) (t)), S G C(n+1) (t)))). # » −1 (n) The right side of the equation = ¬bz Thly A ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), # » bz (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t)))) = ¬ (A ((U # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # » (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) # » # » ((μ(n+1) (t), p (n+1) (t)), G (n+1) (t))) ∧ B (n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), A # » (t))) ∧ T (n+1) (t), L (n+1) (t)), y (n+1) (t) = f (n+1) ((ν (n+1) (t), p (n+1) (t)),G (n+1) B # » C (n+1)Az B ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), z (n+1) (t) = # » f (n+1) ((ω(n+1) (t), p (n+1) (t)), G C(n+1) (t)))). Left side = right side. Proof is completed. # » Proposition 4.243 Suppose that A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), # » th x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) order error logical variable A (t))) is the n −1 ( ( (A(n) ((Φ (n) , defined in domain Φ n) under G n) A (t) the rules for judging errors, Thly # » # » S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) (A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n) (t)), # » n (n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), G (n) A (t))), where x ∈ ; suppose that A # » (t))) is the (n + 1)th L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), G (n+1) A order error logical variable defined in domain Φ (n+1) under G (n+1) (t) the rules A # » −1 (A(n+1) ((Φ (n+1) , S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), for judging errors, Thly # (n+1) » x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (t)), G (n+1) (t)))) = A(n+1) ((U (n+1) (t), A # » (n+1) (n+1) (n+1) (n+1) (n+1) (t), p (t), T (t), L (t)), x (t) = f (n+1) ((μ(n+1) (t), S # (n+1) » (n+1) p (t)), G A (t))), where x n+1 ∈ ; it is assumed that C (n)Az B(n+1) ((Φ (n)(n+1) ,

462

4 Transformation Connectives in Error Logic

# » S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) # » ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))) is the mediator variable for A(n+1) ((Φ (n+1) , # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) and A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n+1) A # (n) » # » −1 p (t)), G (n) Thly (C (n)Az B(n+1) ((Φ (n)(n+1) (t),S (n)(n+1) (t), p (n)(n+1) (t), A (t))), # (n)(n+1) » T (n)(n+1) (t),L (n)(n+1) (t)),z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (t)), # (n)(n+1) » (n)(n+1) (n)(n+1) (n)Az B(n+1) (n)(n+1) (n)(n+1) (t)))) = C ((U (t), S (t), p (t), T (t), GC # » L (n)(n+1) (t)),z (n)(n+1) (t)= f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t))), where z (n)(n+1) ∈  the following relationship holds: # » −1 Thly (¬bx A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # (n) » bx −1 (n) (n) (n) (n) p (t)), G (n) (t), L (n) (t)), A (t)))) = ¬ Thly (A ((Φ , S (t), p (t), T # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » −1 Proof The left side of the equation = Thly (¬bx A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), # » −1 (n+1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((Φ (n+1) , A (t)))) = Thly (A # » # » S (n+1) (t), p (n+1) (t), T (n+1) (t),L (n+1) (t)), x (n+1) (t) = f (n+1) ((μ(n+1) (t), p (n+1) (t)), # » (t))) ∧ A(n) ((Φ (n) ,S (n) (t), p (n) (t),T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n+1) A # (n) » # » (n)Az B(n+1) p (t)), G (n) ((Φ (n)(n+1) , S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), A (t))) ∧C # » L (n)(n+1) (t)), z (n)(n+1) (t) = f (n)(n+1) ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))) = # » (A(n+1) ((U (n+1) (t), S (n+1) (t), p (n+1) (t), T (n+1) (t), L (n+1) (t)), x (n+1) (t) = # » # » (t))) ∧ A(n) ((U (n) (t),S (n) (t), p (n) (t), T (n) (t), f (n+1) ((μ(n+1) (t), p (n+1) (t)),G (n+1) A # » (n)Az B(n+1) ((U (n)(n+1) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧C # » S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t)= f (n)(n+1) # » ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))). # » −1 And the right side of the equation = ¬bx Thly (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), # » bx (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t)))) = ¬ (A ((U # (n) » # (n) » (n) (n) (n) (n) (n) p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G (n) A (t)))) = # (n+1) » (n+1) (n+1) (n+1) (n+1) (n+1) ((U (t), S (t), p (t), T (t), L (t)), x (n+1) (t) = (A # (n+1) » # » (n+1) (n+1) (n+1) (n) (n) (n) ((μ (t), p (t)), G A (t))) ∧ A ((U (t), S (t), p (n) (t), T (n) (t), f # » (n)Az B(n+1) ((U (n)(n+1) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧C # » S (n)(n+1) (t), p (n)(n+1) (t), T (n)(n+1) (t), L (n)(n+1) (t)), z (n)(n+1) (t)= f (n)(n+1) # » ((ω(n)(n+1) (t), p (n)(n+1) (t)), G C(n)(n+1) (t)))). Left side = right side. Proof is completed.

4.5 Destruction Transformation Connectives in Error Logic

463

# » Proposition 4.244 Suppose that an error logical variable A(n) ((Φ (n) , S (n) (t), p (n) (t), # » th T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order error log( ( ical variable defined in domain Φ n) under G n) A (t) the rules for judging errors, # » # » −1 Thly (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), # » (n) (n) G (n) (t), S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), A (t)))) = A ((U # (n) » # » p (t)), where x n ∈ ; suppose that A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th A (n−1) order error logical variable defined in domain Φ (t) under G (n−1) (t) the rules A # » (n−1) −1 (n−1) (n−1) (n−1) (n−1) for judging errors, Thly (A ((Φ (t), S (t), p (t), T (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = A(n−1) ((U (n−1) (t), A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » where x n−1 ∈ ; suppose that B (n−1) ((Φ (n−1) ), S (n−1) (t), p (n−1) (t), T (n−1) (t), # » L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th B # » order error logical complementary variable of A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), # » T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) defined A (n−1) −1 (n−1) (n−1) (t) under G B (t) the rules for judging errors, Thly (B ((Φ (n−1) , in domain Φ # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t)))) = B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # » (t))), where y n−1 ∈ ; y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » suppose that C (n−1)Az B ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t))) is the mediator variable for # » A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) # » # » ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) and B (n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), A # » T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) (t))), B # (n−1) » −1 (n−1)Az B (n−1) (n−1) (n−1) (n−1) (n−1) ((Φ , S (t), p (t), T (t), L (t)), z (t) = Thly (C # (n−1) » # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) f ((ω (t), p (t)), G C (t)))) = C ((U (t), S (t), p (t), # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) T (t), L (t)), z (t) = f ((ω (t), p (t)), G C (t))), where z n−1 ∈ , the following relationship holds: # » −1 bj Thly ¬ (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t),L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # (n) » (n) bj −1 (n) (n) (n) p (t)), G (n) (t),L (n) (t)), x (n) (t) = A (t)))) = ¬ Thly (A ((Φ , S (t), p (t),T # » f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » −1 bj Proof The left side of the equation = Thly ¬ (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), # » −1 (n−1) L (n) (t)), x (n) (t)= f (n) ((μ(n) (t), p (n) (t)), G (n) ((Φ (n−1) , S (n−1) (t), A (t)))) = Thly (A # (n−1) » # » p (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ B (n−1) ((Φ (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) A # » y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)),G (n−1) (t))) ∧ C (n−1)Az B ((Φ (n−1) , S (n−1) B

464

4 Transformation Connectives in Error Logic

# » # » (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), # » G C(n−1) (t)))) = (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), A # (n−1) » # » p (t), T (n−1) (t), L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), # » (t))) ∧ C (n−1)Az B ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), G (n−1) B # » z (n−1) (t) = f (n−1) ((ω(n−1) (t), p (n−1) (t)), G C(n−1) (t)))). # » −1 And the right side of the equation =¬bj Thly (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), # » bj (n) (n) L (n) (t)),x (n) (t) = f (n) ((μ(n) (t), p (n) (t)),G (n) (t), S (n) (t), A (t))))= ¬ A ((U # (n) » # (n) » (n) (n) (n) (n) (n) (n) p (t), T (t), L (t)), x (t) = f ((μ (t), p (t)), G A (t))) = (A(n−1) # » ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), # (n−1) » # » p (t)), G (n−1) (t))) ∧ B (n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), A # » (t))) ∧ C (n−1)Az B L (n−1) (t)), y (n−1) (t) = f (n−1) ((ν (n−1) (t), p (n−1) (t)), G (n−1) B # » ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t),L (n−1) (t)), z (n−1) (t) = f (n−1) ((ω(n−1) (t), # (n−1) » p (t)), G C(n−1) (t)))). Left side = right side. Proof is completed. # » Proposition 4.245 Suppose that A(n) ((Φ (n) ,S (n) (t), p (n) (t),T (n) (t),L (n) (t)), x (n) # » th (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) is the n order error logical variable defined −1 ( ( (A(n) ((Φ (n) , S (n) (t), in domain Φ n) under G n) A (t) the rules for judging errors, Thly # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))) = # (n) » (n) # » (n) (n) (n) (n) (n) (n) (n) A ((U (t), S (t), p (t), T (t), L (t)), x (t) = f ((μ (t), p (n) (t)), # » where x n ∈ ; suppose that A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t))) is the (n − 1)th order error A (n−1) under G (n−1) (t) the rules for judging logical variable defined in domain Φ A # » (n−1) −1 (n−1) (n−1) (n−1) (n−1) ((Φ ,S (t), p (t), T (t), L (n−1) (t)), x (n−1) (t) = errors, Thly (A # » # » f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) (t)))) = A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), A # » (t))), where T (n−1) (t), L (n−1) (t)), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)),G (n−1) A # » x n−1 ∈ ; suppose that C (n)Az B(n−1) ((Φ (n)(n−1) , S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), # » L (n)(n−1) (t)),z (n)(n−1) (t)= f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))) is the # » mediator variable for A(n−1) ((Φ (n−1) , S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), # » # » (t))) and A(n) ((Φ (n) , S (n) (t), p (n) (t), x (n−1) (t) = f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » −1 (n)Az B(n−1) T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)),G (n) A (t))), and Thly (C # » ((Φ (n)(n−1) , S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = # » f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = C (n)Az B(n−1) ((U (n)(n−1) (t), # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1)

4.5 Destruction Transformation Connectives in Error Logic

465

# » ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t))), where z (n)(n−1) ∈ , the following relationship holds: # » −1 Thly (¬bd A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), # (n) » # (n) » bd −1 (n) (n) (n) (n) p (t)), G (n) (t), L (n) (t)), A (t)))) = ¬ Thly (A ((Φ , S (t), p (t), T # » x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t)))). # » −1 bd ¬ (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), Proof The left side of the equation = Thly # » −1 (n−1) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) ((Φ (n−1) , A (t)))) = Thly (A # » # » S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t)= f (n−1) ((μ(n−1) (t), p (n−1) (t)), # » (t))) ∧ A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), G (n−1) A # (n) » # » (n)Az B(n−1) p (t)), G (n) ((Φ (n)(n−1) , S (n)(n−1) (t), p (n)(n−1) (t),T (n)(n−1) (t), A (t))) ∧ C # » L (n)(n−1) (t)),z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = # » (A(n−1) ((U (n−1) (t), S (n−1) (t), p (n−1) (t), T (n−1) (t), L (n−1) (t)), x (n−1) (t) = # » # » (t))) ∧ A(n) ((U (n) (t), S (n) (t), p (n) (t), T (n) (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # » (n)Az B(n−1) ((U (n)(n−1) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) ∧ C # » S (n)(n−1) (t), p (n)(n−1) (t), T (n)(n−1) (t), L (n)(n−1) (t)), z (n)(n−1) (t) = f (n)(n−1) ((ω(n)(n−1) # » (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))). # » −1 And the right side of the equation = ¬bd Thly (A(n) ((Φ (n) , S (n) (t), p (n) (t), T (n) (t), # » bd (n) (n) L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) (t), S (n) (t), A (t)))) = ¬ A ((U # (n) » # » p (t), T (n) (t), L (n) (t)), x (n) (t) = f (n) ((μ(n) (t), p (n) (t)), G (n) A (t))) = # (n−1) » (n−1) (n−1) (n−1) (n−1) (n−1) (n−1) ((Φ , S (t), p (t), T (t), L (t)), x (t) = (A # » # (n) » (n) (n) (n) (n) (t))) ∧ A ((Φ , S (t), p (t), T (t), f (n−1) ((μ(n−1) (t), p (n−1) (t)), G (n−1) A # (n) » (n) (n) (n) (n) (n) (n)Az B(n−1) (n)(n−1) ((Φ , L (t)), x (t) = f ((μ (t), p (t)), G A (t))) ∧ C # (n)(n−1) » (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (n)(n−1) (t), p (t), T (t), L (t)), z (t) = f ((ω S # » (t), p (n)(n−1) (t)), G C(n)(n−1) (t)))) = Φ. Left side = right side. Proof is completed.

Chapter 5

Mathematical Error Propositional Logic

Mathematical error propositional logic is a subject that uses mathematics and semantics to examine the reasoning forms and laws of error compound propositions and logical relationships between error propositions, which is a thinking form that uses object and rules to judge and evaluate erroneous structure of the entity of interests. This chapter mainly discusses concept, connectives, truth value table, logical forms, and effective reasoning methods related to mathematical error propositional logic.

5.1 Concept of Mathematical Error Propositional Logic # » Definition 5.1 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}, where U (t) is the domain # » # » of μ(t) = (U (t), S(t), p(t), T (t), L(t)), S(t) is the thing or subject, p(t) is the # » spatial location and direction of μ(t) = (U (t), S(t), p(t), T (t) is the property or # » predicate of μ(t) = (U (t), S(t), p(t), L(t) is the value of the property or attribute # » of μ(t) = (U (t), S(t), p(t), x(t) = f (μ(t), G(t)) is the truth value or truth value # » function of A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), G(t) is the rule # » for judging error defined in domain U (t), A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))) is the error logical variable defined in U (t) under the rule of judging errors G(t). # » Definition 5.2 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), x(t) ∈ {{0, 1}, [0, 1], (−∞, +∞)}(G(t) is the rule for judging error, U (t) is domain ) is the error logical variable defined in domain U (t) under the rule of judging error G(t), then the set C composed of all error logical variables is called the error logical variable set defined in domain U (t) under the rule of judging error G(t).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Liu and K. Guo, Error Logic: Paving Pathways for Intelligent Error Identification and Management, Studies in Systems, Decision and Control 442, https://doi.org/10.1007/978-3-031-00820-7_5

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5.2 Error Logical System 5.2.1 Existential Quantifiers (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

∃LY , there exists certain domain, ∃SW , there exists certain thing, ∃S J , there exists certain time, ∃K J , there exists certain spatial location, ∃T J , there exists certain constraint, ∃T Z , there exists certain property, ∃L Z , there exists certain value of the property or attribute, ∃C Z , there exists certain error value, ∃H S, there exists certain function, ∃G Z , there exists certain group of rules, ∃F S, there exists certain decomposition method.

5.2.1.1 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

∀LY , for each domain, ∀SW , for each thing, ∀S J , for the whole time, ∀K J , for each spatial location, ∀T J , for each constraint, ∀T Z , for each property, ∀L Z , for each value of the property or attribute, ∀C Z , for each error value, ∀H S, for each function, ∀G Z , for each group of rules, ∀F S, for each decomposition method, Θ there exists certain universal, Ψ for all the universals

5.2.1.2 (1) (2) (3) (4) (5) (6) (7) (8)

Universal Quantifiers

Denotative Connectives

¬ negation, ∧ conjunction, ∨ disjunction, ∨bxr exclusive disjunction: or → material implication: if · · · then, ← inverse material implication: only · · · if, ↔ if and only if-biconditional logical connective, = equal,

5.2 Error Logical System

(9) (10) (11) (12) (13) (14)

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˙ small AND operation, ∧ ¨ equal AND operation, ∧ ∧ large AND operation, ˙ small OR operation, ∨ ¨ equal OR operation, ∨ ∨ large OR operation.

5.2.1.3

Transformation Connectives

(1) Similarity connective, Tx ⊆ {Txly , Txsw , Txk j , Txt z , Txlz , Txcz , Txhs , Txs j , Txgz , Tx zh } (similarity), Tx−1 : inverse similarity connective; (2) Displacement connective, Tz ⊆ {Tzly , Tzsw , Tzk j , Tzt z , Tzlz , Tzcz , Tzhs , Tzs j , Tzgz , Tzzh } (similarity), Tz−1 : inverse displacement connective; (3) Increase connective, Tz n ⊆ {Tznly , Tznsw , Tznk j , Tznt z , Tznlz , Tzncz , Tznhs , Tzns j , Tzngz , Tznzh } (addition), Tz n −1 : inverse increase connective; (4) Decomposition connective, T f ⊆ {T f ly , T f sw , T f k j , T f t z , T f lz , T f cz , T f hs , T f s j , T f gz , T f zh } (decomposition), T f−1 : inverse decomposition connective; (5) Destruction connective, Th ⊆ {Thly , Thsw , Thk j , Tht z , Thlz , Thcz , Thhs , Ths j , Thgz , Thzh } (destruction), Th−1 : inverse destruction connective; (6) Unit transformation connective, Td (unit),Td−1 : inverse unit transformation connective; (7) Quantifier system of error logic.

5.2.1.4

Connotative Connectives

(1) ¬bz , not only negation: there exists error that can be negated before being decomposed; (2) ¬bj , unfinished negation: there exists error that can be negated after being decomposed; (3) ¬bx , unconstrained negation: for the error being negated, there exists its opposite side before being decomposed; (4) ¬bd , uninterrupted negation: for the error being negated, there exists its opposite side after being decomposed; (5) ∧n , connotative conjunction: different errors are compatible; (6) ∨n , connotative disjunction: different errors have mutual infiltration; (7) −n , connotative difference: errors were removed or reduced; (8) |n f l , connotative separation: different errors coexist; (9) |n f h , connotative differentiation: errors with critical points do not exist; (10) nhb , connotative complement: correctness, error with critical points, and error coexist; (11) nhdl , connotative antithesis: correctness and error coexist; (12) →nhy , connotative possibility implication: if . . . , then it is possible;

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(13) →nby , connotative necessity implication, if . . . , then it must be or it could not be otherwise; (14) →nsy , connotative isness implication: if . . . , then it is . . . ; (15) ↔nhdz , connotative equivalence: if...then it is equivalent; (16) =nhdt , connotative same connective: properties are the same or two errors are the same.

5.2.1.5 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19)

Concept of Predicate Logic

D (C) , rare case; D (T ) , special case; D (Y B Q) , generic case; D (Y BG) , generic concept; D (C W Q) , erroneous case; D (FC W Q) , non-erroneous case; D (Z Q Q) , correct case; D (C W L Q) , case of error with critical points; D (CC W Q) , pure error case (without critical points); D (RY C W J ) , set of arbitrary error logical variables; D (QC W J ) , set of all error logical variables; D (B FC W J ) , set of partial error logical variables; D (C W J ) , set of error logical variables; D (FC W J ) , set of non-erroneous logical variables; D (L J C W J ) , set of logical variables for error with critical points ; D (Z Q J ) , set of correct logical variables; D (W QC W J ) , set of complete error logical variables; D (C Z L J C W J ) , set of pure positive critical error logical variables; D (C Z Q J ) , set of absolute correct (without critical points)logical variables.

5.3 Atomic Propositions # » Definition 5.3 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) and B(ν(t), y(t)) = B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domains U (t) and V (t) under judging rules G A (t) and G B (t),respectively, # » if ω(t) in C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is constructed by equivalent parts or certain parts of μ(t) and ν(t), where W (t) is the domain of ω(t) and G C (t) is judging rule for errors defined on W (t). Then C((W (t), SC (t), # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is called the intermediary variable of # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((V (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) defined in different domains noted by # » C Azy B ((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))).

5.3 Atomic Propositions

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# » Definition 5.4 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, if ω(t) # » in C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is constructed by equivalent parts or certain parts of μ(t) and ν(t), where U (t) is the domain of ω(t) # » and G C (t) is judging rule for errors. Then C((U (t), SC (t), pC (t), TC (t), L C (t)), # » z(t) = f (ω(t), G C (t))) is called the intermediary variable of A((U (t), S A (t), p A (t), # » T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) defined in the same domain noted by C Azt B ((U (t), SC (t), # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))). # » Definition 5.5 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, the # » errors in A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) have connotative inclusion rela# » tionships, i.e., T A (t) ⊃nhb TB (t), then C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = # » f (ω(t), G C (t))) = A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » ⊃n B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) is called the con# » notative inclusion variable of A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) with respect to B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G B (t))) denoted by C G C (t))). # » Definition 5.6 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t),respectively, the errors # » in A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) are the same, i.e., T A (t) =nhdt TB (t), # » then C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) = A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) =n B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) is called the connotative same variable with # » the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) denoted by # » C Andt B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))). # » Definition 5.7 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, the errors in # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) are equivalent, i.e., T A (t) ⇐⇒nhd j

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# » TB (t), then C((U (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t)))=A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ⇐⇒nhd j B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) is called the connotative equivalence variable # » with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) expressed # » by C And j B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))).

5.4 Basic Operations # » Definition 5.8 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))  =

# » ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), x(t)  y(t) # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t)  y(t)

# » Then ∨ is called the denotative disjunction of A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))). Please refer to Table 5.1 for the truth value table of disjunction operation on A and B. # » Definition 5.9 Suppose that A(μ(t), x(t)) = A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∧ B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))  =

# » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), x(t) ≤ y(t) # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t) ≥ y(t)

Table 5.1 Truth value table of disjunction operation on A and B A B Relationship Between x(t) and y(t) x(t) x(t) x(t)

y(t) y(t) y(t)

x(t) > y(t) x(t) < y(t) x(t) = y(t)

A∨B A B A or B

5.4 Basic Operations

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Table 5.2 Truth value table of conjunction operation on A and B A B Relationship Between x(t) and y(t) x(t) x(t) x(t)

y(t) y(t) y(t)

x(t) > y(t) x(t) < y(t) x(t) = y(t)

Table 5.3 Truth value table of negation operation on A A Logical value range of x(t) x(t) x(t) x(t) x(t)

x(t) ∈ {0, 1} x(t) ∈ {0, 1} x(t) ∈ [0, 1] x(t) ∈ (−∞, +∞)

A∧B B A A or B

¬A if x(t) = 1, y(t) = 0 if x(t) = 0, y(t) = 1 y(t) = 1 − x(t) y(t) = −x(t)

# » Then ∧ is called the denotative conjunction of A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))). Please refer to Table 5.2 for the truth value table of conjunction operation on A and B. # » Definition 5.10 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t)= f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, if ¬A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), where ⎧ y(t) = 0, x(t) = 1, x(t) ∈ {0, 1}; ⎪ ⎪ ⎨ y(t) = 1, x(t) = 0, x(t) ∈ {0, 1}; = y(t) = 1 − x(t), x(t) ∈ [0, 1]; ⎪ ⎪ ⎩ y(t) = −x(t), x(t) ∈ (−∞, +∞); # » Then ¬ is called the negation operation on A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))). Please refer to Table 5.3 for the truth value table of negation operation on A. # » Definition 5.11 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t)= f (μ(t), G A (t))), B((ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C AnhbB ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative inclusion # » variable with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) =

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5 Mathematical Error Propositional Logic

# » f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), # » if A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) →nsy B((U (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), # » T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) # » = f (ν(t), G B (t))) ∨C AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t)))) ∧(¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ∨ C AnhbB ((U (t), SC (t), # » # » pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t)= f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = # » f (ν(t), G B (t))) ∨ C AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then →nsy is called the connotative isness implication of A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “if . . . , then it is . . . ;”. Please refer to Table 5.4 for the truth value table of connotative isness implication. # » Definition 5.12 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))= B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Anhbhd B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative inclusion # » or equivalence variable with the same domain for A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = # » f (ν(t), G B (t))), if A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » →nsy B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), # » L B (t)), y(t)= f (ν(t), G B (t))) ∨ C Anhbhd B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) # » = f (ω(t), G C (t)))) ∧ (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) # » ∨ ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ∨ C Anhbhd B ((U (t), # » # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), # » T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), # » y(t) = f (ν(t), G B (t))) ∨ C Anhbhd B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f nsy is called the connotative possibility implication of (ω(t), G C (t)))), then → # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “if . . . then it is possible”. Please refer to Table 5.5 for the truth value table of connotative possibility implication. # » Definition 5.13 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t)= f (μ(t), G A (t))), B(ν(t), y(t)) = B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Anhbhd j B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative inclusion # » or equivalence variable with the same domain for A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) =

5.4 Basic Operations

475

Table 5.4 Truth value table of connotative isness implication A

B

C AnhbB

Relationships among x(t), y(t), and z(t)

A →nsy B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≥ 0

C AnhbB

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≥ 0,−y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≥ 0,−y(t) ≤ z(t)

C AnhbB

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≤ 0,−x(t) ≤ y(t)

B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≤ 0,−x(t) ≥ y(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0

C AnhbB

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C AnhbB

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≥ 0

C AnhbB

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, y(t) ≥ −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≤ z(t)

C AnhbB

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≤ y(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≥ 0

C AnhbB

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C AnhbB

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≥ −x(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≤ −x(t)

B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t)

C AnhbB

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t)

C AnhbB

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C AnhbB

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5 Mathematical Error Propositional Logic

Table 5.5 Truth value table of connotative possibility implication A

B

C Anhbhd B

Relationships among x(t), y(t), and z(t)

A →nsy B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≥ 0

C Anhbhd B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, x(t) ≥ 0,−x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, x(t) ≥ 0,−x(t) ≤ z(t)

C Anhbhd B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, x(t) ≤ 0,−y(t) ≤ x(t)

A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, x(t) ≤ 0,−y(t) ≥ x(t)

¬B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0

C Anhbhd B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≤ z(t)

C Anhbhd B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≥ 0

C Anhbhd B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, x(t) ≥ −y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, −y(t) ≤ z(t)

C Anhbhd B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, −y(t) ≥ x(t)

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −y(t) ≥ x(t)

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −y(t) ≤ x(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≥ 0

C Anhbhd B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≤ z(t)

C Anhbhd B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≥ −y(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≤ −y(t)

A

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t)

C Anhbhd B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t)

C Anhbhd B

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C Anhbhd B

5.4 Basic Operations

477

# » f (ν(t), G B (t))), if A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » nsy → B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), # » Anhbhd j B L B (t)), y(t) = f (ν(t), G B (t))) ∨ C ((U (t), SC (t), pC (t), TC (t), L C (t)), # » z(t)= f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) ∨B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ∨ C Anhbhd j B # » ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), p B (t), TB (t), # » L B (t)), y(t)= f (ν(t), G B (t))) ∨ C Anhbhd j B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then →nsy is called the connotative necessity implication of # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “if . . . , then it must be or it could not be otherwise”. Please refer to Table 5.6 for the truth value table of connotative necessity implication. # » Definition 5.14 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Andt B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative same vari# » able with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))=nhdt B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t)))=(A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t))) ∨ C Andt B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ # » (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬B((U (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Andt B ((U (t), SC (t), pC (t), nhdt is called the connotative same of TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then = # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “properties are the same or two errors are the same”. Please refer to Table 5.7 for the truth value table of connotative same. # » Definition 5.15 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))= B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Anhdthd j B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative equivalence # » variable with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), # » if A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ←→nhdz B((U (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t)))=(A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) ∨B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f

478

5 Mathematical Error Propositional Logic

Table 5.6 Truth value table of connotative necessity implication A

B

C Anhbhd j B

Relationships among x(t), y(t), and z(t)

A →nsy B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≥ 0

C Anhbhd j B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≥ 0,−y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≥ 0,−y(t) ≤ z(t)

C Anhbhd j B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≤ 0,−x(t) ≤ y(t)

B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, y(t) ≤ 0,−x(t) ≥ y(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0

C Anhbhd j B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C Anhbhd j B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≥ 0

C Anhbhd j B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, y(t) ≥ −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≤ z(t)

C Anhbhd j B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≤ y(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≥ 0

C Anhbhd j B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C Anhbhd j B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≥ −x(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≤ −x(t)

B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t)

C Anhbhd j B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t)

C Anhbhd j B

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C Anhbhd j B

5.4 Basic Operations

479

Table 5.7 Truth value table of connotative same A

B

C Andt B

Relationships Among x(t), y(t), and z(t)

A =nhdt B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≥ 0

C Andt B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0,

A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −y(t) ≤ z(t)

C Andt B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, z(t) ≤ −y(t) ≤ x(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −y(t) ≥ x(t)

A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, z(t) ≤ −x(t) ≤ y(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≤ z(t)

C Andt B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≥ 0

C Andt B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≥ 0

C Andt B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≤ z(t)

C Andt B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, y(t) ≥ −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≥ y(t)

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −x(t) ≤ y(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≥ 0

C Andt B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, −y(t) ≤ z(t)

C Andt B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, x(t) ≥ −y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, −y(t) ≥ x(t)

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0,

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −y(t) ≥ x(t)

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −y(t) ≤ x(t)

¬B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t)

C Andt B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t)

C Andt B

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C Andt B

480

5 Mathematical Error Propositional Logic

# » (ν(t), G B (t))) ∨ C Anhdthd j B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬ # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Anhdthd j B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then ←→nhdz is called the # » connotative equivalence of A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “if...then it is equivalent”. Please refer to Table 5.8 for the truth value table of connotative equivalence.

