R-Calculus, V: Description Logics 9789819964598, 9789819964604

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Perspectives in Formal Induction, Revision and Evolution

Wei Li Yuefei Sui

R-Calculus, V: Description Logics

Perspectives in Formal Induction, Revision and Evolution Editor-in-Chief Wei Li, Beijing, China Series Editors Jie Luo, Beijing, China Yuefei Sui, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

Perspectives in Formal Induction, Revision and Evolution is a book series focusing on the logics used in computer science and artificial intelligence, including but not limited to formal induction, revision and evolution. It covers the fields of formal representation, deduction, and theories or meta-theories of induction, revision and evolution, where the induction is of the first level, the revision is of the second level, and the evolution is of the third level, since the induction is at the formula stratum, the revision is at the theory stratum, and the evolution is at the logic stratum. In his book “The Logic of Scientific Discovery”, Karl Popper argues that a scientific discovery consists of conjecture, theory, refutation, and revision. Some scientific philosophers do not believe that a reasonable conjecture can come from induction. Hence, induction, revision and evolution have become a new territory for formal exploration. Focusing on this challenge, the perspective of this book series differs from that of traditional logics, which concerns concepts and deduction. The series welcomes proposals for textbooks, research monographs, and edited volumes, and will be useful for all researchers, graduate students, and professionals interested in the field.

Wei Li · Yuefei Sui

R-Calculus, V: Description Logics

Wei Li State Key Laboratory of Software Development Environment Beihang University Beijing, China

Yuefei Sui Institute of Computing Technology Chinese Academy of Sciences Beijing, China

ISSN 2731-3689 ISSN 2731-3697 (electronic) Perspectives in Formal Induction, Revision and Evolution ISBN 978-981-99-6459-8 ISBN 978-981-99-6460-4 (eBook) https://doi.org/10.1007/978-981-99-6460-4 Jointly published with Science Press The print edition is not for sale in China mainland. Customers from China mainland please order the print book from: Science Press. © Science Press 2024 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 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 publishers, 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 publishers 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 publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface to the Series

Classical mathematical logics (propositional logic, first-order logic, and modal logic) and applied logics (temporal logic, dynamic logic, situation calculus, etc.) formalize deduction, a logical process from universal statements to particular statements. Description logics formalize concepts which and deduction compose of general logics. Induction and belief revision are the topics of general logics and philosophical logics, and evolution is a new research area in computer science. To formalize induction, revision, and evolution is a goal of this series. Revision is omnipresent in sciences. A new theory usually is a revision of an old theory or several old theories. Copernicus’ heliocentric theory is a revision of the Tychonic system; the theory of relativity is a revision of the classical theory of movement; the quantum theory is a revision of classical mechanics; etc. The AGM postulates is a set of conditions a reasonable revision operator should satisfy. Professor Li proposed a calculus for first-order logic, called R-calculus, which is sound and complete with respect to maximal consistent subsets. R-calculus has several variants which can be used in other logics, such as nonmonotonic logics, can propose new problems in the classical logics, and will be used in big data. Popper proposed in his book The Logic of Scientific Discovery that a scientific discovery consists of four aspects: conjecture, theory, refutation, and revision. Some scientific philosophers refuted that a reasonable conjecture should come from induction. Hence, induction, revision, and evolution become a new territory to be discovered in a formal way. An induction process is from particular statements to universal statements. A typical example is the mathematical induction, which is a set of nontrivial axioms in Peano arithmetic which makes Peano arithmetic incomplete with respect to the standard model of Peano arithmetic. A logic for induction is needed to guide datamining in artificial intelligence. Data-mining is a canonical induction process, which mines rules from data. In biology, evolution is a change in the heritable traits of biological populations over successive generations. In sciences, Darwin’s evolution theory is an evolution of intuitive theories of plants and animals. In logic, an evolution is a generating process v

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Preface to the Series

of combining two logics into a new logic, where the new logic should have the traits of two logics. Hence, we should define the corresponding heritable traits of logics, sets of logics, and sequences of logics. Simply speaking, predicate modal logic is an evolution of propositional modal logic and predicate logic, and there are many new problems in predicate modal logic to be solved, such as the constant domain semantics and the variant domain semantics. The series should focus on formal representation, deduction, theories or metatheories of induction, revision, and evolution, where induction is at the first level, revision is at the second level, and evolution is at the third level, because induction is at the formula stratum, revision is at the theory stratum, and evolution is at the logic stratum. The books in the series differ in level: some are overviews and some are highly specialized. Here, the books of overviews are for undergraduate students; and the highly specialized ones are for graduate students and researchers in computer science and mathematical logic. Beijing, China October 2016

Li Wei Luo Jie Song Fangming Sui Yuefei Wang Ju Zhu Wujia

Preface

Traditional logics are to formalize deductions, and description logics are to formalize concepts, where deductions and concepts are two main ingredients of logic. Even though description logics are fragments of first-order logic, some are decidable and some are not. Hence, this book consists of two parts: the first part is for decidable description logics and the second for undecidable description logics. Firstly, we will consider four description logics, B2 -valued, Post L3 -valued, B22 -valued, and L4 -valued ones, and for each logic, we build sound and complete deduction systems  M∗ , N∗ , L∗ , K∗ , for M∗ -valid, N∗ -valid, L∗ -valid, and K∗ -valid theories (called 1 n-multisequents)/2/n − /i/n-multisequents, respectively, where i/n  ∗ = {i/n ,  : n = 2, 3, 22 , 4; i ≤ n} and  ∈ {i : i ≤ n} ∪ {ij : i < j ≤ n} ∪ {=, =}. Correspondingly, sound and complete R-calculi R∗ , S∗ , Q∗ , P∗ for M∗ -valid, N∗ -valid, L∗ -valid, K∗ -valid 1/n − /2/n − /i/n-reduction, respectively, will be established. Secondly, we will consider B2 -valued, Post L3 -valued, B22 -valued, and L4 -valued description logics with role constructors R1 ∗ R2 , R1 ◦ R2 , R ∗ , R ◦ , which are undecidable. Hence, there are sound and complete deduction systems X∗ for X∗ -valid multisequents, but no sound and complete deduction systems X∗ for X∗ -valid multii/n i/n ∗ ∗ , Ni/n } and X∗ ∈ {L∗ , K∗ }. Deduction systems X∗ sequents, where X∗ ∈ {Mi/n are sound and recursively enumerable, even though deduction systems X∗ are not complete. We will give sound and complete R-calculi Y∗ , Y∗ for Y∗ /Y∗ -valid reductions, i/n i/n ∗ ∗ , Si/n } and Y∗ ∈ {Q∗ , P∗ }. respectively, where Y∗ ∈ {Ri/n

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Preface

We have the following deduction systems and R-calculi:

deduction systems:

Mi/n Ni/n i/n i/n L K

R-calculi:

Ri/n Si/n i/n i/n Q P

where n = {2, 3, 22 , 4}, i ≤ n,  ∈ {i : i ≤ n} ∪ {ij : i < j ≤ n} ∪ {ijk : i < j < k ≤ n} ∪ {=, =}. Beijing, China December 2022

Wei Li Yuefei Sui

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Many-Valued Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Deduction Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 R-Calculus for Quantifier Constructors . . . . . . . . . . . . . . . . . . . . . . 1.7 R-Calculus for Undecidable DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

1 1 4 6 7 9 10 12 16 17

Decidable DLs

R-Calculus for Binary-Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Binary-Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 1/2-Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Deduction System Mt1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 R-Calculus Rt1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Deduction System Nt1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 R-Calculus St1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 1/2-Co-Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2 2.3.1 Deduction System Lt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2 2.3.2 R-Calculus Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2 2.3.3 Deduction System Kt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2 2.3.4 R-Calculus Pt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 2/2-Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Deduction System M= 2/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 2.4.2 R-Calculus R= 2/2

21 23 25 25 27 29 30 32 32 33 35 36 38 39 40

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2.4.3 Deduction System N= 2/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 2.4.4 R-Calculus S= 2/2 2.5 2/2-Co-Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ 2.5.1 Deduction System L2/2 = . . . . . . . ........................... 2.5.2 R-Calculus Q2/2 = . ........................... 2.5.3 Deduction System K2/2 = . . . . . . . . .......................... 2.5.4 R-Calculus P2/2 = 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 45 48 49 50 54 55 59 60

3

R-Calculus for Post Three-Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Post Three-Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 1/3-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Deduction System Mt1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 R-Calculus Rt1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 2/3-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Deduction System Mtm 2/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 3.3.2 R-Calculi Rtm 2/3 3.4 3/3-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Deduction System M= 3/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 3.4.2 R-Calculus R= 3/3 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 63 66 67 68 73 73 75 79 79 81 86 86

4

R-Calculus for B22 -Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 B22 -Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 1/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/22 ........................... 4.2.1 Deduction System Lt 1/22 ................................. 4.2.2 R-Calculus Qt 4.3 2/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Deduction System L2t . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/22 4.3.2 R-Calculus Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 3/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3/22 4.4.1 Deduction System Lt⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3/2 4.4.2 R-Calculus Qt⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4.5 4/2 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ........................... 4.5.1 Deduction System L4/2 = 2 . . . . . . . .......................... 4.5.2 R-Calculus Q4/2 = 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 91 94 95 97 104 104 106 113 114 117 124 125 127 135 136

Contents

5

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R-Calculi for Post L4 -Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Post L4 -Valued Description Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 1/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Deduction System Nt1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 R-Calculus St1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 2/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ 5.3.1 Deduction System Nt 2 . . . . . . . ........................... 5.3.2 R-Calculus St 2/4 5.4 3/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... 5.4.1 Deduction System Nt⊥ 3/4 t⊥ 5.4.2 R-Calculus S3/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 4/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Deduction System N= 4/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 R-Calculus S= 4/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II

137 141 144 145 147 154 155 158 166 167 170 178 179 181 191 192

Undecidable DLs

6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Undecidable DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 R-Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Post L3 -Valued DL with Role Constructors . . . . . . . . . . . . . . . . . . 6.4 B22 -Valued DL with Role Constructors . . . . . . . . . . . . . . . . . . . . . . . 6.5 Injury . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Arrangement of This Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195 197 199 202 204 207 208

7

Role R-Calculus for Binary-Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Binary-Valued DL with Role Constructors . . . . . . . . . . . . . . . . . . . 7.2 1/2-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Deduction System Mt1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 R-Calculus Rt1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Deduction System Nt1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 R-Calculus St1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 1/2-Co-multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2 7.3.1 Incomplete Deduction System Lt . . . . . . . . . . . . . . . . . . 1/2 7.3.2 R-Calculus Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/2 7.3.3 Incomplete Deduction System Kt . . . . . . . . . . . . . . . . . . 1/2 7.3.4 R-Calculus Pt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 2/2-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Deduction System M= 2/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 7.4.2 R-Calculi R= 2/2 7.4.3 Deduction System N= 2/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 7.4.4 R-Calculi S= 2/2

211 213 214 214 215 217 218 220 220 221 222 223 225 225 226 227 229

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7.5

2/2-Co-multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Incomplete Deduction System L2/2 = .................. . . . . . . . . . . . . . . ..................... 7.5.2 R-Calculi Q2/2 = 7.5.3 Incomplete Deduction System K2/2 ................. = . . . . . . . . . . . . . . . . . . . ................. 7.5.4 R-Calculi P2/2 = 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

230 231 232 233 234 236 238

8

Role R-Calculus for Post Three-Valued DL . . . . . . . . . . . . . . . . . . . . . . 8.1 Post Three-Valued DL with Role Constructors . . . . . . . . . . . . . . . . 8.2 1/3-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Deduction System Nt1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 R-Calculus St1/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/3 8.2.3 Incomplete Deduction System Kt . . . . . . . . . . . . . . . . . . 1/3 8.2.4 R-Calculus Pt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 2/3-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Deduction System Ntm 2/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . tm 8.3.2 R-Calculus S2/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/3 8.3.3 Incomplete Deduction System Ktm . . . . . . . . . . . . . . . . . . 2/3 8.3.4 R-Calculus Ptm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 3/3-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Deduction System N= 3/3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 8.4.2 R-Calculus S= 3/3 8.4.3 Incomplete Deduction System K3/3 = .................. . . . . . . . . . . . . . ..................... 8.4.4 R-Calculus P3/3 = 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

239 240 242 242 244 247 248 252 252 253 257 258 262 262 264 267 269 272 273

9

Role R-Calculus for B22 -Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 B22 -Valued DL with Role Constructors . . . . . . . . . . . . . . . . . . . . . . . 9.2 1/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Deduction System Mt1/22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 R-Calculus Rt1/22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275 277 279 279 281

1/22

9.3

9.4

................. 9.2.3 Incomplete Deduction System Lt 1/22 ................................. 9.2.4 R-Calculus Qt 2/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Deduction System Mt 2/22 . . . . . . . . . . . . . . . . . . . . . . . . . . . t 9.3.2 R-Calculus R2/22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Deduction System Mt⊥ 3/22 . . . . . . . . . . . . . . . . . . . . . . . . . . t⊥ 9.4.2 R-Calculus R3/22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

286 288 292 293 295 299 300 302

Contents

xiii

9.5

4/22 -Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 9.5.1 Deduction System M= 4/22 . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 9.5.2 R-Calculus R= 2 4/2 2

................. 9.5.3 Incomplete Deduction System L4/2 = 2 . . . . . . . . . . . . . . . . ................. 9.5.4 R-Calculus Q4/2 = 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 318 322 324

10 Role R-Calculus for Post L4 -Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Post L4 -Valued DL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 1/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Deduction System Mt1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 R-Calculus Rt1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Deduction System Nt1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 R-Calculus St1/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 2/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Deduction System Nt 2/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 10.3.2 R-Calculus St 2/4 10.4 3/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... 10.4.1 Deduction System Nt⊥ 3/4 . . . . . . . .......................... 10.4.2 R-Calculus St⊥ 3/4 10.5 4/4-Multisequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Deduction System M= 4/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 10.5.2 R-Calculus R= 4/4 10.5.3 Deduction System N= 4/4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................... 10.5.4 R-Calculus S= 4/4 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

325 326 329 330 332 337 339 344 345 347 353 353 356 362 362 365 370 373 379 379

Appendix: Finite Injury Priority Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Chapter 1

Introduction

1.1 Many-Valued Logics Traditional binary-valued logic Fitting (1991), Gottwald (2001), Hähnle (2001), Malinowski (2009) has two values, corresponding to true and false. For many-valued logics, it is not obvious to give the intuitive understanding of the other values than true and false. For three-valued logic Avron (1991), Avron (1993), Bochvar (1938), Łukasiewicz (1970), Post (1920), Post (1921), some explained the third value as unknown, or uncertain, and daily life does not intuitively use the third value, not to say to deduce or reason in three-valued logic. For description logics Baader et al. (2003), Baader et al. (2007), it seems natural to use many-valued logics Urquhart (2001), Wronski (1987), Zach (2023). For example, let U be a universe, and subsets X 1 , X 2 , X 3 , X 3 compose of a partition of U. In binaryvalued DLs, we need use four concepts C1 , C2 , C3 , C4 to represent X 1 , X 2 , X 3 , X 4 , respectively, even though ¬C1 ≡ C2  C3  C4 ¬C2 ≡ C1  C3  C4 ¬C3 ≡ C1  C2  C4 ¬C4 ≡ C1  C2  C3 . In a B22 -valued DL, we use one concept C to represent partition {X 1 , X 2 , X 3 , X 4 } and use concepts ∼1 C, ∼2 C, C to represent others: C = (X 1 , X 2 , X 3 , X 4 ) ∼1 C = (X 2 , X 1 , X 4 , X 3 ) ∼2 C = (X 3 , X 4 , X 1 , X 2 ) C = (X 4 , X 3 , X 2 , X 1 ).

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_1

1

2

1 Introduction

For B22 -valued DL, given two concepts C = (X 1 , X 2 , X 3 , X 4 ) and D = (Y1 , Y2 , Y3 , Y4 ), we have C  D = (X 1 ∩ Y1 , (X 2 ∩ Y2 ) ∪ (X 1 ∩ Y2 ) ∪ (X 2 ∩ Y1 ), (X 3 ∩ Y3 ) ∪ (X 1 ∩ Y3 ) ∪ (X 3 ∩ Y1 ), X 4 ∪ Y4 ∪ (X 2 ∩ Y3 ) ∪ (X 3 ∩ Y2 )) C  D = (X 1 ∪ Y1 ∪ (X 2 ∩ Y3 ) ∪ (X 3 ∩ Y2 ), (X 2 ∩ Y2 ) ∪ (X 4 ∩ Y2 ) ∪ (X 2 ∩ Y4 ), (X 3 ∩ Y3 ) ∪ (X 4 ∩ Y3 ) ∪ (X 3 ∩ Y4 ), X 4 ∩ Y4 ). As for quantifier constructors (∀R.C) and (∃R.C), given a concept C = (X 1 , X 2 , X 3 , X 4 ) and a role R = (R1 , R2 , R3 , R4 ), ∀R.C = (∀R.X 1 ∪ ∀R4 .X ∪ ∀R2 .X 2 ∪ ∀R3 .X 3 , ∀R1 .X 2 ∪ ∀R3 .X 4 , ∀R1 .X 3 ∪ ∀R2 .X 4 , ∀R4 .X 1 ) ∃R.C = (∀R1 .X 1 , ∀R2 .X 2 ∪ ∀R2 .X 1 ∪ ∀R1 .X 2 , ∀R3 .X 3 ∪ ∀R1 .X 3 ∪ ∀R3 .X 1 , ∀R.X 1 ∪ ∀R4 .X ∪ ∀R2 .X 3 ∪ ∀R3 .X 2 ), where the truth tables for logical connectives are given as follows: ∩ t

⊥ f

t t

⊥ f

⊥

⊥

f f⊥ f f

f f f f f

∪ t

⊥ f

t t t t t

⊥ t t

t t⊥

⊥

f → t ⊥f t t t t tt

t t ⊥ ⊥ ⊥ tt f f f ⊥ t

Taking ∀R.C = (Y1 , Y2 , Y3 , Y4 ) and ∃R.C = (Z 1 , Z 2 , Z 3 , Z 4 ), we have ∀R.C = (Y1 , Y2 , Y3 , Y4 ) ∼1 ∀R.C = (Y2 , Y1 , Y4 , Y3 ) ∼2 ∀R.C = (Y3 , Y4 , Y1 , Y2 ) ∀R.C = (Y4 , Y3 , Y2 , Y1 ); and ∃R.C = (Z 1 , Z 2 , Z 3 , Z 4 ) ∼1 ∃R.C = (Z 2 , Z 1 , Z 4 , Z 3 ) ∼2 ∃R.C = (Z 3 , Z 4 , Z 1 , Z 2 ) ∃R.C = (Z 4 , Z 3 , Z 2 , Z 1 ).

1.1 Many-Valued Logics

3

For Post L4 -valued logic, given two concepts C = (X 1 , X 2 , X 3 , X 4 ) and D = (Y1 , Y2 , Y3 , Y4 ), we have C  D = (X 1 ∩ Y1 , (X 2 ∩ Y2 ) ∪ (X 1 ∩ Y2 ) ∪ (X 2 ∩ Y1 ), (X 3 ∩ Y3 ) ∪ (X 2 ∩ Y3 ) ∪ (X 1 ∩ Y3 ) ∪ (X 3 ∩ Y2 ) ∪ (X 3 ∩ Y1 ), X 4 ∪ Y4 ) C  D = (X 1 ∪ Y1 , (X 2 ∩ Y2 ) ∪ (X 3 ∩ Y2 ) ∪ (X 4 ∩ Y2 ) ∪ (X 2 ∩ Y3 ) ∪ (X 2 ∩ Y4 ), (X 3 ∩ Y3 ) ∪ (X 4 ∩ Y3 ) ∪ (X 3 ∩ Y4 ), X 4 ∩ Y4 ). As for quantifier constructors (∀R.C) and (∃R.C), given a concept C = (X 1 , X 2 , X 3 , X 4 ) and a role R = (R1 , R2 , R3 , R4 ), ∀R.C = (∀R.X 1 ∪ ∀R4 .X ∪ ∀R2 .X 2 ∪ ∀R3 .X 3 ∪ ∀R3 .X 2 , ∀R1 .X 2 ∪ ∀R2 , X 3 ∪ ∀R3 .X 4 , ∀R1 .X 3 ∪ ∀R2 .X 4 , ∀R4 .X 1 ) ∃R.C = (∀R1 .X 1 , ∀R2 .X 2 ∪ ∀R2 .X 1 ∪ ∀R1 .X 2 , ∀R3 .X 3 ∪ ∀R2 .X 3 ∪ ∀R1 .X 3 ∪ ∀R3 .X 2 ∪ ∀R3 .X 1 , ∀R.X 1 ∪ ∀R4 .X ), where the truth tables for logical connectives are given as follows: ∩ t

⊥ f

t t

⊥ f

⊥

⊥

⊥ ⊥⊥ f f

f f f f f

∪ t

⊥ f

t t t t t

⊥ t t



⊥

⊥

f → t ⊥f t t t t tt

t tt ⊥ ⊥ ⊥ t t f f f ⊥ t

Taking ∀R.C = (Y1 , Y2 , Y3 , Y4 ) and ∃R.C = (Z 1 , Z 2 , Z 3 , Z 4 ), we have ∀R.C = (Y1 , Y2 , Y3 , Y4 ) ∼ ∀R.C = (Y2 , Y3 , Y4 , Y1 ) ∼2 ∀R.C = (Y3 , Y4 , Y1 , Y2 ) ∼3 ∀R.C = (Y4 , Y1 , Y2 , Y3 ); and ∃R.C = (Z 1 , Z 2 , Z 3 , Z 4 ) ∼ ∃R.C = (Z 2 , Z 3 , Z 4 , Z 1 ) ∼2 ∃R.C = (Z 3 , Z 4 , Z 1 , Z 2 ) ∼3 ∃R.C = (Z 4 , Z 1 , Z 2 , Z 3 ).

4

1 Introduction

1.2 Multisequents We will consider four many-valued DLs: B2 -valued, Post L3 -valued Post (1920), Post (1921), B22 -valued Pynko (1995), Wronski (1987), Zach (2023) and Post L4 -valued, and with/without role constructors. We will define i/n-multisequents and discuss the following deduction systems, Ginsberg (1987), Reiter (1980) ∗ ∗ Ni/n Mi/n i/n i/n L∗ K∗ where n ∈ {2, 3, 22 , 4}, i ≤ n, ∗ = ∗1 ∗2 ∗i , and ∗1 , ∗2 , ∗i ∈ Ln , where L2 = B2 , L22 = B22 . There are four kinds of validity for theories: • Mi : A theory  is Mi -valid if for any interpretation I, there is a statement A(a) ∈  such that I (A(a)) = i; • Ni : A theory  is Ni -valid if for any interpretation I, there is a statement A(a) ∈  such that I (A(a)) = i; • Li : A theory  is Li -valid if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = i; • Ki : A theory  is Ki -valid if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = i. The Mi -validity is complementary to Li -validity, and Ni -validity is to Ki validity. There are four kinds of validity for 2/n-multisequents: • Mij : A 2/n-multisequent | is Mij -valid if for any interpretation I, either for some statement A(a) ∈ , I (A(a)) = i or for some statement B(b) ∈ , I (B(b)) = j; • Nij : A 2/n-multisequent | is Nij -valid if for any interpretation I, either for some statement A(a) ∈ , I (A(a)) = i or for some statement B(b) ∈ , I (B(b)) = j; • Lij : A 2/n-multisequent | is Lij -valid if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = i and for any statement B(b) ∈ , I (B(b)) = j; • Kij : A 2/n-multisequent | is Kij -valid if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = i and for any statement B(b) ∈ , I (B(b)) = j. There are four kinds of validity for 3/n-multisequents: • Mijk : A 3/n-multisequent || is Mijk -valid if for any interpretation I, either for some statement A(a) ∈ , I (A(a)) = i, or for some statement B(b) ∈ , I (B(b)) = j, or for some statement C(c) ∈ , I (C(c)) = k; • Nijk : A 3/n-multisequent || is Nijk -valid if for any interpretation I, either for some statement A(a) ∈ , I (A(a)) = i, or for some statement B(b) ∈ , I (B(b)) = j, or for some statement C(c) ∈ , I (C(c)) = k;

1.2 Multisequents

5

• Lijk : A 3/n-multisequent || is Lijk -valid if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = i, for any statement B(b) ∈ , I (B(b)) = j, and for any statement C(c) ∈ , I (C(c)) = k; • Kijk : A 3/n-multisequent || is Kijk -valid if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = i, for any statement B(b) ∈ , I (B(b)) = j, and for any statement C(c) ∈ , I (C(c)) = j. For i = 4, we have the following kinds of multisequents: • A multisequent ||| is M= -valid, denoted by |== |||, if for any interpretation I, either I (A(a)) = t for some A(a) ∈ , or I (B(b)) = for some B(b) ∈ , or I (C(c)) =⊥ for some C(c) ∈ , or I (D(d)) = f for some D(d) ∈ ; • A multisequent ||| is N= -valid, denoted by |== |||, if for any interpretation I, either I (A(a)) = t for some A(a) ∈ , or I (B(b)) = for some B(b) ∈ , or I (C(c)) =⊥ for some C(c) ∈ , or I (D(d)) = f for some D(d) ∈ ; • A multisequent ||| is L= -valid, denoted by |== |||, if there is an interpretation I such that I (A(a)) = t for each A(a) ∈ , I (B(b)) = for each B(b) ∈ , I (C(c)) =⊥ for each C(c) ∈ , and I (D(d)) = f for each D(d) ∈ . • A multisequent ||| is K= -valid, denoted by |== |||, if there is an interpretation I such that I (A(a)) = t for each A(a) ∈ , I (B(b)) = for each B(b) ∈ , I (C(c)) =⊥ for each C(c) ∈ , and I (D(d)) = f for each D(d) ∈ . Hence, M= /N= -validity is complementary to L= /K= -validity, respectively. 2/i

Remark 1.1 Kt -valid 2/i-multisequent | is different from Gt -valid sequent  ⇒ , where  ⇒  is Gt -valid if there is an interpretation I such that I (A(a)) = t for each A(a) ∈ , and I (B(b)) = t for each B(b) ∈ . Hence, |=t  ⇒  is 2/i  not equivalent to |=t |. The following multisequents are equivalent: ◦ B2 -valued DL: 2/2-multisequent | 1/2-multisequent , ¬ ◦ L3 -valued DL: 3/3-multisequent || , ∼2 | 2/3-multisequent |, ∼  2 1/3-multisequent , ∼ , ∼  ◦ B22 -valued DL: 4/4-multisequent ||| 3/4-multisequent ||, ∼1 ; |, ∼2 |; , || 2/4-multisequent |, , ∼2 ; , , ∼2 |; , |, ; , ∼2 |, ∼2  1/4-multisequent , ∼1 , ∼2 , 

6

1 Introduction

◦ L4 -valued DL: 4/4-multisequent ||| 3/4-multisequent ||, ∼ ; |, ∼2 |; , ∼3 || 2/4-multisequent |, ∼ , ∼2 ; , ∼3 , ∼2 |; , ∼3 |, ∼ ; , ∼2 |, ∼2  1/4-multisequent , ∼ , ∼2 , ∼3 

1.3 Deduction Systems We will consider the following deduction systems: • B2 -valued DL: for each validity of i/2-multisequents Li (2010), Li and Sui (2013), Li and Sui (2017), Takeuti (1987) there is a sound and complete deduction i/2 i/2 ∗ ∗ /Ni/2 /L∗ /K∗ : system Mi/2 t f 1/2 M1/2 , M1/2 1/2 1/2 Lt , Lf = 2/2 M2/2 2/2 L=

t f N1/2 , N1/2 1/2 1/2 Kt , Kf = N2/2 2/2 K=

We will give all the deduction systems for B2 -valued DL. • Post L3 -valued DL: for each validity of i/3-multisequents there is a sound and i/3 i/3 ∗ ∗ /Ni/3 /L∗ /K∗ : complete deduction system Mi/3 t m f , M1/3 , M1/3 1/3 M1/3 1/3 1/3 1/3 Lt , Lm , Lf tm tf mf 2/3 M2/3 , M2/3 , M2/3 2/3 2/3 2/3 Ltm , Ltf , Lmf = 3/3 M3/3 3/3 L=

t m f N1/3 , N1/3 , N1/3 1/3 1/3 1/3 Kt , Km , Kf tm tf mf N2/3 , N2/3 , N2/3 2/3 2/3 2/3 Ktm , Ktf , Kmf = N3/3 3/3 K=

We will give the following deduction systems instead of all the systems: deduction system t 1/3 M1/3 tm 2/3 M2/3 = 3/3 M3/3 • B22 -valued DL: for each validity of i/22 -multisequents there is a sound and i/22 i/22 ∗ ∗ complete deduction system Mi/2 /K∗ : 2 /Ni/22 /L∗

1.4 Reductions

7

t

⊥ f 1/22 M1/2 2 , M1/22 , M1/22 , M1/22

t

⊥ f N1/2 2 , N1/22 , N1/22 , N1/22

1/22 1/22 1/22 1/22 Lt , L , L⊥ , Lf

1/22

1/22

1/22

1/22

Kt , K , K⊥ , Kf

t

t⊥ tf

⊥

f ⊥f 2/22 M2/2 2 , M2/22 , M2/22 , M2/22 , M2/22 , M2/22 t

t⊥ tf

⊥

f ⊥f N2/22 , N2/22 , N2/22 , N2/22 , N2/22 , N2/22 2/22

2/22

2/22

2/22

2/22

2/22

Lt , Lt⊥ , Ltf , L ⊥ , L f , L⊥f 2/22 2/22 2/22 2/22 2/22 2/22 Kt , Kt⊥ , Ktf , K ⊥ , K f , K⊥f t ⊥ t f

⊥f 3/22 M3/2 2 , M3/22 , M3/22 3/2

2

3/2

2

3/2

t ⊥ t f

⊥f N3/2 2 , N3/22 , N3/22

2

3/22

Lt ⊥ , Lt f , L ⊥f

3/22

3/22

Kt ⊥ , Kt f , K ⊥f

= 4/22 M4/2 2 4/22 L=

= N4/2 2 4/22 K=

We will give the following deduction systems instead of all the systems: deduction system 2

1/22

1/2 Lt

2/22

2/22 Lt

3/22

3/22 Lt ⊥ 4/22 4/22 L= • Post L4 -valued DL Artale and Franconi (2005), Hájek (2005): we use the same language for L4 -valued logic as for B22 -valued logic Pynko (1995), the same notations for deduction systems, and consider the following deduction systems:

1/4 2/4 3/4 4/4

deduction system t N1/4 t

N2/4 t ⊥ N3/4 = N4/4

1.4 Reductions R-reductions Li (2007) are simply called reductions. We will define i/n-reductions and R-calculi ∗ ∗ Si/n Ri/n i/n i/n Q∗ P∗

8

1 Introduction

where n ∈ {2, 3, 22 , 4}, i ≤ n, ∗ = ∗1 ∗2 ∗i , and ∗1 , ∗2 , ∗i ∈ Ln , where L2 = B2 , L22 = B22 . There are four kinds of validity for 1/n-reductions: • Ri /Si : Given A(a) ∈ , a 1/n-reduction  ↑ A(a) ⇒ [A (a)] is Ri /Si -valid if Mi /Ni -validity of  implies that of [A (a)], where A = λ/A; • Qi /Pi : Given statement A(a), a 1/n-reduction  ↑ A(a) ⇒ (A (a)) is Qi /Pi -valid if Li /Ki -validity of  implies that of (A (a)). There are four kinds of validity for 2/n-reductions: • Rij /Sij : Given A(a) ∈  and B(b) ∈ , a 2/n-reduction | ↑ (A(a), B(b)) ⇒ [A (a)]|[B  (b)] is Rij /Sij -valid if Mij /Nij -validity of | implies that of [A (a)]|[B  (b)]; where B  = λ/B; • Qij /Pij : Given statements A(a), B(b), a 2/n-reduction | ↑ (A(a), B(b)) ⇒ (A (a))|(B  (b)) is Qij /Pij -valid if Lij /Kij -validity of | implies that of (A (a))|(B  (b)). There are four kinds of validity for 3/n-multisequents: • Rijk /Sijk : Given A(a) ∈ , B(b) ∈  and C(c) ∈ , a 3/n-reduction || ↑ (A(a), B(b), C(c)) ⇒ [A (a)]|[B  (b)]|[C  (c)] is Rijk /Sijk -valid if Mijk /Nijk -validity of || implies that of [A (a)]| [B  (b)]|[C  (c)], where C  = λ/C; • Qijk /Pijk : Given statements A(a), B(b), C(c), a 3/n-reduction || ↑ (A(a), B(b), C(c)) ⇒ (A (a))|(B  (b))|(C  (c)) is Qijk /Pijk -valid if Lijk/Kijk -validity of || implies that of (A (a))| (B  (b))|(C  (c)). For i = 4, we have the following kinds of reductions: • Given A(a) ∈ , B(b) ∈ , C(c) ∈  and D(d) ∈ , a reduction δ = ||| ↑ (A(a), B(b), C(c), D(d)) ⇒ [A (a)]|[B  (b)]|[C  (c)]|[D  (d)]

is R= /S= -valid, denoted by |== / |== δ, if M= /N= -validity of ||| implies that of [A (a)]|[B  (b)]|[C  (c)]|[D  (d)], where D  = λ/D; • Given statements A(a), B(b), C(c) and D(d), a reduction δ = ||| ↑ (A(a), B(b), C(c), D(d)) ⇒ (A (a))|(B  (b))|(C  (c))|(D  (d))

is Q= /P= -valid, denoted by |== / |== δ, if L= /K= -validity of ||| implies that of (A (a))|(B  (b))|(C  (c))|(D  (d)). i/n i/n ∗ ∗ ∗ ∗ /Si/n is to preserve the Mi/n /Ni/n -validity, respectively, and Q∗ /P∗ Here, Ri/n i/n

i/n

is to preserve the L∗ /K∗ -validity, respectively.

1.5 R-Calculi

9

1.5 R-Calculi We consider the following R-calculi Li (2007), Li and Sui (2013), Li and Sui (2017): • B2 -valued DL: for each validity of i/2-reductions there is a sound and complete i/2 i/2 ∗ ∗ /Si/2 /Q∗ /P∗ : R-calculi Ri/2 t f 1/2 R1/2 , R1/2 1/2 1/2 Qt , Qf = 2/2 R2/2 Q2/2 =

St1/2 , Sf1/2 1/2 1/2 Pt , Pf = S2/2 2/2 P=

We will give all the R-calculi for B2 -valued DL. • Post L3 -valued DL: for each validity of i/3-reductions there is a sound and i/3 i/3 ∗ ∗ /Si/3 /Q∗ /P∗ Alchourrón et al. (1985), Alchourrón and complete R-calculi Ri/3 Makinson (1981), Alchourrón and Makinson (1982), Bochman (1999), Doyle (1979): t m f 1/3 R1/3 , R1/3 , R1/3 1/3 1/3 1/3 Qt , Qm , Qf tm tf mf 2/3 R2/3 , R2/3 , R2/3 2/3 2/3 2/3 Qtm , Qtf , Qmf = 3/3 R3/3 Q3/3 =

St1/3 , Sm1/3 , Sf1/3 1/3 1/3 Pt , Pm1/3 , Pf tf mf Stm 2/3 , S2/3 , S2/3 2/3 2/3 2/3 Ptm , Ptf , Pmf = S3/3 3/3 P=

We will give the following R-calculi instead of all the systems: R-calculus t 1/3 R1/3 tm 2/3 R2/3 = 3/3 R3/3 • B22 -valued DL: for each validity of reductions there is a sound and complete i/22 i/22 ∗ ∗ R-calculi Ri/2 /P∗ Dalal (1988), Darwiche and Pearl (1997), Fermé 2 /Si/22 /Q∗ and Hansson (2011), Gärdenfors and Rott (1995), Lang and van der Torre (2008):

10

1 Introduction t

⊥ f 1/22 R1/2 2 , R1/22 , R1/22 , R1/22

1/22 1/22 1/22 1/22 Qt , Q , Q⊥ , Qf t

t⊥ tf

⊥

f ⊥f 2/22 R2/2 2 , R2/22 , R2/22 , R2/22 , R2/22 , R2/22 t

t⊥ tf

⊥

f ⊥f S2/22 , S2/22 , S2/22 , S2/22 , S2/22 , S2/22 2/22 2/22 2/22 2/22 2/22 2/22 Pt , Pt⊥ , Ptf , P ⊥ , P f , P⊥f 2/22 2/22 2/22 2/22 2/22 2/22 Qt , Qt⊥ , Qtf , Q ⊥ , Q f , Q⊥f t ⊥ t f

⊥f 3/22 R3/2 2 , R3/22 , R3/22 2 2 3/2 3/2 3/22 Qt ⊥ , Qt f , Q ⊥f = 4/22 R4/2 2 2 Q4/2 =

⊥ f St1/22 , S

1/22 , S1/22 , S1/22 1/22

1/22

1/22

1/22

Pt , P , P⊥ , Pf

t f

⊥f St ⊥ 3/22 , S3/22 , S3/22 3/22

3/22

3/22

Pt ⊥ , Pt f , P ⊥f S= 4/22 4/22 P=

We will give the following R-calculi instead of all the systems: R-calculus 2

1/22

1/2 Qt

2/22

2/22 Qt

3/22

3/22 Qt ⊥ 2 4/22 Q4/2 = • Post L4 -valued DL Artale and Franconi (2005), Hájek (2005): we use the same language for L4 -valued logic as for L3 -valued DL Pynko (1995), the same notations for R-calculi, and consider the following R-calculi:

1/4 2/4 3/4 4/4

R-calculus St1/4 St

2/4 St ⊥ 3/4 S= 4/4

1.6 R-Calculus for Quantifier Constructors To consider R-calculi in many-valued DL Fensel et al. (2001), Horrocks (2008), Horrocks and Sattler (2001), Li (2010), we use Post L3 -valued DL as an example to show variants of validity of reductions, where ¬R(a, c) means ∼ R(a, c)∨ ∼2 R(a, c). For a decidable DLSoare (1987), there are the following sound and complete R-calculi: ◦ Rt : given any theory  and statement A(a) ∈ , a reduction |A(a) ⇒ [A (a)] is Rt -valid iff |A(a) ⇒ [A (a)] is Rt -provable, where Rt is taken as a combination of Mt and Lt .

1.6 R-Calculus for Quantifier Constructors

11

E A (a) = A(a)A∗ ∈ {λ, ∼, ∼2 }( t ∗A (a)) |A(a) ⇒  − {A(a)} AA (a) = A(a)E∗ ∈ {λ, ∼, ∼2 }( t ∗A (a)) ; (0) |A(a) ⇒ 

(−)

where (−) represents extracting A(a) from , and (0) doing nothing; Let (A1  A2 )(a) ∈ . To extract (A1  A2 )(a) from a theory , if Mt -validity of  implies those of [A1 (a)] and [A2 (a)] implies that of [A1 (a), A2 (a)] then Mt -validity of  implies that of [(A1  A2 )(a)]; and if Mt -validity of  does not imply that of either [A1 (a)] or [A2 (a)] then Mt -validity of  does not imply that of [(A1  A2 )(a)]. Hence, we have the following deduction rules: 

|A1 (a) ⇒ [A1 (a)] [A1 (a)]|A2 (a) ⇒ [A1 (a), A2 (a)] |(A 1  A2 )(a) ⇒ [(A1  A2 )(a)]  |A1 (a) ⇒  (0 ) [A1 (a)]|A2 (a) ⇒ [A1 (a)] |(A1  A2 )(a) ⇒  (− )

Let (∀R.A)(a) ∈ . To extract ∀x A(x) from a theory , if Mt -validity of  implies that of either [¬R(a, c)] or [A(c)] for any new constant c not occurring in  then Mt -validity of  implies that of [(∀R.C)(a)]; and if Mt -validity of  does not imply those of [¬R(a, d)] and [A(d)] for some constant d then Mt validity of  does not implies that of [(∀R.C)(a)]. Hence, we have the following deduction rules:   |¬R(a, c) ⇒ [¬R(a, c)] |¬R(a, d) ⇒  (∀− ) |A(c) ⇒ [A(c)] (∀0 ) |A(d) ⇒  |(∀R.A)(a) ⇒ [(∀R.A)(a)] |(∀R.A)(a) ⇒  Let (∃R.A)(a) ∈ . To extract (∃R.A)(a) from a theory , if Mt -validity of  implies those of [R(a, d)] and [A(d)] for some constant d then Mt -validity of  implies that of [(∃R.A)(a)]; and if Mt -validity of  does not imply that of either [R(a, c)] or [A(c)] for a new constant c then Mt -validity of  does not imply that of [(∃R.A)(a)]; Hence, we have the following deduction rules: 

 |R(a, d) ⇒ [R(a, d)] |R(a, c) ⇒  (∃− ) |A(d) ⇒ [A(d)] (∃0 ) |A(c) ⇒  |(∃R.A)(a) ⇒ [(∃R.A)(a)] |(∃R.A)(a) ⇒  ◦ St : similar to Rt and omitted here. ◦ Qt : given any theory  and statement A(a), a reduction |A(a) ⇒ (A (a)) is Qt -valid iff |A(a) ⇒ (A (a)) is provable in Qt , where Qt is taken as a combination of Lt and Mt .

12

1 Introduction

(+)

 =t ∼ A(a)∧ ∼2 A(a)  =t ∼ A(a)∧ ∼2 A(a) (0) . |A(a) ⇒ (A(a)) |A(a) ⇒ 

Given a statement (A1  A2 )(a), to enumerate (A1  A2 )(a) into a theory , if Lt validity of  implies that of either (A1 (a)) or (A2 (a)) then Lt -validity of  implies that of ((A1  A2 )(a)); and if Lt -validity of  does not imply those of (A1 (a)) and (A2 (a)) then Lt -validity of  does not imply that of ((A1  A2 )(a)). Hence, we have the following deduction rules: 

 |A1 (a) ⇒ (A1 (a)) |A1 (a) ⇒  (+ ) |A2 (a) ⇒ (A2 (a)) (0 ) |A2 (a) ⇒  |(A1  A2 )(a) ⇒ ((A1  A2 )(a)) |(A1  A2 )(a) ⇒  Given a statement (∀R.A)(a), to enumerate (∀R.A)(a) in a theory , if Lt validity of  implies those of (¬R(a, d)) and (A(d)) for any constant d then Lt -validity of  implies that of ((∀R.C)(a)); and if Lt -validity of  does not imply that of either (¬R(a, c)) or (A(c)) for some new constant c then Lt validity of  does not implies that of ((∀R.C)(a)). Hence, we have the following deduction rules:   |¬R(a, d) ⇒ (¬R(a, d)) |¬R(a, c) ⇒  (∀+ ) |A(d) ⇒ (A(d)) (∀0 ) |A(c) ⇒  |(∀R.A)(a) ⇒ ((∀R.A)(a)) |(∀R.A)(a) ⇒  Given a statement (∃R.A)(a), to enumerate (∃R.A)(a) in a theory , if Lt validity of  implies that of either (R(a, c)) or (A(c)) for some new constant c then Lt -validity of  implies that of ((∃R.A)(a)); and if Lt -validity of  does not imply those of (R(a, d)) and (A(d)) for a constant d then Lt -validity of  does not imply that of ((∃R.A)(a)). Hence, we have the following deduction rules: 

 |R(a, c) ⇒ (R(a, c)) |R(a, d) ⇒  (∃+ ) |A(c) ⇒ (A(c)) (∃0 ) |A(d) ⇒  |(∃R.A)(a) ⇒ ((∃R.A)(a)) |(∃R.A)(a) ⇒  ◦ Pt : similar to Qt and omitted here.

1.7 R-Calculus for Undecidable DL For undecidable description logics Baader et al. (2003), Baader et al. (2007) (such as with role constructors R1 ◦ R2 , R2 ∗ R2 ), Mt and Nt are sound and complete; and Lt and Kt are sound and incomplete. Hence, there are no sound and complete R-calculi Rt , St , Qt , Pt .

1.7 R-Calculus for Undecidable DL

13

We know that Mt , Nt , t , =t are recursively enumerable; and Lt , Kt , =t , t are not. We have the following R-calculi with soundness and completeness/incompleteness theorems. • Rt : |=t |R(a, b) ⇒ [R(a, b)] iff t |R(a, b) ⇒ [R(a, b)]; and

t |R(a, b) ⇒  implies |=t |R(a, b) ⇒ ;

and

|=t |R(a, b) ⇒  may not imply t |R(a, b) ⇒ ;

• St : =t |R(a, b) ⇒ [R(a, b)] iff |==t |R(a, b) ⇒ [R(a, b)]; and

=t |R(a, b) ⇒  implies |==t |R(a, b) ⇒ ;

and

|==t |R(a, b) ⇒  may not imply =t |R(a, b) ⇒ ;

• Qt : =t |R(a, b) ⇒ (R(a, b)) implies |==t |R(a, b) ⇒ (R(a, b)); and |==t |R(a, b) ⇒ (R(a, b)) may not imply =t |R(a, b) ⇒ (R(a, b)); and =t |R(a, b) ⇒  iff |==t |R(a, b) ⇒ ; • Pt : t |R(a, b) ⇒ (R(a, b)) implies |=t |R(a, b) ⇒ (R(a, b)); and |=t |R(a, b) ⇒ (R(a, b)) may not imply t |R(a, b) ⇒ (R(a, b)); and t |R(a, b) ⇒  iff |=t |R(a, b) ⇒ .

14

1 Introduction

For Qt and Pt , we enumerate R(a, b) into  and use , R(a, b) to revise R(a, b). Hence, Qt and Pt become the following forms: • Qt : =t (R(a, b))|R(a, b) ⇒ (R(a, b)) ⇒|==t (R(a, b))|R(a, b) ⇒ (R(a, b));

and |==t (R(a, b))|R(a, b) ⇒ (R(a, b))  ⇒=t (R(a, b))|R(a, b) ⇒ (R(a, b));

and =t , R(a, b)|R(a, b) ⇒  iff |==t , R(a, b)|R(a, b) ⇒ ; • Pt : t (R(a, b))|R(a, b) ⇒ (R(a, b)) ⇒|=t (R(a, b))|R(a, b) ⇒ (R(a, b)); and |=t (R(a, b))|R(a, b) ⇒ (R(a, b)) ⇒t (R(a, b))|R(a, b) ⇒ (R(a, b)); and t (R(a, b))|R(a, b) ⇒  iff |=t |R(a, b) ⇒ . We take Post L3 -valued DL as an example. For R-calculus Rt , to extract R(a, b) from a theory , if [R(a, b)] t R  (a  , b )∧ ∼ R  (a  , b )∧ ∼2 R  (a  , b ) for some R  (a  , b ) = R(a, b) then extract R(a, b) from ; and if there is no such R  (a  , b ) = R(a, b) then we do nothing. Hence, we have the following deduction rules: ER  (a  , b ) = R(a, b)A∗ ∈ {λ, ∼, ∼2 }([R(a, b)] t ∗R  (a  , b )) |R(a, b) ⇒ [R(a, b)] AR  (a  , b ) = R(a, b)E∗ ∈ {λ, ∼, ∼2 }([R(a, b)] t ∗R  (a  , b )) (0) |R(a, b) ⇒ .

(−)

Hence, extracting a statement R(a, b) from  is recursively enumerable, and not extracting is not recursively enumerable Friedberg (1957), Kleene (1938), Muchnik (1956). For R-calculus St , to extract R(a, b) from a theory , if  t ∗1 R  (a  , b ) ∧ ∗2 R  (a  , b )

1.7 R-Calculus for Undecidable DL

15

for some R  (a  , b ) and ∗1 , ∗2 ∈ {λ, ∼, ∼2 } with ∗1 = ∗2 then extract R(a, b) from ; and if there is no R  (a  , b ) = R(a, b) then we do nothing. Hence, we have the following deduction rules: ER  (a  , b )E∗1 = ∗2 ( t ∗1 R  (a  , b ) ∧ ∗2 R  (a  , b )) |R(a, b) ⇒ [R(a, b)] AR  (a  , b )A∗1 = ∗2 ( t ∗1 R  (a  , b ) ∧ ∗2 R  (a  , b )) (0) |R(a, b) ⇒ .

(−)

Hence, extracting R(a, b) from  is recursively enumerable, and not extracting is not recursively enumerable Friedberg (1957), Kleene (1938), Muchnik (1956). Hence, for Qt and Pt we enumerate R(a, b) into  and use (R(a, b)) to revise R(a, b), that is, • Qt : to extract R(a, b) from a theory (R(a, b)), if (R(a, b)) t ∼ R(a, b)∧ ∼2 R(a, b) then extract R(a, b) from (R(a, b)); otherwise we do nothing. Hence, we have the following deduction rules: (R(a, b)) t ∼ R(a, b)∧ ∼2 R(a, b) (R(a, b))|R(a, b) ⇒  (R(a, b)) t ∼ R(a, b)∧ ∼2 R(a, b) (0) (R(a, b))|R(a, b) ⇒ (R(a, c)). (−)

Hence, extracting R(a, b) from (R(a, b)) is recursively enumerable, and not extracting is not recursively enumerable. • Pt : to extract R(a, b) from a theory (R(a, b)), if (R(a, b)) t ∼ R(a, b)∨ ∼2 R(a, b) then extract R(a, b) from (R(a, b)); otherwise, do nothing. Hence, we have the following deduction rules: (R(a, b)) t ∼ R(a, b)∨ ∼2 R(a, b) (R(a, b))|R(a, b) ⇒  (R(a, b)) t ∼ R(a, b)∨ ∼2 R(a, b) (0) (R(a, b))|R(a, b) ⇒ (R(a, b)). (−)

Hence, extracting a statement R(a, b) from (R(a, b)) is recursively enumerable, and not extracting is not recursively enumerable.

16

1 Introduction

1.8 Notions Our notation is standard and defined as needed. We use , , ,  to denote theories; A(a), B(b), C(c), D(d), R(a, b), Q(a, b), P(a, b), O(a, b) statements in description logics; a, b, c, d, e, f constants,  ↑ A a configuration which is equivalent to A ◦  in AGM postulates. A reduction is of form  ↑ A ⇒   , where   is a theory which may be or may not be equal to . In meta-language, we use ∼, &, or, A, E, corresponding to ¬, ∧, ∨, ∀, ∃ in logical languages, respectively. Notation for multisequents:  X 1/i-multisequent 2/i-multisequent | X|Y 3/i-multisequent || X 4/i-multisequent ||| X Position notation: X|i X denotes the following notations: X| X = X|(X, λ) X|2 X = X|(λ, X ) 1

X|1 X = X|(X, λ, λ) X|2 X = X|(λ, X, λ) X|3 X = X|(λ, λ, X )

and X[1 X ] =  − {X }| X[2 X ] = | − {X }

X|1 X X|2 X X|3 X X|4 X

= X|(X, λ, λ, λ) = X|(λ, X, λ, λ) = X|(λ, λ, X, λ) = X|(λ, λ, λ, X )

X[1 X ] =  − {X }|| X[2 X ] = | − {X }| X[3 X ] = || − {X }

X[1 X ] =  − {X }||| X[2 X ] = | − {X }|| X(1 X ) = , X | X[3 X ] = || − {X }| X(2 X ) = |, X X[4 X ] = ||| − {X } X(1 X ) = , X ||| X(1 X ) = , X || X(2 X ) = |, X || X(2 X ) = |, X | X(3 X ) = ||, X | X(3 X ) = ||, X X(4 X ) = |||, X and atom literal statements Atomic role statements Role statements

: : : : :

s(a), t (b), u(c), v(d) l(a), m(b), n(c), o(d) A(a), B(b), C(c), D(d) r (a, b), q(a, b), p(a, b), o(a, b) R(a, b), Q(a, b), P(a, b), O(a, b)

References

17

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Logic 50, 510–530 (1985) Alchourrón, C.E., Makinson, D.: Hierarchies of Regulation and Their Logic. In: Hilpinen, R. (ed.) New Studies in Deontic Logic, pp. 125–148. D. Reidel Publishing Company, Dordrecht (1981) Alchourrón, C.E., Makinson, D.: On the logic of theory change: contraction functions and their associated revision functions. Theoria 48, 14–37 (1982) Avron, A.: Natural 3-valued logics: characterization and proof theory. J. Symb. Logic 56, 276–294 (1991) Avron, A.: Gentzen-type systems, resolution and tableaux. J. Autom. Reason. 10, 265–281 (1993) Artale, A., Franconi, E.: Temporal description logics. In: Handbook of Temporal Reasoning in Artificial Intelligence (2005) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation. Applications. Cambridge University Press, Cambridge, UK (2003) Baader, F., Horrocks, I., Sattler, U.: Chapter 3 description logics. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.), Handbook of Knowledge Representation. Elsevier (2007) Bochman, A.: A foundational theory of belief and belief change. Artif. Intell. 108, 309–352 (1999) Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Logic 2, 87–112 (1938) Doyle, J.: A truth maintenance system. Artif. Intell. 12, 231–272 (1979) Dalal, M.: Investigations into a theory of knowledge base revision: preliminary report. In: Proceedings os AAAI-88, St. Paul, MN, pp. , 475–479 (1988) Darwiche, A., Pearl, J.: On the logic of iterated belief revision. Artif. Intell. 89, 1–29 (1997) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Fensel, D., van Harmelen, F., Horrocks, I., McGuinness, D., Patel-Schneider, P.F.: OIL: an ontology infrastructure for the semantic web. IEEE Intell. Syst. 16, 38–45 (2001) Fitting, M.C.: Many-valued modal logics (I,II). Fundamenta Informaticae 15, , 55–73 (1991); 17, 235–254 (1992) Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957) Gärdenfors, P., Rott, H.: Belief revision. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp. 35–132. Oxford Science Pub, Epistemic and Temporal Reasoning (1995) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Gottwald, S.: A Treatise on Many-Valued Logics (Studies in Logic and Computation, vol. 9). Research Studies Press Ltd., Baldock (2001) Hähnle, R.: Advanced many-valued logics. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, pp. 297–395. Kluwer, Dordrecht (2001) Hájek, P.: Making fuzzy description logic more general. Fuzzy Sets Syst. 154, 1–15 (2005) Horrocks, I.: Ontologies and the semantic web. Commun. ACM 51, 58–67 (2008) Horrocks, I., Sattler, U.: Ontology reasoning in the SHOQ(D) description logic. In: Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence (2001) Kleene, S.C.: On notation for ordinal numbers. J. Symb. Logic 3, 150–155 (1938) Lang, J., van der Torre, L.: From belief change to preference change. In: Ghallab, M., Spyropoulos, C.D., Fakotakis, N., Avouris, N.M. (eds.) ECAI 2008 -Proceedings of the 18th European Conference on Artificial Intelligence, Patras, Greece, (Frontiers in Artificial Intelligence and Applications, vol. 178), pp. 351–355. Accessed from 21–25 July 2008 Li, W.: Mathematical Logic, Foundations for Information Science. Progress in Computer Science and Applied Logic, vol. 25. Birkhäuser (2010)

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Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Li, W., Sui, Y.: The R-calculus and the finite injury priority method. J. Comput. 12, 127–134 (2017) Łukasiewicz, J.: Selected works. In: Borkowski, L. (ed.) North-Holland and Warsaw, PWN, Amsterdam (1970) Malinowski, G.: Many-valued Logic and its Philosophy. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic. The Many Valued and Nonmonotonic Turn in Logic, vol. 8. Elsevier (2009) Muchnik, A.A.: On the separability of recursively enumerable sets (in Russian). Dokl. Akad. Nauk SSSR, N.S. 109, 29–32 (1956) Post, E.L.: Determination of all closed systems of truth tables. Bull. Am. Math. Soc. 26, 437 (1920) Post, E.L.: Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185 (1921) Pynko, A.P.: Characterizing Belnap’s logic via De Morgan’s laws. Math. Logic Quart. 41(4), 442– 454 (1995) Reiter, R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980) Soare, R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Functions and Computably Generated Sets. Springer (1987) Sowa, J.F.: Semantic networks. In: Shapiro, S.C. (ed.) Encyclopedia of Artificial Intelligence (1987) Takeuti, G.: Proof theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.), Handbook of Philosophical Logic, vol. 2, 2nd edn., pp. 249–295. Kluwer, Dordrecht (2001) Wronski, A.: Remarks on a survey article on many valued logic by A. Urquhart, Studia Logica 46, 275–278 (1987) Zach, R.: Proof theory of finite-valued logics, Tech. Report TUW-E185.2-Z.1-93

Part I

Decidable DLs

Chapter 2

R-Calculus for Binary-Valued DL

∗ ∗ Mi/2 Ni/2 i/2 i/2 L∗ K∗

∗ ∗ Ri/2 Si/2 i/2 i/2 Q∗ P∗ =∗

∗ ∗ Let ∗ ∈ {t, f}. A 1/2-sequent X is M1/2 /N1/2 -valid, denoted by |=∗1/2 / |=1/2 X, if for any interpretation I, there is a statement X ∈ X such that I (X ) = ∗/I (X ) = ∗. 1/2 1/2 1/2 1/2 A 1/2-co-sequent Y is L∗ /K∗ -valid, denoted by |==∗ / |=∗ Y, if there is an interpretation I such that for each statement Y ∈ Y, I (Y ) = ∗/I (Y ) = ∗. = = = /N2/2 -valid, denoted by |== A 2/2-sequent X ⇒ Y is M2/2 2/2 / |=2/2 X ⇒ Y, if for any interpretation I, either I (X ) = t/I (X ) = t for some X ∈ X or I (Y ) = f/I (Y ) = f for some Y ∈ Y. 2/2 2/2 2/2 /K= -valid, denoted by |== / |=2/2 A 2/2-co-sequent X → Y is L= = X  → Y, if there is an interpretation I such that I (X ) = t/I (X ) = t for each X ∈ X and I (Y ) = f/I (Y ) = f for each Y ∈ Y. 1/2 1/2 ∗ ∗ , L∗ , N1/2 , K∗ (Takeuti There are sound and complete deduction systems M1/2 1/2 ∗ -valid sequents, L∗ -valid co-sequents, 1987; Urquhart 2001; Zach 2023) for M1/2 1/2 ∗ -valid sequents and K∗ -valid co-sequents, respectively. N1/2 Given a 1/2-sequent X and a statement X ∈ X, a 1/2-reduction X ↑ X ⇒ X[X ] =∗ ∗ /S∗1/2 -valid, denoted by |== is R1/2 1/2 / |=1/2 X ↑ X ⇒ X[X ], if

X =



∗ ∗ /N1/2 -valid X if X[X ] is M1/2 λ otherwise.

Given a 1/2-co-sequent X and a statement X, a 1/2-co-reduction X ↑ X ⇒ X, X 1/2 1/2 1/2 1/2 is Q∗ /P∗ -valid, denoted by |==∗ / |=∗ X ↑ X ⇒ X, X , if

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_2

21

22

2 R-Calculus for Binary-Valued DL

X =



1/2

1/2

X if X, X is L∗ /K∗ -valid λ otherwise.

Given a 2/2-sequent X ⇒ Y and statements X ∈ X, Y ∈ Y, a reduction X ⇒ Y ↑ = = = (X, Y ) ⇒ X ⇒ Y is R2/2 /S= 2/2 -valid, denoted by |=2/2 / |=2/2 X ⇒ Y ↑ (X, Y ) ⇒ X ⇒ Y , if  = = /N2/2 -valid X[X ] if X[X ] → Y is M2/2 X = X otherwise;  = = Y[Y ] if X ⇒ Y[Y ] is M2/2 /N2/2 -valid Y = Y otherwise. Given a 2/2-co-sequent X → Y and statements X, Y, a 2/2-co-reduction X → 2/2 2/2 2/2 Y ↑ (X, Y ) ⇒ X → Y is Q2/2 = /P= -valid, denoted by |== / |== X  → Y ↑ (X, Y ) ⇒ X → Y , if 

X, X X  Y, Y Y = Y

X =

2/2 2/2 /K= -valid if X, X → Y is L= otherwise; 2/2 2/2 if X → Y, Y is L= /K= -valid otherwise.

There are sound and complete R-calculi (Alchourrón et al. 1985; Avron 2003; Baader et al. 2003, 2007; Fermé and Hansson 2011; Fensel et al. 2001; Gärdenfors = = = 2/2 2/2 2/2 2/2 , S= and Rott 1995) R2/2 2/2 , Q= , P= for preserving M2/2 -, N2/2 -, L= - and K= validity of 2/2-sequents and 2/2-co-sequents, respectively. Hence, we have the following deduction systems:

1/2-sequents 1/2-co-sequents 2/2-sequents 2/2-co-sequents

deduction R-calculus t t M1/2 N1/2 1/2 1/2 Lt Kt = = M2/2 N2/2 2/2 2/2 L= K=

and R-calculi (Li 2007; Li and Sui 2013):

1/2-sequents 1/2-co-sequents 2/2-sequents 2/2-co-sequents

deduction R-calculus t R1/2 St1/2 1/2 1/2 Qt Pt = = R2/2 S2/2 2/2 Q2/2 P = =

2.1 Binary-Valued DL

23

2.1 Binary-Valued DL Let B2 be the smallest Boolean algebra. The logical language of binary-valued DL contains the following symbols: • atomic concepts: S0 , S1 , . . . ; • roles: R0 , R1 , . . . ; • concept constructors: ¬, , , ∀, ∃; and • logical connectives: ¬, ∧, ∨. Concepts are defined inductively as follows: C:: = S ↑ ¬C ↑ C1 C2 ↑ C1  C2 ↑ ∀R.C ↑ ∃R.C, where S is an atomic concept, and R is a role. Statements are defined as follows: ϕ:: = C(a) ↑ R(a, b) ↑ ¬R(a, b) ↑ ¬ϕ ↑ ϕ1 ∧ ϕ2 ↑ ϕ1 ∨ ϕ2 , where S(a), ¬S(a) (denoted by l(a)),R(a, b), ¬R(a, b) are called literals, and S(a), R(a, b) are called atoms. A model M is a pair (U, I ), where U is a non-empty set, and I is an interpretation such that ◦ for any atomic concept S, I (S) : U → B2 ; ◦ for any role R, I (R) : U 2 → B2 . Given an atomic concept S and a role R, we define interpretation of concepts ¬S and roles ¬R as follows: for any x, y ∈ U, S(x) ¬S(x) ¬2 S(x) t f t t f f

R(x, y) ¬R(x, y) ¬2 R(x, y) t f t t f f

The interpretation I (C) of a concept C is a function from U to B2 such that for any x ∈ U, ⎧ I (S)(x) ⎪ ⎪ ⎨ 1 − I (C)(x) I (C)(x) = ⎪ min{I (C1 )(x), I (C2 )(x)} ⎪ ⎩ max{I (C1 )(x), I (C2 )(x)} and

if if if if

C C C C

=S = ¬C1 = C1 C2 = C1  C2

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2 R-Calculus for Binary-Valued DL

x ∈ I (∀R.C) iff Ay(I (R)(x, y) = t ⇒ I (C)(y) = t) iff Ay(I (¬R)(x, y) = t or I (C)(y) = t) x ∈ I (¬∀R.C) iff Ey(I (R)(x, y) = t&I (C)(y) = f) iff Ey(I (R)(x, y) = t&I (¬C)(y) = t) x ∈ I (∃R.C) iff Ey(I (R)(x, y) = t&I (C)(y) = t) x ∈ I (¬∃R.C) iff Ay(I (R)(x, y) = t ⇒ I (C)(y) = f) iff Ay(I (R)(x, y) = f or I (C)(y) = f) iff Ay(I (¬R)(x, y) = t or I (¬C)(y) = t).

◦t

◦t

◦f

◦f

∀[B2 ]

∃[B2 ]

A statement C(a) is satisfied in I , denoted by I |= C(a), if I (C(a)) = t; and C(a) is valid, denoted by |= C(a), if I |= C(a) for any interpretation I. Lemma 2.1.1 For I (C(a)) = t, we have the following equivalences: (C1 C2 )(a) ≡ C1 (a)∧C2 (a) ¬(C1 C2 )(a) ≡ ¬C1 (a)∨¬C2 (a); (C1  C2 )(a) ≡ C1 (a)∨C2 (a) ¬(C1  C2 )(a) ≡ ¬C1 (a)∧¬C2 (a) (∀R.C)(a) ≡ ∀b(¬R(a, b)∨C(b)) (¬∀R.C)(a) ≡ ∃b(R(a, b)∧¬C(b)) (∃R.C)(a) ≡ ∃b(R(a, b)∧C(b)) (¬∃R.C)(a) ≡ ∀b(¬R(a, b)∨¬C(b)).  Lemma 2.1.2 For I (C(a)) = f, we have the following equivalences:

2.2 1/2-Sequents

25

(C1 C2 )(a) ≡ C1 (a)∨C2 (a) ¬(C1 C2 )(a) ≡ ¬C1 (a)∧¬C2 (a); (C1  C2 )(a) ≡ C1 (a)∧C2 (a) ¬(C1  C2 )(a) ≡ ¬C1 (a)∨¬C2 (a) (∀R.C)(a) ≡ ∃b(¬R(a, b)∧C(b)) (¬∀R.C)(a) ≡ ∀b(R(a, b)∨¬C(b)) (∃R.C)(a) ≡ ∀b(R(a, b)∨C(b)) (¬∃R.C)(a) ≡ ∃b(¬R(a, b)∧¬C(b)). 

2.2 1/2-Sequents Given a 1/2-sequent X of literals, define val(X) : El(a)(l(a), ¬l(a) ∈ X) inval(X) :∼ El(a)(l(a), ¬l(a) ∈ X) con(X) :∼ El(a)(l(a), ¬l(a) ∈ X) incon(X) : El(a)(l(a), ¬l(a) ∈ X). Hence, val(X) ≡ incon(X) and inval(X) ≡ con(X). t -valid, denoted by |=t1/2 X, if for any interpretation I, for A 1/2-sequent X is M1/2 some X ∈ X, I (X ) = t. t -valid, denoted by |=f1/2 X, if for any interpretation I, for A 1/2-sequent X is N1/2 some X ∈ X, I (X ) = f. 1/2 1/2 A 1/2-co-sequent X is Lt -valid, denoted by |=f X, if there is an interpretation I such that for every X ∈ X, I (X ) = f. 1/2 1/2 A 1/2-co-sequent X is Kt -valid, denoted by |=t X, if there is an interpretation I such that for every X ∈ X, I (X ) = t.

2.2.1 Deduction System Mt1/2 t Lemma 2.2.1 Let  be a set of literals.  is M1/2 -valid if and only if val().

Proof Assume that val(). Let l(a) be a literal such that l(a), ¬l(a) ∈ . Then, for any interpretation I, either I (l(a)) = t, or I (¬l(a)) = t, that is, I |=t . Conversely, assume that con(). Define an interpretation I such that U is the set of all the constants occurring in , and for any atomic concept S, role R and individual a,

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2 R-Calculus for Binary-Valued DL

⎧ ⎨ f if S(a) ∈  I (S(a)) = t if ¬S(a) ∈  ⎩ f otherwise.

⎧ ⎨ f if R(a, b) ∈  I (R(a, b)) = t if ¬R(a, b) ∈  ⎩ f otherwise.

Then, I is well-defined and I |=t .



t Deduction system M1/2 consists of the following axiom and deduction rules: • Axiom:  ∩ ¬−  = ∅ , (At1/2 ) 

where  is a set of literals. • Deduction rules: , C(a) (¬2 ) ,  ¬¬C(a)  , C1 (a) , C1 (a) ( ) , C2 (a) () , C2 (a) ,(C1 C2 )(a) ,(C1  C2 )(a) , ¬C1 (a) , ¬C1 (a) (¬ ) , ¬C2 (a) (¬) , ¬C2 (a) , ¬(C1 C2 )(a) , ¬(C1  C2 )(a) and



 , ¬R(a, e) , R(a, f ) (∀) , C(e) (¬∀) , ¬C( f ) , (∀R.C)(a) , (¬∀R.C)(a)   , R(a, f ) , ¬R(a, e) (∃) , C( f ) (¬∃) , ¬C(e) , (∃R.C)(a) , (¬∃R.C)(a)

where f is a constant, and e is a new constant. t Definition 2.2.2 A 1/2-sequent  is provable in M1/2 , denoted by t1/2 , if there is a sequence {δ1 , . . . , δn } of 1/2-sequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-sequents by one of the deduction t . rules in M1/2

Theorem 2.2.3 (Soundness and completeness theorem) For any 1/2-sequent , t1/2  if and only if |=t1/2 . 