Table 5.8 Truth value table of connotative equivalence A

B

C Andt B

Relationships Among x(t), y(t), and z(t)

A ←→nhdz B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≥ 0

C Anhdthd j B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0,

A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0,

C Anhdthd j B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0,

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0,

A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0,

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0,

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0,

C Anhdthd j B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≥ 0

C Anhdthd j B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≥ 0

C Anhdthd j B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0

C Anhdthd j B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0

¬A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0

¬A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≥ 0

C Anhdthd j B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0

C Anhdthd j B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0

¬B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0,

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0

¬B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t)

C Anhdthd j B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t)

C Anhdthd j B

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C Anhdthd j B

5.4 Basic Operations

481

# » Definition 5.16 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C AnhbB ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the connotative inclusion # » variable with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), # » if A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) −n B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t)))=(A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t))) ∨ C AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ # » (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ∨ C AnhbB ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then −n is called the connotative difference of # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “errors were removed or reduced”. Please refer to Table 5.9 for the truth value table of connotative difference. # » Definition 5.17 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable # » with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t),G B (t))), # » if A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨n B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), T A (t), # » x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= L A (t)), # » f (ν(t), G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t)))) # » ∧(A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t), pC (t), TC (t), # » L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ # » C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨¬B((U (t), S B (t), p B (t), # » TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), # » z(t) = f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), # » G A (t))) ∨¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Az B # » ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨B((U (t), S B (t), p B (t), TB (t), L B (t)), # » y(t) = f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t)))) ∧ (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨¬ # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t),

482

5 Mathematical Error Propositional Logic

Table 5.9 Truth value table of connotative difference A B C AnhbB Relationships Among x(t), y(t), and z(t) x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t) x(t) x(t) x(t) x(t)

y(t) y(t) y(t) y(t) y(t)

z(t) z(t) z(t) z(t) z(t)

x(t) ≥ −y(t) ≥ z(t), z(t) ≥ 0 x(t) ≥ −y(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, x(t) ≥ −y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ −y(t) x(t) ≥ −y(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ −y(t) x(t) ≥ z(t) ≥ −y(t), z(t) ≥ 0, x(t) ≥ z(t) ≥ −y(t), z(t) ≤ 0, x(t) ≤ 0 x(t) ≥ z(t) ≥ −y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t) x(t) ≥ z(t) ≥ −y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t) −y(t) ≥ x(t) ≥ z(t) −y(t) ≥ z(t) ≥ x(t) z(t) ≥ x(t) ≥ −y(t) z(t) ≥ −y(t) ≥ x(t) x(t) = y(t) = z(t)

A −n B ¬B A ¬B

¬A

C AnhbB A C AnhbB

¬A

¬B ¬B C AnhbB C AnhbB A or B or C AnhbB

# » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then ∨n is called the connotative # » disjunction on A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “different errors have mutual infiltration”. Please refer to Table 5.10 for the truth value table of connotative disjunction. # » Definition 5.18 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable # » with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∧n B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), T A (t),

5.4 Basic Operations

483

Table 5.10 Truth value table of connotative disjunction A

B

C Az B

Relationships Among x(t), y(t), and z(t)

A ∨n B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≥ 0

C Az B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≥ z(t),

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≥ −y(t)

B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≤ −y(t), x(t) ≥ 0

¬A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≤ −y(t), x(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≥ 0

B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, −y(t) ≤ z(t)

B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≥ 0, y(t) ≤ 0, −y(t) ≥ z(t)

C Az B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≥ 0

A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≤ z(t)

A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≥ 0, x(t) ≤ 0, −x(t) ≥ z(t)

C Az B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≥ 0

C Az B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≥ −x(t)

A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≤ −x(t), y(t) ≥ 0

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≤ −x(t), y(t) ≤ 0

B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≥ 0, y(t) ≤ −x(t),

¬A

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≥ 0, y(t) ≥ −x(t),

B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≤ 0, x(t) ≥ −z(t),

A

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≤ 0, x(t) ≤ −z(t),

¬C Az B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t), z(t) ≤ 0

C Az B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≥ 0, x(t) ≤ −y(t),

¬B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≥ 0, x(t) ≥ −y(t),

A

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≤ 0, y(t) ≥ −z(t),

B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≤ 0, y(t) ≤ −z(t),

¬C Az B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t), z(t) ≤ 0

C Az B

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C Az B

484

5 Mathematical Error Propositional Logic

# » L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= # » Az B f (ν(t), G B (t))) ∨ C ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) # » ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), # » TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t)= f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ # » (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬B((U (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then ∧n is called the connotative conjunc# » tion on A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “different errors are compatible”. Please refer to Table 5.11 for the truth value table of connotative conjunction. # » Definition 5.19 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable # » with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) nhb B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = # » f (ν(t), G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » nhb G C (t)))), then is called the connotative complement on A((U (t), S A (t), p A (t), # » T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “correctness, error with critical points, and error coexist”. Please refer to Table 5.12 for the truth value table of connotative complement. # » Definition 5.20 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable # » having same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) nhdl B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = # » f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t),

5.4 Basic Operations

485

Table 5.11 Truth value table of connotative conjunction A

B

C Az B

Relationships Among x(t), y(t), and z(t)

A ∧n B

x(t)

y(t)

z(t)

z(t) ≥ 0

C Az B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≥ z(t),

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≥ −y(t)

B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≤ −y(t), x(t) ≥ 0

¬A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t), z(t) ≤ 0, y(t) ≤ 0, x(t) ≤ −y(t), x(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

x(t) ≥ z(t) ≥ y(t), z(t) ≤ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≥ z(t)

¬B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≥ 0, −y(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

y(t) ≥ z(t) ≥ x(t), z(t) ≤ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≥ z(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≥ 0, −x(t) ≤ z(t)

C Az B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≥ −x(t)

A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≥ −x(t), y(t) ≥ 0

¬B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ z(t), z(t) ≤ 0, x(t) ≤ 0, y(t) ≤ −x(t), y(t) ≥ 0

B

x(t)

y(t)

z(t)

z(t) ≥ x(t) ≥ y(t), z(t) ≤ 0

C Az B

x(t)

y(t)

z(t)

z(t) ≥ y(t) ≥ x(t), z(t) ≤ 0

C Az B

x(t)

y(t)

z(t)

x(t) = y(t) = z(t)

A or B or C Az B

# » G C (t)))), then nhdl is called the connotative antithesis on A((U (t), S A (t), p A (t), # » T A (t), L A (t)), x(t) = f (μ(t), G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “correctness and error coexist”. Please refer to Table 5.13 for the truth value table of connotative antithesis. # » Definition 5.21 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined

486

5 Mathematical Error Propositional Logic

Table 5.12 Truth value table of connotative complement A B C Az B Relationships Among x(t), y(t), and z(t) x(t) x(t) x(t) x(t) x(t) x(t) x(t)

y(t) y(t) y(t) y(t) y(t) y(t) y(t)

z(t) z(t) z(t) z(t) z(t) z(t) z(t)

x(t) ≥ y(t) ≥ z(t) x(t) ≥ z(t) ≥ y(t) y(t) ≥ x(t) ≥ z(t) y(t) ≥ z(t) ≥ x(t) z(t) ≥ y(t) ≥ x(t) z(t) ≥ x(t) ≥ y(t) x(t) = y(t) = z(t)

Table 5.13 Truth value table of connotative antithesis A B C Az B Relationships Among x(t), y(t), and z(t) x(t) x(t) x(t) x(t) x(t) x(t) x(t)

y(t) y(t) y(t) y(t) y(t) y(t) y(t)

z(t) z(t) z(t) z(t) z(t) z(t) z(t)

x(t) ≥ y(t) ≥ −z(t) x(t) ≥ −z(t) ≥ y(t) y(t) ≥ x(t) ≥ −z(t) y(t) ≥ −z(t) ≥ x(t) −z(t) ≥ y(t) ≥ x(t) −z(t) ≥ x(t) ≥ y(t) x(t) = y(t) = −z(t)

A nhb B

A A B B C Az B C Az B A or B or C Az B

A nhb B

A A B B ¬C Az B ¬C Az B A or B or C Az B

in domain U (t) under judging rules G A (t) and G B (t), respectively, C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable # » with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) |n f l B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = # » f (ν(t), G B (t))) ∨ C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t)))) ∧ (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then |n f l is called the conno# » tative separation on A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means that “different errors coexist”. Please refer to Table 5.14 for the truth value table of connotative separation. # » Definition 5.22 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), B(ν(t), y(t))=B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)

5.4 Basic Operations

487

Table 5.14 Truth value table of connotative separation A B C Az B Relationships Among x(t), y(t), and z(t) x(t) x(t) x(t) x(t) x(t)

y(t) y(t) y(t) y(t) y(t)

z(t) z(t) z(t) z(t) z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t) x(t)

y(t) y(t)

z(t) z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t) x(t)

y(t) y(t)

z(t) z(t)

x(t) ≥ y(t) ≥ z(t) x(t) ≥ z(t) ≥ y(t) y(t) ≥ x(t) ≥ z(t) y(t) ≥ z(t) ≥ x(t) z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≥ 0 z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≤ 0, x(t) ≥ −z(t) z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0, x(t) ≤ 0, x(t) ≤ −z(t) z(t) ≥ x(t) ≥ y(t), z(t) ≤ 0 z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≥ 0 z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≤ 0, y(t) ≥ −z(t) z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0, y(t) ≤ 0, y(t) ≤ −z(t) z(t) ≥ y(t) ≥ x(t), z(t) ≤ 0 x(t) = y(t) = z(t)

A |n f l B A A B B A A ¬C Az B C Az B B B ¬C Az B C Az B A or B or C Az B

= f (ν(t), G B (t))), x(t), y(t) ∈ (−∞, +∞), are the error logical variables defined in domain U (t) under judging rules G A (t) and G B (t), respectively, C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable # » with the same domain for A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), if # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) |n f h B((U (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) = (A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = # » f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ # » B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t), # » # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t)= # » f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), # » G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ # » ¬B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ ¬C Az B ((U (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))), then |n f h is called the connota# » tive differentiation on A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » and B((U (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))), which means

488

5 Mathematical Error Propositional Logic

Table 5.15 Truth value table of connotative differentiation A

B

C Az B

Relationships Among x(t), y(t), and z(t)

A |n f h B

x(t)

y(t)

z(t)

z(t) ≤ 0

¬C Az B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ −z(t), z(t) ≥ 0, y(t) ≥ 0, y(t) ≤ z(t)

¬B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ −z(t), z(t) ≥ 0, y(t) ≥ 0, y(t) ≥ z(t)

¬C Az B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ −z(t), z(t) ≥ 0, y(t) ≤ 0, x(t) ≥ −y(t)

B

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ −z(t), z(t) ≥ 0, y(t) ≤ 0, x(t) ≤ −y(t)x(t) ≥ 0

¬A

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ −z(t), z(t) ≥ 0, y(t) ≤ 0, x(t) ≤ −y(t)x(t) ≤ 0

A

x(t)

y(t)

z(t)

x(t) ≥ −z(t) ≥ y(t), z(t) ≥ 0, x(t) ≥ 0, x(t) ≤ z(t)

¬A

x(t)

y(t)

z(t)

x(t) ≥ −z(t) ≥ y(t), z(t) ≥ 0, x(t) ≥ 0, x(t) ≥ z(t)

¬C Az B

x(t)

y(t)

z(t)

x(t) ≥ −z(t) ≥ y(t), z(t) ≥ 0, x(t) ≤ 0

A

x(t)

y(t)

z(t)

y(t) ≥ −z(t) ≥ x(t), z(t) ≥ 0, y(t) ≥ 0, z(t) ≥ y(t)

¬B

x(t)

y(t)

z(t)

y(t) ≥ −z(t) ≥ x(t), z(t) ≥ 0, y(t) ≥ 0, z(t) ≤ y(t)

¬C Az B

x(t)

y(t)

z(t)

y(t) ≥ −z(t) ≥ x(t), z(t) ≥ 0, y(t) ≤ 0

B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ −z(t), z(t) ≥ 0, x(t) ≥ 0, z(t) ≥ x(t)

¬A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ −z(t), z(t) ≥ 0, x(t) ≥ 0, z(t) ≤ x(t)

¬C Az B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ −z(t), z(t) ≥ 0, x(t) ≤ 0, y(t) ≥ −x(t)

A

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ −z(t), z(t) ≥ 0, x(t) ≤ 0, y(t) ≤ −x(t), y(t) ≥ 0

¬B

x(t)

y(t)

z(t)

y(t) ≥ x(t) ≥ −z(t), z(t) ≥ 0, x(t) ≤ 0, y(t) ≤ −x(t), y(t) ≤ 0

B

x(t)

y(t)

z(t)

−z(t) ≥ x(t) ≥ y(t), z(t) ≥ 0

¬C Az B

x(t)

y(t)

z(t)

−z(t) ≥ y(t) ≥ x(t), z(t) ≥ 0

¬C Az B

x(t)

y(t)

z(t)

x(t) = y(t) = −z(t)

A or B or ¬C Az B

that “errors with critical points do not exist”. Please refer to Table 5.15 for the truth value table of connotative differentiation. # » Definition 5.23 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), x(t) ∈ (−∞, +∞), is the error logical variable defined in # » domain U (t) under judging rules G A (t), A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)),

5.4 Basic Operations

489

Table 5.16 Truth value table of connotative unconstrained negation (n) z A(n+1)

A(n)

A(n+1)

CA

Relationships x (n) , x (n+1) , z(t)

¬bx A(n)

x (n) (t)

x (n+1) (t)

z(t)

x (n) (t) ≥ x (n+1) (t) ≥ z(t) x (n) (t) ≥ z(t) ≥ x (n+1) (t) x (n+1) (t) ≥ x (n) (t) ≥ z(t) x (n+1) (t) ≥ z(t) ≥ x (n) (t)

A(n)

x (n) (t)

x (n+1) (t)

z(t)

x (n) (t)

x (n+1) (t)

z(t)

x (n) (t)

x (n+1) (t)

z(t)

x (n) (t)

x (n+1) (t)

z(t)

z(t) ≥ x (n+1) (t) ≥ x (n) (t)

CA

x (n) (t)

x (n+1) (t)

z(t)

CA

x (n+1) (t)

z(t)

z(t) ≥ x (n) (t) ≥ x (n+1) (t) x (n) (t) = x (n+1) (t) = z(t)

x (n) (t)

A(n) A(n+1) A(n+1) (n) z A(n+1)

(n) z A(n+1)

A(n) (t) or A(n+1) (t) or (n) (n+1) C A zA

# » x(t) = f (μ(t), G A (t))) and A(n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) are respectively the nth and (n + 1)th layer error logical vari# » (n) (n+1) ables of A(μ(t), x(t)), C A z A ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t))) is the intermediary variable with the same domain for A(n) ((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) and A(n+1) ((U (t), S A (t), p A (t), T A (t), # » bx (n) L A (t)), x(t) = f (μ(t), G A (t))), if ¬ A ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) # » = f (μ(t), G A (t))) = A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » (n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ G A (t))) ∨ A # » (n) (n+1) C A z A ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), then ¬bx is # » called the connotative unconstrained negation on A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), which means that “for the error being negated, there exists its opposite side before being decomposed”. Please refer to Table 5.16 for the truth value table of connotative unconstrained negation. # » Definition 5.24 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), x(t) ∈ (−∞, +∞), is an error logical variable defined in # » domain U (t) under judging rules G A (t), A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) and A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) are respectively the nth and (n − 1)r mth layer error logical vari# » (n) (n−1) ables of A(μ(t), x(t)), C A z A ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the intermediary variable having same domain for A(n) ((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) and A(n−1) ((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t)= f (μ(t), G A (t))), if ¬bd A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) # » = f (μ(t), G A (t)))=A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))

490

5 Mathematical Error Propositional Logic

Table 5.17 Truth value table of connotative uninterrupted negation (n) z A(n−1)

A(n)

A(n−1)

CA

Relationship x (n) , x (n−1) , z(t)

¬bd A(n)

x (n) (t)

x (n−1) (t)

z(t)

x (n) (t) ≥ x (n−1) (t) ≥ z(t) x (n) (t) ≥ z(t) ≥ x (n−1) (t) x (n−1) (t) ≥ x (n) (t) ≥ z(t) x (n−1) (t) ≥ z(t) ≥ x (n) (t)

A(n)

x (n) (t)

x (n−1) (t)

z(t)

x (n) (t)

x (n−1) (t)

z(t)

x (n) (t)

x (n−1) (t)

z(t)

x (n) (t)

x (n−1) (t)

z(t)

z(t) ≥ x (n−1) (t) ≥ x (n) (t)

CA

x (n) (t)

x (n−1) (t)

z(t)

CA

x (n−1) (t)

z(t)

z(t) ≥ x (n) (t) ≥ x (n−1) (t) x (n) (t) = x (n−1) (t) = z(t)

x (n) (t)

A(n) A(n−1) A(n−1) (n) z A(n−1)

(n) z A(n−1)

x (n) (t) or x (n−1) (t) or (n) (n−1) C A zA

# » (n) (n−1) ∨ A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ C A z A # » ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), then ¬bd is called the # » connotative uninterrupted negation on A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), which means that “for the error being negated, there exists its opposite side after being decomposed”. Please refer to Table 5.17 for the truth value table of connotative uninterrupted negation. # » Definition 5.25 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), x(t) ∈ (−∞, +∞), is an error logical variable defined in # » domain U (t) under judging rules G A (t), A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) and A(n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) are respectively the nth and (n + 1)th layer error logical vari# » ables of A(μ(t), x(t)), ¬A(n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) is the (n + 1)th layer error logical complement variable of A(μ(t), x(t)), # » C (n+1)Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the inter# » mediary variable with the same domain for A(n+1) ((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))) and ¬A(n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))), if ¬bz A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t)))=A(n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) # » ∨ ¬A(n+1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ C (n+1)Az B # » ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), then ¬bz is called the # » connotative “not only” negation on A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), which means that “there exists error that can be negated before

5.4 Basic Operations

491

Table 5.18 Truth value table of connotative “not only” negation A(n+1) ¬ C (n+1)Az B Relationship among x(t), y(t), z(t) x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t)

y(t)

z(t)

x(t) ≥ y(t) ≥ z(t) x(t) ≥ z(t) ≥ y(t) y(t) ≥ x(t) ≥ z(t) y(t) ≥ z(t) ≥ x(t) z(t) ≥ y(t) ≥ x(t) z(t) ≥ x(t) ≥ y(t) x(t) = y(t) = z(t)

¬bz A(n) A(n+1) A(n+1) ¬A(n+1) ¬A(n+1) C (n+1)Az B C (n+1)Az B A(n+1) or ¬A(n+1) or C (n+1)Az B

being decomposed”. Please refer to Table 5.18 for the truth value table of connotative “not only” negation. # » Definition 5.26 Suppose that A(μ(t), x(t))= A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), x(t) ∈ (−∞, +∞), is an error logical variable defined in # » domain U (t) under judging rules G A (t), A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) and A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) are respectively the nth and (n − 1)th layer error logical vari# » ables of A(μ(t), x(t)), ¬A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) is the (n − 1)th layer error logical complement variable of A(μ(t), x(t)), # » C (n−1)Az B ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) is the inter# » mediary variable with the same domain for A(n−1) ((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t)= f (μ(t), G A (t))) and ¬A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) # » = f (μ(t), G A (t))), if ¬bj A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) = A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬ # » A(n−1) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ C (n−1)Az B # » ((U (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), then ¬bj is called the # » connotative unfinished negation on A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))), which means that “there exists error that can be negated after being decomposed”. Please refer to Table 5.19 for the truth value table of connotative unfinished negation.

492

5 Mathematical Error Propositional Logic

Table 5.19 Truth value table of connotative unfinished negation A(n−1) ¬A(n−1) C (n+1)Az B Relationship among x(t), y(t), z(t) x(t) x(t) x(t) x(t) x(t) x(t) x(t)

y(t) y(t) y(t) y(t) y(t) y(t) y(t)

z(t) z(t) z(t) z(t) z(t) z(t) z(t)

x(t) ≥ y(t) ≥ z(t) x(t) ≥ z(t) ≥ y(t) y(t) ≥ x(t) ≥ z(t) y(t) ≥ z(t) ≥ x(t) z(t) ≥ y(t) ≥ x(t) z(t) ≥ x(t) ≥ y(t) x(t) = y(t) = z(t)

¬bj A(n) A(n−1) A(n−1) ¬A(n−1) ¬A(n−1) C (n−1)Az B C (n−1)Az B A(n−1) or ¬A(n−1) or C (n−1)Az B

5.5 Compound Proposition 5.5.1 Atomic Proposition Definition 5.27 Error logical atomic propositions are generally composed of the error logical propositional constants and propositional variables that only include subject and predicate. The error logical propositional constant is expressed # » by A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 , S0 (t0 ), p0 (t), T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f (μ0 (t0 ), # » G 0 (t0 ))), where U0 , S0 (t0 ), p0 (t), T0 (t0 ), L 0 (t0 ), and x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )) are all constants. The error logical propositional variable is expressed by A((U (t), S A (t), # » p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) in which there is at least one variable. # » The error logical propositional array is {A1 ((U1 (t), S1 (t1 ), p1 (t), T1 (t1 ), L 1 (t1 )), # » x1 (t1 ) = f 1 (μ1 (t1 ), G 1 (t1 ))), A2 ((U2 (t), S2 (t2 ), p2 (t), T2 (t2 ), L 2 (t2 )), x2 (t2 ) = f 2 # » (μ2 (t2 ), G 2 (t2 ))), . . . , Ak ((Uk (t), Sk (tk ), pk (t), Tk (tk ), L k (tk )), xk (tk ) = f k (μk (tk ), G k (tk )))}.

5.5.2 Error Logical Compound Proposition Definition 5.28 Error logical atomic proposition equations have the following structure: 1. Atomic propositions are formulas; # » 2. If A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t) = f (μ(t) A , G A (t))), and B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t) B , G B (t))) are formulas, then: # » (1) ∃LY (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))));

5.5 Compound Proposition

(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)

493

# » ∀LY (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃SW (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀SW (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃S J (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀S J (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃K J (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀K J (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃T J (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀T J (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃T Z (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀T Z (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃L Z (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀L Z (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃H S(A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀H S(A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃G Z (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀G Z (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∃F S(A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ∀F S(A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » Θ(A((U (t), S(t), p(t), T (t), L(t)),x(t) = f (μ(t), G(t)))); # » Ψ (A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » ¬(A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∧ B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∨ B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) ∨bxr B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) → B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); (A((U (t), S A (t), #p»A , T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ← B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) ←→ # » B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) = B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » Tx ⊆ {Txl , Txd ,Txc ,Txh ,Txs ,Txg }(A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); −1 −1 −1 −1 , Txc , Txh , Txs−1 ,Txg }(A((μ(t), x(t)) = A((U (t), S(t), Tx−1 ⊆ {Txl−1 , Txd # » p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » T f ⊆ {T f l , T f d ,T f c ,T f h ,T f s ,T f g }(A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); −1 −1 −1 −1 −1 T f−1 ⊆ {T f−1 l , T f d , T f c , T f h , T f s , T f g }(A((μ(t), x(t)) = A((U (t), S(t), # » p(t), T (t), L(t)), x(t) = f (μ(t), G(t))));

494

5 Mathematical Error Propositional Logic

# » (35) Tzn ⊆ {Tznl , Tznd ,Tznc ,Tznh ,Tzns ,Tzng }(A((μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); −1 −1 −1 −1 −1 −1 , Tznd , Tznc , Tznh , Tzns , Tzng }(A((μ(t), x(t)) = A((U (t), S(t), (36) Tzn−1 ⊆ {Tznl # » p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » (37) Tz ⊆ {Tzl , Tzd ,Tzc ,Tzh ,Tzs ,Tzg }(A((μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); (38) Tz−1 ⊆ {Tzl−1 , Tzd−1 , Tzc−1 , Tzh−1 , Tzs−1 , Tzg−1 }(A((μ(t), x(t)) = A((U (t), S(t), # » p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » (39) Th ⊆ {Thl , Thd ,Thc ,Thh ,Ths ,Thg }(A((μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); −1 −1 −1 −1 , Thc , Thh , Ths−1 , Thg }(A((μ(t), x(t)) = A((U (t), S(t), (40) Th−1 ⊆ {Thl−1 , Thd # » p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))); # » (41) Td (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t)))); # » (42) Td−1 (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t)))); # » (43) ¬bz (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t)))); # » (44) ¬bj (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t)))); # » bx (45) ¬ (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t)))); # » (46) ¬bd (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), x(t)= f (μ(t), G(t)))); # » (47) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∧n B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (48) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∨n B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (49) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) −n B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (50) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) |n f l B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (51) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) |n f h B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (52) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) nhb B((V (t), → S B (t), p B , TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (53) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) nhdl B((V (t), S B (t), #p» B , TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (54) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) →nhy B((V (t), S B (t), #p » B , TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (55) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) →nby B((V (t), S B (t), #p » B , TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (56) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) →nsy B((V (t), S B (t), #p » B , TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » (57) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) ←→nhd B((V (t), S B (t), #p B», TB (t), L B (t)), y(t) = f (ν(t), G B (t)));

# » (58) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))))=nhd B((V (t), # » S B (t), p B , TB (t), L B (t)), y(t) = f (ν(t), G B (t))) are also formulas;

5.5 Compound Proposition

495

(59) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∧ {B1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Bk ((μk (t), xk (t)) = f k ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} = {C1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), C2 ((μ2 (t), x2 (t)) = f 2 ((U2 , S2 (t), #p», T (t), L (t)), x (t)= f (μ (t), G (t))), . . . , C ((μ (t), x (t)) = f ((U , 2 2 2 2 2 2 2 k k k k k Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))}; (60) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∨{B1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Bk ((μk (t), xk (t)) = f k ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} = {C1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), C2 ((μ2 (t), x2 (t)) = f 2 ((U2 , S2 (t), #p», T (t), L (t)), x (t)= f (μ (t), G (t))), . . . , C ((μ (t), x (t)) = f ((U , 2 2 2 2 2 2 2 k k k k k Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))}; (61) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∧ {B1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bm ((μm (t), xm (t))= f m ((Um , Sm (t), #pm», Tm (t), L m (t)), xm (t) = f m (μm (t), G m (t)))}; (62) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∧ &{B1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bm ((μm (t), xm (t))= f m ((Um , Sm (t), #pm», Tm (t), L m (t)), xm (t) = f m (μm (t), G m (t)))}; (63) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∧ &&{B1 ((μ1 (t), x1 (t))= f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bm ((μm (t), xm (t))= f m ((Um , Sm (t), #pm», Tm (t), L m (t)), xm (t) = f m (μm (t), G m (t)))}; (64) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∨ {B1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)),

496

5 Mathematical Error Propositional Logic

x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bm ((μm (t), xm (t))= f m ((Um , Sm (t), #pm», Tm (t), L m (t)), xm (t) = f m (μm (t), G m (t)))}; (65) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∨ &{B1 ((μ1 (t), x1 (t)) = f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bm ((μm (t), xm (t))= f m ((Um , Sm (t), #pm», Tm (t), L m (t)), xm (t) = f m (μm (t), G m (t)))}; (66) {A1 ((μ1 (t), x1 (t)) = A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((μ2 (t), x2 (t))=A2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ak ((μk (t), xk (t)) = Ak ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))} ∨ &&{B1 ((μ1 (t), x1 (t))= f 1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((μ2 (t), x2 (t))= f 2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bm ((μm (t), xm (t))= f m ((Um , Sm (t), #pm», Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))}. 3. The results obtained through limited use of formulas in “1 and “2 are formulas; 4. The results obtained through using legitimate error logic symbols are also formulas. The error logical formula is expressed by f i (A1 , A2 , . . . , An ) “operation symbol” gi (B1 , B2 , . . . , Bm ) “relationship/operation symbol” h(C1 , C2 , . . . , Ck ).

5.6 Basic Rules for Error Logical Reasoning 5.6.1 Axiom Set The axiom set here is the set composed of the axioms and permanently true formulas obtained through exerting correct logical reasoning on proven premises. Apparently, this set is open because newly obtained tautological formulas can be continually added to this set.

5.6.2 Reasoning Rules In the process of logical reasoning, starting from the true premises, correct logical reasoning mechanisms are used to obtain the true conclusion, where correct logical reasoning is the necessary condition for obtaining true conclusion(s). Correct logical reasoning forms are the reasoning rules. This section focuses on the discussion of reasoning rules.

5.6 Basic Rules for Error Logical Reasoning

497

# » 1. Separation rules (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) # » ∧ (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) → B((V (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))))) → B((V (t), S B (t), p B (t), # » TB (t), L B (t)), y(t) = f (ν(t), G B (t))), then (A((U (t), S A (t), p A (t), T A (t), # » L A (t)), x(t) = f (μ(t), G A (t))), A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t))) → B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t)))) ⇒ B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))); # » 2. Elimination disjunction rules ((A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t))) ∨ B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t)))) ∧ (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) → # » C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t)))) ∧ (B((V (t), # » # » S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) → C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t))))) → C((W (t), SC (t), #pC», TC (t), L C (t)), # » z(t) = f (ω(t), G C (t))), then (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = # » f (μ(t), G A (t))) ∨ B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t))), A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) → # » C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t))), B((V (t), S B (t), # » # » p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) → C((W (t), SC (t), pC (t), TC (t), # » L C (t)), z(t)= f (ω(t), G C (t)))) ⇒ C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))); 3. Premises can be introduced at any time, T rule (condition rules); 4. If formulas f (A1 , A2 , . . . , An ) can be effectively derived from the combination of some formulas, f (A1 , A2 , . . . , An ), J rule is applied (conclusion rule); # » 5. Displacement rule A1 ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), # » G A (t))) = B1 ((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))), # » B1 ((V (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) is used to replace # » A1 ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) in f 1 (A1 , A2 , . . . , An ) to obtain f 2 (B1 , A2 , A3 , . . . , An ), then f 1 (A1 , A2 , . . . , An ) = f 2 (B1 , A2 , A3 , . . . , An ); 6. Morgan rule; # » (1) ¬(A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t)))) = (¬A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∧ ¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))); # » (2) ¬(A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∧ B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t)))) = (¬A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ ¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))); 7. Denotative negation rule # » ¬(¬(A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))))=(A((U (t), # » S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))); 8. Denotative conjunction rule # » If A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ⇒ C((W (t), # » # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))) and B((V (t), S B (t), p B (t),

498

5 Mathematical Error Propositional Logic

# » TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ⇒ C((W (t), SC (t), pC (t), TC (t), # » L C (t)), z(t) = f (ω(t), G C (t))), then (A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t) = f (μ(t), G A (t))) ∧ B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), # » G B (t)))) ⇒ C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))); 9. Denotative disjunction rule # » If (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))) ⇒ C((W (t), SC (t), # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))); # » (1) If ¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ⇒ # » C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), then # » B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ⇒ C((W (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))); # » (2) If ¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ⇒ # » C((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), then A((U (t), S A (t), #p»A , T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ⇒ C((W (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))); 10. Connotative necessity implication rule # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) →nby C((W (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), this is demonstrated in the truth table (Table 5.6) of connotative necessity implication; 11. Connotative possibility implication rule # » A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t)))) →nhy C((W (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t))), this is demonstrated in the truth table (Table 5.5) of connotative possibility implication; 12. Decomposition rule; 13. Combination rule; 14. Similarity rule; 15. Increase rule; 16. Reduction rule; 17. Displacement rule; 18. Destruction rule; 19. Production rule; Items 12 through 19 all abide by the following 4 rules: (1) (2) (3) (4)

Actual needs; Feasible conditions; Costs; Logical values of error are changing along with the objectives.

5.7 Forms of Error Logical Proposition

499

5.7 Forms of Error Logical Proposition 1. Simple conjunction and disjunction on error logical propositions # » (1) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∧ (B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))) ∧ (C((W (t), SC (t), # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))); # » (2) (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∨ (B((V (t), # » S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t)))) ∨ (C((W (t), SC (t), # » pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))); # » (3) (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))))) ∧ # » (¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))))∧¬(C((W (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))); # » (4) (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))))) ∨ # » (¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), G B (t))))∨¬(C((W (t), # » SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))). 2. Simple conjunction and disjunction on error logical propositional arrays 

(1) ({A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  xk (t) = f k (μk (t), G k (t)))}) ∧ ({B1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), L k (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ∧ ({C1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), C2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (2) ({A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  L k (t)), xk (t) = f k (μk (t), G k (t)))}) ∨ ({B1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), ,  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ∨ ({C1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), C2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (3) (¬{A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  L k (t)), xk (t) = f k (μk (t), G k (t)))}) ∧ (¬{B1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ∧ (¬{C1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), C2 ((U2 , S2 (t),

500

5 Mathematical Error Propositional Logic 

(4)

(5)

(6)

(7)

(8)



p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (¬{A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  L k (t)), xk (t) = f k (μk (t), G k (t)))}) ∨ (¬{B1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t)= f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ∨ (¬{C1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), C2 ((U2 ,   S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  ({A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ˙ L k (t)), xk (t) = f k (μk (t), G k (t)))})∧({B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t)  = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))),  ˙ . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})∧({C 1   ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), C2 ((U2 , S2 (t), p2 ,  T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  ({A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ˙ L k (t)), xk (t) = f k (μk (t), G k (t)))})∨({B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t)  = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))),  ˙ . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t) = f m (μm (t), G m (t)))})∨({C 1   ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), C2 ((U2 , S2 (t), p2 ,  T2 (t), L 2 (t)), x2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (¬{A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ˙ L k (t)), xk (t) = f k (μk (t), G k (t)))})∧(¬{B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ˙ ∧(¬{C 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t) = f 1 (μ1 (t), G 1 (t))), C 2 ((U2 ,   S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (¬{A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ˙ L k (t)), xk (t) = f k (μk (t), G k (t)))})∨(¬{B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),

5.7 Forms of Error Logical Proposition

501 

(9)

(10)

(11)

(12)

G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ˙ ∨(¬{C 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t) = f 1 (μ1 (t), G 1 (t))), C 2 ((U2 ,   S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  ({A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ¨ L k (t)), xk (t) = f k (μk (t), G k (t)))})∧({B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ¨ ∧({C 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t) = f 1 (μ1 (t), G 1 (t))), C 2 ((U2 ,   S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  ({A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ¨ p1 , T1 (t), L 1 (t)), L k (t)), xk (t) = f k (μk (t), G k (t)))})∨({B 1 ((U1 , S1 (t),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ¨ ∨({C 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t) = f 1 (μ1 (t), G 1 (t))), C 2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (¬{A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ¨ L k (t)), xk (t) = f k (μk (t), G k (t)))})∧(¬{B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ¨ ∧(¬{C 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t)= f 1 (μ1 (t), G 1 (t))), C 2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))});  (¬{A1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t)= f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t),  ¨ L k (t)), xk (t) = f k (μk (t), G k (t)))})∨(¬{B 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)),  x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t),  G 2 (t))), . . . , Bm ((Um , Sm (t), pm , Tm (t), L m (t)), xm (t)= f m (μm (t), G m (t)))})  ¨ ∨(¬{C 1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x 1 (t)= f 1 (μ1 (t), G 1 (t))), C 2 ((U2 , S2 (t),   p2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ck ((Uk , Sk (t), pk , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))}).

Definition 5.29 Simple conjunction or disjunction on limited number of error logical propositions and error logical propositional arrays is called the conjunction or disjunction form for error logical propositions.

502

5 Mathematical Error Propositional Logic

Theorem 5.1 For any error logical propositional formula, there always exists its equivalent error logical propositional conjunction or disjunction form. Proof For any error logical propositional formula f i (A1 , A2 , . . . . . . , An ), it can be changed to error logical propositional conjunction or disjunction form through: (1) Using transformation connectives; (2) Using the basic equations of connotative and denotative connectives for error logical propositional formulas, error logical propositional expressions with connotative connectives are transformed into error logical propositional expressions with denotative connectives; the elementary equivalence formulas in error logical propositional formulas are used to remove those connectives such as ∨bxr , →, ←, and ↔; (3) Employing the De Morgan’s laws of logical reasoning and rules of denotative double negation, the denotative negation in front of the non-error logical proposition being changed is removed and the denotative double negation in error logical proposition being transformed is removed; (4) In the error logical proposition being changed, f i (A1 , A2 , . . . , An ) is changed to conjunction or disjunction form of error logical proposition by using the dis˙ ∧, ¨ ∧, ∨, ∨, ˙ and ∨ ¨ Proof is finished! tributive laws to reasoning of connectives ∧, # » Example 1: ((A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)))) →nby # » (B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))))) ∨ Th (¬C((W (t), # » S(t), p(t), T (t), L(t)), z(t) = f (ω(t), G(t)))). Solving example 1

# » (1) Using transformation connective: ((A((U (t), S(t), p(t), T (t), L(t)), x(t) = # » f (μ(t), G(t)))) →nby (B((U (t), S(t), p(t), T (t), L(t)), y(t)= f (ν(t), G(t))))) (2) Using the basic equations of connotative and denotative connectives for error logical propositional formulas, error logical propositional expressions with connotative connectives are transformed into error logical propositional expressions with denotative connectives; the elementary equivalence formulas in error logical propositional formulas are used to remove those connectives such as ∨bxr , # » →, ←, and ↔: ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ # » B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C Anhbhd j B # » ((W (t), SC (t), pC (t), TC (t), L C (t)), z(t) = f (ω(t), G C (t)))) ∧ (¬A((U (t), # » # » S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ B((V (t), S B (t), p B (t), # » TB (t), L B (t)), y(t)= f (ν(t), G B (t))) ∨ C Anhbhd j B ((W (t), SC (t), pC (t), TC (t), # » L C (t)), z(t)= f (ω(t), G C (t)))) ∧ (¬A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) # » = f (μ(t), G A (t))) ∨ ¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), # » G B (t))) ∨ C Anhbhd j B ((W (t), SC (t), pC (t), TC (t), L C (t)), z(t)= f (ω(t), G C (t)))). # » Example 2: ((A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∧ # » bx (n) (¬ (A ((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))))) ∨ T f d (C((W (t), # » S(t), p(t), T (t), L(t)), z(t) = f (ω(t), G(t)))).