2.2 1/2-Sequents

27

t 2.2.2 R-Calculus R1/2

Given a 1/2-sequent  and a statement C(a) ∈ , a 1/2-reduction  ↑ C(a) ⇒ t -valid, denoted by |=t1/2  ↑ C(a) ⇒ [C (a)], if [C (a)] is R1/2

C (a) =



t C(a) if [C(a)] is M1/2 -valid λ otherwise.

t R-calculus R1/2 consists of the following axioms and deduction rules: • Axioms: ¬El (a) = l(a)(l (a), ¬l (a) ∈ ) (At0 )  ↑ l(a) ⇒  t El (a)  = l(a)(l (a), ¬l (a) ∈ ) , (A− )  ↑ l(a) ⇒ [l(a)]

where  is a set of literals. • Deduction rules:  ↑ C(a) ⇒   ↑ ¬2 C(a) ⇒   ↑ C(a) ⇒ [C(a)] (¬− ) 2 2  ↑  ¬ C(a) ⇒ [¬ C(a)]  ↑ C1 (a) ⇒  ( 0 ) [C1 (a)] ↑ C2 (a) ⇒ [C1 (a)]  ↑ (C1 C2 )(a) ⇒   ↑ C1 (a) ⇒ [C1 (a)] ( − ) [C1 (a)] ↑ C2 (a) ⇒ [C1 (a), C2 (a)]  ↑ (C1 C2 )(a) ⇒ [(C1 C2 )(a)]  ↑ C1 (a) ⇒  (0 )  ↑ C2 (a) ⇒   ↑ (C1  C2 )(a) ⇒   ↑ C1 (a) ⇒ [C1 (a)] (− )  ↑ C2 (a) ⇒ [C2 (a)]  ↑ (C1  C2 )(a) ⇒ [(C1  C2 )(a)] (¬0 )

and

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 ↑ ¬R(a, f ) ⇒   ↑ C( f ) ⇒   ↑ (∀R.C)(a) ⇒   ↑ ¬R(a, e) ⇒ [¬R(a, e)] (∀− )  ↑ C(e) ⇒ [C(e)]  ↑ (∀R.C)(a) ⇒ [(∀R.C)(a)]  ↑ R(a, e) ⇒  (∃0 )  ↑ C(e) ⇒   ↑ (∃R.C)(a) ⇒   ↑ R(a, f ) ⇒ [R(a, f )] (∃− )  ↑ C( f ) ⇒ [C( f )]  ↑ (∃R.C)(a) ⇒ [(∃R.C)(a)] (∀0 )

and



 ↑ ¬C1 (a) ⇒   ↑ ¬C2 (a) ⇒   ↑ ¬(C1 C2 )(a) ⇒   ↑ ¬C1 (a) ⇒ [¬C1 (a)] (¬ − )  ↑ ¬C2 (a) ⇒ [¬C2 (a)]  ↑ ¬(C1 C2 )(a) ⇒ [¬(C1 C2 )(a)]  ↑ ¬C1 (a) ⇒  0 [¬C (¬ ) 1 (a)] ↑ ¬C 2 (a) ⇒ [¬C 1 (a)]  ↑ ¬(C1  C2 )(a) ⇒   ↑ ¬C1 (a) ⇒ [¬C1 (a)] (¬− ) [¬C1 (a)] ↑ ¬C2 (a) ⇒ [¬C1 (a), ¬C2 (a)]  ↑ ¬(C1  C2 )(a) ⇒ [¬(C1  C2 )(a)]

(¬ 0 )

and



 ↑ R(a, e) ⇒   ↑ ¬C(e) ⇒   ↑ (¬∀R.C)(a) ⇒   ↑ R(a, f ) ⇒ [R(a, f )] (¬∀− )  ↑ ¬C( f ) ⇒ [¬C( f )]  ↑ (¬∀R.C)(a) ⇒ [(¬∀R.C)(a)]  ↑ ¬R(a, f ) ⇒  (¬∃0 )  ↑ ¬C( f ) ⇒   ↑ (¬∃R.C)(a) ⇒   ↑ ¬R(a, e) ⇒ [¬R(a, e)] (¬∃− )  ↑ ¬C(e) ⇒ [¬C(e)]  ↑ (¬∃R.C)(a) ⇒ [(¬∃R.C)(a)] (¬∀0 )

where e is a new constant and f is a constant. Definition 2.2.4 Given a 1/2-sequent  and a statement C(a) ∈ , a 1/2-reduction t  ↑ C(a) ⇒  is provable in R1/2 , denoted by t1/2  ↑ C(a) ⇒  , if there is a sequence {δ1 , . . . , δn ] of 1/2-reductions such that δn =  ↑ C(a) ⇒  , and for each

2.2 1/2-Sequents

29

1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-reductions by one t . of the deduction rules in R1/2 Theorem 2.2.5 (Soundness and completeness theorem) For any 1/2-reduction  ↑ C(a) ⇒  with C(a) ∈ , t1/2  ↑ C(a) ⇒  if and only if |=t1/2  ↑ C(a) ⇒  .  t 2.2.3 Deduction System N1/2 t A 1/2-sequent  is N1/2 -valid, denoted by |=f1/2 , if for any interpretation I, I (A(a)) = f for some statement A(a) ∈ . t consists of the following axiom and deduction rules: Deduction system N1/2 • Axiom:  ∩ ¬−  = ∅ (Af1/2 ) , 

where  is a set of literals. • Deduction rules: , C(a) (¬2 ) ,  ¬¬C(a)  , C1 (a) , C1 (a) ( ) , C2 (a) () , C2 (a) ,(C1 C2 )(a) ,(C1  C2 )(a) , ¬C1 (a) , ¬C1 (a) (¬ ) , ¬C2 (a) (¬) , ¬C2 (a) , ¬(C1 C2 )(a) , ¬(C1  C2 )(a) and



 , ¬R(a, f ) , R(a, e) (∀) , C( f ) (¬∀) , ¬C(e) , (∀R.C)(a) , (¬∀R.C)(a)   , R(a, e) , ¬R(a, f ) (∃) , C(e) (¬∃) , ¬C( f ) , (∃R.C)(a) , (¬∃R.C)(a)

where f is a constant, and e is a new constant. t Definition 2.2.6 A 1/2-sequent  is provable in N1/2 , denoted by f1/2 , if there is a sequence {δ1 , . . . , δn } of 1/2-sequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-sequents by one of the deduction t . rules in N1/2

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Theorem 2.2.7 (Soundness and completeness theorem) For any 1/2-sequent , f1/2  if and only if |=f1/2 . 

2.2.4 R-Calculus St1/2 Given a 1/2-sequent  and a statement C(a) ∈ , a 1/2-reduction  ↑ C(a) ⇒ [C (a)] is St1/2 -valid, denoted by |=f1/2  ↑ C(a) ⇒ [C (a)], if

C (a) =



t -valid C(a) if [C(a)] is N1/2 λ otherwise.

R-calculus St1/2 consists of the following axioms and deduction rules: • Axioms: ¬El (a) = l(a)(l (a), ¬l (a) ∈ ) (At0 )  ↑ l(a) ⇒  t El (a)  = l(a)(l (a), ¬l (a) ∈ ) , (A− )  ↑ l(a) ⇒ [l(a)] where , l(a) is a set of literals. • Deduction rules:  ↑ C(a) ⇒ [C(a)]  ↑ ¬2 C(a) ⇒ [¬2 C(a)]  ↑ C(a) ⇒  (¬0 ) 2   ↑ ¬ C(a) ⇒   ↑ C1 (a) ⇒ [C1 (a)] ( − )  ↑ C2 (a) ⇒ [C2 (a)]  ↑ (C1 C2 )(a) ⇒ [(C1 C2 )(a)]  ↑ C1 (a) ⇒  ( 0 )  ↑ C2 (a) ⇒   ↑ (C1 C2 )(a) ⇒   ↑ C1 (a) ⇒ [C1 (a)] (− ) [C1 (a)] ↑ C2 (a) ⇒ [C1 (a), C2 (a)]  ↑ (C1  C2 )(a) ⇒ [(C1  C2 )(a)]  ↑ C1 (a) ⇒  (0 ) [C1 (a)] ↑ C2 (a) ⇒ [C1 (a)]  ↑ (C1  C2 )(a) ⇒  (¬− )

and

2.2 1/2-Sequents

31



 ↑ ¬R(a, f ) ⇒ [¬R(a, f )]  ↑ C( f ) ⇒ [C( f )]   ↑ (∀R.C)(a) ⇒ [(∀R.C)(a)]  ↑ ¬R(a, e) ⇒  (∀0 )  ↑ C(e) ⇒   ↑ (∀R.C)(a) ⇒   ↑ R(a, e) ⇒ [R(a, e)] (∃− )  ↑ C(e) ⇒ [C(e)]  ↑ (∃R.C)(a) ⇒ [(∃R.C)(a)]  ↑ R(a, f ) ⇒  (∃0 )  ↑ C( f ) ⇒   ↑ (∃R.C)(a) ⇒  (∀− )

and



 ↑ ¬C1 (a) ⇒ [¬C1 (a)] [¬C1 (a)] ↑ ¬C2 (a) ⇒ [¬C1 (a), ¬C2 (a)]   ↑ ¬(C1 C2 )(a) ⇒ [¬(C1 C2 )(a)]  ↑ ¬C1 (a) ⇒  (¬ 0 ) [¬C1 (a)] ↑ ¬C2 (a) ⇒ [¬C1 (a)]  ↑ ¬(C1 C2 )(a) ⇒   ↑ ¬C1 (a) ⇒ [¬C1 (a)] − (¬ )  ↑ ¬C2 (a) ⇒ [¬C2 (a)]  ↑ ¬(C1  C2 )(a) ⇒ [¬(C1  C2 )(a)]  ↑ ¬C1 (a) ⇒  (¬0 )  ↑ ¬C2 (a) ⇒   ↑ ¬(C1  C2 )(a) ⇒ 

(¬ − )

and



 ↑ R(a, e) ⇒ [R(a, e)]  ↑ ¬C(e) ⇒ [¬C(e)]   ↑ (¬∀R.C)(a) ⇒ [(¬∀R.C)(a)]  ↑ R(a, f ) ⇒  (¬∀0 )  ↑ ¬C( f ) ⇒   ↑ (¬∀R.C)(a) ⇒   ↑ ¬R(a, f ) ⇒ [¬R(a, f )] (¬∃− )  ↑ ¬C( f ) ⇒ [¬C( f )]  ↑ (¬∃R.C)(a) ⇒ [¬∃R.C)(a)]  ↑ ¬R(a, e) ⇒  (¬∃0 )  ↑ ¬C(e) ⇒   ↑ (¬∃R.C)(a) ⇒  (¬∀− )

where e is a new constant and f is a constant. Definition 2.2.8 Given a 1/2-sequent  and a statement C(a) ∈ , a 1/2-reduction  ↑ C(a) ⇒ [C (a)] is provable in St1/2 , denoted by f1/2  ↑ C(a) ⇒ [C (a)], if there is a sequence {δ1 , . . . , δn } of 1/2-reductions such that δn =  ↑ C(a) ⇒

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[C (a)], and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-reductions by one of the deduction rules in St1/2 . Theorem 2.2.9 (Soundness and completeness theorem) For any 1/2-reduction  ↑ C(a) ⇒ [C (a)], f1/2  ↑ C(a) ⇒ [C (a)] if and only if |=f1/2  ↑ C(a) ⇒ [C (a)]. 

2.3 1/2-Co-Sequents 1/2

2.3.1 Deduction System Lt 1/2

1/2

A 1/2-co-sequent  is Lt -valid, denoted by |=f , if there is an interpretation I such that for each statement A(a) ∈ , I (A(a)) = f. 1/2 Deduction system Lt consists of the following axiom and deduction rules: • Axiom: − 1/2  ∩ ¬  = ∅ , (Af )  where  is a set of literals. • Deduction rules: , C(a) (¬2 ) , ¬¬C(a)  , C1 (a) , C1 (a) ( ) , C2 (a) () , C2 (a) ,(C1 C2 )(a) ,(C1  C2 )(a) , ¬C1 (a) , ¬C1 (a) (¬ ) , ¬C2 (a) (¬) , ¬C2 (a) , ¬(C1 C2 )(a) , ¬(C1  C2 )(a) and



 , ¬R(a, f ) , R(a, e) (∀) , C( f ) (¬∀) , ¬C(e) , (∀R.C)(a) , (¬∀R.C)(a)   , R(a, e) , ¬R(a, f ) (∃) , C(e) (¬∃) , ¬C( f ) , (∃R.C)(a) , (¬∃R.C)(a)

where e is a new constant and f is a constant.

2.3 1/2-Co-Sequents

33 1/2

1/2

Definition 2.3.1 A 1/2-co-sequent  is provable in Lt , denoted by f , if there is a sequence {δ1 , . . . , δn } of 1/2-co-sequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the 1/2 deduction rules in Lt . Theorem 2.3.2 (Soundness and completeness theorem) For any 1/2-co-sequent , 1/2

1/2

f  if and only if |=f .  1/2

2.3.2 R-Calculus Qt

Given a 1/2-co-sequent  and a statement C(a), a 1/2-co-reduction  ↑ C(a) ⇒ 1/2 1/2 (C (a)) is Qt -valid, denoted by |=f  ↑ C(a) ⇒ (C (a)), if C (a) =



1/2

C(a) if (C(a)) is Lt -valid λ otherwise.

1/2

R-calculus Qt consists of the following axioms and deduction rules: • Axioms: ¬l(a) ∈  (At0 )  ↑ l(a) ⇒  ¬l(a) ∈ / t , (A+ )  ↑ l(a) ⇒ (l(a)) where  is a set of literals. • Deduction rules:  ↑ C(a) ⇒ , C(a)  ↑ ¬2 C(a) ⇒ , ¬2 C(a)  ↑ C(a) ⇒  (¬0 ) 2   ↑ ¬ C(a) ⇒   ↑ C1 (a) ⇒ , C1 (a) ( + )  ↑ C2 (a) ⇒ , C2 (a)  ↑ (C1 C2 )(a) ⇒ , (C1 C2 )(a)  ↑ C1 (a) ⇒  ( 0 )  ↑ C2 (a) ⇒   ↑ (C1 C2 )(a) ⇒   ↑ C1 (a) ⇒ , C1 (a) (+ ) , C1 (a) ↑ C2 (a) ⇒ , C1 (a), C2 (a)  ↑ (C1  C2 )(a) ⇒ , (C1  C2 )(a)  ↑ C1 (a) ⇒  (0 ) , C1 (a) ↑ C2 (a) ⇒ , C1 (a)  ↑ (C1  C2 )(a) ⇒  (¬+ )

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2 R-Calculus for Binary-Valued DL



and

 ↑ ¬R(a, e) ⇒   ↑ C(e) ⇒   ↑ (∀R.C)(a) ⇒   ↑ ¬R(a, f ) ⇒ , ¬R(a, f ) (∀− )  ↑ C( f ) ⇒ , C( f )  ↑ (∀R.C)(a) ⇒ , (∀R.C)(a)  ↑ R(a, f ) ⇒  (∃0 )  ↑ C( f ) ⇒   ↑ (∃R.C)(a) ⇒   ↑ R(a, e) ⇒ , R(a, e) (∃− )  ↑ C(e) ⇒ , C(e)  ↑ (∃R.C)(a) ⇒ , (∃R.C)(a) (∀0 )

and



 ↑ ¬C1 (a) ⇒ , ¬C1 (a) , ¬C1 (a) ↑ ¬C2 (a) ⇒ , ¬C1 (a), ¬C2 (a)   ↑ ¬(C1 C2 )(a) ⇒ , ¬(C1 C2 )(a)  ↑ ¬C1 (a) ⇒  (¬ 0 ) , ¬C1 (a) ↑ ¬C2 (a) ⇒ , ¬C1 (a)  ↑ ¬(C1 C2 )(a) ⇒   ↑ ¬C1 (a) ⇒ , ¬C1 (a) (¬+ )  ↑ ¬C2 (a) ⇒ , ¬C2 (a)  ↑ ¬(C1  C2 )(a) ⇒ , ¬(C1  C2 )(a)  ↑ ¬C1 (a) ⇒  (¬0 )  ↑ ¬C2 (a) ⇒   ↑ ¬(C1  C2 )(a) ⇒ 

(¬ + )

and



 ↑ R(a, f ) ⇒   ↑ ¬C( f ) ⇒   ↑ (¬∀R.C)(a) ⇒   ↑ R(a, e) ⇒ , R(a, e) (¬∀− )  ↑ ¬C(e) ⇒ , ¬C(e)  ↑ (¬∀R.C)(a) ⇒ , (¬∀R.C)(a)  ↑ ¬R(a, e) ⇒  (¬∃0 )  ↑ ¬C(e) ⇒   ↑ (¬∃R.C)(a) ⇒   ↑ ¬R(a, f ) ⇒ , ¬R(a, f ) (¬∃− )  ↑ ¬C( f ) ⇒ , ¬C( f )  ↑ (¬∃R.C)(a) ⇒ , ¬∃R.C)(a) (¬∀0 )

where e is a new constant and f is a constant. Definition 2.3.3 Given a 1/2-co-sequent  and a statement C(a), a 1/2-reduction 1/2 1/2  ↑ C(a) ⇒ , C (a) is provable in Qt , denoted by f  ↑ C(a) ⇒ , C (a),

2.3 1/2-Co-Sequents

35

if there is a sequence {δ1 , . . . , δn } of 1/2-reductions such that δn =  ↑ C(a) ⇒  , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/21/2 reductions by one of the deduction rules in Qt . Theorem 2.3.4 (Soundness and completeness theorem) For any 1/2-reduction  ↑ C(a) ⇒ , C (a), f  ↑ C(a) ⇒ , C (a) if and only if |=f1/2  ↑ C(a) ⇒ , C (a). 1/2



1/2

2.3.3 Deduction System Kt 1/2

1/2

A 1/2-co-sequent  is Kf -valid, denoted by |=t , if there is an interpretation I such that for each statement A(a) ∈ , I (A(a)) = t. 1/2 Deduction system Kt consists of the following axiom and deduction rules: • Axiom: − 1/2  ∩ ¬  = ∅ , (At )  where  is a set of literals. • Deduction rules: , C(a) (¬2 ) , ¬¬C(a)  , C1 (a) , C1 (a) ( ) , C2 (a) () , C2 (a) ,(C1 C2 )(a) ,(C1  C2 )(a) , ¬C1 (a) , ¬C1 (a) (¬ ) , ¬C2 (a) (¬) , ¬C2 (a) , ¬(C1 C2 )(a) , ¬(C1  C2 )(a) and



 , ¬R(a, e) , R(a, f ) (∀) , C(e) (¬∀) , ¬C( f ) , (∀R.C)(a) , (¬∀R.C)(a)   , R(a, f ) , ¬R(a, e) (∃) , C( f ) (¬∃) , ¬C(e) , (∃R.C)(a) , (¬∃R.C)(a)

where f is a constant, and e is a new constant. 1/2

1/2

Definition 2.3.5 A 1/2-co-sequent  is provable in Kt , denoted by t , if there is a sequence {δ1 , . . . , δn } of 1/2-co-sequents such that δn = , and for each

36

2 R-Calculus for Binary-Valued DL

1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-co-sequents by 1/2 one of the deduction rules in Kt . Theorem 2.3.6 (Soundness and completeness theorem) For any 1/2-co-sequent , 1/2

1/2

t  if and only if |=t . 

1/2

2.3.4 R-Calculus Pt

Given a 1/2-co-sequent  and a statement C(a), a 1/2-co-reduction  ↑ C(a) ⇒ 1/2 1/2 , C (a) is Pt -valid, denoted by |=t  ↑ C(a) ⇒ , C (a), if

C (a) =



1/2

C(a) if , C(a) is Kt -valid λ otherwise.

1/2

R-calculus Pt consists of the following axioms and deduction rules: • Axioms: ¬l(a) ∈ / ¬l(a) ∈  (A0 ) , (A+ t)  ↑ l(a) ⇒ , l(a) t  ↑ l(a) ⇒  where , l(a) is a set of literals. • Deduction rules:  ↑ C(a) ⇒ , C(a)  ↑ ¬2 C(a) ⇒ , ¬2 C(a)  ↑ C(a) ⇒  (¬0 ) 2   ↑ ¬ C(a) ⇒   ↑ C1 (a) ⇒ , C1 (a) ( + ) , C1 (a) ↑ C2 (a) ⇒ , C1 (a), C2 (a)  ↑ (C1 C2 )(a) ⇒ , (C1 C2 )(a)  ↑ C1 (a) ⇒  ( 0 ) , C1 (a) ↑ C2 (a) ⇒ , C1 (a)  ↑ (C1 C2 )(a) ⇒   ↑ C1 (a) ⇒ , C1 (a) (+ )  ↑ C2 (a) ⇒ , C2 (a)  ↑ (C1  C2 )(a) ⇒ , (C1  C2 )(a)  ↑ C1 (a) ⇒  (0 )  ↑ C2 (a) ⇒   ↑ (C1  C2 )(a) ⇒  (¬+ )

and

2.3 1/2-Co-Sequents

37



 ↑ ¬R(a, f ) ⇒   ↑ C( f ) ⇒   ↑ (∀R.C)(a) ⇒   ↑ ¬R(a, e) ⇒ , ¬R(a, e) (∀− )  ↑ C(e) ⇒ , C(e)  ↑ (∀R.C)(a) ⇒ , (∀R.C)(a)  ↑ R(a, e) ⇒  (∃0 )  ↑ C(e) ⇒   ↑ (∃R.C)(a) ⇒   ↑ R(a, f ) ⇒ , R(a, f ) (∃− )  ↑ C( f ) ⇒ , C( f )  ↑ (∃R.C)(a) ⇒ , (∃R.C)(a) (∀0 )

and



 ↑ ¬C1 (a) ⇒ , ¬C1 (a)  ↑ ¬C2 (a) ⇒ , ¬C2 (a)   ↑ ¬(C1 C2 )(a) ⇒ , ¬(C1 C2 )(a)  ↑ ¬C1 (a) ⇒  (¬ 0 )  ↑ ¬C2 (a) ⇒   ↑ ¬(C1 C2 )(a) ⇒   ↑ ¬C1 (a) ⇒ , ¬C1 (a) + (¬ ) , ¬C1 (a) ↑ ¬C2 (a) ⇒ , ¬C1 (a), ¬C2 (a)  ↑ ¬(C1  C2 )(a) ⇒ , ¬(C1  C2 )(a)  ↑ ¬C1 (a) ⇒  (¬0 ) , ¬C1 (a) ↑ ¬C2 (a) ⇒ , ¬C1 (a)  ↑ ¬(C1  C2 )(a) ⇒ 

(¬ + )

and



 ↑ R(a, e) ⇒   ↑ ¬C(e) ⇒   ↑ (¬∀R.C)(a) ⇒   ↑ R(a, f ) ⇒ , R(a, f ) (¬∀− )  ↑ ¬C( f ) ⇒ , ¬C( f )  ↑ (¬∀R.C)(a) ⇒ , (¬∀R.C)(a)  ↑ ¬R(a, f ) ⇒  (¬∃0 )  ↑ ¬C( f ) ⇒   ↑ (¬∃R.C)(a) ⇒   ↑ ¬R(a, e) ⇒ , ¬R(a, e) (¬∃− )  ↑ ¬C(e) ⇒ , ¬C(e)  ↑ (¬∃R.C)(a) ⇒ , ¬∃R.C)(a) (¬∀0 )

Definition 2.3.7 Given a 1/2-sequent  and a statement C(a), a 1/2-co-reduction 1/2 1/2  ↑ C(a) ⇒ , C (a) is provable in Pt , denoted by t  ↑ C(a) ⇒ , C (a), if there is a sequence {δ1 , . . . , δn } of 1/2-co-reductions such that δn =  ↑ C(a) ⇒

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2 R-Calculus for Binary-Valued DL

, C (a), and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2 1/2-co-reductions by one of the deduction rules in Pt . Theorem 2.3.8 (Soundness and completeness theorem) For any 1/2-co-reduction  ↑ C(a) ⇒ , C (a), t  ↑ C(a) ⇒ , C (a) if and only if |=t  ↑ C(a) ⇒ , C (a). 1/2

1/2



2.4 2/2-Sequents = A 2/2-sequent X ⇒ Y is M2/2 -valid, denoted by |== 2/2 X ⇒ Y, if for any interpretation I, either for some X ∈ X, I (X ) = t, or for some Y ∈ Y, I (Y ) = f. = = -valid, denoted by |=2/2 X ⇒ Y, if for any interA 2/2-sequent X ⇒ Y is N2/2 pretation I, either for some X ∈ X, I (X ) = f, or for some Y ∈ Y, I (Y ) = t. Given statements X ∈ X, Y ∈ Y, a 2/2-reduction X ⇒ Y ↑ (X, Y ) ⇒ X ⇒ Y = = is R2/2 /S= 2/2 -valid, denoted by |=2/2 X ⇒ Y ↑ (X, Y ) ⇒ X ⇒ Y , if



X[X ] X  Y[Y ] Y = Y X =

= = /N2/2 -valid if X[X ] ⇒ Y is M2/2 otherwise; = = /N2/2 -valid if X ⇒ Y[Y ] is M2/2 otherwise.

= = /N2/2 -valid if and only Lemma 2.4.1 Let ,  be a set of literals.  ⇒  is M2/2 if val() or incon() or  ∩  = ∅.

Proof Assume that val(). Let l(a) be a literal such that l(a), ¬l(a) ∈ . Then, for any interpretation I, either I (l(a)) = t, or I (¬l(a)) = t, that is, I |=t  ⇒ . Assume that  ∩  = ∅. Let l(a) be a literal such that l(a) ∈  ∩ . Then, for any interpretation I, either I (l(a)) = t, or I (l(a)) = t. Either implies I |=t  ⇒ . Conversely, assume that inval()&con()& ∩  = ∅. Define an model M = (U, I ) such that U is the set of all the constants occurring in  and , and for atomic concept S and role R, I (S) = {e : S(e) ∈  or ¬S(e) ∈ } I (R) = {(e, f ) : R(e, f ) ∈  or ¬R(e, f ) ∈ }. By assumption, I is well-defined and I |=t  ⇒ .



2.4 2/2-Sequents

39

= 2.4.1 Deduction System M2/2 = A 2/2-sequent  ⇒  is M2/2 -valid, denoted by |== 2/2  ⇒ , if for any interpretation I, either for some A ∈ , I (A) = t, or for some D ∈ , I (D) = f. = consists of the following deduction rules and axiom: Deduction system M2/2 • Axiom: ⎧ ⎨  ∩ ¬−  = ∅  ∩ ¬−  = ∅ (A= 2/2 ) ⎩  ∩   = ∅  ⇒ ,

where ,  are sets of literals. • Deduction rules: , C(a) ⇒  (¬2L ) ,  ¬¬C(a) ⇒  , C1 (a) ⇒  ( L ) , C2 (a) ⇒  ,  (C1 C2 )(a) ⇒  , C1 (a) ⇒  ( L ) , C2 (a) ⇒  ,(C1  C2 )(a) ⇒  , ¬C1 (a) ⇒  (¬ L ) , ¬C2 (a) ⇒  ,  ¬(C1 C2 )(a) ⇒  , ¬C1 (a) ⇒  (¬ L ) , ¬C2 (a) ⇒  , ¬(C1  C2 )(a) ⇒  and



, ¬R(a, e) ⇒  (∀ L ) , C(e) ⇒  ,(∀R.C)(a) ⇒  , R(a, f ) ⇒  (¬∀ L ) , ¬C( f ) ⇒   , (¬∀R.C)(a) ⇒  , R(a, f ) ⇒  L (∃ ) , C( f ) ⇒  ,(∃R.C)(a) ⇒  , ¬R(a, e) ⇒  (¬∃ L ) , ¬C(e) ⇒  , (¬∃R.C)(a) ⇒ 

 ⇒ D(b),  (¬2R )   ⇒ ¬¬D(b),   ⇒ D1 (b),  ( R )  ⇒ D2 (b),    ⇒ (D1 D2 )(b),   ⇒ D1 (b),  ( R )  ⇒ D2 (b),   ⇒ (D1  D2 )(b),   ⇒ ¬D1 (b),  (¬ R )  ⇒ ¬D2 (b),    ⇒ ¬(D1 D2 )(b),   ⇒ ¬D1 (b),  (¬ R )  ⇒ ¬D2 (b),   ⇒ ¬(D1  D2 )(b),  

 ⇒ ¬R(b, f ),   ⇒ D( f ),   ⇒ (∀R.D)(b),   ⇒ R(b, e),  (¬∀ R )  ⇒ ¬D(e),    ⇒ (¬∀R.D)(b),   ⇒ R(b, e),  R (∃ )  ⇒ D(e),   ⇒ (∃R.D)(b),   ⇒ ¬R(b, f ),  (¬∃ R )  ⇒ ¬D( f ),   ⇒ (¬∃R.D)(b),  (∀ R )

where e is a new constant and f is a constant.

40

2 R-Calculus for Binary-Valued DL

= Definition 2.4.2 A 2/2-sequent  ⇒  is provable in M2/2 , denoted by = 2/2  ⇒ , if there is a sequence {δ1 , . . . , δn } of 2/2-sequents such that δn =  ⇒ , and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-sequents by one of the deduction = rules in M2/2 .

Theorem 2.4.3 (Soundness and completeness theorem) For any 2/2-sequent  ⇒ , = |== 2/2  ⇒  iff 2/2  ⇒ .  = 2.4.2 R-Calculus R2/2 = R-calculus R2/2 consists of the following deduction rules and axioms: • Axioms: ⎧ ⎡ [l(a)] ∩ ¬− [l(a)] = ∅ ⎨ [l(a)] ∩ ¬− [l(a)] = ∅ − ⎣  ∩ ¬  = ∅  ∩ ¬−  = ∅ = L = L (A− ) ⎩ (A0 ) [l(a)] ∩  = ∅ [l(a)] ∩  = ∅  ⇒  ↑ l(a) ⇒ [l(a)] ⇒   ⇒⇒ ⎧ ⎡ ⇒  ↑−l(a)  ∩ ¬  = ∅ ⎨  ∩ ¬−  = ∅ − [m(b)] ∩ ¬− [m(b)] = ∅ = R = R ⎣ [m(b)] ∩ ¬ [m(b)] = ∅ (A− ) ⎩ (A0 )  ∩ [m(b)] = ∅  ∩ [m(b)] = ∅  ⇒  ↑ m(b) ⇒  ⇒ [m(b)]  ⇒  ↑ m(b) ⇒  ⇒ 

where  ⇒  is literal, and l(a) ∈ , m(b) ∈ . • Deduction rules consist of two parts E L and E R : ◦ EL :  ⇒  ↑ C(a) ⇒ [C(a)] ⇒   ⇒  ↑ ¬2 C(a) ⇒ [¬2 C(a)] ⇒   ⇒  ↑ C(a) ⇒  ⇒  (¬2L 0 ) 2   ⇒  ↑ ¬ C(a) ⇒  ⇒   ⇒  ↑ C1 (a) ⇒ [C1 (a)] ⇒  L ) [C1 (a)] ⇒  ↑ C2 (a) ⇒ [C1 (a), C2 (a)] ⇒  ( −  ⇒  ↑ (C1 C2 )(a) ⇒ [(C1 C2 )(a)] ⇒    ⇒  ↑ C1 (a) ⇒  ⇒  ( 0L ) [C1 (a)] ⇒  ↑ C2 (a) ⇒ [C1 (a)] ⇒    ⇒  ↑ (C1 C2 )(a) ⇒  ⇒   ⇒  ↑ C1 (a) ⇒ [C1 (a)] ⇒  L (− )  ⇒  ↑ C2 (a) ⇒ [C2 (a)] ⇒   ⇒  ↑ (C1  C2 )(a) ⇒ [(C1  C2 )(a)] ⇒   ⇒  ↑ C1 (a) ⇒  ⇒  (0L )  ⇒  ↑ C2 (a) ⇒  ⇒   ⇒  ↑ (C1  C2 )(a) ⇒  ⇒  (¬2L − )

2.4 2/2-Sequents

41



and

 ⇒  ↑ R(a, e) ⇒  ⇒   ⇒  ↑ ¬C(e) ⇒  ⇒    ⇒  ↑ (∀R.C)(a) ⇒  ⇒   ⇒  ↑ R(a, f ) ⇒ [R(a, f )] ⇒  L (∀− )  ⇒  ↑ ¬C( f ) ⇒ [¬C( f )] ⇒   ⇒  ↑ (∀R.C)(a) ⇒ [(∀R.C)(a)] ⇒   ⇒  ↑ R(a, f ) ⇒  ⇒  (∃0L )  ⇒  ↑ C( f ) ⇒  ⇒    ⇒  ↑ (∃R.C)(a) ⇒  ⇒   ⇒  ↑ R(a, e) ⇒ [R(a, e)] ⇒  L (∃− )  ⇒  ↑ C(e) ⇒ [C(e)] ⇒   ⇒  ↑ (∃R.C)(a) ⇒ [(∃R.C)(a)] ⇒ 

(∀0L )

and 

 ⇒  ↑ ¬C1 (a) ⇒ [¬C1 (a)] ⇒   ⇒  ↑ ¬C2 (a) ⇒ [¬C2 (a)] ⇒    ⇒  ↑ ¬(C1 C2 )(a) ⇒ [¬(C1 C2 )(a)] ⇒   ⇒  ↑ ¬C1 (a) ⇒  ⇒  (¬ 0L )  ⇒  ↑ ¬C2 (a) ⇒  ⇒    ⇒  ↑ ¬(C1 C2 )(a) ⇒  ⇒   ⇒  ↑ ¬C1 (a) ⇒ [¬C1 (a)] ⇒  L ) [¬C1 (a)] ⇒  ↑ ¬C2 (a) ⇒ [¬C1 (a), ¬C2 (a)] ⇒  (¬−  ⇒  ↑ ¬(C1  C2 )(a) ⇒ [¬(C1  C2 )(a)] ⇒    ⇒  ↑ ¬C1 (a) ⇒  ⇒  (¬0L ) [¬C1 (a)] ⇒  ↑ ¬C2 (a) ⇒ [¬C1 (a)] ⇒   ⇒  ↑ ¬(C1  C2 )(a) ⇒  ⇒ 

L (¬ − )

and



 ⇒  ↑ R(a, f ) ⇒  ⇒   ⇒  ↑ ¬C( f ) ⇒  ⇒    ⇒  ↑ (¬∀R.C)(a) ⇒  ⇒   ⇒  ↑ R(a, e) ⇒ [R(a, e)] ⇒  L (¬∀− )  ⇒  ↑ ¬C(e) ⇒ [¬C(e)] ⇒   ⇒  ↑ (¬∀R.C)(a) ⇒ [(¬∀R.C)(a)] ⇒    ⇒  ↑ ¬R(a, f ) ⇒  ⇒  (¬∃0L )  ⇒  ↑ ¬C( f ) ⇒  ⇒    ⇒  ↑ (¬∃R.C)(a) ⇒  ⇒   ⇒  ↑ ¬R(a, e) ⇒ [¬R(a, e)] ⇒  L )  ⇒  ↑ ¬C(e) ⇒ [¬C(e)] ⇒  (¬∃−  ⇒  ↑ (¬∃R.C)(a) ⇒ [(¬∃R.C)(a)] ⇒  (¬∀0L )

where e is a new constant and f is a constant.

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2 R-Calculus for Binary-Valued DL

◦ ER :

 ⇒  ↑ D(b) ⇒  ⇒ [D(b)]  ⇒  ↑ ¬2 D(b) ⇒  ⇒ [¬2 D(b)]  ⇒  ↑ D(b) ⇒  ⇒  (¬2R 0 )  ⇒  ↑ ¬2 D(b) ⇒  ⇒    ⇒  ↑ D1 (b) ⇒  ⇒ [D1 (b)] R ( − )  ⇒  ↑ D2 (b) ⇒  ⇒ [D2 (b)]  ⇒  ↑ (D1 D2 )(b) ⇒  ⇒ [(D1 D2 )(b)]   ⇒  ↑ D1 (b) ⇒  ⇒  ⇒  ↑ D (b) ⇒  ⇒  R  ( 0 ) 2 ⇒  ↑ (D D )(b) ⇒  ⇒   1 2   ⇒  ↑ D1 (b) ⇒  ⇒ [D1 (b)] R )  ⇒ [D (b)] ↑ D (b) ⇒  ⇒ [D (b), D (b)] (− 1 2 1 2 ⇒  ↑ (D  D )(b) ⇒  ⇒ [(D  D )(b)]  1 2 1 2   ⇒  ↑ D1 (b) ⇒  ⇒  R (0 )  ⇒ [D1 (b)] ↑ D2 (b) ⇒  ⇒ [D1 (b)]  ⇒  ↑ (D1  D2 )(b) ⇒  ⇒  (¬2R − )

and (∀0R ) R) (∀−

(∃0R ) R) (∃−

and R) (¬ −

(¬ 0R ) R) (¬−

(¬0R )

  ⇒  ↑ ¬R(b, e) ⇒  ⇒   ⇒  ↑ D(e) ⇒  ⇒    ↑ (∀R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ ¬R(b, f ) ⇒  ⇒ [¬R(b, f )]  ⇒  ↑ D( f ) ⇒  ⇒ [D( f )]  ↑ (∀R.D)(b) ⇒  ⇒ [(∀R.D)(b)]  ⇒  ⇒  ↑ R(b, f ) ⇒  ⇒   ⇒  ↑ D( f ) ⇒  ⇒    ↑ (∃R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ R(b, e) ⇒  ⇒ [R(b, e)]  ⇒  ↑ D(e) ⇒  ⇒ [D(e)]  ⇒  ↑ (∃R.D)(b) ⇒  ⇒ [(∃R.D)(b)]

  ⇒  ↑ ¬D1 (b) ⇒  ⇒ [¬D1 (b)]  ⇒ [¬D1 (b)] ↑ ¬D2 (b) ⇒  ⇒ [¬D1 (b), ¬D2 (b)]  ⇒  ↑ ¬(D1 D2 )(b) ⇒  ⇒ [¬(D1 D2 )(b)]   ⇒  ↑ ¬D1 (b) ⇒  ⇒   ⇒ [¬D1 (b)] ↑ ¬D2 (b) ⇒  ⇒ [¬D1 (b)] ⇒  ↑ ¬(D D )(b) ⇒  ⇒   1 2   ⇒  ↑ ¬D1 (b) ⇒  ⇒ [¬D1 (b)]  ⇒  ↑ ¬D2 (b) ⇒  ⇒ [¬D2 (b)]  ⇒  ↑ ¬(D1  D2 )(b) ⇒  ⇒ [¬(D1  D2 )(b)]   ⇒  ↑ ¬D1 (b) ⇒  ⇒   ⇒  ↑ ¬D2 (b) ⇒  ⇒   ⇒  ↑ ¬(D1  D2 )(b) ⇒  ⇒ 

2.4 2/2-Sequents

and

43



 ⇒  ↑ R(b, f ) ⇒  ⇒   ⇒  ↑ ¬D( f ) ⇒  ⇒    ↑ (¬∀R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ R(b, e) ⇒  ⇒ [R(b, e)] R (¬∀− )  ⇒  ↑ ¬D(e) ⇒  ⇒ [¬D(e)]  ⇒  ↑ (¬∀R.D)(b) ⇒  ⇒ [(¬∀R.D)(b)]   ⇒  ↑ ¬R(b, e) ⇒  ⇒  R (¬∃0 )  ⇒  ↑ ¬D(e) ⇒  ⇒    ↑ (¬∃R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ ¬R(b, f ) ⇒  ⇒ [¬R(b, f )] R (¬∃− )  ⇒  ↑ ¬D( f ) ⇒  ⇒ [¬D( f )]  ⇒  ↑ (¬∃R.D)(b) ⇒  ⇒ [(¬∃R.D)(b)] (¬∀0R )

where e is a new constant and f is a constant. Definition 2.4.4 Given C(a) ∈  and D(b) ∈ , a 2/2-reduction δ =  ⇒  ↑ = (C(a), D(b)) ⇒  ⇒  is provable in R2/2 , denoted by = 2/2 δ, if there is a sequence {δ1 , . . . , δn } of 2/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/2-reductions by one of the deduc= . tion rules in R2/2 Theorem 2.4.5 (Soundness and completeness theorem) Let C(a) ∈  and D(b) ∈ . For any 2/2-reduction δ =  ⇒  ↑ (C(a), D(b)) ⇒  ⇒  , = |== 2/2 δ iff 2/2 δ.

= 2.4.3 Deduction System N2/2 = Deduction system N2/2 consists of the following deduction rules and axiom: • Axiom: ⎧ ⎨  ∩ ¬−  = ∅  ∩ ¬−  = ∅ ) (A= ⎩ 2/2  ∩  = ∅  ⇒ ,

where ,  are sets of literals. • Deduction rules:

44

2 R-Calculus for Binary-Valued DL

, C(a) ⇒  (¬2,L )  , ¬¬C(a) ⇒  , C1 (a) ⇒  L , C2 (a) ⇒  ( ) ,  (C1 C2 )(a) ⇒  , C1 (a) ⇒  ( L ) , C2 (a) ⇒  ,(C1  C2 )(a) ⇒  , ¬C1 (a) ⇒  (¬ L ) , ¬C2 (a) ⇒  ,  ¬(C1 C2 )(a) ⇒  , ¬C1 (a) ⇒  (¬ L ) , ¬C2 (a) ⇒  , ¬(C1  C2 )(a) ⇒  and



, ¬R(a, f ) ⇒  (∀ L ) , C( f ) ⇒  ,(∀R.C)(a) ⇒  , R(a, e) ⇒  (¬∀ L ) , ¬C(e) ⇒   , (¬∀R.C)(a) ⇒  , R(a, e) ⇒  (∃ L ) , C(e) ⇒  ,(∃R.C)(a) ⇒  , ¬R(a, f ) ⇒  (¬∃ L ) , ¬C( f ) ⇒  , (¬∃R.C)(a) ⇒ 

 ⇒ D(b),  (¬2,R )   ⇒ ¬¬D(b),   ⇒ D1 (b),  R  ⇒ D2 (b),  ( )   ⇒ (D1 D2 )(b),   ⇒ D1 (b),  ( R )  ⇒ D2 (b),   ⇒ (D1  D2 )(b),   ⇒ ¬D1 (b),  (¬ R )  ⇒ ¬D2 (b),    ⇒ ¬(D1 D2 )(b), )  ⇒ ¬D1 (b),  (¬ R )  ⇒ ¬D2 (b),   ⇒ ¬(D1  D2 )(b), ) 

 ⇒ ¬R(b, e),   ⇒ D(e),   ⇒ (∀R.D)(b),   ⇒ R(b, f ),  (¬∀ R )  ⇒ ¬D( f ),    ⇒ (¬∀R.D)(b),   ⇒ R(b, f ),  (∃ R )  ⇒ D( f ),   ⇒ (∃R.D)(b),   ⇒ ¬R(b, e),  (¬∃ R )  ⇒ ¬D(e),   ⇒ (¬∃R.C)(b),  (∀ R )

where e is a new constant and f is a constant. =

= Definition 2.4.6 A 2/2-sequent  ⇒  is provable in N2/2 , denoted by 2/2  ⇒ , if there is a sequence {δ1 , . . . , δn } of 2/2-sequents such that δn =  ⇒ , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/2-sequents = . by one of the deduction rules in N2/2

Theorem 2.4.7 (Soundness and completeness theorem) For any 2/2-sequent  ⇒ , = = |=2/2  ⇒  iff 2/2  ⇒ . 

2.4 2/2-Sequents

2.4.4 R-Calculus S= 2/2 R-calculus S= 2/2 consists of the following deduction rules and axioms: • Axioms: ⎧ ⎨ [l(a)] ∩ ¬− [l(a)] = ∅  ∩ ¬−  = ∅ (A=L − ) ⎩ [l(a)] ∩   = ∅  ⎡ ⇒  ↑ l(a)−⇒ [l(a)] ⇒  [l(a)] ∩ ¬ [l(a)] = ∅ ⎣  ∩ ¬−  = ∅ (A=L 0 ) [l(a)] ∩  = ∅  ⇒⇒ ⎧ ⇒  ↑−l(a) ⎨  ∩ ¬  = ∅ [m(b)] ∩ ¬− [m(b)] = ∅ (A=R − ) ⎩  ∩ [m(b)]  = ∅   ↑ m(b) ⇒  ⇒ [m(b)] ⎡ ⇒  ∩ ¬−  = ∅ ⎣ [m(b)] ∩ ¬− [m(b)] = ∅ (A=R 0 )  ∩ [m(b)] = ∅  ⇒  ↑ m(b) ⇒  ⇒  where  ⇒  is literal, and l(a) ∈ , m(b) ∈ . • Deduction rules consist of two parts E L and E R : ◦ EL :  ⇒  ↑ C(a) ⇒ [C(a)] ⇒   ⇒  ↑ ¬2 C(a) ⇒ [¬2 C(a)] ⇒   ⇒  ↑ C(a) ⇒  ⇒  (¬2L 0 ) 2   ⇒  ↑ ¬ C(a) ⇒  ⇒   ⇒  ↑ C1 (a) ⇒ [C1 (a)] ⇒  L )  ⇒  ↑ C2 (a) ⇒ [C2 (a)] ⇒  ( −  ⇒  ↑ (C1 C2 )(a) ⇒ [(C1 C2 )(a)] ⇒   ⇒  ↑ C1 (a) ⇒  ⇒  ( 0L )  ⇒  ↑ C2 (a) ⇒  ⇒    ⇒  ↑ (C1 C2 )(a) ⇒  ⇒   ⇒  ↑ C1 (a) ⇒ [C1 (a)] ⇒  L (− ) [C1 (a)] ⇒  ↑ C2 (a) ⇒ [C1 (a), C2 (a)] ⇒   ⇒  ↑ (C1  C2 )(a) ⇒ [(C1  C2 )(a)] ⇒    ⇒  ↑ C1 (a) ⇒  ⇒  (0L ) [C1 (a)] ⇒  ↑ C2 (a) ⇒ [C1 (a)] ⇒   ⇒  ↑ (C1  C2 )(a) ⇒  ⇒  (¬2L − )

and

45

46

2 R-Calculus for Binary-Valued DL



 ⇒  ↑ R(a, f ) ⇒  ⇒   ⇒  ↑ ¬C( f ) ⇒  ⇒    ⇒  ↑ (∀R.C)(a) ⇒  ⇒   ⇒  ↑ R(a, e) ⇒ [R(a, e)] ⇒  L (∀− )  ⇒  ↑ ¬C(e) ⇒ [¬C(e)] ⇒   ⇒  ↑ (∀R.C)(a) ⇒ [(∀R.C)(a)] ⇒    ⇒  ↑ R(a, e) ⇒  ⇒  (∃0L )  ⇒  ↑ C(e) ⇒  ⇒    ⇒  ↑ (∃R.C)(a) ⇒  ⇒   ⇒  ↑ R(a, f ) ⇒ [R(a, f )] ⇒  L (∃− )  ⇒  ↑ C( f ) ⇒ [C( f )] ⇒   ⇒  ↑ (∃R.C)(a) ⇒ [(∃R.C)(a)] ⇒  (∀0L )

and 

 ⇒  ↑ ¬C1 (a) ⇒ [¬C1 (a)] ⇒  [¬C1 (a)] ⇒  ↑ ¬C2 (a) ⇒ [¬C1 (a), ¬C2 (a)] ⇒    ⇒  ↑ ¬(C1 C2 )(a) ⇒ [¬(C1 C2 )(a)] ⇒   ⇒  ↑ ¬C1 (a) ⇒  ⇒  (¬ 0L ) [¬C1 (a)] ⇒  ↑ ¬C2 (a) ⇒ [C1 (a)] ⇒    ⇒  ↑ ¬(C1 C2 )(a) ⇒  ⇒   ⇒  ↑ ¬C1 (a) ⇒ [¬C1 (a)] ⇒  L (¬− )  ⇒  ↑ ¬C2 (a) ⇒ [¬C2 (a)] ⇒   ⇒  ↑ ¬(C1  C2 )(a) ⇒ [¬(C1  C2 )(a)] ⇒   ⇒  ↑ ¬C1 (a) ⇒  ⇒  (¬0L )  ⇒  ↑ ¬C2 (a) ⇒  ⇒   ⇒  ↑ ¬(C1  C2 )(a) ⇒  ⇒ 

L (¬ − )

and



 ⇒  ↑ R(a, e) ⇒  ⇒   ⇒  ↑ ¬C(e) ⇒  ⇒    ⇒  ↑ (¬∀R.C)(a) ⇒  ⇒   ⇒  ↑ R(a, f ) ⇒ [R(a, f )] ⇒  L (¬∀− )  ⇒  ↑ ¬C( f ) ⇒ [¬C( f )] ⇒   ⇒  ↑ (¬∀R.C)(a) ⇒ [(¬∀R.C)(a)] ⇒   ⇒  ↑ ¬R(a, e) ⇒  ⇒  (¬∃0L )  ⇒  ↑ ¬C(e) ⇒  ⇒    ⇒  ↑ (¬∃R.C)(a) ⇒  ⇒   ⇒  ↑ ¬R(a, f ) ⇒ [¬R(a, f )] ⇒  L )  ⇒  ↑ ¬C( f ) ⇒ [¬C( f )] ⇒  (¬∃−  ⇒  ↑ (¬∃R.C)(a) ⇒ [(¬∃R.C)(a)] ⇒  (¬∀0L )

where e is a new constant and f is a constant.

2.4 2/2-Sequents

◦ ER :

47

 ⇒  ↑ D(b) ⇒  ⇒ [D(b)]  ⇒  ↑ ¬2 D(b) ⇒  ⇒ [¬2 D(b)]  ⇒  ↑ D(b) ⇒  ⇒  (¬2R 0 )  ⇒  ↑ ¬2 D(b) ⇒  ⇒    ⇒  ↑ D1 (b) ⇒  ⇒ [D1 (b)] R ( − )  ⇒ [D1 (b)] ↑ D2 (b) ⇒  ⇒ [D1 (b), D2 (b)]  ⇒  ↑ (D1 D2 )(b) ⇒  ⇒ [(D1 D2 )(b)]   ⇒  ↑ D1 (b) ⇒  ⇒  ⇒ [D (b)] ↑ D (b) ⇒  ⇒ [D (b)] R  ( 0 ) 1 2 1 ⇒  ↑ (D D )(b) ⇒  ⇒   1 2   ⇒  ↑ D1 (b) ⇒  ⇒ [D1 (b)] R (− )  ⇒  ↑ D2 (b) ⇒  ⇒ [D2 (b)]  ⇒  ↑ (D1  D2 )(b) ⇒  ⇒ [(D1  D2 )(b)]   ⇒  ↑ D1 (b) ⇒  ⇒  (0R )  ⇒  ↑ D2 (b) ⇒  ⇒   ⇒  ↑ (D1  D2 )(b) ⇒  ⇒  (¬2R − )

and (∀0R ) R) (∀−

(∃0R ) R) (∃−

  ⇒  ↑ ¬R(b, f ) ⇒  ⇒   ⇒  ↑ D( f ) ⇒  ⇒    ↑ (∀R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ ¬R(b, e) ⇒  ⇒ [¬R(b, e)]  ⇒  ↑ D(e) ⇒  ⇒ [D(e)]  ⇒  ↑ (∀R.D)(b) ⇒  ⇒ [(∀R.D)(b)]   ⇒  ↑ R(b, e) ⇒  ⇒   ⇒  ↑ D(e) ⇒  ⇒    ↑ (∃R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ R(b, f ) ⇒  ⇒ [R(b, f )]  ⇒  ↑ D( f ) ⇒  ⇒ [D( f )]  ⇒  ↑ (∃R.D)(b) ⇒  ⇒ [(∃R.D)(b)]

and 

 ⇒  ↑ ¬D1 (b) ⇒  ⇒ [¬D1 (b)]  ⇒  ↑ ¬D2 (b) ⇒  ⇒ [¬D2 (b)]  ⇒  ↑ ¬(D1 D2 )(b) ⇒  ⇒ [¬(D1 D2 )(b)]   ⇒  ↑ ¬D1 (b) ⇒  ⇒  R (¬ 0 )  ⇒  ↑ ¬D2 (b) ⇒  ⇒    ↑ ¬(D1 D2 )(b) ⇒  ⇒   ⇒  ⇒  ↑ ¬D1 (b) ⇒  ⇒ [¬D1 (b)] R (¬− )  ⇒ [¬D1 (b)] ↑ ¬D2 (b) ⇒  ⇒ [¬D1 (b), ¬D2 (b)]  ⇒  ↑ ¬(D1  D2 )(b) ⇒  ⇒ [¬(D1  D2 )(b)]   ⇒  ↑ ¬D1 (b) ⇒  ⇒  R (¬0 )  ⇒ [¬D1 (b)] ↑ ¬D2 (b) ⇒  ⇒ [¬D1 (b)]  ⇒  ↑ ¬(D1  D2 )(b) ⇒  ⇒ 

(¬ −R )

48

and

2 R-Calculus for Binary-Valued DL



 ⇒  ↑ R(b, e) ⇒  ⇒   ⇒  ↑ ¬D(e) ⇒  ⇒    ↑ (¬∀R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ R(b, f ) ⇒  ⇒ [R(b, f )] R (¬∀− )  ⇒  ↑ ¬D( f ) ⇒  ⇒ [¬D( f )]  ↑ (¬∀R.D)(b) ⇒  ⇒ [(¬∀R.D)(b)]  ⇒  ⇒  ↑ ¬R(b, f ) ⇒  ⇒  R (¬∃0 )  ⇒  ↑ ¬D( f ) ⇒  ⇒    ↑ (¬∃R.D)(b) ⇒  ⇒   ⇒  ⇒  ↑ ¬R(b, e) ⇒  ⇒ [¬R(b, e)] R (¬∃− )  ⇒  ↑ ¬D(e) ⇒  ⇒ [¬D(e)]  ⇒  ↑ (¬∃R.D)(b) ⇒  ⇒ [(¬∃R.D)(b)] (¬∀0R )

where e is a new constant and f is a constant. Definition 2.4.8 A 2/2-reduction δ =  ⇒  ↑ (C(a), D(b)) ⇒  ⇒  is prov= able in S= 2/2 , denoted by 2/2 δ, if there is a sequence {δ1 , . . . , δn } of 2/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/2-reductions by one of the deduction rules in S= 2/2 . Theorem 2.4.9 (Soundness and completeness theorem) Let C(a) ∈  and D(b) ∈ . For any 2/2-reduction δ =  ⇒  ↑ (C(a), D(b)) ⇒  ⇒  , =

=

|=2/2 δ iff 2/2 δ.

2.5 2/2-Co-Sequents 2/2

2/2 A 2/2-co-sequent X → Y is L= -valid, denoted by |== X → Y, if there is an interpretation I such that for each X ∈ X, I (X ) = f, and for each Y ∈ Y, I (Y ) = t. 2/2 -valid, denoted by |=2/2 A 2/2-co-sequent X → Y is K= = X  → Y, if there is an interpretation I such that for each X ∈ X, I (X ) = t, and for each Y ∈ Y, I (Y ) = f. Given statements X, Y, a 2/2-co-reduction X → Y ↑ (X, Y ) ⇒ X → Y is 2/2 2/2 Q= /P=2/2 -valid, denoted by |==/= X → Y ↑ (X, Y ) ⇒ X → Y , if



X = 

Y =

2/2 2/2 /K= -valid X, X if X, X → Y is L= X otherwise; 2/2 2/2 /K= -valid Y, Y if X → Y, Y is L= Y otherwise.

2.5 2/2-Co-Sequents

49

2/2

2.5.1 Deduction System L=

2/2 Deduction system L= consists of the following deduction rules and axiom: • Axiom: ⎡  ∩ ¬−  = ∅ − 2/2 ⎣  ∩ ¬  = ∅ (A= ) ∩=∅  → ,

where ,  are sets of literals. • Deduction rules: , C(a) →  (¬2L ) ,  ¬¬C(a) →  , C1 (a) →  ( L ) , C2 (a) →  ,  (C1 C2 )(a) →  , C1 (a) →  ( L ) , C2 (a) →  ,(C1  C2 )(a) →  , ¬C1 (a) →  (¬ L ) , ¬C2 (a) →  ,  ¬(C1 C2 )(a) →  , ¬C1 (a) →  (¬ L ) , ¬C2 (a) →  , ¬(C1  C2 )(a) ⇒  and



, ¬R(a, f ) →  L (∀ ) , C( f ) →  ,(∀R.C)(a) →  , R(a, e) →  (¬∀ L ) , ¬C(e) →   , (¬∀R.C)(a) →  , R(a, e) →  (∃ L ) , C(e) →  ,(∃R.C)(a) →  , ¬R(a, f ) →  (¬∃ L ) , ¬C( f ) →  , (¬∃R.C)(a) → 

 → D(b),  (¬2R )   → ¬¬D(b),   → D1 (b),  ( R )  → D2 (b),    → (D1 D2 )(b),   → D1 (b),  ( R )  → D2 (b),   → (D1  D2 )(b),   → ¬D1 (b),  (¬ R )  → ¬D2 (b),    → ¬(D1 D2 )(b),   → ¬D1 (b),  (¬ R )  → ¬D2 (b),   → ¬(D1  D2 )(b),  

 → ¬R(b, e),  (∀ )  → D(e),   → (∀R.D)(b),   → R(b, f ),  (¬∀ R )  → ¬D( f ),    → (¬∀R.D)(b),   → R(b, f ),  (∃ R )  → D( f ),   → (∃R.D)(b),   → ¬R(b, e),  (¬∃ R )  → ¬D(e),   → (¬∃R.D)(b),  R

where e is a new constant and f is a constant.

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2 R-Calculus for Binary-Valued DL 2/2

2/2 Definition 2.5.1 A 2/2-co-sequent  →  is provable in L= , denoted by =  → , if there is a sequence {δ1 , . . . , δn } of 2/2-co-sequents such that δn =  → , and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-co-sequents by one 2/2 of the deduction rules in L= .

Theorem 2.5.2 (Soundness and completeness theorem) For any 2/2-co-sequent  → , 2/2 2/2 |==  →  iff =  → . 

2/2

2.5.2 R-Calculus Q=

R-calculus Q2/2 = consists of the following deduction rules and axioms: • Axioms: ⎡ (l(a)) ∩ ¬− (l(a)) = ∅ ⎣  ∩ ¬−  = ∅ = L (A+ ) (l(a)) ∩  = ∅  ⎧ →  ↑ l(a)−⇒ (l(a)) →  ⎨ (l(a)) ∩ ¬ (l(a)) = ∅  ∩ ¬−  = ∅ = L (A0 ) ⎩ (l(a)) ∩  = ∅  ⇒  →  ⎡ →  ↑− l(a)  ∩ ¬  = ∅ − = R ⎣ (m(b)) ∩ ¬ (m(b)) = ∅ (A+ )  ∩ (m(b)) = ∅   ↑ m(b) ⇒  → (m(b)) ⎧ → ⎨  ∩ ¬−  = ∅ (m(b)) ∩ ¬− (m(b)) = ∅ (AtR 0 ) ⎩  ∩ (m(b))  = ∅  →  ↑ m(b) ⇒  →  where  →  is literal, and l(a), m(b) are literals. • Deduction rules consist of two parts E L and E R : ◦ EL :

2.5 2/2-Co-Sequents

51

 →  ↑ C(a) ⇒ , C(a) →   →  ↑ ¬2 C(a) ⇒ , ¬2 C(a) →   →  ↑ C(a) ⇒  →  (¬2L 0 ) 2  →  ↑ ¬ C(a) ⇒  →   →  ↑ C1 (a) ⇒ , C1 (a) →  L ) , C1 (a) →  ↑ C2 (a) ⇒ , C1 (a), C2 (a) →  ( +  →  ↑ (C1 C2 )(a) ⇒ , (C1 C2 )(a) →    →  ↑ C1 (a) ⇒  →  ( 0L ) , C1 (a) →  ↑ C2 (a) ⇒ , C1 (a) →    →  ↑ (C1 C2 )(a) ⇒  →   →  ↑ C1 (a) ⇒ , C1 (a) →  L (+ )  →  ↑ C2 (a) ⇒ , C2 (a) →   →  ↑ (C1  C2 )(a) ⇒ , (C1  C2 )(a) →   →  ↑ C1 (a) ⇒  →  L (0 )  →  ↑ C2 (a) ⇒  →   →  ↑ (C1  C2 )(a) ⇒  → 

(¬2L + )



and

 →  ↑ ¬R(a, f ) ⇒  →   →  ↑ C( f ) ⇒  →    →  ↑ (∀R.C)(a) ⇒  →   →  ↑ ¬R(a, e) ⇒ , ¬R(a, e) →  L )  →  ↑ C(e) ⇒ , C(e) →  (∀+  →  ↑ (∀R.C)(a) ⇒ , (∀R.C)(a) →   →  ↑ R(a, e) ⇒  →  (∃0L )  →  ↑ C(e) ⇒  →    →  ↑ (∃R.C)(a) ⇒  →   →  ↑ R(a, f ) ⇒ , R(a, f ) →  L (∃+ )  →  ↑ C( f ) ⇒ , C( f ) →   →  ↑ (∃R.C)(a) ⇒ , (∃R.C)(a) →  (∀0L )

and 

 →  ↑ ¬C1 (a) ⇒ , ¬C1 (a) →   →  ↑ ¬C2 (a) ⇒ , ¬C2 (a) →    →  ↑ ¬(C1 C2 )(a) ⇒ , ¬(C1 C2 )(a) →   →  ↑ ¬C1 (a) ⇒  →  (¬ 0L )  →  ↑ ¬C2 (a) ⇒  →    →  ↑ ¬(D1 D2 )(a)) ⇒  →   →  ↑ ¬C1 (a) ⇒ , ¬C1 (a) →  L (¬+ ) , ¬C1 (a) →  ↑ ¬C2 (a) ⇒ , ¬C1 (a), ¬C2 (a) →   →  ↑ ¬(C1  C2 )(a) ⇒ , ¬(C1  C2 )(a) →    →  ↑ ¬C1 (a) ⇒ , ¬C1 (a) →  L (¬0 ) , ¬C1 (a) →  ↑ ¬C2 (a) ⇒ , ¬C1 (a) →   →  ↑ ¬(C1  C2 )(a) ⇒  → 

L (¬ + )

52

2 R-Calculus for Binary-Valued DL

and



 →  ↑ R(a, e) ⇒  →   →  ↑ ¬C(e) ⇒  →    →  ↑ (¬∀R.C)(a) ⇒  →   →  ↑ R(a, f ) ⇒ , R(a, f ) →  L (¬∀− )  →  ↑ ¬C( f ) ⇒ , ¬C( f ) →   →  ↑ (¬∀R.C)(a) ⇒ , (¬∀R.C)(a) →   →  ↑ ¬R(a, f ) ⇒  →  (¬∃0L )  →  ↑ ¬C( f ) ⇒  →    →  ↑ (¬∃R.C)(a) ⇒  →   →  ↑ ¬R(a, e) ⇒ , ¬R(a, e) →  L (¬∃+ )  →  ↑ ¬C(e) ⇒ , ¬C(e) →   →  ↑ (¬∃R.C)(a) ⇒ , (¬∃R.C)(a) →  (¬∀0L )

◦ ER :  →  ↑ D(b) ⇒  → , D(b)  →  ↑ ¬2 D(b) ⇒  → , ¬2 D(b)  →  ↑ D(b) ⇒  →  (¬2R 0 ) 2  →  ↑ ¬ D(b) ⇒  →   →  ↑ D1 (b) ⇒  → , D1 (b) ( +R )  →  ↑ D2 (b) ⇒  → , D2 (b)  →  ↑ (D1 D2 )(b) ⇒  → , (D1 D2 )(b)   →  ↑ D1 (b) ⇒  →  R ( 0 )  →  ↑ D2 (b) ⇒  →    ↑ (D1 D2 )(b) ⇒  →   →  →  ↑ D1 (b) ⇒  → , D1 (b) R (+ )  → , D1 (b) ↑ D2 (b) ⇒  → , D1 (b), D2 (b)  →  ↑ (D1  D2 )(b) ⇒  → , (D1  D2 )(b)   →  ↑ D1 (b) ⇒  →  (0R )  → , D1 (b) ↑ D2 (b) ⇒  → , D1 (b)  →  ↑ (D1  D2 )(b) ⇒  → 

(¬2R + )

and

2.5 2/2-Co-Sequents

53



 →  ↑ ¬R(b, e) ⇒  →   →  ↑ D(e) ⇒  →    ↑ (∀R.D)(b) ⇒  →   →  →  ↑ ¬R(b, f ) ⇒  → , ¬R(b, f ) R (∀+ )  →  ↑ D( f ) ⇒  → , D( f )  ↑ (∀R.D)(b) ⇒  → , (∀R.D)(b)  →  →  ↑ R(b, f ) ⇒  →  R (∃0 )  →  ↑ D( f ) ⇒  →    ↑ (∃R.D)(b) ⇒  →   →  →  ↑ R(b, e) ⇒  → , R(b, e) R (∃+ )  →  ↑ D(e) ⇒  → , D(e)  →  ↑ (∃R.D)(b) ⇒  → , (∃R.D)(b)

(∀0R )

and 

 →  ↑ ¬D1 (b) ⇒  → , ¬D1 (b)  → , ¬D1 (b) ↑ ¬D2 (b) ⇒  → , ¬D1 (b), ¬D2 (b)  →  ↑ ¬(D1 D2 )(b) ⇒  → , ¬(D1 D2 )(b)   →  ↑ ¬D1 (b) ⇒  →  (¬ 0R )  → , ¬D1 (b) ↑ ¬D2 (b) ⇒  → , ¬D1 (b)   ↑ ¬(D1 D2 )(b) ⇒  →   →  →  ↑ ¬D1 (b) ⇒  → , ¬D1 (b) (¬+R )  →  ↑ ¬D2 (b) ⇒  → , ¬D2 (b)  →  ↑ ¬(D1  D2 )(b) ⇒  → , ¬(D1  D2 )(b)   →  ↑ ¬D1 (b) ⇒  → , ¬D1 (b) R (¬0 )  →  ↑ ¬D2 (b) ⇒  → , ¬D2 (b)  →  ↑ ¬(D1  D2 )(b) ⇒  → , ¬(D1  D2 )(b)

(¬ +R )

and



 →  ↑ R(b, e) ⇒  →   →  ↑ ¬D(e) ⇒  →    ↑ (¬∀R.D)(b) ⇒  →   →  →  ↑ R(b, f ) ⇒  → , R(b, f ) R (¬∀+ )  →  ↑ ¬D( f ) ⇒  → , ¬D( f )  ↑ (¬∀R.D)(b) ⇒  → , (¬∀R.D)(b)}  →  →  ↑ ¬R(b, f ) ⇒  →  R (¬∃0 )  →  ↑ ¬D( f ) ⇒  →    ↑ (¬∃R.D)(b) ⇒  →   →  →  ↑ ¬R(b, e) ⇒  → , ¬R(b, e) R (¬∃+ )  →  ↑ ¬D(e) ⇒  → , ¬D(e)  →  ↑ (¬∃R.D)(b) ⇒  → , (¬∃R.D)(b)}

(¬∀0R )

where e is a new constant and f is a constant. Definition 2.5.3 A 2/2-reduction δ =  →  ↑ (C(a), D(b)) ⇒  →  is prov2/2 able in Q2/2 = , denoted by = δ, if there is a sequence {δ1 , . . . , δn } of 2/2-co-

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2 R-Calculus for Binary-Valued DL

reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/2-co-reductions by one of the deduction rules in Q2/2 = . Theorem 2.5.4 (Soundness and completeness theorem) For any 2/2-co-reduction δ =  →  ↑ (C(a), D(b)) ⇒ , C (a) → , D (b), 2/2

2/2

|== δ iff = δ. 

2/2

2.5.3 Deduction System K=

2/2 2/2 Lemma 2.5.5 Let ,  be a set of literals. A 2/2-co-sequent  →  is K= /K= valid if and only if inval()&con()& ∩  = ∅.  2/2 Deduction system K= consists of the following deduction rules and axiom: • Axiom: ⎧ ⎨  ∩ ¬−  = ∅ ∩=∅ (A2/2 = ) ⎩  ∩ ¬−  = ∅  → ,

where ,  are sets of literals. • Deduction rules: , C(a) →  (¬2L ) ,  ¬¬C(a) →  , C1 (a) →  ( L ) , C2 (a) →  ,  (C1 C2 )(a) →  , C1 (a) →  ( L ) , C2 (a) →  ,(C1  C2 )(a) →  , ¬C1 (a) →  (¬ L ) , ¬C2 (a) →  ,  ¬(C1 C2 )(a) →  , ¬C1 (a) →  (¬ L ) , ¬C2 (a) →  , ¬(C1  C2 )(a) ⇒  and

 → D(b),  (¬2R )   → ¬¬D(b),   → D1 (b),  ( R )  → D2 (b),    → (D1 D2 )(b),   → D1 (b),  ( R )  → D2 (b),   → (D1  D2 )(b),   → ¬D1 (b),  (¬ R )  → ¬D2 (b),    → ¬(D1 D2 )(b),   → ¬D1 (b),  (¬ R )  → ¬D2 (b),   → ¬(D1  D2 )(b), 

2.5 2/2-Co-Sequents

55



, ¬R(a, e) →  (∀ L ) , C(e) →  ,(∀R.C)(a) →  , R(a, f ) →  (¬∀ L ) , ¬C( f ) →   , (¬∀R.C)(a) →  , R(a, f ) →  (∃ L ) , C( f ) →  ,(∃R.C)(a) →  , ¬R(a, e) →  (¬∃ L ) , ¬C(e) →  , (¬∃R.C)(a) → 



 → ¬R(b, f ),   → D( f ),   → (∀R.D)(b),   → R(b, e),  (¬∀ R )  → ¬D(e),    → (¬∀R.D)(b),   → R(b, e),  (∃ R )  → D(e),   → (∃R.D)(b),   → ¬R(b, f ),  (¬∃ R )  → ¬D( f ),   → (¬∃R.D)(b),  (∀ R )

where e is a new constant and f is a constant. 2/2 Definition 2.5.6 A 2/2-co-sequent  →  is provable in K= , denoted by 2/2 =  → , if there is a sequence {δ1 , . . . , δn } of 2/2-co-sequents such that δn =  → , and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-co-sequents by one 2/2 . of the deduction rules in K=

Theorem 2.5.7 (Soundness and completeness theorem) For any 2/2-co-sequent  → , 2/2 |=2/2 =   →  iff =   → . 

2/2

2.5.4 R-Calculus P=

2/2 A 2/2-reduction δ =  ⇒  ↑ (C(a), D(b)) ⇒ , C (a) ⇒ , D (b), is P= 2/2 -valid, denoted by |== δ, if



2/2 -valid , C(a) if , C(a) ⇒  is L=  otherwise;  2/2 -valid , D(b) if  ⇒ , D(b), is L=  =  otherwise.