5.7 Forms of Error Logical Proposition

503

Solving example 2

# » (1) Using transformation connective: ((A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) # » = f (μ(t), G A (t)))) ∧ (¬bx (A(n) ((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))))) ∨ {C1 ((W1 (t), S1 (t), #p»1 , T1 (t), L 1 (t)), z 1 (t) = f 1 (ω1 (t), G 1 (t))), C2 ((W2 (t), S2 (t), #p»2 , T2 (t), L 2 (t)), z 2 (t) = f 2 (ω2 (t), G 2 (t))), . . . , Ck ((Wk (t), Sk (t), #p»k , Tk (t), L k (t)), z k (t) = f k (ωk (t), G k (t)))} (2) Using the basic equations of connotative and denotative connectives for error logical propositional formulas, error logical propositional expressions with connotative connectives are transformed into error logical propositional expressions with denotative connectives; the elementary equivalence formulas in error logical propositional formulas are used to remove those connectives such as ∨bxr , # » →, ←, and ↔: ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t)))) ∧ # » (A(n) ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ A(n+1) # » ((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) ∨ C A(n)z A(n+1) # » ((W (t), S A (t), p A (t), T A (t), L A (t)), z(t) = f (ω(t), G A (t))))) ∨ {C1 ((W1 (t), # » S1 (t), p1 , T1 (t), L 1 (t)), z 1 (t) = f 1 (ω1 (t), G 1 (t))), C2 ((W2 (t), S2 (t), #p»2 , T2 (t), L 2 (t)), z 2 (t)= f 2 (ω2 (t), G 2 (t))), . . . , Ck ((Wk (t), Sk (t), #p»k , Tk (t), L k (t)), z k (t) = f k (ωk (t), G k (t)))}. # » Example 3: (∃S J (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » TB (t), L B (t)), y(t) = f (ν(t), G B (t))))) ∨ Tzl −n B((V (t), S B (t), p B (t), # » (C((W p rime(t), S(t), p(t), T (t), L(t)), z(t) = f (ω p rime(t), G(t)))). Solving example 3

# » (1) Using transformation connective: (∃S J (A((U (t), S A (t), p A (t), T A (t), L A (t)), # » x(t)= f (μ(t), G A (t))) −n B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t)= f (ν(t), # » G B (t))))) ∨ C((W  (t), S(t), p(t), T (t), L(t)), z(t) = f (ω (t), G(t)))). (2) Using the basic equations of connotation and extension connectives for error logical propositional formulas, error logical propositional expressions with connotation connectives are transformed into error logical propositional expressions with extension connectives; the elementary equivalence formulas in error logical propositional formulas are used to remove those connectives such as ∨bxr , →, # » ←, and ↔: S J (A((U (t), S A (t), p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) # » ∨ ¬B((V (t), S B (t), p B (t), TB (t), L B (t)), y(t) = f (ν(t), G B (t))) ∨ C AnhbB # » ((W (t), S(t), p(t), T (t), L(t)), z(t) = f (ω(t), G(t)))) ∧ (¬A((U (t), S A (t), # » # » p A (t), T A (t), L A (t)), x(t)= f (μ(t), G A (t))) ∨ ¬B((V (t), S B (t), p B (t), TB (t), # » L B (t)), y(t) = f (ν(t), G B (t))) ∨ C AnhbB ((W (t), S(t), p(t), T (t), L(t)), z(t) # » = f (ω(t), G(t))))) ∨ C((W  (t), S(t), p(t), T (t), L(t)), z(t)= f (ω (t), G(t))).

504

5 Mathematical Error Propositional Logic

5.8 Error Predicate Logic Mathematical error predicate logic is the subject that uses mathematics and semantics to examine the forms and laws for error proposition reasoning, which contain value of the property or attribute, domain, thing, space, property, logical value, function, time, error value, and rules for judging error. It is a kind of thinking that employs object and judging rules to evaluate and justify the erroneous structure of certain phenomenon and/or problem of interests. For the mathematical error predicate logic, we mainly discuss its concept, parameters, subject, value of the property or attributes, the semantic structure and explanation of predicate, connectives, truth table, and logical forms and its effective reasoning forms and judging methods.

5.8.1 Forms of Error Predicate Logic 5.8.1.1

Concept of Error Predicate Logic

# » Definition 5.30 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), x(t) ∈ (−∞, +∞), where U (t) is the domain or universe # » of discourse for μ(t) = (U (t), S(t), p(t), T (t), L(t)), S(t) is the thing or subject, # » p(t) is the spatial location and direction, T (t) is the property or predicate, L(t) is the predicate, x(t) = f (μ(t), G A (t)) is the truth value or truth value function of # » μ(t) = (U (t), S(t), p(t), G(t) is the rule for judging error defined in domain U (t); domain is called the universe of discourse, thing is called the subject; property, value # » of the property or attribute, and rules are called predicate, A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))) is called the error logic predicate variable defined in U (t) under the rule of judging errors G(t). # » Definition 5.31 Suppose that A(μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), x(t) ∈ (−∞, +∞), is the error logic predicate variable defined in U (t) (domain) under the rule of judging errors G(t). The set composed of all error logic predicate variables defined in U (t) under the rule of judging errors G(t) is called variable set of error logic predicate, which is noted by C.

5.8.1.2

Subject, Predicate, and Quantifier of Error Logic

Subject, predicate, and quantifier in error logical proposition are three principal elements in forming error logical simple propositions, which each has different types and logical characteristics. The subject in error logical proposition is the thing that is in the objects being judged, which could be single element or set, unitary or pluralistic. Taking postgraduates in 2003 at X X X University as the object of interests, in A(μ(t), x(t)) = # » A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), U (t) = {postgraduates in

5.8 Error Predicate Logic

505

2003 at X X X University }, S(t) = postgraduates A in College of Business at X X X # » University in 2003, p(t) = College of Business at X X X University, Guangzhou, China, T (t) = { health status (in2004), GPA (in2004)}, L(t) = { healthy (in2004), 3.8 (in2004)}, f ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i }, G(t) = rules for evaluating outstanding postgraduates (in 2004). For postgraduate A in 2003 College of Business at X X X University, error logical propositional variable becomes an error logical proposition if an array of explanations is given. Given the definition, the proposition is stated as: postgraduate A in School of Economics and Management at Guangdong University of Technology in 2003 is the outstanding postgraduate. If # » S(t) = {S1 (t), S2 (t), . . . , Sk (t)}, then A((μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))) is called the error logical propositional variable with pluralistic predicate. Predicate in error logical proposition is the term that reflects the relationship between attribute of thing and the rules for judging errors of the object being # » evaluated and justified. It includes p(t), T (t), L(t)), x(t), f (μ(t), G(t)), and G(t) # » in A((μ(t), x(t)) = A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))). Like classic mathematical logic, fuzzy logic, and dialectic logic, the predicates in error logical proposition also have unitary, pluralistic, first-order, and high-order predicate. Moreover, there are classic predicate, fuzzy predicate, and predicate with critical points for error logical proposition.

5.8.1.3

Constant, Variable, Propositional Function and Quantified Formula of Error Predicate Logical Proposition

1. Constant of error predicate logical proposition: in A0 (μ0 (t0 ), x0 (t0 ))=A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 ))), items U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )) are all constants. 2. Function of error predicate logical proposition: in A(μ(t), x(t)) = A((U (t), # » S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t))), at least one of items in U (t), # » S(t), p(t), T (t), L(t)), x(t) = f (μ(t), G(t)) is variable. 3. Function array of error predicate logical proposition: it is in the form of {A1 ((U1 (t), S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((U2 (t), S2 (t), #» 2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Ak ((Uk (t), Sk (t), #p»k , Tk (t), L k (t)), xk (t) = f k (μk (t), G k (t)))}. 4. Quantified formulas of error predicate logical proposition are demonstrated as follows: (1) Quantified formula in constant (A) Existential quantifier: subject-predicate form (a) ∃LY (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain domain; (b) ∃SW (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain thing;

506

5 Mathematical Error Propositional Logic

(c) ∃S J (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain time; (d) ∃K J (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain spatial location; (e) ∃T J (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain constraint; (f) ∃T Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain property; (g) ∃L Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain value of the property or attribute; (h) ∃C Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain error value; (i) ∃H S(A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain function; (j) ∃G Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain group of rules; (k) ∃F S(A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain decomposition mechanism. (B) Universal quantifier: subject-predicate form (a) ∀LY (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for all domain; (b) ∀SW (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each thing; (c) ∀S J (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for all the time; (d) ∀K J (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each spatial location; (e) ∀T J (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each constraint; (f) ∀T Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each property; (g) ∀L Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each value of the property or attribute; (h) ∀C Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each error value; (i) ∀H S(A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each function; (j) ∀G Z (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each group of rules; (k) ∀F S(A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), for each decomposition mechanism; (l) Θ(A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 )))), there exists certain universal;

5.8 Error Predicate Logic

(m) Ψ (A0 ((μ0 (t0 ), x0 (t0 )) = A0 ((U0 (t0 ), S0 (t0 ), #p»0 , f 0 (μ0 (t0 ), G 0 (t0 )))), for all universals;

507

T0 (t0 ), L 0 (t0 )),

(2) Quantified formulas in function (A) Existential quantifier: subject-predicate form # » (a) ∃LY (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain domain; # » (b) ∃SW (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain thing; # » (c) ∃S J (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain time; # » (d) ∃K J (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain spatial location; # » (e) ∃T J (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain constraint; # » (f) ∃T Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain property; # » (g) ∃L Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain value of the property or attribute; # » (h) ∃C Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain error value; # » (i) ∃H S(A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain function; # » (j) ∃G Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain group of rules; # » (k) ∃F S(A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), L(t)), G(t)))), there exists certain decomposition mechanism.

x0 (t0 ) =

x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t), x(t)= f (μ(t),

(A) Universal quantifier: subject-predicate form

# » (a) ∀LY (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each domain; # » (b) ∀SW (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each thing; # » (c) ∀S J (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for all the time; # » (d) ∀K J (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each spatial location; # » (e) ∀T J (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each constraint; # » (f) ∀T Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each property; # » (g) ∀L Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each value of the property or attribute; # » (h) ∀C Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each error value;

L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t),

508

5 Mathematical Error Propositional Logic

# » (i) ∀H S(A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each function; # » (j) ∀G Z (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each group of rules; # » (k) ∀F S(A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for each decomposition mechanism; # » (l) Θ(A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), there exists certain universal; # » (m) Ψ (A((μ(t), x(t))=A((U (t), S(t), p(t), T (t), G(t)))), for all universals;

L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t), L(t)), x(t)= f (μ(t),

5.8.2 Form Language of Error Predicate Logic Form language of error predicate logic is the single-meaning synthetic language that uses various logical forms in error predicate logic.

5.8.2.1

Individual Symbols in Error Predicate Logic

Constant elements: U0 (t), S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ), f 0 (μ0 (t0 ), G 0 (t0 )) # » Variable elements: U (t), S(t), p(t), T (t), L(t)), x(t) f (μ(t), G(t))

5.8.2.2

Term Symbols in Error Predicate Logic

1. Propositional constant terms in error predicate logic: U0 , S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ), f 0 (μ0 (t0 ), G 0 (t0 )), A0 ((U0 , S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), G 0 (t0 ))), B0 ((U0 , S0 (t0 ), #p»0 , T0 (t0 ), L 0 (t0 )), x0 (t0 ) = f 0 (μ0 (t0 ), # (n) » (n) (n) (n) (n) (n) (n) G 0 (t0 ))), A(n) 0 ((U0 , S0 (t0 ), p0 , T0 (t0 ), L 0 (t0 )), x 0 (t0 ) = f 0 (μ0 (t0 ), # » (n) (n) (n) (n) (n) (n) (n) (n) G (n) 0 (t0 ))), and B0 ((U0 , S0 (t0 ), p0 , T0 (t0 ), L 0 (t0 )), x 0 (t0 )= f 0 (μ0 (t0 ), (n) G 0 (t0 ))) are all constant terms. # » 2. Propositional function terms in error predicate logic: in U (t), S(t), p(t), T (t), # » L(t)), x(t) f (μ(t), G(t)), A((U (t), S(t), p(t), T (t), L(t)), x(t) = f (μ(t), # » G(t))), A(n) ((U (n) , S (n) (t), p (n) , T (n) (t), L (n) (t)), x (n) (t) = f (n) (μ(n) (t), # » G (n) (t))), and B (n) ((U (n) , S (n) (t), p (n) , T (n) (t), L (n) (t)), x (n) (t) = f (μ(n) (t), G (n) (t))), at least individual term is variable. 3. Propositional constant term array in error predicate logic: they are in the form of: {A1 ((U1 , S1 (t), #p»1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 ((U2 , S2 (t), #p», T (t), L (t)), x (t) = f (μ (t), G (t))), . . . , A ((U , S (t), #p», T (t), 2 2 2 2 2 2 2 k #k(n) » k (n) k (n) k (n) (n) ((U , S (t), p , T1 (t), L 1 (t)), L k (t)), xk (t) = f k (μk (t), G k (t)))}, {A(n) 1 1 1 # (n) »1 (n) (n) (n) (n) (n) (n) (n) x1 (t) = f 1 (μ1 (t), G 1 (t))), A2 (U2 , S2 (t), p2 , T2(n) (t), L (n) 2 (t)),

5.9 Semantic Explanation of Error Predicate Logical Expression

509

# (n) » (n) (n) (n) (n) (n) (n) x2(n) (t)= f 2(n) (μ(n) 2 (t), G 2 (t))), . . . , Ak ((Uk , Sk (t), pk , Tk (t), L k (t)), (n) (n) (n) (n) # » xk (t) = f k (μk (t), G k (t)))}, {B1 ((U1 , S1 (t), p1 , T1 (t), L 1 (t)), x1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , S2 (t), #p»2 , T2 (t), L 2 (t)), x2 (t) = f 2 (μ2 (t), G 2 (t))), . . . , Bk ((Uk , Sk (t), #p»k , Tk (t), L k (t)), xk (t)= f k (μk (t), G k (t)))}, {B1(n) ((U1(n) , # » (n) (n) (n) (n) (n) (n) S1(n) (t), p1(n) , T1(n) (t), L (n) 1 (t)), x 1 (t) = f 1 (μ1 (t), G 1 (t))), B2 ((U2 , # » (n) (n) (n) (n) (n) (n) S2(n) (t), p2(n) , T2(n) (t), L (n) 2 (t)), x 2 (t)= f 2 (μ2 (t), G 2 (t))), . . . , Bk ((Uk , # » (n) (n) (n) (n) Sk(n) (t), pk(n) , Tk(n) (t), L (n) k (t)), x k (t) = f k (μk (t), G k (t)))}, 5.8.2.3

Formula Symbols in Error Predicate Logic

f i (A1 ), A2 , . . . , An ), G j (B1 , B2 , . . . . . . , Bm ), . . . , h(C1 , C2 , . . . , Ck ), etc.

5.9 Semantic Explanation of Error Predicate Logical Expression This part offers the semantic explanation of error predicate logical expression. Thing, property, property(attribute) value, function, time, error value, rules for judging errors, and the universe of discourse, i.e., {U d , S d , T Z d , L d , F d ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i }, S J d , C W Z d ⊆ {{0, 1}, [0, 1], (−∞, +∞)}, G Z d } are defined as follows. U d is the universe of discourse; S d is set composed of the individuals of all things; T Z d is the set composed of all properties of all elements; L d is the set composed of property (attribute) values of all things; F d ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i } is the function from domain U (t) to C W Z d ⊆ {{0, 1}, [0, 1], (−∞, +∞)}; C W Z d ⊆ {{0, 1}, [0, 1], (−∞, +∞)} is the domain for error values, i.e., truth value domain of error predicate logic; S J d is the time range; and G Z d is the set of rules for judging errors in the domain Ud . Ic indicates the value of constant element, which is the element in {U cl , cl , T Z cl , cl L , F cl ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i }, S J cl , C W Z cl ⊆{{0, 1}, [0, 1], (−∞, +∞), G Z cl }. Id indicates the value of variable element, which is the element in {U bl , S bl , T Z bl , L bl , F bl ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i }, S J bl , C W Z bl ⊆{{0, 1}, [0, 1], (−∞, +∞), G Z bl }. Transformation: (1) Similarity connectives, Tx ⊆ {Txly , Txsw , Txk j , Txt z , Txlz , Txcz , Txhs , Txs j , Txgz , Tx zh } (similarity), Tx−1 : inverse similarity connective; (2) Displacement connectives, Tz ⊆ {Tzly , Tzsw , Tzk j , Tzt z , Tzlz , Tzcz , Tzhs , Tzs j , Tzgz , Tzzh } (similarity), Tz−1 : inverse displacement connective;

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5 Mathematical Error Propositional Logic

(3) Addition connectives, Tz n ⊆ {Tznly , Tznsw , Tznk j , Tznt z , Tznlz , Tzncz , Tznhs , Tzns j , Tzngz , Tznzh } (addition), Tz n −1 : inverse addition connective; (4) Decomposition connectives, T f ⊆ {T f ly , T f sw , T f k j , T f t z , T f lz , T f cz , T f hs , T f s j , T f gz , T f zh } (decomposition), T f−1 : inverse decomposition connective; (5) Destruction connectives, Th ⊆ {Thly , Thsw , Thk j , Tht z , Thlz , Thcz , Thhs , Ths j , Thgz , Thzh } (destruction), Th−1 : inverse destruction connective; (6) Unit transformation connectives, Td (unit),Td−1 : inverse unit transformation connective; (7) Quantifier system of error logic. The value range of transformation {Vbz , Sbz , T Z bz , L bz , Fbz ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i }, S Jbz , C W Z bz ⊆ {{0, 1}, [0, 1], (−∞, +∞), G Z bz }. V bz is the domain after transformation; S bz is the set composed of all variables in all things of interest after transformation; T Z bz is the set composed of all properties of all variables; L bz is the set composed of property (attribute) values after transformation; F bz ∈ {χ (u), μ(u), f 1 , f 2 , . . . , f i } is the function from domain obtained through element transformation on C W Z bz ⊆ {{0, 1}, [0, 1], (−∞, +∞)}; C W Z bz ⊆ {{0, 1}, [0, 1], (−∞, +∞)} is the range of error values, i.e., truth value of variables of error predicate logic elements obtained through transformation; S J d is the time range of the elements obtained through transformation; and G Z d is the set of rules for judging errors of variables in the domain obtained through rule transformation.

Chapter 6

Applications of Error Logic

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion Many cities around the world have been affected by traffic congestion. In order to resolve this problem, some cities have resorted to the development of intelligent transportation systems(ITS) which help analyze the issue and identify the root cause and offer feasible solution for the problem. Error matrix equation provides an operable means to find the solution sets for addressing traffic congestion issues by solving relevant equation. For an error system, in order to establish a decision support system for avoiding or eliminating errors, the necessary steps include: 1. To find out all error types in this error system 2. To identify the conditions under which each type of error occurs (1) External conditions; (2) Internal conditions 3. To find out the causes for each type of error (1) Causes from external environments; (2) Causes within internal environments; 4. For each type of error, according to their corresponding conditions and evaluation indicators, “15-6-3” methodology in our error theory or error matrix equation is employed to find out the solution sets for avoiding or eliminating this type of error and then solution subset for meeting predefined assessment indicators are chosen and ordered in certain sequence. Let’s consider the specific steps for building a decision support system to help reduce traffic congestion.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Liu and K. Guo, Error Logic: Paving Pathways for Intelligent Error Identification and Management, Studies in Systems, Decision and Control 442, https://doi.org/10.1007/978-3-031-00820-7_6

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6 Applications of Error Logic

1. To find out all possible factors that might contribute to traffic congestion; (1) Hardware (1) Roads, bridges, tunnels; (2) Signal systems; (3) Road signs (2) Software (1) Laws, regulations, and rules; (2) Human behaviors; (3) Administration and operations; (3) Emergencies (1) Disaster; (2) Accidents; (3) Road maintenance; (4) Intended or unintended damage; (4) Other factors 2. To identify the conditions under which each type of traffic jam occurs; (1) Environmental conditions; (2) Infrastructural conditions 3. To find out the causes for each type of error; (1) Causes from external environments; (2) Causes within internal environments; 4. For each type of traffic congestion, according to their corresponding conditions and evaluation indicators, “15-6-3” methodology in our error theory or error matrix equation is employed to find out the solution sets for avoiding or eliminating this type of error and then solution subset for meeting predefined assessment indicators are chosen and ordered in certain sequence.

6.1.1 Problem Statement Let’s take the transportation system in a city as an example. Suppose that U (t) = { All factors related to urban transportation }, S(t) = { urban transportation from −−→ 2007 till now }, p(t) = { spatial location where the urban transportation system resides }, T (t) = { traffic congestion }, L(t) = { a% all lanes, b% traffic junctions −−→ }, x(t) = f ((μ(t), p(t)), G U (t)), G U (t) = { traffic laws, rules, and regulations }, −−→ then the error logical variable A((U (t), S(t), p(t), T (t), L(t)), x(t) = f ((μ(t), −−→ p(t)), G(t))) = A((U (t) = { All factors related to urban transportation }, S(t) = −−→ { urban transportation from 2007 till now }, p(t) = { spatial location where the urban transportation system resides }, T (t) = { traffic congestion }, L(t) = { a% all

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion

513

−−→ lanes, b% traffic junctions }, x(t) = f ((μ(t), p(t)), G U (t) = { traffic laws, rules, and regulations })).

6.1.2 Method for Finding Solutions Suppose that   a11 a12 A = a21 a22 

where, −−−−−−−−−−−−−→ a11 = A((U201 (t), S201 (t), p201 (Ψ1 , Ψ2 , Λ, Ψn ), −−−−→ ((μ201 (t), p201 (t)), G 201 (t))), −−−−−−−−−−−−−→ a12 = A((U202 (t), S202 (t), p202 (Ψ1 , Ψ2 , Λ, Ψn ), −−−−→ ((μ202 (t), p202 (t)), G 202 (t))), −−−−−−−−−−−−−→ a21 = A((U211 (t), S211 (t), p211 (Ψ1 , Ψ2 , Λ, Ψn ), −−−−→ ((μ211 (t), p211 (t)), G 211 (t))), −−−−−−−−−−−−−→ a22 = A((U212 (t), S212 (t), p212 (Ψ1 , Ψ2 , Λ, Ψn ), −−−−→ ((μ212 (t), p212 (t)), G 212 (t))).

T201 (t), L 201 (t)), x201 (t) = f 201 T202 (t), L 202 (t)), x202 (t) = f 202 T211 (t), L 211 (t)), x211 (t) = f 211 T212 (t), L 212 (t)), x212 (t) = f 212

The elements in a11 , a12 , a21 , and a22 are all sets. For example, domain U201 (t) is a set with certain number of elements, which is denoted by U201 (t)={u 201 , u 202 , . . ., u 20n }. The elements in a11 , a12 , a21 , and a22 are represented as follows. −−−−−−−−−−−−−−→ −→ p201 , In a11 , U201 (t)={u 201 , u 202 , . . ., u 20n }, S201 (t)={s201 , s202 , . . ., s20n }, p201 (Ψ1 , Ψ2 , Λ, Ψn )={−

−−→ − → p− 202 , . . ., p20n }, T201 (t)={t201 , t202 , . . ., t20n }, L 201 (t)={l201 , l202 , . . ., l20n }, x 201 (t)= f 20 ((μ201 (t), −−−−→ p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n }, G 201 (t)={g201 , g202 , . . ., g20n }, n=5; −−−−−−−−−−−−−−→ −→ In a12 , U202 (t)={u 201 , u 202 , . . ., u 20n }, S202 (t)={s201 , s202 , . . ., s20n }, p202 (Ψ1 , Ψ2 , Λ, Ψn )={− p201 , − − → − − → p202 , . . ., p20n }, T202 (t)={t201 , t202 , . . ., t20n }, L 202 (t)={l201 , l202 , . . ., l20n }, x202 (t)= f 20 ((μ201 (t), −−−−→ p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n }, G 202 (t)={g201 , g202 , . . ., g20n }, n=6; −−−−−−−−−−−−−−→ −→ In a21 , U211 (t)={u 201 , u 202 , . . ., u 20n }, S211 (t)={s201 , s202 , . . ., s20n }, p211 (Ψ1 , Ψ2 , Λ, Ψn )={− p201 , − − → − − → p202 , . . ., p20n }, T211 (t)={t201 , t202 , . . ., t20n }, L 211 (t)={l201 , l202 , . . ., l20n }, x211 (t)= f 21 ((μ201 (t), −−−−→ p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n }, G 211 (t)={g201 , g202 , . . ., g20n }, n=6; −−−−−−−−−−−−−−→ −→ In a22 , U212 (t)={u 201 , u 202 , . . ., u 20n }, S212 (t)={s201 , s202 , . . ., s20n }, p212 (Ψ1 , Ψ2 , Λ, Ψn )={− p201 , − − → − − → p202 , . . ., p20n }, T212 (t)={t201 , t202 , . . ., t20n }, L 212 (t)={l201 , l202 , . . ., l20n }, x212 (t)= f 21 ((μ201 (t), −−−−→ p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n }, G 212 (t)={g201 , g202 , . . ., g20n }, n=7. Suppose that X = (x1 , x1 ), −−−−−−−−−−−−−−→ x1 = A((U10X (t), S10X (t), p10X (Ψ1 , Ψ2 , Λ, Ψn ), T10X (t), L 10X (t)), x10X (t) = f 10X ((μ10X (t), −−−−→ p10X (t)), G 10X (t))), −−−−−−−−−−−−−−→ x2 = A((U11X (t), S11X (t), p11X (Ψ1 , Ψ2 , Λ, Ψn ), T11X (t), L 11X (t)), x11X (t) = f 11X ((μ11X (t), −−−−→ p11X (t)), G 11X (t))).

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Suppose that 

b b B = 11 12 b21 b22 



−−−−−−−−−−−−−−−→ b11 = B((V201 (t), SV 201 (t), pV 201 (Ψ1 , Ψ2 , Λ, Ψn ), TV 201 (t), L V 201 (t)), yV 201 (t) = −−−−−→ pV 201 (t)), G V 201 (t))), −−−−−−−−−−−−−−−→ b12 = B((V202 (t), SV 202 (t), pV 202 (Ψ1 , Ψ2 , Λ, Ψn ), TV 202 (t), L V 202 (t)), yV 202 (t) = −−−−−→ pV 202 (t)), G V 202 (t))), −−−−−−−−−−−−−−−→ b21 = B((V211 (t), SV 211 (t), pV 211 (Ψ1 , Ψ2 , Λ, Ψn ), TV 211 (t), L V 211 (t)), yV 211 (t) = −−−−−→ pV 211 (t)), G V 211 (t))), −−−−−−−−−−−−−−−→ b22 = B((V212 (t), SV 212 (t), pV 212 (Ψ1 , Ψ2 , Λ, Ψn ), TV 212 (t), L V 212 (t)), yV 212 (t) = −−−−−→ pV 212 (t)), G V 212 (t))).

f V 201 ((μV 201 (t), f V 202 ((μV 202 (t), f V 211 ((μV 211 (t), f V 212 ((μV 212 (t),

Similar to those element representation in a11 , a12 , a21 , and a22 , the elements in b11 , b12 , b21 , and b22 can be expressed as below. −−−−−−−−−−−−→ −−−→ −−−→ (Ψ1 , Ψ2 , Λ, Ψn )={ pv201 , pv202 , . . ., In b11 , V201 (t)={v201 , v202 , . . ., v20k }, SV 201 (t)={sv201 , sv202 , . . ., sv20k }, −p−V−201

−−−−−→ − −→ p− v20k }, TV 201 (t)={tv201 , tv202 , . . ., tv20k }, L V 201 (t)={lv201 , lv202 , . . ., lv20k }, yV 20 (t)= f v201 ((μv201 (t), pv201 (t)), G v201 (t))={yv201 , yv202 , . . ., yv20k }, G V 201 (t)={gv201 , gv202 , . . ., gv20k }, k=5; −−−−−−−−−−−−−−−→ −−→ −−−→ In b12 , V202 (t)={v201 , v202 , . . ., v20n }, SV 202 (t)={sv201 , sv202 , . . ., sv20n }, pV 202 (Ψ1 , Ψ2 , Λ, Ψn )={− pv201 , pv202 , . . ., −−−−−→ − − − → pv20n }, TV 202 (t)={tv201 , tv202 , . . ., tv20n }, L V 202 (t)={lv201 , lv202 , . . ., lv20n }, yV 202 (t)= f v20 ((μv201 (t), pv201 (t)), G v201 (t))={yv201 , yv202 , . . ., yv20n }, G V 202 (t)={gv201 , gv202 , . . ., gv20n }, k=6; −−−−−−−−−−−−−−−→ −−→ −−−→ In b21 , V211 (t)={v201 , v202 , . . ., v20k }, SV 211 (t)={sv201 , sv202 , . . ., sv20k }, pV 211 (Ψ1 , Ψ2 , Λ, Ψn )={− pv201 , pv202 , . . ., −−−−−→ − − − → pv20k }, TV 211 (t)={tv201 , tv202 , . . ., tv20k }, L V 211 (t)={lv201 , lv202 , . . ., lv20k }, yV 211 (t)= f v21 ((μv201 (t), pv201 (t)), G v201 (t))={yv201 , yv202 , . . ., yv20k }, G V 211 (t)={gv201 , gv202 , . . ., gv20k }, k=6; −−−−−−−−−−−−−−−→ −−→ −−−→ In b22 , V212 (t)={v201 , v202 , . . ., v20k }, SV 212 (t)={sv201 , sv202 , . . ., sv20k }, pV 212 (Ψ1 , Ψ2 , Λ, Ψn )={− pv201 , pv202 , . . ., −−−−−→ − − − → pv20k }, TV 212 (t)={tv201 , tv202 , . . ., tv20k }, L V 212 (t)={lv201 , lv202 , . . ., lv20k }, yV 212 (t)= f v21 ((μv201 (t), pv201 (t)), G v201 (t))={yv201 , yv202 , . . ., yv20k }, G V 212 (t)={gv201 , gv202 , . . ., gv20k }, k=7.

In practice, the problem of interest may be affected objective conditions kg, people’s decision r w, and actual needs xq. Suppose that: −−−→ −−−→ kg : (Ukg (t), Skg (t), pkg (t), Tkg (t), L kg (t)), xkg (t) = f ((μkg (t), pkg (t)), G U (t)). Ukg (t)={u 201 , u 202 , . . ., u 20n , v201 , v202 , . . ., v20n j }, Skg (t)={s201 , s202 , . . ., s20n }, −−−−−−−−−−−−→ −−→ −−→ →}, T (t)={t , t , . . ., t }, L (t)={l , p−20n pkg (Ψ1 , Ψ2 , Λ, Ψn )={ p201 , p202 , . . ., − kg 201 202 20n kg 201 −−−−→ l202 , . . ., l20n }, ykg (t)= f 20 ((μ201 (t), p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n , y211 , y212 , . . ., y21 j }, G U (t)={g201 , g202 , . . ., g20n }, n=8, j=10; −−−→ −−−→ r w : (Ur w (t), Sr w (t), pr w (t), Tr w (t), L r w (t)), xr w (t) = f ((μr w (t), pr w (t)), G U (t)). Ur w (t)={u 201 , u 202 , . . ., u 20n , v201 , v202 , . . ., v20n j }, Sr w (t)={s201 , s202 , . . ., s20n }, −−−−−−−−−−−−−→ −−→ −−→ →}, T (t)={t , t , . . ., t }, L (t)={l , p−20n pr w (Ψ1 , Ψ2 , Λ, Ψn )={ p201 , p202 , . . ., − rw 201 202 20n rw 201 −−−−→ l202 , . . ., l20n }, yr w (t)= f 20 ((μ201 (t), p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n , y211 , y212 , . . ., y21 j }, G U (t)={g201 , g202 , . . ., g20n }, n=9, j=10; −−−→ −−−→ xq : (Ur w (t), Sr w (t), pr w (t), Tr w (t), L r w (t)), xr w (t) = f ((μr w (t), pr w (t)), G U (t)). Uxq (t)={u 201 , u 202 , . . ., u 20n , v201 , v202 , . . ., v20n j }, Sxq (t)={s201 , s202 , . . ., s20n },

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion

515

−−−−−−−−−−−−−→ −−→ −−→ →}, T (t)={t , t , . . ., t }, L (t)={l , p−20n pxq (Ψ1 , Ψ2 , Λ, Ψn )={ p201 , p202 , . . ., − xq 201 202 20n xq 201 −−−−→ l202 , . . ., l20n }, yxq (t)= f 20 ((μ201 (t), p201 (t)), G 201 (t))={x201 , x202 , . . ., x20n , y211 , y212 , . . ., y21 j }, G U (t)={g201 , g202 , . . ., g20n }, n=9, j=10; According to the existence theorem of solution in error matrix equation, the solutions for X A = B are: −−−−−−−−−−−−−→ U10X (t) = {u 201 , u 202 , . . ., u 20n }, S10X (t) = {s201 , s202 , . . ., s20n }, p10X (Ψ1 , Ψ2 , Λ, Ψn ) →, . . ., − →}, T (t)={t , t , . . ., t }, L (t)={l , l , . . ., l }, →, − p−202 p−20n = {− p−201 10X 201 202 20n 10X 201 202 20n x10X (t) = {y201 , y202 , . . ., y20 j }, G U 10X (t) = {g201 , g202 , . . ., g20n }, n=6; −−−−−−−−−−−−−→ U11X (t) = {u 201 , u 202 , . . ., u 20n }, S11X (t) = {s201 , s202 , . . ., s20n }, p11X (Ψ1 , Ψ2 , Λ, Ψn ) →, . . ., − →}, T (t) = {t , t , . . ., t }, L (t) = {l , l , . . ., l }, →, − p−202 p−20n = {− p−201 11X 201 202 20n 11X 201 202 20n x11X (t) = {y211 , y212 , . . ., y21 j }, G U 11X (t) = {g211 , g212 , . . ., g21n }, n=8; When considering three conditions kg, r w, and xq, then the solutions meeting the condition of X ∪ kg ∪ r w ∪ xq are: −−−−−−−−−−−−−→ U10X (t) = {u 201 , u 202 , . . ., u 20n }, S10X (t) = {s201 , s202 , . . ., s20n }, p10X (Ψ1 , Ψ2 , Λ, Ψn ) →, . . ., − →}, T (t) = {t , t , . . ., t }, L (t) = {l , l , . . ., l }, →, − p−202 p−20n = {− p−201 10X 201 202 20n 10X 201 202 20n x10X (t) = {y201 , y202 , . . ., y20 j }, G U 10X (t) = {g201 , g202 , . . ., g20n }, n=6; −−−−−−−−−−−−−→ U11X (t) = {u 201 , u 202 , . . ., u 20n }, S11X (t) = {s201 , s202 , . . ., s20n }, p11X (Ψ1 , Ψ2 , Λ, Ψn ) →, . . ., − →}, T (t) = {t , t , . . ., t }, L (t) = {l , l , . . ., l }, →, − p−202 p−20n = {− p−201 11X 201 202 20n 11X 201 202 20n x11X (t) = {y211 , y212 , . . ., y21 j }, G U 11X (t)={g211 , g212 , . . ., g21n }, n=8.