 =

2/2 consists of the following deduction rules and axioms: R-calculus P= • Axioms:

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(l(a)) ∩ ¬− (l(a)) ∩  ∩ ¬−  = ∅  →  ↑ l(a) ⇒ , l(a) →  − − = L (l(a)) ∩ ¬ (l(a)) ∩  ∩ ¬   = ∅ (A0 )  →  ↑ l(a) ⇒  →  − − = R  ∩ ¬  ∩ (m(b)) ∩ ¬ (m(b)) = ∅ (A+ )  →  ↑ m(b) ⇒  → , m(b)  ∩ ¬−  ∩ (m(b)) ∩ ¬− (m(b)) = ∅ (AtR 0 )   →  ↑ m(b) ⇒   →  = L

(A+ )

where  →  is literal, and l(a), m(b) are atoms. • Deduction rules consist of two parts E L and E R : ◦ EL :  →  ↑ C(a) ⇒ , C(a) →   →  ↑ ¬2 C(a) ⇒ , ¬2 C(a) →   →  ↑ C(a) ⇒  →  (¬2L 0 ) 2  →  ↑ ¬ C(a) ⇒  →   →  ↑ C1 (a) ⇒ , C1 (a) →  L )  →  ↑ C2 (a) ⇒ , C2 (a) →  ( +  →  ↑ (C1 C2 )(a) ⇒ , (C1 C2 )(a) →   →  ↑ C1 (a) ⇒  →  ( 0L )  →  ↑ C2 (a) ⇒  →    →  ↑ (C1 C2 )(a) ⇒  →   →  ↑ C1 (a) ⇒ , C1 (a) →  L (+ ) , C1 (a) →  ↑ C2 (a) ⇒ , C1 (a), C2 (a) →   →  ↑ (C1  C2 )(a) ⇒ , (C1  C2 )(a) →    →  ↑ C1 (a) ⇒  →  (0L ) , C1 (a) →  ↑ C2 (a) ⇒ , C1 (a) →   →  ↑ (C1  C2 )(a) ⇒  → 

(¬2L + )

and



 →  ↑ ¬R(a, e) ⇒  →   →  ↑ C(e) ⇒  →    →  ↑ (∀R.C)(a) ⇒  →   →  ↑ ¬R(a, f ) ⇒ , ¬R(a, f ) →  L (∀+ )  →  ↑ C( f ) ⇒ , C( f ) →   →  ↑ (∀R.C)(a) ⇒ , (∀R.C)(a) →    →  ↑ R(a, f ) ⇒  →  (∃0L )  →  ↑ C( f ) ⇒  →    →  ↑ (∃R.C)(a) ⇒  →   →  ↑ R(a, e) ⇒ , R(a, e) →  L (∃+ )  →  ↑ C(e) ⇒ , C(e) →   →  ↑ (∃R.C)(a) ⇒ , (∃R.C)(a) →  (∀0L )

and

2.5 2/2-Co-Sequents

57



 →  ↑ ¬C1 (a) ⇒ , ¬C1 (a) →  , ¬C1 (a) →  ↑ ¬C2 (a) ⇒ , ¬C1 (a), ¬C2 (a) →    →  ↑ ¬(C1 C2 )(a) ⇒ , ¬(C1 C2 )(a) →   →  ↑ ¬C1 (a) ⇒  →  (¬ 0L ) , ¬C1 (a) →  ↑ ¬C2 (a) ⇒ , ¬C1 (a) →    →  ↑ ¬(D1 D2 )(a)) ⇒  →   →  ↑ ¬C1 (a) ⇒ , ¬C1 (a) →  L (¬+ )  →  ↑ ¬C2 (a) ⇒ , ¬C2 (a) →   →  ↑ ¬(C1  C2 )(a) ⇒ , ¬(C1  C2 )(a) →   →  ↑ ¬C1 (a) ⇒  →  (¬0L )  →  ↑ ¬C2 (a) ⇒  →   →  ↑ ¬(C1  C2 )(a) ⇒  → 

L (¬ + )

and



 →  ↑ R(a, f ) ⇒  →   →  ↑ ¬C( f ) ⇒  →    →  ↑ (¬∀R.C)(a) ⇒  →   →  ↑ R(a, e) ⇒ , R(a, e) →  L (¬∀− )  →  ↑ ¬C(e) ⇒ , ¬C(e) →   →  ↑ (¬∀R.C)(a) ⇒ , (¬∀R.C)(a) →    →  ↑ ¬R(a, e) ⇒  →  L (¬∃0 )  →  ↑ ¬C(e) ⇒  →    →  ↑ (¬∃R.C)(a) ⇒  →   →  ↑ ¬R(a, f ) ⇒ , ¬R(a, f ) →  L (¬∃+ )  →  ↑ ¬C( f ) ⇒ , ¬C( f ) →   →  ↑ (¬∃R.C)(a) ⇒ , (¬∃R.C)(a) →  (¬∀0L )

◦ ER :  →  ↑ D(b) ⇒  → , D(b)  →  ↑ ¬2 D(b) ⇒  → , ¬2 D(b)  →  ↑ D(b) ⇒  →  (¬2R 0 ) 2  →  ↑ ¬ D(b) ⇒  →   →  ↑ D1 (b) ⇒  → , D1 (b) ( +R )  → , D1 (b) ↑ D2 (b) ⇒  → , D1 (b), D2 (b)  →  ↑ (D1 D2 )(b) ⇒  → , (D1 D2 )(b)   →  ↑ D1 (b) ⇒  →  R ( 0 )  → , D1 (b) ↑ D2 (b) ⇒  → , D1 (b)   ↑ (D1 D2 )(b) ⇒  →   →  →  ↑ D1 (b) ⇒  → , D1 (b) R (+ )  →  ↑ D2 (b) ⇒  → , D2 (b)  →  ↑ (D1  D2 )(b) ⇒  → , (D1  D2 )(b)   →  ↑ D1 (b) ⇒  →  R (0 )  →  ↑ D2 (b) ⇒  →   →  ↑ (D1  D2 )(b) ⇒  → 

(¬2R + )

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2 R-Calculus for Binary-Valued DL



and

 →  ↑ ¬R(b, f ) ⇒  →   →  ↑ D( f ) ⇒  →    ↑ (∀R.D)(b) ⇒  →   →  →  ↑ ¬R(b, e) ⇒  → , ¬R(b, e) R (∀+ )  →  ↑ D(e) ⇒  → , D(e)  →  ↑ (∀R.D)(b) ⇒  → , (∀R.D)(b)   →  ↑ R(b, e) ⇒  →  R (∃0 )  →  ↑ D(e) ⇒  →    ↑ (∃R.D)(b) ⇒  →   →  →  ↑ R(b, f ) ⇒  → , R(b, f ) R (∃+ )  →  ↑ D( f ) ⇒  → , D( f )  →  ↑ (∃R.D)(b) ⇒  → , (∃R.D)(b)

(∀0R )

and 

 →  ↑ ¬D1 (b) ⇒  → , ¬D1 (b)  →  ↑ ¬D2 (b) ⇒  → , ¬D2 (b)  →  ↑ ¬(D1 D2 )(b) ⇒  → , ¬(D1 D2 )(b)   →  ↑ ¬D1 (b) ⇒  →  R (¬ 0 )  →  ↑ ¬D2 (b) ⇒  →    ↑ ¬(D1 D2 )(b) ⇒  →   →  →  ↑ ¬D1 (b) ⇒  → , ¬D1 (b) R (¬+ )  → , ¬D1 (b) ↑ ¬D2 (b) ⇒  → , ¬D1 (b), ¬D2 (b)  →  ↑ ¬(D1  D2 )(b) ⇒  → , ¬(D1  D2 )(b)   →  ↑ ¬D1 (b) ⇒  →  R (¬0 )  → , ¬D1 (b) ↑ ¬D2 (b) ⇒  → , ¬D1 (b)  →  ↑ ¬(D1  D2 )(b) ⇒  → , ¬(D1  D2 )(b)

(¬ +R )

and



 →  ↑ R(b, f ) ⇒  →   →  ↑ ¬D( f ) ⇒  →    ↑ (¬∀R.D)(b) ⇒  →   →  →  ↑ R(b, e) ⇒  → , R(b, e) R (¬∀+ )  →  ↑ ¬D(e) ⇒  → , ¬D(e)  →  ↑ (¬∀R.D)(b) ⇒  → , (¬∀R.D)(b)}   →  ↑ ¬R(b, e) ⇒  →  R (¬∃0 )  →  ↑ ¬D(e) ⇒  →    ↑ (¬∃R.D)(b) ⇒  →   →  →  ↑ ¬R(b, f ) ⇒  → , ¬R(b, f ) R (¬∃+ )  →  ↑ ¬D( f ) ⇒  → , ¬D( f )  →  ↑ (¬∃R.D)(b) ⇒  → , (¬∃R.D)(b)}

(¬∀0R )

where e is a new constant and f is a constant.

2.6 Conclusions

59

Definition 2.5.8 A 2/2-reduction δ =  →  ↑ (C(a), D(b)) ⇒  →  is prov2/2 , denoted by 2/2 able in L= = δ, if there is a sequence {δ1 , . . . , δn } of 2/2-coreductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced 2/2 from the previous 2/2-co-reductions by one of the deduction rules in L= . Theorem 2.5.9 (Soundness and completeness theorem) For any 2/2-reduction δ =  →  ↑ (C(a), D(b)) ⇒ , C (a) → , D (b), 2/2 |=2/2 = δ iff = δ.



2.6 Conclusions Let Tt and Tf be sets of deduction rules for truth and false, respectively. Then t M1/2 = Tt + (At1/2 ) t N1/2 = Tf + (At1/2 ) 1/2 1/2 Lt = Tf + (At ) 1/2 1/2 Kt = Tt + (At ).

Let Rt and Rf be sets of deduction rules for truth and false in R-calculi, respectively. Then t R1/2 = Rt + (St1/2 ) t S1/2 = Rf + (St1/2 ) 1/2 1/2 Qt = Rf + (St ) 1/2 1/2 Pt = Rt + (St ), 1/2

t where (St1/2 ), (St ) are sets of axioms in R1/2 , St1/2 , respectively. Similar for 2/2-multisequents. Let Gt and Gf be sets of deduction rules for truth and false, respectively. Then t = Gt + (At2/2 ) M2/2 t N2/2 = Gf + (At2/2 ) 2/2 2/2 Lt = Gf + (At ) 2/2 2/2 Kt = Gt + (At ).

Let Rt and Rf be sets of deduction rules for truth and false in R-calculi, respectively. Then t R2/2 = Rt + (Rt2/2 ) t S2/2 = Rf + (Rt2/2 ) 2/2 2/2 Qt = Rf + (Rt ) 2/2 2/2 Pt = Rt + (Rt ), 2/2

2/2

2/2

t where (Rt2/2 ), (Rt ) are sets of axioms in R2/2 /Pt , St2/2 /Qt , respectively.

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References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Log. 50, 510–530 (1985) Avron, A.: Gentzen-type systems, resolution and tableaux. J. Autom. Reason. 10, 265–281 (1993) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation, Applications. Cambridge University Press, Cambridge, UK (2003) Baader, F., Horrocks, I., Sattler, U.: Description logics (Chap. 3). In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.) Handbook of Knowledge Representation. Elsevier (2007) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Log. 40, 295–331 (2011) Fensel, D., van Harmelen, F., Horrocks, I., McGuinness, D., Patel-Schneider, P.F.: OIL: an ontology infrastructure for the semantic web. IEEE Intell. Syst. 16, 38–45 (2001) Gärdenfors, P., Rott, H.: Belief revision. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, Epistemic and Temporal Reasoning Belief revision, pp. 35–132. Oxford Science Publish (1995) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to Pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Takeuti, G.: Proof Theory. In: Barwise, J. (ed.) Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, 2nd edn., pp. 249–295. Kluwer, Dordrecht (2001) Zach, R.: Proof theory of finite-valued logics. Technical Report TUW-E185.2-Z.1-93

Chapter 3

R-Calculus for Post Three-Valued DL

∗ ∗ Mi/3 Ni/3 i/3

L∗

i/3

K∗

∗ ∗ Ri/3 Si/3 i/3

Q∗

i/3

P∗

Let ∗, ∗1 , ∗2 ∈ {t, m, f]. ∗ ∗ /N1/3 -valid if for any interpretation I, there is a A 1/3-multisequent X is M1/3 statement X ∈ X such that I (X ) = ∗/I (X ) = ∗. ∗ ∗ /K1/3 -valid if there is an interpretation I such that A 1/3-multisequent X is L1/3 for each statement X ∈ X, I (X ) = ∗/I (X ) = ∗. ∗1 ∗2 ∗1 ∗2 /N2/3 -valid if for any interpretation I, either A 2/3-multisequent X ⇒ Y is M2/3 I (X ) = ∗1 /I (X ) = ∗1 for some statement X ∈ X or I (Y ) = ∗2 /I (Y ) = ∗2 for some statement Y ∈ Y. ∗1 ∗2 ∗1 ∗2 /K2/3 -valid if there is an interpretation I such A 2/3-multisequent X ⇒ Y is L2/3 that I (X ) = ∗1 /I (X ) = ∗1 for each statement X ∈ X and I (Y ) = ∗2 /I (Y ) = ∗2 for each statement Y ∈ Y. = = /N3/3 -valid if for any interpretation I, either A multisequent X = || is M3/3 I (A) = t/I (A) = t for some A ∈  or I (B) = m/I (B) = m for some B ∈ , or I (C) = f/I (C) = f for some C ∈ . 3/3 3/3 /K= -valid if there is an interpretation I such A multisequent X = || is L= that I (A) = t/I (A) = t for each A ∈ , I (B) = m/I (B) = m for each B ∈ , and I (C) = f/I (C) = f for each C ∈ . 1/3 1/3 ∗ ∗ , N1/3 , L∗ , K∗ Post There are sound and complete deduction systems M1/3 1/3 ∗ ∗ -valid, N1/3 -valid, L∗ -valid, and (1921), Takeuti (1987), Urquhart (2001) for M1/3 1/2 K∗ -valid multisequents, respectively.

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_3

61

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Given a 1/3-multisequent X and a statement X ∈ X, a reduction X ↑ X ⇒ X[X  ] =∗ ∗ /S∗1/3 -valid, denoted by |=∗1/3 / |=1/3 X ↑ X ⇒ X[X  ], if is R1/3 X =



∗ ∗ /N1/3 -valid X if X[X ] is M1/3 λ otherwise.

Given a 1/3-multisequent X and a statement X, a reduction X ↑ X ⇒ X(X  ) is 1/3 1/3 denoted by |==∗ / |=∗ X ↑ X ⇒ X(X  ), if

1/3 1/3 Q∗ /P∗ -valid,



X =



1/3

1/3

X if X(X ) is L∗ /K∗ -valid λ otherwise.

Given a statement X ∈ X and a statement Y ∈ Y, a reduction X ⇒ Y ↑ (X, Y ) ⇒ =∗ ∗ ∗1 ∗2 1 ∗2 1 ∗2 /S∗2/3 -valid, denoted by |=∗2/3 / |=2/31 2 X ⇒ Y ↑ (X, Y ) ⇒ X ⇒ X ⇒ Y is R2/3  Y , if  ∗1 ∗2 ∗1 ∗2 /N2/3 -valid X[X ] if X[X ] ⇒ Y is M2/3 X = X otherwise;  ∗1 ∗2 ∗1 ∗2 Y[Y ] if X ⇒ Y[Y ] is M2/3 /N2/3 -valid  Y = Y otherwise. Given statements X, Y, a reduction X ⇒ Y ↑ (X, Y ) ⇒ X ⇒ Y is Q∗1 ∗2 /P∗1 ∗2 2/3 2/3 valid, denoted by |==∗1 ∗2 / |=∗1 ∗2 X ⇒ Y ↑ (X, Y ) ⇒ X ⇒ Y , if 2/3



2/3

2/3

2/3

X(X ) if X(X ) ⇒ Y is L∗1 ∗2 /K∗1 ∗2 -valid X otherwise;  2/3 2/3 Y(Y ) if X ⇒ Y(Y ) is L∗1 ∗2 /K∗1 ∗2 -valid  Y = Y otherwise. X =

Given a multisequent X = || and statements A ∈ , B ∈ , and C ∈ , a = reduction X ↑ (A, B, C) ⇒ X =   | |  is R3/3 /S= 3/3 -valid, denoted by = =  |=3/3 / |=3/3 X ↑ (A, B, C) ⇒ X , if

 



= /N= -valid [A] if [A]|| is M3/3 3/3  otherwise;  = /N= -valid [B] if   |[B]| is M3/3 3/3 =  otherwise;  = /N= -valid [C] if   | |[C] is M3/3 3/3 =  otherwise.

 =

Given statements A, B, and C, a reduction X ↑ (A, B, C) ⇒ X =   | |  is 3/3  denoted by |== / |=3/3 = X ↑ (A, B, C) ⇒ X , if

3/3 Q3/3 = /P= -valid,

3.1 Post Three-Valued DL

63

  =  =  =

 

3/3

3/3

(A) if (A)|| is L= /K= -valid  otherwise; (B) if   |(B)| is L= /K= -valid  otherwise; 3/3

3/3

(C) if   | |(C) is L= /K= -valid  otherwise. 3/3

3/3

There are sound and complete R-calculi Alchourrón et al. (1985); Avron (1991); Baader et al. (2003); Bochvar (1938); Fermé and Hansson (2011); Gärdenfors and Rott (1995); Ginsberg (1987) 1/3 1/3 1/3 1/3 ∗ ∗ , S∗1/3 , Q∗ , P∗ for preserving R1/3 , S∗1/3 , Q∗ , P∗ -validity of 1/3(i) R1/3 multisequents; 2/3 2/3 2/3 2/3 ∗1 ∗2 ∗1 ∗2 1 ∗2 1 ∗2 , S∗2/3 , Q∗1 ∗2 , P∗1 ∗2 for preserving R2/3 , S∗2/3 , Q∗1 ∗2 , P∗1 ∗2 -validity of (ii) R2/3 2/3-multisequents; = = = 3/3 3/3 3/3 3/3 , S= (iii) R3/3 3/3 , Q= , P= for preserving M3/3 -, N3/3 -, L= -, and K= -validity of 3/3-multisequents, respectively. We consider the following deduction systems and R-calculi (Li Li 2007; Li and Sui 2013): t tm = , M2/3 , M3/3 Deduction system M1/3 t tm = R-calculus R1/3 , R2/3 , R3/3 .

3.1 Post Three-Valued DL The logical language of post three-valued DL contains the following symbols: • atomic concepts: S0 , S1 , ...; • roles: R0 , R1 , ...; • concept constructors: ∼, , , ∀, ∃. Concepts are defined inductively as follows: C:: = S| ∼ C|C1 C2 |C1 C2 |∀R.C|∃R.C, where S is an atomic concept, and R is a role. Statements are defined as follows: ϕ:: = C(a)|R(a, b)| ∼ ϕ, where ∼2 =∼∼ and ∼3 =∼∼∼ . We call the statements of forms S(a)| ∼ S(a)| ∼2 S(a)|R(a, b)| ∼ R(a, b)| ∼2 R(a, b) literals, and S(a)|R(a, b) atoms.

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3 R-Calculus for Post Three-Valued DL

A model M is a pair (U, I ), where U is a non-empty set, and I is an interpretation such that ◦ for any atomic concept S, I (S) : U → L3 ; ◦ for any role R, I (R) : U 2 → L3 . Given an atomic concept S and a role R, we define interpretation of concepts ∼ S and roles ∼ R as follows: for any x, y ∈ U, S(x) ∼ S(x) ∼2 S(x) t f m t f m m t f

R(x, y) ∼ R(x, y) ∼2 t f t m m f

R(x, y) m f t

The interpretation I (C) of a concept C is a function from U to L3 such that for any x ∈ U, ⎧ ⎪ ⎪ I (S)(x) ⎨ f ∼ (I (C))(x) I (C)(x) = min{I (C1 )(x), I (C2 )(x)} ⎪ ⎪ ⎩ max{I (C1 )(x), I (C2 )(x)} and

x ∈ I (∀R.C)

iff

x ∈ I (∼ ∀R.C)

iff

x ∈ I (∼2 ∀R.C) x ∈ I (∃R.C) x ∈ I (∼ ∃R.C)

iff iff iff

x ∈ I (∼2 ∃R.C)

iff

where

→ t m f

if C if C if C if C

=S =∼ C1 = C1 C2 = C1 C2

Ay(I (R)(x, y) = f or I (C)(y) = t or (I (R)(x, y) = m&I (C)(y) = m)) Ay((I (R)(x, y) = t&I (C)(y) = m) or (I (R)(x, y) = m&I (C)(y) = f) Ey(I (R)(x, y) = t&I (C)(y) = f)) Ey(I (R)(x, y) = t&I (C)(y) = t) Ey((I (R)(x, y) = m&I (C)(y) = m) or (I (R)(x, y) = t&I (C)(y) = m) or (I (R)(x, y) = m&I (C)(y) = f)) Ay(I (R)(x, y) = f∨I (C)(y) = f) t t m f

m t t m

f t t t

∃ t m f

(3.1)

tmf tmf mmf fff

Therefore, we have the following equivalences: I ((∀R.C)(a)) = t

iff

I ((∼ ∀R.C)(a)) = t

iff

I ((∼2 ∀R.C)(a)) = t

iff

Ab(I (∼2 R(a, b)) = t∨I (C(b)) = t ∨(I (∼ R(a, b)) = t∧I (∼ C(b)) = t) Ab((I (R(a, b)) = t∧I (∼ C(b)) = t) ∨(I (∼ R(a, b)) = t∧I (∼2 C(b)) = t)) Eb(I (R(a, b)) = t∧I (∼2 C(b)) = t)

3.1 Post Three-Valued DL

65

and I ((∃R.C)(a)) = t I ((∼ ∃R.C)(a)) = t

iff iff

I ((∼2 ∃R.C)(a)) = t

iff

Eb(I (R(a, b)) = t∧I (C(b)) = t) Eb((I (∼ R(a, b)) = t&I (∼ C(b)) = t) ∨(I (R(a, b)) = t&I (∼ C(b)) = t) ∨(I (∼ R(a, b)) = t&I (C(b)) = t)) Ab(I (∼2 R(a, b)) = t∨I (∼2 C(b)) = t) ◦t

◦t

◦m

◦m

◦f

◦f

∀

∃

Lemma 3.1.1 For I (C) = t, we have the following equivalences: (C1 C2 )(a) ≡ C1 (a)∧C2 (a) ∼ (C1 C2 )(a) ≡ (C1 (a)∧ ∼ C2 (a))∨(∼ C1 (a)∧C2 (a)) ∨(∼ C1 (a)∧ ∼ C2 (a)) ∼ (C1 C2 )(a) ≡ ∼2 C1 (a)∨ ∼2 C2 (a) 2

(C1 C2 )(a) ≡ C1 (a)∨C2 (a) ∼ (C1 C2 )(a) ≡ (∼ C1 (a)∧ ∼ C2 (a))∨(∼ C1 (a)∧ ∼2 C2 (a)) ∨(∼2 C1 (a)∧ ∼ C2 (a)) ∼2 (C1 C2 )(a) ≡ ∼2 C1 (a)∧ ∼2 C2 (a) and (∀R.C)(a) ≡ ∀b(∼2 R(a, b)∨C(b)∨(∼ R(a, b)∧ ∼ C(b)) (∼ ∀R.C)(a) ≡ ∀b((R(a, b)∧ ∼ C(b))∨∧(∼ R(a, b)∧ ∼2 C(b)) (∼2 ∀R.C)(a) ≡ ∃b(R(a, b)∧ ∼2 C(b)) (∃R.C)(a) ≡ ∃b(R(a, b)∧C(b)) (∼ ∃R.C)(a) ≡ ∀b((∼ R(a, b)∧ ∼ C(b))∨(R(a, b)∧ ∼ C(b)) ∨(∼ R(a, b)∧C(b))) (∼2 ∃R.C)(a) ≡ ∀b(∼2 R(a, b)∨ ∼2 C(b)). 

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3 R-Calculus for Post Three-Valued DL

Lemma 3.1.2 For I (C) = t, we have the following equivalences: (C1 C2 )(a) ≡ C1 (a)∨C2 (a) ∼ (C1 C2 )(a) ≡ (C1 (a)∨ ∼ C2 (a))∧(∼ C1 (a)∨C2 (a)) ∧(∼ C1 (a)∨ ∼ C2 (a)) ∼2 (C1 C2 )(a) ≡ ∼2 C1 (a)∧ ∼2 C2 (a) (C1 C2 )(a) ≡ C1 (a)∧C2 (a) ∼ (C1 C2 )(a) ≡ (∼ C1 (a)∨ ∼ C2 (a))∧(∼ C1 (a)∨ ∼2 C2 (a)) ∧(∼2 C1 (a)∨ ∼ C2 (a)) ∼ (C1 C2 )(a) ≡ ∼2 C1 (a)∨ ∼2 C2 (a) 2

and (∀R.C)(a) ≡ ∃b(∼2 R(a, b)∧C(b)∧(∼ R(a, b)∨ ∼ C(b)) (∼ ∀R.C)(a) ≡ ∃b((R(a, b)∨ ∼ C(b))∧∨(∼ R(a, b)∨ ∼2 C(b)) (∼2 ∀R.C)(a) ≡ ∀b(R(a, b)∨ ∼2 C(b)) (∃R.C)(a) ≡ ∀b(R(a, b)∨C(b)) (∼ ∃R.C)(a) ≡ ∃b((∼ R(a, b)∨ ∼ C(b))∧(R(a, b)∨ ∼ C(b)) ∧(∼ R(a, b)∨C(b))) (∼ ∃R.C)(a) ≡ ∃b(∼2 R(a, b)∧ ∼2 C(b)). 2



3.2 1/3-Multisequents Given a 1/3-multisequent  of literals, define val() : Es(a)(s(a), ∼ s(a), ∼2 s(a) ∈ ) inval() : ¬Es(a)(s(a), ∼ s(a), ∼2 s(a) ∈ ). t A 1/3-multisequent  is M1/3 -valid, denoted by |=t1/3 , if for any interpretation I, for some A(a) ∈ , I (A(a)) = t. t Lemma 3.2.1 Let  be a set of literals.  is M1/3 -valid if and only if val().

Proof Assume that val(). Let s(a) be a literal such that s(a), ∼ s(a), ∼2 s(a) ∈ . Then, for any interpretation I, either I (s(a)) = t, or I (∼ s(a)) = t, or I (∼2 s(a)) = t, that is, I |=t1/3 . Conversely, assume that inval(). Define an interpretation I such that U is the set of all the constants occurring in , for any atomic concept S,

3.2 1/3-Multisequents

67

I (S) = {e : S(e) ∈ / } I (∼ S) = {e :∼ S(e) ∈ / } / }, I (∼2 S) = {e :∼2 S(e) ∈ and for any role R, I (R) = {(e, f ) : R(e, f ) ∈ / } I (∼ R) = {(e, f ) :∼ R(e, f ) ∈ / } I (∼2 R) = {(e, f ) :∼2 R(e, f ) ∈ / }. Then, I is well-defined and I |=t . 

3.2.1 Deduction System Mt1/3 t Deduction system M1/3 consists of the following axiom and deduction rules: • Axiom: ∩ ∼−  ∩  −2 = ∅ (At1/3 ) ,

where  is a set of literals. • Deduction rules: , A(a) (∼3 ) 3 , ∼ A(a)  , A1 (a) , A1 (a) ( ) , A2 (a) ( ) , A2 (a) , ⎧ (A1 A2 )(a) , ⎧ (A1 A2 )(a) , ∼ A1 (a) ⎪ , ∼ A1 (a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ , ∼ A2 (a) , ∼ A2 (a) ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ , A1 (a) , ∼2 A1 (a) (a) , ∼ A , (∼ ) ⎪ (∼ ) ⎪ 2 ⎪ ⎪   ∼ A2 (a) ⎪ ⎪ ⎪ ⎪ (a) , ∼ A , ∼ A1 (a) ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , A2 (a) , ∼2 A2 (a) , ,  ∼ (A21 A2 )(a)  ∼ (A21 A2 )(a) , ∼ A1 (a) , ∼ A1 (a) (∼2 ) , ∼2 A2 (a) (∼2 ) , ∼2 A2 (a) , ∼2 (A1 A2 )(a) , ∼2 (A1 A2 )(a)

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3 R-Calculus for Post Three-Valued DL

and

⎧ ⎧ , A(e) , R(a, e) ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ 2 , ∼ R(a, e)   , ∼ A(e) , ∼ A(e) , ∼ R(a, e) (∀) ⎪ (∼ ∀) ⎪ ⎪ ⎪ ⎩ ⎩ , ∼ R(a, e) , ∼2 A(e) , (∀R.A)(a)   , (∼ ∀R.A)(a) , R(a, f ) , R(a, f ) (∼2 ∀) , ∼2 A( f ) (∃) , A( f ) 2 , (∃R.A)(a) ⎧, (∼ ∀R.A)(a) , ∼ R(a, f ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  , ∼ A( f )  ⎨ , R(a, f ) , ∼2 R(a, e) 2 (∼ ∃) , ∼2 A(e) (∼ ∃) ⎪ ⎪  , ∼ A( f ) ⎪ ⎪ , ∼ R(a, f ) , (∼2 ∃R.A)(a) ⎪ ⎪ ⎩ , A( f ) , (∼ ∃R.A)(a)

where e is a new constant and f is a constant. t Definition 3.2.2 A 1/3-multisequent  is provable in M1/3 , denoted by t1/3 , if there is a sequence {δ1 , ..., δn } of 1/3-multisequent such that δn = , and for each 1 ≤ i ≤ n, δi is deduced from the previous 1/3-multisequents by one of the deduction t . rules in M1/3

Theorem 3.2.3 (Soundness and completeness theorem) For any 1/3-multisequent , t1/3  if and only if |=t1/3 . 

t 3.2.2 R-Calculus R1/3

Given a 1/3-multisequent  and a statement A(a) ∈ , a reduction  ↑ A(a) ⇒ t -valid, denoted by |=t1/3  ↑ A(a) ⇒ [A (a)], if [A (a)] is R1/3 A (a) =



t -valid A(a) if [A(a)] is M1/3 λ otherwise.

t R-calculus R1/3 consists of the following axioms and deduction rules:

3.2 1/3-Multisequents

• Axioms:

69

[s(a)]∩ ∼− ∩ ∼−2  = ∅  ↑ s(a) ⇒ [s(a)] [s(a)]∩ ∼− ∩ ∼−2  = ∅ t0 (A1/3 )  ↑ s(a) ⇒  − −2  = ∅ ∼t− ∩ ∼ [∼ s(a)]∩ ∼ (A1/3 )  ↑∼ s(a) ⇒ [∼ s(a)] ∩ ∼− [∼ s(a)]∩ ∼−2  = ∅ (A∼t0 1/3 )  ↑∼ s(a) ⇒  − −2 2 [∼2 s(a)] = ∅ t− ∩ ∼ ∩ ∼ (A∼ 1/3 )  ↑∼2 s(a) ⇒ [∼2 s(a)] ∩ ∼− ∩ ∼−2 [∼2 s(a)] = ∅ (At0 1/3 )  ↑∼2 s(a) ⇒ 

(At− 1/3 )

where  is a set of literals, s(a) is atomic, and ∼−  = {A(a) :∼ A(a) ∈ } ∼−2  = {A(a) :∼2 A(a) ∈ }. • Deduction rules:  ↑ A(a) ⇒   ↑∼3 A(a) ⇒   ↑ A(a) ⇒ [A(a)] (∼3− ) 3 3  ↑∼ A(a) ⇒ [∼ A(a)]  ↑ A1 (a) ⇒  ( 0 ) [A1 (a)] ↑ A2 (a) ⇒ [A1 (a)]  ↑ (A1 A2 )(a) ⇒   ↑ A1 (a) ⇒ [A1 (a)] ( − ) [A1 (a)] ↑ A2 (a) ⇒ [A1 (a), A2 (a)]  ↑ (A1 A2 )(a) ⇒ [(A1 A2 )(a)]  ↑ A1 (a) ⇒  ( 0 )  ↑ A2 (a) ⇒   ↑ (A1 A2 )(a) ⇒   ↑ A1 (a) ⇒ [A1 (a)] ( − )  ↑ A2 (a) ⇒ [A2 (a)]  ↑ (A1 A2 )(a) ⇒ [(A1 A2 )(a)] (∼30 )

70

and

and

and

3 R-Calculus for Post Three-Valued DL

⎡

 ↑∼2 R(a, f ) ⇒  ⎢  ↑ A( f ) ⇒  ⎢ (∀0 ) ⎣  ↑∼ R(a, f ) ⇒  [∼ R(a, f )] ↑∼ A( f ) ⇒ [∼ R(a, f )] ⎧↑ (∀R.A)(a) ⇒  ⎪  ↑∼2 R(a, e) ⇒ [∼2 R(a, e)] ⎪ ⎨   ↑ A(e) ⇒ [A(e)] (∀− ) ⎪ ⎪  ↑∼ R(a, e) ⇒ [∼ R(a, e)] ⎩ [∼ R(a, e)] ↑∼ A(e) ⇒ [∼ R(a, e), ∼ A(e)]  ↑  (∀R.A)(a) ⇒ [(∀R.A)(a)]  ↑ R(a, e) ⇒  (∃0 ) [R(a, e)] ↑ A(e) ⇒ [R(a, e)]  ↑ (∃R.A)(a) ⇒   ↑ R(a, f ) ⇒ [R(a, f )] (∃− ) [R(a, f ] ↑ A( f ) ⇒ [R(a, f ), A( f )]  ↑ (∃R.A)(a) ⇒ [(∃R.A)(a)] ⎡

 ↑∼ A1 (a) ⇒  ⎢ [∼ A1 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)] ⎢ ⎢  ↑ A1 (a) ⇒  ⎢ (∼ 0 ) ⎢ ⎢  [A1 (a)] ↑∼ A2 (a) ⇒ [A1 (a)] ⎣  ↑∼ A1 (a) ⇒  [∼ A1 (a)] ↑ A2 (a) ⇒ [∼ A1 (a)] ⎧↑∼  (A1 A2 )(a) ⇒   ↑∼ A1 (a) ⇒ [∼ A1 (a)] ⎪ ⎪ ⎪ ⎪ [∼ A1 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a), ∼ A2 (a)] ⎪ ⎪ ⎨  ↑ A1 (a) ⇒ [A1 (a)] (∼ − ) ⎪ ⎪  [A1 (a)] ↑∼ A2 (a) ⇒ [A1 (a), ∼ A2 (a)] ⎪ ⎪  ↑∼ A1 (a) ⇒ [∼ A1 (a)] ⎪ ⎪ ⎩ [∼ A1 (a)] ↑ A2 (a) ⇒ [∼ A1 (a), A2 (a)]  ↑∼ (A1 A2 )(a) ⇒ [∼ (A1 A2 )(a)]

3.2 1/3-Multisequents

⎡

 ↑∼ A1 (a) ⇒  ⎢ [∼ A1 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)] ⎢ ⎢  ↑∼2 A1 (a) ⇒  ⎢ 2 2 (∼ 0 ) ⎢ ⎢  [∼ A1 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)] ⎣  ↑∼ A1 (a) ⇒  [∼ A1 (a)] ↑∼2 A2 (a) ⇒ [∼ A1 (a)] ⎧↑∼  (A1 A2 )(a) ⇒   ↑∼ A1 (a) ⇒ [∼ A1 (a)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  [∼ A2 1 (a)] ↑∼ A2 (a)2 ⇒ [∼ A1 (a), ∼ A2 (a)] ⎨  ↑∼ A1 (a) ⇒ [∼ A1 (a)] 2 2 (∼ − ) ⎪ ⎪  [∼ A1 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a), ∼ A2 (a)] ⎪ ⎪  ↑∼ A1 (a) ⇒ [∼ A1 (a)] ⎪ ⎪ ⎩ [∼ A1 (a)] ↑∼2 A2 (a) ⇒ [∼ A1 (a), ∼2 A2 (a)]  ↑∼ (A1 A2 )(a) ⇒ [∼ (A1 A2 )(a)] and

and

⎡

 ↑ R(a, f ) ⇒  ⎢ [R(a, f )] ↑∼ A( f ) ⇒ [R(a, f )] ⎢ (∼ ∀0 ) ⎣  ↑∼ R(a, f ) ⇒  [∼ R(a, f )] ↑∼2 A( f ) ⇒ [∼ R(a, f )] ⎧↑ (∼ ∀R.A)(a) ⇒   ↑ R(a, e) ⇒ [R(a, e)] ⎪ ⎪ ⎨  [R(a, e)] ↑∼ A(e) ⇒ [R(a, e), ∼ A(e)]  ↑∼ R(a, e) ⇒ [∼ R(a, e)] (∼ ∀− ) ⎪ ⎪ ⎩ [∼ R(a, e)] ↑∼2 A(e) ⇒ [∼ R(a, e), ∼2 A(e)] ⎡↑ (∼ ∀R.A)(a) ⇒ [(∼ ∀R.A)(a)]  ↑∼ R(a, e) ⇒  ⎢ [∼ R(a, e)] ↑∼ A(e) ⇒ [∼ R(a, e)] ⎢ ⎢  ↑ R(a, e) ⇒  ⎢ (∼ ∃0 ) ⎢ ⎢  [R(a, e)] ↑∼ A(e) ⇒ [R(a, e)] ⎣  ↑∼ R(a, e) ⇒  [∼ R(a, e)] ↑ A(e) ⇒ [∼ R(a, e)] ⎧↑ (∼ ∃R.A)(a) ⇒   ↑∼ R(a, f ) ⇒ [∼ R(a, f )] ⎪ ⎪ ⎪ ⎪ [∼ R(a, f )] ↑∼ A( f ) ⇒ [∼ R(a, f ), ∼ A( f )] ⎪ ⎪ ⎨  ↑∼ R(a, f ) ⇒ [∼ R(a, f )] (∼ ∃− ) ⎪ ⎪  [∼ R(a, f )] ↑ A( f ) ⇒ [∼ R(a, f ), A( f )] ⎪ ⎪  ↑ R(a, f ) ⇒ [R(a, f )] ⎪ ⎪ ⎩ [R(a, f )] ↑∼ A( f ) ⇒ [R(a, f ), ∼ A( f )]  ↑ (∼ ∃R.A)(a) ⇒ [(∼ ∃R.A)(a)]

71

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3 R-Calculus for Post Three-Valued DL



(∼

2

(∼2 (∼2 (∼2 and

 ↑∼2 A1 (a) ⇒  0 )  ↑∼2 A2 (a) ⇒   ↑∼2 (A1 A2 )(a) ⇒   ↑∼2 A1 (a) ⇒ [∼2 A1 (a)] − )  ↑∼2 A2 (a) ⇒ [∼2 A2 (a)] 2 2  ↑∼ 2(A1 A2 )(a) ⇒ [∼ (A1 A2 )(a)]  ↑∼ A1 (a) ⇒ 

0 ) [∼2 A1 (a)] ↑∼2 A2 (a) ⇒ [∼2 A1 (a)]  ↑∼2 (A1 A2 )(a) ⇒   ↑∼2 A1 (a) ⇒ [∼2 A1 (a)]

− ) [∼2 A1 (a)] ↑∼2 A2 (a) ⇒ [∼2 A1 (a), ∼2 A2 (a)]  ↑∼2 (A1 A2 )(a) ⇒ [∼2 (A1 A2 )(a)] 

 ↑ R(a, e) ⇒  [R(a, e)] ↑∼2 A(e) ⇒ [R(a, e)]  ↑ (∼2 ∀R.A)(a) ⇒   ↑ R(a, f ) ⇒ [R(a, f )] (∼2 ∀− ) [R(a, f )] ↑∼2 A( f ) ⇒ [R(a, f ), ∼2 A( f )] 2 2  ↑ (∼2 ∀R.A)(a) ⇒ [(∼ ∀R.A)(a)]  ↑∼ R(a, f ) ⇒  (∼2 ∃0 )  ↑∼2 A( f ) ⇒   ↑ (∼2 ∃R.A)(a) ⇒   ↑∼2 R(a, e) ⇒ [∼2 R(a, e)] 2 − (∼ ∃ )  ↑∼2 A(e) ⇒ [∼2 A(e)]  ↑ (∼2 ∃R.A)(a) ⇒ [(∼2 ∃R.A)(a)] (∼2 ∀0 )

where e is a new constant and f is a constant. Definition 3.2.4 Given a 1/3-multisequent  and a statement A(a) ∈ , a reduction  ↑ A(a) ⇒ [A (a)] is provable in Rt , denoted by t1/3  ↑ A(a) ⇒ [A (a)], if there is a sequence {δ1 , ..., δn } of reductions such that δn =  ↑ A(a) ⇒ [A (a)], and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions t . by one of the deduction rules in R1/3 Theorem 3.2.5 (Soundness and completeness theorem) For any reduction  ↑ A(a) ⇒ [A (a)], t1/3  ↑ A(a) ⇒ [A (a)] if and only if |=t1/3  ↑ A(a) ⇒ [A (a)]. 

3.3 2/3-Multisequents

73

3.3 2/3-Multisequents A 2/3-multisequent | is Mtm -valid, denoted by |=tm 2/3 |, if for any interpretation I, either for some A(a) ∈ , I (A(a)) = t, or for some B(b) ∈ , I (B(b)) = m. Lemma 3.3.1 Let ,  be a set of literals. | is Gtm -valid iff there is an atom s(a) ∈  ∩  such that ∼ s(a) ∈ . Proof Assume that there is an atom s(a) ∈  ∩  such that ∼ s(a) ∈ . Then, for any interpretation I, either I (s(a)) = t, or I (s(a)) = m, or I (s(a)) = f, that is, I |= , or I |= , or I (∼ s(a)) = m, ∼ s(a) ∈ , I |= . Hence, I |=tm |. Conversely, assume that there is no such atom s(a). Define an interpretation I such that U is the set of all the constants occurring in  and , and (1) for any constant symbol e, I (e) = e; (2) for any concept symbol S, ⎧ ⎨ m if S(e) ∈  I (S)(e) = f if S(e) ∈  ⎩ t if ∼ S(e) ∈ ; (3) for any role symbol R, ⎧ ⎨ m if R(e, f ) ∈  I (R)(e, f ) = f if R(e, f ) ∈  ⎩ t if ∼ R(e, f ) ∈ . Then, I is well-defined and I |=tm |.  We have the following equivalences: , ∼ A(a)| ≡ |, A(a) |, ∼2 B(b) ≡ , B(b)|.

tm 3.3.1 Deduction System M2/3 tm Deduction system M2/3 consists of the following deduction rules and axiom: • Axiom:  ∩ ∩ ∼−  = ∅ (Atm 2/3 ) |,

where  is a set of atom, and  is a set of literals.

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3 R-Calculus for Post Three-Valued DL

• Deduction rules: , B(b)| (∼2B ) 2 ⎧|,  ∼ B(b) |, B1 (b) ⎪ ⎪ ⎪ ⎪ |, B ⎪ 2 (b) ⎪  ⎨ , B1 (b)| , A1 (a)| ( A ) , A2 (a)| ( B ) ⎪ ⎪  |, B2 (b) ⎪ ⎪ |, B1 (b) , (A1 A2 )(a)| ⎪ ⎪ ⎩ , B2 (b)| |(B1 B2 )(b),    |, ∼ B1 (b) , A1 (a)| (∼ B ) |, ∼ B2 (b) ( A ) , A2 (a)| , (A1 A2 )(a)| ⎧ |, ∼ (B1 B2 )(b) |, B (b) ⎪ 1 ⎪ ⎪ ⎪ |, B2 (b) ⎪ ⎪   ⎨ |, ∼ B1 (b) |, ∼ B1 (b) B (b) |, B |, ∼ B2 (b) ) ( B ) ⎪ (∼

2 ⎪  ⎪ ⎪ (b) |, B |, ∼ (B1 B2 )(b) ⎪ 1 ⎪ ⎩ |, ∼ B2 (b) |, (B1 B2 )(b) (∼ A )

and

|, A(a) , ∼ A(a)|

⎧ ⎧ , ∼2 R(a, e)| , R(b, e)| ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ , A(e)|  |, B(e)  |, R(b, e) , ∼ R(a, e)| (∀ B ) ⎪ (∀ A ) ⎪ ⎪ ⎪ ⎩ ⎩ |, ∼ B(e) , ∼ A(e)| , (∀R.A)(a)| |, (∀R.B)(b)  , R(b, f )| , R(a, f )| (∼ ∀ B ) |, ∼ B( f ) (∃ A ) , A( f )| , (∃R.A)(a)| ⎧ |, (∼ ∀R.B)(b) |, R(b, f ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  |, B( f )  ⎨ , R(b, f )| |, ∼ R(b, e) B |, B( f ) |, ∼ B(e) ) (∃ B ) ⎪ (∼ ∃ ⎪ ⎪ ⎪ |, R(b, f ) |, (∼ ∃R.B)(b) ⎪ ⎪ ⎩ , B( f )| |, (∃R.B)(b)

where e is a new constant and f is a constant.

3.3 2/3-Multisequents

75

tm Definition 3.3.2 A 2/3-multisequent | is provable in M2/3 , denoted by tm 2/3 |, if there is a sequence {δ1 , ..., δn } of 2/3-multisequents such that δn = |, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/3-multisequents tm by one of the deduction rules in M2/3 .

Theorem 3.3.3 (Soundness and completeness theorem) For any 2/3-multisequent |, tm |=tm 2/3 | iff 2/3 |. 

tm 3.3.2 R-Calculi R2/3

Let A(a) ∈  and B(b) ∈ . A reduction δ = | ↑ (A(a), B(b)) ⇒ [A (a)]| tm -valid, denoted by |=tm [B  (b)] is R2/3 2/3 δ, if 

tm -valid [A(a)] if [A(a)]| is M2/3  otherwise;  tm -valid [B(b)] if   |[B(b)] is M2/3   =  otherwise. 

 =

tm consists of the following deduction rules and axioms: R-calculus R2/3 • Axioms: − A− [s(a)] ∩ ∩ ∼   = ∅ ) (A2/3 | ↑ s(a) ⇒ [s(a)]| [s(a)] ∩ ∩ ∼−  = ∅ A0 ) (A2/3 | ↑ s(a) ⇒ | − B−  ∩ [t (b)]∩ ∼   = ∅ ) (A2/3 | ↑ t (b) ⇒ |[t (b)]  ∩ [t (b)]∩ ∼−  = ∅ B0 ) (A2/3 | ↑ t (b) ⇒ |  ∩ ∩ ∼− [∼ t (b)] = ∅ ) (A∼B− 2/3 | ↑∼ t (b) ⇒ |[∼ t (b)]  ∩ ∩ ∼− [∼ t (b)] = ∅ ) (A∼B0 2/3 | ↑∼ t (b) ⇒ |

where , s(a), t (b) is a set of atoms, and  a set of literals.

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3 R-Calculus for Post Three-Valued DL

• Deduction rules: | ↑ 2 A(a) ⇒ |[A(a)] | ↑ 2 A(a) ⇒ | (∼0A ) | ↑∼ A(a) ⇒ [∼ A(a)]| | ↑∼ A(a) ⇒  1 | ↑ 1 B(b) ⇒ [B(b)]| 2B | ↑ B(b) ⇒ | 2B (∼0 ) (∼− ) 2 2 | ↑∼ B(b) ⇒ |[∼ B(b)] | ↑∼2 B(b) ⇒ 

(∼−A )

and



| ↑ A1 (a) ⇒ [A1 (a)]| [A1 (a)]| ↑ A2 (a) ⇒ [A1 (a), A2 (a)]| |  ↑ (A1 A2 )(a) ⇒ [(A1 A2 )(a)]| | ↑ A1 (a) ⇒ | ( 0A ) [A1 (a)]| ↑ A2 (a) ⇒ [A1 (a)]| |  ↑ (A1 A2 )(a) ⇒ | | ↑ A1 (a) ⇒ [A1 (a)]| ( −A ) | ↑ A2 (a) ⇒ [A2 (a)]| | ↑ (A1 A2 )(a) ⇒ [(A1 A2 )(a)|  | ↑ A1 (a) ⇒ | ( 0A ) | ↑ A2 (a) ⇒ | | ↑ (A1 A2 )(a) ⇒ |

( −A )

⎡

and (∀0A )

(∀−A )

(∃0A ) (∃−A ) and

| ↑ 2 ∼ R(a, f ) ⇒ | ⎢ | ↑ A( f ) ⇒ | ⎢ ⎣ | ↑ 2 R(a, f ) ⇒ | |[R(a, f )] ↑ 2 A( f ) ⇒ |[R(a, f )] | ⇒ | ⎧ ↑ (∀R.A)(a) 2 ∼ R(a, e) ⇒ |[∼ R(a, e)] | ↑ ⎪ ⎪ ⎨ | ↑ A(e) ⇒ [A(e)]|  | ↑ 2 R(a, e) ⇒ |[R(a, e)] ⎪ ⎪ ⎩ |[R(a, e)] ↑ 2 A(e) ⇒ |[R(a, e), A(e)] | ↑ (∀R.A)(a) ⇒ [(∀R.A)(a)]|  | ↑ 1 R(a, e) ⇒ | [R(a, e)]| ↑ A(e) ⇒ [R(a, e)]| | ⇒ |  ↑ (∃R.A)(a) | ↑ 1 R(a, f ) ⇒ [R(a, f )] [R(a, f )]| ↑ A( f ) ⇒ [R(a, f ), A( f )]| | ↑ (∃R.A)(a) ⇒ [(∃R.A)(a)]|

3.3 2/3-Multisequents

⎡

  | ↑ B1 (b) ⇒   | ⎢   |[B1 (b)] ↑ B2 (b) ⇒   |[B1 (b)] ⎢  ⎢  | ↑ 1 B1 (b) ⇒   | ⎢ B ⎢   [B (b)]| ↑ B (b) ⇒   [B (b)]| ( 0 ) ⎢  1 2 1 ⎣   | ↑ B1 (b) ⇒   |   |[B1 (b)] ↑ 1 B2 (b) ⇒   |[B1 (b)]   (B1 B2 )(b) ⇒   ⎧ |  ↑   | ↑ B1 (b) ⇒   |[B1 (b)] ⎪ ⎪ ⎪  ⎪   |[B1 (b), B2 (b)] ⎪ ⎪    |[B11(b)]|B2 (b) ⇒ ⎨   | ↑ B1 (b) ⇒  [B1 (b)]|   ( −B ) ⎪ ⎪    [B1 (b)]||B2 (b) ⇒  [B1 (b)]|[B2 (b)] ⎪ ⎪  | ↑ B1 (b) ⇒  |[B1 (b)] ⎪ ⎪ ⎩   |[B1 (b)]|1 B2 (b) ⇒   [B2 (b)]|[B1 (b)]   | ↑ (B1 B2 )(b) ⇒   |[(B1 B2 )(b)] and

and

⎡

  | ↑ B1 (b) ⇒   | ⎢   |[B1 (b)] ↑ B2 (b) ⇒   |[B1 (b)] ⎢  ⎢  | ↑∼ B1 (b) ⇒   | ⎢ B ⎢   |[∼ B (b)] ↑ B (b) ⇒   |[∼ B (b)] ( 0 ) ⎢  1 2 1 ⎣   | ↑ B1 (b) ⇒   |   |[B1 (b)] ↑∼ B2 (b) ⇒   |[B1 (b)]   (B1 B2 )(b) ⇒   ⎧ |  ↑   | ↑ B1 (b) ⇒   |[B1 (b)] ⎪ ⎪ ⎪ ⎪   |[B1 (b)] ↑ B2 (b) ⇒   |[B1 (b), B2 (b)] ⎪ ⎪ ⎨   | ↑∼ B1 (b) ⇒   |[∼ B1 (b)]   B ( − ) ⎪ ⎪    |[∼ B1 (b)] ↑ B2 (b) ⇒  |[∼ B1 (b), B2 (b)] ⎪ ⎪  | ↑ B1 (b) ⇒  |[B1 (b)] ⎪ ⎪ ⎩   |[B1 (b)] ↑∼ B2 (b) ⇒   |[B1 (b), ∼ B2 (b)]   | ↑ (B1 B2 )(b) ⇒   |[(B1 B2 )(b)] 

  | ↑∼ B1 (b) ⇒   |   | ↑∼ B2 (b) ⇒   | (∼   ↑∼ (B1 B2 )(b) ⇒   |  |   | ↑∼ B1 (b) ⇒   |[∼ B1 (b)] (∼ −B )   | ↑∼ B2 (b) ⇒   |[∼ B2 (b)]   | ↑∼ (B1 B2 )(b) ⇒   |[∼ (B1 B2 )(b)]    | ↑∼ B1 (b) ⇒   | B (∼ 0 )   |[∼ B1 (b)] ↑∼ B2 (b) ⇒   |[∼ B1 (b)]   ↑∼ (B1 B2 )(b) ⇒   |  |   | ↑∼ B1 (b) ⇒   |[∼ B1 (b)] B (∼ − )   |[∼ B1 (b)] ↑∼ B2 (b) ⇒   |[∼ B1 (b), ∼ B2 (b)]   | ↑∼ (B1 B2 )(b) ⇒   |[∼ (B1 B2 )(b)] 0B )

77

78

and

and

and

3 R-Calculus for Post Three-Valued DL

⎡

  | ↑ 1 R(b, f ) ⇒   | ⎢   [R(b, f )]| ↑ B( f ) ⇒   [R(b, f )]| ⎢ (∀0B ) ⎣   | ↑ 2 R(b, f ) ⇒   |   |[R(b, f )] ↑∼ B( f ) ⇒   |[R(b, f )]  ⎧| ⇒   |  ↑ (∀R.B)(b) 1 ⎪  | ↑ R(b, e) ⇒   [R(b, e)]| ⎪ ⎨      [R(b,2e)]| ↑ B(e)  ⇒  [R(b, e)]|[B(e)] B− (∀ ) ⎪ ⎪  | ↑ R(b, e) ⇒  |[R(b, e)] ⎩   |[R(b, e)] ↑∼ B(e) ⇒   |[R(b, e), ∼ B(e)]   | ↑ (∀R.B)(b) ⇒   |[(∀R.B)(b)] ⎡

  | ↑ 2 R(b, e) ⇒   | ⎢   |[R(b, e)] ↑ B(e) ⇒   |[R(b, e)] ⎢  ⎢  | ↑ 1 R(b, e) ⇒   | ⎢ B ⎢   [R(b, e)]| ↑ B(e) ⇒   [R(b, e)]| (∃0 ) ⎢  ⎣   | ↑ 2 R(b, e) ⇒   |   |[R(b, e)] ↑ 1 B(e) ⇒   |[R(b, e)]   ⎧|  ↑ (∃R.B)(b) ⇒  |  | ↑ R(b, f ) ⇒  |[R(b, f )] ⎪ ⎪ ⎪  ⎪ f )] ↑ B( f ) ⇒   |[R(b, f ), B( f )] ⎪ ⎪    |[R(b, ⎨ 1  | ↑ R(b, f ) ⇒   [R(b, f )]|   B− (∃ ) ⎪ ⎪    [R(b, f )]| ↑ B( f ) ⇒  [R(b, f )]|[B( f )] ⎪ ⎪  | ↑ R(b, f ) ⇒  |[R(b, f )] ⎪ ⎪ ⎩   |[R(b, f )] ↑ 1 B( f ) ⇒   [B( f )]|[R(b, f )]   | ↑ (∃R.B)(b) ⇒   |[(∃R.B)(b)] 

  | ↑ 1 R(b, e) ⇒   | (∼ ∀ )   [R(b, e)]| ↑∼ B(e) ⇒   [R(b, e)]|  | ↑ (∼ ∀R.B)(b) ⇒   |   | ↑ 1 R(b, f ) ⇒   [R(b, f )]| B− (∼ ∀ )   [R(b, f )]| ↑∼ B( f ) ⇒   [R(b, f )]|[∼ B( f )]  ↑ (∼ ∀R.B)(b) ⇒   |[(∼ ∀R.B)(b)]  |   | ↑ 2 ∼ R(b, f ) ⇒   | B0 (∼ ∃ )   | ↑∼ B( f ) ⇒   |  | ↑ (∼ ∃R.B)(b) ⇒   |   | ↑ 2 ∼ R(b, e) ⇒   |[2 ∼ R(b, e)] B− (∼ ∃ )   | ↑∼ B(e) ⇒   |[∼ B(e)]   | ↑ (∼ ∃R.B)(b) ⇒   |[(∼ ∃R.B)(b)] B0

where e is a new constant and f is a constant.

3.4 3/3-Multisequents

79

Definition 3.3.4 A 2/3-reduction δ = | ↑ (A(a), B(b)) ⇒   | is provable in tm , denoted by tm R2/3 2/3 δ, if there is a sequence {δ1 , ..., δn } of 2/3-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/3-reductions by tm one of the deduction rules in R2/3 . Theorem 3.3.5 (Soundness and completeness theorem) Let A(a) ∈  and B(b) ∈ . For any reduction δ = | ↑ (A(a), B(b)) ⇒   | , tm |=tm 2/3 δ iff 2/3 δ.



3.4 3/3-Multisequents = Let , ,  be sets of statements. A 3/3-multisequent || is M3/3 -valid, denoted = by |=3/3 ||, if for any interpretation I,

⎧ ⎨ for some statement A(a) ∈ , I (A(a)) = t for some statement B(b) ∈ , I (B(b)) = m ⎩ for some statement C(c) ∈ , I (C(c)) = f.

= 3.4.1 Deduction System M3/3 = Deduction system M3/3 consists of the following axiom and deduction rules: • Axiom:

(A= 3/3 )

 ∩  ∩  = ∅ ||,

where , ,  are sets of atomic statements.

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3 R-Calculus for Post Three-Valued DL

• Deduction rules: (∼ A )

|, A(a)| , ∼ A(a)||

||, B(b) |, ∼ B(b)| , A1 (a)|| ( A ) , A1 (a)|| , (A1 A2 )(a)|| (∼ B )

, C(c)|| ||, ∼ C(c) ⎧ |, B1 (b)| ⎪ ⎪ ⎪ ⎪ |, B2 (b)| ⎪ ⎪  ⎨ , B1 (b)|| ||, C1 (c) C (b)| |, B ||, C2 (c) ) ( ( B ) ⎪ 2 ⎪  ⎪ ⎪ (b)| |, B ||, (C ⎪ 1 1 C 2 )(c) ⎪ ⎩ , B2 (b)|| |, (B1 B2 )(b)| ⎧ ⎪ |, B1 (b)| ⎪ ⎪ ⎪ |, B2 (b)| ⎪ ⎪  ⎨ ||, B1 (b) , A1 (a)|| |, B2 (b)| ( A ) , A2 (a)|| ( B ) ⎪ ⎪  ⎪ ⎪ |, B1 (b)| , (A1 A2 )(a)|| ⎪ ⎪ ⎩ ||, B2 (b) |, (B1 B2 )(b)|  ||, C1 (c) ( C ) ||, C2 (c) ||, (C1 C2 )(c) (∼C )

and

⎧ ||, R(a, e) ⎪ ⎪ ⎨ ,  A(e)|| |, R(a, e)| (∀ A ) ⎪ ⎪ ⎩ |, A(e)| , (∀R.A)(a)||  , R(c, f )|| (∀C ) ||, C( f ) ||, (∀R.C)(c) ⎧  |, R(b, f )| ⎪ ⎪ ⎪ ⎪ |, B( f )| ⎪ ⎪ ⎨ , R(b, f )|| (∃ B ) ⎪ ⎪  |, B( f )| ⎪ ⎪ |, R(b, f )| ⎪ ⎪ ⎩ , B( f )|| |, (∃R.B)(b)|

⎧ , R(b, e)|| ⎪ ⎪ ⎨  |, B(e)| |, R(b, e)| (∀ B ) ⎪ ⎪ ⎩ ||, B(e) |, (∀R.B)(b)|  , R(a, f )|| (∃ A ) , A( f )|| , (∃R.A)(a)|| 

||, R(c, e) (∃C ) ||, C(e) ||, (∃R.C)(c)

where e is a new constant and f is a constant.

3.4 3/3-Multisequents

81

= Definition 3.4.1 A 3/3-multisequent || is provable in M3/3 , denoted by = 3/3 ||, if there is a sequence {γ1 , ..., γn } of 3/3-multisequents such that γn = ||, and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous = 3/3-multisequents by one of the deduction rules in M3/3 .

Theorem 3.4.2 (Soundness and completeness theorem) For any 3/3-multisequent ||, = |== 3/3 || iff 3/3 ||. 

= 3.4.2 R-Calculus R3/3

Given a 3/3-multisequent || and a triple (A(a), B(b), C(c)) of statements such that A(a) ∈ , B(b) ∈ , and C(c) ∈ , a reduction δ = || ↑ (A(a), B(b), = -valid, denoted by |== C(c)) ⇒   | |  is R3/3 3/3 δ, if 

= -valid [A(a)] if [A(a)]|| is M3/3  otherwise;  = -valid [B(b)] if   |[B(b)]| is M3/3   =  otherwise;  = -valid [C(c)] if   | |[C(c)] is M3/3   =  otherwise. 

 =

Let X =   || X =   | | X = || X =   | |  X[A(a)] =  − {A(a)}|| X[A(a)] =  − {A(a)}|| X[B(b)] = | − {B(b)}| X[2 A(a)] = | − {A(a)}| X[C(c)] = || − {C(c)} X[3 A(a)] = || − {A(a)} X[1 B(b)] =  − {B(b)}|| X[1 C(c)] =  − {C(c)}|| X[B(b)] = | − {B(b)}| X[2 C(c)] = | − {C(c)}| X[3 B(b)] = || − {B(b)} X[C(c)] = || − {C(c)}. = R-calculus R3/3 consists of the following axioms and deduction rules:

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3 R-Calculus for Post Three-Valued DL

• Axioms:

[s(a)] ∩  ∩  = ∅ || ↑ s(a) ⇒ [s(a)]|| [s(a)] ∩  ∩  = ∅ (A=A0 3/3 ) || ↑ s(a) ⇒ ||  ∩ [t (b)] ∩  = ∅ (A=B− 3/3 ) || ↑ t (b) ⇒ |[t (b)]|  ∩ [t (b)] ∩  = ∅ (A=B0 3/3 ) || ↑ t (b) ⇒ ||  ∩  ∩ [u(c)] = ∅ (A=C− 3/3 ) || ↑ u(c) ⇒ ||[u(c)]  ∩  ∩ [u(c)] = ∅ (A=C0 3/3 ) || ↑ u(c) ⇒ ||

(A=A− 3/3 )

where X and s(a), t (b), u(c) are atomic. • Deduction rules: X ↑ 2 A(a) ⇒ X[2 A(a)] X ↑ 2 A(a) ⇒ X (∼ A0 ) X ↑∼ A(a) ⇒ X[∼ A(a)] X ↑∼ A(a) ⇒ X 3 3 X ↑ X ↑ 3 B(b) ⇒ X B(b) ⇒ X[ B(b)] (∼ B0 ) (∼ B− ) X ↑∼ B(b) ⇒ X[∼ B(b)] X ↑∼ B(b) ⇒ X 1 1 1 C− X ↑ C(c) ⇒ X[ C(c)] C0 X ↑ C(c) ⇒ X (∼ ) (∼ ) X ↑∼ C(c) ⇒ X[∼ C(c)] X ↑∼ C(c) ⇒ X (∼ A− )

and



X ↑ A1 (a) ⇒ X[A1 (a)] X[A1 (a)] ↑ A2 (a) ⇒ X[A1 (a), A2 (a)] X  ↑ (A1 A2 )(a) ⇒ X[(A1 A2 )(a)] X ↑ A1 (a) ⇒ X ( A0 ) X[A1 (a)] ↑ A2 (a) ⇒ X[A1 (a)] X  ↑ (A1 A2 )(a) ⇒ X X ↑ A1 (a) ⇒ X[A1 (a)] ( A− ) X ↑ A2 (a) ⇒ X[A2 (a)] X ↑ (A1 A2 )(a) ⇒ X[(A1 A2 )(a)] X ↑ A1 (a) ⇒ X ( A0 ) X ↑ A2 (a) ⇒ X X ↑ (A1 A2 )(a) ⇒ X

( A− )

and

3.4 3/3-Multisequents

83

⎡

X ↑ 3 R(a, f ) ⇒ X ⎢ X ↑ A( f ) ⇒ X ⎢ (∀ A0 ) ⎣ X ↑ 2 R(a, f ) ⇒ X X[2 R(a, f )] ↑ 2 A( f ) ⇒ X[2 R(a, f )] X ⇒X ⎧↑ (∀R.A)(a) 3 X ↑ R(a, e) ⇒ X[3 R(a, e)] ⎪ ⎪ ⎨ X ↑ A(e) ⇒ X[A(e)]  X ↑ 2 R(a, e) ⇒ X[2 R(a, e)] (∀ A− ) ⎪ ⎪ ⎩ X ↑ 2 A(e) ⇒ X[2 R(a, e), 2 A(e)] ⇒ X[(∀R.A)(a)] X ↑ (∀R.A)(a) X ↑ 1 R(a, e) ⇒ X (∃ A0 ) X[1 R(a, e)] ↑ A(e) ⇒ X[1 R(a, e)] X ⇒X  ↑ (∃R.A)(a) X ↑ 1 R(a, f ) ⇒ X[1 R(a, f )] (∃ A− ) X[1 R(a, f )] ↑ A( f ) ⇒ X[1 R(a, f ), A( f )] X ↑ (∃R.A)(a) ⇒ X[(∃R.A)(a)] where e is a new constant and f is a constant, and ⎧  X ↑ B1 (b) ⇒ X [B1 (b)] ⎪ ⎪ ⎪ ⎪ X [B1 (b)] ↑ B2 (b) ⇒ X [B1 (b), B2 (b)] ⎪ ⎪ ⎨  1 X ↑ B1 (b) ⇒ X [1 B1 (b)]  1 B− ⇒ X [1 B1 (b), B2 (b)] ( ) ⎪ ⎪  X [ B1 (b)] ↑ B2 (b) ⎪  ⎪ X ↑ B1 (b) ⇒ X [B1 (b)] ⎪ ⎪ ⎩ X [B1 (b)] ↑ 1 B2 (b) ⇒ X [B1 (b), 1 B2 (b)]  X ↑ (B1 B2 )(b) ⇒ X [(B1 B2 )(b)] ⎡ X ↑ B1 (b) ⇒ X ⎢ X [B1 (b)] ↑ B2 (b) ⇒ X [B1 (b)] ⎢  1 ⎢ X ↑ B1 (b) ⇒ X ⎢ B0 ⎢ X [1 B (b)] ↑ B (b) ⇒ X [1 B (b)] ( ) ⎢  1 2 1 ⎣ X ↑ B1 (b) ⇒ X X [B1 (b)] ↑ 1 B2 (b) ⇒ X [B1 (b)]  X ↑ (B1 B2 )(b) ↑ X ⇒ X and

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3 R-Calculus for Post Three-Valued DL

⎧   ⎪ ⎪ X ↑ B1 (b) ⇒ X [B1 (b)] ⎪ ⎪ X [B1 (b)] ↑ B2 (b) ⇒ X [B1 (b), B2 (b)] ⎪ ⎪ ⎨  3 X ↑ B1 (b) ⇒ X [3 B1 (b)]  3 B− ⇒ X [3 B1 (b), B2 (b)] ( ) ⎪ ⎪  X [ B1 (b)] ↑ B2 (b) ⎪  ⎪ X ↑ B1 (b) ⇒ X [B1 (b)] ⎪ ⎪ ⎩ X [B1 (b)] ↑ 3 B2 (b) ⇒ X [B1 (b), 3 B2 (b)]  X ↑ (B1 B2 )(b) ⇒ X [(B1 B2 )(b)] ⎡ X ↑ B1 (b) ⇒ X ⎢ X [B1 (b)] ↑ B2 (b) ⇒ X [B1 (b)] ⎢  3 ⎢ X ↑ B1 (b) ⇒ X ⎢ B0 ⎢ X [3 B (b)] ↑ B (b) ⇒ X [3 B (b)] ( ) ⎢  1 2 1 ⎣ X ↑ B1 (b) ⇒ X X [B1 (b)] ↑ 3 B2 (b) ⇒ X [B1 (b)]  X ↑ (B1 B2 )(b) ↑ X ⇒ X and

⎡

X ↑ 1 R(b, f ) ⇒ X ⎢ X [1 R(b, f ) ↑ B( f ) ⇒ X [1 R(b, f )] ⎢ (∀ B0 ) ⎣ X ↑ 2 R(b, f ) ⇒ X X [2 R(b, f )] ↑ 3 B( f ) ⇒ X [2 R(b, f )]  X ⇒ X ⎧ ↑ (∀R.B)(b)  1 X ↑ R(b, e) ⇒ X [1 R(b, e)] ⎪ ⎪ ⎨  1 e)] ↑ B(e) ⇒ X [1 R(b, e), B(e)]  X [ R(b, 2 B− X ↑ R(b, e) ⇒ X [2 R(b, e)] (∀ ) ⎪ ⎪ ⎩ X [2 R(b, e)] ↑ 3 B(e) ⇒ X [2 R(b, e), 3 B(e)]  X↑ (∀R.B)(b) ⇒ X [(∀R.B)(b)] ⎡ X ↑ 2 R(b, e) ⇒ X ⎢ X [2 R(b, e)] ↑ B(e) ⇒ X [2 R(b, e)] ⎢  1 ⎢ X ↑ R(b, e) ⇒ X ⎢ B0 ⎢ X [1 R(b, e)] ↑ B(e) ⇒ X [1 R(b, e)] (∃ ) ⎢  ⎣ X ↑ 2 R(b, e) ⇒ X X [2 R(b, e)] ↑ 1 B(e) ⇒ X [2 R(b, e)]  X ⇒ X ⎧ ↑ (∃R.B)(b)  2 X ↑ R(b, f ) ⇒ X [2 R(b, f )] ⎪ ⎪ ⎪ ⎪ X [2 R(b, f )] ↑ B( f ) ⇒ X [2 R(b, f ), B( f )] ⎪ ⎪ ⎨  1 X ↑ R(b, f ) ⇒ X [1 R(b, f )]  1 B− f )] ↑ B( f ) ⇒ X [1 R(b, f ), B( f )] (∃ ) ⎪ ⎪  X [ R(b, ⎪ 2 ⎪ X ↑ R(b, f ) ⇒ X [2 R(b, f )] ⎪ ⎪ ⎩ X [2 R(b, f )] ↑ 1 B( f ) ⇒ X [2 R(b, f ), 1 B( f )]  X ↑ (∃R.B)(b) ⇒ X [(∃R.B)(b)]

3.4 3/3-Multisequents

85

where e is a new constant and f is a constant, and 

X ↑ C1 (c) ⇒ X [C1 (c)] ( ) X ↑ C2 (c) ⇒ X [C2 (c)]   X ↑ (C1 C2 )(c)⇒ X [(C1 C2 )(c)] X ↑ C1 (c) ⇒ X ( C0 ) X ↑ C2 (c) ⇒ X   X  ↑ (C1 C2 )(c) ⇒ X X ↑ C1 (c) ⇒ X [C1 (c)] ( C− ) X [C1 (c)] ↑ C2 (c) ⇒ X [C1 (c), C2 (c)]   X ↑ (C1 C2 )(c)⇒ X [(C1 C2 )(c)] X ↑ C1 (c) ⇒ X ( C0 ) X [C1 (c)] ↑ C2 (c) ⇒ X [C1 (c)] X ↑ (C1 C2 )(c) ⇒ X C−

and



X ↑ 1 R(c, e) ⇒ X (∀ ) X [1 R(c, e)] ↑ C(e) ⇒ X [1 R(c, e)]  X ⇒ X  ↑ (∀R.C)(c) 1 X ↑ R(c, f ) ⇒ X [1 R(c, f )] C− (∀ ) X [1 R(c, f )] ↑ C( f ) ⇒ X [1 R(c, f ), C( f )]  ⇒ X [(∀R.C)(c)] X ↑ (∀R.C)(c) 3 X ↑ R(c, f ) ⇒ X C0 (∃ ) X ↑ C( f ) ⇒ X  X ⇒ X  ↑ (∃R.C)(c) 3 X ↑ R(c, e) ⇒ X [3 R(c, e)] C− (∃ ) X ↑ C(e) ⇒ X [C(e)] X ↑ (∃R.C)(c) ⇒ X [(∃R.C)(c)] C0

where e is a new constant and f is a constant. Definition 3.4.3 A reduction δ = X ↑ (A(a), B(b), C(c)) ⇒ X is provable in = , denoted by = R3/3 3/3 δ, if there is a sequence {δ1 , ..., δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous = . reductions by one of the deduction rules in R3/3 Theorem 3.4.4 (Soundness and completeness theorem) Let A(a) ∈ , B(b) ∈  and C(c) ∈ . For any reduction δ = X ↑ (A(a), B(b), C(c)) ⇒ X , = |== 3/3 δ if and only if 3/3 δ.



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3 R-Calculus for Post Three-Valued DL

3.5 Conclusions There are transformations σ and τ such that t m t f σ(M1/3 ) = M1/3 τ (M1/3 ) = M1/3 tm mf tf mf σ(M2/3 ) = M2/3 τ (M2/3 ) = M2/3 = 3/3 σ(M3/3 ) = K=

and

t m t f σ(R1/3 ) = R1/3 τ (R1/3 ) = R1/3 tm mf tf mf σ(R2/3 ) = R2/3 τ (R2/3 ) = R2/3 = 3/3 σ(R3/3 ) = P= .

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Logic 50, 510–530 (1985) Avron, A.: Natural 3-valued logics: characterization and proof theory. J. Symb. Logic 56, 276–294 (1991) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation, Applications. Cambridge University Press, Cambridge, UK (2003) Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Logic 2, 87–112 (1938) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Gärdenfors, P., Rott, H.: Belief revision. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp. 35–132. Oxford Science Pub, Epistemic and Temporal Reasoning (1995) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Post, E.L.: Determination of all closed systems of truth tables. Bull. Am. Math. Soc. 26, 437 (1920) Post, E.L.: Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185 (1921) Takeuti, G.: Proof theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, 2nd edn., pp. 249–295. Kluwer, Dordrecht (2001) Zach, R.: Proof theory of finite-valued logics, Tech. Report TUW-E185.2-Z.1-93

Chapter 4

R-Calculus for B22 -Valued DL

∗ ∗ Mi/2 2 Ni/22 2

i/2

Li/2 K∗ ∗

2

∗ ∗ Ri/2 2 Si/22 i/22

Q∗

i/22

P∗

=t

t t t A 1/22 -multisequent  is M1/2 2 /N1/22 -valid, denoted by |=1/22 / |=1/22 , if for any interpretation I, there is a statement A(a) ∈  such that I (A(a)) = t/I (A(a)) = t. 1/22 1/22 1/22 1/22 A 1/22 -multisequent  is Lt /Kt -valid, denoted by |==t / |=t , if there is an interpretation I such that for each statement A(a) ∈ , I (A(a)) = t/I (A(a)) = t. =t t t t A 2/22 -multisequent | is M2/2 2 /N2/22 -valid, denoted by |=2/22 / |=2/22 |, if for any interpretation I, either for some statement A(a) ∈ , I (A(a)) = t/I (A(a)) = t, or for some statement B(b) ∈ , I (B(b)) = /I (B(b)) = . 2/22 2/22 2/22 2/22 A 2/22 -multisequent | is Lt /Kt -valid, denoted by |==t / |=t |, if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = t/I (A(a)) = t, and for any statement B(b) ∈ , I (B(b)) = /I (B(b)) = . =t⊥ t⊥ t⊥ t⊥ A 3/22 -multisequent || is M3/2 2 /N3/22 -valid, denoted by |=3/22 / |=3/22 ||, if for any interpretation I, either for some statement A(a) ∈ , I (A(a)) = t/I (A(a)) = t, for some statement B(b) ∈ , I (B(b)) = /I (B(b)) = , or for some statement C(c) ∈ , I (C(c)) =⊥ /I (C(c)) =⊥ . 3/22 3/22 3/22 3/22 A 3/22 -multisequent || is Lt⊥ /Kt⊥ -valid, denoted by |==t⊥ / |=t⊥ ||, if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = t/I (A(a)) = t, for any statement B(b) ∈ , I (B(b)) = /I (B(b)) = , and for any statement C(c) ∈ , I (C(c)) =⊥ /I (C(c)) =⊥ . 2 4/22 4/22 4/22 /K= -valid, denoted by |== / |=4/2 A 4/22 -multisequent ||| is L= = |||, if there is an interpretation I such that for any statement A(a) ∈

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_4

87

4 R-Calculus for B22 -Valued DL

88

, I (A(a)) = t/I (A(a)) = t, for any statement B(b) ∈ , I (B(b)) = / I (B(b)) = , for any statement C(c) ∈ , I (C(c)) =⊥ /I (C(c)) =⊥, and for any statement D(d) ∈ , I (D(d)) = f/I (D(d)) = f. There are sound and complete deduction systems Arieli and Avron (2000), Belnap (1997), Bergstra and Ponse (2000), Font (1997), Gottwald (2001), Zach (2023): 1/22 1/22 1/22 1/2 ∗ ∗ ∗ ∗ , N1/2 for M1/2 -valid, N1/2 (i) M1/2 2 , L∗ 2 , K∗ 2 -valid, L∗ 2 -valid, and K∗ valid multisequents, respectively; 2/22 2/22 2/22 2/2 ∗1 ∗2 ∗1 ∗2 ∗1 ∗2 ∗1 ∗2 (ii) M2/2 2 , L∗1 ∗2 , N2/22 , K∗1 ∗2 for M2/22 -valid, L∗1 ∗2 -valid, N2/22 -valid, and K∗1 ∗2 valid multisequents, respectively; 3/22 3/22 3/22 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 , L∗1 ∗2 ∗3 , N3/2 , K∗1 ∗2 ∗3 for M3/2 -valid, L∗1 ∗2 ∗3 -valid, N3/2 (iii) M3/2 2 2 2 2 3/22

valid, and K∗1 ∗2 ∗3 -valid multisequents, respectively; = = = = 4/22 4/22 4/22 4/2 , N4/2 for M4/2 -valid, N4/2 (iv) M4/2 2 , L= 2 , K= 2 -valid, L= 2 -valid, and K= valid multisequents, respectively. Given a 1/22 -multisequent  and a statement A ∈ , a 1/22 -reduction  ↑ A ⇒ =t t t t  [A ] is R1/2 2 /S1/22 -valid, denoted by |=1/22 / |=1/22  ↑ A ⇒ [A ], if A =



t t A if [A] is M1/2 2 /N1/22 -valid λ otherwise.

Given a 1/22 -multisequent  and a statement A, a 1/22 -reduction  ↑ A ⇒ , A 1/22 1/22 1/22 1/22 is Qt /Pt -valid, denoted by |==t / |=t  ↑ A ⇒ , A , if 

A =



1/22

1/22

A if , A is Lt /Kt -valid λ otherwise.

Given a 2/22 -multisequent | and two statements A ∈  and B ∈ , a 2/22 t t t reduction | ↑ (A, B) ⇒ [A ]|[B  ] is R2/2 2 /S2/22 -valid, denoted by |=2/22 =t

/ |=2/22 | ↑ (A, B) ⇒ [A ]|[B  ], if 

A λ  B B = λ 

A =

t t if [A]| is M2/2 2 /N2/22 -valid otherwise. t t if [A ]|[B] is M2/2 2 /N2/22 -valid otherwise.

Given a 2/22 -multisequent | and two statements A and B, a 2/22 -reduction 2/22 2/22 2/22 2/22 | ↑ (A, B) ⇒ , A |, B  is Qt /Kt -valid, denoted by |==t / |=t |  ↑ (A, B) ⇒ , A |, B  , if

4 R-Calculus for B22 -Valued DL



A λ  B = B λ 

A =

89 2/22

2/22

if , A| is Lt /Kt -valid otherwise. 2/22 2/22 if , A |, B is Lt /Kt -valid otherwise.

Given a 3/22 -multisequent || and three statements A ∈ , B ∈ , C ∈ , t⊥ t⊥ a 3/22 -reduction || ↑ (A, B, C) ⇒ [A ]|[B  ]|[C  ] is R3/2 2 /S3/22 -valid, =t⊥

denoted by |=t⊥ 3/22 / |=3/22 

A λ  B  B = λ  C  C = λ A =

|| ↑ (A, B, C) ⇒ [A ]|[B  ]|[C  ], if

t⊥ t⊥ if [A]|| is M3/2 2 /N3/22 -valid otherwise; t⊥ t⊥ if [A ]|[B]| is M3/2 2 /N3/22 -valid otherwise; t⊥ t⊥ if [A ]|[B  ]|[C] is M3/2 2 /N3/22 -valid otherwise.

Given a 3/22 -multisequent || and three statements A, B, C, a 3/22 -reduction 3/22 || ↑ (A, B, C) ⇒ , A |, B  |, C  is Lt⊥ /Pt⊥ -valid, denoted by |==t⊥ 3/22

/ |=t⊥ || ↑ (A, B, C) ⇒ , A |, B  |, C  , if 

A λ   B = B λ  C C = λ A =

3/22

3/22

if , A|| is Lt⊥ /Kt⊥ -valid otherwise; 3/22 3/22 if , A |, B| is Lt⊥ /Kt⊥ -valid otherwise; 3/22 3/22 if , A |, B  |, C is Lt⊥ /Kt⊥ -valid otherwise.

Given a 4/22 -multisequent ||| and four statements A ∈ , B ∈ , C ∈ , D ∈ , a 4/22 -reduction δ = ||| ↑ (A, B, C, D) ⇒ [A ]|[B  ]|[C  ] = = |[D  ] is M4/2 2 /N4/22 -valid, denoted by =

|== 4/22 / |=4/22 δ, if



= = A if [A]||| is M4/2 2 /N4/22 -valid  λ otherwise; = = B if [A ]|[B]|| is M4/2 2 /N4/22 -valid B =  λ otherwise; = = C if [A ]|[B  ]|[C]| is M4/2 2 /N4/22 -valid C =  λ otherwise; = = D if [A ]|[B  ]|[C  ]|[D] is M4/2 2 /N4/22 -valid  D = λ otherwise.

A =

4 R-Calculus for B22 -Valued DL

90

Given a 4/22 -multisequent ||| and four statements A, B, C, D, a 4/22 2 4/22 reduction ||| ↑ (A, B, C, D) ⇒ , A |, B  |, C  |, D  is Q4/2 = /P= -valid, denoted by 4/22

|==

/ |=4/2 ||| ↑ (A, B, C, D) ⇒ , A |, B  |, C  |, D  , = 2



if

2

2

4/2 4/2 /K= -valid A if , A||| is L= λ otherwise;  4/22 4/22 /K= -valid B if , A |, B|| is L=  B = λ otherwise;  4/22 4/22 /K= -valid C if , A |, B  |, C| is L= C = λ otherwise;  4/22 4/22 /K= -valid D if , A |, B  |, C  |, D is L= D = λ otherwise. 

A =

There are sound and complete R-calculi Arieli and Avron (1996), Belnap (1997), Ginsberg (1998), Ponse and van der Zwaag (2006), Pynko (1995), Takeuti (1975), Urquhart (2001): 1/22 1/22 1/22 1/2 ∗ ∗ , S∗1/22 , P∗ for R1/2 -valid, S∗1/22 -valid, and P∗ (i) R1/2 2 , Q∗ 2 -valid, Q∗ valid multisequents, respectively; 2/22 2/22 2/22 2/2 ∗1 ∗2 ∗1 ∗2 ∗1 ∗2 ∗1 ∗2 (ii) R2/2 2 , Q∗1 ∗2 , S2/22 , P∗1 ∗2 for R2/22 -valid, Q∗1 ∗2 -valid, S2/22 -valid, and P∗1 ∗2 valid multisequents, respectively; 3/22 3/22 3/22 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 1 ∗2 ∗3 1 ∗2 ∗3 , Q∗1 ∗2 ∗3 , S∗3/2 , P∗1 ∗2 ∗3 for R3/2 -valid, Q∗1 ∗2 ∗3 -valid, S∗3/2 -valid, (iii) R3/2 2 2 2 2 3/22

and P∗1 ∗2 ∗3 -valid multisequents, respectively; = = 4/22 4/22 4/22 4/2 , S= for R4/2 -valid, S= (iv) R4/2 2 , Q= 2 -valid, Q= 4/22 , P= 4/22 -valid, and P= valid multisequents, respectively. We consider the following deduction systems and R-calculi: 1/22

2/22

3/22

4/2 Deduction system Lt , Lt , Lt⊥ , L=

R-calculus and

1/22

Lt 2/22 Lt 3/22 Lt⊥ 4/22 L=

1/22

2/22

3/22

2

Qt , Qt , Qt⊥ , Q4/2 =

||| ≡ , ∼1 , ∼2 ,  ||| ≡ , |,  ||| ≡ ||, ∼1  ||| ≡ |||.

2

4.1 B22 -Valued DL

91

4.1 B22 -Valued DL Let U be a universe and B22 = ({t, , ⊥, f}, ∼1 , ∼2 , , ∪, ∩, →), where t  ⊥ f ∪ t  ⊥ f

∼1 ∼2   ⊥ f t f ⊥ f t  ⊥  t t⊥ f tt t t t t  tt⊥⊥ t⊥ f

∩ t ⊥f t t ⊥f ff ⊥⊥f⊥f f f f ff → t ⊥f t t t tt  t t ⊥ ⊥ tt f f ⊥t

The logical language of B22 -valued DL contains the following symbols: • atomic concepts: S0 , S1 , ...; • roles: R0 , R1 , ...; • concept constructors: ∼1 , ∼2 , , , , ∀. Concepts are defined inductively as follows: C:: = S| ∼1 C| ∼2 C|  C|C1  C2 |C1  C2 |∀R.C|∃R.C, where S is an atomic concept, and R is a role. Statements are defined as follows: ϕ:: = C(a)|R(a, b)| ∼1 ϕ| ∼2 ϕ|  ϕ. A model M is a pair (U, I ), where U is a non-empty set, and I is an interpretation such that ◦ for any atomic concept S, I (S) : U → B22 ; ◦ for any role R, I (R) : U 2 → B22 . Given an atomic concept S and a role R, we define concepts S, ∼1 S, ∼2 S, P and roles R, ∼1 R, ∼2 R, R as follows: for any x, y ∈ U, S(x) R(x, y) t  ⊥ f

∼1 S(x) ∼2 S(x) S(x) ∼1 R(x, y) ∼2 R(x, y) R(x, y)  ⊥ f t f ⊥ f t  ⊥  t

The interpretation C I of a concept C is a function from U to B22 such that for any x ∈ U,

4 R-Calculus for B22 -Valued DL

92

⎧ ⎪ ⎪ I (A)(x) ⎨ f ∗ (C I )(x) I C (x) = C I (x) ∩ C2I (x) ⎪ ⎪ ⎩ 1I C1 (x) ∪ C2I (x)

if C if C if C if C

=A = ∗C1 = C1  C2 = C1  C2

where ∗ ∈ {∼1 , ∼2 , }, and (∀R.C1 )(x) = t ≡ Ay ∈ U (I (R)(x, y) = t or I (C1 )(y) = t or (I (R)(x, y) = & I (C1 ) I (y) = ) or (I (R)(x, y) =⊥ & I (C1 )(y) =⊥)) (∼1 ∀R.C1 )(x) =  ≡ Ay ∈ U ((I (R)(x, y) = t&I (C1 )(y) = ) or I (R)(x, y) =⊥ & C1I (y) = f) (∼2 ∀R.C1 )(x) ≡ Ay ∈ U ((I (R)(x, y) = t&I (C1 )(y) =⊥) or (I (R)(x, y) = & I (C1 )(y) = f)) (∀R.C1 )(x) ≡ Ey ∈ U (I (R)(x, y) = t&I (C1 )(y) = f) and (∃R.C1 )(x) = t ≡ Ey ∈ U (I (R)(x, y) = t&I (C1 )(y) = t) (∼1 ∃R.C1 )(x) =  ≡ Ey ∈ U ((I (R)(x, y) = &I (C1 )(y) = ) or (I (R)(x, y) = t& I (C1 )(y) = ) or (I (R)(x, y) = & I (C1 )(y) = t)) (∼2 ∃R.C1 )(x) ≡ Ey ∈ U ((I (R)(x, y) =⊥ &I (C1 )(y) =⊥) or (I (R)(x, y) = t& I (C1 )(y) =⊥) or (I (R)(x, y) =⊥ & I (C1 )(y) = t)) (∃R.C1 )(x) ≡ Ay ∈ U (I (R)(x, y) = f or I (C1 )(y) = f)

t ◦   

t ◦                   ◦ ◦ ⊥  ◦ ◦ ⊥         ◦ ◦ f f ∀ = t, we have the following equivalences: For v(C(a))

∃

4.1 B22 -Valued DL

∼1 ∼1 C1 (a) ≡ C1 (a) ∼1 ∼2 C1 (a) ≡ C1 (a) ∼1 C1 (a) ≡∼2 C1 (a) ∼2 ∼1 C1 (a) ≡ C1 (a) ∼2 ∼2 C1 (a) ≡ C1 (a) ∼2 C1 (a) ≡∼1 C1 (a)  ∼1 C1 (a) ≡∼2 C1 (a)  ∼2 C1 (a) ≡∼1 C1 (a)   C1 (a) ≡ C1 (a); and ∼1 (C1  C2 )(a) ≡ (∼1 C1 (a)∧ ∼1 C2 (a))∨(∼1 C1 (a)∧C2 (a)) ∨(C1 (a) ∼1 C2 (a)) ∼2 (C1  C2 )(a) ≡ (∼2 C1 (a)∧ ∼2 C2 (a))∨(∼2 C1 (a)∧C2 (a)) ∨(C1 (a)∧ ∼2 C2 (a)) (C1  C2 )(a) ≡  C1 ∨  C2 ∨(∼1 C1 (a)∧ ∼2 C2 (a)) ∨(∼2 C1 (a)∧ ∼1 C2 (a)); (C1  C2 )(a) ≡ C1 (a)∨C2 (a)∨(∼1 C1 (a)∧ ∼2 C2 (a)) ∨(∼2 C1 (a)∧ ∼1 C2 (a)) ∼1 (C1  C2 )(a) ≡ (∼1 C1 (a)∧ ∼1 C2 (a))∨(∼1 C1 (a)∧  C2 (a)) ∨(C1 (a)∧ ∼1 C2 (a)) ∼2 (C1  C2 )(a) ≡ (∼2 C1 (a)∧ ∼2 C2 (a))∨(∼2 C1 (a)∧  C2 (a)) ∨(C1 (a)∧ ∼2 C2 (a)) (C1  C2 )(a) ≡  C1 (a)  C2 (a) and (∀R.C)(a) ≡ Ae(R(a, e)∨C(e)∨(∼1 R(a, e)∧ ∼1 C(e)) ∨(∼2 R(a, e)∧ ∼2 C(e)) (∼1 ∀R.C)(a) ≡ Ae(R(a, e)∧ ∼1 C(e))∨(∼1 R(a, e)∧ ∼2 C(e)) (∼2 ∀R.C)(a) ≡ Ae(R(a, e)∧ ∼2 C(e))∨(∼1 R(a, e)∧  C(e)) (∀R.C)(a) ≡ E f (R(a, f )∧ ∼3 C( f )) (∃R.C)(a) ≡ E f (R(a, f )∧C( f )) (∼1 ∃R.C)(a) ≡ E f ((∼1 R(a, f )∧ ∼1 C( f ))∨(R(a, f )∧ ∼1 C( f )) ∨(∼1 R(a, f )∧C( f )) (∼2 ∃R.C)(a) ≡ E f ((∼2 R(a, f )∧ ∼2 C( f ))∨(R(a, f )∧ ∼2 C( f )) ∨(∼2 R(a, f )∧C( f )) (∃R.C)(a) ≡ Ae(R(a, e)∨  C(e)). For v(C(a)) = t, we have the following equivalences: ∼1 ∼1 C1 (a) ≡ C1 (a) ∼1 ∼2 C1 (a) ≡ C1 (a) ∼1 C1 (a) ≡∼2 C1 (a) ∼2 ∼1 C1 (a) ≡ C1 (a) ∼2 ∼2 C1 (a) ≡ C1 (a) ∼2 C1 (a) ≡∼1 C1 (a)  ∼1 C1 (a) ≡∼2 C1 (a)  ∼2 C1 (a) ≡∼1 C1 (a)   C1 (a) ≡ C1 (a);

93

4 R-Calculus for B22 -Valued DL

94

and ∼1 (C1  C2 )(a) ≡ (∼1 C1 (a)∨ ∼1 C2 (a))∧(∼1 C1 (a)∨C2 (a)) ∧(C1 (a) ∼1 C2 (a)) ∼2 (C1  C2 )(a) ≡ (∼2 C1 (a)∨ ∼2 C2 (a))∧(∼2 C1 (a)∨C2 (a)) ∧(C1 (a)∨ ∼2 C2 (a)) (C1  C2 )(a) ≡  C1 ∧  C2 ∧(∼1 C1 (a)∨ ∼2 C2 (a)) ∧(∼2 C1 (a)∨ ∼1 C2 (a)); (C1  C2 )(a) ≡ C1 (a)∧C2 (a)∧(∼1 C1 (a)∨ ∼2 C2 (a)) ∧(∼2 C1 (a)∨ ∼1 C2 (a)) ∼1 (C1  C2 )(a) ≡ (∼1 C1 (a)∨ ∼1 C2 (a))∧(∼1 C1 (a)∨  C2 (a)) ∧(C1 (a)∨ ∼1 C2 (a)) ∼2 (C1  C2 )(a) ≡ (∼2 C1 (a)∨ ∼2 C2 (a))∧(∼2 C1 (a)∨  C2 (a)) ∧(C1 (a)∨ ∼2 C2 (a)) (C1  C2 )(a) ≡  C1 (a)  C2 (a) and (∀R.C)(a) ≡ E f (R(a, f )∧C( f )∧(∼1 R(a, f )∨ ∼2 C( f )) ∧(∼2 R(a, f )∨ ∼1 C( f )) (∼1 ∀R.C)(a) ≡ E f ((R(a, f )∨ ∼1 C( f ))(∼1 R(a, f )∨ ∼2 C( f )) (∼2 ∀R.C)(a) ≡ E f ((R(a, f )∨ ∼2 C( f ))(∼1 R(a, f )∨  C( f )) (∀R.C)(a) ≡ Ae(R(a, e)∨ ∼3 C(e)) (∃R.C)(a) ≡ Ae(R(a, e)∨C(e)) (∼1 ∃R.C)(a) ≡ Ae((∼1 R(a, e)∨ ∼1 C(e))∧(R(a, e)∨ ∼1 C(e)) ∧(∼1 R(a, e)∨C(e)) (∼2 ∃R.C)(a) ≡ Ae((∼2 R(a, e)∨ ∼2 C(e))∧(R(a, e)∨ ∼2 C(e)) ∧(∼2 R(a, e)∨C(e)) (∃R.C)(a) ≡ E f (R(a, f )∧  C( f )).

4.2 1/22 -Multisequents 1/22

1/22

A 1/22 -multisequent  is Lt -valid, denoted by t , if there is an interpretation I such that for any statement A(a) ∈ , I (A(a)) = t. Given a 1/22 -multisequent  of literals, define

4.2 1/22 -Multisequents

95

con() : ¬Es(a)E≥3 ∗ ∈ {λ, ∼1 , ∼2 , }(∗s(a) ∈ ) incon() : Es(a)A∗ ∈ {λ, ∼1 , ∼2 , }(∗s(a) ∈ ).

1/22

4.2.1 Deduction System Lt 1/22

Deduction system Lt • Axiom:

consists of the following axiom and deduction rules: 1/22

(At )

− − ∩ ∼− 1 ∩ ∼2  ∩   = ∅ ,

where  is a set of literals and ∼− 1  = {A(a) :∼1 A(a) ∈ } ∼− 2  = {A(a) :∼2 A(a) ∈ } −  = {A(a) : A(a) ∈ }. • Deduction rules for logical constructors: (∼21 ) and

and

, A(a) , A(a) , A(a) (∼22 ) (2 ) 2 2 , 2 A(a) , ∼1 A(a) , ∼2 A(a) ⎡

, ∼1 A1 (a) ⎢ , ∼1 A2 (a) ⎢  ⎢ , A1 (a) , A1 (a) ⎢ () , A2 (a) (∼1 ) ⎢ ⎢  , ∼1 A2 (a) ⎣ , ∼1 A1 (a) , (A1  A2 )(a) , A2 (a) , ∼ 1 (A1  A2 )(a)) ⎡ ⎡ , ∼2 A1 (a) , A1 (a) ⎢ , ∼2 A2 (a) ⎢ , A2 (a) ⎢ ⎢ ⎢ , A1 (a) ⎢ , ∼1 A1 (a) ⎢ ⎢ ⎢ () ⎢ (∼2 ) ⎢  , ∼2 A2 (a) ⎢  , ∼2 A2 (a) ⎣ , ∼2 A1 (a) ⎣ , ∼2 A1 (a) , A2 (a) , ∼1 A2 (a) , ∼2 (A1  A2 )(a)) , (A1  A2 )(a)

4 R-Calculus for B22 -Valued DL

96

⎡ , ∼1 A1 (a) , A1 (a) ⎢ , A2 (a) ⎢ , ∼1 A2 (a) ⎢ ⎢ ⎢ , ∼1 A1 (a) ⎢ , ∼1 A1 (a) ⎢ ⎢ ⎢ () ⎢  , ∼2 A2 (a) (∼1 ) ⎢ ⎢  , A2 (a) ⎣ , ∼2 A1 (a) ⎣ , A1 (a) , ∼1 A2 (a) , ∼1 A2 (a) , (A  A )(a) , ∼ 1 2 1 (A1  A2 )(a) ⎡ , ∼2 A1 (a) ⎢ , ∼2 A2 (a) ⎢  ⎢ , ∼2 A1 (a) , A1 (a) ⎢ (a) , A , A2 (a) () (∼2 ) ⎢ 2 ⎢ ⎣ , A1 (a) , (A1  A2 )(a) , ∼2 A2 (a) , ∼2 (A1  A2 )(a) ⎡

• Deduction rules for quantifier constructors: ⎡

, R(a, f ) ⎡ ⎢ , A( f ) , R(a, f ) ⎢ ⎢ , ∼1 R(a, f ) ⎢ , ∼1 A( f ) ⎢ ⎢ (∀) ⎢ (∼1 ∀) ⎣ , ∼2 R(a, f ) ⎢  , ∼1 A( f ) ⎣ , ∼2 R(a, f ) , A( f ) , ∼2 A( f ) , (∼1 ∀R.A)(a) , (∀R.A)(a) ⎡ , R(a, f )  ⎢ , ∼2 A( f ) , R(a, e) ⎢ (∼2 ∀) ⎣ , ∼1 R(a, f ) (∀) , A(e) , A( f ) , (∀R.A)(a) , (∼2 ∀R.A)(a) and

⎡

, ∼1 R(a, e) ⎢ , ∼1 A(e) ⎢  ⎢ , R(a, e) , R(a, e) ⎢ (∃) , A(e) (∼1 ∃) ⎢ ⎢  , ∼1 A(e) ⎣ , ∼1 R(a, e) , (∃R.A)(a) , A(e) , ∼ 1 (∃R.A)(a)) ⎡ ⎡ , ∼2 R(a, e) , R(a, f ) ⎢ , ∼2 A(e) ⎢ , A( f ) ⎢ ⎢ ⎢ , R(a, e) ⎢ , ∼1 R(a, f ) ⎢ ⎢ ⎢ (∼2 ∃) ⎢  , ∼2 A(e) (∃) ⎢ ⎢  , ∼2 A( f ) ⎣ , ∼2 R(a, e) ⎣ , ∼2 R(a, f ) , A(e) , ∼1 A( f ) , ∼2 (∃R.A)(a)) , (∃R.A)(a)

4.2 1/22 -Multisequents

97

where e is a new constant and f is a constant. 1/22

1/22

Definition 4.2.1 A 1/22 -multisequent  is provable in Lt , denoted by =t , if there is a sequence {1 , ..., n } of 1/22 -multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/22 -multisequents 1/22 by one of the deduction rules in Lt . Theorem 4.2.2 For any 1/22 -multisequent , 1/22

1/22

|==t  iff =t .  1/22

4.2.2 R-Calculus Qt

Given a 1/22 -multisequent  and a statement A, a 1/22 -reduction  ↑ A ⇒ , A 1/22 1/22 is Qt -valid, denoted by |==t  ↑ A ⇒ , A , if 

A = 1/22

R-calculus Qt • Axioms:



1/22

A if , A is Lt -valid λ otherwise.

consists of the following axioms and deduction rules:

− − (s(a))∩ ∼− 1 ∩ ∼2  ∩   = ∅  ↑ s(a) ⇒ (s(a)) − − − 1/22 (s(a))∩ ∼1 ∩ ∼2  ∩    = ∅ (A At0 )  ↑ s(a) ⇒  − − ∩ ∼− 1/22 1 (∼1 s(a))∩ ∼2  ∩   = ∅ (A∼1 At+ )  ↑∼1 s(a) ⇒ (∼1 s(a)) − − ∩ ∼− 1/22 1 (∼1 s(a))∩ ∼2  ∩    = ∅ (A∼1 At0 )  ↑∼1 s(a) ⇒  − − ∩ ∼− 1/22 1 ∩ ∼2 (∼2 s(a)) ∩   = ∅ (A∼2 At+ )  ↑∼2 s(a) ⇒ (∼2 s(a)) − − ∩ ∼− 1/22 1 ∩ ∼2 (∼2 s(a)) ∩    = ∅ (A∼2 At0 )  ↑∼2 s(a) ⇒  − − ∩ ∼− 1/22 1 ∩ ∼2  ∩  (s(a)) = ∅ (AAt+ )  ↑ s(a) ⇒ (s(a)) − − 2 ∩ ∼− 1/2 1 ∩ ∼2  ∩  (s(a))  = ∅ (AAt0 )  ↑ s(a) ⇒  1/22

(A At+ )

where  is a set of literals and s(a) is atomic.

4 R-Calculus for B22 -Valued DL

98

• Deduction rules:  ↑ A(a) ⇒ , A(a)  ↑ A(a) ⇒  (∼210 ) 2 2  ↑∼1 A(a) ⇒ , ∼1 A(a)  ↑∼21 A(a) ⇒   ↑ A(a) ⇒ , A(a)  ↑ A(a) ⇒  (∼22+ ) (∼220 ) 2 2  ↑∼2 A(a) ⇒ , ∼2 A(a)  ↑∼22 A(a) ⇒   ↑ A(a) ⇒ , A(a)  ↑ A(a) ⇒  (2+ ) (20 ) 2 2  ↑  A(a) ⇒ ,  A(a)  ↑ 2 A(a) ⇒ 

(∼21+ )



and

 ↑ A1 (a) ⇒ , A1 (a)  ↑ A2 (a) ⇒ , A2 (a) 

↑ (A1  A2 )(a) ⇒ , (A1  A2 )(a)  ↑ A1 (a) ⇒  (0 )  ↑ A2 (a) ⇒   ↑ (A1  A2 )(a) ⇒  (+ )

and ⎡

 ↑ A1 (a) ⇒ , A1 (a) ⎢ , A1 (a) ↑ A2 (a) ⇒ , A1 (a), A2 (a) ⎢ ⎢ , A1 (a), A2 (a) ↑∼1 A1 (a) ⇒ , A1 (a), A2 (a), ∼1 A1 (a) ⎢ (+ ) ⎢ ⎢  , A1 (a), A2 (a) ↑∼2 A2 (a) ⇒ , A1 (a), A2 (a), ∼2 A2 (a) ⎣ , Z 1 ↑∼2 A1 (a) ⇒ , Z 1 , ∼2 A1 (a) , Z 1 ↑∼1 A2 (a) ⇒ , Z 1 , ∼1 A2 (a)  ↑ ⎧ (A1  A2 )(a) ⇒ , (A1  A2 )(a)  ↑ A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪ , A1 (a) ↑ A2 (a) ⇒ , A1 (a) ⎪ ⎪

⎨ , A1 (a), A2 (a) ↑∼1 A1 (a) ⇒ , A1 (a), A2 (a) (0 ) ⎪ ⎪ , A1 (a), A2 (a) ↑∼2 A2 (a) ⇒ , A1 (a), A2 (a) ⎪ ⎪ , Z 1 ↑∼2 A1 (a) ⇒ , Z 1 ⎪ ⎪ ⎩ , Z 1 ↑∼1 A2 (a) ⇒ , Z 1  ↑ (A1  A2 )(a) ⇒  where Z 1 = A1 (a), A2 (a), ∼1 A1 (a)∨ ∼2 A2 (a), and

4.2 1/22 -Multisequents

99

⎡

 ↑ R(a, f ) ⇒ , R(a, f ) ⎢ , R(a, f ) ↑ A( f ) ⇒ , R(a, f ), A( f ) ⎢ ⎢ , R(a, f ), A( f ) ↑ R(a, f ) ⇒ , R(a, f ), A( f ), R(a, f ) ⎢ + ⎢ , R(a, f ), A( f ) ↑∼ A( f ) ⇒ , R(a, f ), A( f ), ∼ A( f ) (∀ ) ⎢  1 1 ⎣ , Z 2 ↑∼2 R(a, f ) ⇒ , Z 2 , ∼2 R(a, f ) , Z 2 ↑ A( f ) ⇒ , Z 2 , A( f )  ↑ ⎧ (∀R.A)(a) ⇒ , (∀R.A)(a)  ↑ R(a, e) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ,

R(a, e) ↑ A(e) ⇒ , R(a, e) ⎨ , R(a, e), A(e) ↑ R(a, e) ⇒ , R(a, e), A(e) e), A(e) ↑∼1 A(e) ⇒ , R(a, e), A(e) (∀0 ) ⎪ ⎪ , R(a, ⎪  ⎪ ↑∼ R(a, e) ⇒ , Z 2 , Z ⎪ 2 2 ⎪ ⎩  , Z 2 ↑ A(e) ⇒ , Z 2  ↑ (∀R.A)(a) ⇒  where

Z 2 = R(a, f ), A( f ), R(a, f )∨ ∼1 A( f ), and Z 2 = R(a, e), A(e), R(a, e)∨ ∼1 A(e), 

 ↑ R(a, e) ⇒ , R(a, e)  ↑ A(e) ⇒ , A(e) 

↑ (∃R.A)(a) ⇒ , (∃R.A)(a)  ↑ R(a, f ) ⇒  0 (∃ )  ↑ A( f ) ⇒   ↑ (∃R.A)(a) ⇒ 

(∃− )

and

⎡

 ↑∼1 A1 (a) ⇒ , ∼1 A1 (a) ⎢  ↑∼1 A2 (a) ⇒ , ∼1 A2 (a) ⎢ ⎢ , Z 3 ↑ A1 (a) ⇒ , Z 3 , A1 (a) ⎢ (∼1 + ) ⎢ ⎢  , Z 3 ↑∼1 A2 (a) ⇒ , Z 3 , ∼1 A2 (a) ⎣ , Z 4 ↑∼1 A1 (a) ⇒ , Z 4 , ∼1 A1 (a) , Z 4 ↑ A2 (a) ⇒ , Z 4 , A2 (a)  ↑∼ ⎧ 1 (A1  A2 )(a) ⇒ , ∼1 (A1  A2 )(a)  ↑∼1 A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ↑∼1 A2 (a) ⇒  ⎨ , Z 3 ↑ A1 (a) ⇒ , Z 3 , (∼1 0L ) ⎪ ⎪ Z 3 ↑∼1 A2 (a) ⇒ , Z 3 ⎪ ⎪ , Z 4 ↑∼1 A1 (a) ⇒ , Z 4 ⎪ ⎪ ⎩ , Z 4 ↑ A2 (a) ⇒ , Z 4  ↑∼1 (A1  A2 )(a) ⇒ 

where Z 3 =∼1 A1 (a)∨ ∼1 A2 (a); Z 4 = Z 3 , A1 (a)∨ ∼1 A2 (a), and

4 R-Calculus for B22 -Valued DL

100

⎡

 ↑∼1 A1 (a) ⇒ , ∼1 A1 (a) ⎢  ↑∼1 A2 (a) ⇒ , ∼1 A2 (a) ⎢ ⎢ , Z 3 ↑ A1 (a) ⇒ , Z 3 , A1 (a) ⎢ (∼1 + ) ⎢ ⎢  , Z 3 ↑∼1 A2 (a) ⇒ , Z 3 , ∼1 A2 (a) ⎣ , Z 5 ↑∼1 A1 (a) ⇒ , Z 5 , ∼1 A1 (a) , Z 5 ↑ A2 (a) ⇒ , Z 5 , A2 (a)  ↑∼ ⎧ 1 (A1  A2 )(a) ⇒ , ∼1 (A1  A2 )(a)  ↑∼1 A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ↑∼1 A2 (a) ⇒  ⎨ , Z 3 ↑ A1 (a) ⇒ , Z 3 , (∼1 0 ) ⎪ ⎪ Z 3 ↑∼1 A2 (a) ⇒ , Z 3 ⎪ ⎪ , Z 5 ↑∼1 A1 (a) ⇒ , Z 5 ⎪ ⎪ ⎩ , Z 5 ↑ A2 (a) ⇒ , Z 5  ↑∼1 (A1  A2 )(a) ⇒  where Z 5 = Z 3 , A1 (a)∨ ∼1 A2 (a), and ⎡

 ↑ R(a, f ) ⇒ , R(a, f ) ⎢  ↑∼1 A( f ) ⇒ , ∼1 A( f ) ⎢ (∼1 ∀+ ) ⎣ , R(a, f ), ∼1 A( f ) ↑∼2 R(a, e) ⇒ , R(a, f ), ∼1 A( f ), ∼2 R(a, e) , R(a, f ), ∼1 A( f ) ↑ A(e) ⇒ , R(a, f ), ∼1 A( f ), A(e)  ↑ ⎧ (∼1 ∀R.A)(a) ⇒ , (∼1 ∀R.A)(a) ⎪  ↑ R(a, e) ⇒  ⎪ ⎨

 ↑∼1 A(e) ⇒  0 , R(a, e), ∼1 A(e) ↑∼2 R(a, e) ⇒ , R(a, e), ∼1 A(e) (∼1 ∀ ) ⎪ ⎪ ⎩ , R(a, e), ∼1 A(e) ↑ A(e) ⇒ , R(a, e), ∼1 A(e)  ↑ (∼1 ∀R.A)(a) ⇒ 

and ⎡

 ↑∼1 R(a, e) ⇒ , ∼1 R(a, e) ⎢  ↑∼1 A(e) ⇒ , ∼1 A(e) ⎢ ⎢ , ∼1 R(a, e)∨ ∼1 A(e) ↑ R(a, e) ⇒ , ∼1 R(a, e)∨ ∼1 A(e), R(a, e) ⎢ + (∼1 ∃ ) ⎢ ⎢  , ∼1 R(a, e)∨ ∼1 A(e) ↑∼1 A(e) ⇒ , ∼1 R(a, e)∨ ∼1 A(e), ∼1 A(e) ⎣ , Z 6 ↑∼1 R(a, e) ⇒ , Z 6 , ∼1 R(a, e) , Z 6 ↑ A(e) ⇒ , Z 6 , A(e)  ↑ ⎧ (∼1 ∃R.A)(a) ⇒ , (∼1 ∃R.A)(a)  ↑∼1 R(a, f ) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪  ↑∼1 A( f ) ⇒  ⎪ ⎨ , ∼1 R(a, f )∨ ∼1 A( f ) ↑ R(a, f ) ⇒ , ∼1 R(a, f )∨ ∼1 A( f ) 0 (∼1 ∃ ) ⎪ ⎪ , ∼1 R(a, f )∨ ∼1 A( f ) ↑∼1 A( f ) ⇒ , ∼1 R(a, f )∨ ∼1 A( f ) ⎪   ⎪ ⎪ ⎪ , Z 6 ↑∼1 R(a, f ) ⇒ , Z 6 ⎩   , Z 6 ↑ A( f ) ⇒ , Z 6  ↑ (∼1 ∃R.A)(a) ⇒ 

where

Z 6 =∼1 R(a, e)∨ ∼1 A(e), R(a, e)∨ ∼1 A(e), and Z 6 =∼1 R(a, f )∨ ∼1 A( f ), R(a, f )∨ ∼1 A( f ),

4.2 1/22 -Multisequents

101

⎡

 ↑∼2 A1 (a) ⇒ , ∼2 A1 (a) ⎢  ↑∼2 A2 (a) ⇒ , ∼2 A2 (a) ⎢ ⎢ , Y1 ↑ A1 (a) ⇒ , Y1 , A1 (a) ⎢ (∼2 + ) ⎢ ⎢  , Y1 ↑∼2 A2 (a) ⇒ , Y1 , ∼2 A2 (a) ⎣ , Y2 ↑∼2 A1 (a) ⇒ , Y2 , ∼2 A1 (a) , Y2 ↑ A2 (a) ⇒ , Y2 , A2 (a)  ↑∼ ⎧ 2 (A1  A2 )(a) ⇒ , ∼2 (A1  A2 )(a)  ↑∼2 A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ↑∼2 A2 (a) ⇒  ⎨ , Y1 ↑ A1 (a) ⇒ , Y1 , (∼2 0 ) ⎪ ⎪ Y1 ↑∼2 A2 (a) ⇒ , Y1 ⎪ ⎪ , Y2 ↑∼2 A1 (a) ⇒ , Y2 ⎪ ⎪ ⎩ , Y2 ↑ A2 (a) ⇒ , Y2  ↑∼2 (A1  A2 )(a) ⇒  where Y1 =∼2 A1 (a)∨ ∼2 A2 (a); Y2 = Y1 , A1 (a)∨ ∼2 A2 (a), and ⎡

 ↑∼2 A1 (a) ⇒ , ∼2 A1 (a) ⎢  ↑∼2 A2 (a) ⇒ , ∼2 A2 (a) ⎢ ⎢ , Y1 ↑ A1 (a) ⇒ , Y1 , A1 (a) ⎢ (∼2 + ) ⎢ ⎢  , Y1 ↑∼2 A2 (a) ⇒ , Y1 , ∼2 A2 (a) ⎣ , Y3 ↑∼2 A1 (a) ⇒ , Y3 , ∼2 A1 (a) , Y3 ↑ A2 (a) ⇒ , Y3 , A2 (a)  ↑∼2 (A1  A2 )(a) ⇒ , ∼2 (A1  A2 )(a) ⎧ ⎪  ↑∼2 A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪  ↑∼2 A2 (a) ⇒  ⎪ ⎨ , Y1 ↑ A1 (a) ⇒ , Y1 (∼2 0 ) ⎪ ⎪ , Y1 ↑∼2 A2 (a) ⇒ , Y1 ⎪ ⎪ ⎪ , Y3 ↑∼2 A1 (a) ⇒ , Y3 ⎪ ⎩ , Y3 ↑ A2 (a) ⇒ , Y3  ↑∼2 (A1  A2 )(a) ⇒ } where Y3 = Y2 , A1 (a)∨ ∼2 A2 (a), and ⎡

 ↑ R(a, f ) ⇒ , R(a, f ) ⎢  ↑∼2 A( f ) ⇒ , ∼2 A( f ) ⎢ (∼2 ∀+ ) ⎣ , R(a, f )∨ ∼2 A( f ) ↑∼1 R(a, f ) ⇒ , R(a, f )∨ ∼2 A( f ), ∼1 R(a, f ) , R(a, f )∨ ∼2 A( f ) ↑ A(e) ⇒ , R(a, f )∨ ∼2 A( f ), A(e)  ↑ ⎧ (∼2 ∀R.A)(a) ⇒ , (∼2 ∀R.A)(a) ⎪  ↑ R(a, e) ⇒  ⎪ ⎨

, R(a, e) ↑∼2 A(e) ⇒ , R(a, e) 0 (∼2 ∀ ) ⎪ ⎪ , R(a, e)∨ ∼2 A(e) ↑∼1 R(a, e) ⇒ , R(a, e)∨ ∼2 A(e) ⎩ , R(a, e)∨ ∼2 A(e) ↑ A(e) ⇒ , R(a, e)∨ ∼2 A(e)  ↑ (∼2 ∀R.A)(a) ⇒ 

4 R-Calculus for B22 -Valued DL

102

⎡

and

where

 ↑∼2 R(a, e) ⇒ , ∼2 R(a, e) ⎢  ↑∼2 A(e) ⇒ , ∼2 A(e) ⎢ ⎢ , Y4 ↑ R(a, e) ⇒ , Y4 , R(a, e) ⎢ + ⎢ , Y ↑∼ A(e) ⇒ , Y , ∼ A(e) (∼2 ∃ ) ⎢  4 2 4 2 ⎣ , Y5 ↑∼2 R(a, e) ⇒ , Y5 , ∼2 R(a, e) , Y5 ↑ A(e) ⇒ , Y5 , A(e)  ↑ (∼2 ∃R.A)(a) ⇒ , (∼2 ∃R.A)(a) ⎧ ⎪  ↑∼2 R(a, f ) ⇒  ⎪ ⎪ ⎪ ⎪  ↑∼2 A( f ) ⇒  ⎪ ⎨ , Y4 ↑ R(a, f ) ⇒ , Y4 0 (∼2 ∃ ) ⎪ ⎪ , Y4 ↑∼2 A( f ) ⇒ , Y4 ⎪ ⎪ ⎪ , Y5 ↑∼2 R(a, f ) ⇒ , Y5 ⎪ ⎩ , Y5 ↑ A( f ) ⇒ , Y5  ↑ (∼2 ∃R.A)(a) ⇒  Y4 =∼2 R(a, e)∨ ∼2 A(e); Y5 = Y4 , R(a, e)∨ ∼2 A(e), and Y4 =∼2 R(a, f )∨ ∼2 A( f ); Y5 = Y4 , R(a, f )∨ ∼2 A( f ), ⎡

 ↑ A1 (a) ⇒ , A1 (a) ⎢ , A1 (a) ↑ A2 (a) ⇒ , A1 (a), A2 (a) ⎢ ⎢ , A1 (a), A2 (a) ↑∼1 A1 (a) ⇒ , A1 (a), A2 (a), ∼1 A1 (a) ⎢ (+ ) ⎢ ⎢  , A1 (a), A2 (a) ↑∼2 A2 (a) ⇒ , A1 (a), A2 (a), ∼2 A2 (a) ⎣ , Y6 ↑∼2 A1 (a) ⇒ , Y6 , ∼2 A1 (a) , Y6 ↑∼1 A2 (a) ⇒ , Y6 , ∼1 A2 (a) 2  ↑∼ (A1  A2 )(a)) ⇒ , ∼2 (A1  A2 )(a) ⎧  ↑ A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪ , A1 (a) ↑ A2 (a) ⇒ , A1 (a) ⎪ ⎪ ⎨

, A1 (a), A2 (a) ↑∼1 A1 (a) ⇒ , A1 (a), A2 (a) (0 ) ⎪ ⎪

, A1 (a), A2 (a) ↑∼2 A2 (a) ⇒ , A1 (a), A2 (a) ⎪ ⎪ , Y6 ↑∼2 A1 (a) ⇒ , Y6 ⎪ ⎪ ⎩ , Y6 ↑∼1 A2 (a) ⇒ , Y6  ↑ (A1  A2 )(a) ⇒  where Y6 = A1 (a), A2 (a), ∼1 A1 (a)∨ ∼2 A2 (a), and

4.2 1/22 -Multisequents

103



 ↑ A1 (a) ⇒ , A1 (a)  ↑ A2 (a) ⇒ , A2 (a)

 ↑ (A1  A2 )(a) ⇒ , (A1  A2 )(a)  ↑ A1 (a) ⇒  (0L )  ↑ A2 (a) ⇒   ↑ (A1  A2 )(a) ⇒    ↑ R(a, e) ⇒ , R(a, e) (∀+ )  ↑ A(e) ⇒ , A(e)

 ↑ (∀R.A)(a) ⇒ , (∀R.A)(a)  ↑ R(a, f ) ⇒  (∀0 )  ↑ A( f ) ⇒   ↑ (∀R.A)(a) ⇒ 

(+ )

and

⎡

 ↑ R(a, f ) ⇒ , R(a, f ) ⎢ , R(a, f ) ↑ A( f ) ⇒ , R(a, f ), A( f ) ⎢ ⎢ , Y7 ↑∼1 R(a, f ) ⇒ , Y7 , ∼1 R(a, f ) ⎢ (∃− ) ⎢ ⎢  , Y7 ↑∼2 A( f ) ⇒ , Y7 , ∼2 A( f ) ⎣ , Y8 ↑∼2 R(a, f ) ⇒ , Y8 , ∼2 R(a, f ) , Y8 ↑∼1 A( f ) ⇒ , Y8 , ∼1 A( f )  ↑ ⎧ (∃R.A)(a) ⇒ , (∃R.A)(a)  ↑ R(a, e) ⇒  ⎪ ⎪ ⎪ ⎪ , R(a, e) ↑ A(e) ⇒ , R(a, e) ⎪ ⎪ ⎨ , Y7 ↑∼1 R(a, e) ⇒ , Y7   0 (∃ ) ⎪ ⎪  , Y7 ↑∼2 A(e) ⇒ , Y7  ⎪ ⎪ , Y8 ↑∼2 R(a, e) ⇒ , Y8 ⎪ ⎪ ⎩ , Y8 ↑∼1 A(e) ⇒ , Y8  ↑ (∃R.A)(a) ⇒ 

Y7 = R(a, f ), A( f ); Y8 = Y7 , ∼1 R(a, f )∨ ∼2 A( f ), and e is a new Y7 = R(a, e), A(e); Y8 = Y7 , ∼1 R(a, e)∨ ∼2 A(e), constant and f is a constant.

where

Definition 4.2.3 Given a 1/22 -multisequent  and a statement A(a), a reduction 1/22 1/22  ↑ A(a) ⇒   is provable in Q=t , denoted by t  ↑ A(a) ⇒   , if there is a sequence {δ1 , ..., δn } of 1/22 -reductions such that δn =  ↑ A(a) ⇒   , and for each 1 ≤ i ≤ n, δi is deduced from the previous 1/22 -reductions by one of the deduction 1/22 rules in Qt . Theorem 4.2.4 (Soundness and completeness theorem) For any 1/22 -reduction  ↑ A(a) ⇒ , A (a), 1/22

1/22

=t  ↑ A(a) ⇒ , A (a) iff |==t  ↑ A(a) ⇒ , A (a). 

4 R-Calculus for B22 -Valued DL

104

4.3 2/22 -Multisequents 2/22

2/22

A 2/22 -multisequent | is Lt -valid, denoted by |=t |, if there is an interpretation I such that (i) I (A(a)) = t for each A(a) ∈ , and (ii) I (B(b)) =  for each B(b) ∈ . We have the following equivalences: , ∼1 A(a)| ≡ |, A(a) , ∼2 A(a)| ≡ , ∼2 A(a)| , A(a)| ≡ , A(a)| |, ∼1 B(b) ≡ , B(b)| |, ∼2 B(b) ≡ , B(b)| |, B(b) ≡ , ∼2 B(b)|. Given two sets  and  of literals, we move each statement beginning with ∼1 from  to  and move each statement with ∼1 , ∼2 ,  from  to  so that  is a set of literals beginning with ∼2 ,  and  is a set of atoms.     ∼1 A(a) → A(a) B(b) ←∼1 B(b) B(b) ←∼2 B(b) ∼2 A(a) ∼2 B(b) ← B(b) A(a)

2 4.3.1 Deduction System Lt 2/22

Deduction system Lt contains the following axiom and deduction rules. • Axiom: − − 2/22  ∩ ∩ ∼2  ∩   = ∅ (At ) | where  is a set of literals of forms s(a), ∼2 s(a), s(a) and  is a set of atoms. • Deduction rules for unary connectives: |, A(a) , B(b)| (∼1B ) , ∼1 A(a)| |, ∼1 B(b) , A(a)| , B(b)| (∼2B ) (∼22 A ) , ∼22 A(a)| |, ∼2 B(b) , A(a)| , ∼2 B(b)| ( B ) (2 A ) , 22 A(a)| |, B(b) (∼1A )

• Deduction rules for logical connectives:

4.3 2/22 -Multisequents

105

⎡

|, B1 (b) ⎢ |, B2 (b) ⎢  ⎢ , B1 (b)| , A1 (a)| ⎢ A B ( ) , A2 (a)| ( ) ⎢ ⎢  |, B2 (b) ⎣ |, B1 (b) , (A1  A2 )(a)| , B2 (b)| |, ⎡ ⎡ (B1  B2 )(b) , ∼2 A1 (a)| , A1 (a)| ⎢ , ∼2 A2 (a)| ⎢ , A2 (a)| ⎢ ⎢ ⎢ , A1 (a)| ⎢ , ∼2 A1 (a)| ⎢ ⎢ A ⎢ , ∼ A (a)| A ⎢ |, A (a) ( ) ⎢  (∼2  ) ⎢  2 2 2 ⎣ , ∼2 A1 (a)| ⎣ |, A1 (a) , A2 (a)| , ∼2 A2 (a)| , ∼2 (A1  A2 )(a)| , (A1  A2 )(a)| and

⎡ |, B1 (b) , A1 (a)| ⎢ , A2 (a)| ⎢ |, B2 (b) ⎢ ⎢ ⎢ |, A1 (a) ⎢ , B1 (b)| ⎢ ⎢ A ⎢ , ∼ A (a)| B ⎢ |, B (b) ( ) ⎢  ( ) ⎢  2 2 2 ⎣ , ∼2 A1 (a)| ⎣ |, B1 (b) |, A2 (a) , B2 (b)| , (A  A )(a)| |, (B1  B2 )(b) 1 2 ⎡ , ∼2 A1 (a)| ⎢ , ∼2 A2 (a)| ⎢  ⎢ , ∼2 A1 (a)| , A1 (a)| ⎢ A (a)| , A , A2 (a)| ) (∼2  A ) ⎢ ( 2 ⎢ ⎣ , A1 (a)| , (A1  A2 )(a)| , ∼2 A2 (a)| , ∼2 (A1  A2 )(a)| ⎡

• Deduction rules for quantifier constructors: ⎡

, R(a, f )| ⎡ ⎢ , A( f )| , R(b, f )| ⎢ ⎢ |, R(a, f ) ⎢ |, B( f ) ⎢ ⎢ (∀ A ) ⎢ (∀ B ) ⎣ , ∼2 R(b, f )| ⎢  |, A( f ) ⎣ , ∼2 R(a, f )| , B( f )| , ∼2 A( f )| |, (∀R.B)(b) , (∀R.A)(a)| ⎡ , R(a, f )|  ⎢ , ∼2 A( f )| , R(a, e)| ⎢ (∼2 ∀ A ) ⎣ |, R(a, f ) (∀ A ) , A(e)| , A( f )| , (∀R.A)(a)| , ∼2 (∀R.A)(a)|

4 R-Calculus for B22 -Valued DL

106

⎡

and 

(∃ A )

, R(a, e)| , A(e)| , (∃R.A)(a)| ⎡

|, R(b, e) ⎢ |, B(e) ⎢ ⎢ , R(b, e)| ⎢ B ⎢ |, B(e) (∃ ) ⎢  ⎣ |, R(b, e) , B(e)| |, (∃R.B)(b)

, ∼2 R(a, e)| ⎢ , ∼2 A(e)| ⎢

⎢ , R(a, e)| , R(a, f )| ⎢ (∼2 ∃ A ) ⎢ (∃ A ) , A( f )| ⎢  , ∼2 A(e)| ⎣ , ∼2 R(a, e)| , (∃R.A)(a)| , A(e)| , ∼2 (∃R.A)(a)| where e is a constant and f is a new constant. 2/22

2/22

Definition 4.3.1 A 2/22 -multisequent | is proof in Lt , denoted by =t |, if there is a sequence {1 |1 , ..., n |n } of 2/22 -multisequents such that n |n = |, and for each 1 ≤ i ≤ n, i |i is either an axiom or deduced from the previous 2/22 2/22 -multisequents by one of the deduction rules in Lt . Theorem 4.3.2 (Soundness and completeness theorem) For any 2/22 -multisequent |, 2/22 2/22 |==t | iff =t |.  2/22

4.3.2 R-Calculus Qt

Given a 2/22 -multisequent | and two statements A(a), B(b), a 2/22 -reduction 2/22 2/22 δ = | ↑ (A(a), B(b)) ⇒ , A (a)|, B  (b) is Qt -valid, denoted by |==t δ, if  2/22 A (a) = A(a) if , A(a)| is Lt -valid λ otherwise.  2/22  B  (b) = B(b) if , A (a)|, B(b) is Lt -valid λ otherwise. 2/22

R-calculus Qt consists of the following deduction rules and axioms: • Axioms:

4.3 2/22 -Multisequents

107

(s(a)) ∩  ∩ − ∩ ∼− 2  =∅ | ↑ s(a) ⇒ (s(a))| − (s(a)) ∩ ∩ ∼− 2/22 2  ∩   = ∅ (A At0 ) | ↑ s(a) ⇒ | −  ∩ (t (b))∩ ∼− 2/22 2 ∩  =∅ (A Bt+ ) | ↑ t (b) ⇒ |(t (b)) −  ∩ (t (b))∩ ∼− 2/22 2  ∩   = ∅ (A Bt0 ) | ↑ t (b) ⇒ | − 2  ∩ ∩ ∼− 2/2 2 (∼2 s(a)) ∩   = ∅ (A∼2 At+ ) | ↑∼2 s(a) ⇒ (∼2 s(a))| − 2  ∩ ∩ ∼− 2/2 2 (∼2 s(a)) ∩    = ∅ (A∼2 At0 ) | ↑∼2 s(a) ⇒ | − 2  ∩ ∩ ∼− 2/2 2  ∩  (s(a)) = ∅ (AAt+ ) | ↑ s(a) ⇒ (s(a))| −  ∩ ∩ ∼− 2/22 2  ∩  (s(a))  = ∅ (A∼2 At0 ) | ↑ s(a) ⇒ | 2/22

(A At+ )

where  is a set of literals and , s(a), t (b) is a set of atoms. • Deduction rules: | ↑ 2 A(a) ⇒ |, A(a) | ↑∼1 A(a) ⇒ , ∼1 A(a)| | ↑ A(a) ⇒ |, A(a) 2 A+ (∼2 ) | ↑∼22 A(a) ⇒ , ∼22 A(a)| | ↑ A(a) ⇒ , A(a)| (2 A+ ) | ↑ 2 A(a) ⇒ , 2 A(a)| | ↑ 1 B(b) ⇒ , B(b)| (∼1B+ ) | ↑∼1 B(b) ⇒ |, ∼1 B(b) 1 B+ | ↑  B(b) ⇒ , B(b)| (∼2 ) | ↑∼2 B(b) ⇒ |, ∼2 B(b) 1 B+ | ↑ ∼2 B(b) ⇒ , ∼2 B(b)| ( ) | ↑ B(b) ⇒ , ∼2 B(b)| (∼1A+ )

and



| ↑ A1 (a) ⇒ , A1 (a)| | ↑ A2 (a) ⇒ , A2 (a)| | ↑ (A1  A2 )(a) ⇒ , (A1  A2 )(a)|

| ↑ A1 (a) ⇒ | ( A0 ) | ↑ A2 (a) ⇒ | | ↑ (A1  A2 )(a) ⇒ |

( A+ )

and

| ↑ 2 A(a) ⇒ | | ↑∼1 A(a) ⇒ | | ↑ A(a) ⇒ | 2 A0 (∼2 ) | ↑∼22 A(a) ⇒ | | ↑ A(a) ⇒ | (2 A− ) | ↑ 2 A(a) ⇒ | | ↑ 1 B(b) ⇒ | (∼1B0 ) | ↑∼1 B(b) ⇒ | 1 B0 | ↑  B(b) ⇒ | (∼2 ) | ↑∼2 B(b) ⇒ | 1 B− | ↑ ∼2 B(b) ⇒ | ( ) | ↑ B(b) ⇒ | (∼1A0 )

4 R-Calculus for B22 -Valued DL

108

⎡

| ↑ A1 (a) ⇒ , A1 (a)| ⎢ , A1 (a)| ↑ A2 (a) ⇒ , A1 (a), A2 (a)| ⎢ ⎢ , A1 (a), A2 (a)| ↑ 2 A1 (a) ⇒ , A1 (a), A2 (a)|, A1 (a) ⎢ A+ ⎢ , A (a), A (a)| ↑∼ A (a) ⇒ , A (a), A (a), ∼ A (a)| ( ) ⎢  1 2 2 2 1 2 2 2 ⎣ , Z 1 | ↑∼2 A1 (a) ⇒ , Z 1 , ∼2 A1 (a)| , Z 1 | ↑ 2 A2 (a) ⇒ , Z 1 |, A2 (a) | ⎧ ↑ (A1  A2 )(a) ⇒ , (A1  A2 )(a)| | ↑ A1 (a) ⇒ | ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ,

A1 (a)| ↑ A2 (a) ⇒2, A1 (a)| ⎨ , A1 (a), A2 (a)| ↑ A1 (a) ⇒ , A1 (a), A2 (a)| , ( A− ) ⎪ ⎪ A1 (a), A2 (a)| ↑∼2 A2 (a) ⇒ , A1 (a), A2 (a)| ⎪ ⎪ , Z 1 | ↑∼2 A1 (a) ⇒ , Z 1 | ⎪ ⎪ ⎩ , Z 1 | ↑ 2 A2 (a) ⇒ , Z 1 | | ↑ (A1  A2 )(a) ⇒ | where Z 1 = A1 (a), A2 (a), ∼1 A1 (a)∨ ∼2 A2 (a), and ⎡

| ↑∼2 A1 (a) ⇒ , ∼2 A1 (a)| ⎢ | ↑∼2 A2 (a) ⇒ , ∼2 A2 (a)| ⎢ ⎢ , Z  | ↑ A1 (a) ⇒ , Z  , A1 (a)| 2 2 ⎢ A+ ⎢ , Z  | ↑∼ A (a) ⇒ , Z  , ∼ A (a)| (∼2  ) ⎢  2 2 2 2 2 2 ⎣ , Z 3 | ↑∼2 A1 (a) ⇒ , Z 3 , ∼2 A1 (a)| , Z 3 | ↑ A2 (a) ⇒ , Z 3 , A2 (a)| | ⎧

↑∼2 (A1  A2 )(a) ⇒ , ∼2 (A1  A2 )(a)| ⎪ | ↑∼2 A1 (a) ⇒ | ⎪ ⎪ ⎪ ⎪ | ↑∼2 A2 (a) ⇒ | ⎪ ⎨ , Z 2 | ↑ A1 (a) ⇒ , Z 2 | A0 (∼2  ) ⎪ ⎪ , Z 2 | ↑∼2 A2 (a) ⇒ , Z 2 | ⎪ ⎪ ⎪ , Z 3 | ↑∼2 A1 (a) ⇒ , Z 3 | ⎪ ⎩ , Z 3 | ↑ A2 (a) ⇒ , Z 3 | | ↑∼2 (A1  A2 )(a) ⇒ | where Z 2 =∼2 A1 (a)∨ ∼2 A2 (a); Z 3 = Z 2 , A1 (a)∨ ∼2 A2 (a), and

4.3 2/22 -Multisequents

109

⎡

| ↑∼2 A1 (a) ⇒ , ∼2 A1 (a)| ⎢ | ↑∼2 A2 (a) ⇒ , ∼2 A2 (a)| ⎢ ⎢ , Z  | ↑ A1 (a) ⇒ , Z  , A1 (a)| 2 2 ⎢ A+ ⎢ , Z  | ↑∼ A (a) ⇒ , Z  , ∼ A (a)| (∼2  ) ⎢  2 2 2 2 2 2 ⎣ , Z 5 | ↑∼2 A1 (a) ⇒ , Z 5 , ∼2 A1 (a)| , Z 5 | ↑ A2 (a) ⇒ , Z 5 , A2 (a)| | ⎧

↑∼2 (A1  A2 )(a) ⇒ , ∼2 (A1  A2 )(a)| | ↑∼2 A1 (a) ⇒ | ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ | ↑∼2 A2 (a) ⇒ |  ⎨ , Z 2 | ↑ A1 (a) ⇒ , Z 2 |   , (∼2  A0 ) ⎪ ⎪ Z 2 | ↑∼2 A2 (a) ⇒ , Z 2 | ⎪ ⎪ ⎪ ⎪ , Z 5 | ↑∼2 A1 (a) ⇒ , Z 5 | ⎩ , Z 5 | ↑ A2 (a) ⇒ , Z 5 | | ↑∼2 (A1  A2 )(a) ⇒ | where Z 5 = Z 2 , A1 (a)∨ ∼2 A2 (a), and ⎡

| ↑ A1 (a) ⇒ , A1 (a)| ⎢ , A1 (a)| ↑ A2 (a) ⇒ , A1 (a), A2 (a)| ⎢ ⎢ , Z 2 | ↑ 2 A1 (a) ⇒ , Z 2 |, A1 (a) ⎢ A+ ⎢ , Z | ↑∼ A (a) ⇒ , Z , ∼ A (a)| ( ) ⎢  2 2 2 2 2 2 ⎣ , Z 3 | ↑∼2 A1 (a) ⇒ , Z 3 , ∼2 A1 (a)| , Z 3 | ↑ 2 A2 (a) ⇒ , Z 3 |, A2 (a) | ↑ (A1  A2 )(a) ⇒ , (A1  A2 )(a)| ⎧ | ↑ A1 (a) ⇒ | ⎪ ⎪ ⎪ ⎪ , A1 (a)| ↑ A2 (a) ⇒ , A1 (a)| ⎪ ⎪ ⎨

, Z 2 | ↑ 2 A1 (a) ⇒ , Z 2 | A0 ( ) ⎪ ⎪

, Z 2 | ↑∼2 A2 (a) ⇒ , Z 2 | ⎪ ⎪ , Z 3 | ↑∼2 A1 (a) ⇒ , Z 3 | ⎪ ⎪ ⎩ , Z 3 | ↑ 2 A2 (a) ⇒ , Z 3 | | ↑ (A1  A2 )(a) ⇒ | where Z 2 = A1 (a), A2 (a); Z 3 = Z 2 , ∼1 A1 (a)∨ ∼2 A2 (a), and 

| ↑ A1 (a) ⇒ , A1 (a)| | ↑ A2 (a) ⇒ , A2 (a)|

| ↑ (A1  A2 )(a) ⇒ , (A1  A2 )(a)| | ↑ A1 (a) ⇒ | ( A0 ) | ↑ A2 (a) ⇒ | | ↑ (A1  A2 )(a) ⇒ | ( A+ )

and

4 R-Calculus for B22 -Valued DL

110

⎡

| ↑ R(a, f ) ⇒ , R(a, f )| ⎢ , R(a, f )| ↑ A( f ) ⇒ , R(a, f ), A( f )| ⎢ ⎢ , R(a, f ), A( f )| ↑ 2 R(a, f ) ⇒ , R(a, f ), A( f )|, R(a, f ) ⎢ A+ ⎢ , R(a, f ), A( f )| ↑ 2 A( f ) ⇒ , R(a, f ), A( f )|, A( f ) (∀ ) ⎢  ⎣ , Y4 | ↑∼2 R(a, f ) ⇒ , Y4 , ∼2 R(a, f )| , Y4 | ↑∼2 A( f ) ⇒ , Y4 , ∼2 A( f )| | ⎧ ↑ (∀R.A)(a) ⇒ , (∀R.A)(a)| | ↑ R(a, e) ⇒ , R(a, e)| ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ,

R(a, e)| ↑ A(e) ⇒2, R(a, e), A(e)| ⎨ , R(a, e), A(e)| ↑ R(a, e) ⇒ , R(a, e), A(e)| , e), A(e)| ↑ 2 A(e) ⇒ , R(a, e), A(e)| (∀ A0 ) ⎪ ⎪ R(a, ⎪  ⎪ , Z 4 | ↑∼2 R(a, e) ⇒ , Z 4 | ⎪ ⎪ ⎩ , Z 4 | ↑∼2 A(e) ⇒ , Z 4 | | ↑ (∀R.A)(a) ⇒ | where

Z 4 = R(a, f ), A( f ), ∼1 R(a, f )∨ ∼1 A( f ), and Z 4 = R(a, e), A(e), ∼1 R(a, e)∨ ∼1 A(e), 

| ↑ R(a, e) ⇒ , R(a, e)| | ↑ A(e) ⇒ , A(e)| | ↑ (∃R.A)(a) ⇒ , (∃R.A)(a)|

| ↑ R(a, f ) ⇒ | (∃ A0 ) | ↑ A( f ) ⇒ | | ↑ (∃R.A)(a) ⇒ |

(∃ A+ )

and ⎡

| ↑ R(a, f ) ⇒ , R(a, f )| ⎢ | ↑∼2 A( f ) ⇒ , ∼2 A( f )| ⎢ (∼2 ∀ A+ ) ⎣ , R(a, f )∨ ∼2 A( f )| ↑ 2 R(a, f ) ⇒ , R(a, f )∨ ∼2 A( f )|, R(a, f ) , R(a, f )∨ ∼2 A( f )| ↑ A( f ) ⇒ , R(a, f )∨ ∼2 A( f ), A( f )| | ⎧ ↑ (∼2 ∀R.A)(a) ⇒ , (∼2 ∀R.A)(a)| | ↑ R(a, e) ⇒ | ⎪ ⎪ ⎨ ,

R(a, e)| ↑∼2 A(e) ⇒2, R(a, e)| , R(a, e)∨ ∼2 A(e)| ↑ R(a, e) ⇒ , R(a, e)∨ ∼2 A(e)| (∼2 ∀ A0 ) ⎪ ⎪ ⎩ , R(a, e)∨ ∼2 A(e)| ↑ A(e) ⇒ , R(a, e)∨ ∼2 A(e)| | ↑ (∼2 ∀R.A)(a) ⇒ |

and

4.3 2/22 -Multisequents

111

⎡

| ↑∼2 R(a, e) ⇒ , ∼2 R(a, e)| ⎢ | ↑∼2 A(e) ⇒ , ∼2 A(e)| ⎢ ⎢ , Y5 | ↑ R(a, e) ⇒ , Y5 , R(a, e)| ⎢ A+ (∼2 ∃ ) ⎢ ⎢  , Y5 | ↑∼2 A(e) ⇒ , Y5 , ∼2 A(e)| ⎣ , Y6 | ↑∼2 R(a, e) ⇒ , Y6 , ∼2 R(a, e)| , Y6 | ↑ A(e) ⇒ , Y6 , A(e)| | ⎧ ↑ (∼2 ∃R.A)(a) ⇒ , (∼2 ∃R.A)(a)| ⎪ | ↑∼2 R(a, f ) ⇒ | ⎪ ⎪ ⎪ | ↑∼2 A( f ) ⇒ | ⎪ ⎪ ⎨

, Y5 | ↑ R(a, f ) ⇒ , Y5 |   , (∼2 ∃ A0 ) ⎪ ⎪

Y5 | ↑∼2 A( f ) ⇒ , Y5 | ⎪ ⎪ | ↑∼ R(a, f ) ⇒ , Y6 | , Y ⎪ 2 ⎪ 6 ⎩ , Y6 | ↑ A( f ) ⇒ , Y6 | | ↑ (∼2 ∀R.A)(a) ⇒ |

where

Y5 =∼2 R(a, e)∨ ∼2 A(e); Y6 = Y5 , R(a, e)∨ ∼2 A(e), and Y5 =∼2 R(a, f )∨ ∼2 A( f ); Y6 = Y5 , R(a, f )∨ ∼2 A( f ), 

| ↑ R(a, e) ⇒ , R(a, e)| | ↑ A(e) ⇒ , A(e)|

| ↑ (∀R.A)(a) ⇒ , (∀R.A)(a)| | ↑ R(a, f ) ⇒ | (∀ A0 ) | ↑ A( f ) ⇒ | |

↑ (∀R.A)(a) ⇒ | | ↑ R(a, f ) ⇒ , R(a, f )| (∃ A+ ) , R(a, f )| ↑ A( f ) ⇒ , R(a, f ), A( f )| | ↑ (∃R.A)(a) ⇒ , (∃R.A)(a)| | ↑ R(a, e) ⇒ | (∃ A0 ) , R(a, e)| ↑ A(e) ⇒ , R(a, e)| | ↑ (∀R.A)(a) ⇒ | (∀ A+ )

and

⎡

  | ↑ B1 (b) ⇒   |, B1 (b) ⎢   | ↑ B2 (b) ⇒   |, B2 (b) ⎢  ⎢  |, Z 7 ↑ 1 B1 (b) ⇒   , B1 (b)|, Z 7 ⎢   ( B+ ) ⎢ ⎢    |, Z 7 ↑ B2 (b) ⇒   |, Z 7 , B2 (b) ⎣  |, Z 8 ↑ B1 (b) ⇒  |, Z 8 , B1 (b)   |, Z 8 ↑ 1 B2 (b) ⇒   , B2 (b)|, Z 8 , B1 (b)   | ↑ (B1  B2 )(b) ⇒   |, (B1  B2 )(b) ⎧ ⎪   | ↑ B1 (b) ⇒   | ⎪ ⎪ ⎪ ⎪   | ↑ B2 (b) ⇒   | ⎪ ⎨   |, Z 7 ↑ 1 B1 (b) ⇒   |, Z 7 B0  ( ) ⎪ ⎪  |, Z 7 ↑ B2 (b) ⇒   |, Z 7 ⎪ ⎪ ⎪   |, Z 8 ↑ B1 (b) ⇒   |, Z 8 ⎪ ⎩   |, Z 8 ↑ 1 B2 (b) ⇒   |, Z 8   | ↑ (B1  B2 )(b) ⇒   |

where Z 7 = B1 (b) ∨ B2 (b); Z 8 = Z 7 , B1 (b)∨ ∼1 B2 (b), and

4 R-Calculus for B22 -Valued DL

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⎡

  | ↑ B1 (b) ⇒   |, B1 (b) ⎢   | ↑ B2 (b) ⇒   |, B2 (b) ⎢  ⎢  |, Z 7 ↑ 1  B1 (b) ⇒   , B1 (b)|, Z 7 ⎢ B+ ⎢   |, Z ↑ B (b) ⇒   |, Z , B (b) ( ) ⎢  7 2 7 2 ⎣   |, Z 8 ↑ B1 (b) ⇒   |, Z 8 , B1 (b)   |, Z 8 ↑ 1  B2 (b) ⇒   , B2 (b)|, Z 8 , B1 (b)   | ↑ (B1  B2 )(b) ⇒   |, (B1  B2 )(b) ⎧   | ↑ B1 (b) ⇒   | ⎪ ⎪ ⎪  ⎪ ⇒   | ⎪ ⎪   | ↑ B2 (b) ⎨ 1  |, Z 7 ↑  B1 (b) ⇒   |, Z 7   B0 ( ) ⎪ ⎪   |, Z 7 ↑ B2 (b) ⇒   |, Z 7 ⎪ ⎪  |, Z 8 ↑ B1 (b) ⇒  |, Z 8 ⎪ ⎪ ⎩   |, Z 8 ↑ 1  B2 (b) ⇒   |, Z 8   | ↑ (B1  B2 )(b) ⇒   | where Z 8 = Z 7 , ∼2 B1 (b) ∨ B2 (b), and ⎡

  | ↑ 1 R(b, f ) ⇒   , R(b, f )| ⎢   | ↑ B( f ) ⇒   |, B( f ) ⎢ (∀ B+ ) ⎣   |, Z 9 ↑∼2 R(b, f ) ⇒   , ∼2 R(b, f )|, Z 9   |, Z 9 ↑ 1  B( f ) ⇒   , B( f )|, Z 9   | ↑ (∀R.B)(b) ⇒   |, (∀R.B)(b) ⎧   | ↑ 1 R(b, e) ⇒   | ⎪ ⎪ ⎨  ⇒   |

  | ↑ B(e)  B0  |, Z 9 ↑∼2 R(b, e) ⇒   |, Z 9 (∀ ) ⎪ ⎪ ⎩   |, Z 9 ↑ 1  B(e) ⇒   |, Z 9   | ↑ (∀R.B)(b) ⇒   | where Z 9 = R(b, f )∨ ∼1 B( f ), Z 9 = R(b, e)∨ ∼1 B(e), and ⎡

  | ↑ R(b, e) ⇒   |, R(b, e) ⎢   | ↑ B(e) ⇒   |, B(e) ⎢  ⎢  |, Z 10 ↑ 1 R(b, e) ⇒   , R(b, e)|, Z 10 ⎢  B+ ⎢   |, Z (∃ ) ⎢  10 ↑ B(e) ⇒  |, Z 10 , B(e)  ⎣  |, Z 11 ↑ R(b, e) ⇒   |, Z 11 , R(b, e)   |, Z 11 ↑ 1 B(e) ⇒   , B(e)|, Z 11 , R(b, e)   | ↑ (∃R.B)(b) ⇒   |, (∃R.B)(b) ⎧   | ↑ R(b, f ) ⇒   | ⎪ ⎪ ⎪ ⎪   | ↑ B( f ) ⇒   |, B( f ) ⎪ ⎪ ⎨    ↑ 1 R(b, f ) ⇒   |, Z 10  |, Z 10     B0 ↑ B( f ) ⇒  |, Z 10 (∃ ) ⎪ ⎪

  |, Z 10 ⎪   ⎪ |, Z ↑ R(b, f ) ⇒   |, Z 11  ⎪ 11 ⎪ ⎩    1   |, , Z 11 ↑ B( f ) ⇒  |, , Z 11    | ↑ (∃R.B)(b) ⇒  |

4.4 3/22 -Multisequents

113

Z 10 = R(b, e) ∨ B(e); Z 11 = Z 9 , ∼1 R(b, e) ∨ B(e), and where e is a   = R(b, f ) ∨ B( f ); Z 11 = Z 9 , ∼1 R(b, f ) ∨ B( f ), Z 10 new constant and f is a constant. Remark: Notice that Z 5 = B1 (b) ∨ B2 (b) is with respect to , and with respect to t it becomes ∼1 B1 (b)∨ ∼1 B2 (b), that is, where

wrt  B1 (b) ∨ B2 (b) ∼1 B1 (b) ∨ B2 (b) wrt t ∼1 B1 (b)∨ ∼1 B2 (b) B1 (b)∨ ∼1 B2 (b). As for Z 5 = B1 (b) ∨ B2 (b) is with respect to , and with respect to t it becomes ∼2 B1 (b)∨ ∼2 B2 (b), that is, wrt  B1 (b) ∨ B2 (b) ∼1 B1 (b) ∨ B2 (b) wrt t ∼2 B1 (b)∨ ∼2 B2 (b) B1 (b)∨ ∼2 B2 (b).  