6.1.3 Modeling Error-elimination for a Urban Traffic Intersection Figure 6.1 presents the illustration for a typical urban traffic intersection. 1. Average traffic volume per hour Table 6.1 tabulates the data at the road intersection. 2. Calculation for the time spent of eastbound vehicles Table 6.2 tabulates the time spent of eastbound vehicles at the road intersection. Based on the observation in Table 6.1, we can find that the waiting time spent at the straight lanes are the longest since the traffic volumes for east/westbound directions are very large and the signal-light time is longer than that of south/northbound traffic lights. Since right turn is not affected by traffic light, the waiting time for right turn is the shortest among three types of lanes. 3. Traffic capacity at the intersection (1) Straight lane capacity (a) Eastbound capacity: C E S = 1560v/ h (b) Westbound capacity: C W S = 1820v/ h (c) South/northbound capacity: C SS = 782v/ h

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6 Applications of Error Logic

Fig. 6.1 Illustration for a typical urban traffic intersection Table 6.1 Average traffic volume per hour at certain road intersection Volume Left turn Straight Right turn Total Bicycle VE S = 1391 Westbound VL W = 250 VW S = 1020 Northbound VL N = 233 VN S = 358 Southbound VL S = 434 VSS = 1161 Eastbound

VL E = 240

Pedestrian

V R E = 468 2099

23

370

V RW = 124 1394

33

274

V R N = 205 796 V RS = 220 1815

15 16

666 364

(2) Left turn capacity (a) East/westbound left turn capacity: C L E W = 224v/ h (b) South/northbound left turn capacity: C L S N = 500v/ h (3) Right turn capacity: C R = 1032v/ h (4) Total capacity: C (a) C Eastbound =C L E + C E S + C R =224 + 1560 + 1032 = 2816v/ h (b) C W estbound =C L W + C W S + C R =224 + 1820 + 1032 = 3076v/ h (c) C N or thbound =C L N + C SS + C R =500 + 782 + 1032 = 2314v/ h (d) C Southbound =C L S + C SS + C R =500 + 782 + 1032 = 2314v/ h 4. Level of service at the intersection Based on the capacity and actual volumes defined in Table 6.1, we calculate the LOS through volume/capacity (Table 6.3). The LOS is defined based on the relevant categories defined in Highway Capacity Manual (HCM) in the USA. Table 6.5 provides the LOS for each side fo the intersection in discussion.

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion Table 6.2 Time spent of eastbound vehicles at the road intersection Waiting time Left turn Straight Right turn Straight Left turn

43.88 112.15 67.22

25.20 15.75 15.75

Table 6.3 Level of service (LOS) at the road intersection LOS Left turn-V/S Straight-V/S Eastbound Westbound Northbound Southbound

240/224 = 1.07 250/221 = 1.116 233/500 = 0.466 434/500 = 0.868

1391/1560 = 0.892 1020/1820 = 0.560 358/782 = 0.458 1161/782 = 1.485

Table 6.4 Level of service defined in HCM in North America LOS Average V /C speed(miles/hours) A B C D E F

 42  39  35  28 < 28

0.45 0.6 0.76 1.00

Right turn 18.68 96.40 51.47

Right turn-V/S 468/1032 = 0.453 124/1032 = 0.120 205/1032 = 0.199 220/1032 = 0.213

MSF (cars/hour/lane)

850 1150 1450 1900

Note LOS-Level of Service; MSF-Maximum Service Flow. Table 6.5 Level of service for each side of the intersection LOW Left turn Straight Eastbound Westbound Northbound Southbound

F F B D

D B B F

Right turn B A A A

6.1.4 Flowchart Representing Process of Finding Solutions See Fig. 6.2.

517

518

6 Applications of Error Logic

Fig. 6.2 Flowchart for the process of finding solutions

6.1.5 Modeling Building and Analysis Data for the case of traffic congestion in discussion. Suppose that A((U (t), S A (t), −−−→ p A (t), T A (t), L A (t)), x(t) = f (μ(t), G A (t))) is an error logical variable defined in U (t) under G A (t) the rules for judging error, which is used to represent the characteristics of the object in discussion (i.e., traffic congestion problem at certain urban traffic intersection). 1. Eastbound lanes (1) Current states A E −−−→ (a) Location p A E (t): eastbound road at the intersection; (b) Eastbound lanes S A E (t): left turn, straight, right turn; (c) Traffic volume V: VL E = 240v/ h, VE S = 1391v/ h,VR E = 468v/ h; (d) Traffic capacity C: C L E = 224v/ h, C E S = 1560v/ h, C R = 1032v/ h;

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(e) Property of the object T A E (t) = V /C: the LOS of eastbound left turn= V /C = 240/224 = 1.07, the LOS of eastbound straight = V /C = 1391/1560 = 0.892, the LOS of eastbound right turn= V /C = 468/ 1032 = 0.453 are property values L A E (t) for T A E (t); (f) Suppose that Y = {(μ, G | μ ∈ U }, f : V → R, then f is called the error function defined within U under the G rules for judging errors, which is denoted by x = f (G  μ) or f (μ), where R is the set of real number, x is the error value of object μ; if Ran ( f ) = {0, 1}, then f is called the classic error function defined within U , it is a discontinous function. According to the criteria for LOS, let M > C L O S, M is better than C LOS. The classic error function is:  0 G⇒M x = f (G, M) = 1 GM Based on above analysis, error matrix A E for current state of eastbound lanes is presented as below:

⎡ U E Lanes E L = ⎣U E Lanes E S U E Lanes E R

⎤ − → p− A E1 T A E1 L A E1 x A E1 G A E1 − − → p A E2 T A E2 L A E2 x A E2 G A E2 ⎦ − → p− A E3 T A E3 L A E3 x A E3 G A E3 ⎤ I nter section L O S E L 1.07 1 L O S − in − H C M I nter section L O S E S 0.892 1 L O S − in − H C M ⎦ I nter section L O S E R 0.453 0 L O S − in − H C M ⎡ U A E1 S A E1 A E = ⎣U A E2 S A E2 U A E3 S A E3

Where the EL, ES, and ER represent eastbound left turn, eastbound straight, and eastbound right turn lanes. (2) Target states for the eastbound lanes According to LOS categories in HCM, we build the target state matrix, where the threshold of C level LOS is V /C  0.6. The target state matrix BE :

B E = B E11 B E12 B E13 where, B E11 = (U E11 Lanes E L Intersection L O S E L (0, 0.6] 0 LOS-in-HCM) B E12 = (U E12 Lanes E S Intersection L O S E S (0, 0.6] 0 LOS-in-HCM) B E13 = (U E13 Lanes E R Intersection L O S E R (0, 0.6] 0 LOS-in-HCM) Let

X E = U X E SX E − p→ X E TX E L X E x X E G X E For matrix equation X E AE =B E

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6 Applications of Error Logic ⎡

U A E1 S A E1 − − → U X E S X E p X E TX E L X E x X E G X E ⎣U A E2 S A E2 U A E3 S A E3

⎤ − T A E1 L A E1 x A E1 G A E1 p−A−→ E1 − − − → p A E2 T A E2 L A E2 x A E2 G A E2 ⎦ − T A E3 L A E3 x A E3 G A E3 p−A−→ E3

= B E 11 B E 12 B E 13

where, TBE1 L BE1 x BE1 G BE1 ) p−B→ B E11 = (U BE1 S BE1 − E1 − TBE2 L BE2 x BE2 G BE2 ) B E12 = (U BE2 S BE2 p−B→ E2 p−B→ T B E13 = (U BE3 S BE3 − B E3 L B E3 x B E3 G B E3 ) E3 p−X→ ∧− p−A→ =− p−→ In equation X E AE =B E , there exist S X Ei ∧ S A Ei = S BEi , − B Ei , Ei Ei TX Ei ∧ T A Ei = TBEi , L X Ei ∧ L A Ei = L BEi , x X Ei ∧ x A Ei = x BEi , and G X Ei ∧ G A Ei = G BEi . (3) Solving equation (a) U X Ei ∧ U A Ei = U BEi , i.e., U X Ei ∧ U = U , U X Ei = U and the domain is the designated urban transportation, U X Ei = U . (b) S X Ei ∧ S A Ei = S B Ei , i.e., S X Ei ∧ {Lanes E L } = {Lanes E L }, S X Ei = {Lanes E L }. −−→ −−→ (c) − p−X→ ∧− p−A→ =− p−→ B Ei , i.e., p X Ei ∧ { Intersection } = { Intersection }, p X Ei Ei Ei ={ Intersection } . (d) TX Ei ∧ T A Ei = TBEi , i.e., TX Ei ∧ {L O S E = V /C} = {L O S E = V /C}, TX Ei = {L O S E = V /C}. (e) G X Ei ∧ G A Ei = G BEi , i.e., G X Ei ∧ { LOS-in-HCM } = { LOS-in-HCM }, G X Ei = { LOS-in-HCM }. (f) x X E1 ∧ x A E1 = x BE1 , i.e., x X E1 ∧ 1 = 0, then x X E1 = 0, and the condition of L X E1 ∧ L A E1 = L BE1 must be met, i.e., L X E1 ∧ 1.07 = (0, 0.6], therefore L X E1 ∈ (0, 0.6]. (g) x X E2 ∧ x A E2 = x BE2 , i.e., x X E2 ∧ 1 = 0, then x X E2 = 0, and the condition of L X E2 ∧ L A E2 = L BE2 must be met, i.e., L X E2 ∧ 0.892 = (0, 0.6], therefore L X E2 ∈ (0, 0.6]. (h) x X E3 ∧ x A E3 = x BE3 , i.e., x X E3 ∧ 0 = 0, then x X E3 = 0, since it is nonerroneous, the current state is kept. Then the solutions are as follows: X E1 = (U Lanes E L Intersection L O S E L (0, 0.6] 0 LOS-in-HCM) X E2 = (U Lanes E S Intersection L O S E S (0, 0.6] 0 LOS-in-HCM) X E3 = (U Lanes E R Intersection L O S E R 0.453 0 LOS-in-HCM) For the intersection in discussion, the expected state (target state), i.e., LOS-V/C of A E1 : eastbound left turn and A E2 : eastbound straight should be lowered to the range of (0, 0.6]. While the LOS of eastbound right turn A E3 can be kept current state. 2. Westbound lanes (1) Current states A W −−−−→ (a) Location p AW (t): westbound road at the intersection; (b) Westbound lanes S AW (t): left turn, straight, right turn; (c) Traffic volume V: VW L = 250v/ h, VSW = 1020v/ h,VW R = 124v/ h; (d) Traffic capacity C: C E L = 224v/ h, C E S = 1820v/ h, C R = 1032v/ h;

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion

521

(e) Property of the object T AW (t) = V /C: the LOS of westbound left turn= V /C = 250/224 = 1.16, the LOS of westbound straight = V /C = 1020/1820 = 0.560, the LOS of westbound right turn= V /C = 124/ 1032 = 0.120 are property values L AW (t) for T AW (t); (f) Suppose that Y = {(μ, G | μ ∈ U }, f : V → R, then f is called the error function defined within U under the G rules for judging errors, which is denoted by x = f (G  μ) or f (μ), where R is the set of real number, x is the error value of object μ; if Ran ( f ) = {0, 1}, then f is called classic error function defined within U , it is a discontinous function. According to the criteria for LOS, let M > C L O S, M is better than C LOS. The classic error function is:  0 G⇒M x = f (G, M) = 1 GM Based on above analysis, error matrix A W for current state of westbound lanes is presented as below:

⎡ UW = ⎣UW UW

⎤ − → p− AW 1 TAW 1 L AW 1 x AW 1 G AW 1 − − → p AW 2 TAW 2 L AW 2 x AW 2 G AW 2 ⎦ AW − → p− AW 3 TAW 3 L AW 3 x AW 3 G AW 3 ⎤ LanesW L I nter section L O SW L 1.16 1 L O S − in − H C M LanesW S I nter section L O SW S 0.56 0 L O S − in − H C M ⎦ LanesW R I nter section L O SW R 0.12 0 L O S − in − H C M ⎡ U AW 1 S AW 1 = ⎣U A W 2 S A W 2 U AW 3 S AW 3

where the WL, WS, and WR represent westbound left turn, westbound straight, and westbound right turn lanes. (2) Target states for the westbound lanes According to LOS categories in HCM, we build the target state matrix, where the threshold of C level LOS is V /C  0.6. The target state matrix BW :

BW = BW11 BW12 BW13 Where, BW11 = (UW11 LanesW L Intersection L O SW L (0, 0.6] 0 LOS-in-HCM) BW12 = (UW12 LanesW S Intersection L O SW S (0, 0.6] 0 LOS-in-HCM) BW13 = (UW13 LanesW R Intersection L O SW R (0, 0.6] 0 LOS-in-HCM) Let

p−→ X W = U X W SX W − X W TX W L X W x X W G X W For matrix equation X W AW =BW

522

6 Applications of Error Logic ⎡  U AW 1 S AW 1 ⎢ − − − → U X W S X W p X W TX W L X W x X W G X W ⎣U A W 2 S A W 2 U AW 3 S AW 3

⎤ − −→ p− AW 1 TAW 1 L AW 1 x AW 1 G AW 1 ⎥ − − − → p AW 2 TAW 2 L AW 2 x AW 2 G AW 2 ⎦ − −→ p− A W 3 T A W 3 L A W 3 x A W 3 G Aw3  = BW11 BW12 BW13

Where, → p− BW11 = (U BW 1 S BW 1 − BW 1 T BW 1 L BW 1 x BW 1 G BW 1 ) − − T BW 2 L BW 2 x BW 2 G BW 2 ) BW12 = (U BW 2 S BW 2 p B→ W2 → p− BW13 = (U BW 3 S BW 3 − BW 3 T BW 3 L BW 3 x BW 3 G BW 3 ) → −−→ p− In equation X W AW =BW , there exist S X W i ∧ S AW i = S BW i , − X W i ∧ p AW i = − − → p BW i , TX W i ∧ T AW i = TBW i , L X W i ∧ L AW i = L BW i , x X W i ∧ x AW i = x BW i , and G X W i ∧ G A W i = G BW i . (3) Solving equation (a) U X W i ∧ U AW i = U BW i , i.e., U X W i ∧ U = U , U X W i = U and the domain is the designated urban transportation, U X W i = U . (b) S X W i ∧ S AW i = S BW i , i.e., S X W i ∧ {LanesW L } = {LanesW L }, S X W i = {LanesW L }. → −−→ −−→ −−→ −−→ (c) − p− X W i ∧ p A W i = p BW i , i.e., p X W i ∧ { Intersection } = { Intersection }, p X W i ={ Intersection } . (d) TX W i ∧ T AW i = TBW i , i.e., TX W i ∧ {L O SW = V /C} = {L O SW = V /C}, TX W i = {L O SW = V /C}. (e) G X W i ∧ G AW i = G BW i , i.e., G X W i ∧ { LOS-in-HCM } = { LOS-in-HCM }, G X W i = { LOS-in-HCM }. (f) x X W 1 ∧ x AW 1 = x BW 1 , i.e., x X W 1 ∧ 1 = 0, then x X W 1 = 0, and the condition of L X W 1 ∧ L AW 1 = L BW 1 must be met, i.e., L X W 1 ∧ 1.16, therefore L X W 1 ∈ (0, 0.6]. (g) x X W 2 ∧ x AW 2 = x BW 2 , i.e., x X W 2 ∧ 0 = 0, then x X W 2 = 0, since it is nonerroneous, the current state is kept. (h) x X W 3 ∧ x AW 3 = x BW 3 , i.e., x X W 3 ∧ 0 = 0, then x X W 3 = 0, since it is nonerroneous, the current state is kept. Then the solutions are as follows: X W 1 = (U LanesW L Intersection L O SW L (0, 0.6] 0 LOS-in-HCM) X W 2 = (U LanesW S Intersection L O SW S 0.56 0 LOS-in-HCM) X W 3 = (U LanesW R Intersection L O SW R 0.12 0 LOS-in-HCM) For the intersection in discussion, the expected state (target state), i.e., LOS-V/C of A W 1 : westbound left turn should be lowered to the range of(0, 0.6]. While the LOS of westbound straight A W 2 and westbound right turn A W 3 can be kept current state. 3. Southbound lanes (1) Current states A S −−−→ (a) Location p A S (t): southbound road at the intersection; (b) Southbound lanes S A S (t): left turn, straight, right turn; (c) Traffic volume V: VSL = 434v/ h, VSS = 1161v/ h,VS R = 220v/ h; (d) Traffic capacity C: C SL = 500v/ h, C SS = 782v/ h, C R = 1032v/ h; (e) Property of the object T A S (t) = V /C: the LOS of southbound left turn= V /C = 434/500 = 0.868, the LOS of southbound straight = V /C =

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion

523

1161/782 = 1.485, the LOS of southbound right turn= V /C = 220/ 1032 = 0.213 are property values L A S (t) for T A S (t); (f) Suppose that Y = {(μ, G | μ ∈ U }, f : V → R, then f is called the error function defined within U under the G rules for judging errors, which is denoted by x = f (G  μ) or f (μ), where R is the set of real number, x is the error value of object μ; if Ran ( f ) = {0, 1}, then f is called the classic error function defined within U , it is a discontinous function. According to the criteria for LOS, let M > C L O S, M is better than C LOS. The classic error function is:  0 G⇒M x = f (G, M) = 1 GM Based on above analysis, error matrix A S for current state of southbound lanes is presented as below:

⎡ US = ⎣U S US

⎡ U A S1 S A S1 A S = ⎣U A S2 S A S2 U A S3 S A S3

⎤ T A S1 L A S1 x A S1 G A S1 T A S2 L A S2 x A S2 G A S2 ⎦ A S3 T A S3 L A S3 x A S3 G A S3 ⎤ Lanes S L I nter section L O SS L 0.868 1 L O S − in − H C M Lanes SS I nter section L O SSS 1.485 1 L O S − in − H C M ⎦ Lanes S R I nter section L O SS R 0.213 0 L O S − in − H C M − → p− A S1 − → p− A S2 − − → p

where the SL, SS, and SR represent southbound left turn, southbound straight, and southbound right turn lanes. (2) Target states for the southbound lanes According to LOS categories in HCM, we build the target state matrix, where the threshold of C level LOS is V /C  0.6. The target state matrix BS :

BS = BS11 BS12 BS13 where, BS11 = (U S11 Lanes SL Intersection L O SSL (0, 0.6] 0 LOS-in-HCM) BS12 = (U S12 Lanes SS Intersection L O SSS (0, 0.6] 0 LOS-in-HCM) BS13 = (U S13 Lanes S R Intersection L O SS R (0, 0.6] 0 LOS-in-HCM) Let

p→ X S = U X S SX S − X S TX S L X S x X S G X S For matrix equation X S AS =BS

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6 Applications of Error Logic

⎡ U A S1 S A S1

⎣U A S2 S A S2 U X S SX S − p→ X S TX S L X S x X S G X S U A S3 S A S3

⎤ − p−→ A S1 T A S1 L A S1 x A S1 G A S1 − ⎦ p−→ A S2 T A S2 L A S2 x A S2 G A S2 − − → p A S3 T A S3 L A S3 x A S3 G A S3

= BS11 BS12 BS13

where, p−→ BS11 = (U BS1 S BS1 − BS1 TBS1 L BS1 x BS1 G BS1 ) − BS12 = (U BS2 S BS2 p−→ BS2 TBS2 L BS2 x BS2 G BS2 ) p−→ BS13 = (U BS3 S BS3 − BS3 TBS3 L BS3 x BS3 G BS3 ) −−→ −−→ p−→ In equation X S AS =BS , there exist S X Si ∧ S A Si = S BSi , − X Si ∧ p A Si = p BSi , TX Si ∧ T A Si = TBSi , L X Si ∧ L A Si = L BSi , x X Si ∧ x A Si = x BSi , and G X Si ∧ G A Si = G BSi . (3) Solving equation (a) U X Si ∧ U A Si = U BSi , i.e., U X Si ∧ U = U , U X Si = U and the domain is the designated urban transportation, U X Si = U . (b) S X Si ∧ S A Si = S BSi , i.e., S X Si ∧ {Lanes S L } = {Lanes S L }, S X Si = {Lanes S L }. −−→ −−→ −−→ −−→ (c) − p−→ X Si ∧ p A Si = p BSi , i.e., p X Si ∧ { Intersection } = { Intersection }, p X Si ={ Intersection } . (d) TX Si ∧ T A Si = TBSi , i.e., TX Si ∧ {L O SS = V /C} = {L O SS = V /C}, TX Si = {L O SS = V /C}. (e) G X Si ∧ G A Si = G BSi , i.e., G X Si ∧ { LOS-in-HCM } = { LOS-in-HCM }, G X Si = { LOS-in-HCM }. (f) x X S1 ∧ x A S1 = x BS1 , i.e., x X S1 ∧ 1 = 0, then x X S1 = 0, and the condition of L X S1 ∧ L A S1 = L BS1 must be met, i.e., L X S1 ∧ 0.868 = (0, 0.6], therefore L X S1 ∈ (0, 0.6]. (g) x X S2 ∧ x A S2 = x BS2 , i.e., x X S2 ∧ 1 = 0, then x X S2 = 0, and the condition of L X S2 ∧ L A S2 = L BS2 must be met, i.e., L X S2 ∧ 1.485 = (0, 0.6], therefore L X S2 ∈ (0, 0.6]. (h) x X S3 ∧ x A S3 = x BS3 , i.e., x X S3 ∧ 0 = 0, then x X S3 = 0, since it is nonerroneous, the current state is kept. Then the solutions are as follows: X S1 =(U Lanes SL Intersection L O SSL (0, 0.6] 0 LOS-in-HCM) X S2 =(U Lanes SS Intersection L O SSS (0, 0.6] 0 LOS-in-HCM) X S3 =(U Lanes S R Intersection L O SS R 0.453 0 LOS-in-HCM) For the intersection in discussion, the expected state (target state), i.e., LOS-V/C of A S1 : southbound left turn and A S2 : southbound straight should be lowered to the range of (0, 0.6]. While the LOS of southbound right turn A S3 can be kept current state. 4. Northbound lanes (1) Current states A N −−−→ (a) Location p A N (t): northbound road at the intersection; (b) Northbound lanes S A N (t): left turn, straight, right turn; (c) Traffic volume V: VN L = 233v/ h, VN S = 358v/ h,VN R = 205v/ h; (d) Traffic capacity C: C N L = 500v/ h, C N S = 782v/ h, C R = 1032v/ h;

6.1 Applying Error Matrix Equation to Investigate Urban Traffic Congestion

525

(e) Property of the object T A N (t) = V /C: the LOS of northbound left turn= V /C = 233/500 = 0.466, the LOS of northbound straight = V /C = 358/782 = 0.458, the LOS of northbound right turn= V /C = 205/1032 = 0.199 are property values L A N (t) for T A N (t); (f) Suppose that Y = {(μ, G | μ ∈ U }, f : V → R, then f is called the error function defined within U under the G rules for judging errors, which is denoted by x = f (G  μ) or f (μ), where R is the set of real number, x is the error value of object μ; if Ran ( f ) = {0, 1}, then f is called the classic error function defined within U , it is a discontinuous function. According to the criteria for LOS, let M > C L O S, M is better than C LOS. The classic error function is:  0 G⇒M x = f (G, M) = 1 GM Based on above analysis, error matrix A N for current state of northbound lanes is presented as below: ⎡ U AN1 SAN1 A N = ⎣U A N 2 S A N 2 U AN3 SAN3

⎤ TA N 1 L A N 1 x A N 1 G A N 1 TA N 2 L A N 2 x A N 2 G A N 2 ⎦ A N 3 TA N 3 L A N 3 x A N 3 G A N 3 ⎡ ⎤ U N Lanes N L I nter section L O S N L 1.07 1 L O S − in − H C M = ⎣U N Lanes N S I nter section L O S N S 0.892 1 L O S − in − H C M ⎦ U N Lanes N R I nter section L O S N R 0.453 0 L O S − in − H C M − p−A−→ N1 − p−A−→ N2 − p−−→

where the NL, NS, and NR represent northbound left turn, northbound straight, and northbound right turn lanes. (2) Target states for the eastbound lanes According to LOS categories in HCM, we build the target state matrix, where the threshold of C level LOS is V /C  0.6. The target state matrix B N :

B N = B N11 B N12 B N13 where, B N11 = (U N11 Lanes N L Intersection L O S N L (0, 0.6] 0 LOS-in-HCM) B N12 = (U N12 Lanes N S Intersection L O S N S (0, 0.6] 0 LOS-in-HCM) B N13 = (U N13 Lanes N R Intersection L O S N R (0, 0.6] 0 LOS-in-HCM) Let

p−→ X N = U X N SX N − X N TX N L X N x X N G X N For matrix equation X N AN =B N

526

6 Applications of Error Logic ⎡

U AN1 SAN1 − − → U X N S X N p X N TX N L X N x X N G X N ⎣U A N 2 S A N 2 U AN3 SAN3

⎤ − → p− A N 1 TA N 1 L A N 1 x A N 1 G A N 1 − − → p A N 2 TA N 2 L A N 2 x A N 2 G A N 2 ⎦ − → p− A N 3 TA N 3 L A N 3 x A N 3 G A N 3

= B N11 B N12 B N13

where, p−B→ TB N 1 L B N 1 x B N 1 G B N 1 ) B N11 = (U BN 1 S BN 1 − N1 − TB N 2 L B N 2 x B N 2 G B N 2 ) B N12 = (U BN 2 S BN 2 p−B→ N2 TB N 3 L B N 3 x B N 3 G B N 3 ) p−B→ B N13 = (U BN 3 S BN 3 − N3  p−X→ ∧− p−A→ =− p−B→ , In equation X N A N =B N , there exist S X N i ∧ S A N i = S BN i , − Ni Ni Ni TX N i ∧ T A N i = TBN i , L X N i ∧ L A N i = L BN i , x X N i ∧ x A N i = x BN i , and G X N i ∧ G A N i = G BN i . (3) Solving equation (a) U X N i ∧ U A N i = U BN i , i.e., U X N i ∧ U = U , U X N i = U and the domain is the designated urban transportation, U X N i = U . (b) S X N i ∧ S A N i = S B N i , i.e., S X N i ∧ {Lanes N L } = {Lanes N L }, S X N i = {Lanes N L }. ∧− p−A→ =− p−B→ , i.e., − p−X→ ∧ { Intersection } = { Intersection }, − p−X→ (c) − p−X→ Ni Ni Ni Ni Ni ={ Intersection } . (d) TX N i ∧ T A N i = TBN i , i.e., TX N i ∧ {L O S N = V /C} = {L O S N = V /C}, TX N i = {L O S N = V /C}. (e) G X N i ∧ G A N i = G BN i , i.e., G X N i ∧ { LOS-in-HCM } = { LOS-in-HCM }, G X N i = { LOS-in-HCM }. (f) x X N 1 ∧ x A N 1 = x BN 1 , i.e., x X N 1 ∧ 0 = 0, then x X N 1 = 0, since it is nonerroneous, the current state is kept. (g) x X N 2 ∧ x A N 2 = x BN 2 , i.e., x X N 2 ∧ 0 = 0, then x X N 2 = 0, since it is nonerroneous, the current state is kept. (h) x X N 3 ∧ x A N 3 = x BN 3 , i.e., x X N 3 ∧ 0 = 0, then x X N 3 = 0, since it is nonerroneous, the current state is kept. Then the solutions are as follows: X N 1 = (U Lanes N L Intersection L O S N L 0.466 0 LOS-in-HCM) X N 2 = (U Lanes N S Intersection L O S N S 0.458 0 LOS-in-HCM) X N 3 = (U Lanes N R Intersection L O S N R 0.199 0 LOS-in-HCM) For the intersection in discussion, the LOSs of northbound left turn, northbound straight, and northbound right turn can be kept current state. The solutions obtained from the above process are as follows:

6.2 Computerized Error Logical Reasoning

⎡ U ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎢U ⎢ ⎣U U

LanesW L LanesW S LanesW R Lanes E L Lanes E S Lanes E R Lanes SL Lanes SS Lanes S R Lanes N L Lanes N S Lanes N R

I nter section I nter section I nter section I nter section I nter section I nter section I nter section I nter section I nter section I nter section I nter section I nter section

L O SW L L O SW S L O SW R L O SE L L O SE S L O SE R L O SSL L O SSS L O SS R L O SN L L O SN S L O SN R

527

(0, 0.6] 0.56 0.12 (0, 0.6] (0, 0.6] 0.453 (0, 0.6] (0, 0.6] 0.453 0.466 0.458 0.199

0 0 0 0 0 0 0 0 0 0 0 0

⎤ L O S − in − H C M L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎥ ⎥ L O S − in − H C M ⎦ L O S − in − H C M

According solution-finding process for the four-direction lanes, for the first-layer (in terms of target system) decision set, the target state is to lower the V /C for right turn lanes of eastbound, westbound, and southbound directions and the straight lanes of eastbound and southbound directions to be within the range of (0, 0.6] which is the C level LOS. This example is an application of searching for error-elimination (avoidance) solution set at the first layer of the target system. Regarding the first-layer solution set, properties for the issue of interest can be further divided (or decomposed) to define second-layer target state and current matrices and to find solutions for the newly-constructed matrix equations. Finally, the opportune solution sets of the issue in discussion are confirmed and presented to inform decision. In our error logical transformation system (Chap. 4), six principal transformations and their inverse transformations can be employed to solve for matrix equations, we can have more satisfactory solutions, which can be ordered based on given criteria. Since this example is just a demonstration on how to use error matrix equations to tackle practical issues, we admit that finding satisfactory solutions for an intersection of a complicated urban transportation system does not necessarily reduce traffic congestion at the system level because any action on the element or subsystem of a system may cause new states or imbalance. Of course, we can consider using error system theory to build system-level model and address the issues accordingly.

6.2 Computerized Error Logical Reasoning As a recently developed original theory, the objective of error theory is to employ theories, approaches, and research literature in mathematics and logic to examine various errors in different fields and to consequently establish a unique theoretic system for error studies. In recent years, error theory has been recognized as one of the five emerging fundamental subjects related to artificial intelligence developed in China. Since error theory’s advent in 1980s, it had developed “15-6-3” and “error matrix” approaches. These approaches laid the theoretic foundation for the expert

528

6 Applications of Error Logic

system that employs computer to help eliminate and remove errors in socioeconomic systems. In the development and application of artificial intelligence (AI), expert system is a pivotal part and knowledge database is the core and foundation for expert system. The research contents of expert system mainly include acquisition of knowledge, expression of knowledge, and application and handling of knowledge. The commonly-used expression approaches of knowledge in AI include: predicate logic representation, production rule representation, semantic net, frame knowledge representation, object-oriented knowledge representation, relation model, and ontologybased knowledge representation. Knowledge expression, to a great extent, determines whether the problems can be solved or program can be executed. It is necessary to choose and design a practical expression model as correct knowledge structure is extremely critical. The factors need to considered include but not limited to: (1) if the expression is accurate and efficient; (2) if the expression rules are simple, understandable, and applicable; (3) easy to carry knowledge expansion and maintenance; (4) easy to realize knowledge access and program execution; (5) support top-down step-wise refinement design principles; (6) meet the universal thinking models of human beings. The error logic and error matrix in error theory provide a new model for knowledge expression and reasoning in AI. This section offers the computerization for knowledge expression based on error logic and error matrix in error theory. We also provide a case in transportation management that demonstrates the computerization process for the knowledge reasoning using error logical equations.

6.2.1 Error Theory-Based Expert System Structure Error-eliminating expert system is to analyze the meaning of each unit in an error matrix using identifiers. Then, computer is used to find the solutions for error matrix based on error-eliminating methods in error matrix. Finally, knowledge reasoning is achieved through solving the transformation matrix equations.

6.2.1.1

Process of System Implementation

Figure 6.3 provides the flowchart of system implementation. (1) Inputs: according to the structure and storage format of error logic, existing state matrix and expected state matrix are produced by obtaining data inputs from database or user interaction terminals; (2) Applying rule: rules in rulebase are called; new rules will be generated by users in model when rules in current rulebase do not meet the requirements of users and the newly generated rules will then be added to the rulebase;

6.2 Computerized Error Logical Reasoning

529

Fig. 6.3 Error matrix-based error-eliminating expert system

(3) Analyzing error matrix: character strings stored in each unit in error matrix are split using systematic reasoning mechanism, and then the stored contents are identified and analyzed according to the identifiers; (4) Computing and reasoning: having done the analysis on the error matrix, the error matrix solution-finding methods in generic matrix multiplication are used to compute and deduce the results.

6.2.1.2

Database

The database in error-elimination expert system is mainly used to store status information for the convenience of writing/reading. The major item in a database is the access table, which has a “name” column differentiating the status information. “Position” in the table is used to differentiate the current state matrix and expected state matrix that are stored in the storage structure of the model. With the 7 factors in error matrix, the table is stated in Fig. 6.4. When calling the data in the table, the system first obtains the appropriate row in the table by matching the “Name”; “Position” is then used to determine if the current row should be allocated to current state matrix or expected state matrix; finally the 7 factors are referred to read the character string into the model.

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6 Applications of Error Logic

Fig. 6.4 Information on storage state of database

6.2.1.3

User Interaction

User interaction is another form to input information into model and construct the current state matrix or expected state matrix. The key step in the process is to translate an object in reality into error matrix and decompose object into the 7 factors that constitute error matrix.

6.2.1.4

Rulebase

Rule is used to store the criteria for deducing error function value in which error function value is derived through applying logical judgment of rule on property (or attribute) value of object in discussion. And rulebase is a database that is used store those rules, which is connected to the system when it is activated and the pertinent rules are called accordingly. 1. Construction of rulebase: Rulebase is used to store rules. Two dimensions are used to profile the rules with the first one describing the practical meaning of rules, i.e., “qualitative description” and the second one describing the mathematical meaning of rules,i.e., “logical expression”. “Qualitative description” works as an identifier to differentiate rules, which has uniqueness without permitting duplication of name. And “logical expression” is the identifier that translates the qualitative description of rules into the logical judgment directly recognized by computer. Simply put, practical meaning of rule is to intuitively label the rule; logical meaning of rule

6.2 Computerized Error Logical Reasoning

531

Fig. 6.5 Information on storage status of database

is used by computer to conduct logical reasoning. Although rules are stored in a format of character string in rulebase, they also have particular forms because they represent a realistic rule. Therefore, in the system design, the “rule type” is used to express the rules. Please refer to Fig. 6.5. 2 New rules: As there are numerous rules in the nature, it is not possible to store all rules in the rulebase. In this rulebase, a rule-generating mechanism is employed to generate new rules that are pertinent to the objects of interest. User can create a piece of rule in the rulebase when current rules can not meet needs and requirements of users. The newly created rule is stored in the rulebase for future use.