Definition 4.3.3 A 2/2 -reduction δ = | ↑ (A(a), B(b)) ⇒  | is provable in 2/22 2/22 Qt , denoted by t δ, if there is a sequence {δ1 , ..., δn } of 2/22 -reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/22 -reductions 2/22 by one of the deduction rules in Qt . 2

Theorem 4.3.4 (Soundness and completeness theorem) For any reduction δ = | ↑ (A(a), B(b)) ⇒   | , 2/22

2/22

|=t δ iff t δ. 

4.4 3/22 -Multisequents 3/22

3/22

A 3/22 -multisequent || is Lt⊥ -valid, denoted by |=t⊥ ||, if there is an interpretation I such that • I (A(a)) = t for each A(a) ∈ , • I (B(b)) =  for each B(b) ∈ , and • I (C(c)) =⊥ for each C(c) ∈ . We have the following equivalences: |, ∼1 B(b)| ≡ , B(b)|| , ∼1 A(a)|| ≡ |, A(a)| ||, ∼1 C(c) ≡ ||, ∼1 C(c) |, ∼2 B(b)| ≡ ||, ∼1 B(b) , ∼2 A(a)|| ≡ ||, A(a) ||, ∼2 C(c) ≡ , C(c)|| , A(a)|| ≡ ||, ∼1 A(a) |, B(b)| ≡ ||, B(b) ||, C(c) ≡ |, C(c)|

4 R-Calculus for B22 -Valued DL

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and given three sets , ,  of literals, we move each statement in  beginning with ∼1 from  to  with ∼2 from  to  move each statement in  beginning with ∼1 from  to  with ∼2 from  to  with ∼1 with  from  to  and move each statement in  with ∼1 from  to  with ∼1 with ∼2 from  to  with  from  to  so that ,  become sets of atoms and  is a set of literals beginning with ∼1 only.       ∼1 A(a) → A(a) B(b) ←∼1 B(b) A(a) ∼2 B(b) → ∼1 B(b) ∼2 A(a) → B(b) → B(b) A(a) → ∼1 A(a)    ∼1 C(c) C(c) ←∼2 C(c) C(c) ← C(c)

3/22

4.4.1 Deduction System Lt⊥ 3/22

Deduction system Lt⊥ contains the following axiom and deduction rules: • Axiom:  ∩  ∩ ∩ ∼− 1  =∅ (A3t⊥ ) ||, where  is a set of literal statements, and ,  are sets of atomic statements. • Deduction rules for unary logical connectives:

4.4 3/22 -Multisequents

115

|, A(a)| , B(b)|| ||, ∼1 C(c) (∼1B ) (∼C1 ) , ∼1 A(a)|| |, ∼1 B(b)| ||, ∼1 C(c) ||, A(a) , C(c)|| C A B ||, ∼1 B(b) (∼2 ) (∼2 ) (∼2 ) , ∼2 A(a)|| |, ∼2 B(b)| ||, ∼2 C(c) ||, B(b) |, C(c)| A ||, ∼1 A(a) B C ( ) ( ) ( ) , A(a)|| |, B(b)| ||, C(c)

(∼1A )

• Deduction rules for binary logical connective : ⎡

|, B1 (b)| ⎢ |, B2 (b)| ⎢   ⎢ , B1 (b)|| , A1 (a)|| ⎢ ( A ) , A2 (a)|| ( B ) ⎢ ⎢  |, B2 (b)| ⎣ |, B1 (b)| , (A1  A2 )(a)|| , B2 (b)|| |, ⎡ ⎡ (B1  B2 )(a)| ||, C1 (c) ||, ∼1 C1 (a) ⎢ ||, C2 (c) ⎢ || ∼1 C2 (a) ⎢ ⎢ ⎢ , C1 (c)|| ⎢ |, C1 (a)| ⎢ ⎢ C ⎢ ||, C (c) C ⎢ ||, C (a) ( ) ⎢  (∼1  ) ⎢  2 2 ⎣ ||, C1 (c) ⎣ ||, C1 (a) , C2 (c)|| |, C2 (a)| ||, (C1  C2 )(a) ||, ∼1 (C1  C2 )(a) • Deduction rules for binary logical connective : ⎡ |, B1 (b)| , A1 (a)|| ⎢ , A2 (a)|| ⎢ |, B2 (b)| ⎢ ⎢ ⎢ ||, A1 (a) ⎢ , B1 (b)|| ⎢ ⎢ B ⎢ |, B (b)| (a)| |, A ( A ) ⎢ ) ( 2 2 ⎢ ⎢ ⎣ |, A1 (a)| ⎣ |, B1 (b)| ||, A2 (a) , B2 (b)|| , (A  A )(a)|| |, (B1  B2 )(a)| 2 ⎡ 1 ||, C1 (c) ⎢ ||, C2 (c) ⎢  ⎢ , C1 (c)|| ||, ∼1 C1 (c) ⎢ C (c) ||, C ||, ∼1 C2 (c)  ) (C ) ⎢ (∼ 2 1 ⎢ ⎣ ||, C1 (c) ||, ∼1 (C1  C2 )(c) , C2 (c)|| ||, (C1  C2 )(c) ⎡

4 R-Calculus for B22 -Valued DL

116

• Deduction rules for quantifier ∀: ⎡

, R(a, f )|| ⎢ , A( f )|| ⎢ ⎢ |, R(a, f )| ⎢ A ⎢ |, A( f )| (∀ ) ⎢  ⎣ ||, R(a, f ) ||, A( f ) , (∀R.A)(a)|| ⎡ , R(c, f )|| ⎢ ||, C( f ) ⎢ (∀C ) ⎣ |, R(c, f )| , C( f )|| ||, (∀R.C)(c) and

⎡

, R(b, f )|| ⎢ |, B( f )| ⎢ (∀ B ) ⎣ ||, R(b, f ) , B( f )|| |, (∀R.B)(b)| 

(∼1 ∀C )

||, R(c, e) ||, ∼1 C(e) ||, ∼1 (∀R.C)(c)

⎡

|, R(b, e)| ⎢ |, B(e)| ⎢  ⎢ , R(b, e)|| , R(a, e)|| ⎢ (∃ A ) , A(e)|| (∃ B ) ⎢ ⎢  |, B(e)| ⎣ |, R(b, e)| , (∃R.A)(a)|| , B(e)|| |, ⎡ ⎡ (∃R.B)(b)| ||, R(c, e) ||, ∼1 R(c, f ) ⎢ ||, C(e) ⎢ ||, ∼1 A( f ) ⎢ ⎢

⎢ , R(c, e)|| ⎢ |, R(c, f )| ⎢ ⎢ C ⎢ ||, A( f ) ||, C(e) ∃ ) (∃C ) ⎢ (∼ 1 ⎢ ⎢

⎣ ||, R(c, e) ⎣ ||, R(c, f ) , C(e)|| |, A( f )| ||, (∃R.C)(c) ||, ∼1 (∃R.C)(c)

where f is a constant and e is a new constant. 3/22

3/22

Definition 4.4.1 A 3/22 -multisequent || is provable in Lt⊥ , denoted by t⊥ ||, if there is a sequence {1 |1 |1 , ..., n |n |n } of 3/22 -multisequents such that n |n |n = ||, and for each 1 ≤ i ≤ n, i |i |i is either an axiom or 3/22 deduced from the previous 3/22 -multisequents by one of the deduction rules in Lt⊥ . Theorem 4.4.2 For any 3/22 -multisequent ||, 3/22

3/22

|=t⊥ || iff t⊥ ||.

4.4 3/22 -Multisequents

117

3/22

4.4.2 R-Calculus Qt⊥ Given a 3/22 -multisequent || and two statements A(a), B(b), C(c), a 3/22 reduction δ = || ↑ (A(a), B(b), C(c)) ⇒ , A (a)|, B  (b)|, C  (c) 3/22

3/22

is Qt⊥ -valid, denoted by |=t⊥ δ, if 

3/22

A(a) if , A(a)|| is Lt⊥ -valid λ otherwise;  3/22 B(b) if , A (a)|, B(b)| is Lt⊥ -valid B  (b) = λ otherwise;  3/22   C  (c) = C(c) if , A (a)|, B (b)|, C(c) is Lt⊥ -valid λ otherwise. 

A (a) =

Given a statement triple X = (A(a), B(b), C(c)), let X = || and X = , A (a)|| X = , A (a)|, B  (b)| X = , A (a)|, B  (b)|, C  (c) X(1 A(a)) = , A(a)|| X(2 A(a)) = |, A(a)| X(3 A(a)) = ||, A(a);

X(A(a)) = , A(a)|| X(B(b)) = |, B(b)| X(C(c)) = ||, C(c); X(1 R(a, b)) = , R(a, b)|| X(2 R(a, b)) = |, R(a, b)| X(3 R(a, b)) = ||, R(a, b).

3/22

R-calculus Qt⊥ consists of the following axioms and deduction rules: • Axioms: (s(a)) ∩  ∩ ∩ ∼− 1  =∅ || ↑ s(a) ⇒ (s(a))|| (s(a)) ∩  ∩ ∩ ∼− 3/22 1  = ∅ (A At⊥0 ) || ↑ s(a) ⇒ ||  ∩ (t (b)) ∩ ∩ ∼− 3/22 1  =∅ (A Bt⊥+ ) || ↑ t (b) ⇒ |(t (b))|  ∩ (t (b)) ∩ ∩ ∼− 3/22 1  = ∅ (A Bt⊥0 ) || ↑ t (b) ⇒ ||  ∩  ∩ (u(c))∩ ∼− 3/22 1  =∅ (ACt⊥+ ) || ↑ u(c) ⇒ ||(u(c))  ∩  ∩ (u(c))∩ ∼− 3/22 1  = ∅ (ACt⊥0 ) || ↑ u(c) ⇒ ||  ∩  ∩ ∩ ∼− 3/22 1 (∼1 u(c)) = ∅ (A∼1 Ct⊥+ ) || ↑∼1 u(c) ⇒ ||(∼1 u(c))  ∩  ∩ ∩ ∼− 3/22 1 (∼1 u(c))  = ∅ (A∼1 Ct⊥0 ) || ↑∼1 u(c) ⇒ || 3/22

(A At⊥+ )

4 R-Calculus for B22 -Valued DL

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where  is a set of literals, ,  are sets of atoms, and s(a), t (b), u(c) are atoms. • Deduction rules consist of three parts E A , E B , EC . ◦ EA: X ↑ 2 A(a) ⇒ X (∼1A+ ) X ↑∼1 A(a) ⇒ X X ↑ 3 A(a) ⇒ X (∼2A+ ) (∼2A0 ) X ↑∼2 A(a) ⇒ X X ↑ 3 ∼1 A(a) ⇒ X ( A+ ) ( A0 ) X ↑ A(a) ⇒ X

(∼1A0 )

and

X ↑ 2 A(a) ⇒ X(2 A(a)) X ↑∼1 A(a) ⇒ X(∼1 A(a)) X ↑ 3 A(a) ⇒ X(3 A(a)) X ↑∼2 A(a) ⇒ X(∼2 A(a)) X ↑ 3 ∼1 A(a) ⇒ X(3 ∼1 A(a)) X ↑ A(a) ⇒ X(A(a))



X ↑ A1 (a) ⇒ X(A1 (a)) X ↑ A2 (a) ⇒ X(A2 (a)) X

↑ (A1  A2 )(a) ⇒ X((A1  A2 )(a)) X ↑ A1 (a) ⇒ X ( A0 ) X ↑ A2 (a) ⇒ X X ↑ (A1  A2 )(a) ⇒ X ( A+ )

and

⎡

X ↑ A1 (a) ⇒ X(A1 (a)) ⎢ X(A1 (a)) ↑ A2 (a) ⇒ X(A1 (a), A2 (a)) ⎢ ⎢ X(Y7 ) ↑ 3 A1 (a) ⇒ X(Y7 , 3 A1 (a)) ⎢ A+ ⎢ X(Y ) ↑ 2 A (a) ⇒ X(Y , 2 A (a)) ( ) ⎢  7 2 7 2 ⎣ X(Y8 ) ↑ 2 A1 (a) ⇒ X(Y8 , 2 A1 (a)) X(Y8 ) ↑ 3 A2 (a) ⇒ X(Y8 , 3 A2 (a)) X ↑ (A1  A2 )(a) ⇒ X((A1  A2 )(a)) ⎧ X ↑ A1 (a) ⇒ X ⎪ ⎪ ⎪ ⎪ X(A1 (a)) ↑ A2 (a) ⇒ X(A1 (a)) ⎪ ⎪ ⎨

X(Y7 ) ↑ 3 A1 (a) ⇒ X(Y7 ) 2 A0 ( ) ⎪ ⎪

X(Y7 ) ↑ 2 A2 (a) ⇒ X(Y7 ) ⎪ ⎪ X(Y8 ) ↑ A1 (a) ⇒ X(Y8 ) ⎪ ⎪ ⎩ X(Y8 ) ↑ 3 A2 (a) ⇒ X(Y8 ) X ↑ (A1  A2 )(a) ⇒ X

where Y7 = A1 (a), A2 (a); Y8 = Y7 , ∼2 A1 (a)∨ ∼1 A2 (a), and

4.4 3/22 -Multisequents

119

⎡

X ↑ 3 ∼1 R(a, f ) ⇒ X(3 ∼1 R(a, f )) ⎢ X(3 ∼1 R(a, f )) ↑ A( f ) ⇒ X(3 ∼1 R(a, f ), A( f )) ⎢ ⎢ X(Z 1 ) ↑ 2 R(a, f ) ⇒ X(Z 1 , 2 R(a, f )) ⎢ A+ ⎢ X(Z ) ↑ 2 A( f ) ⇒ X(Z , 2 A( f )) (∀ ) ⎢  1 1 ⎣ X(Z 2 ) ↑ 3 R(a, f ) ⇒ X(Z 2 , 3 R(a, f )) X(Z 2 ) ↑ 3 A( f ) ⇒ X(Z 2 , 3 A( f )) X ↑ (∀R.A)(a) ⇒ X((∀R.A)(a)) ⎧ X ↑ 3 ∼1 R(a, e) ⇒ X ⎪ ⎪ ⎪ 3 ⎪ e)) ↑ A(e) ⇒ X(3 ∼1 R(a, e)) ⎪ ⎪ X(

∼1 R(a, ⎨ 2 X(Z 1 ) ↑ R(a, e) ⇒ X(Z 1 )   2 A0 (∀ ) ⎪ ⎪ X(Z 1 ) ↑ 3 A(e) ⇒ X(Z 1 )  ⎪ ⎪ X(Z 2 ) ↑ R(a, e) ⇒ X(Z 2 ) ⎪ ⎪ ⎩ X(Z 2 ) ↑ 3 A(e) ⇒ X(Z 2 ) X ↑ (∀R.A)(a) ⇒ X where

Z 1 = R(a, f ), A( f ); Z 2 = Z 1 , ∼1 R(a, f )∨ ∼1 A( f ), and Z 1 = R(a, e), A(e); Z 2 = Z 1 , ∼1 R(a, e)∨ ∼1 A(e), 

X ↑ R(a, e) ⇒ X(R(a, e)) X ↑ A(e) ⇒ X(A(e)) X

↑ (∃R.A)(a) ⇒ X X ↑ R(a, f ) ⇒ X A0 (∃ ) X ↑ A( f ) ⇒ X X ↑ (∃R.A)(a) ⇒ X

(∃ A+ )

where e is a new constant and f is a constant. ◦ EB : X X X (∼2B0 )  X  X ( B0 )  X

(∼1B0 )

and

↑ 1 B(b) ⇒ X (∼1B+ ) ↑∼1 B(b) ⇒ X ↑ 3 ∼1 B(b) ⇒ X (∼2B+ ) ↑∼2 B(b) ⇒ X ↑ 3 B(b) ⇒ X ( B+ ) ↑ B(b) ⇒ X

X X X X X X

↑ 1 B(b) ⇒ X (1 B(b)) ↑∼1 B(b) ⇒ X (∼1 B(b)) ↑ 3 ∼1 B(b) ⇒ X (3 ∼1 B(b)) ↑∼2 B(b) ⇒ X (∼2 B(b)) ↑ 3 B(b) ⇒ X (3 B(b)) ↑ B(b) ⇒ X (B(b))

4 R-Calculus for B22 -Valued DL

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⎡

X ↑ B1 (b) ⇒ X (B1 (b)) ⎢ X ↑ B2 (b) ⇒ X (B2 (b)) ⎢  ⎢ X (Y1 ) ↑ 1 B1 (b) ⇒ X (Y1 , 1 B1 (b)) ⎢ B+ ⎢ X (Y ) ↑ B (b) ⇒ X (Y , B (b)) ( ) ⎢  1 2 1 2 ⎣ X (Y2 ) ↑ B1 (b) ⇒ X (Y2 , B1 (b)) X (Y2 ) ↑ 1 B2 (b) ⇒ X (Y2 , 1 B2 (b))  X ↑ (B1  B2 )(b) ⇒ X ((B1  B2 )(b)) ⎧ X ↑ B1 (b) ⇒ X ⎪ ⎪ ⎪  ⎪ ⇒ X ⎪ ⎪ X ↑ B2 (b) ⎨ 1 X (Y1 ) ↑ B1 (b) ⇒ X (Y1 )   B0 ( ) ⎪ ⎪ X (Y1 ) ↑ B2 (b) ⇒ X (Y1 ) ⎪ ⎪ X (Y2 ) ↑ B1 (b) ⇒ X (Y2 ) ⎪ ⎪ ⎩ X (Y2 ) ↑ 1 B2 (b) ⇒ X (Y2 )  X ↑ (B1  B2 )(b) ⇒ X where Y1 = B1 (b) ∨ B2 (b), Y2 = Y1 , ∼1 B1 (b) ∨ B2 (b), and ⎡

X ↑ B1 (b) ⇒ X (B1 (b)) ⎢ X ↑ B2 (b) ⇒ X (B2 (b)) ⎢  ⎢ X (Y9 ) ↑ 3 ∼1 B1 (b) ⇒ X (Y9 , 3 ∼1 B1 (b)) ⎢ B+ ⎢ X (Y ) ↑ B (b) ⇒ X (Y , B (b)) ( ) ⎢  9 2 9 2 ⎣ X (Y10 ) ↑ B1 (b) ⇒ X (Y10 , B1 (b)) X (Y10 ) ↑ 3 ∼1 B2 (b) ⇒ X (Y10 , 3 ∼1 B2 (b))  X ↑ (B1  B2 )(b) ⇒ X ((B1  B2 )(b)) ⎧ ⎪ X ↑ B1 (b) ⇒ X ⎪ ⎪ ⎪ ⎪ X ↑ B2 (b) ⇒ X ⎪ ⎨ X (Y9 ) ↑ 3 ∼1 B1 (b) ⇒ X (Y9 ) B0 ( ) ⎪ ⎪ X (Y9 ) ↑ B2 (b) ⇒ X (Y9 ) ⎪ ⎪ ⎪ X (Y10 ) ↑ B1 (b) ⇒ X (Y10 ) ⎪ ⎩ X (Y10 ) ↑ 3 ∼1 B2 (b) ⇒ X (Y10 )  X ↑ (B1  B2 )(b) ⇒ X where Y9 = B1 (b) ∨ B2 (b); Y10 = Y9 , ∼2 B1 (b) ∨ B2 (b), and ⎡

X ↑ 1 R(b, f ) ⇒ X (1 R(b, f )) ⎢ X (1 R(b, f )) ↑ B( f ) ⇒ X (1 R(b, f ), B( f )) ⎢ B+ ⎣ X (Z ) ↑ 3 R(b, f ) ⇒ X (Z , 3 R(b, f )) (∀ ) 3 3 X (Z 3 ) ↑ 3 ∼1 B( f ) ⇒ X (Z 3 , 3 ∼1 B( f )) X ↑ (∀R.B)(b) ⇒ X ((∀R.B)(b)) ⎧ X ↑ 1 R(b, e) ⇒ X ⎪ ⎪ ⎨  1 e)) ↑ B(e) ⇒ X (1 R(b, e))  X ( R(b,  B0 X (Z 3 ) ↑ 3 R(b, e) ⇒ X (Z 3 ) (∀ ) ⎪ ⎪ ⎩ X (Z 3 ) ↑ 3 ∼1 B(e) ⇒ X (Z 3 )  X ↑ (∀R.B)(b) ⇒ X

4.4 3/22 -Multisequents

where

121

Z 3 =∼1 R(b, f ) ∨ B( f ), and Z 3 =∼1 R(b, e) ∨ B(e), ⎡  2 X ↑ R(b, e) ⇒ X (2 R(b, e)) ⎢ X ↑ B(e) ⇒ X (B(e)) ⎢  ⎢ X (Z 4 ) ↑ 1 R(b, e) ⇒ X (Z 4 , 1 R(b, e)) ⎢ B+   (∃ ) ⎢ ⎢  X (Z 4 ) ↑ B(e) ⇒ X (Z 4 , B(e)) ⎣ X (Z ) ↑ 2 R(b, e) ⇒ X (Z , 2 R(b, e)) 5 5 X (Z 5 ) ↑ 1 B(e) ⇒ X (Z 5 , 1 B(e)) X ↑ (∃R.B)(b) ⇒ X ((∃R.B)(b))

where Z 4 = R(b, e) ∨ B(e); Z 5 = Z 4 , ∼1 R(b, e) ∨ B(e), and ⎡  2 X ↑ R(b, f ) ⇒ X ⎢ X ↑ B( f ) ⇒ X ⎢   ⎢ X (Z ) ↑ 1 R(b, f ) ⇒ X (Z  ) 4 4 ⎢     B0 (∃ ) ⎢ ⎢  X (Z 4 ) ↑ B( f ) ⇒ X (Z 4 ) ⎣ X (Z  ) ↑ 2 R(b, f ) ⇒ X (Z  ) 5 5 X (Z 5 ) ↑ 1 B( f ) ⇒ X (Z 5 ) X ↑ (∃R.B)(b) ⇒ X

where Z 4 = R(b, f ) ∨ B( f ); Z 5 = Z 4 , ∼1 R(b, f ) ∨ B( f ). ◦ EC : X X X (∼C0 2 ) X X (C0 )  X

(∼C0 1 )

and

↑∼1 C(c) ⇒ X (∼C+ 1 ) ↑∼1 C(c) ⇒ X ↑ 1 C(c) ⇒ X (∼C+ 2 ) ↑∼2 C(c) ⇒ X 2  ↑ C(c) ⇒ X (C+ ) ↑ C(c) ⇒ X ⎡

X X X X X X

↑∼1 C(c) ⇒ X (1  C(c)) ↑∼1 C(c) ⇒ X (∼1 C(c)) ↑ 1 C(c) ⇒ X (1 C(c)) ↑∼2 C(c) ⇒ X (∼2 C(c)) ↑ 2 C(c) ⇒ X (2 C(c)) ↑ C(c) ⇒ X (C(c))

X ↑ C1 (c) ⇒ X (C1 (c)) ⎢ X ↑ C2 (c) ⇒ X (C2 (c)) ⎢   ⎢ X (Y3 ) ↑ 1 C1 (c) ⇒ X (Y3 , 1 C1 (c)) ⎢ C+ ⎢ X (Y ) ↑ C (c) ⇒ X (Y , C (c)) ( ) ⎢  3 2 3 2 ⎣ X (Y4 ) ↑ C1 (c) ⇒ X (Y3 , C1 (c)) X (Y4 ) ↑ 1 C2 (c) ⇒ X (Y3 , 1 C2 (c)) X ↑ (C1  C2 )(c) ⇒ X ((C1  C2 )(c)) ⎧ ⎪ X ↑ C1 (c) ⇒ X ⎪ ⎪ ⎪ X ↑ C2 (c) ⇒ X ⎪ ⎪ ⎨  X (Y3 ) ↑ 1 C1 (c) ⇒ X (Y3 ) (C0 ) ⎪ ⎪ X (Y3 ) ↑ C2 (c) ⇒ X (Y3 ) ⎪ ⎪ X (Y4 ) ↑ C1 (c) ⇒ X (Y4 ) ⎪ ⎪ ⎩ X (Y4 ) ↑ 1 C2 (c) ⇒ X (Y4 ) X ↑ (C1  C2 )(c) ⇒ X

4 R-Calculus for B22 -Valued DL

122

where Y3 = C1 (c) ∨ C2 (c), Y4 = Y3 , ∼2 C1 (c) ∨ C2 (c), and ⎡

X ↑ C1 (c) ⇒ X (C1 (c)) ⎢ X ↑ C2 (c) ⇒ X (C2 (c)) ⎢    ⎢ X (Y ) ↑∼1 C1 (c) ⇒ X (Y  , ∼1 C1 (c)) 9 9 ⎢ C+ ⎢ ( ) ⎢  X (Y9 ) ↑ C2 (c) ⇒ X (Y9 , C2 (c))   ⎣ X (Y10 ) ↑ C1 (c) ⇒ X (Y10 , C1 (c))     , ∼1 C2 (c)) X (Y10 ) ↑∼1 C2 (c) ⇒ X (Y10   X ↑ (C1  C2 )(c) ⇒ X ((C1  C2 )(c))  = Y9 , ∼1 C1 (c) ∨ C2 (c), and where Y9 = C1 (c) ∨ C2 (c); Y10

⎧  X ↑ C1 (c) ⇒ X ⎪ ⎪ ⎪ ⎪ X ↑ C2 (c) ⇒ X ⎪ ⎪ ⎨   X (Y9 ) ↑∼1 C1 (c) ⇒ X (Y9 ) C0     ( ) ⎪ ⎪

X (Y9 ) ↑ C2 (c) ⇒ X (Y9 ) ⎪ ⎪ X (Y10 ) ↑ C1 (c) ⇒ X (Y10 ) ⎪ ⎪ ⎩   )(C1 (c)) ↑∼1 C2 (c) ⇒ X (Y10 ) X (Y10   X ↑ (C1  C2 )(c) ⇒ X and

⎡

X ↑∼1 C1 (c) ⇒ X(∼1 C1 (c)) ⎢ X(∼1 C1 (c)) ↑∼1 C2 (c) ⇒ X(∼1 C1 (c), ∼1 C2 (c)) ⎢ ⎢ X(Y5 ) ↑ 2 C1 (c) ⇒ X(Y5 , 2 C1 (c)) ⎢ C+ ⎢ X(Y ) ↑ 3 C (c) ⇒ X(Y , 3 C (c)) (∼1  ) ⎢  5 2 5 2 ⎣ X(Y6 ) ↑ 3 C1 (c) ⇒ X(Y6 , 3 C1 (c)) X(Y6 ) ↑ 2 C2 (c) ⇒ X(Y6 , 2 C2 (c)) X ↑∼1 (C1  C2 )(c) ⇒ X(∼1 (C1  C2 )(c)) ⎧ X ↑∼1 C1 (c) ⇒ X ⎪ ⎪ ⎪ ⎪ X(∼1 C1 (c) ↑∼1 C2 (c) ⇒ X(∼1 C1 (c)) ⎪ ⎪ ⎨

X(Y5 ) ↑ 2 C1 (c) ⇒ X(Y5 ) 3 C0 (∼1  ) ⎪ ⎪

X(Y5 ) ↑ 3 C2 (c) ⇒ X(Y5 ) ⎪ ⎪ X(Y6 ) ↑ C1 (c) ⇒ X(Y6 ) ⎪ ⎪ ⎩ X(Y6 ) ↑ 2 C2 (c) ⇒ X(Y6 ) X ↑∼1 (C1  C2 )(c) ⇒ X

where Y5 =∼1 C1 (c), ∼1 C2 (c), Y6 = Y5 , ∼1 C1 (c)∨ ∼2 C2 (c), and

X ↑∼1 C1 (c) ⇒ X(∼1 C1 (c)) X(∼1 C1 (c)) ↑∼1 C2 (c) ⇒ X(∼1 C1 (c), ∼1 C2 (c)) X ↑∼1 (C1  C2 )(c) ⇒ X(∼1 (C1  C2 )(c)) X ↑∼1 C1 (c) ⇒ X (∼1 C0 ) X(∼1 C1 (c)) ↑∼1 C2 (c) ⇒ X(∼1 C1 (c)) X ↑∼1 (C1  C2 )(c) ⇒ X

(∼1 C+ )

4.4 3/22 -Multisequents

and

where

123

⎡

X ↑ 1 R(c, f ) ⇒ X (1 R(c, f )) ⎢ X (1 R(c, f )) ↑ C( f ) ⇒ X (1 R(c, f ), C( f )) ⎢ (∀C+ ) ⎣ X (Z 3 ) ↑ 2 R(c, f ) ⇒ X (Z 3 , 2 R(c, f )) X (Z 3 ) ↑∼1 C( f ) ⇒ X (Z 3 , ∼1 C( f ))  X ↑ (∀R.C)(c) ⇒ X (∀R.C)(c) ⎧ ⎪ X ↑ 1 R(c, e) ⇒ X ⎪ ⎨  1 e)) ↑ C(e) ⇒ X (1 R(c, e))

X ( R(c,  C0 (∀ ) ⎪ ⎪ X (Z 3 ) ↑ 2 R(c, e) ⇒ X (Z 3 ) ⎩ X (Z 3 ) ↑∼1 C(e) ⇒ X (Z 3 )  X ↑ (∀R.C)(c) ⇒ X Z 3 =∼2 R(c, f ) ∨ C( f ), and Z 3 =∼2 R(c, e) ∨ C(e), ⎡

X ↑ 3 R(c, e) ⇒ X (3 R(c, e)) ⎢ X ↑ C(e) ⇒ X (C(e)) ⎢    ⎢ X (Z ) ↑ 1 R(c, e) ⇒ X (Z  , 1 R(c, e)) 4 4 ⎢ C+ ⎢ (∃ ) ⎢  X (Z 4 ) ↑ C(e) ⇒ X (Z 4 , C(e)) ⎣ X (Z 5 ) ↑ 3 R(c, e) ⇒ X (Z 5 , 3 R(c, e)) X (Z 5 ) ↑ 1 C(e) ⇒ X (Z 5 , 1 C(e))  X ↑ (∃R.C)(c) ⇒ X ((∃R.C)(c)) where Z 4 = R(c, e) ∨ C(e); Z 5 = Z 4 , ∼2 R(c, e) ∨ C(e), and ⎧  3 X ↑ R(c, f ) ⇒ X ⎪ ⎪ ⎪ ⎪ X ↑ C( f ) ⇒ X ⎪ ⎪ ⎨   X (Z 4 ) ↑ 1 R(c, f ) ⇒ X (Z 4 ) C0   f ) ⇒ X (Z 4 ) (∃ ) ⎪ ⎪

X (Z 4 ) ↑ C( ⎪ 3 ⎪ X (Z 5 ) ↑ R(c, f ) ⇒ X (Z 5 ) ⎪ ⎪ ⎩ X (Z 5 ) ↑ 1 C( f ) ⇒ X (Z 5 )  X ↑ (∃R.C)(c) ⇒ X where Z 4 = R(c, f ) ∨ C( f ); Z 5 = Z 4 , ∼2 R(c, f ) ∨ C( f ), and 

X ↑ R(c, e) ⇒ X(R(c, e)) X ↑∼1 C(e) ⇒ X(∼1 C(e))

X ↑∼1 (∀R.C)(c) ⇒ X(∼1 (∀R.C)(c)) X ↑ R(c, f ) ⇒ X (∼1 ∀C0 ) X ↑∼1 C( f ) ⇒ X X ↑∼1 (∀R.C)(c) ⇒ X

(∼1 ∀C+ )

and

4 R-Calculus for B22 -Valued DL

124



X ↑∼1 R(c, f ) ⇒ X(∼1 R(c, f )) X(∼1 R(c, f )) ↑∼1 C( f ) ⇒ X(∼1 R(c, f ), ∼1 C( f )) X

↑∼1 (∃R.C)(c) ⇒ X(∼1 (∃R.C)(c)) X ↑∼1 R(c, e) ⇒ X (∼1 ∃C0 ) X(∼1 R(c, e)) ↑∼1 C(e) ⇒ X X ↑∼1 (∃R.C)(c) ⇒ X (∼1 ∃C+ )

Definition 4.4.3 A 3/22 -reduction δ = || ↑ (A(a), B(b), C(c)) ⇒   | |  3/22

3/22

is provable in Qt⊥ , denoted by t⊥ δ, if there is a sequence {δ1 , ..., δn } of 3/22 reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced 3/22 from the previous 3/22 -reductions by one of the deduction rules in Qt⊥ . Theorem 4.4.4 (Soundness and completeness theorem) For any reduction δ = ||  ↑ (A(a), B(b), C(c)) ⇒   | |  , 3/22

3/22

|=t⊥ δ iff t⊥ δ. 

4.5 4/22 -Multisequents 2

2

4/2 A 4/22 -multisequent ||| is L= -valid, denoted by |=4/2 |||, if there = is an interpretation I such that (i) I (A) = t for each A ∈ , (ii) I (B) =  for each B ∈ , (iii) I (C) =⊥ for each C ∈ , and (iv) I (D) = f for each D ∈ . Given sets , , ,  of literals, we (i) move each statement in  beginning with ∼1 , ∼2 ,  to , , , respectively; (ii) move each statement in  beginning with ∼1 , ∼2 ,  to , , , respectively, (iii) move each statement in  beginning with ∼1 , ∼2 ,  to , , , respectively, and (iv) move each statement in  beginning with ∼1 , ∼2 ,  to , , , respectively, so that , , ,  become sets of atoms.

4.5 4/22 -Multisequents

125

4/22

4.5.1 Deduction System L= 2

4/2 Deduction system L= contains the following axiom and deduction rules: • Axiom: 4/22  ∩  ∩  ∩  = ∅ (A= ) |||

where , , ,  are sets of atomic statements. • Deduction rules for unary logical connectives: (∼1A ) (∼C1 ) (∼2A ) (∼C2 ) ( A ) (C )

|, A(a)|| , ∼1 A(a)||| |||, C(c) ||, ∼1 C(c)| ||, A(a)| , ∼2 A(a)||| , C(c)||| ||, ∼2 C(c)| |||, A(a) , A(a)||| |, C(c)|| ||, C(c)|

(∼1B ) (∼1D ) (∼2B ) (∼2D ) ( B ) ( D )

, B(b)||| |, ∼1 B(b)|| ||, D(d)| |||, ∼1 D(d) |||, B(b) |, ∼2 B(b)|| |, D(d)|| |||, ∼2 D(d) ||, B(b)| |, B(b)|| , D(d)||| |||, D(d)

• Deduction rules for binary logical connective : ⎡

|, B1 (b)|| ⎢ |, B2 (b)|| ⎢  ⎢ , B1 (b)||| , A1 (a)||| ⎢ ( A ) , A2 (a)||| ( B ) ⎢ ⎢  |, B2 (b)|| ⎣ |, B1 (b)|| , (A1  A2 )(a)||| , B2 (b)||| |, (B1  B2 )(b)|| ⎡ ⎡ |||, D1 (d) ||, C1 (c)| ⎢ ||, C2 (c)| ⎢ |||, D2 (d) ⎢ ⎢ ⎢ , C1 (c)||| ⎢ |, D1 (d)|| ⎢ ⎢ D ⎢ ||, D (d),  (c)| ||, C ) (C ) ⎢ ( 2 2 ⎢ ⎢ ⎣ ||, C1 (c)| ⎣ ||, D1 (d)| , C2 (c)||| |, D2 (d)|| ||, (C1  C2 )(c)| |||, (D1  D2 )(d) • Deduction rules for binary logical connective :

4 R-Calculus for B22 -Valued DL

126

⎡ |, B1 (b)|| , A1 (a)||| ⎢ , A2 (a)||| ⎢ |, B2 (b)|| ⎢ ⎢ ⎢ ||, A1 (a)| ⎢ |||, B1 (b) ⎢ ⎢ A ⎢ |, A (a)|| B ⎢ |, B (b)|| ( ) ⎢  ( ) ⎢  2 2 ⎣ |, A1 (a)|| ⎣ |, B1 (b)|| ||, A2 (a)| |||, B2 (b) , (A  A )(a)||| |, (B1  B2 )(b)|| 1 2 ⎡ ||, C1 (c)| ⎢ ||, C2 (c)| ⎢  ⎢ |||, C1 (c) |||, D1 (d) ⎢ D (c)| ||, C |||, D2 (d) ) ( (C ) ⎢ 2 ⎢ ⎣ ||, C1 (c)| |||, (D1  D2 )(d) |||, C2 (c) ||, (C1  C2 )(c)| ⎡

• Deduction rules for quantifier ∀: ⎡

, A( f )||| ⎡ ⎢ |||, R(a, f ) , R(b, f )||| ⎢ ⎢ |, A( f )|| ⎢ |, B( f )|| ⎢ ⎢ B ⎣ ||, R(b, f )| (∀ A ) ⎢ ⎢  |, R(a, f )|| (∀ ) ⎣ ||, A( f )| |||, B( f ) ||, R(a, f )| |, (∀R.B)(b)|| , ⎡ (∀R.A)(a)||| , R(c, f )|||  ⎢ ||, C( f )| , R(d, e)||| ⎢ (∀C ) ⎣ |, R(c, f )|| (∀ D ) |||, D(e) |||, C( f ) |||, (∀R.D)(d) ||, (∀R.C)(c)| and

⎡

|, R(b, e)|| ⎢ |, B(e)|| ⎢  ⎢ , R(b, e)||| , R(a, e)||| ⎢ A B (∃ ) , A(e)||| (∃ ) ⎢ ⎢  |, B(e)|| ⎣ |, R(b, e)|| , (∃R.A)(a)||| , B(e)||| |, (∃R.B)(b)|| ⎡ ⎡ ||, R(c, e)| |||, R(d, f ) ⎢ ||, C(e)| ⎢ |||, D( f ) ⎢ ⎢ ⎢ , R(c, e)||| ⎢ |, R(d, f )|| ⎢ ⎢ C ⎢ ||, C(e)| D ⎢ ||, D( f )| (∃ ) ⎢  (∃ ) ⎢  ⎣ ||, R(c, e)| ⎣ ||, R(d, f )| , C(e)||| |, D( f )|| ||, (∃R.C)(c)| |||, (∃R.D)(d)

4.5 4/22 -Multisequents

127

where e is a constant and f is a new constant. 2

4/2 Definition 4.5.1 A 4/22 -multisequent ||| is provable in L= , denoted by 4/22 2 = |||, if there is a sequence {δ1 , ..., δn } of 4/2 -multisequents such that δn = |||, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the 4/22 . previous 4/22 -multisequents by one of the deduction rules in L=

Theorem 4.5.2 For any 4/22 -multisequent |||, 4/22

|==

4/22

||| iff =

|||. 

4/22

4.5.2 R-Calculus Q=

Given a 4/22 -multisequent ||| and four statements A(a), B(b), C(c), D(d), a 4/22 -reduction δ = ||| ↑ (A(a), B(b), C(c), D(d)) ⇒ , A (a)|, B  (b)|, C  (c)|, D  (d) 2

2

4/2 is Q4/2 δ, if = -valid, denoted by |==



2

4/2 -valid A(a) if , A(a)||| is L= λ otherwise;  4/22 -valid B(b) if , A (a)|, B(b)|| is L= B  (b) = λ otherwise;  4/22 -valid C(c) if , A (a)|, B  (b)|, C(c)| is L= C  (c) = λ otherwise;  4/22 -valid D(d) if , A (a)|, B  (b)|, C  (c)|, D(d) is L=  D (d) = λ otherwise. 

A (a) =

Let Y = ||| and Y = , A (a)||| Y = , A (a)|, B  (b)|| Y = , A (a)|, B  (b)|, C  (c)| Y(4) = , A (a)|, B  (b)|, C  (c)|, D  (d) Y(1 A(a)) = , A(a)||| Y(2 A(a)) = |, A(a)|| Y(3 A(a)) = ||, A(a)| Y(4 A(a)) = |||, A(a)

Y(A(a)) = , A(a)||| Y(B(b)) = |, B(b)|| Y(C(c)) = ||, C(c)| Y(D(d)) = |||, D(d) Y(1 R(a, b)) = , R(a, b)||| Y(2 R(a, b)) = |, R(a, b)|| Y(3 R(a, b)) = ||, R(a, b)| Y(4 R(a, b)) = |||, R(a, b).

4 R-Calculus for B22 -Valued DL

128 2

R-calculus Q4/2 consists of the following axioms and deduction rules: = • Axioms: (s(a)) ∩  ∩  ∩  = ∅ 4/22 (A A=+ ) Y ↑ s(a) ⇒ Y(s(a)) 4/22 (s(a)) ∩  ∩  ∩   = ∅ (A A=0 ) Y ↑ s(a) ⇒ Y  ∩ (t (b)) ∩  ∩  = ∅ 4/22 (A B=+ ) Y ↑ t (b) ⇒ Y(t (b)) 4/22  ∩ (t (b)) ∩  ∩   = ∅ (A B=0 ) Y ↑ t (b) ⇒ Y  ∩  ∩ (u(c)) ∩  = ∅ 4/22 (AC=+ ) Y ↑ u(c) ⇒ Y(u(c)) 4/22  ∩  ∩ (u(c)) ∩   = ∅ (AC=0 ) Y ↑ u(c) ⇒ Y  ∩  ∩  ∩ (v(d)) = ∅ 4/22 (A D=+ ) Y ↑ v(d) ⇒ Y(v(d)) 4/22  ∩  ∩  ∩ (v(d))  = ∅ (A D=0 ) Y ↑ v(d) ⇒ Y where , , ,  are sets of atoms, and s(a), t (b), u(c), v(d) are atoms. • Deduction rules consist of four parts E A , E B , EC , E D . ◦ EA: Y ↑ 2 A(a) ⇒ Y(2 A(a)) Y ↑ 2 A(a) ⇒ Y (∼1A0 ) Y ↑∼1 A(a) ⇒ Y(∼1 A(a)) Y ↑∼1 A(a) ⇒ Y Y ↑ 3 A(a) ⇒ Y(3 A(a)) Y ↑ 3 A(a) ⇒ Y A+ A0 (∼2 ) (∼2 ) Y ↑∼2 A(a) ⇒ Y(∼2 A(a)) Y ↑∼2 A(a) ⇒ Y 4 4 4 A+ Y ↑ A(a) ⇒ Y( A(a)) A0 Y ↑ A(a) ⇒ Y ( ) ( ) Y ↑ A(a) ⇒ Y(A(a)) Y ↑ A(a) ⇒ Y

(∼1A+ )

and



Y ↑ A1 (a) ⇒ Y(A1 (a)) Y ↑ A2 (a) ⇒ Y(A2 (a)) Y

↑ (A1  A2 )(a) ⇒ Y((A1  A2 )(a)) Y ↑ A1 (a) ⇒ Y ( A0 ) Y ↑ A2 (a) ⇒ Y Y ↑ (A1  A2 )(a) ⇒ Y

( A+ )

and

4.5 4/22 -Multisequents

129

⎡

Y ↑ A1 (a) ⇒ Y(A1 (a)) ⎢ Y(A1 (a)) ↑ A2 (a) ⇒ Y(A1 (a), A2 (a)) ⎢ ⎢ Y(Y5 ) ↑ 3 A1 (a) ⇒ Y(Y5 , 3 A1 (a)) ⎢ A+ ⎢ Y(Y ) ↑ 2 A (a) ⇒ Y(Y , 2 A (a)) ( ) ⎢  5 2 5 2 ⎣ Y(Y6 ) ↑ 2 A1 (a) ⇒ Y(Y6 , 2 A1 (a)) Y(Y6 ) ↑ 3 A2 (a) ⇒ Y(Y6 , 3 A2 (a)) Y ↑ (A1  A2 )(a) ⇒ Y((A1  A2 )(a)) ⎧ Y ↑ A1 (a) ⇒ Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y(A

1 (a)) ↑3 A2 (a) ⇒ Y(A1 (a)) ⎨ Y(Y1 ) ↑ A1 (a) ⇒ Y(Y5 ) 2 Y(Y ( A0 ) ⎪ 1 ) ↑ A2 (a) ⇒ Y(Y5 ) ⎪

⎪ ⎪ Y(Y2 ) ↑ 2 A1 (a) ⇒ Y(Y6 ) ⎪ ⎪ ⎩ Y(Y2 ) ↑ 3 A2 (a) ⇒ Y(Y6 ) Y ↑ (A1  A2 )(a) ⇒ Y where Y1 = A1 (a), A2 (a); Y2 = Y1 , ∼2 A1 (a)∨ ∼1 A2 (a), and ⎡

Y ↑ 4 R(a, f ) ⇒ Y(4 R(a, f )) ⎢ Y(4 R(a, e)) ↑ A( f ) ⇒ Y(4 R(a, e, A( f )) ⎢ ⎢ Y(Z 1 ) ↑ 2 R(a, f ) ⇒ Y(Z 1 , 2 R(a, f )) ⎢ A+ ⎢ Y(Z ) ↑ 2 A( f ) ⇒ Y(Z , 2 A( f )) (∀ ) ⎢  1 1 ⎣ Y(Z 2 ) ↑ 3 R(a, f ) ⇒ Y(Z 2 , 3 R(a, f )) Y(Z 2 ) ↑ 3 A( f ) ⇒ Y(Z 2 , 3 A( f )) Y ↑ ((∀R.A)(a)) ⇒ Y((∀R.A)(a)) ⎧ ⎪ Y ↑ 4 R(a, e) ⇒ Y ⎪ ⎪ ⎪ Y(4 R(a, e)) ↑ A(e) ⇒ Y(4 R(a, e)) ⎪

⎪ ⎨ Y(Z 1 ) ↑ 2 R(a, e) ⇒ Y(Z 1 ) A0 (∀ ) ⎪ ⎪ Y(Z 1 ) ↑ 2 A(e) ⇒ Y(Z 1 ) ⎪ ⎪ ⎪ Y(Z 2 ) ↑ 3 R(a, e) ⇒ Y(Z 2 ) ⎪ ⎩ Y(Z 2 ) ↑ 3 A(e) ⇒ Y(Z 2 ) Y ↑ ((∀R.A)(a)) ⇒ Y where Z 1 = R(a, f ), A( f ); Z 2 = Z 1 , ∼1 R(a, f )∨ ∼1 A( f ), and 

Y ↑ R(a, e) ⇒ Y(R(a, e)) Y ↑ A(e) ⇒ Y(A(e)) Y

↑ (∃R.A)(a) ⇒ Y((∃R.A)(a)) Y ↑ R(a, f ) ⇒ Y (∃ A0 ) Y ↑ A( f ) ⇒ Y Y ↑ (∃R.A)(a) ⇒ Y (∃ A+ )

where e is a new constant and f is a constant. ◦ EB :

4 R-Calculus for B22 -Valued DL

130

Y ↑ 1 B(b) ⇒ Y (1 B(b)) Y ↑ 1 B(b) ⇒ Y B0 (∼ ) 1 Y ↑∼1 B(b) ⇒ Y (∼1 B(b)) Y ↑∼1 B(b) ⇒ Y Y ↑ 4 B(b) ⇒ Y (4 B(b)) Y ↑ 4 B(b) ⇒ Y B+ B0 (∼ (∼2 )  ) 2 Y ↑∼2 B(b) ⇒ Y (∼2 B(b)) Y ↑∼2 B(b) ⇒ Y  3  3  3  ↑ B(b) ⇒ Y ( B(b)) Y B0 Y ↑ B(b) ⇒ Y ( ( B+ )  ) Y ↑ B(b) ⇒ Y (B(b)) Y ↑ B(b) ⇒ Y (∼1B+ )

and

⎡

Y ↑ B1 (b) ⇒ Y (B1 (b)) ⎢ Y ↑ B2 (b) ⇒ Y (B2 (b)) ⎢  ⎢ Y (Y3 ) ↑ 1 B1 (b) ⇒ Y (Y3 , 1 B1 (b)) ⎢ B+ ⎢ Y (Y ) ↑ B (b) ⇒ Y (Y , B (b)) ( ) ⎢  3 2 3 2 ⎣ Y (Y4 ) ↑ B1 (b) ⇒ Y (Y4 , B1 (b)) Y (Y4 ) ↑ 1 B2 (b) ⇒ Y (Y4 , 1 B2 (b))  Y ↑ (B1  B2 )(b) ⇒ Y ((B1  B2 )(b)) ⎧ Y ↑ B1 (b) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ B2 (b) ⇒ Y ⎪ ⎪ ⎨  Y (Y3 ) ↑ 1 B1 (b) ⇒ Y (Y3 )   B0 ( ) ⎪ ⎪

Y (Y3 ) ↑ B2 (b) ⇒ Y (Y3 ) ⎪ ⎪ Y (Y4 ) ↑ B1 (b) ⇒ Y (Y4 ) ⎪ ⎪ ⎩ Y (Y4 ) ↑ 1 B2 (b) ⇒ Y (Y4 ) Y ↑ (B1  B2 )(b) ⇒ Y

where Y3 = B1 (b) ∨ B2 (b); Y4 = Y3 , ∼1 B1 (b) ∨ B2 (b), and ⎡

Y ↑ B1 (b) ⇒ Y (B1 (b)) ⎢ Y ↑ B2 (b) ⇒ Y (B2 (b)) ⎢  ⎢ Y (Y5 ) ↑ 4 B1 (b) ⇒ Y (Y5 , 4 B1 (b)) ⎢ B+ ⎢ ( ) ⎢  Y (Y5 ) ↑ B2 (b) ⇒ Y (Y5 , B2 (b)) ⎣ Y (Y6 ) ↑ B1 (b) ⇒ Y (Y6 , B1 (b)) Y (Y6 ) ↑ 4 B2 (b) ⇒ Y (Y6 , 4 B2 (b))  Y ↑ (B1  B2 )(b) ⇒ Y ((B1  B2 )(b)) ⎧  Y ↑ B1 (b) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ B2 (b) ⇒ Y ⎪ ⎪ ⎨  Y (Y5 ) ↑ 4 B1 (b) ⇒ Y (Y5 ) B0   ( ) ⎪ ⎪

Y (Y5 ) ↑ B2 (b) ⇒ Y (Y5 ) ⎪ ⎪ Y (Y6 ) ↑ B1 (b) ⇒ Y (Y6 ) ⎪ ⎪ ⎩ Y (Y6 ) ↑ 4 B2 (b) ⇒ Y (Y6 ) Y ↑ (B1  B2 )(b) ⇒ Y where Y5 = B1 (b) ∨ B2 (b); Y6 = Y5 , ∼2 B1 (b) ∨ B2 (b), and

4.5 4/22 -Multisequents

131

⎡

Y ↑ R(b, f ) ⇒ Y (R(b, f )) ⎢ Y ↑ 2 B( f ) ⇒ Y (2 B( f )) ⎢ (∀ B+ ) ⎣ Y (Z 3 ) ↑ 3 R(b, f ) ⇒ Y (Z 3 , 3 R(b, f )) Y (Z 3 ) ↑ 4 B( f ) ⇒ Y (Z 3 , 4 B( f ))  Y ↑ ((∀R.B)(b)) ⇒ Y ((∀R.B)(b)) ⎧ Y ↑ R(b, e) ⇒ Y ⎪ ⎪ ⎨  2 

Y ↑  B(e)3 ⇒ Y   B0 (∀ ) ⎪ 3) ⎪ Y (Z 3 ) ↑ 4 R(b, e) ⇒ Y (Z ⎩  Y (Z 3 ) ↑ B(e) ⇒ Y (Z 3 ) Y ↑ ((∀R.B)(b)) ⇒ Y where

Z 3 = R(b, f ) ∨ B( f ), and Z 3 = R(b, e) ∨ B(e), ⎡

Y ↑ R(b, e) ⇒ Y (R(a, e)) ⎢ Y ↑ B(e) ⇒ Y (B(e)) ⎢  ⎢ Y (Z 4 ) ↑ R(b, e) ⇒ Y (Z 4 , R(b, e)) ⎢  1  1 (∃ B+ ) ⎢ ⎢  Y (Z 4 ) ↑ 1 B(e) ⇒ Y (Z 4 , B(e)) ⎣ Y (Z 5 ) ↑ R(b, e) ⇒ Y (Z 5 , 1 R(b, e)) Y (Z 5 ) ↑ B(e) ⇒ Y (Z 5 , B(e)) Y ↑ ((∃R.B)(b)) ⇒ Y ((∃R.B)(b)) ⎧ ⎪ Y ↑ R(b, f ) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ B( f ) ⇒ Y ⎪ ⎨ Y (Z 4 ) ↑ R(b, f ) ⇒ Y (Z 4 ) (∃ B0 ) ⎪ ⎪ Y (Z 4 ) ↑ 1 B( f ) ⇒ Y (Z 4 ) ⎪ ⎪ ⎪ Y (Z 5 ) ↑ 1 R(b, f ) ⇒ Y (Z 5 ) ⎪ ⎩ Y (Z 5 ) ↑ B( f ) ⇒ Y (Z 5 ) Y ↑ ((∃R.B)(b)) ⇒ Y Z 4 = R(b, e) ∨ B(e); Z 5 = Z 4 , R(b, e)∨ ∼1 B(e), and e is a new conZ 4 = R(b, f ) ∨ B( f ); Z 5 = Z 4 , R(b, f )∨ ∼1 B( f ), stant, f is a constant. ◦ EC : where

Y ↑ 4 C(c) ⇒ Y (4 C(c)) Y ↑ 4 C(c) ⇒ Y C0 (∼ ) 1 Y ↑∼1 C(c) ⇒ Y (∼1 C(c)) Y ↑∼1 C(c) ⇒ Y  1  1 ↑ C(c) ⇒ Y ( C(c)) Y Y ↑ 1 C(c) ⇒ Y C0 (∼ (∼C+ ) 2 ) 2 Y ↑∼2 C(c) ⇒ Y (∼2 C(c)) Y ↑∼2 C(c) ⇒ Y  2  2  2  ↑ C(c) ⇒ Y ( C(c)) Y C0 Y ↑ C(c) ⇒ Y ( (C+ )  ) Y ↑ C(c) ⇒ Y (C(c)) Y ↑ C(c) ⇒ Y (∼C+ 1 )

and

4 R-Calculus for B22 -Valued DL

132

⎡

Y ↑ C1 (c) ⇒ Y (C1 (c)) ⎢ Y ↑ C2 (c) ⇒ Y (C2 (c)) ⎢   ⎢ Y (Y7 ) ↑ 1 C1 (c) ⇒ Y (Y7 , 1 C1 (c)) ⎢ C+ ⎢ Y (Y ) ↑ C (c) ⇒ Y (Y , C (c)) ( ) ⎢  7 2 7 2 ⎣ Y (Y8 ) ↑ C1 (c) ⇒ Y (Y8 , C1 (c)) Y (Y8 ) ↑ 1 C2 (c) ⇒ Y (Y8 , 1 C2 (c))  Y ↑ (C1  C2 )(c) ⇒ Y ((C1  C2 )(c)) ⎧ Y ↑ C1 (c) ⇒ Y ⎪ ⎪ ⎪  ⎪ ⇒ Y ⎪ ⎪ Y ↑ C2 (c) ⎨ 1 Y (Y7 ) ↑ C1 (c) ⇒ Y (Y7 )   C0 ( ) ⎪ ⎪ Y (Y7 ) ↑ C2 (c) ⇒ Y (Y7 ) ⎪ ⎪ Y (Y8 ) ↑ C1 (c) ⇒ Y (Y8 ) ⎪ ⎪ ⎩ Y (Y8 ) ↑ 1 C2 (c) ⇒ Y (Y8 ) Y ↑ (C1  C2 )(c) ⇒ Y where Y7 = C1 (c) ∨ C2 (c); Y8 = Y7 , ∼2 C1 (c) ∨ C2 (c), and ⎡

Y ↑ C1 (c) ⇒ Y (C1 (c)) ⎢ Y ↑ C2 (c) ⇒ Y (C2 (c)) ⎢    ⎢ Y (Y ) ↑ 4 C1 (c) ⇒ Y (Y  , 4 C1 (c)) 7 7 ⎢ C+ ⎢ ( ) ⎢  Y (Y7 ) ↑ C2 (c) ⇒ Y (Y7 , C2 (c)) ⎣ Y (Y8 ) ↑ C1 (c) ⇒ Y (Y8 , C1 (c)) Y (Y8 ) ↑ 4 C2 (c) ⇒ Y (Y8 , 4 C2 (c))  Y ↑ (C1  C2 )(c) ⇒ Y ((C1  C2 )(c)) ⎧  Y ↑ C1 (c) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ C2 (c) ⇒ Y ⎪ ⎪ ⎨   Y (Y7 ) ↑ 3 C1 (c) ⇒ Y (Y7 ) C0     ( ) ⎪ ⎪ Y (Y7 ) ↑ C2 (c) ⇒ Y (Y7 ) ⎪ ⎪ Y (Y8 ) ↑ C1 (c) ⇒ Y (Y8 ) ⎪ ⎪ ⎩ Y (Y8 ) ↑ 3 C2 (c) ⇒ Y (Y8 ) Y ↑ (C1  C2 )(c) ⇒ Y where Y7 = C1 (c) ∨ C2 (c); Y8 = Y7 , ∼1 C1 (c) ∨ C2 (c), and ⎡

Y ↑ 1 R(c, f ) ⇒ Y (1 R(c, f )) ⎢ Y ↑ 3 C( f ) ⇒ Y (3 C( f )) ⎢ (∀C+ ) ⎣ Y (Z 6 ) ↑ 2 R(c, f ) ⇒ Y (Z 6 , 2 R(c, f )) Y (Z 6 ) ↑ 4 C( f ) ⇒ Y (Z 6 , 4 C( f ))  Y ↑ ((∀R.C)(c)) ⇒ Y ((∀R.C)(c)) ⎧ Y ↑ 1 R(c, e) ⇒ Y ⎪ ⎪ ⎨  3 ⇒ Y

Y ↑ C(e) 2 C0 Y (Z 6 ) ↑ R(c, e) ⇒ Y (Z 6 ) (∀ ) ⎪ ⎪ ⎩ Y (Z 6 ) ↑ 4 C(e) ⇒ Y (Z 6 )  Y ↑ ((∀R.C)(c)) ⇒ Y

4.5 4/22 -Multisequents

133

where Z 6 =∼2 R(c, f ) ∨ C( f ), and ⎡

Y ↑ 3 R(c, e) ⇒ Y (3 R(c, e)) ⎢ Y ↑ C(e) ⇒ Y (C(e)) ⎢   ⎢ Y (Z 7 ) ↑ 1 R(c, e) ⇒ Y (Z 7 , 1 R(c, e)) ⎢ C+ ⎢ Y (Z ) ↑ C(e) ⇒ Y (Z , , C(e)) (∃ ) ⎢  7 7 ⎣ Y (Z 8 ) ↑ 3 R(c, e) ⇒ Y (Z 8 , 3 R(c, e)) Y (Z 8 ) ↑ 1 C(e) ⇒ Y (Z 8 , 1 C(e))  Y ↑ ((∃R.C)(c)) ⇒ Y ((∃R.C)(c)) ⎧ Y ↑ 3 R(c, f ) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ C( f ) ⇒ Y ⎪ ⎪ ⎨   Y (Z 7 ) ↑ 1 R(c, f ) ⇒ Y (Z 7 )   C0 f ) ⇒ Y (Z 7 ) (∃ ) ⎪ ⎪

Y (Z 7 ) ↑ C( ⎪ 3 ⎪ Y (Z 8 ) ↑ R(c, f ) ⇒ Y (Z 8 ) ⎪ ⎪ ⎩ Y (Z 8 ) ↑ 1 C( f ) ⇒ Y (Z 8 )  Y ↑ ((∃R.C)(c)) ⇒ Y Z 7 = R(c, e) ∨ C(e); Z 8 = Z 7 , ∼2 R(c, e) ∨ C(e), and e is a new conZ 7 = R(c, f ) ∨ C( f ); Z 8 = Z 7 , ∼2 R(c, f ) ∨ C( f ), stant, f is a constant. ◦ ED :

where

Y ↑ 3 D(d) ⇒ Y (3 D(d)) Y ↑ 3 D(d) ⇒ Y (∼1D0 )   ↑∼1 D(d) ⇒ Y (∼1 D(d)) Y ↑∼1 D(d) ⇒ Y  2  2 ↑ D(d) ⇒ Y ( D(d)) Y Y ↑ 2 D(d) ⇒ Y (∼2D0 )  (∼2D+ )   Y ↑∼2 D(d) ⇒ Y (∼2 D(d)) Y ↑∼2 D(d) ⇒ Y  1  1 Y ↑ D(d) ⇒ Y ( D(d)) Y ↑ 1 D(d) ⇒ Y ( D0 )  ( D+ )   Y ↑ D(d) ⇒ Y (D(d)) Y ↑ D(d) ⇒ Y (∼1D+ )

and

Y

⎡

Y ↑ D1 (d) ⇒ Y (D1 (d)) ⎢ Y (D1 (d)) ↑ D2 (d) ⇒ Y (D1 (d), D2 (d)) ⎢   ⎢ Y (Y9 ) ↑ 2 D1 (d) ⇒ Y (Y9 , 2 D1 (d)) ⎢ D+ ⎢ Y (Y ) ↑ 3 D (d) ⇒ Y (Y , 3 D (d)) ( ) ⎢  9 2 9 2 ⎣ Y (Y10 ) ↑ 3 D1 (d) ⇒ Y (Y10 , 3 D1 (d)) Y (Y10 ) ↑ 2 D2 (d) ⇒ Y (Y10 , 2 D2 (d))  Y ↑ (D1  D2 )(d) ⇒ Y ((D1  D2 )(d)) ⎧ Y ↑ D1 (d) ⇒ Y ⎪ ⎪ ⎪   ⎪ ⎪ 1 (d)) ↑ D2 (d) ⇒ Y (D1 (d)) ⎪Y

(D ⎨  2  Y (Y9 ) ↑ D1 (d) ⇒ Y (Y9 )  3  Y ( D0 ) ⎪ ⎪  (Y9 ) ↑ 3D2 (d) ⇒ Y (Y9 ) ⎪ ⎪ Y (Y10 ) ↑ D1 (d) ⇒ Y (Y10 ) ⎪ ⎪ ⎩ Y (Y10 ) ↑ 2 D2 (d) ⇒ Y (Y10 )  Y ↑ (D1  D2 )(d) ⇒ Y

4 R-Calculus for B22 -Valued DL

134

where Y9 = D1 (d), D2 (d); Y10 = Y9 , ∼2 D1 (d)∨ ∼1 D2 (d), and

Y ↑ D1 (d) ⇒ Y (D1 (d)) ( ) Y (D1 (d)) ↑ D2 (d) ⇒ Y (D1 (d), D2 (d))   Y ↑ (D1  D2 )(d)⇒ Y ((D1  D2 )(d)) Y ↑ D1 (d) ⇒ Y ( D0 ) Y (D1 (d)) ↑ D2 (d) ⇒ Y (D1 (d)) Y ↑ (D1  D2 )(d) ⇒ Y D+

and



Y ↑ 1 R(d, e) ⇒ Y (1 R(d, e)) (∀ ) Y ↑ D(e) ⇒ Y (D(e))  ⇒ Y ((∀R.D)(d))

Y ↑ ((∀R.D)(d)) 1 Y ↑ R(d, f ) ⇒ Y (∀ D0 ) Y ↑ D( f ) ⇒ Y Y ↑ ((∀R.D)(d)) ⇒ Y D+

and

⎡

Y ↑ 4 R(d, f ) ⇒ Y (4 R(d, f )) ⎢ Y ↑ D( f ) ⇒ Y (D( f )) ⎢   ⎢ Y (Y11 ) ↑ 2 R(d, f ) ⇒ Y (Y11 , 3 R(d, f )) ⎢ D+ ⎢ Y (Y ) ↑ 3 D( f ) ⇒ Y (Y , 3 D( f )) (∃ ) ⎢  11 11 ⎣ Y (Y12 ) ↑ 3 R(d, f ) ⇒ Y (Y12 , 3 R(d, f )) Y (Y12 ) ↑ 2 D( f ) ⇒ Y (Y12 , 2 D( f ))  Y ↑ (∃R.D)(d) ⇒ Y ((∃R.D)(d)) ⎧ Y ↑ 4 R(d, e) ⇒ Y ⎪ ⎪ ⎪  ⎪ ⇒ Y ⎪ ⎪Y

↑ D(e) ⎨   ) Y (Y11 ) ↑ 2 R(d, e) ⇒ Y (Y11    3  D0 (∃ ) ⎪ ⎪ Y (Y11 ) ↑ D(e) ⇒ Y (Y11 ) ⎪   ⎪ ) ↑ 3 R(d, e) ⇒ Y (Y12 ) ⎪ Y (Y12 ⎪ ⎩    2  Y (Y12 ) ↑ D(e) ⇒ Y (Y12 ) Y ↑ ((∃R.D)(d)) ⇒ Y

Y11 = R(d, f ) ∨ D( f ); Y12 = Z 11 , ∼1 R(d, f )∨ ∼1 D( f ), and e is a new    = R(d, e) ∨ D(e); Y12 = Z 11 , ∼1 R(d, e)∨ ∼1 D(e), Y11 constant, f is a constant.

where

Definition 4.5.3 A 4/22 -reduction δ = Y ↑ (A(a), B(b), C(c), D(d)) ⇒ Y(4) is 2 4/22 provable in Q4/2 δ, if there is a sequence {δ1 , ..., δn } of 4/22 = , denoted by = reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2 4/22 -reductions by one of the deduction rules in Q4/2 = .

4.6 Conclusions

135

Theorem 4.5.4 (Soundness and completeness theorem) For any 4/22 -reduction δ = Y ↑ (A(a), B(b), C(c), D(d)) ⇒ Y(4) , 4/22

 =

4/22

δ iff |==

δ. 

4.6 Conclusions 1/22

1/22

2/22

1/22

There are transformations τ1 , τ2 , τ3 from Lt /Qt to Lt /Qt , from 2/22 1/22 3/22 3/22 3/22 3/22 3/22 3/22 Lt /Qt to Lt⊥ /Qt⊥ , and from Lt⊥ /Qt⊥ to Lt⊥ /Qt⊥ , respectively, such that 1/22 2/22 1/22 2/22 τ1 (Lt ) = Lt τ1 (Qt ) = Qt , 2/22 3/22 2/22 3/22 τ2 (Lt ) = Lt⊥ τ2 (Qt ) = Qt⊥ , 3/22 3/22 = = τ3 (Lt⊥ ) = L4/2 2 τ3 (Qt⊥ ) = Q4/22 , where

and

and

τ1 (∪ ∼− 1 ) = 2/22 τ1 (∼−  ∪ − ) = 2/22 2 τ1 (∼1 A1 (a)) = B1 (a) τ1 (∼2 A1 (a)) = B1 (a) τ1 (A1 (a)) =∼2 B1 (a) 2

2

τ2 ( 2/2 ) =  3/2 3/22 τ2 (∼− 1 2/22 ) =  2 2 2/2 τ2 (∼− ) =  3/2 2  τ2 (− 2/22 ) =∼− 3/22 τ2 (2/22 ) = 3/22 τ2 (∼− 1 2/22 ) = 3/22 − τ2 (∼− 2 2/22 ) =∼1 3/22 τ3 (3/22 ) = 4/22 τ3 (∼− 1 3/22 ) = 4/22 τ3 (∼− 2 3/22 ) = 4/22 τ3 (− 3/22 ) = 4/22 τ3 (∼− 1 3/22 ) = 4/22 τ3 (∼− 2 3/22 ) = 4/22 τ3 (− 3/22 ) = 4/22 τ3 (∼− 1 3/22 ) = 4/22 τ3 (∼− 2 3/22 ) = 4/22 τ3 (− 3/22 ) = 4/22

τ2 (∼1 A1 (a)) = B1 (a) τ2 (∼2 A1 (a)) = C1 (a) τ2 (A1 (a)) =∼ C1 (a) τ2 (∼1 B1 (b)) = C1 (b) τ2 (∼2 B1 (b)) =∼1 C1 (b)

τ3 (∼− 1 3/22 ) = 4/22 τ3 (∼− 2 3/22 ) = 4/22 τ3 (− 3/22 ) = 4/22 τ3 (∼1 A1 (a)) = B1 (a) τ3 (B1 (b)) = C1 (b) τ3 (∼2 C1 (c)) = D1 (c) τ3 (D1 (d)) = A1 (d).

136

4 R-Calculus for B22 -Valued DL

References Arieli, O., Avron, A.J.: Reasoning with logical bilattices. Logic Lang. Inf. 5, 25–63 (1996) Arieli, O., Avron, A.: Bilattices and paraconsistency. Front. Paraconsistent Logic Stud. Logic Comput. 8, 11–27 (2000) Belnap, N.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (Eds.) Modern Uses of Multiplevalued Logic, pp. 8–37. D. Reidel (1997) Bergstra, J.A., Ponse, A.: Process algebra with four-valued logic. J. Appl. Non-Class. Logics 10, 27–53 (2000) Font, J.M.: Belnap’s four-valued logic and De Morgan lattices. Logic J. I.G.P.L. 5, 413–440 (1997) Ginsberg, M.L.: Multi-valued logics: a uniform approach to reasoning in artificial intelligence. Comput. Intell. 4, 256–316 (1998) Gottwald, S.: A Treatise on Many-Valued Logics. (Studies in Logic and Computation, vol. 9). Research Studies Press Ltd., Baldock (2001) Li, W.: Mathematical Logic, Foundations for Information Science, Progress in Computer Science and Applied Logic, vol.25, Birkhäuser (2010) Ponse, A., van der Zwaag, M.B.: A generalization of ACP using Belnap’s logic. Electron Notes Theor. Comput. Sci. 162, 287–293 (2006) Pynko, A.P.: Characterizing Belnap’s logic via De Morgan’s laws. Math. Logic Quarter. 41, 442–454 (1995) Takeuti, G.: Proof Theory. North-Holland Pub, Co (1975) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, 2nd edn., pp. 249–295. Kluwer, Dordrecht (2001) Zach, R.: Proof theory of finite-valued logics, Tech. Report TUW-E185.2-Z.1-93

Chapter 5

R-Calculi for Post L4 -Valued DL

∗ ∗ ∗ ∗ Ri/4 Si/4 Mi/4 N i/4 i/4 i/4 i/4 i/4 L∗ K∗ Q∗ P∗ =t

t t A 1/4-multisequent  is M1/4 /N1/4 -valid, denoted by |=t1/4 / |=1/4 , if for any interpretation I , there is a statement A ∈  such that I (A) = t/I (A) = t. 1/4 1/4 1/4 1/4 A 1/4-multisequent  is Lt /Kt -valid, denoted by |=t / |==t , if there is an interpretation I such that for each statement A ∈ , I (A) = t/I (A) = t. =t t t /N2/4 -valid, denoted by |=t A 2/4-multisequent | is M2/4 2/4 / |=2/4 |, if for any interpretation I , either for some statement A ∈ , I (A) = t/I (A)=t, or for some statement B ∈ , I (B) = /I (B)=. 2/4 2/4 2/4 2/4 A 2/4-multisequent | is Lt /Kt -valid, denoted by |==t / |=t |, if there is an interpretation I such that for each statement A ∈ , I (A) = t/I (A) = t, and for each statement B ∈ , I (B) = /I (B) = . =t⊥ t⊥ t⊥ /N3/4 -valid, denoted by |=t⊥ A 3/4-multisequent || is M3/4 3/4 / |=3/4 ||, if for any interpretation I , either for some statement A ∈ , I (A) = t/I (A)=t, or for some statement B ∈ , I (B) = /I (B)=, or for some statement C ∈ , I (C) =⊥ /I (C)= ⊥. 3/4 3/4 3/4 3/4 A 3/4-multisequent || is Lt⊥ /Kt⊥ -valid, denoted by |==t⊥ / |=t⊥ ||, if there is an interpretation I such that for each statement A ∈ , I (A) = t/I (A) = t, for each statement B ∈ , I (B) = /I (B) = , and for each statement C ∈ , I (C) =⊥ /I (C) =⊥. = = = /N4/4 -valid, denoted by |== A 4/4-multisequent ||| is M4/4 4/4 / |=4/4 |||, if for any interpretation I , either for some statement A ∈ , I (A) = t/I (A)=t, or for some statement B ∈ , I (B) = /I (B)=, or for some statement C ∈ , I (C) =⊥ /I (C)= ⊥, or for some statement D ∈ , I (D) = f/I (D)=f.

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_5

137

138

5 R-Calculi for Post L4 -Valued DL 4/4

4/4 4/4 A 4/4-multisequent ||| is L= /K= -valid, denoted by |== / |=4/4 = |||, if there is an interpretation I such that for each statement A ∈ , I (A) = t/I (A) = t, for each statement B ∈ , I (B) = /I (B) = , for each statement C ∈ , I (C) =⊥ /I (C) =⊥, and for each statement D ∈ , I (D) = f/I (D) = f. There are sound and complete deduction systems (Arieli and Avron 1996; 2000; Belnap 1977; Bochvar 1938; Font 1997; Post 1920, 1921; Pynko 1995; Urquhart 2001; Zach 2023). 1/4 1/4 1/4 1/2 ∗ ∗ ∗ ∗ , L∗ , N1/4 , K∗ for M1/4 -valid, L∗ -valid, N1/4 -valid and K∗ -valid (i) M1/4 multisequents, respectively; 2/4 2/4 2/4 2/2 ∗1 ∗2 ∗1 ∗2 ∗1 ∗2 ∗1 ∗2 , L∗1 ∗2 , N2/4 , K∗1 ∗2 for M2/4 -valid, L∗1 ∗2 -valid, N2/4 -valid and K∗1 ∗2 (ii) M2/4 valid multisequents, respectively; 3/4 3/4 3/4 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 , L∗1 ∗2 ∗3 , N3/4 , K∗1 ∗2 ∗3 for M3/4 -valid, L∗1 ∗2 ∗3 -valid, N3/4 -valid (iii) M3/4 3/4 and K∗1 ∗2 ∗3 -valid multisequents, respectively; = = = = 4/4 4/4 4/4 4/2 , L= , N4/4 , K= for M4/4 -valid, L= -valid, N4/4 -valid and K= -valid (iv) M4/4 multisequents, respectively. Given a 1/4-multisequent  and a statement A ∈ , a 1/4-reduction |A ⇒ [A ] =t t /St1/4 -valid, denoted by |=t1/4 / |=1/4 |A ⇒ [A ], if is R1/4 



t t A if [A] is M1/4 /N1/4 -valid λ otherwise.

A =

Given a 1/4-multisequent  and a statement A, a 1/4-reduction |A ⇒ (A ) is 1/4 1/4 denoted by |==t / |=t |A ⇒ (A ), if

1/4 1/4 Qt /Pt -valid,

A =



1/4

1/4

A if (A) is Lt /Kt -valid λ otherwise.

Given a 2/4-multisequent | and two statements A ∈  and B ∈ , a 2/4t t /St reduction | ↑ (A, B) ⇒ [A ]|[B  ] is R2/4 2/4 -valid, denoted by |=2/4 / =t |=2/4 | ↑ (A, B) ⇒ [A ]|[B  ], if 

A λ  B  B = λ A =

t t /N2/4 -valid if [A]| is M2/4 otherwise. t t /N2/4 -valid if [A ]|[B] is M2/4 otherwise.

Given a 2/4-multisequent | and two statements A, B, a 2/4-reduction δ = =t t t /St | ↑ (A, B) ⇒ (A )|(B  ) is R2/4 2/4 -valid, denoted by |=2/4 / |=2/4 δ, if

5 R-Calculi for Post L4 -Valued DL



A λ  B  B = λ A =

139 2/4

2/4

if (A)| is Lt /Kt -valid otherwise. 2/4 2/4 if (A )|(B) is Lt /Kt -valid otherwise.

Given a 3/4-multisequent || and two statements A ∈ , B ∈ , C ∈ , a t⊥ /St⊥ 3/4-reduction δ = || ↑ (A, B, C) ⇒ [A ]|[B  ]|[C  ] is R3/4 3/4 -valid, =t⊥ t⊥ denoted by |=3/4 / |=3/4 δ, if 

A λ  B B = λ C  C = λ 

A =

t⊥ t⊥ /N3/4 -valid if [A]|| is M3/4 otherwise; t⊥ t⊥ /N3/4 -valid if [A ]|[B]| is M3/4 otherwise; t⊥ t⊥ if [A ]|[B]|[C] is M3/4 /N3/4 -valid otherwise.

Given a 3/4-multisequent || and two statements A, B, C, a 3/4-reduction 3/4 3/4 δ = || ↑ (A, B, C) ⇒ (A )|(B  )|(C  ) is Qt⊥ /Pt⊥ -valid, denoted by 3/4 3/4 |==t⊥ / |=t⊥ δ, if 

A λ  B B = λ  C  C = λ 

A =

3/4

3/4

if (A)|| is Lt⊥ /Kt⊥ -valid otherwise; 3/4 3/4 if (A )|(B)| is Lt⊥ /Kt⊥ -valid otherwise; 3/4 3/4 if (A )|(B  )|(C) is Lt⊥ /Kt⊥ -valid otherwise.

Given a 4/4-multisequent ||| and two statements A ∈ , B ∈ , C ∈ , D ∈ , a 4/4-reduction δ = ||| ↑ (A, B, C, D) ⇒ [A ]|[B  ]|[C  ]| = = = /S= [D  ] is R4/4 4/4 -valid, denoted by |=4/4 / |=4/4 δ, if 

A λ B  B = λ  C C = λ D  D = λ 

A =

= = /N4/4 -valid if [A]||| is M4/4 otherwise; = = /N4/4 -valid if [A ]|[B]|| is M4/4 otherwise; = = /N4/4 -valid if [A ]|[B  ]|[C]| is M4/4 otherwise; = = /N4/4 -valid if [A ]|[B  ]|[C  ]|[D] is M4/4 otherwise.

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5 R-Calculi for Post L4 -Valued DL

Given a 4/4-multisequent ||| and statements A, B, C, D, a 4/4-reduction 4/4 δ = ||| ↑ (A, B, C, D) ⇒ (A )|(B  )|(C  )|(D  ) is Q4/4 = /P= -valid, 4/4 denoted by |== / |=4/4 = δ, if 

A λ  B B = λ  C C = λ D  D = λ A =

4/4 4/4 /K= -valid if (A)||| is L= otherwise; 4/4 4/4 /K= -valid if (A )|(B)|| is L= otherwise; 4/4 4/4 /K= -valid if (A )|(B  )|(C)| is L= otherwise; 4/4 4/4 /K= -valid if (A )|(B  )|(C  )|(D) is L= otherwise.

There are sound and complete R-calculi (Alchourrón et al. 1985; Baader et al. 2003; Fermé and Hansson 2011; Ginsberg 1988; Gottwald 2001; Li 2007; Li and Sui 2013; Ponse and van der Zwaag 2006). 1/4 1/4 1/4 1/2 ∗ ∗ , Q∗ , S∗1/4 , P∗ for R1/4 -valid, Q∗ -valid, S∗1/4 -valid and P∗ -valid mul(i) R1/4 tisequents, respectively; 2/4 2/4 2/4 2/2 ∗1 ∗2 ∗1 ∗2 1 ∗2 1 ∗2 , Q∗1 ∗2 , S∗2/4 , P∗1 ∗2 for R2/4 -valid, Q∗1 ∗2 -valid, S∗2/4 -valid and P∗1 ∗2 (ii) R2/4 valid multisequents, respectively; 3/4 3/4 3/4 ∗1 ∗2 ∗3 ∗1 ∗2 ∗3 1 ∗2 ∗3 1 ∗2 ∗3 , Q∗1 ∗2 ∗3 , S∗3/4 , P∗1 ∗2 ∗3 for R3/4 -valid, Q∗1 ∗2 ∗3 -valid, S∗3/4 -valid (iii) R3/4 3/4 and P∗1 ∗2 ∗3 -valid multisequents, respectively; = = = = 4/4 4/4 4/2 , Q4/4 (iv) R4/4 = , S4/4 , P= for R4/4 -valid, Q= -valid, S4/4 -valid and P= -valid multisequents, respectively. We consider the following deduction systems and R-calculi: t t t⊥ , N2/3 , N3/3 , N= Deduction system N1/4 t t t⊥ R-calculus S1/4 , S2/4 , S3/4 , S=

and

t N1/4 t N2/4 t⊥ N3/4 = N4/4

, ∼1 , ∼2 ,  , |,  ||, ∼1  |||

5.1 Post L4 -Valued Description Logic

141

5.1 Post L4 -Valued Description Logic Let L4 = ({t, , ⊥, f], ∼, ∩, ∪, →), where t  ⊥ f ∪ t  ⊥ f

∼ ∼2 ∼3  ⊥ f ⊥ f t f t  t  ⊥ t⊥ f tt t t t t⊥⊥ t⊥ f

∩ t ⊥f t t ⊥f ⊥f ⊥⊥⊥⊥f f f f ff → t ⊥f t t t tt  t tt ⊥ ⊥tt f f ⊥t

The logical language of Post L4 -valued description logic contains the following symbols: • atomic concepts: S0 , S1 , . . . ; • roles: R0 , R1 , . . . ; • concept constructors: ∼, , , ∀, ∃. Concepts are defined inductively as follows: C:: = S| ∼ C|C1 C2 |C1  C2 |∀R.C|∃R.C, where S is an atomic concept, and R is a role. Statements are defined as follows: ϕ:: = C(a)|R(a, b)| ∼ ϕ. A model M is a pair (U, I ), where U is a non-empty set, and I is an interpretation such that ◦ for any atomic concept S, I (S) : U → L4 ; ◦ for any role R, I (R) : U 2 → L4 . Given an atomic concept S and a role R, we define concepts ∼ S and roles ∼ R as follows: for any x, y ∈ U, ∼ S(x) ∼2 S(x) ∼3 S(x) S(x) 2 R(x, y) ∼ R(x, y) ∼ R(x, y) ∼3 R(x, y) t  ⊥ f ⊥ f t  f t  ⊥ f  ⊥ f

142

5 R-Calculi for Post L4 -Valued DL

The interpretation C I of a concept C is a function from U to L4 such that for any x ∈ U, ⎡ I (S)(x) if C = S ⎢ ∼ (C I )(x) if C =∼ C1 I C (x) = ⎢ ⎣ C1I (x) ∩ C2I (x) if C = C1 C2 C1I (x) ∪ C2I (x) if C = C1  C2 and (∀R.C)(a) ≡ Ab(R(a, b)∨C(b)∨(∼ R(a, b)∧ ∼ C(b)) ∨(∼2 R(a, b)∧ ∼ C(b))∨(∼2 R(a, b)∧ ∼2 C(b))) ∼ ∀R.C(a) ≡ Ab((R(a, b)∧ ∼ C(b))∨(∼ R(a, b)∧ ∼2 C(b)) ∨(∼2 R(a, b)∧ ∼3 C(b))) ∼2 ∀R.C(a) ≡ Ab((R(a, b)∧ ∼2 C(b))∨(∼ R(a, b)∧ ∼3 C(b))) ∼3 ∀R.C(a) ≡ Eb(R(a, b)∧ ∼3 C(b)) ∃R.C(a) ≡ Eb(R(a, b)∧C(b)) ∼ ∃R.C(a) ≡ Eb((∼ R(a, b)∧ ∼ C(b))∨(R(a, b)∧ ∼ C(b)) ∨(∼ R(a, b)∧C(b))) ∼ ∃R.C(a) ≡ Eb((∼2 R(a, b)∧ ∼2 C(b))∨(∼ R(a, b)∧ ∼2 C(b)) 2

∨(R(a, b)∧ ∼2 C(b))∨(∼2 R(a, b)∧ ∼ C(b)) ∨(∼2 R(a, b)∧C(b))) ∼3 ∃R.C(a) ≡ Ab(∼3 R(a, b)∨ ∼3 C(b)). ◦t

◦t

◦

◦

◦⊥

◦⊥

◦f

◦f

∀

∃

5.1 Post L4 -Valued Description Logic

For v(C(a)) = t, we have the following equivalences: (C1 C2 )(a) ≡ C1 (a)∧C2 (a) ∼ (C1 (a) C2 (a)) ≡ (∼ C1 (a)∧ ∼ C2 (a))∨(∼ C1 (a)∧C2 (a)) ∨(C1 (a)∧ ∼ C2 (a)) ∼2 (C1 C2 )(a) ≡ (∼2 C1 (a)∧ ∼2 C2 (a))∨(∼2 C1 (a)∧ ∼ C2 (a)) ∨(∼2 C1 (a)∧C2 (a))∨(∼ C1 (a)∧ ∼2 C2 (a)) ∨(C1 (a)∧ ∼2 C2 (a)) ∼3 (C1 C2 )(a) ≡ ∼3 C1 (a)∨ ∼3 C2 (a); (C1 ∨C2 )(a) ≡ C1 (a)∨C2 (a) ∼ (C1  C2 )(a) ≡ (∼ C1 (a)∧ ∼ C2 (a))∨(∼ C1 (a)∧ ∼2 C2 (a)) ∨(∼ C1 (a)∧ ∼3 C2 (a))∨(∼2 C1 (a)∧ ∼ C2 (a)) ∨(∼3 C1 (a)∧ ∼ C2 (a)) ∼2 (C1  C2 )(a) ≡ (∼2 C1 (a)∧ ∼2 C2 (a))∨(∼2 C1 (a)∧ ∼3 C2 (a)) ∨(∼3 C1 (a)∧ ∼2 C2 (a)) ∼3 (C1  C2 )(a) ≡ ∼3 C1 (a)∧ ∼3 C2 (a) and (∀R.C)(a) ≡ Ae(∼3 R(a, e)∨C(a)∨(∼ R(a, e)∧ ∼ C(e)) ∨(∼2 R(a, e)∧ ∼ C(e))∨(∼2 R(a, e)∧ ∼2 C(e)) ∼ (∀R.C)(a) ≡ Ae(R(a, e)∧ ∼ C(e))∨(∼ R(a, e)∧ ∼2 C(e)) ∨(∼2 R(a, e)∧ ∼3 C(e)) ∼2 (∀R.C)(a) ≡ Ae(R(a, e)∧ ∼2 C(e))∨(∼ R(a, e)∧ ∼3 C(e)) ∼3 (∀R.C)(a) ≡ E f (R(a, f )∧ ∼3 C( f )) (∃R.C)(a) ≡ E f (R(a, f )∧C( f )) ∼ (∃R.C)(a) ≡ E f ((∼ R(a, f )∧ ∼ C( f ))∨(R(a, f )∧ ∼ C( f )) ∨(∼ R(a, f )∧C( f ))) ∼2 (∃R.C)(a) ≡ E f ((∼2 R(a, f )∧ ∼2 C( f ))∨(∼ R(a, f )∧ ∼2 C( f )) ∨(R(a, f )∧ ∼2 C( f )))∨(∼2 R(a, f )∧ ∼ C( f )) ∨(∼2 R(a, f )∧C( f ))) ∼ (∃R.C)(a) ≡ Ae(∼3 R(a, e)∨ ∼3 C(e)). 3

143

144

5 R-Calculi for Post L4 -Valued DL

For v(C(a)) = t, we have the following equivalences: (C1 C2 )(a) ≡ C1 (a)∨C2 (a) ∼ (C1 (a) C2 (a)) ≡ (∼ C1 (a)∨ ∼ C2 (a))∧(∼ C1 (a)∨C2 (a)) ∧(C1 (a)∨ ∼ C2 (a)) ∼2 (C1 C2 )(a) ≡ (∼2 C1 (a)∨ ∼2 C2 (a))∧(∼2 C1 (a)∨ ∼ C2 (a)) ∧(∼2 C1 (a)∨C2 (a)) ∧(∼ C1 (a)∨ ∼2 C2 (a))∧(C1 (a)∨ ∼2 C2 (a)) ∼3 (C1 C2 )(a) ≡ ∼3 C1 (a)∧ ∼3 C2 (a); (C1  C2 )(a) ≡ C1 (a)∧C2 (a) ∼ (C1  C2 )(a) ≡ (∼ C1 (a)∨ ∼ C2 (a))∧(∼ C1 (a)∨ ∼2 C2 (a)) ∧(∼ C1 (a)∨ ∼3 C2 (a))∧(∼2 C1 (a)∨ ∼ C2 (a)) ∧(∼3 C1 (a)∨ ∼ C2 (a)) ∼2 (C1  C2 )(a) ≡ (∼2 C1 (a)∨ ∼2 C2 (a))∧(∼2 C1 (a)∨ ∼3 C2 (a)) ∧(∼3 C1 (a)∨ ∼2 C2 (a)) ∼3 (C1  C2 )(a) ≡ ∼3 C1 (a)∨ ∼3 C2 (a) and (∀R.C)(a) ≡ E f (∼3 R(a, f )∧C(a)∧(∼ R(a, f )∨ ∼ C( f )) ∧(∼2 R(a, f )∨ ∼ C( f ))∧(∼2 R(a, f )∨ ∼2 C( f )) ∼ (∀R.C)(a) ≡ E f (R(a, f )∨ ∼ C( f ))∧(∼ R(a, f )∨ ∼2 C( f )) ∧(∼2 R(a, f )∨ ∼3 C( f )) ∼2 (∀R.C)(a) ≡ E f (R(a, f )∨ ∼2 C( f ))∧(∼ R(a, f )∨ ∼3 C( f )) ∼3 (∀R.C)(a) ≡ Ae(R(a, e)∨ ∼3 C(e)) (∃R.C)(a) ≡ Ae(R(a, e)∨C(e)) ∼ (∃R.C)(a) ≡ Ae((∼ R(a, e)∨ ∼ C(e))∧(R(a, e)∨ ∼ C(e)) ∧(∼ R(a, e)∨C(e))) ∼2 (∃R.C)(a) ≡ Ae((∼2 R(a, e)∨ ∼2 C(e))∧(∼ R(a, e)∨ ∼2 C(e)) ∧(R(a, e)∨ ∼2 C(e)))∧(∼2 R(a, e)∨ ∼ C(e)) ∧(∼2 R(a, e)∨C(e))) ∼ (∃R.C)(a) ≡ E f (∼3 R(a, f )∧ ∼3 C( f )). 3

5.2 1/4-Multisequents =t

t A 1/4-multisequent  is N1/4 -valid, denoted by |=1/4 , if for any interpretation I , there is a statement A(a) ∈  such that I (A(a)) = t.

5.2 1/4-Multisequents

145

t 5.2.1 Deduction System N1/4 t Deduction system N1/4 consists of the following axiom and deduction rules: • Axiom: ⎧ ∩ ∼−  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎪ ⎨ ∩ ∼−3  = ∅ t ∼− ∩ ∼−2  = ∅ (A1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− ∩ ∼−3  = ∅ ⎪ ⎪ ⎩ −2 ∼ ∩ ∼−3  = ∅ ,

where  is a set of literals and ∼−  = {A(a) :∼ A(a) ∈ } ∼−2  = {A(a) :∼2 A(a) ∈ } ∼−3  = {A(a) :∼3 A(a) ∈ }. • Deduction rules for logical constructors: (∼4 )

, A(a) , ∼4 A(a)



( )

, A1 (a) , A2 (a) , (A1 A2 )(a) ⎡

⎡

, ∼ A1 (a) ⎢ , ∼ A2 (a) ⎢ ⎢ , A1 (a) ⎢ (∼ ) ⎢ ⎢  , ∼ A2 (a) ⎣ , ∼ A1 (a) , A2 (a) , ∼ (A1 A2 )(a))

, ∼2 A1 (a) ⎢ , ∼2 A2 (a) ⎢ ⎢ , ∼ A1 (a) ⎢ ⎢ , ∼2 A2 (a) ⎢

⎢ , A1 (a) , ∼3 A1 (a) ⎢ 2 2 3 ⎢ (∼ ) ⎢  , ∼ A2 (a) (∼ ) , ∼3 A2 (a) ⎢ , ∼2 A1 (a) , ∼3 (A1 A2 )(a) ⎢ ⎢ , ∼ A2 (a) ⎢ ⎣ , ∼2 A1 (a) , A2 (a) , ∼2 (A1 A2 )(a))

146

5 R-Calculi for Post L4 -Valued DL

⎡

and



, A1 (a) () , A2 (a) , (A1  A2 )(a)

⎡

, ∼ A1 (a) ⎢ , ∼ A2 (a) ⎢ ⎢ , ∼ A1 (a) ⎢ ⎢ , ∼2 A2 (a) ⎢ ⎢ , ∼ A1 (a) ⎢ 3 (∼ ) ⎢ ⎢  , ∼2 A2 (a) ⎢ , ∼ A1 (a) ⎢ ⎢ , ∼ A2 (a) ⎢ ⎣ , ∼3 A1 (a) , ∼ A2 (a) , ∼ (A1  A2 )(a)

, ∼2 A1 (a) ⎢ , ∼2 A2 (a) ⎢  ⎢ , ∼2 A1 (a) , ∼3 A1 (a) ⎢ 3 2 3 ⎢ (∼ ) ⎢  , ∼ A2 (a) (∼ ) , ∼3 A2 (a) ⎣ , ∼3 A1 (a) , ∼3 (A1  A2 )(a) , ∼2 A2 (a) , ∼2 (A1  A2 )(a) • Deduction rules for quantifier constructors: ⎡

, ∼3 R(a, f ) ⎡ ⎢ , A( f ) , R(a, f ) ⎢ ⎢ , ∼ R(a, f ) ⎢ , ∼ A( f ) ⎢ ⎢ ⎢ , ∼ A( f ) ⎢ , ∼ R(a, f ) ⎢ ⎢ 2 ⎢ , ∼2 A( f ) R(a, f ) , ∼ (∀) ⎢ (∼ ∀) ⎢ ⎢ ⎢ , ∼ A( f ) ⎣ , ∼2 R(a, f ) ⎢ 2 ⎣ , ∼ R(a, f ) , ∼3 A( f ) 2 , ∼ A( f ) , (∼ ∀R.A)(a) , (∀R.A)(a) ⎡ , R(a, f )  ⎢ , ∼2 A( f ) , R(a, e) ⎢ (∼2 ∀) ⎣ , ∼ R(a, f ) (∼3 ∀) , ∼3 A(e) , ∼3 A( f ) , (∼ ∀R.A)(a) , (∼2 ∀R.A)(a)

5.2 1/4-Multisequents

147

⎡

and

, ∼ R(a, e) ⎢ , ∼ A(e) ⎢ ⎢ , R(a, e) ⎢ (∼ ∃) ⎢ ⎢  , ∼ A(e) ⎣ , ∼ R(a, e) , A(e) , ∼ (∃R.A)(a))

 (∃)

, R(a, e) , A(e) , (∃R.A)(a) ⎡

, ∼2 R(a, e) ⎢ , ∼2 A(e) ⎢ ⎢ , ∼ R(a, e) ⎢ ⎢ , ∼2 A(e) ⎢

⎢ , R(a, e) , ∼3 R(a, f ) ⎢ 2 2 3 ⎢ (∼ ∃) ⎢  , ∼ A(e) (∼ ) , ∼3 A( f ) ⎢ , ∼2 R(a, e) , ∼3 (∃R.A)(a) ⎢ ⎢ , ∼ A(e) ⎢ ⎣ , ∼2 R(a, e) , A(e) , ∼2 (∃R.A)(a)) where e is a new constant, and f is a constant. =t

t Definition 5.2.1 A 1/4-multisequent  is provable in N1/4 , denoted by 1/4 , if there is a sequence {1 , . . . , n } of 1/4-multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/4-multisequents by t . one of the deduction rules in N1/4

Theorem 5.2.2 For any 1/4-multisequent , =t

=t

|=1/4  iff 1/4 .



5.2.2 R-Calculus St1/4 Given a 1/4-multisequent  and a statement A ∈ , a 1/4-reduction |A ⇒ [A ] =t is St -valid, denoted by |=1/4 |A ⇒ [A ], if A =



t -valid A if [A] is N1/4 λ otherwise.

148

5 R-Calculi for Post L4 -Valued DL

R-calculus St1/4 consists of the following axioms and deduction rules: • Axioms: ⎧ [s(a)]∩ ∼−   = ∅ ⎪ ⎪ ⎪ ⎪ [s(a)]∩ ∼−2   = ∅ ⎪ ⎪ ⎨ [s(a)]∩ ∼−3   = ∅

∼− ∩ ∼−2   = ∅ (At ⎪ −1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− ∩ ∼−3   = ∅ ⎪ ⎩ −2 ∼ ∩ ∼−3   = ∅   s(a) ⇒ [s(a)] ⎧ − ⎪ ∩ ∼ [∼ s(a)]  = ∅ ⎪ ⎪ ⎪ ∩ ∼−2   = ∅ ⎪ ⎪ ⎨ ∩ ∼−3   = ∅ (At∼ ⎪ ∼− [∼ s(a)]∩ ∼−2   = ∅ −1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− [∼ s(a)]∩ ∼−3   = ∅ ⎪ ⎩ −2 ∼ ∩ ∼−3   = ∅  ∼ s(a) ⇒ [∼ s(a)] ⎧ − ⎪ ∩ ∼   = ∅ ⎪ ⎪ ⎪ ∩ ∼−2 [∼2 s(a)]  = ∅ ⎪ ⎪ ⎨ ∩ ∼−3   = ∅ 2 ∼− ∩ ∼−2 [∼2 s(a)]  = ∅ (At∼ ⎪ −1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− ∩ ∼−3   = ∅ ⎪ ⎩ −2 ∼ [∼2 s(a)]∩ ∼−3   = ∅  ∼2 s(a) ⇒ [∼2 s(a)] ⎧ ∩ ∼−   = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−2   = ∅ ⎪ ⎪ ⎨ ∩ ∼−3 [∼3 s(a)]  = ∅ 3 ∼− ∩ ∼−2   = ∅ (At∼ ⎪ −1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− ∩ ∼−3 [∼3 s(a)]  = ∅ ⎪ ⎩ −2 ∼ ∩ ∼−3 [∼3 s(a)]  = ∅  ∼3 s(a) ⇒ [∼3 s(a)]

⎡

(At 01/4 )

(At∼ 01/4 )

2

(At∼ 01/4 )

3

(At∼ 01/4 )

[s(a)]∩ ∼−  = ∅ ⎢ [s(a)]∩ ∼−2  = ∅ ⎢ ⎢ [s(a)]∩ ∼−3  = ∅ ⎢ ⎢ ∼− ∩ ∼−2  = ∅ ⎢ ⎣ ∼− ∩ ∼−3  = ∅ ∼−2 ∩ ∼−3  = ∅   s(a) ⇒  ⎡ ∩ ∼− [∼ s(a)] = ∅ ⎢ ∩ ∼−2  = ∅ ⎢ ⎢ ∩ ∼−3  = ∅ ⎢ ⎢ ∼− [∼ s(a)]∩ ∼−2  = ∅ ⎢ ⎣ ∼− [∼ s(a)]∩ ∼−3  = ∅ ∼−2 ∩ ∼−3  = ∅  ∼ s(a) ⇒  ⎡ ∩ ∼−  = ∅ ⎢ ∩ ∼−2 [∼2 s(a)] = ∅ ⎢ ⎢ ∩ ∼−3  = ∅ ⎢ ⎢ ∼− ∩ ∼−2 [∼2 s(a)] = ∅ ⎢ ⎣ ∼− ∩ ∼−3  = ∅ ∼−2 [∼2 s(a)]∩ ∼−3  = ∅  ∼2 s(a) ⇒  ⎡ ∩ ∼−  = ∅ ⎢ ∩ ∼−2  = ∅ ⎢ ⎢ ∩ ∼−3 [∼3 s(a)] = ∅ ⎢ ⎢ ∼− ∩ ∼−2  = ∅ ⎢ ⎣ ∼− ∩ ∼−3 [∼3 s(a)] = ∅ ∼−2 ∩ ∼−3 [∼3 s(a)] = ∅  ∼3 s(a) ⇒ 

where  is a set of literals and s(a) is atomic. • Deduction rules: (∼4− )

 ↑ A(a) ⇒ [A(a)]  ↑ A(a) ⇒  (∼40 )  ↑∼4 A(a) ⇒ [∼4 A(a)]  ↑∼4 A(a) ⇒ 

5.2 1/4-Multisequents

149

and 

 ↑ A1 (a) ⇒ [A1 (a)]  ↑ A2 (a) ⇒ [A2 (a)]

 ↑ (A1 A2 )(a) ⇒ [(A1 A2 )(a)  ↑ A1 (a) ⇒  ( 0 )  ↑ A2 (a) ⇒   ↑⎡(A  1 A2 )(a) ⇒   ↑∼ A1 (a) ⇒ [∼ A1 (a)] ⎢  ↑∼ A2 (a) ⇒ [∼ A2 (a)] ⎢ ⎢ [∼ A1 (a)∨ ∼ A2 (a)] ↑ A1 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a), A1 (a)] ⎢ (∼ − ) ⎢ ⎢  [∼ A1 (a)∨ ∼ A2 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a), ∼ A2 (a)] ⎣ [Z 1 ] ↑∼ A1 (a) ⇒ [Z 1 , ∼ A1 (a)] [Z 1 ] ↑ A2 (a) ⇒ [Z 1 , A2 (a)]  ↑∼ ⎧ (A1 A2 )(a) ⇒ [∼ (A1 A2 )(a)  ↑∼ A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪  ↑∼ A2 (a) ⇒  ⎪ ⎪ ⎨

[∼ A1 (a)∨ ∼ A2 (a)] ↑ A1 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a)] (∼ 0 ) ⎪ ⎪ [∼ A1 (a)∨ ∼ A2 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a)] ⎪ ⎪ [Z 1 ] ↑∼ A1 (a) ⇒ [Z 1 ] ⎪ ⎪ ⎩ [Z 1 ] ↑ A2 (a) ⇒ [Z 1 ]  ↑∼ (A1 A2 )(a) ⇒  ( − )

where Z 1 =∼ A1 (a)∨ ∼ A2 (a), A1 (a)∨ ∼ A2 (a), and ⎡

 ↑∼2 A1 (a) ⇒ [∼2 A1 (a)] ⎢  ↑∼2 A2 (a) ⇒ [∼2 A2 (a)] ⎢ ⎢ [∼2 A1 (a)∨ ∼2 A2 (a)] ↑ A1 (a) ⇒ [∼2 A1 (a)∨ ∼2 A2 (a), A1 (a)] ⎢ ⎢ [∼2 A1 (a)∨ ∼2 A2 (a)] ↑∼2 A2 (a) ⇒ [∼2 A1 (a)∨ ∼2 A2 (a), ∼2 A2 (a)] ⎢ ⎢ [Z 2 ] ↑∼ A1 (a) ⇒ [Z 2 , ∼ A1 (a)] ⎢ 2 2 (∼2 − ) ⎢ ⎢  [Z 2 ] ↑∼ A2 (a) ⇒ [Z 2 , ∼ A2 (a)] ⎢ [Z 3 ] ↑∼2 A1 (a) ⇒ [Z 3 , ∼2 A1 (a)] ⎢ ⎢ [Z 3 ] ↑ A2 (a) ⇒ [Z 3 , A2 (a)] ⎢ ⎣ [Z 4 ] ↑∼2 A1 (a) ⇒ [Z 4 , ∼2 A1 (a)] [Z 4 ] ↑∼ A2 (a) ⇒ [Z 4 , ∼ A2 (a)]  ↑∼2 (A1 A2 )(a) ⇒ [∼2 (A1 A2 )(a)

⎧ ⎨ Z 2 =∼2 A1 (a)∨ ∼2 A2 (a), A1 (a)∨ ∼2 A2 (a), where Z 3 = Z 2 , ∼ A1 (a)∨ ∼2 A2 (a), and ⎩ Z 4 = Z 3 , ∼2 A1 (a) ∨ A2 (a),

150

5 R-Calculi for Post L4 -Valued DL

⎧

2 ⎪ ⎪  ↑∼2 A1 (a) ⇒  ⎪ ⎪ A2 (a) ⇒  ⎪ ⎪

 ↑∼ ⎪ 2 2 2 2 ⎪ A [∼ ⎪ 1 (a)∨ ∼ A2 (a)] ↑ A1 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a)] ⎪ ⎪ 2 2 2 2 ⎪ [∼ A1 (a)∨ ∼ A2 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼2 A2 (a)] ⎪ ⎪ ⎨

[Z 2 ] ↑∼ A1 (a) ⇒ [Z 2 ] 2 [Z (∼2 0 ) ⎪ 2 ] ↑∼ A2 (a) ⇒ [Z 2 ] ⎪

⎪ 2 ⎪ [Z 3 ] ↑∼ A1 (a) ⇒ [Z 3 ] ⎪ ⎪ ⎪ ⎪ [Z ⎪ 3 ] ↑∼ A2 (a) ⇒ [Z 3 ] ⎪

⎪ 2 ⎪ [Z ⎪ 4 ] ↑∼ A1 (a) ⇒ [Z 4 ] ⎪ ⎩ [Z 4 ] ↑ A2 (a) ⇒ [Z 4 ]  ↑∼2 (A1 A2 )(a) ⇒  and



 ↑∼3 A1 (a) ⇒ [∼3 A1 (a)] (∼ − ) [∼3 A1 (a)] ↑∼3 A2 (a) ⇒ [∼3 A1 (a), ∼3 A2 (a)] 2 2  ↑∼ 3(A1 A2 )(a)) ⇒ [∼ (A1 A2 )(a)  ↑∼ A1 (a) ⇒  (∼3 0 ) [∼3 A1 (a)] ↑∼3 A2 (a) ⇒ [∼3 A1 (a)]  ↑∼3 (A1 A2 )(a) ⇒  3



and

 ↑ A1 (a) ⇒ [A1 (a)] [A1 (a)] ↑ A2 (a) ⇒ [A1 (a), A2 (a)]   ↑ (A1  A2 )(a) ⇒ [(A1  A2 )(a)  ↑ A1 (a) ⇒  (0 ) [A1 (a)] ↑ A2 (a) ⇒ [A1 (a)]  ↑ (A1  A2 )(a) ⇒ 

(− )

and ⎡

 ↑∼ A1 (a) ⇒ [∼ A1 (a)] ⎢  ↑∼ A2 (a) ⇒ [∼ A2 (a)] ⎢ ⎢ [∼ A1 (a)∨ ∼ A2 (a)] ↑∼3 A1 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a), ∼3 A1 (a)] ⎢ ⎢ [∼ A1 (a)∨ ∼ A2 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a), ∼ A2 (a)] ⎢ ⎢ [Z 5 ] ↑∼2 A1 (a) ⇒ [Z 5 , ∼2 A1 (a)] ⎢ (∼ − ) ⎢ ⎢  [Z 5 ] ↑∼ A2 (a) ⇒ [Z 5 , ∼ A2 (a)] ⎢ [Z 6 ] ↑∼ A1 (a) ⇒ [Z 6 , ∼ A1 (a)] ⎢ ⎢ [Z 6 ] ↑∼3 A2 (a) ⇒ [Z 6 , ∼3 A2 (a)] ⎢ ⎣ [Z 7 ] ↑∼ A1 (a) ⇒ [Z 7 , ∼ A1 (a)] [Z 7 ] ↑∼2 A2 (a) ⇒ [Z 7 , ∼2 A2 (a)]  ↑∼ (A1  A2 )(a) ⇒ [∼ (A1  A2 )(a)

⎧ ⎨ Z 5 =∼ A1 (a)∨ ∼ A2 (a), ∼3 A1 (a)∨ ∼ A2 (a) where Z 6 = Z 5 , ∼2 A1 (a)∨ ∼ A2 (a), and ⎩ Z 7 = Z 6 , ∼ A1 (a)∨ ∼3 A2 (a),

5.2 1/4-Multisequents

151

⎧

⎪ ⎪  ↑∼ A1 (a) ⇒  ⎪ ⎪ ⎪ ⎪

 ↑∼ A2 (a) ⇒  ⎪ ⎪ [∼ A1 (a)∨ ∼ A2 (a)] ↑∼3 A1 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a)] ⎪ ⎪ ⎪ ⎪ [∼ A1 (a)∨ ∼ A2 (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼ A2 (a)] ⎪ ⎪ ⎨

[Z 5 ] ↑∼2 A1 (a) ⇒ [Z 5 ] (∼ 0 ) ⎪ ⎪ [Z 5 ] ↑∼ A2 (a) ⇒ [Z 5 ] ⎪ ⎪ [Z 6 ] ↑∼ A1 (a) ⇒ [Z 6 ] ⎪ ⎪ ⎪ 2 ⎪ [Z ⎪ 6 ] ↑∼ A2 (a) ⇒ [Z 6 ] ⎪

⎪ ⎪ [Z 7 ] ↑∼ A1 (a) ⇒ [Z 7 ] ⎪ ⎪ ⎩ [Z 7 ] ↑∼3 A2 (a) ⇒ [Z 7 ]  ↑∼ (A1  A2 )(a) ⇒ ] and ⎡

 ↑∼2 A1 (a) ⇒ [∼2 A1 (a)] ⎢  ↑∼2 A2 (a) ⇒ [∼2 A2 (a)] ⎢ ⎢ [∼2 A1 (a)∨ ∼2 (a)] ↑∼3 A1 (a) ⇒ [∼2 A1 (a)∨ ∼2 (a), ∼3 A1 (a)] ⎢ 2 2 2 2 2 2 2 (∼ − ) ⎢ ⎢  [∼ A1 (a)∨ ∼ (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼ (a), ∼ A2 (a)] ⎣ [Z 8 ] ↑∼2 A1 (a) ⇒ [Z 8 , ∼2 A1 (a)] [Z 8 ] ↑∼3 A2 (a) ⇒ [Z 8 , ∼3 A2 (a)] 2 (A  A )(a) ⇒ [∼2 (A  A )(a)  ↑∼ 1 2 1 2 ⎧

 ↑∼2 A1 (a) ⇒  ⎪ ⎪ ⎪ 2 A (a) ⇒  ⎪ ⎪ 2 ⎪  ↑∼ ⎨ [∼2 A1 (a)∨ ∼2 (a)] ↑∼3 A1 (a) ⇒ [∼2 A1 (a)∨ ∼2 (a)] 2 2 2 2 2 (∼2 0 ) ⎪ ⎪ [∼ A1 (a)∨ ∼ (a)] ↑∼ A2 (a) ⇒ [∼ A1 (a)∨ ∼ (a)] ⎪ 2 ⎪ ⎪ [Z 8 ] ↑∼ A1 (a) ⇒ [Z 8 ] ⎪ ⎩ [Z 8 ] ↑∼3 A2 (a) ⇒ [Z 8 ]  ↑∼2 (A1  A2 )(a) ⇒ ]

where Z 8 =∼2 A1 (a)∨ ∼2 (a), ∼3 A1 (a)∨ ∼2 A2 (a), and 

 ↑∼3 A1 (a) ⇒ [∼3 A1 (a)] (∼ − )  ↑∼3 A2 (a) ⇒ [∼3 A2 (a)] 3 3

 ↑∼ 3(A1  A2 )(a) ⇒ [∼ (A1  A2 )(a)  ↑∼ A1 (a) ⇒  (∼3 0 )  ↑∼3 A2 (a) ⇒   ↑∼3 (A1  A2 )(a) ⇒  3

and

152

5 R-Calculi for Post L4 -Valued DL ⎡

 ↑∼3 R(a, f ) ⇒ [∼3 R(a, f )] ⎢ [∼3 R(a, f )] ↑ A( f ) ⇒ [∼3 R(a, f ), A( f )] ⎢ ⎢ [∼3 R(a, f ), A( f )] ↑∼ R(a, f ) ⇒ [∼3 R(a, f ), A( f ), ∼ R(a, f )] ⎢ ⎢ [∼3 R(a, f ), A( f )] ↑∼ A( f ) ⇒ [∼3 R(a, f ), A( f ), ∼ A( f )] ⎢ 2 2 + (∀ ) ⎢ ⎢ [Y1 ] ↑∼ R(a, f ) ⇒ [Y1 , ∼ R(a, f )] ⎢ [Y1 ] ↑∼ A( f ) ⇒ [Y1 , ∼ A( f )] ⎢ ⎣ [Y  ] ↑∼2 R(a, f ) ⇒ [Y  , ∼2 R(a, f )] 1 1 [Y1 ] ↑∼2 A( f ) ⇒ [Y1 , ∼2 A( f )]  ↑ (∀R.A)(a) ⇒ [(∀R.A)(a)] ⎧ 3 ⎪  ↑∼ R(a, e) ⇒  ⎪ ⎪ 3 ⎪ R(a, e)] ↑ A(e) ⇒ [∼3 R(a, e)] ⎪ ⎪ [∼

⎪ 3 R(a, e), A(e)] ↑∼ R(a, e) ⇒ [∼3 R(a, e), A(e)] ⎪ [∼ ⎪ ⎪ ⎨ 3 3

[∼ R(a,2 e), A(e)] ↑∼ A(e) ⇒ [∼ R(a, e), A(e)] 0 [Y1 ] ↑∼ R(a, e) ⇒ [Y1 ] (∀ ) ⎪ ⎪ ⎪ ⎪ ⎪ [Y1 ] ↑∼ A(e) ⇒ [Y1 ] ⎪

⎪ ⎪ ⎪ [Y1 ] ↑∼2 R(a, e) ⇒ [Y1 ] ⎪ ⎩ [Y1 ] ↑∼2 A(e) ⇒ [Y1 ]  ↑ (∀R.A)(a) ⇒ 

where Y1 =∼3 R(a, f ), A( f ), ∼ R(a, f )∨ ∼ A( f ), Y1 = Y1 , ∼2 R(a, f )∨ ∼ A( f ) and ⎡

 ↑ R(a, f ) ⇒ [R(a, f )] ⎢  ↑∼ A( f ) ⇒ [∼ A( f )] ⎢ ⎢ [R(a, f )∨ ∼ A( f )] ↑∼ R(a, f ) ⇒ [R(a, f )∨ ∼ A( f ), ∼ R(a, f )] ⎢ 2 2 (∼ ∀− ) ⎢ ⎢  [R(a, f )∨ ∼ A( f )] ↑∼ A( f ) ⇒ [R(a, f )∨ ∼ A( f ), ∼ A( f )] ⎣ [Y2 ] ↑∼2 R(a, f ) ⇒ [Y2 , ∼2 R(a, f )] [Y2 ] ↑∼3 A( f ) ⇒ [Y2 , ∼3 A( f )]  ↑ ⎧ (∼ ∀R.A)(a) ⇒ [(∼ ∀R.A)(a)]  ↑ R(a, e) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪  ↑∼ A(e) ⇒  ⎪ ⎨ [R(a, e)∨ ∼ A(e)] ↑∼ R(a, e) ⇒ [R(a, e)∨ ∼ A(e)] 0 [R(a, e)∨ ∼ A(e)] ↑∼2 A(e) ⇒ [R(a, e)∨ ∼ A(e)] (∼ ∀ ) ⎪ ⎪

⎪ 2 R(a, e) ⇒ [Y ] ⎪ ⎪ [Y ] ↑∼ 2 2 ⎪ ⎩ [Y2 ] ↑∼3 A(e) ⇒ [Y2 ]  ↑ (∼ ∀R.A)(a) ⇒ 

where Y2 = R(a, f )∨ ∼ A( f ), ∼ R(a, f )∨ ∼2 A( f ), and ⎡

 ↑ R(a, f ) ⇒ [R(a, f )] ⎢  ↑∼2 A( f ) ⇒ [∼2 A( f )] ⎢ (∼2 ∀− ) ⎣ [R(a, f )∨ ∼2 A( f )] ↑∼ R(a, f ) ⇒ [R(a, f )∨ ∼2 A( f ), ∼ R(a, f )] [R(a, f )∨ ∼2 A( f )] ↑∼3 A( f ) ⇒ [R(a, f )∨ ∼2 A( f ), ∼3 A( f )] 2 2  ↑ ⎧ (∼ ∀R.A)(a) ⇒ [(∼ ∀R.A)(a)]  ↑ R(a, e) ⇒  ⎪ ⎪ ⎨  ↑∼2 A(e) ⇒ 

[R(a, e)∨ ∼2 A(e)] ↑∼ R(a, e) ⇒ [R(a, e)∨ ∼2 A(e)] (∼2 ∀0 ) ⎪ ⎪ ⎩ [R(a, e)∨ ∼2 A(e)] ↑∼3 A(e) ⇒ [R(a, e)∨ ∼2 A(e)]  ↑ (∼2 ∀R.A)(a) ⇒ 

5.2 1/4-Multisequents

153



and

 ↑ R(a, e) ⇒ [R(a, e)  ↑∼3 A(e) ⇒ [∼3 A(e)] 3 3

 ↑ (∼ ∀R.A)(a) ⇒ [(∼ ∀R.A)(a)]  ↑ R(a, f ) ⇒  (∼3 ∀0 )  ↑∼3 A( f ) ⇒  3   ↑ (∼ ∀R.A)(a) ⇒   ↑ R(a, e) ⇒ [R(a, e) (∃− )  ↑ A(e) ⇒ [A(e)]

 ↑ (∃R.A)(a) ⇒ [(∃R.A)(a)]  ↑ R(a, f ) ⇒  (∃0 )  ↑ A( f ) ⇒   ↑ (∃R.A)(a) ⇒ 

(∼3 ∀− )

and ⎡

 ↑∼ R(a, e) ⇒ [∼ R(a, e)] ⎢  ↑∼ A(e) ⇒ [∼ A(e)] ⎢ ⎢ [∼ R(a, e)∨ ∼ A(e)] ↑ R(a, e) ⇒ [∼ R(a, e)∨ ∼ A(e), R(a, e)] ⎢ (∼ ∃− ) ⎢ ⎢  [∼ R(a, e)∨ ∼ A(e)] ↑∼ A(e) ⇒ [∼ R(a, e)∨ ∼ A(e), ∼ A(e)] ⎣ [Y3 ] ↑∼ R(a, e) ⇒ [Y3 , ∼ R(a, e)] [Y3 ] ↑ A(e) ⇒ [Y3 , A(e)]  ↑ ⎧ (∼ ∃R.A)(a) ⇒ [(∼ ∃R.A)(a)]  ↑∼ R(a, f ) ⇒  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  ↑∼ A( f ) ⇒  ⎨ [∼ R(a, f )∨ ∼ A( f )] ↑ R(a, f ) ⇒ [∼ R(a, f )∨ ∼ A( f )] (∼ ∃0 ) ⎪ ⎪ [∼ R(a, f )∨ ∼ A( f )] ↑∼ A( f ) ⇒ [∼ R(a, f )∨ ∼ A( f )] ⎪ ⎪ ⎪ [Y3 ] ↑∼ R(a, f ) ⇒ [Y3 ] ⎪ ⎩ [Y3 ] ↑ A( f ) ⇒ [Y3 ]  ↑ (∼ ∃R.A)(a) ⇒ 

where Y3 =∼ R(a, e)∨ ∼ A(e), R(a, e)∨ ∼ A(e), and ⎡

 ↑∼2 R(a, e) ⇒ [∼2 R(a, e) ⎢  ↑∼2 A(e) ⇒ [∼2 A(e)] ⎢ ⎢ [∼2 R(a, e)∨ ∼2 A(e)] ↑∼ R(a, e) ⇒ [∼2 R(a, e)∨ ∼2 A(e), ∼ R(a, e) ⎢ ⎢ [∼2 R(a, e)∨ ∼2 A(e)] ↑∼2 A(e) ⇒ [∼2 R(a, e)∨ ∼2 A(e), ∼2 A(e)] ⎢ ⎢ [Y4 ] ↑ R(a, e) ⇒ [Y4 , R(a, e) ⎢ 2 2 (∼2 ∃+ ) ⎢ ⎢  [Y4 ] ↑∼ A(e) ⇒ [Y4 , ∼ A(e)] ⎢ [Y5 ] ↑∼2 R(a, e) ⇒ [Y5 , ∼2 R(a, e) ⎢ ⎢ [Y5 ] ↑ A(e) ⇒ [Y5 , A(e)] ⎢ ⎣ [Y6 ] ↑∼2 R(a, e) ⇒ [Y6 , ∼2 R(a, e) [Y6 ] ↑∼ A(e) ⇒ [Y6 , ∼ A(e)]  ↑ (∼2 ∃R.A)(a) ⇒ [(∼2 ∃R.A)(a)]

Y4 =∼2 R(a, e)∨ ∼2 A(e), ∼ R(a, e)∨ ∼2 A(e) where Y5 = Y4 , R(a, e)∨ ∼2 A(e), and Y6 = Y5 , ∼2 R(a, e) ∨ A(e),

154

5 R-Calculi for Post L4 -Valued DL ⎧

 ↑∼2 R(a, f ) ⇒  ⎪ ⎪ ⎪ 2 A( f ) ⇒  ⎪ ⎪

 ↑∼ ⎪ ⎪ 2 ⎪ [∼ R(a, f )∨ ∼2 A( f )] ↑∼ R(a, f ) ⇒ [∼2 R(a, f )∨ ∼2 A( f )] ⎪ ⎪ ⎪ ⎪ [∼2 R(a, f )∨ ∼2 A( f )] ↑∼2 A( f ) ⇒ [∼2 R(a, f )∨ ∼2 A( f )] ⎪ ⎪ ⎨

[Y4 ] ↑ R(a, f ) ⇒ [Y4 ] (∼2 ∃0 ) ⎪ ⎪ [Y4 ] ↑∼2 A( f ) ⇒ [Y4 ] ⎪ ⎪ ⎪ [Y5 ] ↑∼2 R(a, f ) ⇒ [Y5 ] ⎪ ⎪ ⎪ ⎪ f ) ⇒ [Y5 ] ⎪

[Y5 ] ↑ A( ⎪ ⎪ ⎪ [Y6 ] ↑∼2 R(a, f ) ⇒ [Y6 ] ⎪ ⎩ [Y6 ] ↑∼ A( f ) ⇒ [Y6 ]  ↑ (∼2 ∃R.A)(a) ⇒ 

and

 ↑∼3 R(a, f ) ⇒ [∼3 R(a, f )] (∼ ∃ ) [∼3 R(a, f )] ↑∼3 A( f ) ⇒ [∼3 R(a, f ), ∼3 A( f )] 3 3  ↑ (∼3 ∃R.A)(a) ⇒ [(∼ ∃R.A)(a)]  ↑∼ R(a, e) ⇒  (∼3 ∃0 ) [∼3 R(a, e)] ↑∼3 A(e) ⇒ [∼3 R(a, e)]  ↑ (∼3 ∃R.A)(a) ⇒  3

−

where e is a new constant and f is a constant. Definition 5.2.3 Given a 1/4-multisequent  and a statement A(a) ∈ , a reduc=t tion |A(a) ⇒   is provable in St1/4 , denoted by 1/4 |A(a) ⇒   , if there is a sequence {δ1 , . . . , δn } of reductions such that δn = |A(a) ⇒   , and for each 1 ≤ i ≤ n, δi is deduced from the previous reductions by one of the deduction rules in St1/4 . Theorem 5.2.4 (Soundness and completeness theorem) For any reduction |A(a) ⇒ [A (a)] and A(a) ∈ , =t

=t

1/4 |A(a) ⇒ [A (a)] iff |=1/4 |A(a) ⇒ [A (a)]. 

5.3 2/4-Multisequents =t

t A 2/4-multisequent | is N2/4 -valid, denoted by |=2/4 |, if for any interpretation I, either (i) I (A(a)) = t for some A(a) ∈ , or (ii) I (B(b)) =  for some B(b) ∈ . We have the following equivalences:

5.3 2/4-Multisequents

155

, ∼ A(a)| ≡ |, A(a) , ∼2 A(a)| ≡ , ∼2 A(a)| 3 2 , ∼ A(a)| ≡ |, ∼ A(a) |, ∼ B(b) ≡ , ∼2 B(b)| |, ∼2 B(b) ≡ |, ∼2 B(b) |, ∼3 B(b) ≡ , B(b)| and given two sets ,  of literals, we (i) move each statement in  with ∼ from  to , with ∼2 in  remains in , and with ∼3 in  to  with ∼2 ; and (ii) move each statement in  beginning with ∼ from  to  with ∼2 , with ∼2 in  remains in , and with ∼3 in  to  without ∼2 .     ∼ A(a) → A(a) ∼2 B(b) ←∼ B(b) ∼2 B(b) ∼2 A(a) 3 2 ∼ A(a) → ∼ A(a) B(b) ←∼3 B(b)

5.3.1 Deduction System N2t t Deduction system N2/4 contains the following axiom and deduction rules. • Axiom: ⎧ ⎪ ⎪  ∩ −2= ∅ ⎨ ∩ ∼  = ∅ ∼−2  ∩  = ∅ ⎪ ⎪ ⎩ −2 ∼ ∩ ∼−2  = ∅ |

where ,  are sets of literals of forms s(a) or ∼2 s(a). • Deduction rules for unary connectives: |, A(a) , ∼2 A(a)| |, ∼2 A(a) (∼2 A ) (∼3A ) 2 , ∼ A(a)| , ∼ A(a)| , ∼3 A(a)| 2 , A(a)| , ∼ B(b)| |, ∼2 B(b) (∼ B ) (∼2B ) (∼4 A ) 4 , ∼ A(a)| |, ∼ B(b) |, ∼2 B(b) , B(b)| |, B(b) (∼4B ) (∼3B ) |, ∼3 B(b) |, ∼4 B(b) (∼ A )

156

5 R-Calculi for Post L4 -Valued DL

• Deduction rules for logical connectives: ⎡



( A )

, A1 (a)| , A2 (a)| , (A1 A2 )(a)| ⎡

|, B1 (b) ⎢ |, B2 (b) ⎢ ⎢ , B1 (b)| ⎢ B ⎢ |, B (b) ( ) ⎢  2 ⎣ |, B1 (b) , B2 (b)| |, (B1 B2 )(a)

, ∼2 A1 (a)| ⎢ , ∼2 A2 (a)| ⎢ ⎢ |, A1 (a) ⎢ ⎢ , ∼2 A2 (a)| ⎢

⎢ , A1 (a)| |, ∼2 B1 (b) ⎢ 2 A ⎢ , ∼2 A (a)| 2 B (∼ ) ⎢  (∼ ) |, ∼2 B1 (b) 2 ⎢ , ∼2 A1 (a)| |, ∼2 (B1 B2 )(a) ⎢ ⎢ |, A2 (a) ⎢ ⎣ , ∼2 A1 (a)| , A2 (a)| , ∼2 (A1 A2 )(a)| ⎡

and



( A )

, A1 (a)| , A2 (a)| , (A1  A2 )(a)|

⎡

|, B1 (b) ⎢ |, B2 (b) ⎢ ⎢ , ∼2 B1 (b)| ⎢ ⎢ |, B2 (b) ⎢ ⎢ |, ∼2 B1 (b) ⎢ B ⎢ |, B (b) ( ) ⎢  2 ⎢ |, B1 (b) ⎢ ⎢ , ∼2 B2 (b)| ⎢ ⎣ |, B1 (b) |, ∼2 B2 (b) |, (B1  B2 )(a)

, ∼2 A1 (a)| ⎢ , ∼2 A2 (a)| ⎢  ⎢ |, ∼2 A1 (a) |, ∼2 B1 (b) ⎢ 2 A ⎢ , ∼2 A (a)| 2 B (∼  ) |, ∼2 B2 (b) (∼  ) ⎢  2 2 ⎣ , ∼ A1 (a)| |, ∼2 (B1  B2 )(a) 2 |, ∼ A2 (a) , ∼2 (A1  A2 )(a)|

5.3 2/4-Multisequents

157

• Deduction rules for quantifier constructors: ⎡ |, ∼2 R(a, f ) ⎡ ⎢ , A( f )| , R(a, f )| ⎢ ⎢ |, R(a, f ) ⎢ |, B( f ) ⎢ ⎢ ⎢ |, A( f ) ⎢ |, R(a, f ) ⎢ ⎢ A ⎢ , ∼2 R(a, f )| B ⎢ , ∼2 B( f )| (∀ ) ⎢  (∀ ) ⎢ ⎢ |, A( f ) ⎣ , ∼2 R(a, f )| ⎢ ⎣ , ∼2 R(a, f )| |, ∼2 B( f ) , ∼2 A( f )| |, (∀R.B)(a) , (∀R.A)(a)| ⎡ , R(a, f )|  ⎢ , ∼2 A( f )| , R(a, e)| ⎢ (∼2 ∀ A ) ⎣ |, R(a, f ) (∼2 ∀ B ) |, ∼2 B(e) |, ∼2 A( f ) |, ∼2 (∀R.B)(a) 2 , ∼ (∀R.A)(a)| ⎡

and 

(∃ A )

, R(a, e)| , A(e)| , (∃R.A)(a)| ⎡

|, R(a, e) ⎢ |, B(e) ⎢ ⎢ , R(a, e)| ⎢ B ⎢ |, B(e) (∃ ) ⎢  ⎣ |, R(a, e) , B(e)| |, (∃R.B)(a)

, ∼2 R(a, e)| ⎢ , ∼2 A(e)| ⎢ ⎢ |, R(a, e) ⎢ ⎢ , ∼2 A(e)| ⎢

⎢ , R(a, e)| |, ∼2 R(a, f ) ⎢ 2 A ⎢ , ∼2 A(e)| 2 B (∼ ∃ ) ⎢  (∼ ∃ ) |, ∼2 B( f ) ⎢ , ∼2 R(a, e)| |, ∼2 (∃R.B)(a) ⎢ ⎢ |, A(e) ⎢ ⎣ , ∼2 R(a, e)| , A(e)| , ∼2 (∃R.A)(a)| where f is a constant and e is a new constant. =t

t Definition 5.3.1 A 2/4-multisequent | is provable in N2/4 , denoted by 2/4 |, if there is a sequence {1 |1 , . . . , n |n } of 2/4-multisequents such that n |n = |, and for each 1 ≤ i ≤ n, i |i is either an axiom or deduced from t . the previous 2/4-multisequents by one of the deduction rules in N2/4

158

5 R-Calculi for Post L4 -Valued DL

Theorem 5.3.2 (Soundness and completeness theorem) For any 2/4-multisequent |, =t =t |=2/4 | iff 2/4 |. 