6.2.1.5

Model

Model provides the foundation for qualitative reasoning and quantitative computation of the system. In this system, operation of error matrix equation is used to conduct logical reasoning. 1. Error matrix equation Error matrix can not only represent object of interest but also express the transformation connectives proposed in error logic. By combining decomposition transformation, similarity transformation, increase transformation, displacement transformation, destruction transformation, and unit transformation, error matrix can be constructed using error logical variables. For given matrices A and B, X is the method of transforming A into B, then X is called the transformation matrix. By solving the error matrix equations, the specific methods for eliminating errors can be found. In the error matrix-based error-eliminating expert system, system attempts to acquires the meaning of the unit in each row of matrix (i.e., the object of interest) by analyzing the storage type corresponding to particular factor in the 7 factors that constitute error matrix. 2. Storage type of error matrix equation Based on the structure of error matrix, an object can be divided into 7 factors. As each factor has it own specific meaning, the storage types of different factors are different. Therefore, for two matrices representing different meaning, storage types for all factors in the two matrices are of course entirely different. Taking property (attribute) value L(t) as an example, in the current state matrix, L(t)

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6 Applications of Error Logic

is an observable and determined value; in the expected state matrix, L(t) is an expected value and not a determined value, which is a range. Therefore, based on the meanings represented by different factors in different matrices, we introduce 6 storage type in our error-eliminating expert system. Because all types are stored in the form of character string, a particular identifier is added to each type in order to clarify the corresponding type of the content stored. (1) Numeric type Numeric type is used to store the data having types of int and f loat. Numeric data are inputted in the format of character string and are then transformed into int or f loat type that is identifiable for computer through system analysis and translation. Its format is: “D: stored content”. For example, “D: 234.68” represents that “234.68” is stored a numeric type. (2) Character type Character type is defined to store character string. Its format is: “S: stored content”. Character data are inputted in the format of character string and are then transformed into the type obtained through using the same principles in generating other self-defined storage type. For example, “S: adg78” represents that “adg78” is stored in a type of character string. (3) Matrix type Matrix type is used to store matrix. Being different from other storage type, a second identifier is added behind the first identifier because the contents in a matrix also have certain format. The second identifier is used to depict the type for the contents stored in the matrix, which are integer , f loat, and string. Its format is: “S: A : (2nd identifier) {{. . . . . . } . . . . . . } ” For example, “S: A : ( f loat) {{3.124, 3.258, 3.168}, {1.248, 6.354, 87.65}} ” represents that the store matrix is a 2 × 3 matrix containing f loat type elements noted by   3.124 3.258 3.168   1.248 6.354 87.65 (4) Function type Function type exists depending on the existence of the rule type and data type. It is self-determined by the stored data in the data type location and rules in the rule type location. In classic error function, the result is 0 if data meet the rules and 1 if the data are against the rules. In the fuzzy error function, the degree of error is measured by the deviation between the data and the rules. The fuzzy error function value is obtained by dividing the deviation by the rule if the deviation is less than the value provided by the rules and is 1 otherwise. Its format is: (1) classic error function:“F : {0, 1}”; (2) fuzzy error function:“F : [0, 1]”. For example, in the case of classic error function, the error function value is F : 1 if data D : 528 and rule G :> 925.1. In the case of fuzzy error function, the error function value is F : 0.8311124 if the same data and rule apply.

6.2 Computerized Error Logical Reasoning

533

(5) Rule type In order to deal with error matrix, new “rule type” is defined here. However, rule-type data alone can not provide meaningful expression. Rule-type data must be jointly used with numeric and function type data where rules in ruletype data provide judging conditions, numeric data are used to determine if the conditions are met, and final results are returned by the function. Its format is: “identifier G: & data” and & is the symbol representing judgment. Judgment symbols include: >, ≤, 925.66, its meaning is expounded as follows when considering the judging rule of property (attribute) value: (a) when property (attribute) value ≤ 925.66, the returned result is 0; (b) and when property (attribute) value < 925.66, the returned function value needs to confirm the function type, i.e., the returned result is a value belonging to (0, 1] for fuzzy error function and 1 for classic error function. (6) Range type Range type is defined for storing the property (attribute) value in expectation matrix, which is a new type. As the property (attribute) value in expectation matrix is a expected value having uncertainty, therefore, its value is a range instead of a particular point. Its format is: “R : left limit, right limit”. For example “R : (0.7, 8], its meaning is to store a range from 0.7 to 8 but not including 0.7. 3. System analysis of character string In the expert system, all the inputs are character spring whether the data are retrieved in database or the data are inputted in system by converting thing into 7 factors in error matrix through user-system interaction. The meaning represented by error matrix can not be interpreted without building the analysis mechanism for character string. System analysis of character string can well address this issue, which is the process that user translates the inputted natural language into the data representing the meaning the 7 factors in error matrix intending to deliver. And the most important thing is that those data must be understood by computer program. The procedure is that: (1) system obtains a character string from model; (2) the character string is then split into standalone characters; (3) based on the first character(identifier), the type the character string intending to express is identified; (4) the structure for the stored contents is identified; (5) new character string or character string array is formed by restructuring the characters according to the storage structure; (6) finally they are translated into the actual meaning that those character strings intend to express. The flowchart is shown in Fig. 6.6. Taking rule-type character string “G :≤ 0.6” as an example, after getting this character string from matrix, reasoning machine splits this character string into “G”, “:”, “ a ⎨e f (G, u) = 1 si = a or ⎪ ⎩ (si −a) si < a e f (G, u) =

 1 α(si − a)− β si ≥ a α(a − si )− β si < a 1

(6.11)

(6.12)

α is positive constant and β ≥ 1. (6) Judging rules with range approximation indicator Suppose that judging rules of object system presents range approximation indicator for certain state si , where si ∈ / [a, b], and error value gets larger as si approaches to [a, b]. Under this circumstance, error function can take the forms of fuzzy error function and non-negative error function. While for si ∈ (−∞, +∞), following functions can be used: ⎧ (b−si ) ⎪ si > b ⎨e f (G, u) = 1 (6.13) a ≤ si ≤ b ⎪ ⎩ (si −a) si < a e (7) Judging rules with qualitative indicator Suppose that judging rules of object system include qualitative indicator, error function can take the forms of classic error function:  1 Ga f (G, u) = (6.14) 0 G⇒a

6.4 Error Transmission in a System 6.4.1 Critical Factors for Error Transmission in a System Carrier-borne error in a system is transmitted (or transferred) through structural network to certain element, thereby causing intrinsic function of a system to deviate from its objective function. The transfer of error in a system involves three critical factors, namely, error source, error flow, and error carrier (or vector).

6.4 Error Transmission in a System

549

1. Source of errors: it rises when system elements or their interactions produce non-conforming outcomes or behaviors under given rules; it exists within the system and exerts impacts on the system behaviors and its environment as well, which leads to uncertainty for the system to produce desired function. 2. Error flow: Error in a system can be regarded as a kind of energy or force resided in the error source, which can be measured by an error function and this kind of energy or force is produced due to the state change of internal and/or external environments of the system and is transmitted and spread via system structure by bonding with its vector or carrier. The error-related forces bonded to each functional node of the system are enforced, transformed, and lessened during the transmitting process due to the connections among those tightly coupled interacting nodes, which consequently influence the error states of the whole system. We call this type of error-related forces error flow. Error flow has the characteristics of liquidity, dependency, conductivity, and additivity. 3. Carriers or vectors for errors With the state change of internal and external environments, error flow coming out of error sources is transmitted along system structure by bonding to certain tangible and intangible thing(s) which are called the carrier or vector for errors (or expression form). Based on the existence form of error carriers, they can be categorized into material carrier, capital carrier, technology carrier, information carrier, and behavior carrier, etc. The errors bonded to the carriers mentioned above become capital error flow, technology error flow, information error flow, and material error flow, etc. The above-mentioned examples intend to address how different indicators might exert impacts on error functions. In practice, error functions could have millions of forms. Therefore, the construction and form adoption of error function depend on specific issues and situation. We consider not only the judging rules but also actual needs. The major principle for depicting error function is to objectively evaluate to what extent the object of interest violates the judging rules.

6.4.2 Expression Form for a System Suppose that U (t) ⊆ U1 × U2 × · · · × Ui × · · · Un , n-nary object μ(t) = (μ1 (t), μ2 (t), . . ., μi (t), . . ., μn (t)) stands for the object set defined within U (t), μi ∈ Ui i = −−→ (1, 2, . . ., n), the ith object μi = (Ui (t), Si (t), pi (t), Ti (t), L i (t)), where Si (t) is −−→ thing, pi (t) is spatial location, Ti (t) is property or attribute, and L i (t) are the values of Ti (t). For the ease of studying the transmission mechanism of errors in a system, the subsystem SSi at time point t is represented by a trinary object SSi = (μ1 (t), −−−→ μ2 (t), μ3 (t)), μi j = (Ui j (t), Si j , pi j (t), Ti j (t), L i j (t)), where Si j represents thing that has three elements including the state si (t), output relationship flow R F Oi (t) −−→ and error function X i (t); p j (t) denotes the spatial location of thing; T j (t) stands for

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6 Applications of Error Logic

Fig. 6.11 Input and output relationship flows

property vector which mainly includes capital, technology, materials, information, energy, and human resources, etc.; and L j (t) represents the values of T j (t).

6.4.3 Error Transmission Function in a System In general system theory, the interconnections among system elements are bonded by relationship flows such as information flow, material flow, capital flow. As system’s states or behaviors are determined by its structure, the process of realizing system’s function is the process include three primary phases: (1) system receives inputs in the form of materials, energies, and information from external environments; (2) those inputs are processed and transformed; (3) and outputs and outcomes are generated and fed back to the environment in the form of new type of materials, energies, and information. For any node or interacting points of different elements, there must exist input and output flows. Figure 6.11 demonstrates that node μ j receives input R Fi j (t) from node μi at time point t and the output R F jk (t) is obtained through transformation within node μ j , which is then, as an input, fed into node μk . In this process, as far as the μ j is concerned, R Fi j (t) is input relationship flow while R F jk (t) is output relationship flow. Many system-related research have shown that there exists certain functional relationship between system behaviors and system internal structure and system inputs. In our case,the function here can be represented by: R F jk (t) = Ψ j (R Fi j (t), s j (t)) or in a different form as s j (t) = ϕ j (R Fi j (t), R F jk (t)). Error occurred in an element of a system generally is attached to certain relationship flow and transmitted through system structure. Because there exists functional relationship among the connected nodes in a system with errors and matching relationship between heterogeneous errors, different errors in transmission may exert impacts on each other, which consequently changes the flow volume and characteristics of those affected errors. If the errors in input relationship and output relationship can be easily identified via observation while errors in the system’s element are latent (i.e., it is hard to observe or unobservable), the element’s state can be obtained by solving for relationship flow functions and error values can be obtained by building corresponding error functions. In the following example, it is assumed that system is composed of three subsystems SS1 , SS2 , and SS3 in a system with series structure. Errors, by attaching to capital, technology, and materials, etc., become error flows transmitting in SS1 , SS2 , and SS3 . As the transmission of relationship flows follows chronological order, it is assumed that time spent for errors in SS1 and SS2 transmitting to SS2 and SS3 are T12 and

6.4 Error Transmission in a System

551

T23 , respectively (Fig. 6.12). The error values in subsystems SS1 , SS2 , and SS3 are X 1 , X 2 , and X 3 , respectively (Fig. 6.12). 1. Error values of each subsystem from initial time to T12 It is assumed that the system of interests gets a signal(action) at the initial time and the error in the system hasn’t been transmitted along certain structure due to the existence of delay. The error values in the system X i (i = 1, 2, 3) functions with respect to their states and corresponding rules. X q = f q (sq , G q ), where G q = (G i1 ,G i2 , . . ., G in represents the rule vector for judging errors system’s states. 2. Error values after T12 As SS1 is the 1st subsystem in the system with series structure, no other error source was fed into SS1 after T12 and therefore SS1 kept its current state. The error in SS1 was transmitted to SS2 via particular path in the system after T12 . The output relationship flow R F O1 of subsystem SS1 is a function of SS1 ’s state and output relationship flow R F O0 of its environment, i.e., R F O1 = ϕ1 (R F O0 , s1 ). As the output relationship flow R F O1 was transmitted to SS2 , the error value of SS2 was not a result from a simple mechanic addition between its original state and the input from R F O1 but a new error value (could be increased, decreased, or tranformed values) generated from system’s state superposition and forced coupling interactions, which is a function of output relationship flow SS1 and state of SS2 as well as the rules applied in it. Hereby we have X 2 = f 12 (R F O1 , s2 , G 2 ). Similarly, the new error value of SS3 after time T12 + T23 is X 3 = f 23 (R F O2 , s3 , G 3 ). And the output relationship flow of SS2 can be represented by R F O2 = ϕ2 (R F O1 , s2 ). 3. Error values of each subsystem after T12 + T23 Since there is no other input from the environment and SS1 and SS2 are 1st and 2nd subsystems, respectively in the assumed series system structure, the error values in SS1 and SS2 did not change after time T12 + T23 , i.e., X 1 = f 1 (s1 , G 1 ) or s1 = g1 (X 1 , G 1 ) X 2 = f 12 (R F O1 , s2 , G 2 ) or s2 = g12 (R F O1 , X 2 , G 2 ) The error value of SS3 is X 3 = f 23 (R F O2 , s3 , G 3 ) which can also be represented by the function of s3 = g23 (R F O2 , X 3 , G 3 ). As the errors had finished the transmission through the whole system structure, the objective function of the system was realized by producing output relationship flow of SS3 by subsystem SS3 , which is denoted by R F O3 = ϕ3 (R F O2 , s3 ). According to the definition of function, the final error value of the system in discussion can be a function of initial input signals (relationship flow from environment) and rules applied in all the relevant hierarchies of the system). X = f (R F O3 , G O3 ) = f (ϕ3 (R F O2 , s3 ), G O3 ) = f (ϕ3 (ϕ2 (R F O1 , s2 ), s3 ), G O3 ) = f (ϕ3 (ϕ2 (ϕ1 (R F O0 , s1 ), s2 ), s3 ), G O3 ) = f (ϕ3 (ϕ2 (ϕ1 (R F O0 , g1 (X 1 , G 1 )), g12 (ϕ1 (R F O0 , g1 (X 1 , G 1 )), X 2 , G 2 )), g23 (ϕ2 (ϕ1 (R F O0 , g1 (X 1 , G 1 )), g12 (g1 (X 1 , G 1 ), X 2 , G 2 )), X 3 , G 3 )), G O3 ).

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6 Applications of Error Logic

Fig. 6.12 Error transmission in a system with series structure

Fig. 6.13 Error transmission in an enterprise technology innovation system

6.4.4 Application Example for Concept of Error Transmission This section offers a simple example of applying concept of error transmission in enterprise innovation system. An lathe manufacturer signed a technology transfer agreement with university A A A to buy out the patent for a new type of tile press machine with one million RMB (Chinese Yuan). Thereafter, the manufacturer spent 800, 000 RBM to acquire necessary equipment for pilot production of the tile press machine. During the pilot production process, it took 460,000 RMB for the direct costs for production and overhead costs. Due to the existence of flaws in the new technology, the company made revisions and adjustment on the structure and carried out improvement on the production process. The company invested 260, 000 RMB in promotion and produced a batch 20 machine for sale. However, it turned out that most of the pieces were returned for reparation or for refund due to the technological flaws, which ultimately caused the failure of producing this type of product. The failure incurred a loss of three million RMB. This example actually demonstrates an enterprise innovation system having a series structure consisting of R&D subsystem, pilot production subsystem, and sales subsystem (Fig. 6.13). The objective function of this system in discussion is to provide satisfied products for customers. The attainment of the objective function depends on whether the output relationship flows from each of the three subsystems can conform to the rules defined in them. Table 6.6 provides a simple illustration on the rules for judging errors in the output relationship flows. There exists error(s) in the system in discussion since the system did not produce desired results, i.e. satisfied products. According to the system error function mentioned above, the error(s) in the system was determined by the patented technology, error values and rules for judging errors in three subsystems, and the rules for

6.5 Application of Error Logic in Decision Support System for Nanquan Referees Table 6.6 Rules for judging errors in the output relationship flows Subsystem Vector for errors Output relationship flow R&D subsystem

Technology

Technology flow

Pilot production subsystem Sales subsystem

Product

Product flow

Goods for sale

Product flow

553

Rules for judging errors in output relationship flows Mature advanced technology in relevant industry Quality product Customer are satisfied with the product

judging error in the final output relationship flow. With further analysis, the primary reason is that the R&D did not conduct a comprehensive evaluation on the maturity and produciability of the acquired technology, which caused the technology-borne error (flaws) to transmit (transfer) to pilot production subsystem. Although pilot production subsystem conducted improvement on the production process to catering to the needs of new product, the company still produced defective products due to the intrinsic flaws and complexity of the technology. Moreover, with the hasty introduction of new technology, the company did not update their inspection procedures and standards, which caused the defective product-borne error to be transmitted to sales subsystem and consequently to the sites of customers.

6.5 Application of Error Logic in Decision Support System for Nanquan Referees When referees make judgment if there exists foul play in a sport, the referees have to identify the incorrect movements of athlete and consequently impose penalty. For the foul identification and imposition of penalty in incorrect actions of athletes in sports like soccer, basketball, volleyball, rhythmic gymnastics, and martial arts, identification of incorrect actions and wrong movements demands significant work. With the development of high technology, computers and RFID (Radio Frequency Identification) are widely used to assist “computerized referee” to impose penalty, which makes the judgment of all kinds of matches more wonderful, scientific, and impartial. However, the judgment on the artistic proficiency and performance of the movements of athletes in the difficulty beauty sports such as rhythmic gymnastics, diving, and martial arts is conducted by multiple referees where the subjectivity of referees created many misjudgment cases. This challenges the justice of this kind of sport. Adoption of “computerized referee” system will significantly advance the sustainable and healthy development of difficulty beauty sports. With the accelerated improvement and upgrading in computer hardware and the ever-upgrading in

554

6 Applications of Error Logic

tracking techniques and algorithm for human movements, video-based and sensorbased tracking systems have remarkably improved the acquisition of high-quality kinematic data. The extensive investment in R&D of ultra-small sensors and wireless data transmission technologies renders sensor-based human movement tracking system successful. Video-based human movement tracking techniques include the 2D or 3D tracking techniques in monocular or multicular video , which uses the image sequences obtained using 500 frames/s high-speed camera. In two consecutive images, the position of human joint can only change within a very small range. Therefore, the human movement can be accurately tracked and three-dimensional reconstruction can be performed. By continually tracking and positioning human’s movement, the movement trajectory can recorded and kinematic variables such as the 3-D coordinates of joint position and joint angle of human can be obtained. In order to use “computerized referee” to identify and score the foul play or incorrect movement, theory and method in error logic are employed to establish computervision-based model. In martial arts, with continuous improvement and modification in rules of competitive Wushu Taolu (Wushu artistic performance forms), the variation in movement and incorrect movements for each Taolu have been clearly defined and clarified. In the 2005 version of Competition Rules for Competitive Wushu, the scoring rules on the movement quality (Group A) dictate that judgments on movement results only have correct and wrong states with point deduction for wrong movements and no point deduction for correct movements. Each athlete must exercise Taolu in certain pre-determined sequences. This section uses the martial art-Nanquan as object.

6.5.1 Description on the Error Object of Nanquan in Error Logical Matrix 6.5.1.1

Feature Description of Athlete in Error Logical Matrix

In martial art Nanquan, suppose that athlete A has body height h 0 , shoulder height h 1 , thigh length h 2 , calf length h 3 , foot length h 4 , the following is the object matrix:  “optional  “optional  B A = “optional “optional  “optional

N anquan  N anquan  N anquan  N anquan  N anquan 

“athlete A “body height  “athlete A “shoulder height  “athlete A “thigh length  “athlete A “cal f length  “athlete A “ f oot length 

 h 0  h 1  h 2  h 3  h4

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

555

Fig. 6.14 Court for Wushu Taolu competition

6.5.1.2

Construction of 3-D Coordinates for Nanquan

Optional Nanquan competition performs on carpet with a court of 14 M ×8 M. Please refer to Fig. 6.14. In the court, one corner point is set as the origin (0, 0), the length direction of court is x coordinate and the width direction is y coordinate, and vertical direction is z coordinate, x0 = 14 m, y0 = 8 m, and according to the maximum possible height of jumping, we set z 0 = 5 m. Please refer to Fig. 6.15.

Fig. 6.15 Court for Wushu Taolu competition

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6 Applications of Error Logic

6.5.1.3

Athlete Feature Description in 3-D Coordinates

In order to conduct formal description on the Nanquan optional exercises, the 3-D coordinates for key joints and body parts of athlete are labeled. This session mainly demonstrates the 3-D coordinates for head, shoulders, hands, buttocks, knees, thighs, calves, hip joints, heels, and foot toes. 1. 3-D coordinate for head In this case, we mainly care about the distance from the top of the athlete’s head to the court ground, i.e., point(s) on Z axis. Suppose that the top of the athlete’s head is in the same plane, any point in this plane is labeled using the same coordinate point, and coordinate for the central point of head is defined as (xtd , ytd , z td ) ; 2. 3-D coordinate for shoulders The 3-D coordinate for the left shoulder is (xl jb , yl jb , zl jb ) and the right shoulder is (xr jb , yr jb , zr jb ); 3. 3-D coordinate for palms In the 3-coordinate system, suppose that the coordinate expression for distal phalanx is (xsh1 , ysh1 , z sh1 ), the coordinate expression for capitate is (xsh2 , ysh2 , z sh2 ), and the coordinate expression for the center of the palm is (xsh3 , ysh3 , z sh3 ), when using right or left hand, r and l are added as prefix in the above coordinate expressions. Per the requirement of actual studies, it is, sometimes, necessary to portray the planes in which the palms reside. From the above three coordinate points, a plane can obtained and any point must satisfy the following equation:    x − xsh1 y − ysh1 z − z sh1    xsh2 − xsh1 ysh2 − ysh1 z sh2 − z sh1  = 0   xsh3 − xsh1 ysh3 − ysh1 z sh3 − z sh1  The corresponding plane equation for the left palm is:    x − xlsh1 y − ylsh1 z − zlsh1   xlsh2 − xlsh1 ylsh2 − ylsh1 zlsh2 − zlsh1  = 0   xlsh3 − xlsh1 ylsh3 − ylsh1 zlsh3 − zlsh1  The corresponding plane equation for the right palm is:    x − xr sh1 y − yr sh1 z − zr sh1   xr sh2 − xr sh1 yr sh2 − yr sh1 zr sh2 − zr sh1  = 0   xr sh3 − xr sh1 yr sh3 − yr sh1 zr sh3 − zr sh1  4. 3-D coordinate for fingers For example, in the Jump flying kick with the right leg landing (Tengkong Feijiao

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TKFJ), when flapping the foot top, the flapping position is in a plane formed by closed finger. Suppose that (xsz , ysz , z sz ) represents any point in the above plane, the coordinate for the left fingers is (xlsz , ylsz , zlsz ) and the coordinate for the right fingers is (xr sz , yr sz , zr sz ); 5. 3-D coordinate for buttock Suppose that a selected point on left buttock is (xltb , yltb , zltb ); a selected point on right buttock is (xr tb , yr tb , zr tb ), the midpoint of middle line of buttocks is (xmtb , ymtb , z mtb ). In actual application, when describing the whole plane positioned by buttock, it is assumed that the plane is constructed by randomly choosing three points on the buttocks, one of the points must meet the following equation:    x − x1 y − y1 z − z 1    x2 − x1 y2 − y1 z 2 − z 1  = 0   x3 − x1 y3 − y1 z 3 − z 1  6. 3-D coordinate for hip joint Suppose that a selected point of the left hip joint is (xltg , yltg , zltg ); a selected point of the right hip joint is is (xr tg , yr tg , zr tg ); 7. 3-D coordinate for inner thighs Suppose that the inner side of thigh is on a plane, a selected point of the left inner thigh is (xldt2 , yldt2 , zldt2 ); a selected point of the right inner thigh is (xr dt2 , yr dt2 , zr dt2 ); 8. 3-D coordinate for inner calves Suppose that the inner side of calf is on a plane, a selected point of the left inner calf is (xlxt2 , ylxt2 , zlxt2 ); a selected point of the right inner calf is is (xr xt2 , yr xt2 , zr xt2 ); 9. 3-D coordinate for knees Suppose that the coordinate for the left knee is (xlxg , ylxg , zlxg ) and the coordinate for the right knee is (xr xg , yr xg , zr xg ); 10. 3-D coordinate for heels Suppose that three points are selected on the heel, they are (x jg1 , y jg1 , z jg1 ), (x jg2 , y jg2 , z jg2 ), and (x jg3 , y jg3 , z jg3 ). From the above three coordinate points, a plane can be obtained and any point must satisfy the following equation ïijŽ    x − x jg1 y − y jg1 z − z jg1   x jg2 − x jg1 y jg2 − y jg1 z jg2 − z jg1  = 0   x jg3 − x jg1 y jg3 − y jg1 z jg3 − z jg1  The corresponding plane equation for the left foot bottom is:    x − xl jg1 y − yl jg1 z − zl jg1   xl jg2 − xl jg1 yl jg2 − yl jg1 zl jg2 − zl jg1  = 0   xl jg3 − xl jg1 yl jg3 − yl jg1 zl jg3 − zl jg1 

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6 Applications of Error Logic

The corresponding plane equation for the right foot bottom is:    x − xr jg1 y − yr jg1 z − zr jg1   xr jg2 − xr jg1 yr jg2 − yr jg1 zr jg2 − zr jg1  = 0   xr jg3 − xr jg1 yr jg3 − yr jg1 zr jg3 − zr jg1  11. 3-D coordinate for ankles Suppose that the coordinate for the left ankle is (xl j h , yl j h , zl j h ) and the coordinate for the right ankle is (xr j h , yr j h , zr j h ); 12. 3-D coordinate for foot toe tips Taking the thumb toe as an example, suppose that the coordinate for the left great toe tip is (xl j j , yl j j , zl j j ) and the coordinate for the right great toe tip is (xr j j , yr j j , zr j j ); 13. 3-D coordinate for insteps Suppose that the coordinate for the left instep is (xl jm , yl jm , zl jm ) and the coordinate for the right instep is (xr jm , yr jm , zr jm ); 14. 3-D coordinate for sole of foot Suppose that the coordinate for the distal phalanx of great toe is (x j z1 , y j z1 , z j z1 ), the coordinate for the distal phalanx of little toe is (x j z2 , y j z2 , z j z2 ), the coordinate for the central point of the sole of foot is (x j z3 , y j z3 , z j z3 ). From the above three coordinate points, a plane can obtained. The corresponding plane equation for the sole of left foot is:    x − xl j z1 y − yl j z1 z − zl j z1    xl j z2 − xl j z1 yl j z2 − yl j z1 zl j z2 − zl j z1  = 0   xl j z3 − xl jg1 yl j z3 − yl j z1 zl j z3 − zl j z1  The corresponding plane equation for the sole of right foot is:    x − xr j z1 y − yr j z1 z − zr j z1   xr j z2 − xr j z1 yr j z2 − yr j z1 zr j z2 − zr j z1  = 0   xr j z3 − xr j z1 yr j z3 − yr j z1 zr j z3 − zr j z1 

6.5.2 Error Logic-based Object System of Nanquan Movement Rules 6.5.2.1

Error Items and Scoring Criteria in Optional Nanquan

According the judging rules for optional Nanquan, when athlete finishes the Taolu, 0.1 point will be deducted for any movement that is against criteria. The error items and scoring criteria in optional Nanquan is listed in Table 6.7.

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Table 6.7 The error items and scoring criteria in optional Nanquan Type

Movements

Leg techniques

Forward sweep

Errors for deduction

1st code

2nd code

Thigh of supporting leg above level

2

22

Sole of sweeping foot off carpet Sweeping leg bent Horizontal nail kick

Kicking leg not kept straight after

28

Kick No nail kick to opposite side Jumps

Tumbles

Kick in flight

Toes of slapped leg below shoulder

Tornado kick

Level

Outward kick in flight

Slap missing in flight

Cross-leg kick in flight

Swing leg below head level

3

30

4

40

with full twist to land on side

Stances

Kip-up

Lift-up on support of hands

41

Double side kick with

Kicking legs not close together

42

sole in flight

Kicking legs bent

Bow stance

Front leg not bent into near half

5

50

Squat Heel of rear foot off carpet Horse-riding stance

Upper body obviously bent forward

51

Legs not bent into near half squat Thighs below horizontal level Feet not far apart enough Heels off carpet Empty stance

Heel of rear leg off carpet

52

Rear leg not bent to near right angle Crouch stance

Rear leg not bent completely

53

Front leg bent Sole of front foot not turned inward And flat on carpet Butterfly stance

Inner side of calf of kneeling leg not

55

On carpet Inner side of heel of kneeling leg not On carpet Bent-knee stance

Bent knee on carpet

56

Buttocks not on calf of bent leg Dragon-riding stance

Knee of rear leg on carpet Front leg not bent into near half Squat

Note 0.1 points will be deducted if two errors occur in one movement

6

57

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6 Applications of Error Logic

6.5.2.2

Establishing Object System for Optional Nanquan Movement Rules

According error-eliminating theory, the object system is constructed by condition F, conclusion J , intricate features G, and objective features M G using relationship R. Without causing confusion, it is also called system in this book, which is noted by: X ({W }, F( f 1 , f 2 ), J , G, M G, R) or X (W , F,G, M G, R), or X , Y , . . . . . .. The object system for optional Nanquan movement rules is noted by X N QG , i.e., X N QG = (W , F, J , G, M G, R). The set for the object system of optional Nanquan movement rules is: {W } = {errors in optional Nanquan movements } = {u 1 , u 2 , u 3 , . . . . . ., u 32 }. They are listed as follows: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31)

u 1 = Forward sweep: thigh of supporting leg above level; u 2 = Forward sweep: sole of sweeping foot off carpet; u 3 = Forward sweep: sweeping leg bent; u 4 = Horizontal nail kick: kicking leg not kept straight after kick; u 5 = Horizontal nail kick: no nail kick to opposite side; u 6 = Kick in flight: toes of slapped leg below shoulder level; u 7 = Kick in flight: slap missing in flight; u 8 = Tornado kick: toes of slapped leg below shoulder level; u 9 = Tornado kick: slap missing in flight;; u 10 = Outward kick in flight: toes of slapped leg below shoulder level; u 11 = Cross-leg kick in flight with full twist to land on side: swing leg below head level; u 12 = Kip-up: lift-up on support of hands; u 13 = Double side kick with sole in flight: kicking legs not close together; u 14 = Double side kick with sole in flight: kicking legs bent; u 15 = Bow stance: front leg not bent into near half squat; u 16 = Bow stance: heel of rear foot off carpet; u 17 = Horse-riding stance: upper body obviously bent forward; u 18 = Horse-riding stance: legs not bent into near half squat; u 19 = Horse-riding stance: thighs below horizontal level; u 20 = Horse-riding stance: feet not far apart enough; u 21 = Horse-riding stance: heels off carpet; u 22 = Empty stance: heel of rear leg off carpet; u 23 = Empty stance: rear leg not bent to near right angle; u 24 = Crouch stance: rear leg not bent completely; u 25 = Crouch stance: front leg bent; u 26 = Crouch stance: sole of front foot not turned inward and flat on carpet; u 27 = Butterfly stance: Inner side of calf of kneeling leg not on carpet; u 28 = Butterfly stance: Inner side of heel of kneeling leg not on carpet; u 29 = Bent-knee stance: bent knee on carpet; u 30 = Bent-knee stance: buttocks not on calf of bent leg; u 31 = Dragon-riding stance: knee of rear leg on carpet;

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(32) u 32 = Dragon-riding stance: front leg not bent into near half squat; The condition set of object system for optional Nanquan movement rules is F( f 1 , f 2 ) = { f 11 , f 12 , f 21 , f 22 }. Among which, f 1 are constraints and f 1 = { f 11 , f 12 }, where f 11 is time-the applicable year of rules and f 12 is space-the country where the event happens. And f 2 are other conditions and f 2 = { f 21 , f 22 }, where f 21 is the level of technology and f 22 is the skill development of athletes. The conclusion set of object system for optional Nanquan movement rules is J ( j1 , j2 ), where j1 represents that some point is deducted and j1 represents that no point deduction occurs.

6.5.3 Computer Vision-based Error Identification Model for Optional Nanquan 6.5.3.1

Computer Vision-based Error Identification Model for Forward Sweep in Optional Nanquan

1. u 1 = forward sweep: thigh of supporting leg above level Suppose that domain U1 = forward sweep in optional Nanquan; thing S1 = supporting leg; property T1 = the 3-D coordinate of the left knee is (xlxg , ylxg , zlxg ); the 3-D coordinate of the left hip joint (xltg , yltg , zltg ); the 3-D coordinate of the right knee is (xr xg , yr xg , zr xg ); and the 3-D coordinate of the right hip joint (xr tg , yr tg , zr tg ). (1) When the support is provided by the left leg If zltg > zlxg then thigh of supporting leg is above level; while if zltg ≤ zlxg , then thigh of supporting leg is below level or parallel with court ground. Establishing error function:  0, zltg ≤ zlxg f (u 11 ) = 1, zltg > zlxg If zltg > zlxg , thigh of supporting leg is above level and the value of error function is 1; if zltg ≤ zlxg , thigh of supporting leg is below level or parallel with court ground and the value of error function is 0. (2) When the support is provided by the right leg If zr tg > zr xg then thigh of supporting leg is above level; while if zr tg ≤ zr xg , then thigh of supporting leg is below level or parallel with court ground. Establishing error function:  f (u 12 ) =

0, zr tg ≤ zr xg 1, zr tg > zr xg

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6 Applications of Error Logic

If zr tg > zr xg , thigh of supporting leg is above level and the value of error function is 1; If zr tg ≤ zr xg , thigh of supporting leg is below level or parallel with court ground and the value of error function is 0. Therefore, based the results from (1) and (2), the error function for u 1 is: f (u 1 ) = f (u 11 ) ∨ f (u 12 ). 2. u 2 = forward sweep: sole of sweeping foot off carpet Suppose that domain U1 = forward sweep in optional Nanquan; thing S2 = sweeping foot; property T2 = the plane for the sole of right foot, the plane for the sole of left foot. (1) For left sweeping foot The corresponding plane equation for the sole of left foot is:    x − xl j z1 y − yl j z1 z − zl j z1    xl j z2 − xl j z1 yl j z2 − yl j z1 zl j z2 − zl j z1  = 0   xl j z3 − xl jg1 yl j z3 − yl j z1 zl j z3 − zl j z1  Suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the sole of left food is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = yl j z2 zl j z3 − yl j z2 zl j z1 − yl j z1 zl j z3 − yl j z3 zl j z2 + yl j z3 zl j z1 + yl j z1 zl j z2 ; B = xl j z1 zl j z3 − xl j z1 zl j z2 + xl j z2 zl j z1 − xl j z2 zl j z3 + xl j z3 zl j z2 − xl j z3 zl j z1 ; C = xl j z1 zl j z2 − xl j z1 zl j z3 + xl j z2 zl j z3 − xl j z2 zl j z1 + xl j z3 zl j z1 − xl j z3 zl j z2 ; +xl j z2 yl j z1 zl j z3 − xl j z2 yl j z3 zl j z1 + D = xl j z1 yl j z2 zl j z3 + xl j z1 yl j z3 zl j z2 xl j z3 yl j z2 zl j z1 − xl j z3 yl j z1 zl j z2 . dl is the distance between sole of food and court ground: If dl > 0, sole of sweeping foot is off carpet; If dl = 0, sole of sweeping foot is not off carpet. Establishing error function:  0, dl = 0 f (u 21 ) = 1, dl > 0 If dl > 0, sole of sweeping foot is off carpet and the value of error function is 1; If dl = 0, sole of sweeping foot is not off carpet and the value of error function is 0. (2) For right sweeping foot The corresponding plane equation for the sole of right foot is:

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   x − xr j z1 y − yr j z1 z − zr j z1   xr j z2 − xr j z1 yr j z2 − yr j z1 zr j z2 − zr j z1  = 0   xr j z3 − xr j z1 yr j z3 − yr j z1 zr j z3 − zr j z1  Suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the sole of left food is dr where, dr =

|Ax0 + By0 + C z 0 + D| (A, B, C, and D are the same as that in dl ). √ A2 + B 2 + C 2

dr is the distance between sole of food and court ground: If dr > 0, sole of sweeping foot is off carpet; If dr = 0, sole of sweeping foot is not off carpet. Establishing error function:  f (u 22 ) =

0, dr = 0 1, dr > 0

If dr > 0, sole of sweeping foot is off carpet and the value of error function is 1; If dr = 0, sole of sweeping foot is not off carpet and the value of error function is 0. Therefore, the error function for u 2 is: f (u 2 ) = f (u 21 ) ∨ f (u 22 ). A deviation range can be set for d . 3. u 3 = forward sweep: sweeping leg bent Suppose that domain U1 = forward sweep in optional Nanquan; thing S3 = sweeping foot; property T3 = the 3-D coordinate for the heel of left foot (xl jg , yl jg , zl jg ); the 3-D coordinate for the left knee (xlxg , ylxg , zlxg ); the 3-D coordinate for the left hip joint (xltg , yltg , zltg ); the 3-D coordinate for the heel of right foot (xr jg , yr jg , zr jg ); the 3-D coordinate for the right knee (xr xg , yr xg , zr xg ); and the 3-D coordinate for the right hip joint (xr tg , yr tg , zr tg ). (1) For the left sweeping foot Suppose that the 3-D coordinate for the left heel is (xl jg , yl jg , zl jg ) and the 3D coordinate for the left hip joint is (xltg , yltg , zltg ), the equation connecting the two points is: x − xl jg y − yl jg z − zl jg = = xltg − xl jg yltg − yl jg zltg − zl jg If the 3-D coordinate for the left knee (xlxg , ylxg , zlxg ) does not meet the straight line equation, it indicates there exists sweeping leg bent; If the 3-D coordinate for the left knee (xlxg , ylxg , zlxg ) meets the straight line equation, it indicates that the sweeping leg is straight.

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Establishing error function:  f (u 31 ) =

0, 1,

xlxg −xl jg xltg −xl jg xlxg −xl jg xltg −xl jg

= =

ylxg −yl jg yltg −yl jg ylxg −yl jg yltg −yl jg

= =

zlxg −zl jg zltg −zl jg zlxg −zl jg zltg −zl jg

If the 3-D coordinate for the left knee does not meet the straight line equation, it indicates there exists sweeping leg bent and the value of error function is 1; If the 3-D coordinate for the left knee meets the straight line equation, it indicates that the sweeping leg is straight and the value of error function is 0. (2) For the right sweeping foot Suppose that the 3-D coordinate for the right heel is (xl jg , yl jg , zl jg ) and the 3-D coordinate for the right hip joint is (xltg , yltg , zltg ), the equation connecting the two points is: x − xr jg y − yr jg z − zr jg = = xr tg − xr jg yr tg − yr jg zr tg − zr jg If the 3-D coordinate for the right knee (xr xg , yr xg , zr xg ) does not meet the straight line equation, it indicates there exists sweeping leg bent; If the 3-D coordinate for the right knee (xr xg , yr xg , zr xg ) meets the straight line equation, it indicates that the sweeping leg is straight. Establishing error function:  f (u 32 ) =

0, 1,

xr xg −xr jg xr tg −xr jg xr xg −xr jg xr tg −xr jg

= =

yr xg −yr jg yr tg −yr jg yr xg −yr jg yr tg −yr jg

= =

zr xg −zr jg zr tg −zr jg zr xg −zr jg zr tg −zr jg

If the 3-D coordinate for the right knee does not meet the straight line equation, it indicates there exists sweeping leg bent and the value of error function is 1; If the 3-D coordinate for the right knee meets the straight line equation, it indicates there sweeping leg is straight and the value of error function is 0. Therefore, the error function for u 3 is: f (u 3 ) = f (u 31 ) ∨ f (u 32 ). 4. Point deduction model for error in forward sweeping in optional Nanquan Per rules, if there are two or more errors in the same movement, 0.1 is deducted. Therefore, in the forward sweeping movement, based on the error functions for different rules, point deduction model for forward sweeping movement is established hereby. According to the established error function: f (u 1 ) = f (u 11 ) ∨ f (u 12 ); f (u 2 ) = f (u 21 ) ∨ f (u 22 ); f (u 3 ) = f (u 31 ) ∨ f (u 32 ). Suppose that the movement of forward sweeping in optional Nanquan is noted by u qgs , the total points deducted for forward sweeping in optional Nanquan is

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f (u k f qgs ), the point deducted for u 1 is f (u k f qgs1 ), the point deducted for u 2 is f (u k f qgs2 ), and the point deducted for u 3 is f (u k f qgs3 )

f (u k f i j ) =

⎧ ⎪ ⎨0,

0; (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

where i stands for the ith point-deduction movement and j stands for the jth deduction point for forward sweep. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error value is 1. The point deduction model for forward sweeping is f (u k f qgs ) = f (u k f qgs1 ) ∨ f (u k f qgs2 ) ∨ f (u k f qgs3 ). 6.5.3.2

Computer Vision-based Error Identification Model for Horizontal Nail Kick in Optional Nanquan

In the rules for judging optional Nanquan, the point deduction mainly happens when kicking leg is not kept straight after kick and no nail kicks to opposite side. 1. u 4 = Horizontal nail kick: kicking leg not kept straight after kick Suppose that domain U2 = horizontal nail kick in optional Nanquan; thing S3 = nail kick leg; property T3 = the 3-D coordinate of the left heel is (xl jg , yl jg , zl jg ); the 3-D coordinate of the right heel (xr jg , yr jg , zr jg ); the length of thigh is h 2 and the length of leg is h 2 , then the length of whole leg is (h 2 + h 3 ). According to the movement trajectory, the movements of horizontal nail kick can be divided into three stages: (1) the moment when the tiptoe turns upward and lifts up is (thd0 ), 2) the moment when the lifted leg changes from bent state to become straight is (thd0 + 1), and 3) the moment when the nail kick to opposite side ends is (thd0 + 2). (1) When the nail kick is done by the left leg Suppose that the moment when the tiptoe turns upward and lifts up is (thd0 ), the 3-D coordinate for the left heel is (xl jg0 , yl jg0 , zl jg0 ), the moment when the lifted leg changes from bent state to become straight is (thd0 + 1), the 3-D coordinate for the left heel is (xl jg1 , yl jg1 , zl jg1 ). The 3-D coordinates for the left heels at moments (thd0 ) and (thd0 + 1) determines if the left leg is straight. At moment (thd0 + 1), xl jg1 = xl jg0 , or yl jg1 = yl jg0 − (h 2 + h 3 ), or zl jg1 = zl jg0 , the kicking leg is not straight; At moment (thd0 + 1), xl jg1 = xl jg0 − (h 2 + h 3 ), and yl jg1 = yl jg0 and zl jg1 = zl jg0 , the kicking leg is straight. Establishing error function:  1, xl jg1 = xl jg0 , or yl jg1 = yl jg0 − (h 2 + h 3 ), or zl jg1 = zl jg0 ; f (u 41 ) = 0, xl jg1 = xl jg0 − (h 2 + h 3 ), and yl jg1 = yl jg0 , and zl jg1 = zl jg0 .

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6 Applications of Error Logic

If xl jg1 = xl jg0 , or yl jg1 = yl jg0 − (h 2 + h 3 ), or zl jg1 = zl jg0 , kicking leg is not kept straight after kick and the value of error function is 1; If xl jg1 = xl jg0 − (h 2 + h 3 ), and yl jg1 = yl jg0 , and zl jg1 = zl jg0 , kicking leg is kept straight after kick and the value of error function is 0. When the horizontal nail kick is done by the right leg, the error function is f (u 42 ), similar to the process of establishing error function for the left leg, we obtained the following error function for the right leg:  1, xr jg1 = xr jg0 , or yr jg1 = yr jg0 − (h 2 + h 3 ), or zr jg1 = zr jg0 ; f (u 42 ) = 0, xr jg1 = xr jg0 − (h 2 + h 3 ), and yr jg1 = yr jg0 , and zr jg1 = zr jg0 .

If xr jg1 = xr jg0 , or yr jg1 = yr jg0 − (h 2 + h 3 ), or zr jg1 = zr jg0 , kicking leg is not kept straight after kick and the value of error function is 1; If xr jg1 = xr jg0 − (h 2 + h 3 ), and yr jg1 = yr jg0 , and zr jg1 = zr jg0 , kicking leg is kept straight after kick and the value of error function is 0. Therefore, the error function for u 4 is f (u 4 ) = f (u 41 ) ∨ f (u 42 ) . 2. u 5 = horizontal nail kick: no nail kick to opposite side Suppose that domain U2 = horizontal nail kick in optional Nanquan; thing S3 = nail kicking leg; property T3 = the 3-D coordinate of the right heel is (xl jg , yl jg , zl jg ); the 3-D coordinate of the right heel (xr jg , yr jg , zr jg ); the length of thigh is h 2 and the length of leg is h 2 , then the length of whole leg is (h 2 + h 3 ). Based on the movement trajectory the nail kicking leg, the displacement of the 3-D coordinates at moments of (thd0 + 1) and (thd0 + 2) can confirm if the nail kick to opposite side has been completed. Suppose that the 3-D coordinate of left heel is (xl jg1 , yl jg1 , zl jg1 ) at time (thd0 + 1) and its coordinate changes to (xl jg2 , yl jg2 , zl jg2 ). At time (thd0 + 2), If xl jg2 = xl jg1 − (h 2 + h 3 ), and yl jg1 = yl jg1 , and zl jg2 = zl jg1 at time (thd0 + 2), the movement of nail kick to opposite side has been conducted; If xr jg2 = xr jg1 − (h 2 + h 3 ), or yr jg2 = yr jg1 , or zr jg2 = zr jg1 at time (thd0 + 2), there is no nail kick to opposite side. (1) When the nail kick is done by the left leg Establishing error function:  0, xl jg2 = xl jg1 − (h 2 + h 3 ), and yl jg2 = yl jg1 , and zl jg2 = zl jg1 ; f (u 51 ) = 1, xl jg2 = xl jg1 − (h 2 + h 3 ), or yl jg2 = yl jg1 , or zl jg2 = zl jg1 .

If xl jg2 = xl jg1 − (h 2 + h 3 ), and yl jg2 = yl jg1 , and zl jg2 = zl jg1 at time moment (thd0 + 2), the movement of nail kick to opposite side has been conducted and the value of error function is 0; If xl jg2 = xl jg1 − (h 2 + h 3 ), or yl jg2 = yl jg1 , or zl jg2 = zl jg1 at time moment (thd0 + 2), there is no nail kick to opposite side and the value of error function is 1.

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(2) When the nail kick is done by the right leg Establishing error function: f (u 52 ) =

 0, xr jg2 = xr jg1 − (h 2 + h 3 ), and yr jg2 = yr jg1 , or zr jg2 = zr jg1 ; 1, xr jg2 = xr jg1 − (h 2 + h 3 ), or yr jg2 = yr jg1 , or zr jg2 = zr jg1 .

If xr jg2 = xr jg1 − (h 2 + h 3 ), and yr jg2 = yr jg1 , and zr jg2 = zr jg1 at time moment (thd0 + 2), the movement of nail kick to opposite side has been conducted and the value of error function is 0; If xr jg2 = xr jg1 − (h 2 + h 3 ), or yr jg2 = yr jg1 , or zr jg2 = zr jg1 at time moment (thd0 + 2) , there is no nail kick to opposite side and the value of error function is 1. Therefore, the error function for u 5 is f (u 5 ) = f (u 51 ) ∨ f (u 52 ) 3. Point deduction model for error in horizontal nail kick in optional Nanquan Per rules, if there are two or more errors in the same movement, 0.1 is deducted. Therefore,in the horizontal nail kick, based on the error functions for different rules, point deduction model for horizontal nail kick is established hereby. Suppose that the object of horizontal nail kick in optional Nanquan is noted by u hd , the total points deducted for horizontal nail kick in optional Nanquan is f (u k f hdi j ), ⎧ ⎪ ⎨0,

0; f (u k f hdi j ) = (i, j = 1, 2, . . . , n) . ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the horizontal nail kick. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. In f (u k f hdi j4 ), i represents the ith point-deduction movement and j stands for the jth deduction point. The total point deducted for u 4 is f (u k f hd4 ), the point deducted for u 41 is f (u k f hd41 ), and the point deducted for u 4 is f (u k f hd42 ); the total point deducted for u 5 is f (u k f hd5 ), the point deducted for u 51 is f (u k f hd51 ), and the point deducted for u 5 is f (u k f hd52 ). From f (u 4 ) = f (u 41 ) ∨ f (u 42 ), the point deduction model for “horizontal nail kick: kicking leg not kept straight after kick” is f (u k f hd4 ) = f (u k f hd41 ) ∨ f (u k f hd42 ); From f (u 5 ) = f (u 51 ) ∨ f (u 52 ), the point deduction model for “horizontal nail kick: kicking leg not kept straight after kick” is f (u k f hd5 ) = f (u k f hd51 ) ∨ f (u k f hd52 ). Therefore, the comprehensive point deduction model is: f (u k f hd ) = f (u k f hd4 ) ∨ f (u k f hd5 ).

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6 Applications of Error Logic

6.5.3.3

Computer Vision-based Error Identification Model for Kick in Flight in Optional Nanquan

According the rules in optional Nanquan, the correct movements in kick in flight are: (1) move left leg forward and kick upward; (2) drive right leg up; (3) two arms move forward and upward; (4) the back of right hand slaps left palm; (5) right foot kicks forward and upward (left foot is higher than the plane that waist is on) with toe pointed and right hand slapping right instep, (6) left leg bent and gets close to the inner side of the right leg with toe pointed downward; (7) after the right leg touches the ground, the left leg is then landed. After completing the whole set, 0.1 is deducted if toes of slapped leg is below shoulder level or slap is missed in flight. 1. u 6 = Kick in flight: toes of slapped leg below shoulder level Suppose that domain U3 = kick in flight in optional Nanquan; thing S4 = kick in flight:slapping leg or swinging leg; property T4 = the 3-D coordinate of the left foot toe tip is (xl j j , yl j j , zl j j ), the 3-D coordinate of the right foot toe tip is (xr j j , yr j j , zr j j ); the 3-D coordinate of the left shoulder is (xl jb , yl jb , zl jb ), and the 3-D coordinate of the right shoulder is (xr jb , yr jb , zr jb ). In the 3-D coordinate, in order to judge if the toes of slapped leg is below shoulder level, we can get it by comparing the z coordinates of foot toe tip and shoulder. (1) When the slapped leg in kick in flight is the left leg Suppose that the 3-D coordinate of the left foot toe tip (xl j j , yl j j , zl j j ); the 3-D coordinate of the left shoulder (xl jb , yl jb , zl jb ). If zl j j < zl jb , toes of slapped leg is below shoulder level; If zl j j ≥ zl jb , toes of slapped leg is above shoulder level. Establishing error function:  f (u 61 ) =

1, zl j j < zl jb ; 0, zl j j ≥ zl jb .

If zl j j < zl jb , toes of slapped leg is below shoulder level and the movement has mistake, the error function value is 1; If zl j j ≥ zl jb , toes of slapped leg is above shoulder level and the movement has no mistake, the error function value is 0. (2) When the slapped leg in kick in flight is the right leg Suppose that the 3-D coordinate of the right foot toe tip (xr j j , yr j j , zr j j ); the 3-D coordinate of the left shoulder (xr jb , yr jb , zr jb ). When zr j j < zr jb , toes of slapped leg is below shoulder level; When zr j j ≥ zr jb , toes of slapped leg is above shoulder level. Establishing error function:  f (u 62 ) =

1, zr j j < zr jb ; 0, zr j j ≥ zr jb .

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

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If zr j j < zr jb , toes of slapped leg is below shoulder level and the movement has mistake, the error function value is 1; If zr j j ≥ zr jb , toes of slapped leg is above shoulder level and the movement has no mistake, the error function value is 0. Based on the above definition, the error function for u 6 is: f (u 6 ) = f (u 61 ) ∨ f (u 62 ). 2. u 7 = kick in flight: slap missing in flight Suppose that domain U3 = kick in flight in optional Nanquan; thing S5 = left or right hand; property T5 = the 3-D coordinate of the left finger is (xlsz , ylsz , zlsz ), the 3-D coordinate of the right finger is (xr sz , yr sz , zr sz ); the 3-D coordinate of the left instep is (xl jm , yl jm , zl jm ), and the 3-D coordinate of the right instep is (xr jm , yr jm , zr jm ). (1) When the slapping or swinging leg in kick in flight is the left leg Suppose that the 3-D coordinate of the left finger (xlsz , ylsz , zlsz ); the 3-D coordinate of the left instep is (xl jm , yl jm , zl jm ). According to the slapping movement in kick in flight, the movement is being initiated at time t1 : If xlsz = xl jm , or ylsz = yl jm , or zlsz = zl jm , slap is missed in flight; If xlsz = xl jm , and ylsz = yl jm , and zlsz = zl jm , slap is conducted in flight. Establishing error function:  f (u 71 ) =

1, xlsz = xl jm , or ylsz = yl jm , or zlsz = zl jm ; 0, xlsz = xl jm , and ylsz = yl jm , and zlsz = zl jm .

At time t1 when the slap movement is being conducted, if xlsz = xl jm , or ylsz = yl jm or zlsz = zl jm , the slap is missed in flight and the movement has mistake, the error function value is 1; If xlsz = xl jm , and ylsz = yl jm and zlsz = zl jm , slap is conducted in flight and the movement has no mistake, the error function value is 0. (2) When the slapping or swinging leg in kick in flight is the right leg Suppose that the 3-D coordinate of the right finger (xr sz , yr sz , zr sz ); the 3-D coordinate of the right instep is (xr jm , yr jm , zr jm ). According to the slapping movement in kick in flight, the movement is being initiated at time t1 : If xr sz = xr jm , or yr sz = yr jm . or zr sz = zr jm , slap is missed in flight; If xr sz = xr jm , and yr sz = yr jm , and zr sz = zr jm , slap is conducted in flight. Establishing error function:  f (u 72 ) =

1, xr sz = xr jm , or yr sz = yr jm , or zr sz = zr jm ; 0, xr sz = xr jm , andr yr sz = yr jm , and zr sz = zr jm .

At time t1 when the slap movement is being conducted, if xr sz = xr jm , or yr sz = yr jm or zr sz = zr jm , the slap is missed in flight and the movement has

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6 Applications of Error Logic

mistake, the error function value is 1; If xr sz = xr jm , and yr sz = yr jm and zr sz = zr jm , slap is conducted in flight and the movement has no mistake, the error function value is 0. Based on the above definition, the error function for u 7 is: f (u 7 ) = f (u 71 ) ∨ f (u 72 ) 3. Point deduction model for error in kick in flight in optional Nanquan Per rules, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for kick in flight is established hereby. Suppose that the object of kick in flight in optional Nanquan is noted by u t f . Establishing the point deduction model for kick in flight:

f (u k f i j ) =

⎧ ⎪ ⎨0,

0; (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

where i stands for the ith point-deduction movement and j stands for the jth deduction point for the kick in flight. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. From f (u 6 ) = f (u 61 ) ∨ f (u 62 ), the point deduction model for “toe of slapped leg below shoulder level” is f (u k f t f 6 ) = f (u k f t f 61 ) ∨ f (u k f t f 62 ); From f (u 7 ) = f (u 71 ) ∨ f (u 72 ), the point deduction model for “kick in flight: slap missing in flight” is f (u k f t f 7 ) = f (u k f t f 71 ) ∨ f (u k f t f 72 ). Therefore, the comprehensive point deduction model for kick in flight is: f (u k f t f ) = f (u k f t f 6 ) ∨ f (u k f t f 7 ).

6.5.3.4

Computer Vision-based Error Identification Model for Tornado Kick in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in tornado kick , 0.1 is deducted if toes of slapped leg is below shoulder level or slap is missed in flight. 1. u 8 = tornado kick: toes of slapped leg below shoulder level Suppose that domain U4 = tornado kick in optional Nanquan; thing S6 = tornado kick:slapping leg or swinging leg; property T6 = the 3-D coordinate of the left foot toe tip is (xl j j , yl j j , zl j j ), the 3-D coordinate of the right foot toe tip is (xr j j , yr j j , zr j j ); the 3-D coordinate of the left shoulder is (xl jb , yl jb , zl jb ), and the 3-D coordinate of the right shoulder is (xr jb , yr jb , zr jb ). In the 3-D coordinate, in order to judge if the toes of slapped leg is below shoulder level, we can get it by comparing the z coordinates of foot toe tip and shoulder. (1) When the slapped leg in tornado kick is the left leg Suppose that the 3-D coordinate of the left foot toe tip (xl j j , yl j j , zl j j ); the 3-D

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

571

coordinate of the left shoulder (xl jb , yl jb , zl jb ). If zl j j < zl jb , toes of slapped leg is below shoulder level; If zl j j ≥ zl jb , toes of slapped leg is above shoulder level. Establishing error function:  f (u 81 ) =

1, zl j j < zl jb ; 0, zl j j ≥ zl jb .

If zl j j < zl jb , toes of slapped leg is below shoulder level and the movement has mistake, the error function value is 1; If zl j j ≥ zl jb , toes of slapped leg is above shoulder level and the movement has no mistake, the error function value is 0. (2) When the slapped leg in tornado kick is the right leg Suppose that the 3-D coordinate of the right foot toe tip (xr j j , yr j j , zr j j ); the 3-D coordinate of the left shoulder (xr jb , yr jb , zr jb ). When zr j j < zr jb , toes of slapped leg is below shoulder level; When zr j j ≥ zr jb , toes of slapped leg is above shoulder level. Establishing error function:  f (u 82 ) =

1, zr j j < zr jb ; 0, zr j j ≥ zr jb .

If zr j j < zr jb , toes of slapped leg is below shoulder level and the movement has mistake, the error function value is 1; If zr j j ≥ zr jb , toes of slapped leg is above shoulder level and the movement has no mistake, the error function value is 0. Based on the above definition, the error function for u 8 is: f (u 8 ) = f (u 81 ) ∨ f (u 82 ). 2. u 9 = tornado kick: slap missing in flight Suppose that domain U4 = tornado kick in optional Nanquan; thing S7 = left or right hand; property T7 = the 3-D coordinate of the left finger is (xlsz , ylsz , zlsz ), the 3-D coordinate of the right finger is (xr sz , yr sz , zr sz ); the 3-D coordinate of the left instep is (xl jm , yl jm , zl jm ), and the 3-D coordinate of the right instep is (xr jm , yr jm , zr jm ). (1) When the slapping or swinging leg in tornado kick is the left leg Suppose that the 3-D coordinate of the left finger (xlsz , ylsz , zlsz ); the 3-D coordinate of the left instep is (xl jm , yl jm , zl jm ). According to the slapping movement in tornado kick, the movement is being initiated at time t1 : If xlsz = xl jm , or ylsz = yl jm , or zlsz = zl jm , slap is missed in flight; If xlsz = xl jm , and ylsz = yl jm , and zlsz = zl jm , slap is conducted in flight. Establishing error function:  f (u 91 ) =

1, xlsz = xl jm , or ylsz = yl jm , or zlsz = zl jm ; 0, xlsz = xl jm , and ylsz = yl jm , and zlsz = zl jm .

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6 Applications of Error Logic

At time t1 when the slap movement is being conducted, if xlsz = xl jm , or ylsz = yl jm or zlsz = zl jm , the slap is missed in flight and the movement has mistake, the error function value is 1; If xlsz = xl jm , and ylsz = yl jm and zlsz = zl jm , slap is conducted in flight and the movement has no mistake, the error function value is 0. (2) When the slapping or swinging let in tornado kick is the right leg Suppose that the 3-D coordinate of the right finger (xr sz , yr sz , zr sz ); the 3-D coordinate of the right instep is (xr jm , yr jm , zr jm ). According to the slapping movement in tornado kick, the movement is being initiated at time t1 : If xr sz = xr jm , or yr sz = yr jm or zr sz = zr jm , slap is missed in flight; If xr sz = xr jm , and yr sz = yr jm and zr sz = zr jm , slap is conducted in flight. Establishing error function:  f (u 92 ) =

1, xr sz = xr jm , or yr sz = yr jm , or zr sz = zr jm ; 0, xr sz = xr jm , and yr sz = yr jm , and zr sz = zr jm .

At time t1 when the slap movement is being conducted, if xr sz = xr jm , or yr sz = yr jm or zr sz = zr jm , the slap is missed in flight and the movement has mistake, the error function value is 1; If xr sz = xr jm , and yr sz = yr jm and zr sz = zr jm , slap is conducted in flight and the movement has no mistake, the error function value is 0. Based on the above definition, the error function for u 9 is: f (u 9 ) = f (u 91 ) ∨ f (u 92 ) 3. Point deduction model for error in tornado kick in optional Nanquan Per rules, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for tornado kick is established hereby. Suppose that the object of tornado kick in optional Nanquan is noted by u x f . Establishing the point deduction model for tornado kick: ⎧ ⎪ ⎨0,

0; f (u k f i j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the tornado. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. From f (u 8 ) = f (u 81 ) ∨ f (u 82 ), the point deduction model for “toe of slapped leg below shoulder level” is f (u k f x f 8 ) = f (u k f x f 81 ) ∨ f (u k f x f 82 ); From f (u 9 ) = f (u 91 ) ∨ f (u 92 ), the point deduction model for “tornado kick: slap missing in flight” is f (u k f x f 9 ) = f (u k f x f 91 ) ∨ f (u k f x f 92 ). Therefore, the comprehensive point deduction model for tornado kick is: f (u k f x f ) = f (u k f x f 8 ) ∨ f (u k f x f 9 ).

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

6.5.3.5

573

Computer Vision-based Error Identification Model for Outward Kick in Flight in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in outward kick in flight , 0.1 is deducted if toes of slapped leg is below shoulder level or slap is missed in flight. 1. u 10 = outward kick in flight: toes of slapped leg below shoulder level Suppose that domain U5 = outward kick in flight in optional Nanquan; thing S8 = outward kick in flight:slapping leg or swinging leg; property T8 = the 3-D coordinate of the left foot toe tip is (xl j j , yl j j , zl j j ), the 3-D coordinate of the right foot toe tip is (xr j j , yr j j , zr j j ); the 3-D coordinate of the left shoulder is (xl jb , yl jb , zl jb ), and the 3-D coordinate of the right shoulder is (xr jb , yr jb , zr jb ). In the 3-D coordinate, in order to judge if the toes of slapped leg is below shoulder level, we can get it by comparing the z coordinates of foot toe tip and shoulder. (1) When the slapped leg in outward kick in flight is the left leg Suppose that the 3-D coordinate of the left foot toe tip (xl j j , yl j j , zl j j ); the 3-D coordinate of the left shoulder (xl jb , yl jb , zl jb ). If zl j j < zl jb , toes of slapped leg is below shoulder level; If zl j j ≥ zl jb , toes of slapped leg is above shoulder level. Establishing error function:  f (u 101 ) =

1, zl j j < zl jb ; 0, zl j j ≥ zl jb .

If zl j j < zl jb , toes of slapped leg is below shoulder level and the movement has mistake, the error function value is 1; If zl j j ≥ zl jb , toes of slapped leg is above shoulder level and the movement has no mistake, the error function value is 0. (2) When the slapped leg in outward kick in flight is the right leg Suppose that the 3-D coordinate of the right foot toe tip (xr j j , yr j j , zr j j ); the 3-D coordinate of the left shoulder (xr jb , yr jb , zr jb ). When zr j j < zr jb , toes of slapped leg is below shoulder level; When zr j j ≥ zr jb , toes of slapped leg is above shoulder level. Establishing error function:  1, zr j j < zr jb ; f (u 102 ) = 0, zr j j ≥ zr jb . If zr j j < zr jb , toes of slapped leg is below shoulder level and the movement has mistake, the error function value is 1; If zr j j ≥ zr jb , toes of slapped leg is above shoulder level and the movement has no mistake, the error function value is 0.

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6 Applications of Error Logic

Based on the above definition, the error function for u 10 is: f (u 10 ) = f (u 101 )∨ f (u 102 ). 2. Point deduction model for error in tornado kick in optional Nanquan Per rules, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for outward kick in flight is established hereby. Suppose that the object of outward kick in flight in optional Nanquan is noted by u tw . Establishing the point deduction model for outward kick in flight: ⎧ ⎪ ⎨0,

0; f (u k f i j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the tornado kick. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. From f (u 10 ) = f (u 101 ) ∨ f (u 102 ), the point deduction model for “toe of slapped leg below shoulder level” is f (u k f tw ) = f (u k f tw101 ) ∨ f (u k f tw102 ). Therefore, the comprehensive point deduction model for tornado kick is: f (u k f tw ) = f (u k f tw101 ) ∨ f (u k f tw102 ).

6.5.3.6

Computer Vision-based Error Identification Model for Cross-leg Kick in Flight Full Twist to Land on Side in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in cross-leg kick in flight full twist to land on side(TKPTCP), 0.1 is deducted if swing leg is below head level. 1. u 11 = cross-leg kick in flight full twist to land on side: swing leg is below head level Suppose that domain U6 = cross-leg kick in flight full twist to land on side in optional Nanquan; thing S9 = swinging leg; property T9 = the 3-D coordinate of the head is (xtd , ytd , z td ), the 3-D coordinate of the left foot toe tip is (xl j j , yl j j , zl j j ); and the 3-D coordinate of the right foot toe tip is (xr j j , yr j j , zr j j ). In the 3-D coordinate, in order to judge if swing leg is above head level, we can find it by figuring out the z coordinates of foot toe tip and head. (1) When the swing leg is the left leg Suppose that the 3-D coordinate of the left foot toe tip (xl j j , yl j j , zl j j ); the 3-D coordinate of the head is (xtd , ytd , z td ). If zl j j ≤ z td , swing leg is below head level; If zl j j > z td , swing leg is above head level. Establishing error function:

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

 f (u 111 ) =

575

1, zl j j ≤ z td ; 0, zl j j > z td .

If zl j j ≤ z td , swing leg is below head level and the movement has mistake, the error function value is 1; If zl j j > z td , swing leg is above head level and the movement has no mistake, the error function value is 0. (2) When the swing leg is the right leg Suppose that the 3-D coordinate of the right foot toe tip (xr j j , yr j j , zr j j ); the 3-D coordinate of the head is (xtd , ytd , z td ). When zr j j ≤ z td , swing leg is below head level; When zr j j > z td , swing leg is above head level. Establishing error function:  f (u 112 ) =

1, zr j j ≤ z td ; 0, zr j j > z td .

If zr j j ≤ z td , swing leg is below head level and the movement has mistake, the error function value is 1; If zr j j > z td , swing leg is above head level and the movement has no mistake, the error function value is 0. Based on the above definition, the error function for u 1 1 is: f (u 1 1) = f (u 111 ) ∨ f (u 112 ). 2. Point deduction model for error in cross-leg kick in flight full twist to land on side in optional Nanquan Per rules, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for “cross-leg kick in flight full twist to land on side” is established hereby. Suppose that the object of “cross-leg kick in flight full twist to land on side” in optional Nanquan is noted by u t p . Establishing the point deduction model for “cross-leg kick in flight full twist to land on side”:

f (u k f i j ) =

⎧ ⎪ ⎨0,

0; (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

where i stands for the ith point-deduction movement and j stands for the jth deduction point for the tornado kick. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. From f (u 11 ) = f (u 111 ) ∨ f (u 112 ), the comprehensive point deduction model for “cross-leg kick in flight full twist to land on side” is f (u k f t p ) = f (u k f t p111 ) ∨ f (u k f t p112 ).