5.3.2 R-Calculus St 2/4 Given a 2/4-multisequent | and two statements A(a) ∈  and B(b) ∈ , a 2/4reduction δ = | ↑ (A(a), B(b)) ⇒ [A (a)]|[B  (b)] is St -valid, denoted by =t |=2/4 δ, if  t -valid A(a) if [A(a)]| is N2/4  A (a) = λ otherwise.  t -valid B(b) if [A (a)]|[B(b)] is N2/4  B (b) = λ otherwise. R-calculus St 2/4 consists of the following deduction rules and axioms: • Axioms: ⎧ [s(a)] ∩  = ∅ ⎪ ⎪ ⎨ [s(a)]∩ ∼−2  = ∅ A− ∼−2 ([s(a)]) ∩  = ∅ (A2/4 ) ⎪ ⎪ ⎩ −2 ∼ ([s(a)])∩ ∼−2  = ∅ | ↑ s(a) ⇒ [s(a)]| ⎡ [s(a)] ∩  = ∅ ⎢ [s(a)]∩ ∼−2  = ∅ ⎢ A0 ⎣ ∼−2 ([s(a)]) ∩  = ∅ (A2/4 ) ∼−2 ([s(a)])∩ ∼−2  = ∅ | ⎧  ↑ s(a) ⇒ |  ∩ [t (b)] = ∅ ⎪ ⎪ ⎨   ∩ ∼−2 ([t (b)]) = ∅ B− ∼−2   ∩ [t (b)] = ∅ ) ⎪ (A2/4 ⎪ ⎩ −2  ∼  ∩ ∼−2 ([t (b)]) = ∅ | ↑ t (b) ⇒ |[t (b)] ⎡   ∩ [t (b)] = ∅ ⎢   ∩ ∼−2 ([t (b)]) = ∅ ⎢ B0 ⎣ ∼−2   ∩ [t (b)] = ∅ ) (A2/4 ∼−2   ∩ ∼−2 ([t (b)]) = ∅ | ↑ t (b) ⇒ | and

5.3 2/4-Multisequents

159

⎧ 2 ⎪ ⎪ [∼2 s(a)] ∩ −2= ∅ ⎨ [∼ s(a)]∩ ∼  = ∅ A∼2 − −2 ([∼2 s(a)]) ∩  = ∅ ∼ (A2/4 ) ⎪ ⎪ ⎩ −2 ∼ ([∼2 s(a)])∩ ∼−2  = ∅ | ↑∼2 s(a) ⇒ [∼2 s(a)]| ⎡ [∼2 s(a)] ∩  = ∅ ⎢ [∼2 s(a)]∩ ∼−2  = ∅ ⎢ A∼2 0 ⎣ ∼−2 ([∼2 s(a)]) ∩  = ∅ ) (A2/4 ∼−2 ([∼2 s(a)])∩ ∼−2  = ∅ 2 | ⇒ | ⎧  ↑∼ s(a) 2 ∩ [∼ t (b)] = ∅  ⎪ ⎪ ⎨   ∩ ∼−2 ([∼2 t (b)]) = ∅ B∼2 − ∼−2   ∩ [∼2 t (b)] = ∅ (A2/4 ) ⎪ ⎪ ⎩ −2  ∼  ∩ ∼−2 ([∼2 t (b)]) = ∅ | ↑∼2 t (b) ⇒ |[∼2 t (b)] ⎡   ∩ [∼2 t (b)] = ∅ ⎢   ∩ ∼−2 ([∼2 t (b)]) = ∅ ⎢ B∼2 0 ⎣ ∼−2   ∩ [∼2 t (b)] = ∅ ) (A2/4 ∼−2   ∩ ∼−2 ([∼2 t (b)]) = ∅ | ↑∼2 t (b) ⇒ | where | is literal of forms s(a), ∼2 s(a), and s(a), t (b) are atoms. • Deduction rules: | ↑ 2 A(a) ⇒ |[A(a)] | ↑∼ A(a) ⇒ [∼ A(a)]| 2 2 2 3A− | ↑ ∼ A(a) ⇒ |[∼ A(a)] (∼ ) 3 2 | ↑∼ A(a) ⇒ |[∼ A(a)] | ↑ A(a) ⇒ [A(a)]| 4 A− (∼ ) | ↑∼4 A(a) ⇒ [∼4 A(a)]| | ↑ 1 ∼2 B(b) ⇒ [∼2 B(b)]| (∼ B− ) | ↑∼ B(b) ⇒ [∼2 B(b)]| | ↑ 1 ∼2 B(b) ⇒ [∼2 B(b)]| (∼3B− ) | ↑∼3 B(b) ⇒ [∼2 B(b)]| | ↑ A(a) ⇒ [A(a)]| (∼4B− ) | ↑∼4 B(b) ⇒ |[∼4 B(b)] (∼ A− )

and

| ↑ 2 A(a) ⇒ | | ↑∼ A(a) ⇒ | 2 2 3A0 | ↑ ∼ A(a) ⇒ | (∼ ) 3 | ↑∼ A(a) ⇒ | | ↑ A(a) ⇒ | 4 A0 (∼ ) | ↑∼4 A(a) ⇒ | | ↑ 1 ∼2 B(b) ⇒ | (∼ B0 ) | ↑∼ B(b) ⇒ | 1 2 3B0 | ↑ ∼ B(b) ⇒ | (∼ ) | ↑∼3 B(b) ⇒ | | ↑ B(b) ⇒ | (∼4B0 ) | ↑∼4 B(b) ⇒ | (∼ A0 )

160

5 R-Calculi for Post L4 -Valued DL



| ↑ A1 (a) ⇒ [A1 (a)]| | ↑ A2 (a) ⇒ [A2 (a)]| | ↑ (A1 A2 )(a) ⇒ [(A1 A2 )(a)]|

| ↑ A1 (a) ⇒ | ( A0 ) | ↑ A2 (a) ⇒ | |

↑ (A1 A2 )(a) ⇒ | | ↑ A1 (a) ⇒ [A1 (a)]| ( A− ) [A1 (a)]| ↑ A2 (a) ⇒ [A1 (a), A2 (a)]| | ↑ (A1  A2 )(a) ⇒ [(A1  A2 )(a)]| | ↑ A1 (a) ⇒ | ( A0 ) [A1 (a)]| ↑ A2 (a) ⇒ [A1 (a)]| | ↑ (A1  A2 )(a) ⇒ | ( A− )

and

⎡

| ↑∼2 A1 (a) ⇒ [∼2 A1 (a)]| ⎢ | ↑∼2 A2 (a) ⇒ [∼2 A2 (a)]| ⎢ ⎢ [Z 1 ]| ↑ 2 A1 (a) ⇒ |[A1 (a)] ⎢ ⎢ [Z 1 ]| ↑∼2 A2 (a) ⇒ [Z 1 , ∼2 A2 (a)]| ⎢ ⎢ [Z 2 ]| ↑ A1 (a) ⇒ [Z 2 , A1 (a)]| ⎢ 2 2 (∼2 A− ) ⎢ ⎢  [Z 2 ]| ↑∼2 A2 (a) ⇒ [Z 2 , ∼2 A2 (a)]| ⎢ [Z 3 ]| ↑∼ A1 (a) ⇒ [Z 3 , ∼ A1 (a)]| ⎢ ⎢ [Z 3 ]| ↑ 2 A2 (a) ⇒ [Z 3 ]|[A2 (a)] ⎢ ⎣ [Z 4 ]| ↑∼2 A1 (a) ⇒ [Z 4 , ∼2 A1 (a)]| [Z 4 ]| ↑ A2 (a) ⇒ [Z 4 , A2 (a)]| | ↑∼2 (A1 A2 )(a) ⇒ [∼2 (A1 A2 )(a)]|

⎧ Z1 ⎪ ⎪ ⎨ Z2 where ⎪ Z3 ⎪ ⎩ Z4

=∼2 A1 (a)∨ ∼2 A2 (a), = Z 1 , ∼ A1 ∨ ∼2 A2 (a), and = Z 2 , A1 ∨ ∼2 A2 (a), = Z 3 , ∼2 A1 ∨ ∼ A2 (a), ⎧

2 ⎪ ⎪ | ↑∼2 A1 (a) ⇒ | ⎪ ⎪ ⎪ ⎪ | ↑∼ A22 (a) ⇒ | ⎪ ⎪ [Z 1 ]| ↑ A1 (a) ⇒ [Z 1 ]| ⎪ ⎪ ⎪ 2 ⎪ [Z ⎪ 1 ]| ↑∼ A2 (a) ⇒ [Z 1 ]| ⎪ ⎨

[Z 2 ]| ↑ A1 (a) ⇒ [Z 2 ]| 2 [Z (∼2 A0 ) ⎪ 2 ]| ↑∼ A2 (a) ⇒ [Z 2 ]| ⎪

⎪ 2 ⎪ [Z 3 ]| ↑∼ A1 (a) ⇒ [Z 3 ]| ⎪ ⎪ ⎪ 2 ⎪ [Z ⎪ 3 ]| ↑ A2 (a) ⇒ [Z 3 ]| ⎪

⎪ ⎪ [Z 4 ]| ↑∼2 A1 (a) ⇒ [Z 4 ]| ⎪ ⎪ ⎩ [Z 4 ]| ↑ A2 (a) ⇒ [Z 4 ]| | ↑∼2 (A1 A2 )(a) ⇒ |

and

5.3 2/4-Multisequents

161

⎡

| ↑∼2 A1 (a) ⇒ [∼2 A1 (a)]| ⎢ | ↑∼2 A2 (a) ⇒ [∼2 A2 (a)]| ⎢ ⎢ [Z 1 ]| ↑ 2 ∼2 A1 (a) ⇒ [Z 1 ]|[∼2 A1 (a)] ⎢ 2 A− ⎢ (∼  ) ⎢  [Z 1 ]| ↑∼2 A2 (a) ⇒ [Z 1 , ∼2 A2 (a)]| ⎣ [Z 5 ]| ↑∼2 A1 (a) ⇒ [Z 5 , ∼2 A1 (a)]| [Z 5 ]| ↑ 2 ∼2 A2 (a) ⇒ [Z 5 ]|[∼2 A2 (a)] | ↑∼2 (A1  A2 )(a) ⇒ [∼2 (A1  A2 )(a)]| where Z 5 = Z 1 , ∼3 A1 (a)∨ ∼2 A2 (a), and ⎧

| ↑∼2 A1 (a) ⇒ | ⎪ ⎪ ⎪ ⎪ | ↑∼2 A2 (a) ⇒ | ⎪ ⎪ ⎨

[Z 1 ]| ↑∼2 A1 (a) ⇒ [Z 1 ]| 2 A0 2 2 (∼  ) ⎪ ⎪ [Z 1 ]| ↑ 2 ∼2 A2 (a) ⇒ [Z 1 ]| ⎪ ⎪ [Z 5 ]| ↑ ∼ A1 (a) ⇒ [Z 5 ]| ⎪ ⎪ ⎩ [Z 5 ]| ↑∼2 A2 (a) ⇒ [Z 5 ]| | ↑∼2 (A1  A2 )(a) ⇒ | and ⎡

| ↑ B1 (b) ⇒ |[B1 (b)] ⎢ | ↑ B2 (b) ⇒ |[B2 (b)] ⎢ ⎢ [B1 (b) ∨ B2 (b)]| ↑ 1 B1 (b) ⇒ [B1 (b) ∨ B2 (b), B1 (b)]| ⎢ B− ⎢ ( ) ⎢  [B1 (b) ∨ B2 (b)]| ↑ B2 (b) ⇒ [B1 (b) ∨ B2 (b)]|[B2 (b)] ⎣ |[Z 6 ] ↑ B1 (b) ⇒ |[Z 6 , B1 (b)] |[Z 6 ] ↑ 1 B2 (b) ⇒ [B2 (b)]|[Z 6 ] | ↑ (B1 B2 )(a) ⇒ |[(B1 B2 )(a)] where Z 6 = B1 (b) ∨ B2 (b), ∼3 B1 (b) ∨ B2 (b), and ⎧

| ↑ B1 (b) ⇒ | ⎪ ⎪ ⎪ ⎪ | ↑ B2 (b) ⇒ | ⎪ ⎪ ⎨

[B1 (b) ∨ B2 (b)]| ↑ 1 B1 (b) ⇒ [B1 (b) ∨ B2 (b)]| B0 ( ) ⎪ ⎪

[B1 (b) ∨ B2 (b)]| ↑ B2 (b) ⇒ [B1 (b) ∨ B2 (b)]| ⎪ ⎪ |[Z 6 ] ↑ B1 (b) ⇒ |[Z 6 ] ⎪ ⎪ ⎩ |[Z 6 ] ↑ 1 B2 (b) ⇒ |[Z 6 ] | ↑ (B1 B2 )(a) ⇒ | and



| ↑∼2 B1 (b) ⇒ |[∼2 B1 (b)] (∼ ) | ↑∼2 B2 (b) ⇒ |[∼2 B2 (b)] 2 2

| ↑∼ 2(B1 B2 )(a) ⇒ |[∼ (B1 B2 )(a)] | ↑∼ B1 (b) ⇒ | (∼2 B0 ) | ↑∼2 B2 (b) ⇒ | | ↑∼2 (B1 B2 )(a) ⇒ | 2

B−

162

5 R-Calculi for Post L4 -Valued DL

and ⎡

| ↑ B1 (b) ⇒ |[B1 (b)] ⎢ | ↑ B2 (b) ⇒ |[B2 (b)] ⎢ ⎢ |[B1 (b) ∨ B2 (b)] ↑ 1 ∼2 B1 (b) ⇒ [∼2 B1 (b)]|[B1 (b) ∨ B2 (b)] ⎢ ⎢ |[B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ |[B1 (b) ∨ B2 (b), B2 (b)] ⎢ ⎢ |[Y7 ] ↑∼2 B1 (b) ⇒ |[Y7 , ∼2 B1 (b)] ⎢ B− ⎢ ( ) ⎢  |[Y7 ] ↑ B2 (b) ⇒ |[Y7 , B2 (b)] ⎢ |[Y8 ] ↑ B1 (b) ⇒ |[Y8 , B1 (b)] ⎢ ⎢ |[Y8 ] ↑ 1 ∼2 B2 (b) ⇒ [∼2 B2 (b)]|[B1 (b)] ⎢ ⎣ |[Y9 ] ↑ B1 (b) ⇒ |[Y9 , B1 (b)] |[Y9 ] ↑∼2 B2 (b) ⇒ |[Y9 , ∼2 B2 (b)] | ↑ (B1 B2 )(a) ⇒ |[(B1 B2 )(a)] ⎧ ⎨ Y7 = B1 (b) ∨ B2 (b), ∼ B1 (b) ∨ B2 (b), where Y8 = Y7 , ∼2 B1 (b) ∨ B2 (b), and ⎩ Y9 = Y8 , B1 (b)∨ ∼ B2 (b), ⎧

| ↑ B1 (b) ⇒ | ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

| ↑ B2 (b) ⇒ | 1 2 ⎪ ⎪ |[B1 (b) ∨ B2 (b)] ↑ ∼ B1 (b) ⇒ |[B1 (b) ∨ B2 (b)] ⎪ ⎪ ⎪ ⎪ |[B ⎪ 1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ |[B1 (b) ∨ B2 (b)] ⎪ ⎨

|[Y7 ] ↑∼2 B1 (b) ⇒ |[Y7 ] ( B0 ) ⎪ ⎪

|[Y7 ] ↑ B2 (b) ⇒ |[Y7 ] ⎪ ⎪ |[Y8 ] ↑ B1 (b) ⇒ |[Y8 ] ⎪ ⎪ ⎪ 1 2 ⎪ |[Y ⎪ 8 ] ↑ ∼ B2 (b) ⇒ |[Y8 ] ⎪

⎪ ⎪ |[Y9 ] ↑ B1 (b) ⇒ |[Y9 ] ⎪ ⎪ ⎩ |[Y9 ] ↑∼2 B2 (b) ⇒ |[Y9 ] | ↑ (B1 B2 )(a) ⇒ | and



| ↑∼2 B1 (b) ⇒ |[∼2 B1 (b)] (∼  ) | ↑∼2 B2 (b) ⇒ |[∼2 B2 (b)] 2 2

| ↑∼ 2(B1  B2 )(a) ⇒ |[∼ (B1  B2 )(a)] | ↑∼ B1 (b) ⇒ | (∼2  B0 ) | ↑∼2 B2 (b) ⇒ | | ↑∼2 (B1  B2 )(a) ⇒ | 2

and

B−

5.3 2/4-Multisequents

163

⎡

| ↑ 2 ∼2 R(a, f ) ⇒ |[∼2 R(a, f )] ⎢ |[∼2 R(a, f )] ↑ A( f ) ⇒ [A( f )]|[∼2 R(a, f )] ⎢ ⎢ [Z 1 ]| ↑ 2 R(a, f ) ⇒ [Z 1 ]|[R(a, f )] ⎢ ⎢ [Z 1 ]| ↑ 2 A( f ) ⇒ [Z 1 ]|[A( f )] ⎢ A− ⎢ (∀ ) ⎢ [Z 2 ]| ↑∼2 R(a, f ) ⇒ [Z 2 , ∼2 R(a, f )]| ⎢ [Z 2 ]| ↑∼ A( f ) ⇒ [Z 2 , ∼ A( f )]| ⎢ ⎣ [Z 2 ]| ↑∼2 R(a, f ) ⇒ [Z 2 , ∼2 R(a, f )]| [Z 2 ]| ↑∼2 A( f ) ⇒ [Z 2 , ∼2 A( f )]| | ↑ (∀R.A)(a) ⇒ [(∀R.A)(a)]| Z 1 =∼3 R(a, f ), A( f ); where Z 2 = Z 1 , ∼ R(a, f )∨ ∼ A( f ); and Z 2 = Z 2 , ∼2 R(a, f )∨ ∼ A( f ) ⎧ | ↑ 2 ∼2 R(a, e) ⇒ | ⎪ ⎪ ⎪ ⎪ | ↑ A(e) ⇒ | ⎪ ⎪

⎪ 2 ⎪ [Z ⎪ 1 ]| ↑ R(a, e) ⇒ [Z 1 ]| ⎪ ⎨ e)] ↑ 2 A(e) ⇒ [Z 1 ]|[R(a, e)]

[Z 1 ]|[R(a, A0 2 [Z 2 ]| ↑∼ R(a, e) ⇒ [Z 2 ]| (∀ ) ⎪ ⎪ ⎪ ⎪ [Z ⎪ 2 ]| ↑∼ A(e) ⇒ [Z 2 ]| ⎪

⎪ ⎪ [Z  ]| ↑∼2 R(a, e) ⇒ [Z 2 ]| ⎪ 2 ⎪ ⎩ [Z 2 ]| ↑∼2 A(e) ⇒ [Z 2 ]| | ↑ (∀R.A)(a) ⇒ | and



| ↑ R(a, e) ⇒ [R(a, e)]| [R(a, e)]| ↑ A(e) ⇒ [R(a, e), A(e)]|

| ↑ (∃R.A)(a) ⇒ [(∃R.A)(a)]| | ↑ R(a, f ) ⇒ | (∃ A0 ) [R(a, f )]| ↑ A( f ) ⇒ [R(a, f )]| | ↑ (∃R.A)(a) ⇒ |

(∃ A− )

and ⎡

| ↑ R(a, f ) ⇒ [R(a, f )]| ⎢ [R(a, f )]| ↑∼2 A( f ) ⇒ [R(a, f ), ∼2 A( f )]| ⎢ (∼2 ∀ A+ ) ⎣ | ↑ 2 R(a, f ) ⇒ |[R(a, f )] |[R(a, f )] ↑ 2 ∼2 A( f ) ⇒ |[R(a, f ), ∼2 A( f )] 2 2 | ⎧

↑ (∼ ∀R.A)(a) ⇒ [(∼ ∀R.A)(a)]| | ↑ R(a, f ) ⇒ | ⎪ ⎪ ⎨ 2

[R(a,2 f )]| ↑∼ A( f ) ⇒ [R(a, f )]| 2 A0 | ↑ R(a, f ) ⇒ | (∼ ∀ ) ⎪ ⎪ ⎩ |[R(a, f )] ↑ 2 ∼2 A( f ) ⇒ |[R(a, f )] | ↑ (∼2 ∀R.A)(a) ⇒ |

164

5 R-Calculi for Post L4 -Valued DL

⎡

and

| ↑∼2 R(a, e) ⇒ [∼2 R(a, e)]| ⎢ | ↑∼2 A(e) ⇒ [∼2 A(e)]| ⎢ ⎢ [Z 3 ]| ↑∼2 R(a, e) ⇒ [Z 3 , ∼2 R(a, e)]| ⎢ ⎢ [Z 3 ]| ↑ 2 A(e) ⇒ [Z 3 ]|[A(e)] ⎢ ⎢ [Z 4 ]| ↑∼2 R(a, e) ⇒ [Z 4 , ∼2 R(a, e)]| ⎢ 2 A− ⎢ [Z ]| ↑ A(e) ⇒ [Z , A(e)]| (∼ ∃ ) ⎢  4 4 ⎢ [Z 5 ]| ↑ 2 R(a, e) ⇒ [Z 5 ]|[R(a, e)] ⎢ ⎢ [Z 5 ]| ↑∼2 A(e) ⇒ [Z 5 , ∼2 A(e)]| ⎢ ⎣ [Z 6 ]| ↑ R(a, e) ⇒ [Z 6 , R(a, e)]| [Z 6 ]| ↑∼2 A(e) ⇒ [Z 6 , ∼2 A(e)]| 2 | ⇒ [(∼2 ∃R.A)(a)]| ⎧

↑ (∼ ∃R.A)(a) 2 ⎪ | ↑∼ R(a, f ) ⇒ | ⎪ ⎪ ⎪ ⎪ | ↑∼2 A( f ) ⇒ | ⎪ ⎪ ⎪ ⎪ [Z 3 ]| ↑∼2 R(a, f ) ⇒ [Z 3 ]| ⎪ ⎪ ⎪ ⎪ [Z 3 ]| ↑ 2 A( f ) ⇒ [Z 3 ]| ⎪ ⎨ [Z 4 ]| ↑∼2 R(a, f ) ⇒ [Z 4 ]|   2 A0 [Z (∼ ∃ ) ⎪ ⎪

4 ]| ↑ A( f ) ⇒ [Z 4 ]| ⎪   2 ⎪ ]| ↑ R(a, f ) ⇒ [Z [Z ⎪ 5 5 ]| ⎪ ⎪   2 ⎪ ]| ↑∼ A( f ) ⇒ [Z [Z ⎪

5 5 ]| ⎪ ⎪   ⎪ ]| ↑ R(a, f ) ⇒ [Z [Z ⎪ 6 6 ]| ⎪ ⎩ [Z 6 ]| ↑∼2 A( f ) ⇒ [Z 6 ]| | ↑ (∼2 ∃R.A)(a) ⇒ |

Z3 Z4 where Z5 Z6

=∼2 R(a, e)∨ ∼2 A(e); = Z 3 , ∼2 R(a, e)∨ ∼ A(e); = Z 4 , ∼2 R(a, e) ∨ A(e); = Z 5 , R(a, e)∨ ∼2 A(e); ⎡

Z 3 Z 4 Z 5 Z 6

=∼2 R(a, f )∨ ∼2 A( f ); = Z 3 , ∼2 R(a, f )∨ ∼ A( f ); and = Z 4 , ∼2 R(a, f ) ∨ A( f ); = Z 5 , R(a, f )∨ ∼2 A( f );

  | ↑ 1 R(a, f ) ⇒   [R(a, f )]| ⎢   | ↑ B( f ) ⇒   |[B( f )] ⎢  ⎢  |[Z 7 ] ↑ R(a, f ) ⇒   |[Z 7 , R(a, f )] ⎢ B− ⎢   |[Z ] ↑ 1 ∼2 B( f ) ⇒   [∼2 B( f )]|[Z ] (∀ ) ⎢  7 7 ⎣   |[Z 8 ] ↑ 1 ∼2 R(a, f ) ⇒   [∼2 R(a, f )]|[Z 8 ]   |[Z 8 ] ↑∼2 B( f ) ⇒   |[Z 8 , ∼2 B( f )]   | ↑ (∀R.B)(a) ⇒   |[(∀R.B)(a)] ⎧   | ↑ 1 R(a, e) ⇒   | ⎪ ⎪ ⎪ ⎪   | ↑ B(e) ⇒   | ⎪ ⎪ ⎨   |[Z 7 ] ↑ R(a, e) ⇒   |[Z 7 ]   2 B0 B(e) ⇒   |[Z 7 ] (∀ ) ⎪ ⎪

  |[Z 7 ] ↑∼ ⎪ 1 ⎪  |[Z 8 ] ↑ ∼2 R(a, e) ⇒   |[Z 8 ] ⎪ ⎪ ⎩   |[Z 8 ] ↑∼2 B(e) ⇒   |[Z 8 ]   | ↑ (∀R.B)(a) ⇒   |

5.3 2/4-Multisequents

where

165

Z 7 = R(a, f )∨ ∼ B( f ), Z 7 = R(a, e)∨ ∼ B(e), and 2 Z 8 = Z 7 , ∼ R(a, f )∨ ∼ B( f ), Z 8 = Z 7 , ∼ R(a, e)∨ ∼2 B(e), ⎡

  | ↑ R(a, e) ⇒   |[R(a, e)] ⎢   | ↑ B(e) ⇒   |[B(e)] ⎢  ⎢  |[Z 9 ] ↑ 1 R(a, e) ⇒   [R(a, e)]|[Z 9 ] ⎢ B− ⎢   |[Z ] ↑ B(e) ⇒   |[Z , B(e)] (∃ ) ⎢  9 9 ⎣   |[Z 10 ] ↑ R(a, e) ⇒   |[Z 10 , R(a, e)]   |[Z 10 ] ↑ 1 B(e) ⇒   [B(e)]|[Z 10 ]   | ↑ (∃R.B)(a) ⇒   |[(∃R.B)(a)] ⎧   | ↑ R(a, f ) ⇒   | ⎪ ⎪ ⎪ ⎪   | ↑ B( f ) ⇒   |[B( f )] ⎪ ⎪ ⎨   |[Z 9 ] ↑ 1 R(a, f ) ⇒   |[Z 9 ]    B0 f )]|[Z 9 ] (∃ ) ⎪ ⎪

  |[Z 9 ] ↑ B( f ) ⇒  [R(a, ⎪   ⎪ ]  |[Z 10 ] ↑ R(a, f ) ⇒  |[Z 10 ⎪ ⎪ ⎩    1   |[Z 10 ] ↑ B( f ) ⇒  |[Z 10 ]   | ↑ (∃R.B)(a) ⇒   | where

Z 9 =∼ R(a, e) ∨ B(e), Z 9 =∼ R(a, f ) ∨ B( f ) and  = Z 9 , R(a, f )∨ ∼ B( f ), Z 10 = Z 9 , R(a, e)∨ ∼ B(e), Z 10 

  | ↑∼2 R(a, f ) ⇒   |[∼2 R(a, f )] (∼ ∀ )   | ↑∼2 B( f ) ⇒   |[∼2 B( f )]  ↑ (∼2 ∀R.B)(a) ⇒   |[(∼2 ∀R.B)(a)]

 |   | ↑∼2 R(a, e) ⇒   | 2 B0 (∼ ∀ )   | ↑∼2 B(e) ⇒   |   | ↑ (∀R.B)(a) ⇒   | 2

and

B−



  | ↑∼2 R(a, e) ⇒   |[∼2 R(a, e)] (∼ ∃ )   | ↑∼2 B(e) ⇒   |[∼2 B(e)]  ↑ (∼2 ∃R.B)(a) ⇒   |[(∼2 ∃R.B)(a)]

 |   | ↑∼2 R(a, e) ⇒   | 2 B0 (∼ ∃ )   | ↑∼2 B(e) ⇒   |   | ↑ (∼2 ∃R.B)(a) ⇒   | 2

B−

Definition 5.3.3 A 2/4-reduction δ = | ↑ (A(a), B(b)) ⇒   | is provable in =t St 2/4 , denoted by 2/4 δ, if there is a sequence {δ1 , . . . , δn } of 2/4-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/4-reductions by one of the deduction rules in St 2/4 .

166

5 R-Calculi for Post L4 -Valued DL

Theorem 5.3.4 (Soundness and completeness theorem) For any reduction δ = | ↑ (A(a), B(a  )) ⇒   | , =t

=t

|=2/4 δ iff 2/4 δ. 

5.4 3/4-Multisequents =t⊥

t⊥ A 3/4-multisequent || is N3/4 -valid, denoted by |=3/4 ||, if for any interpretation I, either I (A(a)) = t for some A(a) ∈ , or I (B(b)) =  for some B(b) ∈ , or I (C(c)) =⊥ for some C(c) ∈ . We have the following equivalences:

, ∼ A(a)|| ≡ |, A(a)| |, ∼ B(b)| ≡ ||, B(b) ||, ∼ C(c) ≡ ||, ∼ C(c) , C(c)|| ≡ ||, ∼2 C(c) Given three sets , ,  of literals, we move each statement in  beginning with ∼ from  to  with ∼2 from  to  with ∼3 from  to  with ∼ move each statement in  beginning with ∼ from  to  with ∼2 from  to  with ∼ with ∼3 from  to  and move each statement in  beginning with ∼ from  ramaining in  with ∼2 from  to  with ∼3 from  to 

5.4 3/4-Multisequents

167

so that ,  become sets of atoms and  is a set of literals beginning with one ∼ only.       ∼ A(a) → A(a) ∼ B(b) → B(b) A(a) ∼2 B(b) → ∼ B(b) ∼2 A(a) → 3 ∼ A(a) B(b) ←∼3 B(b) ∼ A(a) →    ∼ C(c) C(c) ←∼2 C(c) C(c) ←∼3 C(c)

t⊥ 5.4.1 Deduction System N3/4 t⊥ Deduction system N3/4 contains the following axiom and deduction rules: • Axiom: ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎨ ∩ ∼−  = ∅ t (A3/4 ) ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎩ ∩ ∼−  = ∅ ||,

where ,  are sets of atomic statements, and  is a set of literal statements. • Deduction rules for unary logical connective ∼: (∼ A )

|, A(a)| ||, B(b) , C(c)|| (∼ B ) (∼2C ) , ∼ A(a)|| |, ∼ B(b)| ||, ∼2 C(c)

• Deduction rules for binary logical connective :

168

5 R-Calculi for Post L4 -Valued DL

⎡

|, B1 (b)| ⎢ |, B2 (b)| ⎢  ⎢ , B1 (b)|| , A1 (a)|| ⎢ A B ( ) , A2 (a)|| ( ) ⎢ ⎢  |, B2 (b)| ⎣ |, B1 (b)| , (A1 A2 )(a)|| , B2 (b)|| |, (B1 B2 )(a)| ⎡ ||, C2 (c) ⎢ ||, C2 (c) ⎢ ⎢ |, C1 (c)| ⎢ ⎢ ||, C2 (c) ⎢

⎢ , C1 (c)|| ||, ∼ C2 (c) ⎢ C (c) ||, C ||, ∼ C2 (c) ( C ) ⎢ (∼

) 2 ⎢ ⎢ ||, C1 (c) ||, ∼ (C1 C2 )(a) ⎢ ⎢ , C2 (c)|| ⎢ ⎣ ||, C1 (c) |, C2 (c)| ||, (C1 C2 )(a) • Deduction rules for binary logical connective : ⎡

|, B1 (b)| ⎢ |, B2 (b)| ⎢ ⎢ ||, B1 (b) ⎢ ⎢ |, B2 (b)| ⎢

⎢ ||, ∼ B1 (b) , A1 (a)|| ⎢ A B ⎢ |, B (b)| (a)|| , A ( ) ( ) ⎢  2 2 ⎢ |, B1 (b)| , (A1  A2 )(a)|| ⎢ ⎢ ||, B2 (b) ⎢ ⎣ |, B1 (b)| ||, ∼ B2 (b) |, (B1  B2 )(a)| ⎡ ||, C1 (c) ⎢ ||, C2 (c) ⎢  ⎢ ||, ∼ C1 (c) ||, ∼ C2 (c) ⎢ C ⎢ ||, C (c) C ||, ∼ C2 (c) ( ) ⎢  (∼  ) 2 ⎣ ||, C1 (c) ||, ∼ (C1  C2 )(a) ||, ∼ C2 (c) ||, (C1  C2 )(a) • Deduction rules for quantifier ∀:

5.4 3/4-Multisequents

169

⎡

||, ∼ R(a, f ) ⎡ ⎢ , A( f )|| , R(a, f )|| ⎢ ⎢ |, R(a, f )| ⎢ |, B( f )| ⎢ ⎢ ⎢ |, A( f )| ⎢ |, R(a, f )| ⎢ ⎢ A ⎢ ||, R(a, f ) B ⎢ ||, B( f ) (∀ ) ⎢ (∀ ) ⎢  ⎢ |, A( f )| ⎣ ||, R(a, f ) ⎢ ⎣ ||, R(a, f ) ||, ∼ B( f ) ||, A( f ) |, (∀R.B)(a)| , ⎡ (∀R.A)(a)|| , R(a, f )||  ⎢ ||, C( f ) , R(a, e)|| ⎢ (∀C ) ⎣ |, R(a, f )| (∼ ∀C ) ||, ∼ C(e) ||, ∼ C( f ) ||, ∼ (∀R.C)(a) ||, (∀R.C)(a) ⎡

and 

(∃ A )

, R(a, e)|| , A(e)|| , (∃R.A)(a)|| ⎡

||, R(a, e) ⎢ ||, C(e) ⎢ ⎢ |, R(a, e)| ⎢ ⎢ ||, C(e) ⎢ ⎢ , R(a, e)|| ⎢ C ⎢ ||, C(e) (∃ ) ⎢  ⎢ ||, R(a, e) ⎢ ⎢ |, C(e)| ⎢ ⎣ ||, R(a, e) , C(e)|| ||, (∃R.C)(a)

|, R(a, e)| ⎢ |, B(e)| ⎢ ⎢ |, R(a, e)| ⎢ B ⎢ , B(e)|| (∃ ) ⎢  ⎣ , R(a, e)|| |, B(e)| |, (∃R.B)(a)|



(∼ ∃C )

, R(a, f )|| ||, ∼ C( f ) ||, ∼ (∃R.C)(a)

where f is a constant and e is a new constant not occurring in , , . =t⊥

t⊥ Definition 5.4.1 A 3/4-multisequent || is provable in N3/4 , denoted by 3/4 || if there is a sequence {1 |1 |1 , . . . , n |n |n } of 3/4-multisequents such that n |n |n = ||, and for each 1 ≤ i ≤ n, i |i |i is either an axiom or t⊥ . deduced from the previous 3/4-multisequents by one of the deduction rules in N3/4

170

5 R-Calculi for Post L4 -Valued DL

Theorem 5.4.2 For any 3/4-multisequent ||, =t⊥

|=3/4

=t⊥

|| iff 3/4

||. 

5.4.2 R-Calculus St⊥ 3/4 Given a 3/4-multisequent || and three statements A(a) ∈ , B(b) ∈ , C(c) ∈ , a 3/4-reduction δ = || ↑ (A(a), B(b), C(e)) ⇒ [A (a)]|[B  (b)]|[C  (c)] =t⊥

is St⊥ -valid, denoted by |=3/4 

A(a) λ  B(b)  B (b) = λ  C(c)  C (c) = λ A (a) =

δ, if

t⊥ -valid if [A(a)]|| is N3/4 otherwise; t⊥ if [A (a)]|[B(b)]| is N3/4 -valid otherwise; t⊥ if [A (a)]|[B  (b)]|[C(c)] is N3/4 -valid otherwise.

Let X = ||. Given a statement triple X = (A(a), B(b), C(c)) such that A(a) ∈ , B(b) ∈ , C(c) ∈ , X = [A (a)]|| X = [A (a)]|[B  (b)]| X = [A (a)]|[B  (b)]|[C  (c)] X[1 A(a)] =  − {A(a)}|| X[2 A(a)] = | − {A(a)}| X[3 A(a)] = || − {A(a)};

X[A(a)] =  − {A(a)}|| X[B(b)] = | − {B(b)}| X[C(c)] = || − {C(c)}; X[1 R(a, b)] =  − {R(a, b)}|| X[2 R(a, b)] = | − {R(a, b)}| X[3 R(a, b)] = || − {R(a, b)}

R-calculus St⊥ 3/4 consists of the following axioms and deduction rules: • Axioms:

5.4 3/4-Multisequents

⎡

[s(a)] ∩  = ∅ ⎢ [s(a)] ∩  = ∅ ⎢ ⎢ [s(a)]∩ ∼−  = ∅ ⎢ A0 ⎢  ∩  = ∅ (A3/4 ) ⎢ ⎣ ∩ ∼−  = ∅ ∩ ∼−  = ∅ X|s(a) ⎡  ⇒X  ∩ [t (b)] = ∅ ⎢  ∩  = ∅ ⎢  ⎢  ∩ ∼−  = ∅ ⎢ B0 ⎢ [t (b)] ∩  = ∅ (A3/4 ) ⎢ ⎣ [t (b)]∩ ∼−  = ∅ ∩ ∼−  = ∅   X ⎡ |t(b) ⇒ X  ∩ =∅ ⎢   ∩ [u(c)] = ∅ ⎢  ⎢  ∩ ∼−  = ∅ ⎢ C0 ⎢  ∩ [u(c)] = ∅ (A3/4 ) ⎢ ⎣  ∩ ∼−  = ∅ [u(c)]∩ ∼−  = ∅  X⎡ |u(c) ⇒ X   ∩  = ∅ ⎢  ∩  = ∅ ⎢  ⎢  ∩ ∼− [∼ u(c)] = ∅ ⎢ C∼0 ⎢  ∩  = ∅ (A3/4 ) ⎢ ⎣  ∩ ∼− [∼ u(c)] = ∅ ∩ ∼− [∼ u(c)] = ∅  X | ∼ u(c) ⇒ X

171

⎧ ⎪ ⎪ [s(a)] ∩  = ∅ ⎪ ⎪ [s(a)] ∩  = ∅ ⎪ ⎪ ⎨ [s(a)]∩ ∼−  = ∅ A−  ∩  = ∅ (A3/4 ) ⎪ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎪ ⎩ ∩ ∼−  = ∅  X ⇒ X [s(a)] ⎧ |s(a)   ∩ [t (b)] = ∅ ⎪ ⎪ ⎪  ⎪ ∩  = ∅  ⎪ ⎪ ⎨   ∩ ∼−  = ∅ B− [t (b)] ∩  = ∅ (A3/4 ) ⎪ ⎪ ⎪ ⎪ [t (b)]∩ ∼−  = ∅ ⎪ ⎪ ⎩ ∩ ∼−  = ∅   X ⎧ |t (b) ⇒ X [t (b)]  ∩  = ∅ ⎪ ⎪ ⎪ ⎪   ∩ [u(c)] = ∅ ⎪ ⎪ ⎨   ∩ ∼−  = ∅ C−  ∩ [u(c)] = ∅ (A3/4 ) ⎪ ⎪ ⎪ ⎪  ∩ ∼−  = ∅ ⎪ ⎪ ⎩ [u(c)]∩ ∼−  = ∅  X⎧|u(c) ⇒ X [u(c)]   ∩   = ∅ ⎪ ⎪ ⎪ ⎪   ∩  = ∅ ⎪ ⎪ ⎨   ∩ ∼− [∼ u(c)] = ∅ C∼−  ∩   = ∅ (A3/4 ) ⎪ ⎪ ⎪ ⎪  ∩ ∼− [∼ u(c)] = ∅ ⎪ ⎪ ⎩ ∩ ∼− [∼ u(c)] = ∅  X | ∼ u(c) ⇒ X [∼ u(c)]

where ,  are sets of atoms,  is a set of literals, and s(a), t (b), u(c) are atoms such that s(a) ∈ , t (b) ∈ , u(c), ∼ u(c) ∈ . • Deduction rules consists of three parts E A , E B , EC . ◦ EA:

172

5 R-Calculi for Post L4 -Valued DL

X ↑ 2 A(a) ⇒ X X ↑∼ A(a) ⇒ X X ↑ 2 A(a) ⇒ X[2 A(a)] (∼ A− ) X ↑∼ A(a) ⇒ X[∼ A(a)] X ↑ A1 (a) ⇒ X[A1 (a)] ( A− ) X ↑ A2 (a) ⇒ X[A2 (a)] X ↑ (A1 A2 )(a) ⇒ X[(A1 A2 )(a)] X ↑ A1 (a) ⇒ X ( A0 ) X ↑ A2 (a) ⇒ X X

↑ (A1 A2 )(a) ⇒ X X ↑ A1 (a) ⇒ X[A1 (a)] ( A− ) X[A1 (a)] ↑ A2 (a) ⇒ X[A1 (a), A2 (a)] X ↑ (A1  A2 )(a) ⇒ X[(A1  A2 )(a)] X ↑ A1 (a) ⇒ X ( A0 ) X[A1 (a)] ↑ A2 (a) ⇒ X[A1 (a)] X ↑ (A1  A2 )(a) ⇒ X

(∼ A0 )

and

where

⎡

X ↑ 3 ∼ R(a, f ) ⇒ X[3 ∼ R(a, f )] ⎢ X ↑ A( f ) ⇒ X[A( f )] ⎢ ⎢ X[Z 1 ] ↑ 2 R(a, f ) ⇒ X[Z 1 , 2 R(a, f )] ⎢ ⎢ X[Z 1 ] ↑ 2 A( f ) ⇒ X[Z 1 , 2 A( f )] ⎢ A− ⎢ X[Z ] ↑ 3 R(a, f ) ⇒ X[Z , 3 R(a, f )] (∀ ) ⎢ 2 2 ⎢ X[Z 2 ] ↑ 2 A( f ) ⇒ X[Z 2 , 2 A( f )] ⎢ ⎣ X[Z 3 ] ↑ 3 R(a, f ) ⇒ X[Z 3 , 3 R(a, f )] X[Z 3 ] ↑ 3 A( f ) ⇒ X[Z 3 , 3 A( f )] X ↑ (∀R.A)(a) ⇒ X[(∀R.A)(a)] ⎧ ⎪ X ↑ 3 ∼ R(a, e) ⇒ X ⎪ ⎪ 3 3 ⎪ ∼ R(a, e)] ⎪ X[ ⎪

∼ R(a,2 e)] ↑ A(e) ⇒ X[ ⎪  ⎪ ] ↑ R(a, e) ⇒ X[Z ] X[Z ⎪ 1 1 ⎪ ⎨   2

X[Z 1 ] ↑ 3 A(e) ⇒ X[Z 1 ]  A0 ] ↑ R(a, e) ⇒ X[Z X[Z (∀ ) ⎪ ⎪ 2 2] ⎪   2 ⎪ ] ↑ A(e) ⇒ X[Z ] X[Z ⎪ 2 2 ⎪

⎪ ⎪ X[Z 3 ] ↑ 3 R(a, e) ⇒ X[Z 3 ] ⎪ ⎪ ⎩ X[Z 3 ] ↑ 3 A(e) ⇒ X[Z 3 ] X ↑ (∀R.A)(a) ⇒ X Z 1 = R(a, f ), A( f ); Z 2 = Z 1 , ∼ R(a, f )∨ ∼ A( f ); Z 3 = Z 2 , ∼2 R(a, f )∨ ∼ A( f ), Z 1 = R(a, e), A(e); Z 2 = Z 1 , ∼ R(a, e)∨ ∼ A(e); Z 3 = Z 2 , ∼2 R(a, e)∨ ∼ A(e),

5.4 3/4-Multisequents

173



X ↑ R(a, e) ⇒ X[R(a, e)] X ↑ A(e) ⇒ X[A(e)] X

↑ (∃R.A)(a) ⇒ X[(∃R.A)(a)] X ↑ R(a, f ) ⇒ X (∃ A0 ) X ↑ A( f ) ⇒ X X ↑ (∃R.A)(a) ⇒ X

(∃ A− )

where e is a new constant and f is a constant. ◦ EB : X ↑ 3 B(b) ⇒ X X ↑∼ B(b) ⇒ X X ↑ 3 B(b) ⇒ X [3 B(b)] (∼ B− )  X↑∼ B(b) ⇒ X [∼ B(b)] ⎡ X ↑ B1 (b) ⇒ X [B1 (b)] ⎢ X ↑ B2 (b) ⇒ X [B2 (b)] ⎢  ⎢ X [B1 (b) ∨ B2 (b)] ↑ 1 B1 (b) ⇒ X [B1 (b) ∨ B2 (b), 1 B1 (b)] ⎢ B− ⎢ X [B (b) ∨ B (b)] ↑ B (b) ⇒ X [B (b) ∨ B (b), B (b)] ( ) ⎢  1 2 2 1 2 2 ⎣ X [Y1 ] ↑ B1 (b) ⇒ X [Y1 , B1 (b)] X [Y1 ] ↑ 1 B2 (b) ⇒ X [Y1 , 1 B2 (b)]  X ↑ (B1 B2 )(a) ⇒ X [(B1 B2 )(a)] ⎧ ⎪ X ↑ B1 (b) ⇒ X ⎪ ⎪ ⎪ X ↑ B2 (b) ⇒ X ⎪ ⎪ ⎨  X [B1 (b) ∨ B2 (b)] ↑ 1 B1 (b) ⇒ X [B1 (b) ∨ B2 (b)]   ( B0 ) ⎪ ⎪

X [B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ X [B1 (b) ∨ B2 (b)] ⎪ ⎪ X [Y1 ] ↑ B1 (b) ⇒ X [Y1 ] ⎪ ⎪ ⎩ X [Y1 ] ↑ 1 B2 (b) ⇒ X [Y1 ] X ↑ (B1 B2 )(a) ⇒ X (∼ B0 )

where Y1 = B1 (b) ∨ B2 (b), ∼3 B1 (b) ∨ B2 (b), and ⎡

X ↑ B1 (b) ⇒ X [B1 (b)] ⎢ X ↑ B2 (b) ⇒ X [B2 (b)] ⎢  ⎢ X [B1 (b) ∨ B2 (b)] ↑ 3 B1 (b) ⇒ X [B1 (b) ∨ B2 (b), 3 B1 (b)] ⎢ ⎢ X [B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ X [B1 (b) ∨ B2 (b), B2 (b)] ⎢  ⎢ X [Y4 ] ↑ 3 ∼ B1 (b) ⇒ X [Y4 , 3 ∼ B1 (b)] ⎢ B− ⎢ ( ) ⎢  X [Y4 ] ↑ B2 (b) ⇒ X [Y4 , B2 (b)] ⎢ X [Y5 ] ↑ B1 (b) ⇒ X [Y5 , B1 (b)] ⎢ ⎢ X [Y5 ] ↑ 3 ∼ B2 (b) ⇒ X [Y5 , 3 ∼ B2 (b)] ⎢  ⎣ X [Y6 ] ↑ B1 (b) ⇒ X [Y6 , B1 (b)] X [Y6 ] ↑ 3 B2 (b) ⇒ X [Y6 , 3 B2 (b)]  X ↑ (B1  B2 )(a) ⇒ X [(B1  B2 )(a)]

174

5 R-Calculi for Post L4 -Valued DL

⎧ ⎨ Y4 = B1 (b) ∨ B2 (b), ∼ B1 (b) ∨ B2 (b), where Y5 = Y4 , ∼2 B1 (b) ∨ B2 (b), and ⎩ Y6 = Y5 , B1 (b)∨ ∼ B2 (b), ⎧  X ↑ B1 (b) ⇒ X ⎪ ⎪ ⎪ ⎪ X ↑ B2 (b) ⇒ X ⎪ ⎪ ⎪  ⎪ X [B1 (b) ∨ B2 (b)] ↑ 3 B1 (b) ⇒ X [B1 (b) ∨ B2 (b)] ⎪ ⎪ ⎪ ⎪ X [B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ X [B1 (b) ∨ B2 (b)] ⎪ ⎪ ⎨  X [Y4 ] ↑ 3 ∼ B1 (b) ⇒ X [Y4 ] B0   ( ) ⎪ ⎪

X [Y4 ] ↑ B2 (b) ⇒ X [Y4 ] ⎪ ⎪ X [Y5 ] ↑ B1 (b) ⇒ X [Y5 ] ⎪ ⎪ ⎪  3 ⎪ X [Y5 ] X ⎪ ⎪

 [Y5 ] ↑ ∼ B2 (b) ⇒ ⎪  ⎪ X [Y6 ] ↑ B1 (b) ⇒ X [Y6 ] ⎪ ⎪ ⎩ X [Y6 ] ↑ 3 B2 (b) ⇒ X [Y6 ] X ↑ (B1  B2 )(a) ⇒ X and ⎡

X ↑ 1 R(a, f ) ⇒ X [1 R(a, f )] ⎢ X ↑ B( f ) ⇒ X [B( f )] ⎢  1 ⎢ X [ R(a, f ) ∨ B( f )] ↑ 2 R(a, f ) ⇒ X [1 R(a, f ) ∨ B( f ), 2 R(a, f )] ⎢ B− ⎢ X [1 R(a, f ) ∨ B( f )] ↑ 3 B( f ) ⇒ X [1 R(a, f ) ∨ B( f ), 3 B( f )] (∀ ) ⎢  ⎣ X [Z 3 ] ↑ 3 R(a, f ) ⇒ X [Z 3 , 3 R(a, f )] X [Z 3 ] ↑ 3 ∼ B( f ) ⇒ X [Z 3 , 3 ∼ B( f )]  X↑ (∀R.B)(a) ⇒ X [(∀R.B)(a)] ⎡ X ↑ 1 R(a, e) ⇒ X ⎢ X ↑ B(e) ⇒ X ⎢  1 ⎢ X [ R(a, e) ∨ B(e)] ↑ 2 R(a, e) ⇒ X [1 R(a, e) ∨ B(e)] ⎢ B0 ⎢ X [1 R(a, e) ∨ B(e)][2 R(a, e)] ↑ 3 B(e) ⇒ X [1 R(a, e) ∨ B(e)] (∀ ) ⎢  ⎣ X [Z 3 ] ↑ 3 R(a, e) ⇒ X [Z 3 ] X [Z 3 ] ↑ 3 ∼ B(e) ⇒ X [Z 3 ]  X ↑ (∀R.B)(a) ⇒ X where

Z 3 = 1 R(a, f ) ∨ B( f ), 2 R(a, f ) ∨ 3 B( f ), and Z 3 = 1 R(a, e) ∨ B(e), 2 R(a, e) ∨ 3 B(e),

5.4 3/4-Multisequents

175

⎡

X ↑ 2 R(a, e) ⇒ X [2 R(a, e)] ⎢ X ↑ B(e) ⇒ X [B(e)] ⎢  2 ⎢ X [ R(a, e) ∨ B(e)] ↑ 1 R(a, e) ⇒ X [2 R(a, e) ∨ B(e), 1 R(a, e)] ⎢ B− ⎢ X [2 R(a, e) ∨ B(e)] ↑ B(e) ⇒ X [2 R(a, e) ∨ B(e), B(e)] (∃ ) ⎢  ⎣ X [Z 4 ] ↑ 2 R(a, e) ⇒ X [Z 4 , 2 R(a, e)] X [Z 4 ] ↑ 1 B(e) ⇒ X [Z 4 , 1 B(e)]  X↑ (∃R.B)(a) ⇒ X [(∃R.B)(a)] ⎡ X ↑ 2 R(a, f ) ⇒ X ⎢ X ↑ B( f ) ⇒ X ⎢  2 ⎢ X [ R(a, f ) ∨ B( f )] ↑ 1 R(a, f ) ⇒ X [2 R(a, f ) ∨ B( f )] ⎢ B0 ⎢ X [2 R(a, f ) ∨ B( f )] ↑ B( f ) ⇒ X [2 R(a, f ) ∨ B( f )] (∃ ) ⎢  ⎣ X [Z 4 ] ↑ 2 R(a, f ) ⇒ X [Z 4 ] X [Z 4 ] ↑ 1 B( f ) ⇒ X [Z 4 ]  X ↑ (∃R.B)(a) ⇒ X Z 4 = 2 R(a, e) ∨ B(e), 1 R(a, e) ∨ B(e), and e is a new constant, f is a Z 4 = 2 R(a, f ) ∨ B( f ), 1 R(a, f ) ∨ B( f ), constant. ◦ EC :

where

X ↑ 1 C(c) ⇒ X X ↑∼2 C(c) ⇒ X X ↑ 1 C(c) ⇒ X [1 C(c)] (∼2C− )  2 C(c) ⇒ X [∼2 C(c)] ⎡X ↑∼  X ↑ C1 (c) ⇒ X [C1 (c)] ⎢ X ↑ C2 (c) ⇒ X [C2 (c)] ⎢   ⎢ X [C1 (c) ∨ C2 (c)] ↑ 1 C1 (c) ⇒ X [C1 (c) ∨ C2 (c), 1 C1 (c)] ⎢ ⎢ X [C1 (c) ∨ C2 (c)] ↑ C2 (c) ⇒ X [C1 (c) ∨ C2 (c), C2 (c)] ⎢    ⎢ X [Y ] ↑ 2 C1 (c) ⇒ X [Y  , 2 C1 (c)] 1 1 ⎢ C− ⎢ X [Y  ] ↑ C (c) ⇒ X [Y  , C (c)] ( ) ⎢  2 2 1 1 ⎢ X [Y2 ] ↑ C1 (c) ⇒ X [Y2 , C1 (c)] ⎢ ⎢ X [Y2 ] ↑ 2 C2 (c) ⇒ X [Y2 , 2 C2 (c)] ⎢   ⎣ X [Y3 ] ↑ C1 (c) ⇒ X [Y3 , C1 (c)] X [Y3 ] ↑ 1 C2 (c) ⇒ X [Y3 , 1 C2 (c)]  X ↑ (C1 C2 )(a) ⇒ X [(C1 C2 )(a)] (∼2C0 )

⎧  ⎨ Y1 = C1 (c) ∨ C2 (c), ∼3 C1 (c) ∨ C2 (c), where Y2 = Y1 , ∼4 C1 (c) ∨ C2 (c), and ⎩ Y3 = Y2 , C1 (c)∨ ∼4 C2 (c),

176

5 R-Calculi for Post L4 -Valued DL

⎧   ⎪ ⎪ X ↑ C1 (c) ⇒ X ⎪ ⎪ ⎪ ⎪

X ↑ C2 (c) ⇒ X ⎪ ⎪ X [C1 (c) ∨ C2 (c)] ↑ 1 C1 (c) ⇒ X [C1 (c) ∨ C2 (c)] ⎪ ⎪ ⎪ ⎪ X [C1 (c) ∨ C2 (c)] ↑ C2 (c) ⇒ X [C1 (c) ∨ C2 (c)] ⎪ ⎪ ⎨   X [Y1 ] ↑ 2 C1 (c) ⇒ X [Y1 ] C0     ( ) ⎪ ⎪ X [Y1 ] ↑ C2 (c) ⇒ X [Y1 ] ⎪ ⎪ X [Y2 ] ↑ C1 (c) ⇒ X [Y2 ] ⎪ ⎪ ⎪  2  ⎪ ⎪ ⎪ X [Y2 ] ↑ C2 (c) ⇒ X [Y2 ] ⎪ ⎪ X [Y3 ] ↑ C1 (c) ⇒ X [Y3 ] ⎪ ⎪ ⎩ X [Y3 ] ↑ 1 C2 (c) ⇒ X [Y3 ]  X ↑ (C1 C2 )(a) ⇒ X and



X ↑∼ C1 (c) ⇒ X [∼ C1 (c)] (∼ ) X [∼ C1 (c)] ↑∼ C2 (c) ⇒ X [∼ C1 (c), ∼ C2 (c)]  

X ↑∼ (C1 C2 )(a) ⇒ X [∼ (C1 C2 )(a)] X ↑∼ C1 (c) ⇒ X (∼ C0 ) X [∼ C1 (c)] ↑∼ C2 (c) ⇒ X [∼ C1 (c)] X ↑∼ (C1 C2 )(a) ⇒ X C−

⎡

X ↑ C1 (c) ⇒ X [C1 (c)] ⎢ X ↑ C2 (c) ⇒ X [C2 (c)] ⎢   ⎢ X [C1 (c) ∨ C2 (c)] ↑∼ C1 (c) ⇒ X [C1 (c) ∨ C2 (c), ∼ C1 (c)] ⎢ C− ⎢ X [C (c) ∨ C (c)] ↑ C (c) ⇒ X [C (c) ∨ C (c), C (c)] ( ) ⎢  1 2 2 1 2 2 ⎣ X [Y7 ] ↑ C1 (c) ⇒ X [Y7 , C1 (c)] X [Y7 ] ↑∼ C2 (c) ⇒ X [Y7 , ∼ C2 (c)] X ↑ (C1  C2 )(a) ⇒ X [(C1  C2 )(a)] ⎧ ⎪ X ↑ C1 (c) ⇒ X ⎪ ⎪ ⎪ X ↑ C2 (c) ⇒ X ⎪ ⎪ ⎨  X [C1 (c) ∨ C2 (c)] ↑∼ C1 (c) ⇒ X [C1 (c) ∨ C2 (c)] (C0 ) ⎪ ⎪ X [C1 (c) ∨ C2 (c)] ↑ C2 (c) ⇒ X [C1 (c) ∨ C2 (c)] ⎪ ⎪ X [Y7 ] ↑ C1 (c) ⇒ X [Y7 ] ⎪ ⎪ ⎩ X [Y7 ] ↑∼ C2 (c) ⇒ X [Y7 ] X ↑ (C1  C2 )(a) ⇒ X where Y7 = C1 (c) ∨ C2 (c), ∼ C1 (c) ∨ C2 (c), and

X ↑∼ C1 (c) ⇒ X [∼ C1 (c)] (∼  ) X [∼ C1 (c)] ↑∼ C2 (c) ⇒ X [∼ C1 (c), ∼ C2 (c)]   X ↑∼ (C1  C2 )(a)⇒ X [∼ (C1  C2 )(a)] X ↑∼ C1 (c) ⇒ X (∼ C0 ) X [∼ C1 (c)] ↑∼ C2 (c) ⇒ X [∼ C1 (c)] X ↑∼ (C1  C2 )(a) ⇒ X C−

and

5.4 3/4-Multisequents

⎡   1 X ↑ R(a, f ) ⇒ X [1 R(a, f )] ⎢ X ↑ C( f ) ⇒ X [C( f )] ⎢ (∀C− ) ⎣ X [1 R(a, f ) ∨ C( f )] ↑ 2 R(a, f ) ⇒ X [1 R(a, f ) ∨ C( f ), 2 R(a, f )] X [1 R(a, f ) ∨ C( f )] ↑ 3 ∼ C( f ) ⇒ X [1 R(a, f ) ∨ C( f ), 3 ∼ C( f )]  X ↑ (∀R.C)(a) ⇒ X [(∀R.C)(a)] ⎡   1 X ↑ R(a, e) ⇒ X ⎢ X ↑ C(e) ⇒ X ⎢ (∀C0 ) ⎣ X [1 R(a, e) ∨ C(e)] ↑ 2 R(a, e) ⇒ X [1 R(a, e) ∨ C(e)] X [1 R(a, e) ∨ C(e)] ↑ 3 ∼ C(e) ⇒ X [1 R(a, e) ∨ C(e)] X ↑ (∀R.C)(a) ⇒ X X ↑ 1 R(a, e) ⇒ X [1 R(a, e)] C− (∼ ∀ ) X ↑∼ C(e) ⇒ X [∼ C(e)]  

X ↑∼1(∀R.C)(a) ⇒ X [∼ (∀R.C)(a)] X ↑ R(a, f ) ⇒ X (∼ ∀C0 ) X ↑∼ C( f ) ⇒ X X ↑ (∀R.C)(a) ⇒ X

and ⎡

X ↑ 3 R(a, e) ⇒ X [3 R(a, e)] ⎢ X [3 R(a, e)] ↑ C(e) ⇒ X [3 R(a, e), C(e)] ⎢   3 ⎢ X [ R(a, e) ∨ C(e)] ↑ 2 R(a, e) ⇒ X [3 R(a, e) ∨ C(e), 2 R(a, e)] ⎢ ⎢ X [3 R(a, e) ∨ C(e)] ↑ C(e) ⇒ X [3 R(a, e) ∨ C(e), C(e)] ⎢   ⎢ X [Z 5 ] ↑ 1 R(a, e) ⇒ X [Z 5 , 1 R(a, e)] ⎢  ⇒ X [Z 5 , C(e)] (∃C− ) ⎢ ⎢  X [Z 5 ] ↑ C(e) 3 ⎢ X [Z 6 ] ↑ R(a, e) ⇒ X [Z 6 , 3 R(a, e)] ⎢ ⎢ X [Z 6 ] ↑ 2 C(e) ⇒ X [Z 6 , 2 C(e)] ⎢   ⎣ X [Z 7 ] ↑ 3 R(a, e) ⇒ X [Z 7 , 3 R(a, e)] X [Z 7 ] ↑ 1 C(e) ⇒ X [Z 7 , 1 C(e)]  X ↑ (∃R.C)(a) ⇒ X [(∃R.C)(a)] ⎡ X ↑ 3 R(a, f ) ⇒ X ⎢ X ↑ C( f ) ⇒ X ⎢   3 ⎢ X [ R(a, f ) ∨ C( f )] ↑ 2 R(a, f ) ⇒ X [3 R(a, f ) ∨ C( f )] ⎢ ⎢ X [3 R(a, f ) ∨ C( f )] ↑ C( f ) ⇒ X [3 R(a, f ) ∨ C( f )] ⎢    ⎢ X [Z ] ↑ 1 R(a, f ) ⇒ X [Z  ] 5 5 ⎢   f ) ⇒ X [Z 5 ] (∃C0 ) ⎢ ⎢  X [Z 5 ] ↑ C( ⎢ X [Z ] ↑ 3 R(a, f ) ⇒ X [Z  ] 6 6 ⎢ ⎢ X [Z  ] ↑ 2 C( f ) ⇒ X [Z  ] 6 ⎢   6 ⎣ X [Z 7 ] ↑ 3 R(a, f ) ⇒ X [Z 7 ] X [Z 7 ] ↑ 1 C( f ) ⇒ X [Z 7 ] X ↑ (∃R.C)(a) ⇒ X

177

178

5 R-Calculi for Post L4 -Valued DL

⎧ ⎨ Z5 Z6 ⎩ Z where ⎧ 7 ⎨ Z5 Z ⎩ 6 Z7

= 3 R(a, e) ∨ C(e), 2 R(a, e) ∨ C(e), = Z 5 , 1 R(a, e) ∨ C(e), = Z 6 , 3 R(a, e) ∨ 2 C(e), and = 3 R(a, f ) ∨ C( f ), 2 R(a, f ) ∨ C( f ), = Z 5 , 1 R(a, f ) ∨ C( f ), = Z 6 , 3 R(a, f ) ∨ 2 C( f ),

X ↑ 3 ∼ R(a, f ) ⇒ X [3 ∼ R(a, f )] (∼ ∃ ) X [3 ∼ R(a, f )] ↑∼ C( f ) ⇒ X [3 ∼ R(a, f ), ∼ C( f )]   X ↑∼ 3(∃R.C)(a) ⇒ X [∼ (∃R.C)(a)] X ↑ ∼ R(a, e) ⇒ X (∼ ∃C0 ) X [3 ∼ R(a, e)] ↑∼ C(e) ⇒ X [3 ∼ R(a, e)] X ↑ (∃R.C)(a) ⇒ X C+

where e is a new constant and f is a constant. Definition 5.4.3 A 3/4-reduction δ = || ↑ (A(a), B(b), C(e)) ⇒   | |  is =t⊥ δ, if there is a sequence {δ1 , . . . , δn } of 3/4provable in St⊥ 3/4 , denoted by 3/4 reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 3/4-reductions by one of the deduction rules in St⊥ 3/4 . Theorem 5.4.4 (Soundness and completeness theorem) For any 3/4-reduction δ = || ↑ (A(a), B(b), C(e)) ⇒   | |  , where A(a) ∈ , B(b) ∈ , C(e) ∈ , =t⊥

|=3/4

=t⊥

δ iff 3/4

δ. 

5.5 4/4-Multisequents =

= A 4/4-multisequent ||| is N4/4 -valid, denoted by |=4/4 |||, if for any interpretation I , either for some statement A(a) ∈ , I (A(a)) = t, or for some statement B(b) ∈ , I (B(b)) = , or for some statement C(e) ∈ , I (C(e)) =⊥, or for some statement D( f ) ∈ , I (D( f )) = f.

5.5 4/4-Multisequents

179

= 5.5.1 Deduction System N4/4 = Deduction system N4/4 contains the following axiom and deduction rules: • Axiom: ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎨  ∩  = ∅  ∩  = ∅ (At4/4 ) ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎩  ∩  = ∅ |||

where , , ,  are sets of atoms. • Deduction rules for unary logical connective ∼: |, A(a)|| ||, B(b)| (∼ B ) , ∼ A(a)||| |, ∼ B(b)|| |||, C(c) , D(d)||| C D (∼ ) (∼ ) ||, ∼ C(c)| |||, ∼ D(d) (∼ A )

• Deduction rules for binary logical connective : ⎡

|, B1 (b)|| ⎢ |, B2 (b)|| ⎢  ⎢ , B1 (b)||| , A1 (a)||| ⎢ A B ( ) , A2 (a)||| ( ) ⎢ ⎢  |, B2 (b)|| ⎣ |, B1 (b)|| , (A1 A2 )(a)||| , B2 (b)||| |, (B1 B2 )(a)|| ⎡ ||, C1 (c)| ⎢ ||, C2 (c)| ⎢ ⎢ |, C1 (c)|| ⎢ ⎢ ||, C2 (c)| ⎢

⎢ , C1 (c)||| |||, D1 (d) ⎢ D (c)| ||, C |||, D2 (d) ( C ) ⎢ ( ) 2 ⎢ ⎢ ||, C1 (c)| |||, (D1 D2 )(a) ⎢ ⎢ |, C2 (c)|| ⎢ ⎣ ||, C1 (c)| , C2 (c)||| ||, (C1 C2 )(a)| • Deduction rules for binary logical connective :

180

5 R-Calculi for Post L4 -Valued DL

⎡

|, B1 (b)|| ⎢ |, B2 (b)|| ⎢ ⎢ ||, B1 (b)| ⎢ ⎢ |, B2 (b)|| ⎢

⎢ |||, B1 (b) , A1 (a)||| ⎢ A B ( ) , A2 (a)||| ( ) ⎢ ⎢  |, B2 (b)|| ⎢ |, B1 (b)|| , (A1  A2 )(a)||| ⎢ ⎢ ||, B2 (b)| ⎢ ⎣ |, B1 (b)|| |||, B2 (b) |, (B1  B2 )(a)|| ⎡ ||, C1 (c)| ⎢ ||, C2 (c)| ⎢  ⎢ |||, C1 (c) |||, D1 (d) ⎢ D (c)| ||, C |||, D2 (d) ) (C ) ⎢ ( 2 ⎢ ⎣ ||, C1 (c)| |||, (D1  D2 )(a) |||, C2 (c) ||, (C1  C2 )(a)| • Deduction rules for quantifier ∀: ⎡

|||, R(a, f ) ⎡ ⎢ , A( f )||| , R(a, f )||| ⎢ ⎢ |, R(a, f )|| ⎢ |, B( f )|| ⎢ ⎢ ⎢ |, A( f )|| ⎢ |, R(a, f )|| ⎢ ⎢ A ⎢ ||, R(a, f )| B ⎢ ||, B( f )| (∀ ) ⎢ (∀ ) ⎢  ⎢ |, A( f )|| ⎣ ||, R(a, f )| ⎢ ⎣ ||, R(a, f )| |||, B( f ) ||, A( f )| |, (∀R.B)(a)|| , (∀R.A)(a)||| ⎡ , R(a, f )|||  ⎢ ||, C( f )| , R(a, e)||| ⎢ (∀C ) ⎣ |, R(a, f )|| (∀ D ) |||, D(e) |||, C( f ) |||, (∀R.D)(a) ||, (∀R.C)(a)| and

5.5 4/4-Multisequents

181

⎡



(∃ A )

, R(a, e)||| , A(e)||| , (∃R.A)(a)||| ⎡

|, R(a, e)|| ⎢ |, B(e)|| ⎢ ⎢ |, R(a, e)|| ⎢ B ⎢ , B(e)||| (∃ ) ⎢  ⎣ , R(a, e)||| |, B(e)|| |, (∃R.B)(a)||

||, R(a, e)| ⎢ ||, C(e)| ⎢ ⎢ |, R(a, e)|| ⎢ ⎢ ||, C(e)| ⎢

⎢ , R(a, e)||| , R(a, f )||| ⎢ C ||, C(e)| |||, D( f ) (∃C ) ⎢ (∼ ∃ ) ⎢ ⎢ ||, R(a, e)| |||, (∃R.D)(a) ⎢ ⎢ |, C(e)|| ⎢ ⎣ ||, R(a, e)| , C(e)||| ||, (∃R.C)(a)| where f is a constant and e is a new constant not occurring in , , , . =

= Definition 5.5.1 A multisequent ||| is provable in N4/4 , denoted by 4/4 ||| , if there is a sequence {δ1 , . . . , δn } of 4/4-multisequents such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 4/4-multisequents = . by one of the deduction rules in N4/4

Theorem 5.5.2 (Soundness and completeness theorem) For any 4/4-multisequent δ = |||, = = |=4/4 δ iff 4/4 δ. 

5.5.2 R-Calculus S= 4/4 Given a 4/4-multisequent ||| and four statements A(a) ∈ , B(b) ∈ , C(c) ∈ , D(d) ∈ , a 4/4-reduction δ=||| ↑ (A(a), B(b), C(c), D(d)) ⇒  − = {A (a)}| − {B  (b)}| − {C  (c)}| − {D  (d)} is O= 4/4 -valid, denoted by |=4/4 δ, if

182

5 R-Calculi for Post L4 -Valued DL



= -valid A(a) if  − {A(a)}||| is N4/4 λ otherwise;  = -valid B(b) if  − {A (a)}| − {B(b)}|| isN4/4  B (b) = λ otherwise;  = -valid C(c) if  − {A (a)}| − {B  (b)}| − {C(c)}| is N4/4  C (c) = λ otherwise;  = -valid D(d) if  − {A (a)}| − {B  (b)}| − {C  (c)}| − {D(d)} is N4/4  D (d) = λ otherwise.

A (a) =

Let Y = ||| and Y = [A (a)]||| Y = [A (a)]|[B  (b)]|| Y = [A (a)]|[B  (b)]|[C  (c)]| Y(4) = [A (a)]|[B  (b)]|[C  (c)]|[D  (d)] Y[A(a)] =  − {A(a)}||| Y[2 A(a)] = | − {A(a)}|| Y[3 A(a)] = || − {A(a)}| Y[4 A(a)] = ||| − {A(a)}.