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6.5.3.7

Computer Vision-based Error Identification Model for Kip-up in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in kip-up , 0.1 is deducted if there is lift-up on support of hands. 1. u 12 = kip-up: lift-up on support of hands Suppose that domain U7 = kip-up in optional Nanquan; thing S10 = palms; property T10 = the 3-D coordinate for distal phalanx is (xsh1 , ysh1 , z sh1 ), the 3-D coordinate for capitate is (xsh2 , ysh2 , z sh2 ), and the 3-D coordinate for the center of the palm is (xsh3 , ysh3 , z sh3 ). From the above three coordinate points, a plane can obtained and any point must satisfy the following equation:    x − xsh1 y − ysh1 z − z sh1    xsh2 − xsh1 ysh2 − ysh1 z sh2 − z sh1  = 0   xsh3 − xsh1 ysh3 − ysh1 z sh3 − z sh1  The corresponding plane equation for the left palm is:    x − xlsh1 y − ylsh1 z − zlsh1   xlsh2 − xlsh1 ylsh2 − ylsh1 zlsh2 − zlsh1  = 0   xlsh3 − xlsh1 ylsh3 − ylsh1 zlsh3 − zlsh1  The corresponding plane equation for the right palm is:    x − xr sh1 y − yr sh1 z − zr sh1   xr sh2 − xr sh1 yr sh2 − yr sh1 zr sh2 − zr sh1  = 0   xr sh3 − xr sh1 yr sh3 − yr sh1 zr sh3 − zr sh1  (1) For left palm Suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the left palm is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = ylsh2 zlsh3 − ylsh2 zlsh1 − ylsh1 zlsh3 − ylsh3 zlsh2 + ylsh3 zlsh1 + ylsh1 zlsh2 ; B = xlsh1 zlsh3 − xlsh1 zlsh2 + xlsh2 zlsh1 − xlsh2 zlsh3 + xlsh3 zlsh2 − xlsh3 zlsh1 ; C = xlsh1 zlsh2 − xlsh1 zlsh3 + xlsh2 zlsh3 − xlsh2 zlsh1 + xlsh3 zlsh1 − xlsh3 zlsh2 ; D = xlsh1 ylsh2 zlsh3 + xlsh1 ylsh3 zlsh2 +xlsh2 ylsh1 zlsh3 − xlsh2 ylsh3 zlsh1 + xlsh3 ylsh2 zlsh1 − xlsh3 ylsh3 zlsh2 . dl is the distance between the left palm and court ground: If dl > 0, there is not lift-up on support of hands;

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If dl = 0, there is lift-up on support of hands. Establishing error function:  f (u 121 ) =

1, dl = 0 0, dl > 0

If dl > 0, there is not lift-up on support of hands and the value of error function is 0; If dl = 0, there is lift-up on support of hands and the value of error function is 1. (1) For right palm Similarly, suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the right palm is dr where, Establishing error function:  1, dr = 0 f (u 122 ) = 0, dr > 0 If dr > 0, there is not lift-up on support of hands and the value of error function is 0; If dr = 0, there is lift-up on support of hands and the value of error function is 1. 2. Point deduction model for error in kip-up in optional Nanquan Per rules, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for kip-up is established hereby. Suppose that the object of kip-up in optional Nanquan is noted by u ly . Establishing the point deduction model for kip-up: ⎧ ⎪ ⎨0,

0; f (u k f i j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the kip-up. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. From f (u 12 ) = f (u 121 ) ∨ f (u 122 ), the obtained comprehensive point deduction model for “kip-up: lift-up on support of hands” is f (u k f ly ) = f (u k f ly121 ) ∨ f (u k f ly122 ).

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6.5.3.8

Computer Vision-based Error Identification Model for Double Side Kick with Sole in Flight in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in kip-up , 0.1 is deducted if there is the error of “kicking legs not close together” or “kicking legs bent”. 1. Establishing Function for Errors in Double Side Kick with Sole in Flight . (1) u 13 = double side kick with sole in flight : kicking legs not close together Suppose that domain U8 = double side kick with sole in flight in optional Nanquan; thing S11 = kicking legs; property T11 = the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the center of the left calf is (xlt z , ylt z , zlt z ), and the 3-D coordinate for the center of the left ankle is (xl j h , yl j h , zl j h ); the 3-D coordinate for the right hip joint is (xr tg , yr tg , zr tg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ), the 3-D coordinate for the center of the right calf is (xr t z , yr t z , zr t z ), and the 3-D coordinate for the center of the right ankle is (xr j h , yr j h , zr j h ). The kicking legs are closed together when the coordinates for left and right hip points, knees, centers of calves, and ankles overlap, which means that xltg = xr tg , yltg = yr tg , zltg = zr tg , and xlxg = xr xg , ylxg = yr xg , zlxg = zr xg , and xlt z = xlt z , ylt z = ylt z , zlt z = zlt z , and xl j h = xl j h , yl j h = yl j h , zl j h = zl j h hold. The kicking legs are not closed together when the coordinates for left and right hip points, knees, centers of calves, and ankles overlap, which means that xltg = xr tg , yltg = yr tg , zltg = zr tg , and xlxg = xr xg , ylxg = yr xg , zlxg = zr xg , and xlt z = xlt z , ylt z = ylt z , zlt z = zlt z , and xl j h = xl j h , yl j h = yl j h , zl j h = zl j h hold. Establishing error function: ⎧ 0, xltg = xr tg , yltg = yr tg , zltg = zr tg and xlxg = xr xg , ylxg = yr xg , ⎪ ⎪ ⎪ ⎪ ⎪ zlxg = zr xg and xlt z = xlt z , ylt z = ylt z , zlt z = zlt z and xl j h = xl j h , ⎪ ⎪ ⎪ ⎨ yl j h = yl j h , zl j h = zl j h ; f (u 13 ) = ⎪ 1, x ltg  = xr tg , yltg  = yr tg , z ltg  = z r tg and xlxg  = xr xg , ylxg  = yr xg , ⎪ ⎪ ⎪ ⎪ ⎪ z lxg  = z r xg and xlt z  = xlt z , ylt z  = ylt z , z lt z  = z lt z and xl j h  = xl j h , ⎪ ⎪ ⎩ yl j h = yl j h , zl j h = zl j h .

If xltg = xr tg , yltg = yr tg , zltg = zr tg , and xlxg = xr xg , ylxg = yr xg , zlxg = zr xg , and xlt z = xlt z , ylt z = ylt z , zlt z = zlt z , and xl j h = xl j h , yl j h = yl j h , zl j h = zl j h hold, kicking legs are not closed together and the value of error function is 1; If xltg = xr tg , yltg = yr tg , zltg = zr tg , and xlxg = xr xg , ylxg = yr xg , zlxg = zr xg , and xlt z = xlt z , ylt z = ylt z , zlt z = zlt z , and xl j h = xl j h , yl j h = yl j h , zl j h = zl j h hold and the value of error function is 0.

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(2) u 14 = double side kick with sole in flight : kicking legs bent Suppose that domain U8 = double side kick with sole in flight in optional Nanquan; thing S11 = kicking legs; property T12 = the 3-D coordinate for the sole of the left foot is (xl jg , yl jg , zl jg ), the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ); the 3-D coordinate for the sole of the right foot is (xr jg , yr jg , zr jg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ), the 3- D coordinate for the right hip joint is (xr tg , yr tg , zr tg ). Suppose that the line equation for connecting two points (xr jg , yr jg , zr jg ) and (xr tg , yr tg , zr tg ) is: x − xr jg y − yr jg z − zr jg = = xr tg − xr jg yr tg − yr jg zr tg − zr jg Suppose that the line equation for connecting two points (xl jg , yl jg , zl jg ) and (xltg , yltg , zltg ) is: x − xl jg y − yl jg z − zl jg = = xltg − xl jg yltg − yl jg zltg − zl jg When the 3-D coordinates for both knees meet the line equation, there is no “kicking legs bent” and the value of error function is 0; When the 3-D coordinate for either right or left knee does not meet the line equation, there exists “kicking legs bent” and the value of error function is 1. Establishing the following function:  f (u 14 ) =

0, 1,

xr xg −xr jg xr tg −xr jg xr xg −xr jg xr tg −xr jg

= =

yr xg −yr jg yr tg −yr jg yr xg −yr jg yr tg −yr jg

= =

zr xg −zr jg zr tg −zr jg zr xg −zr jg zr tg −zr jg

xlxg −xl jg ylxg −yl jg zlxg −zl jg xltg −xl jg = yltg −yl jg = zltg −zl jg ; xlxg −xl jg ylxg −yl jg zlxg −zl jg xltg −xl jg  = yltg −yl jg  = zltg −zl jg .

and or

Therefore, the error function for double side kick with sole in flight is f (u tc ) = f (u 13 ) ∨ f (u 14 ). 2. Point deduction model for error in double side kick with sole in flight in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for double side kick with sole in flight is established hereby. Suppose that the object of double side kick with sole in flight in optional Nanquan is noted by u tc . Establishing the point deduction model for double side kick with sole in flight: ⎧ ⎪ ⎨0,

0; f (u k f i j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

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where i stands for the ith point-deduction movement and j stands for the jth deduction point for the kip-up. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. The point deduction model for “double side kick with sole in flight: kicking legs not close together” is f (u k f tc13 ), and the point deduction model for “double side kick with sole in flight: kicking legs bent” is f (u k f tc14 ). Therefore, the comprehensive point deduction model for tornado kick is: f (u k f tc ) = f (u k f tc13 ) ∨ f (u k f tc14 ).

6.5.3.9

Computer Vision-based Error Identification Model for Bow Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in bow stance , 0.1 is deducted if there is the error of “front leg not bent into near half squat” or “heel of rear foot off carpet”. 1. Establishing Function for Errors in Bow Stance. (1) u 15 = Bow stance : front leg not bent into near half squat Suppose that domain U9 = bow stance in optional Nanquan; thing S12 = front leg in bow stance; property T13 = the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ), and the 3-D coordinate for the right hip joint is (xr tg , yr tg , zr tg ). (a) When the front leg in bow stance is the left leg, the error function is f (u 151 ) The difference between z coordinates for hip joint and knee can serve as the criteria for judging if there exist the error of “front leg not bent into near half squat”. Suppose that the upper limit (UL) and lower limit (LL) for judging the difference between the two z coordinates for hip joint and knee are ±Δmm, the maximum value of Δ is determined by the committee for codifying the rules. If zltg = zltg ± Δ, then front leg is bent into near half squat; if zltg > zltg + Δ or zltg < zltg − Δ, then front leg is not bent into near half squat. Establishing error function:  f (u 151 ) =

0, zltg − Δ ≤ zltg ≤ zltg + Δ 1, zltg > zltg + Δ or zltg < zltg − Δ

If zltg − Δ ≤ zltg ≤ zltg + Δ , then front leg is bent into near half squat and the value of error function is 0; if zltg > zltg + Δ or zltg < zltg − Δ , then front leg is not bent into near half squat and the value of error function is 1. .

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(b) When the front leg in bow stance is the right leg, the error function is f (u 152 ) Similarly, we can establish error function for the case of the right leg:  f (u 152 ) =

0, zr tg − Δ ≤ zr tg ≤ zr tg + Δ 1, zr tg > zr tg + Δ or zr tg < zr tg − Δ

If zr tg − Δ ≤ zr tg ≤ zr tg + Δ , then front leg is bent into near half squat and the value of error function is 0; if zr tg > zr tg + Δ or zr tg < zr tg − Δ , then front leg is not bent into near half squat and the value of error function is 1. Hereby, the error function for f (u 15 ) = f (u 151 ) ∨ f (u 152 ) . (2) u 16 = Bow stance : heel of rear foot off carpet Suppose that domain U9 = bow stance in optional Nanquan; thing S13 = rear leg; property T14 = the plane on which the left heel is; the plane on which the right heel is. (a) The rear leg is the left leg Suppose that , the error function is f (u 161 ). The corresponding equation for the plane on which the left heel is:    x − xl jg1 y − yl jg1 z − zl jg1   xl jg2 − xl jg1 yl jg2 − yl jg1 zl jg2 − zl jg1  = 0   xl jg3 − xl jg1 yl jg3 − yl jg1 zl jg3 − zl jg1  Suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the left heel is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = yl jg2 zl jg3 − yl jg2 zl jg1 − yl jg1 zl jg3 − yl jg3 zl jg2 + yl jg3 zl jg1 + yl jg1 zl jg2 ; B = xl jg1 zl jg3 − xl jg1 zl jg2 + xl jg2 zl jg1 − xl jg2 zl jg3 + xl jg3 zl jg2 − xl jg3 zl jg1 ; C = xl jg1 zl jg2 − xl jg1 zl jg3 + xl jg2 zl jg3 − xl jg2 zl jg1 + xl jg3 zl jg1 − xl jg3 zl jg2 ; D = xl jg1 yl jg2 zl jg3 + xl jg1 yl jg3 zl jg2 +xl jg2 yl jg1 zl jg3 − xl jg2 yl jg3 zl jg1 + xl jg3 yl jg2 zl jg1 − xl jg3 yl jg3 zl jg2 . dl is the distance between the left heel and court ground: If dl > 0, the heel of rear foot is off carpet; If dl = 0, the heel of rear foot is not off carpet. Establishing error function:

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6 Applications of Error Logic

 f (u 161 ) =

0, dl = 0 1, dl > 0

If dl = 0, the heel of rear foot is not off carpet and the value of error function is 0; If dl > 0, the heel of rear foot is off carpet and the value of error function is 1. (b) The rear leg is the right leg Suppose that the 3-D coordinate for a point on the supporting court ground is P1 (x1 , y1 , 0), the distance from P1 to the plane for the right heel is dr the error function is f (u 162 ), similarly, Establishing error function:  f (u 162 ) =

0, dr = 0 1, dr > 0

If dr = 0, the heel of rear foot is not off carpet and the value of error function is 0; If dr > 0, the heel of rear foot is off carpet and the value of error function is 1. In conclusion, the error function for f (u 16 ) is: f (u 16 ) = f (u 161 ) ∨ f (u 162 ). 2. Point deduction model for error in bow stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for bow stance is established hereby. Suppose that the object of bow stance in optional Nanquan is noted by u gb . The total deduction function is f (u k f gb ). In f (u k f i j ), i stands for the ith point-deduction movement and j stands for the jth deduction point for the bow stance. The point deduction representations for u 15 , u 151 , and u 152 are f (u k f gb15 ), f (u k f gb151 ), and f (u k f gb152 ) respectively. f (u k f gb16 ), f (u k f gb161 ), and f (u k f gb162 ) represent the point deduction for u 16 , u 161 , and u 162 correspondingly. From f (u 15 ) = f (u 151 ) ∨ f (u 152 ), the point deduction model for “bow stance: front leg not bent into near half squat” is f (u k f gb15 ) = f (u k f gb151 ) ∨ f (u k f gb152 ). From f (u 16 ) = f (u 161 ) ∨ f (u 162 ), the point deduction model for “bow stance: heel of rear foot off carpet” is f (u k f gb16 ) = f (u k f gb161 ) ∨ f (u k f gb162 ). Establishing the general point deduction model for bow stance: ⎧ ⎪ 0; ⎨0, f (u k f gbi j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the bow stance. The point deducted is 0 when error value is

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0, and the point deducted is 0.1 when the error function value is 1. The point deduction model for bow stance is: f (u k f gb ) = f (u k f gb15 ) ∨ f (u k f gb16 ).

6.5.3.10

Computer Vision-based Error Identification Model for Horse-riding Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in horse-riding stance , 0.1 is deducted if there is the error of “upper body obviously bent forward” or “legs not bent into near half squat” or “thighs below horizontal level” or “feet not far apart enough” or “heels off carpet”. The essential movements in horse-riding stance: standing straight, then putting legs apart in a distance of equaling to the length of three feet, having feet facing forward, bending knees and lowering the upper body, keeping knees facing outward and not putting them forward or outward over the toe tips, thighs are parallel with horizontal level, keeping the crotch inward and preventing buttock from protruding, make the crotch form a circle, keeping spine straight and posture flat, holding arms outward and palms parallel to body, and keeping head upward. 1. Establishing Function for Errors in Horse-riding Stance. (1) u 17 = horse-riding stance: upper body obviously bent forward Suppose that domain U1 0 = horse-riding stance in optional Nanquan; thing S14 = torso; property T15 = the 3-D coordinate for the left shoulder is (xl jb , yl jb , zl jb ), the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), and the 3-D coordinate for the right shoulder is (xr jb , yr jb , zr jb ), the 3-D coordinate for the right hip joint is (xr tg , yr tg , zr tg ), the length of thigh is h 2 . According to the rules in horse-riding stance, the correct posture is to keep the torso straight and the error happens when upper body obviously is bent forward. The identification of error is done through investigating the relationship between the y coordinates of shoulder and hip joint. When the torso is straight, the y coordinates for the shoulder and hip joint is very close or have overlap, i.e., yl jb = yltg ± Δ and yr jb = zr tg ± Δ, the ±Δ is upper and lower limits for the difference, the maximum value of Δ is determined by the committee for codifying the rules. When the upper body is obviously bent forward, it is assumed that the “obviously” is defined as exceeding 15 of the thigh length, i.e., h52 . And yltg ≥ yl jb + h52 and yr tg ≥ yr jb + h52 . If yl jb = yltg ± Δ and yr jb = yr tg ± Δ, then there is no “upper body obviously bent forward”; if yltg ≥ yl jb + h52 and yr tg ≥ yr jb + h52 , then there exist “upper body obviously bent forward”. Establishing error function:  f (u 17 ) =

0, yl jb = yltg ± Δ and yr jb = zr tg ± Δ; 1, yltg ≥ yl jb + h52 and yr tg ≥ yr jb + h52 .

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If yl jb = yltg ± Δ and yr jb = zr tg ± Δ, there is no “upper body obviously bent forward” and the value of error function is 0; If yltg ≥ yl jb + h52 and yr tg ≥ yr jb + h52 , then there exist “upper body obviously bent forward” and the value of error function is 1. In conclusion, the error function for u 17 is f (u 17 ). (2) u 18 = horse-riding stance: legs not bent into near half squat; u 19 = thighs below horizontal level According to the description on the erroneous movements, points need to be deducted for both incorrect movements “legs not bent into near half squat” and “thighs below horizontal level”. The “legs not bent into near half squat” means that “thighs below horizontal level” or “thighs above horizontal level”. Therefore, we combine them together in this session. Suppose that domain U10 = horse-riding stance in optional Nanquan; thing S15 = bent legs in horse-riding stance; property T16 = the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ), and the 3-D coordinate for the right hip joint is (xr tg , yr tg , zr tg ). Referring to the rules in horse-riding stance, the legs should be bent into near half squat. It is assumed that the difference for z coordinates of both hip joint and knee should be within ±Δ, the maximum value of Δ is determined by the committee for codifying the rules. If zltg = zlxg ± Δ and zr tg = zr xg ± Δ, then legs are bent into near half squat; If zltg > zlxg + Δ and zltg < zlxg − Δ; zr tg > zr xg + Δ and zr tg < zr xg − Δ, then legs are not bent into near half squat. Establishing error function:  f (u 18 ) =

0, zltg ≤ zlxg + Δ and zltg ≥ zlxg − Δ; zrtg ≤ zr xg + Δ and zrtg ≥ zr xg − Δ 1, zltg > zlxg + Δ and zltg < zlxg − Δ, zrtg > zr xg + Δ and zrtg < zr xg − Δ

If zltg = zlxg ± Δ and zr tg = zr xg ± Δ, then legs are bent into near half squat and the value of error function is 0; If zltg > zlxg + Δ and zltg < zlxg − Δ; zr tg > zr xg + Δ and zr tg < zr xg − Δ, then legs are not bent into near half squat and the value of error function is 1. The error function for u 18 is f (u 18 ). (3) u 20 = horse-riding stance: feet not far apart enough Suppose that domain U10 = horse-riding stance in optional Nanquan; thing S16 = both feet; property T17 = the 3-D coordinate for the left heel is (xl jg , yl jg , zl jg ), and the 3-D coordinate for the right heel is (xr jg , yr jg , zr jg ). Referring to the rules in horse-riding stance, the distance between feet should be the length of three times of the foot. It is assumed that the length of a foot is h 4 : If xl jg =| xr jg ± 3h 4 | , feet are far apart enough; If xl jg 0 and dr > 0. If dl = 0 and dr = 0 , the heels are not off carpet and the value of error function is 0; If dl > 0 and dr > 0, the heel are off carpet and the value of error function is 1. The error function for u 21 is f (u 21 ). Of course,certain difference range can be set for d. . 2. Point deduction model for error in horse-riding stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for horse-riding stance is established hereby. Suppose that the object of horse-riding stance in optional Nanquan is noted by u mb .

f (u k f mbi j ) =

⎧ ⎪ ⎨0,

0; (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

where i stands for the ith point-deduction movement and j stands for the jth deduction point for the horse-riding stance. The total deduction function is f (u k f mb ). In f (u k f i j ), i stands for the ith point-deduction movement and j stands for the jth deduction point for the horse-riding stance. The point deduction representations for u 17 , u 20 , and u 21 are f (u k f mb17 ), f (u k f mb20 ), and f (u k f mb21 ) respectively. And the point deduction representation for u 18 and u 19 is f (u k f mb18 ). The comprehensive point deduction model for horse-riding stance is: f (u mb ) = f (u k f mb17 ) ∨ f (u k f mb18 ) ∨ f (u k f mb20 )∨ f (u k f mb21 ).

6.5.3.11

Computer Vision-based Error Identification Model for Empty Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in empty stance , 0.1 is deducted if there is the error of “rear leg not bent to near right angle” or “heel of rear leg off carpet”. 1. Establishing Function for Errors in Empty Stance. (1) u 22 = empty stance: rear leg not bent to near right angle Suppose that domain U11 = empty stance in optional Nanquan; thing S18 = the bent leg; property T19 = the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ), and the 3-D coordinate for the right hip joint is (xr tg , yr tg , zr tg ).

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(a) When the bent leg in empty stance is the left leg, the error function is f (u 221 ) The difference between z coordinates for hip joint and knee can serve as the criteria for judging if there exist the error of “rear leg not bent to near right angle”. Suppose that the upper limit (UL) and lower limit (LL) for judging the difference between the two z coordinates for hip joint and knee are ±Δmm, the maximum value of Δ is determined by the committee for codifying the rules. If zltg = zlxg ± Δ, rear leg is bent to near right angle; if zltg > zlxg + Δ or zltg < zlxg − Δ, rear leg is not bent to near right angle. Establishing error function:  0, zlxg − Δ ≤ zltg ≤ zlxg + Δ f (u 221 ) = 1, zltg > zlxg + Δ or zltg < zlxg − Δ If zlxg − Δ ≤ zltg ≤ zlxg + Δ , rear leg is bent to near right angle and the value of error function is 0; if zltg > zlxg + Δ or zltg < zlxg − Δ , rear leg is bent to near right angle and the value of error function is 1. (b) When the bent leg in empty stance is the right leg, the error function is f (u 222 ) Similar to the way of establishing error function for the left leg, we can establish error function for the case of the right leg:  f (u 222 ) =

0, zr xg − Δ ≤ zr tg ≤ zr xg + Δ 1, zr tg > zr xg + Δ or zr tg < zr xg − Δ

If zr xg − Δ ≤ zr tg ≤ zr xg + Δ , then front leg is bent into near half squat and the value of error function is 0; if zr tg > zr xg + Δ or zr tg < zr xg − Δ , then front leg is not bent into near half squat and the value of error function is 1. Hereby, the error function for f (u 22 ) = f (u 221 ) ∨ f (u 222 ) . (2) u 23 = empty stance: heel of rear leg off carpet Suppose that domain U11 = empty stance in optional Nanquan; thing S18 = rear leg; property T20 = the planes on which the left and right heels are. (a) The rear leg is the left leg Suppose that , the error function is f (u 231 ). The corresponding equation for the plane on which the left heel is:    x − xl jg1 y − yl jg1 z − zl jg1   xl jg2 − xl jg1 yl jg2 − yl jg1 zl jg2 − zl jg1  = 0   xl jg3 − xl jg1 yl jg3 − yl jg1 zl jg3 − zl jg1  Suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the left heel

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is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = yl jg2 zl jg3 − yl jg2 zl jg1 − yl jg1 zl jg3 − yl jg3 zl jg2 + yl jg3 zl jg1 + yl jg1 zl jg2 ; B = xl jg1 zl jg3 − xl jg1 zl jg2 + xl jg2 zl jg1 − xl jg2 zl jg3 + xl jg3 zl jg2 − xl jg3 zl jg1 ; C = xl jg1 zl jg2 − xl jg1 zl jg3 + xl jg2 zl jg3 − xl jg2 zl jg1 + xl jg3 zl jg1 − xl jg3 zl jg2 ; D = xl jg1 yl jg2 zl jg3 + xl jg1 yl jg3 zl jg2 +xl jg2 yl jg1 zl jg3 − xl jg2 yl jg3 zl jg1 + xl jg3 yl jg2 zl jg1 − xl jg3 yl jg3 zl jg2 . dl is the distance between the left heel and court ground: If dl = 0, heel of rear leg is not off carpet; If dl > 0, the heel of rear foot is off carpet. Establishing error function:  f (u 231 ) =

0, dl = 0 1, dl > 0

If dl = 0, the heel of rear foot is not off carpet and the value of error function is 0; If dl > 0, the heel of rear foot is off carpet and the value of error function is 1. (b) The rear leg is the right leg Suppose that the 3-D coordinate for a point on the supporting court ground is P1 (x1 , y1 , 0), the distance from P1 to the plane for the right heel is dr the error function is f (u 232 ), similarly, Establishing error function:  f (u 232 ) =

0, dr = 0 1, dr > 0

If dr = 0, the heel of rear foot is not off carpet and the value of error function is 0; If dr > 0, the heel of rear foot is off carpet and the value of error function is 1. In conclusion, the error function for f (u 23 ) is: f (u 23 ) = f (u 231 ) ∨ f (u 232 ). Of course,certain difference range can be set for d. 2. Point deduction model for error in empty stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for empty stance is established hereby. Suppose that the object

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of empty stance in optional Nanquan is noted by u xb . Establishing the general point deduction model for bow stance: ⎧ ⎪ ⎨0,

0; f (u k f xbi j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the bow stance. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. The total deduction function is f (u k f xb ). In f (u k f i j ), i stands for the ith pointdeduction movement and j stands for the jth deduction point for the bow stance. The point deduction representations for u 22 , u 221 , and u 222 are f (u k f xb22 ), f (u k f xb221 ), and f (u k f xb222 ) respectively. The point deduction representations for u 23 , u 231 , and u 232 are f (u k f xb23 ), f (u k f xb231 ), and f (u k f xb232 ) correspondingly. From f (u 22 ) = f (u 221 ) ∨ f (u 222 ), the point deduction model for “empty stance: rear leg not bent to near right angle” is f (u k f xb22 ) = f (u k f xb221 ) ∨ f (u k f xb222 ). From f (u 23 ) = f (u 231 ) ∨ f (u 232 ), the point deduction model for “empty stance: heel of rear leg off carpet” is f (u k f xb23 ) = f (u k f xb231 ) ∨ f (u k f xb232 ). The point deduction model for empty stance is: f (u k f xb ) = f (u k f xb22 ) ∨ f (u k f xb23 ).

6.5.3.12

Computer Vision-based Error Identification Model for Crouch Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in crouch stance , 0.1 is deducted if there is the error of “rear leg not bent completely” or “front leg bent” or “sole of front foot not turned inward and flat on carpet”. 1. Establishing Function for Errors in Crouch Stance. (1) u 24 = crouch stance: rear leg not bent completely Suppose that domain U12 = crouch stance in optional Nanquan; thing S19 = the bent leg; property T19 = the planes on which the back sides of the right and left calves, the planes on which the back sides of the right and left thighs are. (a) When the bent leg in crouch stance is the left leg, the error function is f (u 241 ) Suppose that three points are randomly selected from the plane on which the back side of the left calf is (xlxth1 , ylxth1 , zlxth1 ), (xlxth2 , ylxth2 , zlxth2 ), and (xlxth3 , ylxth3 , zlxth3 ). The corresponding plane equation for the back side of the left calf is:

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   x − xlxth1 y − ylxth1 z − zlxth1   xlxth2 − xlxth1 ylxth2 − ylxth1 zlxth2 − zlxth1  = 0   xlxth3 − xlxth1 ylxth3 − ylxth1 zlxth3 − zlxth1  Suppose that the 3-D coordinate for a point from the plane on which the court ground is P0 (x, y, 0), the distance from P0 to the plane for the back side of the left calf is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = ylxth2 zlxth3 − ylxth2 zlxth1 − ylxth1 zlxth3 − ylxth3 zlxth2 + ylxth3 zlxth1 + ylxth1 zlxth2 ; B = xlxth1 zlxth3 − xlxth1 zlxth2 + xlxth2 zlxth1 − xlxth2 zlxth3 + xlxth3 zlxth2 − xlxth3 zlxth1 ; C = xlxth1 zlxth2 − xlxth1 zlxth3 + xlxth2 zlxth3 − xlxth2 zlxth1 + xlxth3 zlxth1 − xlxth3 zlxth2 ; D = xlxth1 ylxth2 zlxth3 + xlxth1 ylxth3 zlxth2 + xlxth2 ylxth1 zlxth3 − xlxth2 ylxth3 zlxth1 + xlxth3 ylxth2 zlxth1 − xlxth3 ylxth3 zlxth2 . dl is the distance between the planes on which the back sides of the left calf and thigh. If dl = 0, rear leg is bent completely in empty stance; If dl > 0, rear leg is not bent completely in empty stance. Establishing error function:  0, dl = 0; f (u 241 ) = 1, dl > 0. If dl = 0, rear leg is bent completely in empty stance and the value of error function is 0; If dl > 0 rear leg is not bent completely in empty stance and the value of error function is 1. (b) When the bent leg in crouch stance is the right leg, the error function is f (u 242 ) Suppose that the 3-D coordinate for a point from the plane on which the court ground is P0 (x1 , y1 , 0), the distance from P1 to the plane for the back side of the right calf is dr . Similar to the way of establishing error function for the left leg, we can establish error function for the case of the right leg:  0, dr = 0; f (u 242 ) = 1, dr > 0.

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If dr = 0, rear leg is bent completely in empty stance and the value of error function is 0; If dr > 0 rear leg is not bent completely in empty stance and the value of error function is 1. Hereby, the error function for f (u 24 ) = f (u 241 ) ∨ f (u 242 ). Of course,certain difference range can be set for d. . (2) u 25 = crouch stance: front leg bent Suppose that domain U11 = empty stance in optional Nanquan; thing S20 = front leg; property T22 = the 3-D coordinate for the left knee is (xl jg , yl jg , zl jg ), the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), the 3-D coordinate for the right knee is (xr jg , yr jg , zr jg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ), and the 3-D coordinate for the right hip joint is (xr tg , yr tg , zr tg ) . (a) When the front leg in crouch stance is the left leg, the error function is f (u 251 ) Suppose that the 3-D coordinate for the left heel is (xl jg , yl jg , zl jg ) and the 3-D coordinate for the left hip joint is (xltg , yltg , zltg ), the equation connecting the two points is: x − xl jg y − yl jg z − zl jg = = xltg − xl jg yltg − yl jg zltg − zl jg If the 3-D coordinate for the left knee (xlxg , ylxg , zlxg ) does not meet the line equation, it indicates there exists “front leg bent”; If the 3-D coordinate for the left knee (xlxg , ylxg , zlxg ) meets the line equation, it indicates that the front leg is not bent. Establishing error function:  f (u 251 ) =

0, 1,

xlxg −xl jg xltg −xl jg xlxg −xl jg xltg −xl jg

= =

ylxg −yl jg yltg −yl jg ylxg −yl jg yltg −yl jg

= =

zlxg −zl jg zltg −zl jg zlxg −zl jg zltg −zl jg

If the 3-D coordinate for the left knee does not meet the line equation, it indicates there exists “front leg bent” and the value of error function is 1; If the 3-D coordinate for the left knee meets the line equation, it indicates that the front leg is not bent and the value of error function is 0. (b) When the front leg in crouch stance is the right leg, the error function is f (u 252 ) Similar to the way of establishing error function for the left leg, we can establish error function for the case of the right leg:  f (u 252 ) =

0, 1,

xr xg −xr jg xr tg −xr jg xr xg −xr jg xr tg −xr jg

= =

yr xg −yr jg yr tg −yr jg yr xg −yr jg yr tg −yr jg

= =

zr xg −zr jg zr tg −zr jg zr xg −zr jg zr tg −zr jg

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If the 3-D coordinate for the right knee does not meet the line equation, it indicates there exists “front leg bent”and the value of error function is 1; If the 3-D coordinate for the right knee meets the line equation, it indicates that the front leg is not bent and the value of error function is 0. Hereby, the error function for f (u 25 ) = f (u 251 ) ∨ f (u 252 ). (3) u 26 = crouch stance: sole of front foot not turned inward and flat on carpet Suppose that domain U12 = empty stance in optional Nanquan; thing S20 = front leg; property T23 = the plane for heel, toe tip, and sole of left foot, the plane for heel, toe tip, and sole of right foot. Suppose that the 3-D coordinate for the toe tip of left foot is (xl j j ,yl j j , zl j j ),the 3-D coordinate for the toe tip of right foot is (xr j j , yr j j , zr j j ), the 3-D coordinate for the left heel is (xl jg , yl jg , zl jg ), the 3-D coordinate for the distal phalanx of great toe is (x j z1 , y j z1 , z j z1 ), the 3-D coordinate for the distal phalanx of little toe is (x j z2 , y j z2 , z j z2 ), the coordinate for the central point of the sole of foot is (x j z3 , y j z3 , z j z3 ). Let’s construct a plane using the 3-D coordinates for the distal phalanx of great toe, the distal phalanx of little toe, and the central point of the sole of foot. Hereby, any point on the sole of the foot will be meet the equation: The general plane equation for the sole of a foot is:    x − x j z1 y − y j z1 z − z j z1    x j z2 − x j z1 y j z2 − y j z1 z j z2 − z j z1  = 0   x j z3 − x j z1 y j z3 − y j z1 z j z3 − z j z1  The corresponding plane equation for the sole of the left foot is:    x − xl j z1 y − yl j z1 z − zl j z1   xl j z2 − xl j z1 yl j z2 − yl j z1 zl j z2 − zl j z1  = 0   xl j z3 − xl j z1 yl j z3 − yl j z1 zl j z3 − zl j z1  The corresponding plane equation for the sole of the right foot is:    x − xr j z1 y − yr j z1 z − zr j z1   xr j z2 − xr j z1 yr j z2 − yr j z1 zr j z2 − zr j z1  = 0   xr j z3 − xr j z1 yr j z3 − yr j z1 zr j z3 − zr j z1  For the case of different legs, please see the following part. (a) When the front leg in crouch stance is the left leg, the error function is f (u 261 ) The corresponding plane equation for the sole of the left foot is:    x − xl j z1 y − yl j z1 z − zl j z1   xl j z2 − xl j z1 yl j z2 − yl j z1 zl j z2 − zl j z1  = 0   xl j z3 − xl j z1 yl j z3 − yl j z1 zl j z3 − zl j z1 

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Suppose that the 3-D coordinate for a point from the plane on which the court ground is P0 (x, y, 0), the distance from P0 to the plane for the sole of the left foot is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = yl j z2 zl j z3 − yl j z2 zl j z1 − yl j z1 zl j z3 − yl j z3 zl j z2 + yl j z3 zl j z1 + yl j z1 zl j z2 ; B = xl j z1 zl j z3 − xl j z1 zl j z2 + xl j z2 zl j z1 − xl j z2 zl j z3 + xl j z3 zl j z2 − xl j z3 zl j z1 ; C = xl j z1 zl j z2 − xl j z1 zl j z3 + xl j z2 zl j z3 − xl j z2 zl j z1 + xl j z3 zl j z1 − xl j z3 zl j z2 ; D = xl j z1 yl j z2 zl j z3 + xl j z1 yl j z3 zl j z2 + xl j z2 yl j z1 zl j z3 − xl j z2 yl j z3 zl j z1 + xl j z3 yl j z2 zl j z1 − xl j z3 yl j z3 zl j z2 . dl is the distance between the planes on which the sole of the left foot and the court ground are. If dl = 0, the sole of front foot is flat on carpet; If dl > 0, the sole of front foot is not flat on carpet. The difference between the y coordinates for heel and toe tip can be used to determine if the sole of front foot is turned inward.Establishing error function:  0, dl = 0 and yl j j ≥ yl jg ; f (u 261 ) = 1, dl > 0 and yl j j < yl jg . If dl = 0 and yl j j ≥ yl jg , sole of front foot is turned inward and flat on carpet and the value of error function is 0; If dl > 0 and yl j j < yl jg , sole of front foot is not turned inward and flat on carpet and the value of error function is 1. (b) When the front leg in crouch stance is the right leg, the error function is f (u 262 ) Similar to the way of establishing error function for the left leg, we can establish error function for the case of the right leg:  f (u 262 ) =

0, dr = 0 and yr j j ≥ yr jg ; 1, dr > 0 and yr j j < yr jg .

If dr = 0 and yr j j ≥ yr jg , sole of front foot is turned inward and flat on carpet and the value of error function is 0; If dr > 0 and yr j j < yr jg , sole of front foot is not turned inward and flat on carpet and the value of error function is 1. Therefore, the error function for u 26 is: f (u 26 ) = f (u 261 ) ∨ f (u 262 ). Of course,certain difference range can be set for d.

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2. Point deduction model for error in crouch stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for crouch stance is established hereby. Suppose that the object of crouch stance in optional Nanquan is noted by u pb . Establishing the general point deduction model for crouch stance: ⎧ ⎪ ⎨0,

0; f (u k f pbi j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the crouch stance. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. The total deduction function is f (u k f pb ). In f (u k f i j ), i stands for the ith pointdeduction movement and j stands for the jth deduction point for the crouch stance. The point deduction representations for u 24 , u 241 , and u 242 are f (u k f pb24 ), f (u k f pb241 ), and f (u k f pb242 ) respectively. The point deduction representations for u 25 , u 251 , and u 252 are f (u k f pb25 ), f (u k f pb251 ), and f (u k f pb252 ) correspondingly. The point deduction representations for u 26 , u 261 , and u 262 are f (u k f pb26 ), f (u k f pb261 ), and f (u k f pb262 ) respectively. From f (u 24 ) = f (u 241 ) ∨ f (u 242 ), the point deduction model for “crouch stance: rear leg not bent completely” is f (u k f pb24 ) = f (u k f pb241 ) ∨ f (u k f pb242 ); From f (u 25 ) = f (u 251 ) ∨ f (u 252 ), the point deduction model for “crouch stance: front leg bent” is f (u k f pb25 ) = f (u k f pb251 ) ∨ f (u k f pb252 ); From f (u 26 ) = f (u 261 ) ∨ f (u 262 ), the point deduction model for “crouch stance: sole of front foot not turned inward and flat on carpet” is f (u k f pb26 ) = f (u k f pb261 ) ∨ f (u k f pb262 ). The point deduction model for crouch stance is: f (u k f pb ) = f (u k f pb24 ) ∨ f (u k f pb25 ) ∨ f (u k f pb26 ).

6.5.3.13

Computer Vision-based Error Identification Model for Butterfly Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in butterfly stance, 0.1 is deducted if there is the error of “inner side of calf of kneeling leg not on carpet” or “inner side of heel of kneeling leg not on carpet”. 1. Establishing function for errors in butterfly stance. (1) u 27 = butterfly stance: inner side of calf of kneeling leg not on carpet Suppose that domain U13 = butterfly stance in optional Nanquan; thing S21 = the kneeling leg; property T24 = the planes on which the inner side calf is, the planes on which the inner side of right calf is.

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(a) When kneeling leg in butterfly stance is the left leg, the error function is f (u 271 ) Suppose that three points from inner side of left calf are selected, they have the 3-D coordinates of (xlxtn1 , ylxtn1 , zlxtn1 ), (xlxtn2 , ylxtn2 , zlxtn2 ), and (xlxtn3 , ylxtn3 , zlxtn3 ). The corresponding plane equation for the inner side of the left calf is:    x − xlxtn1 y − ylxtn1 z − zlxtn1   xlxtn2 − xlxtn1 ylxtn2 − ylxtn1 zlxtn2 − zlxtn1  = 0   xlxtn3 − xlxtn1 ylxtn3 − ylxtn1 zlxtn3 − zlxtn1  Suppose that the 3-D coordinate for a point from the plane on which the court ground is P0 (x, y, 0), the distance from P0 to the plane for the inner side of the left calf is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = ylxtn2 zlxtn3 − ylxtn2 zlxtn1 − ylxtn1 zlxtn3 − ylxtn3 zlxtn2 + ylxtn3 zlxtn1 + ylxtn1 zlxtn2 ; B = xlxtn1 zlxtn3 − xlxtn1 zlxtn2 + xlxtn2 zlxtn1 − xlxtn2 zlxtn3 + xlxtn3 zlxtn2 − xlxtn3 zlxtn1 ; C = xlxtn1 zlxtn2 − xlxtn1 zlxtn3 + xlxtn2 zlxtn3 − xlxtn2 zlxtn1 + xlxtn3 zlxtn1 − xlxtn3 zlxtn2 ; D = xlxtn1 ylxtn2 zlxtn3 + xlxtn1 ylxtn3 zlxtn2 + xlxtn2 ylxtn1 zlxtn3 − xlxtn2 ylxtn3 zlxtn1 + xlxtn3 ylxtn2 zlxtn1 − xlxtn3 ylxtn3 zlxtn2 . dl is the distance between the planes on which the sole of the left foot and the court ground are. If dl = 0, inner side of calf of kneeling leg is on carpet; If dl > 0, inner side of calf of kneeling leg is not on carpet. Establishing error function:  f (u 271 ) =

0, dl = 0; 1, dl > 0.

If dl = 0, inner side of calf of kneeling leg is on carpet and the value of error function is 0; If dl > 0, inner side of calf of kneeling leg is not on carpet and the value of error function is 1. (b) When kneeling leg in butterfly stance is the right leg, the error function is f (u 272 ) Suppose that the 3-D coordinate for a point from the plane on which the court ground is P1 (x1 , y2 , 0), the distance from P1 to the plane for the inner side of the right calf is dr where

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Similar to the way of establishing error function for the left leg, we can establish error function for the case of the right leg:  0, dr = 0; f (u 272 ) = 1, dr > 0. If dr = 0, inner side of calf of kneeling leg is on carpet and the value of error function is 0; If dr > 0, inner side of calf of kneeling leg is not on carpet and the value of error function is 1. Therefore, the error function for u 27 is: f (u 27 ) = f (u 271 ) ∨ f (u 272 ). Of course,certain difference range can be set for d. (2) u 28 = butterfly stance: inner side of heel of kneeling leg not on carpet Suppose that domain U13 = butterfly stance in optional Nanquan; thing S22 = kneeling leg; property T25 = the planes on which inner side of left heel and inner side of right heel. (a) When kneeling leg in butterfly stance is the left leg, the error function is f (u 281 ) Suppose that three points from inner side of left heel are selected, they have the 3-D coordinates of (xl jgn1 , yl jgn1 , zl jgn1 ), (xl jgn2 , yl jgn2 , zl jgn2 ), and (xl jgn3 , yl jgn3 , zl jgn3 ). The corresponding plane equation for the inner side of the left heel is:    x − xl jgn1 y − yl jgn1 z − zl jgn1   xl jgn2 − xl jgn1 yl jgn2 − yl jgn1 zl jgn2 − zl jgn1  = 0   xl jgn3 − xl jgn1 yl jgn3 − yl jgn1 zl jgn3 − zl jgn1  Suppose that the 3-D coordinate for a point from the plane on which the court ground is P0 (x, y, 0), the distance from P0 to the plane for the inner side of the left heel is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = yl jgn2 zl jgn3 − yl jgn2 zl jgn1 − yl jgn1 zl jgn3 − yl jgn3 zl jgn2 + yl jgn3 zl jgn1 + yl jgn1 zl jgn2 ; B = xl jgn1 zl jgn3 − xl jgn1 zl jgn2 + xl jgn2 zl jgn1 − xl jgn2 zl jgn3 + xl jgn3 zl jgn2 − xl jgn3 zl jgn1 ; C = xl jgn1 zl jgn2 − xl jgn1 zl jgn3 + xl jgn2 zl jgn3 − xl jgn2 zl jgn1 + xl jgn3 zl jgn1 − xl jgn3 zl jgn2 ; D = xl jgn1 yl jgn2 zl jgn3 + xl jgn1 yl jgn3 zl jgn2 + xl jgn2 yl jgn1 zl jgn3 − xl jgn2 yl jgn3 zl jgn1 + xl jgn3 yl jgn2 zl jgn1 − xl jgn3 yl jgn3 zl jgn2 . dl is the distance between the planes on which the inner side of the left heel is and the court ground are.

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If dl = 0, the inner side of heel of kneeling leg is on carpet; If dl > 0, the inner side of heel of kneeling leg is not on carpet. Establishing error function:  f (u 281 ) =

0, dl = 0; 1, dl > 0.

If dl = 0, the inner side of heel of kneeling leg is on carpet and the value of error function is 0; If dl > 0, the inner side of heel of kneeling leg is not on carpet and the value of error function is 1. (b) When kneeling leg in butterfly stance is the right leg, the error function is f (u 282 ) Suppose that the 3-D coordinate for a point from the plane on which the court ground is P1 (x1 , y2 , 0), the distance from P1 to the plane for the inner side of the right calf is dr where Similar to the way of establishing error function for the left leg, we can establish error function for the case of the right leg:  f (u 282 ) =

0, dr = 0; 1, dr > 0.

If dr = 0, the inner side of heel of kneeling leg is on carpet and the value of error function is 0; If dr > 0, the inner side of heel of kneeling leg is not on carpet and the value of error function is 1. Therefore, the error function for u 28 is: f (u 28 ) = f (u 281 ) ∨ f (u 282 ). Of course,certain difference range can be set for d. 2. Point deduction model for error in butterfly stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for butterfly stance is established hereby. Suppose that the object of butterfly stance in optional Nanquan is noted by u pb . Establishing the general point deduction model for butterfly stance: ⎧ ⎪ ⎨0,

0; f (u k f dbi j ) = (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1. where i stands for the ith point-deduction movement and j stands for the jth deduction point for the butterfly stance. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1.

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6 Applications of Error Logic

The total deduction function is f (u k f db ). In f (u k f i j ), i stands for the ith pointdeduction movement and j stands for the jth deduction point for the butterfly stance. The point deduction representations for u 27 , u 271 , and u 272 are f (u k f db27 ), f (u k f db271 ), and f (u k f db272 ) respectively. The point deduction representations for u 28 , u 281 , and u 282 are f (u k f db28 ), f (u k f db281 ), and f (u k f db282 ) correspondingly. From f (u 27 ) = f (u 271 ) ∨ f (u 272 ), the point deduction model for “butterfly stance: inner side of calf of kneeling leg not on carpet” is f (u k f db27 ) = f (u k f db271 ) ∨ f (u k f db272 ); From f (u 28 ) = f (u 281 ) ∨ f (u 282 ), the point deduction model for “butterfly stance: inner side of heel of kneeling leg not on carpet” is f (u k f db28 ) = f (u k f db281 ) ∨ f (u k f db282 ). The point deduction model for butterfly stance is: f (u k f db ) = f (u k f db27 ) ∨ f (u k f db28 ).

6.5.3.14

Computer Vision-based Error Identification Model for Bent-knee Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in bent-knee stance, 0.1 is deducted if there is the error of “bent knee on carpet” or “buttocks not on calf of bent leg”. 1. Establishing Function for Errors in Bent-knee Stance. (1) u 29 = bent-knee stance: bent knee on carpet Suppose that domain U14 = bent-knee stance in optional Nanquan; thing S23 = the bent leg; property T26 = the 3-D coordinate of the left knee (xlxg , ylxg , zlxg ), the 3-D coordinate of the right knee (xr xg , yr xg , zr xg ). The z coordinate of knee can be used to determine if the bent knee is on carpet. (a) When kneeling leg in bent-knee stance is the left leg, the error function is f (u 291 ) If zlxg = 0, bent knee is on carpet; If zlxg > 0, bent knee is not on carpet. Establishing error function:  f (u 291 ) =

1, zlxg = 0; 0, zlxg > 0.

If zlxg = 0, bent knee is on carpet and the value of error function is 1; If zlxg > 0, bent knee is not on carpet and the value of error function is 0. (b) When kneeling leg in bent-knee stance is the right leg, the error function is f (u 292 ) If zr xg = 0, bent knee is on carpet; If zr xg > 0, bent knee is not on carpet. Establishing error function:

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

599

 1, zr xg = 0; f (u 292 ) = 0, zr xg > 0. If zr xg = 0, bent knee is on carpet and the value of error function is 1; If zr xg > 0, bent knee is not on carpet and the value of error function is 0. Therefore, the error function for u 29 is: f (u 29 ) = f (u 291 ) ∨ f (u 292 ). (2) u 30 = bent-knee stance: buttocks not on calf of bent leg Suppose that domain U14 = bent-knee stance in optional Nanquan; thing S24 = buttocks; property T27 = the coordinates for buttocks. Suppose that the 3-D coordinate for a point on left buttock is (xltb , yltb , zltb ) , the 3-D coordinate for a point on right buttock is (xr tb , yr tb , zr tb ), the 3-D coordinate for the middle point is (xmtb , ymtb , z mtb ). When studying the plane formed by left buttock, three points on the left buttock are chosen to the plane: (xltb1 , yltb1 , zltb1 ), (xltb2 , yltb2 , zltb2 ), (xltb3 , yltb3 , zltb3 ): The corresponding plane equation for the left buttock is:    x − xltb1 y − yltb1 z − zltb1   xltb2 − xltb1 yltb2 − yltb1 zltb2 − zltb1  = 0   xltb3 − xltb1 yltb3 − yltb1 zltb3 − zltb1  Suppose that the 3-D coordinate for a point from the plane on which the left calf is P0 (x, y, 0), the distance from P0 to the plane for the left buttock is dl where, dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = yltb2 zltb3 − yltb2 zltb1 − yltb1 zltb3 − yltb3 zltb2 + yltb3 zltb1 + yltb1 zltb2 ; B = xltb1 zltb3 − xltb1 zltb2 + xltb2 zltb1 − xltb2 zltb3 + xltb3 zltb2 − xltb3 zltb1 ; C = xltb1 zltb2 − xltb1 zltb3 + xltb2 zltb3 − xltb2 zltb1 + xltb3 zltb1 − xltb3 zltb2 ; xltb2 yltb3 zltb1 + D = xltb1 yltb2 zltb3 + xltb1 yltb3 zltb2 + xltb2 yltb1 zltb3 − xltb3 yltb2 zltb1 − xltb3 yltb3 zltb2 . dl is the distance between the planes on which the left calf is and the left buttock. If dl > 0, buttocks are not on calf of bent leg; If dl = 0, buttocks are on calf of bent leg. Establishing error function:  f (u 30 ) =

0, dl = 0; 1, dl > 0.

If dl > 0, buttocks are not on calf of bent leg and the value of error function is 1;

600

6 Applications of Error Logic

If dl = 0, buttocks are on calf of bent leg and the value of error function is 0. Therefore, the error function for u 30 is: f (u 38 ). Of course,certain difference range can be set for d. 2. Point deduction model for error in bent-knee stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for bent-knee stance is established hereby. Suppose that the object of bent-knee stance in optional Nanquan is noted by u gub . Establishing the general point deduction model for bent-knee stance:

f (u k f i j ) =

⎧ ⎪ ⎨0,

0; (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

where i stands for the ith point-deduction movement and j stands for the jth deduction point for the bent-knee stance. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. The total deduction function is f (u k f gub ). In f (u k f i j ), i stands for the ith pointdeduction movement and j stands for the jth deduction point for the bent-knee stance. The point deduction representation for u 29 is f (u k f gub29 ). The point deduction representations for u 30 is f (u k f gub30 ). From f (u 29 ) = f (u 291 ) ∨ f (u 292 ), the point deduction model for “bent-knee stance: bent knee on carpet” is f (u k f gub29 ) = f (u k f gub291 ) ∨ f (u k f gub292 ); From f (u 30 ), the point deduction model for “bent-knee stance: buttocks not on calf of bent leg” is f (u k f gub30 ). The point deduction model for bent-knee stance is: f (u k f gub ) = f (u k f gub29 ) ∨ f (u k f gub30 ).

6.5.3.15

Computer Vision-based Error Identification Model for Dragon-riding Stance in Optional Nanquan

According the rules in optional Nanquan, after completing the whole movements in bent-knee stance, 0.1 is deducted if there is the error of “knee of rear leg on carpet” or “front leg not bent into near half squat”. 1. Establishing Function for Errors in Dragon-riding Stance. (1) u 31 = dragon-riding stance: knee of rear leg on carpet Suppose that domain U15 = dragon-riding stance in optional Nanquan; thing S25 = the rear leg; property T28 = the 3-D coordinate of the left knee (xlxg , ylxg , zlxg ), the 3-D coordinate of the right knee (xr xg , yr xg , zr xg ). The z coordinate of rear knee can be used to determine if the rear knee is on carpet.

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

601

(a) When rear leg in dragon-riding stance is the left leg, the error function is f (u 311 ) If zlxg = 0, rear knee is on carpet; If zlxg > 0, rear knee is not on carpet. Establishing error function:  1, zlxg = 0; f (u 311 ) = 0, zlxg > 0. If zlxg = 0, rear knee is on carpet and the value of error function is 1; If zlxg > 0, rear knee is not on carpet and the value of error function is 0. (b) When rear leg in dragon-riding stance is the right leg, the error function is f (u 312 ) If zr xg = 0, rear knee is on carpet; If zr xg > 0, rear knee is not on carpet. Establishing error function:  f (u 312 ) =

1, zr xg = 0; 0, zr xg > 0.

If zr xg = 0, rear knee is on carpet and the value of error function is 1; If zr xg > 0, rear knee is not on carpet and the value of error function is 0. Therefore, the error function for u 31 is: f (u 31 ) = f (u 311 ) ∨ f (u 312 ). (2) u 32 = dragon-riding stance: front leg not bent into near half squat Suppose that domain U15 = dragon-riding stance in optional Nanquan; thing S26 = front let; property T29 = the 3-D coordinate for the left knee is (xlxg , ylxg , zlxg ), the 3-D coordinate for the hip joint of the left thigh is (xltg , yltg , zltg ), the 3-D coordinate for the right knee is (xr xg , yr xg , zr xg ) , the 3-D coordinate for the hip joint of the right thigh is (xr tg , yr tg , zr tg ). (a) When front leg in dragon-riding stance is the left leg The error function is f (u 321 ), The difference between z coordinates for hip joint and knee can serve as the criteria for judging if there exist the error of “front leg not bent into near half squat”. Suppose that the upper limit (UL) and lower limit (LL) for judging the difference between the two z coordinates for the hip joint and knee are ±Δmm, the maximum value of Δ is determined by the committee for codifying the rules. If zltg = zltg ± Δ, then front leg is bent into near half squat; If zltg > zltg + Δ or zltg < zltg − Δ, then front leg is not bent into near half squat. Establishing error function:  f (u 321 ) =

0, zltg − Δ ≤ zltg ≤ zltg + Δ 1, zltg > zltg + Δ or zltg < zltg − Δ

602

6 Applications of Error Logic

If zltg − Δ ≤ zltg ≤ zltg + Δ , then front leg is bent into near half squat and the value of error function is 0; If zltg > zltg + Δ or zltg < zltg − Δ , then front leg is not bent into near half squat and the value of error function is 1. (b) When front leg in dragon-riding stance is the right leg, the error function is f (u 322 ) If zr xg = 0, front leg is bent into near half squat; If zr xg > 0, front leg is not bent into near half squat. Establishing error function:  f (u 322 ) =

0, zr tg − Δ ≤ zr tg ≤ zr tg + Δ 1, zr tg > zr tg + Δ or zr tg < zr tg − Δ

If zr tg − Δ ≤ zr tg lezr tg + Δ , then front leg is bent into near half squat and the value of error function is 0; If zr tg > zr tg + Δ or zr tg < zr tg − Δ , then front leg is not bent into near half squat and the value of error function is 1. Therefore, the error function for u 32 is: f (u 32 ) = f (u 321 ) ∨ f (u 322 ). 2. Point deduction model for error in dragon-riding stance in optional Nanquan Per rules in Nanquan, 0.1 is deducted if one error appears for any movement; if there are two or more errors in the same movement, 0.1 is deducted. The point deduction model for dragon-riding stance is established hereby. Suppose that the object of dragon-riding stance in optional Nanquan is noted by u ql . Establishing the general point deduction model for dragon-riding stance:

f (u k f qli j ) =

⎧ ⎪ ⎨0,

0; (i, j = 1, 2, . . . , n) ⎪ ⎩ −0.1, 1.

where i stands for the ith point-deduction movement and j stands for the jth deduction point for the dragon-riding stance. The point deducted is 0 when error value is 0, and the point deducted is 0.1 when the error function value is 1. The total deduction function is f (u k f ql ). In f (u k f i j ), i stands for the ith pointdeduction movement and j stands for the jth deduction point for the dragonriding stance. The point deduction representation for u 31 , u 311 , and u 312 are f (u k f ql31 ), f (u k f ql311 ), and f (u k f ql312 ) respectively. The point deduction representations for u 32 , u 321 , and u 322 are f (u k f ql32 ), f (u k f ql321 ), and f (u k f ql322 ) correspondingly. From f (u 31 ) = f (u 311 ) ∨ f (u 312 ), the point deduction model for “dragonriding stance: knee of rear leg on carpet” is f (u k f ql31 ) = f (u k f ql311 ) ∨ f (u k f ql312 ); From f (u 32 ) = f (u 321 ) ∨ f (u 322 ), the point deduction model for “dragon-riding stance: front leg not bent into near half squat” is f (u k f ql32 ) = f (u k f ql321 ) ∨ f (u k f ql322 ). The point deduction model for dragon-riding stance is: f (u k f ql ) = f (u k f ql31 ) ∨ f (u k f ql32 ).

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

603

The total point deduction model for optional Nanquan is: f (u A ) = f (u k f qgs ) + f (u k f hd ) + f (u k f t f ) + f (u k f x f ) + f (u k f tw ) + f (u k f t p ) + f (u k f ly ) + f (u k f tc ) + f (u k f gb ) + f (u k f mb ) + f (u k f xb ) + f (u k f pb ) + f (u k f db ) + f (u k f gub ) + f (u k f ql ). The above session discusses the point deduction model for 15 movements and 32 deduction points in optional Nanquan (A group). In practice, each movement can be divided into n intervals (e.g. 25 intervals in 1 minute) and a sample in an interval can be picked to compute the total deducted points by using the above-mentioned models. For each movement in the routines of optional Nanquan, an error identification model is established. A complete error identification model can be constructed for arranging the individual error identification models according to the sequences of movements in the routines of optional Nanquan. With the help of electronic display system, the points deducted can be displayed on the screen. Therefore, the computer vision-based decision support system for judging optional Nanquan (A group) is realized.

6.5.4 Application of Computer Vision-based Error Identification Model for Optional Nanquan In this example, point deduction model is employed to calculate total points deducted in performing a complete set of movements in the routines of optional Nanquan. The sequence of athlete A performing the movements in the routines of optional Nanquan is as follows: (1) bring feet together and draw fists, (2) cup one fist in the other hand and shake feet, (3) punching in left bow stance, (4) whip boxing in high empty stance, (5) punching in dragon-riding stance, (6) punching in left bow stance, (7) fist chopping in left bow stance, (8) palm cutting in left bow stance, (9) double fist chopping in horse-riding stance, (10) lightning lift of double palms in horse-riding stance, (11) double index fingers pushing in horse-riding stance, (12) lowering palms and striking out in horse-riding stance, (13) raising both fists in right bow stance, (14) elbow striking in dragon-riding stance, (15) fist chopping in empty stance, 16) pushing palm in dragon-riding stance, (17) tornado kick and outward kick in flight, (18) side sole kick in flight with swooping and leg sweeping, (19) kip-up, (20) crane-beak strike in empty stance, (21) double tiger claws with one standing foot, (22) punching in left bow stance, (23) turn body and strike fist back, (24) punching in front kick, (25) vertical strike with heart fist in kneeling stance, (26) pounding fist in dragon-riding stance, (27) fist chopping in horse-riding stance, (28) turn body and vertical strike with left heart fist, (29) horizontal nail kick and punching in right bow stance, (30) slashing double eagle claws, horizontal stepping, pushing double palms, (31) slapping ground in kneeling stance, (32) fist striking and punching in left bow stance, (33) right throwing fist in left bow stance, (34) left throwing fist in right bow stance, (35) vertical strike with heart fist in left bow stance, (36) whipping fist with cross steps, (37) turn body and vertical strike with heart fist, (38) fist throwing and striking in dragging stance, (39) palms striking in

604

6 Applications of Error Logic

horse-riding stance, (40) fist horizontal sweeping in right bow stance and then fist striking, (41) thrusting fist downward in a seated stance, (42) double-hanging fist in horse-riding stance, (43) double tiger claws in kneeling stance, (44) slapping the instep of right foot, forming half horse-riding stance, punching fist, (45) pressing elbow in single butterfly stance, (46) fist punching in cross steps, (47) fist punching in step forward (48) turn body and vertical strike with palm, (49) fist side punching in horse-riding stance, (50) turn body, fist sweeping around the body, forming right bow stance (51) double index finger pushing in right bow stance, (52) step forward and vertical strike with heart fist, (53) dragging stance and fist throwing and striking, (54) dragging stance and fist punching, (55) cross step, whipping fist, turn body and vertical strike with heart fist, (56) palms lifting in right bow stance, (57) turn body, sweeping fist, step backward, fist punching, (58) double palm pushing in left bow stance, (59) empty stance, palm pushing, fist punching, (60) bring feet together and draw fists. Suppose that there are three errors in the above 60 movements according to the rules in the contest: (1) The 5th movement: the knee of the rear leg is on the carpet in dragon-riding stance and rear leg is right leg; (2) The 19th movement: support with left palm in the kip-up movement; (3) The 42th movement: the distance between feet is too short in horse-riding stance. Based on the model constructed for the routines in optional Nanquan, the point deduction calculation is presented as follows. Several notations need to be clarified: (1) The routines in the optional Nanquan chosen by athlete A should include not only the mandatory 15 movements but also some basic stances, i.e., resting stance (bring feet together), kick forward, seated stance, cross steps, and dragging stance. Point must also be deducted when error appears in one of the basic stances and referee uses the criteria evaluating the technical competency to make judgment. However, in the contest for the routines performance (A group), specific requirements were not listed for exercising point deduction; (2) The routines in the optional Nanquan performed by athlete A do not include the rules for judging hand formations because rules in routines performance of group A do not provide the specific specifications on how much points are deducted in this respect. Referee uses the criteria of evaluating the technical competency to make judgment. In this model, we do not discuss the point deduction for hand formation related errors. Suppose that domain U (t) is the contest for routines performance in optional Nanquan, thing S(t) is athlete A, properties T (t) are the height, the shoulder height, the length of thigh, the length of calf, and the length of feet, property (attribute) values L(t) are h 0 , h 1 , h 2 , h 3 , and h 4 . Suppose that the height of athlete A is 164 cm, the height of shoulder is 140 cm, the length of thigh is 90 cm, the length of calf is 50 cm, then the length of feet is 22 cm, the property logic matrix of athlete A is as follows:

6.5 Application of Error Logic in Decision Support System for Nanquan Referees

 contest o f  contest o f  B A = contest o f contest o f  contest o f

6.5.4.1

optional optional optional optional optional

N anquan N anquan N anquan N anquan N anquan

athelete A athelete A athelete A athelete A athelete A

height shoulder height legth o f thigh length o f cal f length o f f eet

605

 164cm  140cm 40cm  50cm  22cm 

Computer Vision-based Error Identification Model for Dragon-riding Stance in Optional Nanquan

Suppose that the error function for the knee of the rear leg is on the carpet in dragonriding stance is f (u 51 ), where 5 is the 5th movement and 1 is the 1st deduction point in dragon-riding stance; the error function for the support with left palm in the kip-up movement is f (u 191 ), where 19 is the 19th movement and 1 is the 1st deduction point in kip-up movement; the error function for the distance between feet is too short in horse-riding stance is f (u 424 ), where 42 is the 42th movement and 4 is the 4th deduction point in horse-riding stance. 1. u 51 = knee of the rear leg is on the carpet in dragon-riding stance Suppose that domain U15 = dragon-riding stance in optional Nanquan; thing S31 = the rear leg; property T31 = the 3-D coordinate of the left knee (xlxg , ylxg , zlxg ). The z coordinate of rear knee can be used to determine if the rear knee is on carpet. If zlxg = 0, rear knee is on carpet; if zlxg > 0, rear knee is not on carpet. Establishing error function:  f (u 51 ) =

1, zlxg = 0; 0, zlxg > 0.

If zlxg = 0, rear knee is on carpet and the value of error function is 1; If zlxg > 0, rear knee is not on carpet and the value of error function is 0. 2. u 191 = support with left palm in the kip-up movement Suppose that domain U7 = lift-up in kip-up; thing S12 = left palm; property T12 = the 3-D coordinate for distal phalanx is (xsh1 , ysh1 , z sh1 ), the 3-D coordinate for capitate is (xsh2 , ysh2 , z sh2 ), and the 3-D coordinate for the center of the palm is (xsh3 , ysh3 , z sh3 ). From the above three coordinate points, a plane can obtained and any point must satisfy the following equation.    x − xlsh1 y − ylsh1 z − zlsh1   xlsh2 − xlsh1 ylsh2 − ylsh1 zlsh2 − zlsh1  = 0   xlsh3 − xlsh1 ylsh3 − ylsh1 zlsh3 − zlsh1  Suppose that the 3-D coordinate for a point on the supporting court ground is P0 (x, y, 0), the distance from P0 to the plane for the left palm is dl where,

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6 Applications of Error Logic

dl =

| Ax0 + By0 + C z 0 + D | √ A2 + B 2 + C 2

A = ylsh2 zlsh3 − ylsh2 zlsh1 − ylsh1 zlsh3 − ylsh3 zlsh2 + ylsh3 zlsh1 + ylsh1 zlsh2 ; B = xlsh1 zlsh3 − xlsh1 zlsh2 + xlsh2 zlsh1 − xlsh2 zlsh3 + xlsh3 zlsh2 − xlsh3 zlsh1 ; C = xlsh1 zlsh2 − xlsh1 zlsh3 + xlsh2 zlsh3 − xlsh2 zlsh1 + xlsh3 zlsh1 − xlsh3 zlsh2 ; D = xlsh1 ylsh2 zlsh3 + xlsh1 ylsh3 zlsh2 + xlsh2 ylsh1 zlsh3 − xlsh2 ylsh3 zlsh1 + xlsh3 ylsh2 zlsh1 − xlsh3 ylsh3 zlsh2 . dl is the distance between the left palm and court ground: If dl > 0, there is no support with left palm in the kip-up movement; If dl = 0, there is support with left palm in the kip-up movement. Establishing error function:  f (u 191 ) =

1, dl = 0 0, dl > 0

If dl > 0, there is no support with left palm in the kip-up movement and the value of error function is 0; If dl = 0, there is support with left palm in the kip-up movement and the value of error function is 1. 3. u 424 = distance between feet is too short in horse-riding stance Suppose that domain U10 = horse-riding stance in optional Nanquan; thing S20 = both feet; property T20 = the 3-D coordinate for left heel is (xl jg , yl jg , zl jg ), the 3-D coordinate for right heel is (xr jg , yr jg , zr jg ). According to the rules, the distance between feet should be 3 times of the length of foot. Suppose that the length of food is h 4 = 22 cm. If xl jg =| xl jg ± 66 | cm, the distance between feet is appropriate in horse-riding stance; If xl jg