Y[A(a)] =  − {A(a)}||| Y[B(b)] = | − {B(b)}|| Y[C(c)] = || − {C(c)}| Y[D(d)] = ||| − {D(d)}

R-calculus S= 4/4 consists of the following axioms and deduction rules: • Axioms: ⎧ ⎡ [s(a)] ∩  = ∅ [s(a)] ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ⎢ [s(a)] ∩  = ∅ [s(a)] ∩   = ∅ ⎪ ⎪ ⎢ ⎨ ⎢ [s(a)] ∩  = ∅ [s(a)] ∩  = ∅ ⎢ A− A0 ⎢  ∩   = ∅ ) ⎪ ) (A4/4 (A 4/4 ⎪ ⎢ ∩  = ∅ ⎪ ⎪  ∩  = ∅ ⎣ ∩  = ∅ ⎪ ⎪ ⎩  ∩  = ∅ ∩=∅ Y|s(a) Y|s(a) ⇒ Y[s(a)] ⎧  ⎡  ⇒Y   ∩ [t (b)] = ∅ ∩ [t (b)]  = ∅ ⎪ ⎪ ⎪   ⎪ ⎢ ∩   = ∅   ⎪ ⎪ ⎢ ∩ =∅ ⎨  ⎢ ∩  = ∅  ∩  = ∅ ⎢ B− B0 ⎢ [t (b)] ∩  = ∅ [t (b)] ∩   = ∅ (A4/4 )⎪ ) (A 4/4 ⎢ ⎪ ⎪ ⎪ [t (b)] ∩  = ∅ ⎣ [t (b)] ∩  = ∅ ⎪ ⎪ ⎩  ∩  = ∅ ∩=∅ Y |t (b) ⇒ Y [t (b)] Y |t (b) ⇒ Y and

5.5 4/4-Multisequents

⎧   ⎪ ⎪   ∩  = ∅ ⎪ ⎪  ∩ [u(c)] = ∅ ⎪ ⎪ ⎨   ∩  = ∅  ∩ [u(c)] = ∅ (AC0 (AC− 4/4 ) ⎪ 4/4 ) ⎪ ⎪  ⎪ ∩   = ∅  ⎪ ⎪ ⎩ [u(c)] ∩  = ∅  Y ⇒ Y [u(c)] ⎧ |u(c)   ∩   = ∅ ⎪ ⎪ ⎪ ⎪   ∩   = ∅ ⎪ ⎪ ⎨   ∩ [v(d)] = ∅ D− D0  ∩   = ∅ (A4/4 ) ⎪ ) (A4/4 ⎪ ⎪  ⎪ ∩ [v(d)]  = ∅  ⎪ ⎪ ⎩   ∩ [v(d)] = ∅ Y |v(d) ⇒ Y [v(d)]

183

⎡

  ∩  = ∅ ⎢   ∩ [u(c)] = ∅ ⎢  ⎢ ∩  = ∅ ⎢  ⎢  ∩ [u(c)] = ∅ ⎢  ⎣ ∩  = ∅ [u(c)] ∩  = ∅  Y ⇒ Y ⎡ |u(c)   ∩  = ∅ ⎢  ∩  = ∅ ⎢  ⎢  ∩ [v(d)] = ∅ ⎢  ⎢  ∩  = ∅ ⎢  ⎣  ∩ [v(d)] = ∅   ∩ [v(d)] = ∅  Y |v(d) ⇒ Y

where , , ,  are sets of atoms, and s(a), t (b), u(c), v(d) are atoms such that s(a) ∈ , t (b) ∈ , u(c) ∈ , v(d) ∈ . • Deduction rules consists of four parts E A , E B , EC , E D . ◦ EA: Y ↑ 2 A(a) ⇒ Y[2 A(a)] (∼ A− ) Y ↑∼ A(a) ⇒ Y[∼1 A(a)] 2 A0 Y ↑ A(a) ⇒ Y (∼ ) Y  ↑∼ A(a) ⇒ Y Y ↑ A1 (a) ⇒ Y[A1 (a)] ( A− ) Y ↑ A2 (a) ⇒ Y[A2 (a)]

Y ↑ (A1 A2 )(a) ⇒ Y[(A1 A2 )(a)] Y ↑ A1 (a) ⇒ Y ( A0 ) Y ↑ A2 (a) ⇒ Y Y

↑ (A1 A2 )(a) ⇒ Y Y ↑ A1 (a) ⇒ Y[A1 (a)] ( A− ) Y[A1 (a)] ↑ A2 (a) ⇒ Y[A1 (a), A2 (a)] Y ↑ (A1  A2 )(a) ⇒ Y[(A1  A2 )(a)] Y ↑ A1 (a) ⇒ Y ( A0 ) Y[A1 (a)] ↑ A2 (a) ⇒ Y[A1 (a)] Y ↑ (A1  A2 )(a) ⇒ Y and

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5 R-Calculi for Post L4 -Valued DL

⎡

Y ↑ 4 R(a, f ) ⇒ Y[4 R(a, f )] ⎢ Y[4 R(a, f )] ↑ A( f ) ⇒ Y[4 R(a, f ), A( f )] ⎢ 4 ⎢ Y[ R(a, f ), A( f )] ↑ 2 R(a, f ) ⇒ Y[4 R(a, f ), A( f ), 2 R(a, f )] ⎢ ⎢ Y[4 R(a, f ), A( f )] ↑ 2 A( f ) ⇒ Y[4 R(a, f ), A( f ), 2 A( f )] ⎢ A− ⎢ Y[Z ] ↑ 3 R(a, f ) ⇒ Y[Z , 3 R(a, f )] (∀ ) ⎢ 1 1 ⎢ Y[Z 1 ] ↑ 2 A( f ) ⇒ Y[Z 1 , 2 A( f )] ⎢ ⎣ Y[Z 2 ] ↑ 3 R(a, f ) ⇒ Y[Z 2 , 3 R(a, f )] Y[Z 2 ] ↑ 3 A( f ) ⇒ Y[Z 2 , 3 A( f )] Y ↑ ((∀R.A)(a)] ⇒ Y[(∀R.A)(a)] ⎧ Y ↑ 4 R(a, e) ⇒ Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Y

↑ 4A(e) ⇒ Y ⎪ ⎪ Y[ R(a, f ), A( f )] ↑ 2 R(a, e) ⇒ Y[4 R(a, f ), A( f )] ⎪ ⎪ ⎨ 4 f ), A( f )] ↑ 2 A(e) ⇒ Y[4 R(a, f ), A( f )]

Y[ R(a,  3 A0 Y[Z 1 ] ↑ R(a, e) ⇒ Y[Z 1 ] (∀ ) ⎪ ⎪ ⎪   2 ⎪ ⎪ ⎪ Y[Z 1 ] ↑ 3 A(e) ⇒ Y[Z 1 ]  ⎪ ⎪ Y[Z 2 ] ↑ R(a, e) ⇒ Y[Z 2 ] ⎪ ⎪ ⎩ Y[Z 2 ] ↑ 3 A(e) ⇒ Y[Z 2 ] Y ↑ ((∀R.A)(a)] ⇒ Y where

Z 1 = 4 R(a, f ), A( f ), ∼ R(a, f )∨ ∼ A( f ), Z 2 = Z 1 , ∼2 R(a, f )∨ ∼ A( f ), and Z 1 = 4 R(a, e), A(e), ∼ R(a, e)∨ ∼ A(e), Z 2 = Z 1 , ∼2 R(a, e)∨ ∼ A(e),



Y ↑ R(a, e) ⇒ Y Y ↑ A(e) ⇒ Y Y  ↑ (∃R.A)(a) ⇒ Y Y ↑ R(a, f ) ⇒ Y[R(a, f )] (∃ A− ) Y ↑ A( f ) ⇒ Y[A( f )] Y ↑ (∃R.A)(a) ⇒ Y[(∃R.A)(a)] (∃ A0 )

where e is a new constant and f is a constant. ◦ EB :

5.5 4/4-Multisequents

Y ↑ 3 B(b) ⇒ Y [3B(b)) Y ↑∼ B(b) ⇒ Y [∼ B(b)]  ↑ 3 B(b) ⇒ Y Y (∼ B0 )   Y ⎡ ↑∼ B(b) ⇒ Y  Y ↑ B1 (b) ⇒ Y [B1 (b)] ⎢ Y ↑ B2 (b) ⇒ Y [B2 (b)] ⎢  ⎢ Y [B1 (b) ∨ B2 (b)] ↑ 1 B1 (b) ⇒ Y [B1 (b) ∨ B2 (b), 1 B1 (b)) ⎢ B− ⎢ Y [B (b) ∨ B (b)] ↑ B (b) ⇒ Y [B (b) ∨ B (b), B (b)] ( ) ⎢  1 2 2 1 2 2 ⎣ Y [Y1 ] ↑ B1 (b) ⇒ Y [Y1 , B1 (b)] Y [Y1 ] ↑ 1 B2 (b) ⇒ Y [Y1 , 1 B2 (b)]  Y ↑ (B1 B2 )(a) ⇒ Y [(B1 B2 )(a)] ⎧ Y ↑ B1 (b) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ B2 (b) ⇒ Y ⎪ ⎪ ⎨  Y [B1 (b) ∨ B2 (b)] ↑ 1 B1 (b) ⇒ Y [B1 (b) ∨ B2 (b)]   B0 ( ) ⎪ ⎪

Y [B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ Y [B1 (b) ∨ B2 (b)] ⎪ ⎪ Y [Y1 ] ↑ B1 (b) ⇒ Y [Y1 ] ⎪ ⎪ ⎩ Y [Y1 ] ↑ 1 B2 (b) ⇒ Y [Y1 ] Y ↑ (B1 B2 )(a) ⇒ Y (∼ B− )

where Y1 = B1 (b) ∨ B2 (b), 1 B1 (b) ∨ B2 (b), and ⎡

Y ↑ B1 (b) ⇒ Y [B1 (b)] ⎢ Y ↑ B2 (b) ⇒ Y [B2 (b)] ⎢  ⎢ Y [B1 (b) ∨ B2 (b)] ↑ 3 B1 (b) ⇒ Y [B1 (b) ∨ B2 (b), 3 B1 (b)] ⎢ ⎢ Y [B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ Y [B1 (b) ∨ B2 (b), B2 (b)] ⎢  ⎢ Y [Y5 ] ↑ 4 B1 (b) ⇒ Y [Y5 , 4 B1 (b)] ⎢ B− ⎢ ( ) ⎢  Y [Y5 ] ↑ B2 (b) ⇒ Y [Y5 , B2 (b)] ⎢ Y [Y6 ] ↑ B1 (b) ⇒ Y [Y6 , B1 (b)] ⎢ ⎢ Y [Y6 ] ↑ 4 B2 (b) ⇒ Y [Y6 , 4 B2 (b)] ⎢  ⎣ Y [Y7 ] ↑ B1 (b) ⇒ Y [Y7 , B1 (b)] Y [Y7 ] ↑ 3 B2 (b) ⇒ Y [Y7 , 3 B2 (b)]  Y ↑ (B1  B2 )(a) ⇒ Y [(B1  B2 )(a)] and

⎧  Y ↑ B1 (b) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y ↑ B2 (b) ⇒ Y ⎪ ⎪ ⎪  ⎪ Y [B1 (b) ∨ B2 (b)] ↑ 3 B1 (b) ⇒ Y [B1 (b) ∨ B2 (b)] ⎪ ⎪ ⎪ ⎪ Y [B1 (b) ∨ B2 (b)] ↑ B2 (b) ⇒ Y [B1 (b) ∨ B2 (b)] ⎪ ⎪ ⎨  Y [Y5 ] ↑ 4 B1 (b) ⇒ Y [Y5 ]   B0 ( ) ⎪ ⎪

Y [Y5 ] ↑ B2 (b) ⇒ Y [Y5 ] ⎪ ⎪ Y [Y6 ] ↑ B1 (b) ⇒ Y [Y6 ] ⎪ ⎪ ⎪  4  ⎪ Y ⎪ ⎪

 [Y6 ] ↑ B2 (b) ⇒ Y [Y6 ] ⎪ ⎪ Y [Y7 ] ↑ B1 (b) ⇒ Y [Y7 ] ⎪ ⎪ ⎩ Y [Y7 ] ↑ 3 B2 (b) ⇒ Y [Y7 ] Y ↑ (B1  B2 )(a) ⇒ Y

185

186

5 R-Calculi for Post L4 -Valued DL

⎧ ⎨ Y5 = B1 (b) ∨ B2 (b), 3 B1 (b) ∨ B2 (b) where Y6 = Y5 , 4 B1 (b) ∨ B2 (b), and ⎩ Y7 = Y6 , B1 (b) ∨ 4 B2 (b), ⎡

Y ↑ 1 R(a, f ) ⇒ Y [1 R(a, f )] ⎢ Y ↑ B( f ) ⇒ Y [B( f )] ⎢  1 ⎢ Y [ R(a, f ) ∨ B( f )] ↑ 2 R(a, f ) ⇒ Y [1 R(a, f ) ∨ B( f ), 2 R(a, f )] ⎢ B− ⎢ Y [1 R(a, f ) ∨ B( f )] ↑ 3 B( f ) ⇒ Y [1 R(a, f ) ∨ B( f ), 3 B( f )] (∀ ) ⎢  ⎣ Y [Z 3 ] ↑ 3 R(a, f ) ⇒ Y [Z 3 , 3 R(a, f )] Y [Z 3 ] ↑ 4 B( f ) ⇒ Y [Z 3 , 4 B( f )]  Y ↑ ((∀R.B)(a)] ⇒ Y [(∀R.B)(a)] ⎧ Y ↑ 1 R(a, e) ⇒ Y ⎪ ⎪ ⎪ ⎪ Y [1 R(a, e)] ↑ B(e) ⇒ Y [1 R(a, e)] ⎪ ⎪ ⎨  1 Y [ R(a, e) ∨ B(e)] ↑ 2 R(a, e) ⇒ Y [1 R(a, e) ∨ B(e)]  1 B0 e) ∨ B(e)] ↑ 3 B(e) ⇒ Y [1 R(a, e) ∨ B(e)] (∀ ) ⎪ ⎪

Y [ R(a, ⎪  ⎪ Y [Z 3 ] ↑ 3 R(a, e) ⇒ Y [Z 3 ] ⎪ ⎪ ⎩ Y [Z 3 ] ↑ 4 B(e) ⇒ Y [Z 3 ]  Y ↑ ((∀R.B)(a)] ⇒ Y where

Z 3 = 1 R(a, f ) ∨ B( f ), 2 R(a, f ) ∨ 3 B( f ), and Z 3 = 1 R(a, e) ∨ B(e), 2 R(a, e) ∨ 3 B(e), ⎡

Y ↑ 2 R(a, e) ⇒ Y [2 R(a, e)] ⎢ Y ↑ B(e) ⇒ Y [B(e)] ⎢  2 ⎢ Y [ R(a, e) ∨ B(e)] ↑ 1 R(a, e) ⇒ Y [2 R(a, e) ∨ B(e), 1 R(a, e)] ⎢ B− ⎢ (∃ ) ⎢  Y [2 R(a, e) ∨ B(e)] ↑ B(e) ⇒ Y [2 R(a, e) ∨ B(e), B(e)] ⎣ Y [Z 4 ] ↑ 2 R(a, e) ⇒ Y [Z 4 , 2 R(a, e)] Y [Z 4 ] ↑ 1 B(e) ⇒ Y [Z 4 , 1 B(e)]  Y ↑ ((∃R.B)(a)] ⇒ Y [(∃R.B)(a)] and ⎧  2  ⎪ ⎪ Y ↑ R(a, f ) ⇒ Y ⎪ ⎪ Y ↑ B( f ) ⇒ Y ⎪ ⎪ ⎨  2 Y [ R(a, f ) ∨ B( f )] ↑ 2 R(a, f ) ⇒ Y [2 R(a, f ) ∨ B( f )]  2 f ) ∨ B( f )] ↑ 1 B( f ) ⇒ Y [2 R(a, f ) ∨ B( f )] (∃ B0 ) ⎪ ⎪ Y [ R(a, ⎪  1 ⎪ Y [Z 4 ] ↑ R(a, f ) ⇒ Y [Z 4 ] ⎪ ⎪ ⎩ Y [Z 4 ] ↑ B( f ) ⇒ Y [Z 4 ]  Y ↑ ((∃R.B)(a)] ⇒ Y Z 4 = 2 R(a, e) ∨ B(e), 1 R(a, e) ∨ B(e), and e is a new constant and f is Z 4 = 2 R(a, f ) ∨ B( f ), 1 R(a, f ) ∨ B( f ), a constant. ◦ EC :

where

5.5 4/4-Multisequents

Y ↑ 4 C(c) ⇒ Y [4 C(c)] Y ↑∼ C(c) ⇒ Y [∼ C(c)]  ↑ 4 C(c) ⇒ Y Y (∼C0 )   Y ⎡ ↑∼ C(c) ⇒ Y  Y ↑ C1 (c) ⇒ Y [C1 (c)] ⎢ Y ↑ C2 (c) ⇒ Y [C2 (c)] ⎢   ⎢ Y [C1 (c) ∨ C2 (c)] ↑ 1 C1 (c) ⇒ Y [C1 (c) ∨ C2 (c), 1 C1 (c)) ⎢ ⎢ Y [C1 (c) ∨ C2 (c)] ↑ C2 (c) ⇒ Y [C1 (c) ∨ C2 (c), C2 (c)] ⎢   ⎢ Y [Y2 ] ↑ 2 C1 (c) ⇒ Y [Y2 , 2 C1 (c)) ⎢ C− ⎢ Y [Y ] ↑ C (c) ⇒ Y [Y , C (c)] ( ) ⎢  2 2 2 2 ⎢ Y [Y3 ] ↑ C1 (c) ⇒ Y [Y3 , C1 (c)] ⎢ ⎢ Y [Y3 ] ↑ 2 C2 (c) ⇒ Y [Y3 , 2 C2 (c)] ⎢   ⎣ Y [Y4 ] ↑ C1 (c) ⇒ Y [Y4 , C1 (c)] Y [Y4 ] ↑ 1 C2 (c) ⇒ Y [Y4 , 1 C2 (c)]  Y ↑ (C1 C2 )(a) ⇒ Y [(C1 C2 )(a)] (∼C− )

⎧ ⎨ Y2 = C1 (c) ∨ C2 (c), 1 C1 (c) ∨ C2 (c), where Y3 = Y2 , 2 C1 (c) ∨ C2 (c), and ⎩ Y4 = Y3 , C1 (c) ∨ 2 C2 (c), ⎧  Y ↑ C1 (c) ⇒ Y ⎪ ⎪ ⎪   ⎪ ⎪ ⎪

Y ↑ C2 (c) ⇒ Y ⎪ ⎪ Y [C1 (c) ∨ C2 (c)] ↑ 2 C1 (c) ⇒ Y [C1 (c) ∨ C2 (c)] ⎪ ⎪ ⎪ ⎪ Y [C1 (c) ∨ C2 (c)] ↑ C2 (c) ⇒ Y [C1 (c) ∨ C2 (c)] ⎪ ⎪ ⎨  Y [Y2 ] ↑ 1 C1 (c) ⇒ Y [Y2 ] C0   ( ) ⎪ ⎪

Y [Y2 ] ↑ C2 (c) ⇒ Y [Y2 ] ⎪ ⎪ Y [Y3 ] ↑ C1 (c) ⇒ Y [Y3 ] ⎪ ⎪ ⎪  1  ⎪ Y ⎪ ⎪

 [Y3 ] ↑ C2 (c) ⇒ Y [Y3 ] ⎪ ⎪ Y [Y4 ] ↑ C1 (c) ⇒ Y [Y4 ] ⎪ ⎪ ⎩ Y [Y4 ] ↑ 2 C2 (c) ⇒ Y [Y4 ] Y ↑ (C1 C2 )(a) ⇒ Y and

187

188

5 R-Calculi for Post L4 -Valued DL

⎡

Y ↑ C1 (c) ⇒ Y [C1 (c)] ⎢ Y ↑ C2 (c) ⇒ Y [C2 (c)] ⎢   ⎢ Y [C1 (c) ∨ C2 (c)] ↑ 4 C1 (c) ⇒ Y [C1 (c) ∨ C2 (c), 4 C1 (c)] ⎢ C− ⎢ Y [C (c) ∨ C (c)] ↑ C (c) ⇒ Y [C (c) ∨ C (c), C (c)] ( ) ⎢  1 2 2 1 2 2 ⎣ Y [Y8 ] ↑ C1 (c) ⇒ Y [Y8 , C1 (c)] Y [Y8 ] ↑ 4 C2 (c) ⇒ Y [Y8 , 4 C2 (c)]  Y ↑ (C1  C2 )(a) ⇒ Y [(C1  C2 )(a)] ⎧ Y ↑ C1 (c) ⇒ Y ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ Y ↑ C2 (c) ⇒ Y ⎨ Y [C1 (c) ∨ C2 (c)] ↑ 4 C1 (c) ⇒ Y [C1 (c) ∨ C2 (c)]   C0 ( ) ⎪ ⎪ Y [C1 (c) ∨ C2 (c)] ↑ C2 (c) ⇒ Y [C1 (c) ∨ C2 (c)] ⎪ ⎪ Y [Y8 ] ↑ C1 (c) ⇒ Y [Y8 ] ⎪ ⎪ ⎩ Y [Y8 ] ↑ 4 C2 (c) ⇒ Y [Y8 ] Y ↑ (C1  C2 )(a) ⇒ Y where Y8 = C1 (c) ∨ C2 (c), 4 C1 (c) ∨ C2 (c), and ⎡   1 Y ↑ R(a, f ) ⇒ Y [1 R(a, f )] ⎢ Y ↑ 3 C( f ) ⇒ Y [3 C( f )] ⎢ (∀C− ) ⎣ Y [1 R(a, f ) ∨ 3 C( f )] ↑ 2 R(a, f ) ⇒ Y [1 R(a, f ) ∨ 3 C( f ), 2 R(a, f )] Y [1 R(a, f ) ∨ 3 C( f )] ↑ 4 C( f ) ⇒ Y [1 R(a, f ) ∨ 3 C( f ), 4 C( f )] Y ↑ (∀R.C)(a) ⇒ Y [(∀R.C)(a)] ⎧  1  ⎪ Y ↑ R(a, e) ⇒ Y ⎪ ⎨  3  Y ↑ C(e) ⇒ Y

 1 (∀C0 ) ⎪ Y [ R(a, e) ∨ 3 C(e)] ↑ 2 R(a, e) ⇒ Y [1 R(a, e) ∨ 3 C(e)] ⎪ ⎩ Y [1 R(a, e) ∨ 3 C(e)] ↑ 4 C(e) ⇒ Y [1 R(a, e) ∨ 3 C(e)]  Y ↑ (∀R.C)(a) ⇒ Y

and

5.5 4/4-Multisequents

189

⎡

Y ↑ 3 R(a, e) ⇒ Y [3 R(a, e)] ⎢ Y ↑ C(e) ⇒ Y [C(e)] ⎢   3 ⎢ Y [ R(a, e) ∨ C(e)] ↑ 2 R(a, e) ⇒ Y [3 R(a, e) ∨ C(e), 2 R(a, e)] ⎢ ⎢ Y [3 R(a, e) ∨ C(e)] ↑ C(e) ⇒ Y [3 R(a, e) ∨ C(e), C(e)] ⎢   ⎢ Y [Z 5 ] ↑ 1 R(a, e) ⇒ Y [Z 5 , 1 R(a, e)] ⎢ C− ⎢ Y [Z ] ↑ C(e) ⇒ Y [Z , C(e)] (∃ ) ⎢  5 5 ⎢ Y [Z 6 ] ↑ 3 R(a, e) ⇒ Y [Z 6 , 3 R(a, e)] ⎢ ⎢ Y [Z 6 ] ↑ 2 C(e) ⇒ Y [Z 6 , 2 C(e)] ⎢   ⎣ Y [Z 7 ] ↑ 3 R(a, e) ⇒ Y [Z 7 , 3 R(a, e)] Y [Z 7 ] ↑ 1 C(e) ⇒ Y [Z 7 , 1 C(e)]  Y ↑ ((∃R.C)(a)] ⇒ Y [(∃R.C)(a)] ⎧ Y ↑ 3 R(a, f ) ⇒ Y ⎪ ⎪ ⎪  ⎪ C( f ) ⇒ Y ⎪ ⎪ Y ↑ ⎪ 3 ⎪ Y [ R(a, f ) ∨ C( f )] ↑ 2 R(a, f ) ⇒ Y [3 R(a, f ) ∨ C( f )] ⎪ ⎪ ⎪  3 ⎪ f ) ∨ C( f )] ↑ C( f ) ⇒ Y [3 R(a, f ) ∨ C( f )] ⎪ ⎪ Y [ R(a, ⎨  1 Y [Z 5 ] ↑ R(a, f ) ⇒ Y [Z 5 ]   C0 f ) ⇒ Y [Z 5 ] (∃ ) ⎪ ⎪ Y [Z 5 ] ↑ C( ⎪ 3 ⎪ Y [Z 6 ] ↑ R(a, f ) ⇒ Y [Z 6 ] ⎪ ⎪ ⎪    2  ⎪ ⎪ ⎪ Y [Z 6 ] ↑ 3 C( f ) ⇒ Y [Z 6 ]  ⎪ ⎪ Y [Z 7 ] ↑ R(a, f ) ⇒ Y [Z 7 ] ⎪ ⎪ ⎩ Y [Z 7 ] ↑ 1 C( f ) ⇒ Y [Z 7 ]  Y ↑ ((∃R.C)(a)] ⇒ Y ⎧ ⎨ Z 5 = 3 R(a, e) ∨ C(e), 2 R(a, e) ∨ C(e), Z 6 = Z 5 , 1 R(a, e) ∨ C(e), ⎩ Z = Z 6 , 3 R(a, e) ∨ 2 C(e) and e is a new constant and f where ⎧ 7 ⎨ Z 5 = 3 R(a, f ) ∨ C( f ), 2 R(a, f ) ∨ C( f ),   1 Z = Z 5 , R(a, f ) ∨ C( f ), ⎩ 6 Z 7 = Z 6 , 3 R(a, f ) ∨ 2 C( f ). is a constant. ◦ ED :

190

5 R-Calculi for Post L4 -Valued DL

Y ↑ 1 D(d) ⇒ Y [1 D(d)) Y ↑∼ D(d) ⇒ Y [∼ D(d)]  ↑ 1 D(d) ⇒ Y Y (∼ D0 )   Y

↑∼ D(d) ⇒ Y  Y ↑ D1 (d) ⇒ Y [D1 (d)] ( D− ) Y [D1 (d)] ↑ D2 (d) ⇒ Y [D1 (d), D2 (d)]   Y ↑ (D1 D2 )(a)⇒ Y [(D1 D2 )(a)] Y ↑ D1 (d) ⇒ Y ( D0 ) Y [D1 (d)] ↑ D2 (d) ⇒ Y [D1 (d)]  Y (D1 D2 )(a) ⇒ Y  ↑  Y ↑ D1 (d) ⇒ Y [D1 (d)] D− ( ) Y [D1 (d)] ↑ D2 (d) ⇒ Y [D1 (d), D2 (d)]  

Y ↑ (D1  D2 )(a)⇒ Y [(D1  D2 )(a)] Y ↑ D1 (d) ⇒ Y ( D0 ) Y [D1 (d)] ↑ D2 (d) ⇒ Y [D1 (d)] Y ↑ (D1  D2 )(a) ⇒ Y (∼ D− )



and

and

Y ↑ 1 R(a, e) ⇒ Y [1 R(a, e)] (∀ D− ) Y ↑ D(e) ⇒ Y [D(e)]  ⇒ Y [(∀R.D)(a)]

Y ↑ (∀R.D)(a) Y ↑ 1 R(a, f ) ⇒ Y (∀ D0 ) Y ↑ D( f ) ⇒ Y Y ↑ (∀R.D)(a) ⇒ Y

Y ↑ 4 R(a, f ) ⇒ Y [4 R(a, f )] (∃ ) Y [4 R(a, f )] ↑ D( f ) ⇒ Y [4 R(a, f ), D( f )]  ⇒ Y [(∃R.D)(a)] Y ↑ (∃R.D)(a) 4 Y ↑ R(a, e) ⇒ Y D0 (∃ ) Y [4 R(a, e)] ↑ D(e) ⇒ Y [4 R(a, e)] Y ↑ ((∃R.D)(a)] ⇒ Y D−

where e is a new constant and f is a constant. Definition 5.5.3 A 4/4-reduction δ = Y ↑ (A(a), B(b), C(c), D(d)) ⇒ Y(4) is prov= able in S= 4/4 , denoted by 4/4 δ, if there is a sequence {δ1 , . . . , δn } of 2/3-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 4/4-reductions by one of the deduction rules in S= 4/4 . Theorem 5.5.4 (Soundness and completeness theorem) For any reduction δ = Y ↑ (A(a), B(b), C(c), D(d)) ⇒ Y(4) , =

=

4/4 δ iff |=4/4 δ. 

5.6 Conclusions

191

5.6 Conclusions t t t t There are transformations σ1 , σ2 , σ3 from N1/4 /St1/4 to N2/4 /St 2/4 , from N2/4 /S2/4 t⊥ t⊥ t⊥ = = to N3/4 /St⊥ 3/4 , and from N3/4 /S3/4 to N4/4 /S4/4 , respectively, such that t t ) = N2/4 σ1 (St1/4 ) = St σ1 (N1/4 2/4 , t t⊥ t⊥ σ2 (N2/4 ) = N3/4 σ2 (St ) = S 2/4 3/4 , t⊥ = t⊥ σ3 (N3/4 ) = N4/4 σ3 (S3/4 ) = S= 4/4 ,

where

and

and

σ1 (∪ ∼− ) = 2/4 σ1 (∼−2 ∪ ∼−3 ) = 2/4 σ1 (∼ A1 (a)) = B1 (a) σ1 (∼2 A1 (a)) =∼2 A1 (a) σ1 (∼3 A1 (a)) =∼2 B1 (a) σ2 (2/4 ) = 3/4 σ2 (∼− 2/4 ) = 3/4 σ2 (∼−2 2/4 ) = 3/4 σ2 (∼−3 2/4 ) =∼− 3/4 σ2 (2/4 ) = 3/4 σ2 (∼− 2/4 ) = 3/4 σ2 (∼−2 2/4 ) =∼− 3/4 σ3 (3/4 ) = 4/4 σ3 (∼− 3/4 ) = 4/4 σ3 (∼−2 3/4 ) = 4/4 σ3 (∼−3 3/4 ) = 4/4 σ3 (∼− 3/4 ) = 4/4 σ3 (∼−2 3/4 ) = 4/4 σ3 (∼−3 3/4 ) = 4/4 σ3 (∼− 3/4 ) = 4/4 σ3 (∼−2 3/4 ) = 4/4 σ3 (∼−3 3/4 ) = 4/4

σ2 (∼ A1 (a)) = B1 (a) σ2 (∼2 A1 (a)) = C1 (a) σ2 (∼3 A1 (a)) =∼ C1 (a) σ2 (∼ B1 (b)) = C1 (b) σ2 (∼2 B1 (b)) =∼ C1 (b)

σ3 (∼− 3/4 ) = 4/4 σ3 (∼−2 3/4 ) = 4/4 σ3 (∼−3 3/4 ) = 4/4 σ3 (∼ A1 (a)) = B1 (a) σ3 (∼ B1 (b)) = C1 (b) σ3 (∼ C1 (c)) = D1 (c) σ3 (∼ D1 (d)) = A1 (d).

192

5 R-Calculi for Post L4 -Valued DL

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Logic 50, 510–530 (1985) Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inform. 5, 25–63 (1996) Arieli, O., Avron, A.: Bilattices and paraconsistency, frontiers of paraconsistent logic. Stud. Logic Comput. 8, 11–27 (2000) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory. Implementation. Applications. Cambridge University Press, Cambridge, UK (2003) Belnap, N.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiplevalued Logic, pp. 8–37. D. Reidel (1977) Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Logic 2, 87–112 (1938) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Log. 40, 295–331 (2011) Font, J.M.: Belnap’s four-valued logic and De Morgan lattices. Logic J. I.G.P.L. 5, 413–440 (1997) Ginsberg, M.L.: Multi-valued logics: a uniform approach to reasoning in artificial intelligence. Comput. Intell. 4, 256–316 (1988) Gottwald, S.: A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9. Research Studies Press Ltd., Baldock (2001) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Ponse, A., van der Zwaag, M.B.: A generalization of ACP using Belnap’s logic. Electr. Notes Theor. Comput. Sci. 162, 287–293 (2006) Post, E.L.: Determination of all closed systems of truth tables. Bull. Amer. Math. Soc. 26, 437 (1920) Post, E.L.: Introduction to a general theory of elementary propositions. Amer. J. Math. 43, 163–185 (1921) Pynko, A.P.: Characterizing Belnap’s logic via De Morgan’s laws. Math. Log. Q. 41, 442–454 (1995) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 2, 2nd edn., pp. 249–295. Kluwer, Dordrecht (2001) Zach, R.: Proof theory of finite-valued logics. Technical Report TUW-E185.2-Z.1-93

Part II

Undecidable DLs

Chapter 6

Introduction

Most DLs without role constructors are decidable Rogers (1987), Soare (1987), and with role constructors are not. We consider two sets {◦, ∗} of role constructors: one is ◦, ∗ being taken as binary-valued constructors, which are used in binary DL, and anther is ◦, ∗ being taken as unary constructors, which are used in Post L3 -valued, B22 -valued and Post L4 -valued DLs with role constructors. In the latter case, R ∗ is the transitive closure of role R. We take Post L3 - and B22 -valued DLs as examples, to show how different R-calculi we will have from the last part.

6.1 Undecidable DL There are the following four kinds of deduction systems Baader et al. (2003), Horrocks and Sattler (2001), Malinowski (2009), Takeuti (1987), Zach (2023): t t : M1/3 is sound and complete. ◦ M1/3 Definition: |=t1/3  if AI ER(a, b) ∈ (I (R(a, b)) = t) |=t1/3  if EI AR(a, b) ∈ (I (R(a, b)) = t) (Soundness) t1/3  ⇒|=t1/3  (Completeness) |=t1/3  ⇒t1/3 , t1/3 r.e. SC ≡|=t1/3 where SC denotes soundness and completeness, and SI denotes soundness and incompleteness.

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_6

195

196

6 Introduction

t t ◦ N1/3 : N1/3 is sound and complete. =t

Definition: |=1/3  if AI ER(a, b) ∈ (I (R(a, b)) = t) =t |=1/3  if EI AR(a, b) ∈ (I (R(a, b)) = t) =t =t (Soundness) 1/3  ⇒|=1/3  =t =t (Completeness) |=1/3  ⇒1/3 . =t

=t

1/3 r.e. SC ≡|=1/3 1/3

1/3

◦ Lt : Lt is sound and incomplete. 1/3

Definition: |==t  if EI AR(a, b) ∈ (I (R(a, b)) = t) 1/3 |==t  if AI ER(a, b) ∈ (I (R(a, b)) = t) 1/3 1/3 (Soundness) =t  ⇒|==t  1/3 1/3 (Incompleteness) |==t  ⇒=t . 1/3

1/3

=t not r.e. SI ≡|==t 1/3

1/3

◦ Kt : Kt is sound and incomplete. 1/3

Definition: |=t  if EI AR(a, b) ∈ (I (R(a, b)) = t) 1/3 |=t  if AI ER(a, b) ∈ (I (R(a, b)) = t) 1/3 1/3 (Soundness) t  ⇒|=t  1/3 1/3 (Incompleteness) |=t  ⇒t . 1/3

t

1/3

not r.e. SI ≡|=t

Hence, we have the following equivalences: |=t1/3 =t |=1/3 1/3 |==t 1/3 |=t

 iff  iff  iff  iff

1/3

|==t 1/3 |=t |=t1/3 =t |=1/3

, , , .

6.2 R-Calculi

197

6.2 R-Calculi Similar to the first part, we have the following R-calculi: • Rt . Given a theory  and a statement R(a, b) ∈ , we define  ↑ R(a, b) ⇒ [R(a, b)] iff t [R(a, b)]  ↑ R(a, b) ⇒  iff t [R(a, b)]. Because t is recursively enumerable, we have that if t [R(a, b)] (recursively enumerable) then  ↑ R(a, b) ⇒ [R(a, b)]; otherwise, we do nothing. 

[R(a, b)] if t 1/3 [R(a, b)]  if t 1/3 [R(a, b)] t  ↑ R(a, b) ⇒ [R(a, b)] t  ↑ R(a, b) ⇒ [R(a, b)] ⇒|= 1/3 1/3 t |=t 1/3  ↑ R(a, b) ⇒ [R(a, b)] ⇒1/3  ↑ R(a, b) ⇒ [R(a, b)] t t 1/3  ↑ R(a, b) ⇒  ⇒|=1/3  ↑ R(a, b) ⇒  t |=t 1/3  ↑ R(a, b) ⇒   ⇒1/3  ↑ R(a, b) ⇒ ,

Definition: |=t 1/3  ↑ R(a, b) ⇒ (Soundness) (Completeness) (Soundness) (Inompleteness)

t1/3  ↑ R(a, b) ⇒ [R(a, b)] r.e. SC ≡|=t1/3  ↑ R(a, b) ⇒ [R(a, b)] t 1/3  ↑ R(a, b) ⇒  not r.e. SI ≡|=t1/3  ↑ R(a, b) ⇒  where SC denotes soundness and completeness, and SI denotes soundness and incompleteness. • St . Given a theory  and a statement R(a, b) ∈ , we define  ↑ R(a, b) ⇒ [R(a, b)] iff =t [R(a, b)]  ↑ R(a, b) ⇒  iff =t [R(a, b)]. Because =t is recursively enumerable, we have that if =t [R(a, b)] (recursively enumerable) then  ↑ R(a, b) ⇒ [R(a, b)]; otherwise, we do nothing.  =t Definition: |=1/3  ↑ R(a, b) ⇒ =t

=t

[R(a, b)] if 1/3 [R(a, b)] =t  if 1/3 [R(a, b)] =t

(Soundness) 1/3  ↑ R(a, b) ⇒ [R(a, b)] ⇒|=1/3  ↑ R(a, b) ⇒ [R(a, b)] =t

=t

(Completeness) |=1/3  ↑ R(a, b) ⇒ [R(a, b)] ⇒1/3  ↑ R(a, b) ⇒ [R(a, b)] =t =t (Soundness) 1/3  ↑ R(a, b) ⇒  ⇒|=1/3  ↑ R(a, b) ⇒  =t

=t

(Incompleteness) |=1/3  ↑ R(a, b) ⇒   ⇒1/3  ↑ R(a, b) ⇒ ; =t

=t

1/3  ↑ R(a, b) ⇒ [R(a, b)] r.e. SC ≡|=1/3  ↑ R(a, b) ⇒ [R(a, b)] =t =t 1/3  ↑ R(a, b) ⇒  not r.e. SI ≡|=1/3  ↑ R(a, b) ⇒ 

198

6 Introduction

• Qt . Given a theory  and a statement R(a, b), we define  ↑ R(a, b) ⇒ (R(a, b)) iff =t (R(a, b))  ↑ R(a, b) ⇒  iff =t (R(a, b)). Because =t is not recursively enumerable, we enumerate R(a, b) into  and use (R(a, b)) to revise R(a, b), such that (R(a, b)) ↑ R(a, b) ⇒ (R(a, b)) iff =t (R(a, b)) (R(a, b)) ↑ R(a, b) ⇒  iff t (R(a, b)). Hence, if t  (recursively enumerable) then (R(a, b)) ↑ R(a, b) ⇒ ; otherwise, we do nothing.  1/3 Definition: |==t  ↑ R(a, b) ⇒ 1/3

1/3

(R(a, b)) if =t (R(a, b)) 1/3  if =t (R(a, b)) 1/3

(Soundness) =t  ↑ R(a, b) ⇒ (R(a, b)) ⇒|==t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 (Incompleteness) |==t  ↑ R(a, b) ⇒ (R(a, b))  ⇒=t  ↑ R(a, b) ⇒ (R(a, b)) 1/3

1/3

(Soundness) =t  ↑ R(a, b) ⇒  ⇒|==t  ↑ R(a, b) ⇒  1/3 1/3 (Completeness) |==t  ↑ R(a, b) ⇒  ⇒=t  ↑ R(a, b) ⇒ . 1/3

1/3

=t  ↑ R(a, b) ⇒ (R(a, b)) not r.e. SI ≡|==t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 =t  ↑ R(a, b) ⇒  r.e. SC ≡|==t  ↑ R(a, b) ⇒  We enumerate R(a, b) into  and use (R(a, b)) to revise R(a, b), and have 1/3

(Soundness) =t (R(a, b)) ↑ R(a, b) ⇒  1/3 ⇒|==t (R(a, b)) ↑ R(a, b) ⇒  1/3 (Completeness) |==t (R(a, b)) ↑ R(a, b) ⇒  1/3 ⇒=t (R(a, b)) ↑ R(a, b) ⇒ ; • Pt . Given a theory  and a statement R(a, b), we define  ↑ R(a, b) ⇒ (R(a, b)) iff t (R(a, b))  ↑ R(a, b) ⇒  iff t (R(a, b)). Because t is not recursively enumerable, we enumerate R(a, b) into  and use (R(a, b)) to revise R(a, b), such that (R(a, b)) ↑ R(a, b) ⇒ (R(a, b)) iff t (R(a, b)) (R(a, b)) ↑ R(a, b) ⇒  iff t (R(a, b)).

6.3 Post L3 -Valued DL with Role Constructors

199

Hence, if t (R(a, b)) (recursively enumerable) then (R(a, b)) ↑ R(a, b) ⇒ ; otherwise, we do nothing. 

1/3

(R(a, b)) if t (R(a, b)) 1/3  if t (R(a, b)) 1/3 1/3 t  ↑ R(a, b) ⇒ (R(a, b)) ⇒|=t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 |=t  ↑ R(a, b) ⇒ (R(a, b))  ⇒t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 t  ↑ R(a, b) ⇒  ⇒|=t  ↑ R(a, b) ⇒  1/3 1/3 |=t  ↑ R(a, b) ⇒  ⇒t  ↑ R(a, b) ⇒ .

1/3 Definition: |=t  ↑ R(a, b) ⇒

(Soundness) (Incompleteness) (Soundness) (Completeness) 1/3

1/3

t  ↑ R(a, b) ⇒ (R(a, b)) not r.e. SI ≡|=t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 t  ↑ R(a, b) ⇒  r.e. SC ≡|=t  ↑ R(a, b) ⇒  We enumerate R(a, b) into  and use (R(a, b)) to revise R(a, b), and have 1/3

(Soundness) t (R(a, b)) ↑ R(a, b) ⇒  1/3 ⇒|=t (R(a, b)) ↑ R(a, b) ⇒  1/3 (Completeness) |=t (R(a, b)) ↑ R(a, b) ⇒  1/3 ⇒t (R(a, b)) ↑ R(a, b) ⇒ . The reason to do so is because given a recursively enumerable set A, define for any natural number x,  1 if x ∈ A f (x) = 0 if x ∈ / A which is not computable, and for any x,  g(x) =

1 if x ∈ A ↑ if x ∈ / A

which is computable.

6.3 Post L3 -Valued DL with Role Constructors There are four kinds of validity Avron (1991), Post (1920), Post (1921): t M1/3 t N1/3 1/3 Lt 1/3 Kt

: |=t1/3 =t : |=1/3 1/3 : |==t 1/3 : |=t

   

↔ AI ER(a, b) ∈ (I (R(a, b)) = t), ↔ AI ER(a, b) ∈ (I (R(a, b)) = t), ↔ EI AR(a, b) ∈ (I (R(a, b)) = t), ↔ EI AR(a, b) ∈ (I (R(a, b)) = t),

200

6 Introduction 1/3

1/3

t t where M1/3 /N1/3 is complementary to Lt /Kt , respectively. =t

=t

t N1/3 |=t1/3 |=1/3 M1/3 1/3 1/3 1/3 1/3 L=t Kt |==t |=t . =t

For each validity, there is a sound and complete deduction system t1/3 / 1/3 =t t t for M1/3 /N1/3 -validity; and if t1/3 / 1/3 is recursive, then there is a sound and 1/3 1/3 1/3 1/3 complete deduction system =t / t for Lt /Kt -validity, respectively. =t

=t

t1/3 1/3 |=t1/3 |=1/3 iff 1/3 1/3 1/3 1/3 =t t |==t |=t . =t

1/3

1/3

1/3

1/3

If t1/3 / 1/3 is not recursive then deduction system =t / t for Lt /Kt validity is sound and incomplete. =t

=t

=t

t M1/3 N1/3 |=t1/3 |=1/3 t1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 L=t Kt |==t |=t =t t .

There are four kinds of R-calculi Li (2007), Li and Sui (2013): t  ↑ R(a, b) ⇒ [R  (a, b)] ∈R1/3 R(a, b) ↔ R  (a, b) = λ  ↑ R(a, b) ⇒ [R  (a, b)] ∈ St1/3  R(a, b) ↔ R  (a, b) = λ 1/3  ↑ R(a, b) ⇒ (R  (a, b)) ∈ Qt  R(a, b) ↔ R  (a, b) = λ 1/3  ↑ R(a, b) ⇒ (R  (a, b)) ∈ Pt R(a, b) ↔ R  (a, b) = λ =t

if |=t1/3 [R(a, b)] otherwise =t

if |=1/3 [R(a, b)] otherwise 1/3

if |==t (R(a, b)) otherwise 1/3

if |=t (R(a, b)) otherwise

=t

t S1/3 |=t1/3 |=1/3 R1/3 1/3 1/3 1/3 1/3 Q=t Pt |==t |=t .

6.3 Post L3 -Valued DL with Role Constructors

201

If there is a sound and complete R-calculus, then we have t 1/3  ↑ R(a, b) ⇒ [R(a, b)]     2   iff EQ(a  , b )  = R(a, b)( t 1/3 Q(a , b )∧ ∼ Q(a , b )∧ ∼ Q(a , b )) t 1/3  ↑ R(a, b) ⇒      2   iff AQ(a  , b )  = R(a, b)( t 1/3 Q(a , b )∧ ∼ Q(a , b )∧ ∼ Q(a , b )) =t

1/3  ↑ R(a, b) ⇒ [R(a, b)]

=t

iff EQ(a  , b )  = R(a, b)E∗1 , ∗2 (∗1  = ∗2 & 1/3 ∗1 Q(a  , b ) ∧ ∗2 Q(a  , b )) =t 1/3  ↑ R(a, b) ⇒  =t iff AQ(a  , b )  = R(a, b)A∗1 , ∗2 (∗1  = ∗2 ⇒  1/3 ∗1 Q(a  , b ) ∧ ∗2 Q(a  , b )) =t

1/3

=t  ↑ R(a, b) ⇒ (R(a, b)) iff  1/3 ∼ R(a, b)∧ ∼2 R(a, b)) =t 1/3 =t  ↑ R(a, b) ⇒  iff  1/3 ∼ R(a, b)∧ ∼2 R(a, b) 1/3

2  ↑ R(a, b) ⇒ (R(a, b)) iff  t 1/3 ∼ R(a, b)∨ ∼ R(a, b) 1/3 2 t  ↑ R(a, b) ⇒  iff  t 1/3 ∼ R(a, b)∨ ∼ R(a, b).

t

=t

For each R-calculus, if deduction system t1/3 / 1/3 is recursive, then there is a =t 1/3 1/3 1/3 1/3 t sound and complete R-calculus t1/3 / 1/3 / =t / t for R1/3 /St1/3 /Qt /Pt validity, respectively. =t

=t

t1/3 δ 1/3 δ |=t δ |=1/3 δ iff 1/3 1/3 1/3 1/3 1/3 =t δ t δ |==t δ |=t δ, where δ is a reduction of forms  ↑ R(a, b) ⇒ [R  (a, b)] or  ↑ R(a, b) ⇒ , R  (a, b). =t 1/3 1/3 If deduction system t1/3 / 1/3 / =t / t is not recursive then R-calculus =t

1/3

1/3

t1/3 / 1/3 / =t / t complete. That is,

1/3

1/3

t for R1/3 /St1/3 /Qt /Pt -validity is sound and partially

t t1/3  ↑ R(a, b) ⇒ [R(a, b)] iff |=t1/3  ↑ R(a, b) ⇒ [R(a, b)] R1/3 t1/3  ↑ R(a, b) ⇒  ⇒|=t1/3  ↑ R(a, b) ⇒  |=t1/3  ↑ R(a, b) ⇒  ⇒t1/3  ↑ R(a, b) ⇒  =t =t St1/3 1/3  ↑ R(a, b) ⇒ [R(a, b)] iff |=1/3  ↑ R(a, b) ⇒ [R(a, b)] =t =t 1/3  ↑ R(a, b) ⇒  ⇒|=1/3  ↑ R(a, b) ⇒  =t =t |=1/3  ↑ R(a, b) ⇒  ⇒1/3  ↑ R(a, b) ⇒  1/3 1/3 1/3 Qt =t  ↑ R(a, b) ⇒ (R(a, b)) ⇒|==t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 |==t  ↑ R(a, b) ⇒ (R(a, b)) ⇒=t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 =t  ↑ R(a, b) ⇒  iff |==t  ↑ R(a, b) ⇒  1/3 1/3 1/3 Pt t  ↑ R(a, b) ⇒ (R(a, b)) ⇒|=t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 |=t  ↑ R(a, b) ⇒ (R(a, b)) ⇒t  ↑ R(a, b) ⇒ (R(a, b)) 1/3 1/3 t  ↑ R(a, b) ⇒  iff |=t  ↑ R(a, b) ⇒ .

202

6 Introduction

6.4 B22 -Valued DL with Role Constructors We will consider case i = 4 only. There are four kinds of validity Pynko (1995): = ||| ∈ M4/2 2 ↔ AI (ER(a, b) ∈ (I (R(a, b)) = t)

∨EQ(a  , b ) ∈ (I (Q(a  , b )) = ) ∨EP(a  , b ) ∈ (I (P(a  , b )) =⊥) = ||| ∈ N4/2 2

∨EO(a  , b ) ∈ (I (O(a  , b )) = f)) ↔ AI (ER(a, b) ∈ (I (R(a, b)) = t) ∨EQ(a  , b ) ∈ (I (Q(a  , b )) = ) ∨EP(a  , b ) ∈ (I (P(a  , b )) =⊥) ∨EO(a  , b ) ∈ (I (O(a  , b )) = f))

and 2

4/2 ↔ EI (AR(a, b) ∈ (I (R(a, b)) = t) ||| ∈ L=

∧AQ(a  , b ) ∈ (I (Q(a  , b )) = ) ∧AP(a  , b ) ∈ (I (P(a  , b )) =⊥) ∧AO(a  , b ) ∈ (I (O(a  , b )) = f))

2

4/2 ↔ EI (AR(a, b) ∈ (I (R(a, b)) = t) ||| ∈ K=

∧AQ(a  , b ) ∈ (I (Q(a  , b )) = ) ∧AP(a  , b ) ∈ (I (P(a  , b )) =⊥) ∧AO(a  , b ) ∈ (I (O(a  , b )) = f)),

= = 4/2 4/2 /K= , respectively. where M4/2 2 /N4/22 is complementary to L= 2

2

=

=

= R4/2 |== 2 S4/22 4/22 |=4/22 4/22

Q=

4/2 P=

2

4/22

|==

2

|=4/2 . = =

For each validity, there is a sound and complete deduction system = 4/22 / 4/22 for = = M4/2 2 /N4/22 -validity; and deduction system  = sound and incomplete.

4/2

2

2

2

2

4/2 4/2 / 4/2 for L= /K= -validity is =

6.4 B22 -Valued DL with Role Constructors

203

= = = M4/2 2 4/22 ||| iff |=4/22 ||| =

=

= N4/2 2 4/22 ||| iff |=4/22 ||| 2

4/22

4/2 L=  =

4/22

4/2 K=

2

4/22

||| ⇒|==

|||

4/22

|== ||| ⇒= ||| 2 2 4/2 ||| ⇒|=4/2 ||| 2 2 |=4/2 ||| ⇒4/2 |||.

There are four kinds of R-calculi Alchourrón et al. (1985); Fermé and Hansson = 4/22 4/22 (2011); Ginsberg (1987): let ∗ = = 4/22 , 4/22 , = , = , |=∗ ||| ↑ (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b ))            ⇒ [R (a, b)]|[Q (a , b )]|[P (a , b )]|[O (a , b )] ↔ ⎧ [R(a, b)]||| R(a, b) if |= ∗ ⎪ R  (a, b) = ⎪ ⎪ ⎪ λ otherwise ⎪ ⎪  ⎪ ⎪ Q(a  , b ) if |=∗ [R  (a, b)]|[Q(a  , b )]|| ⎪    ⎪ ⎨ Q (a , b ) = λ otherwise   , b ) if |= [R  (a, b)]|[Q  (a  , b )]|[P(a  , b )]| P(a ⎪ ∗    ⎪ P (a , b ) = ⎪ ⎪ λ otherwise ⎪ ⎪  ⎪            ⎪ ⎪ ⎪ O  (a  , b ) = O(a , b ) if |=∗ [R (a, b)]|[Q (a , b )]|[P (a , b )]|[O(a , b )] ⎩ λ otherwise =

=

= R4/2 |== 2 S4/22 4/22 |=4/22 4/22

Q=

4/2 P=

2

4/22

|==

2

|=4/2 . =

= = For R-calculi R4/2 2 /S4/22 , we have that for any reduction δ = ||| ↑ (R(a, b), S(a, b), Q(a, b), P(a, b)) ⇒ , R  (a, b)|, S  (a, b)|, Q  (a, b)|, P  (a, b), where R(a, b) ∈ , S(a, b) ∈ , Q(a, b) ∈ , P(a, b) ∈ , such that one of R  (a, b), S  (a, b), Q  (a, b), P  (a, b) is not the empty string, = = = R4/2 2 4/22 δ iff |=4/22 δ =   4/22 δ implies |== 4/22 δ =   |=4/22 δ may not imply |== 4/22 δ =

=

= N4/2 2 4/22 δ iff |=4/22 δ = = 4/22 δ  implies |=4/22 δ  = = |=4/22 δ  may not imply |=4/22 δ  ,

where δ  = ||| ↑ (R(a, b), S(a, b), Q(a, b), P(a, b)) ⇒ |||, 2

2

4/2 As for Q4/2 = and P= , instead of using multisequent ||| to revise quadruple (R(a, b), S(a, b), Q(a, b), P(a, b)), we use multisequent

204

6 Introduction

(R(a, b))|(S(a, b))|(Q(a, b))|(P(a, b)) to revise quadruple (R(a, b), S(a, b), Q(a, b), P(a, b)), such that (R(a, b))|(S(a, b))|(Q(a, b))|(P(a, b)) ↑ (R(a, b), S(a, b), Q(a, b), P(a, b)) ⇒ (R  (a, b))|(S  (a, b))|(Q  (a, b))|(P  (a, b)) = 4/2     is Q4/2 = /P= -valid if (R (a, b))|(S (a, b))|(Q (a, b))|(P (a, b)) is L4/22 / = K4/22 -valid. 2

2

2

2

4/2 For R-calculi Q4/2 = /P= , we have that for any reduction δ = ||| ↑ (R(a, b), S(a, b), Q(a, b), P(a, b)) ⇒ , R  (a, b)|, S  (a, b)|, Q  (a, b)|, P  (a, b), such that one of R  (a, b), S  (a, b), Q  (a, b), P  (a, b) is not the empty string, 2 4/22 4/22 = δ iff |== δ Q4/2 = 4/22

 = 2

δ

4/22

|== δ  may not imply = δ  2 2 4/2 δ iff |=4/2 δ = = 2  4/22  4/2 δ implies |= δ = = 2 2  |=4/2 δ may not imply 4/2 δ , = = 4/2

4/2 P=

2

4/22

δ  implies |==

where δ  = , R(a, b)|, S(a, b)|, Q(a, b)|, P(a, b) ↑ (R(a, b), S(a, b), Q(a, b), P(a, b)) ⇒ |||,

6.5 Injury = = 4/2 4/2 In R-calculi, R4/2 /P= for undecidable logics, there are injuries in 2 /S4/22 /Q= revising. Let R-calculus consist of four parts 2

2

E R (X, R(a, b)), E Q (X, Q(a  , b )), E P (X, P(a  , b )), E O (X, O(a  , b )), revising R(a, b), Q(a  , b ), P(a  , b ), O(a  , b ), respectively. Assume that ◦ three first-level injuries: let s0 be the first stage at which an R-calculus terminates, say E O (X, O(a  , b )), that is,     = 4/4 ||| ↑ O(a , b ) ⇒s0 |||[O(a , b )];

and at a later stage s1 > s0 another R-calculus, say E P (X, P(a  , b )), terminates, that is,

6.5 Injury

205     = 4/4 ||| ↑ P(a , b ) ⇒s1 ||[P(a , b )]|;

and at a later stage s2 > s1 R-calculus E Q (X, Q(a  , b )) terminates, that is,     = 4/4 ||| ↑ Q(a , b ) ⇒s2 |[Q(a , b )]||;

and at a later stage s3 > s2 R-calculus E R (X, R(a, b)) terminates, that is = 4/4 ||| ↑ R(a, b) ⇒s3 [R(a, b)]|||; where P(a  , b ) extracted from  may injury O(a  , b ) extracted from  (making O(a  , b ) not to be extracted from ); Q(a  , b ) extracted from  may injury P(a  , b ) extracted from ; and R(a, b) extracted from  may injury Q(a  , b ) extracted from . ◦ two second-level injuries: there is a stage t0 > s3 such that     = 4/4 [R(a, b)]||| ↑ O(a , b ) ⇒t0 [R(a, b)]|||[O(a , b )];

and at a later stage t1 > t0 such that     = 4/4 [R(a, b)]||| ↑ P(a , b ) ⇒t1 [R(a, b)]||[P(a , b )]|;

and at a later stage t2 > t1 such that     = 4/4 [R(a, b)]||| ↑ Q(a , b ) ⇒t2 [R(a, b)]|[Q(a , b )]||.

◦ one third-level injury: there is a stage u 0 > t2 such that     = 4/4 [R(a, b)]|[Q(a , b )]|| ↑ O(a , b )   ⇒u 0 [R(a, b)]|[Q(a , b )]||[O(a  , b )];

and at a later stage u 1 > u 0 such that     = 4/4 [R(a, b)]|[Q(a , b )]|| ↑ P(a , b ) ⇒u 1 [R(a, b)]|[Q(a  , b )]|[P(a  , b )]|;

and at a later stage u 2 > u 1 such that       = 4/4 [R(a, b)]|[Q(a , b )]|[P(a , b )]| ↑ O(a , b )     ⇒u 2 [R(a, b)]|[Q(a , b )]|[P(a , b )]|[O(a  , b )].

In this example, there are three-levels of injuries Friedberg (1957); Li and Sui (2017); Muchnik (1956); Rogers (1987); Soare (1987):

206

6 Introduction

• first-level injury: eliminating O(a  , b ) from  at stage s0 + 1 may be injured by eliminating P(a  , b ) from  at stage s1 + 1; eliminating P(a  , b ) from  at stage s1 + 1 may be injured by eliminating Q(a  , b ) from  at stage s2 + 1, and eliminating Q(a  , b ) from  at stage s2 + 1 may be injured by eliminating R(a, b) from  at stage s3 + 1. • second-level injury: eliminating O(a  , b ) from  at stage t0 + 1 may be injured by eliminating P(a  , b ) from  at stage t1 + 1; and eliminating P(a  , b ) from  at stage t1 + 1 may be injured by eliminating Q(a  , b ) from  at stage t2 + 1. • third-level injury: eliminating O(a  , b ) from  at stage u 0 + 1 may be injured by eliminating P(a  , b ) from  at stage u 1 + 1. This is because that we define that given a 4/22 -multisequent X = ||| and statements X = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )), a 4/22 -reduction = X ↑ X ⇒ X = [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )]|[O  (a  , b )], is R4/2 2= valid, denoted by |=4/22 δ, if 

= R(a, b) ifX(R(a, b))is M4/2 2 -valid λ otherwise;  = Q(a  , b ) ifX (Q(a  , b ))is M4/2 2 -valid    Q (a , b ) = λ otherwise;  = P(a  , b ) ifX (P(a  , b ))is M4/2 2 -valid    P (a , b ) = otherwise; λ = O(a  , b ) if X (O(a  , b ))isM4/2 2 -valid    O (a , b ) = λ otherwise,

R  (a, b) =

where

X = [R  (a, b)]|||, X = [R  (a, b)]|[Q  (a  , b )||, X = [R  (a, b)]|[Q  (a  , b )|[P  (a  , b )|.

The deduction proceeds in stages: (i) There is a stage s0 at which X ↑ X ⇒ X[X ], where X ∈ {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )}; and use X = X[X ] to revise the remaining X 1 , X 2 , X 3 , where {X 1 , X 2 , X 3 } = {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )} − {X }. (ii) There is a stage s1 ≥ s0 at which X ↑ X  ⇒ X [X  ], where X  ∈ {X 1 , X 2 , X 3 }; and use X = X [X  ] to revise the remaining X 1 , X 2 only if X ≺ X  ; otherwise, use X[X  ] to revise {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )} − {X  }, where {X 1 , X 2 } = {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )} − {X, X  } and R(a, b) ≺ Q(a  , b ) ≺ P(a  , b ) ≺ O(a  , b ).

There are injuries in such a definition. Assume that we define that given a 4/22 -multisequent X = ||| and statements X = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )), a 4/22 -reduction δ = X ↑ = = X ⇒ X is R4/2 2 -valid, denoted by |=4/22 δ, if

6.6 Arrangement of This Part

207



= R(a, b) ifX(R(a, b))is M4/2 2 -valid λ otherwise;  = Q(a  , b ) ifX(Q(a  , b ))isM4/2 2 -valid    Q (a , b ) = λ otherwise;  = P(a  , b ) if X(P(a  , b ))is M4/2 2 -valid P  (a  , b ) = otherwise; λ = O(a  , b ) if X(O(a  , b ))is M4/2 2 -valid    O (a , b ) = λ otherwise.

R  (a, b) =

The deduction proceeds in stages: (i) There is a stage s0 at which X ↑ X ⇒ X[X ], where X ∈ {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )}; and use X = X[X ] to revise the remaining X 1 , X 2 , X 3 , and {X 1 , X 2 , X 3 } = {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )} − {X }. (ii) There is a stage s1 ≥ s0 at which X ↑ X  ⇒ X [X  ], where X  ∈ {X 1 , X 2 , X 3 }; and use X = X [X  ] to revise the remaining X 1 , X 2 and {X 1 , X 2 } = {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )} − {X, X  }. (iii) There is a stage s2 ≥ s1 at which X ↑ X  ⇒ X [X  ], where X  ∈ {X 1 , X 2 }; and use X = X [X  ] to revise the remaining X 1 ∈ {R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )} − {X, X  , X  }. (iv) There is a stage s3 ≥ s2 at which X ↑ X  ⇒ X [X  ].

6.6 Arrangement of This Part • For DL with role constructors, we give the following deduction systems and R-calculi: ◦ B2 -valued: deduction system R-calculus t t t 1/2 M1/2 , N1/2 R1/2 , St1/2 1/2 1/2 1/2 1/2 Lt , Kf Qt , Pt = = = = 2/2 M2/2 , N2/2 R2/2 , S2/2 2/2 2/2 2/2 L= , K= Q2/2 = , P= ◦ L3 -valued: deduction system R-calculus 1/3 1/3 t 1/3 N1/3 , Kt St1/3 , Pt tm 2/3 N2/3 Stm 2/3 = 3/3 3/3 3/3 N3/3 , K= S= 3/3 , P=

208

6 Introduction

◦ B22 -valued: deduction system R-calculus 2

1/2 2/22 3/22 4/22

1/22

t M1/2 2 , Lt t M2/2 2 t⊥ M3/2 2 = 4/22 M4/2 2 , L=

1/22

t R1/2 2 , Qt t R2/2 2 t⊥ R3/2 2 = 4/22 R4/2 2 , Q=

◦ L4 -valued: 1/4 2/4 3/4 4/4

deduction system R-calculus t t t M1/4 , N1/4 R1/4 , St1/4 t t N2/4 S2/4 t⊥ N3/4 St⊥ 3/4 = = = M4/4 , N4/4 R4/4 , S= 4/4

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Logic 50, 510–530 (1985) Avron, A.: Natural 3-valued logics: characterization and proof theory. J. Symb. Logic 56, 276–294 (1991) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory. Implementation. Applications. Cambridge University Press, Cambridge, UK (2003) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Horrocks, I., Sattler, U.: Ontology reasoning in the SHOQ(D) description logic. In: Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence (2001) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The sound and complete R-calculi with respect to pseudo-revision and pre-revision. Int. J. Intell. Sci. 3, 110–117 (2013) Li, W., Sui, Y.: The R-calculus and the finite injury priority method. J. Comput. 12, 127–134 (2017) Malinowski, G.: Many-valued logic and its philosophy. In: Gabbay, D.M., Woods, J. (eds.) Handbook of the History of Logic, vol. 8. The Many Valued and Nonmonotonic Turn in Logic. Elsevier (2009) Muchnik, A.A.: On the separability of recursively enumerable sets (in Russian). Dokl. Akad. Nauk SSSR, N.S. 109, 29–32 (1956) Post, E.L.: Determination of all closed systems of truth tables. Bull. Am. Math. Soc. 26, 437 (1920) Post, E.L.: Introduction to a general theory of elementary propositions. Am. J. Math. 43, 163–185 (1921) Pynko, A.P.: Characterizing Belnap’s logic via De Morgan’s laws. Math. Logic Quart. 41(4), 442– 454 (1995) Rogers, H.: Theory of Recursive Functions and Effective Computability. The MIT Press (1987)

References

209

Soare, R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Functions and Computably Generated Sets. Springer (1987) Takeuti, G.: Proof theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987) Zach, R.: Proof theory of finite-valued logics, Tech. Report TUW-E185.2-Z.1-93

Chapter 7

Role R-Calculus for Binary-Valued DL

∗ ∗ Mi/2 Ni/2 i/2

L∗

∗ ∗ Ri/2 Si/2

i/2

i/2

K∗

i/2

Q∗ P∗

There are four kinds of validity Baader (2003, 2007), Fensel et al. (2001), Takeuti (1987), Urquhart (2001): t Mt1/2 :  ∈ M1/2 ↔ AI E A ∈ (I (A) = t), t t N1/2 :  ∈ N1/2 ↔ AI E A ∈ (I (A) = t), 1/2 1/2 Lt :  ∈ Lt ↔ EI AA ∈ (I (A) = t), 1/2 1/2 Kt :  ∈ Kt ↔ EI AA ∈ (I (A) = t). 1/2

1/2

t t M1/2 /N1/2 is complementary to Lt /Kt , respectively. =t For each validity, there is a sound and complete R-calculus t1/2 / 1/2 for  = t t R1/2 /St1/2 -validity; and if t1/2 / 1/2 is recursive, then there is a sound and complete 1/2 1/2 1/2 1/2 R-calculus =t / t for Lt /Kt -validity, respectively. =t

t1/2 1/2 1/2

1/2

=t t

=t

iff

|=t1/2 |=1/2 1/2

1/2

|==t |=t .

=t

1/2

1/2

1/2

1/2

If t1/2 / 1/2 is not recursive then the deduction system =t / t for Lt /Kt validity is sound and incomplete.

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_7

211

212

7 Role R-Calculus for Binary-Valued DL t M1/2

t1/2  iff |=t1/2 

=t

1/2  iff |=1/2 

=t

N1/2 1/2

=t

1/2

1/2

1/2

1/2

SC SC

L=t =t  implies |==t  SI 1/2

t  implies |=t  SI.

Kt

There are four kinds of R-calculi Alchourrón et al. (1985), Fermé and Hansson (2011), Gärdenfors and Rott (1995), Li (2007), Li and Sui (2017): 

 ↑ A ⇒ [A ] ∈

t R1/2





↔A =

 ↑ A ⇒ [A ] ∈ St1/2 ↔ A = 

1/2 Qt



1/2 Pt

 ↑ A ⇒ , A ∈  ↑ A ⇒ , A ∈

 



↔A = 



↔A =

A if |=t1/2 [A]) λ otherwise =t

A if |=1/2 [A] λ otherwise 1/2

A if |==t , A λ otherwise 1/2

A if |=t , A λ otherwise

where ∗1 , ∗2 ∈ {λ, ∼, ∼2 ]. =t For each R-calculus, if deduction systems t1/2 / 1/2 are recursive then there is a =t 1/2 1/2 1/2 1/2 t sound and complete R-calculus t1/2 / 1/2 / =t / t for R1/2 /St1/2 /Qt /Pt validity, respectively. =t

=t

t1/2 δ 1/2 δ |=t δ |=1/2 δ iff 1/2 1/2 1/2 1/2 1/2 =t δ t δ |==t δ |=t δ, where δ is a reduction of forms  ↑ A ⇒ [A] or (A) ↑ A ⇒ . =t =t If deduction systems t1/2 / 1/2 are not recursive then R-calculus t1/2 / 1/2 1/2 1/2 1/2 1/2 t / =t / t for R1/2 /St1/2 /Qt /Pt -validity is sound and partially complete. t1/2  ↑ R(a, b) ⇒ [R(a, b)] iff |=t1/2  ↑ R(a, b) ⇒ [R(a, b)] t1/2  ↑ R(a, b) ⇒  implies |=t1/2  ↑ R(a, b) ⇒   =t

 =t

1/2  ↑ R(a, b) ⇒ [R(a, b)] iff |=1/2  ↑ R(a, b) ⇒ [R(a, b)]

 =t 1/2 1/2 =t 1/2 =t 1/2 t 1/2 t

 =t |=1/2

 ↑ R(a, b) ⇒  implies  ↑ R(a, b) ⇒  1/2 , R(a, b) ↑ R(a, b) ⇒  iff |==t , R(a, b) ↑ R(a, b) ⇒  1/2

, R(a, b) ↑ R(a, b) ⇒ , R(a, b) implies |==t , R(a, b) ↑ R(a, b) ⇒ , R(a, b) 1/2 |=t

, R(a, b) ↑ R(a, b) ⇒  iff , R(a, b) ↑ R(a, b) ⇒  1/2 , R(a, b) ↑ R(a, b) ⇒ , R(a, b) implies |=t , R(a, b) ↑ R(a, b) ⇒ , R(a, b).

7.1 Binary-Valued DL with Role Constructors

That is,

t R1/2

St1/2 1/2

Qt

1/2

Pt

213

 ↑ R(a, b) ⇒ [R(a, b)]  ↑ R(a, b) ⇒   ↑ R(a, b) ⇒ [R(a, b)] ⇒  ↑ R(a, b) , R(a, b) ↑ R(a, b) ⇒  , R(a, b) ↑ R(a, b) ⇒ , R(a, b) , R(a, b) ↑ R(a, b) ⇒  , R(a, b) ↑ R(a, b) ⇒ , R(a, b)

SC SI SC SI SC SI SC SI.

7.1 Binary-Valued DL with Role Constructors The logical language of binary-valued description logic is added role constructors as follows: • atomic roles: S0 , S1 , ...; • role constructors: ¬, ∩, ∪, ◦, ∗. Roles are defined inductively as follows: R ::= S|¬R|R1 ∩ R2 |R1 ∪ R2 |R1 ◦ R2 |R1 ∗ R2 , where S is an atomic role. Given a model M = (U, I ), we define interpretation of roles R as functions from U 2 to L3 such that for any x, y ∈ U, ⎧ I (S)(x, y) ⎪ ⎪ ⎪ ⎪ (1 − I (R1 ))(x, y) ⎪ ⎪ ⎨ I (R1 )(x, y)&I (R2 )(x, y) I (R)(x, y) = ⎪ I (R1 )(x, y) or I (R2 )(x, y) ⎪ ⎪ ⎪ Ez(I (R1 )(x, z)&I (R2 )(z, y)) ⎪ ⎪ ⎩ Az(I (¬R1 )(x, z) or I (¬R2 )(z, y))

if if if if if if

R R R R R R

=S = ¬R1 = R1 ∩ R2 = R1 ∪ R2 = R1 ◦ R2 = R1 ∗ R2

Lemma 7.1.1 For I (R(a, b)) = t, we have the following equivalences: (R1 ∩ R2 )(a, b) ≡ R1 (x, y)∧R2 (x, y) ¬(R1 ∩ R2 )(a, b) ≡ ¬R1 (a, b)∨¬R2 (a, b) (R1 ∪ R2 )(a, b) ≡ R1 (a, b)∨R2 (a, b) ¬(R1 ∪ R2 )(a, b) ≡ ¬R1 (a, b)∧¬R2 (a, b)) (R1 ∗ R2 )(a, b) ≡ ∀c((¬R1 )(a, c)∨(¬R2 )(c, b)) (R1 ◦ R2 )(a, b) ≡ ∃d(R1 (a, d)∧R2 (d, b)).

214

Hence,

7 Role R-Calculus for Binary-Valued DL

¬(R1 ∗ R2 )(a, b) ≡ (R1 ◦ R2 )(a, b) ¬(R1 ◦ R2 )(a, b) ≡ (R1 ∗ R2 )(a, b). 

Lemma 7.1.2 For I (R(a, b)) = f, we have the following equivalences:

Hence,

(R1 ∩ R2 )(a, b) ≡ ¬(R1 ∩ R2 )(a, b) ≡

R1 (a, b)∨R2 (a, b) ¬R1 (a, b)∧¬R2 (a, b)

(R1 ∪ R2 )(a, b) ≡ ¬(R1 ∪ R2 )(a, b) ≡

R1 (a, b)∧R2 (a, b) ¬R1 (a, b)∨¬R2 (a, b)

(R1 ∗ R2 )(a, b) ≡ (R1 ◦ R2 )(a, b) ≡

∃d(R1 (a, d)∧R2 (d, b)) ∀c((¬R1 )(a, c)∨(¬R2 )(c, b)).

¬(R1 ∗ R2 )(a, b) ≡ (R1 ◦ R2 )(a, b) ¬(R1 ◦ R2 )(a, b) ≡ (R1 ∗ R2 )(a, b). 

7.2 1/2-Multisequents t A 1/2-multisequent  is M1/2 -valid, denoted by |=t1/2 , if for any interpretation I , there is a statement R(a, b) ∈  such that I (R(a, b)) = t.

7.2.1 Deduction System Mt1/2 t Deduction system M1/2 consists of the following axiom and deduction rules: • Axiom:  ∩ ¬−  = ∅ (At1/2 ) 

where  is a set of atomic role statements.

7.2 1/2-Multisequents

215

• Deduction rules: , R(a, b) (¬2 ) 2 , ¬ R(a, b)  , R1 (a, b) , R1 (a, b) (∩) , R2 (a, b) (∪) , R2 (a, b) ,(R1 ∩ R2 )(a, b) ,(R1 ∪ R2 )(a, b) , ¬R1 (a, b) , ¬R1 (a, b) (¬∩) , ¬R2 (a, b) (¬∪) , ¬R2 (a, b) , ¬(R1 ∩ R2 )(a, b) , ¬(R1 ∪ R2 )(a, b) 

 , R1 (a, d) , ¬R1 (a, c) (d, b) , R , ¬R2 (c, b) (◦) (¬◦) 2 , ,  (R1 ◦ R2 )(a, b)  ¬(R1 ◦ R2 )(a, b) , R1 (a, c) , ¬R1 (a, d) (∗) , R2 (c, b) (¬∗) , ¬R2 (d, b) , (R1 ∗ R2 )(a, b) , ¬(R1 ∗ R2 )(a, b)

and

where c is a new constant and d is a constant. t Definition 7.2.1 A 1/2-multisequent  is provable in M1/2 , denoted by t1/2 , if there is a sequence {δ1 , ..., δn ] of 1/2-multisequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous multisequents by one t . of the deduction rules in M1/2

Theorem 7.2.2 (Soundness and completeness theorem) For any 1/2-multisequent , t1/2  iff |=t1/2 .  t 7.2.2 R-Calculus R1/2 t Given a theory X and a statement X ∈ X, a reduction X ↑ X ⇒ X[X  ] is R1/2 -valid, denoted by |=t1/2 X ↑ X ⇒ X[X  ], if

X =



t -valid X if X[X ] is M1/2 λ otherwise.

That is, (0)

EY = X (X t Y, ¬Y ) AY = X (X t Y or X t ¬X ) (−) X ↑ X ⇒ X[X ] X↑X ⇒X

216

7 Role R-Calculus for Binary-Valued DL

Because X t ¬X is not recursively enumerable, we cannot deduce X ↑ X ⇒ X and deduce X ↑ X ⇒ X[X ] only. t consists of the following axioms and deduction rules: R-calculus R1/2 • Axioms: [r (a, b)] ∩ ¬−  = ∅ (At− 1/2 )  ↑ r (a, b) ⇒ [r (a, b)]  ∩ ¬− [¬r (a, b)] = ∅ (A¬t− 1/2 )  ↑ ¬r (a, b) ⇒ [¬r (a, b)], where  is a set of literal statements, and r (a, b) is atomic. • Deduction rules:  ↑ R(a, b) ⇒ [R(a, b)] (¬2− ) 2 2   ↑ ¬ R(a, b) ⇒ [¬ R(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∩− ) [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]   ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∪− )  ↑ R2 (a, b) ⇒ [R2 (a, b)]  ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)] and



 ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)]  ↑ ¬R2 (a, b) ⇒ [¬R2 (a, b)]   ↑ ¬(R1 ∩ R2 )(a, b) ⇒ [¬(R1 ∩ R2 )(a, b)]  ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] (¬∪− ) [¬R1 (a, b)] ↑ ¬R2 (a, b) ⇒ [¬R1 (a, b), ¬R2 (a, b)]  ↑ ¬(R1 ∪ R2 )(a, b) ⇒ [¬(R1 ∪ R2 )(a, b)] (¬∩− )

and



 ↑ R1 (a, d) ⇒ [R1 (a, d)] , R1 (a, d) ↑ R2 (d, b) ⇒ [R1 (a, d), R2 (d, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, c) ⇒ [¬R1 (a, c)] (¬◦− )  ↑ ¬R2 (c, b) ⇒ [¬R2 (c, b)]   ↑ ¬(R1 ◦ R2 )(a, b) ⇒ [¬(R1 ◦ R2 )(a, b)]  ↑ R1 (a, c) ⇒ [R1 (a, c)] (∗− )  ↑ R2 (c, b) ⇒ [R2 (c, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, d) ⇒ [¬R1 (a, d)] (¬∗− ) [¬R1 (a, d)] ↑ ¬R2 (d, b) ⇒ [¬R1 (a, d), ¬R2 (d, b)]  ↑ ¬(R1 ∗ R2 )(a, b) ⇒ [¬(R1 ∗ R2 )(a, b)]

(◦− )

Definition 7.2.3 A 1/2-reduction δ =  ↑ R(a, b) ⇒ , R  (a, b) is provable in =t t , denoted by 1/2 δ, if there is a sequence {δ1 , ..., δn } of 1/2-reductions such R1/2

7.2 1/2-Multisequents

217

that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the pret . vious 2/3-reductions by one of the deduction rules in R1/2 Theorem 7.2.4 (Soundness and completeness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ [R(a, b)], t1/2 δ iff |=t1/2 δ.  Theorem 7.2.5 (Soundness and incompleteness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ , t1/2 δ implies |=t1/2 δ; and

|=t1/2 δ may not imply t1/2 δ.

t 7.2.3 Deduction System N1/2 t Deduction system N1/2 consists of the following axiom and deduction rules: • Axiom:  ∩ ¬−  = ∅ (At1/2 ) ,

where  is a set of atomic role statements. • Deduction rules: , R(a, b) (¬2 ) 2 , ¬ R(a, b)  , R1 (a, b) , R1 (a, b) (∩) , R2 (a, b) (∪) , R2 (a, b) ,(R1 ∩ R2 )(a, b) ,(R1 ∪ R2 )(a, b) , ¬R1 (a, b) , ¬R1 (a, b) (a, b) , ¬R , ¬R2 (a, b) (¬∩) (¬∪) 2 , ¬(R1 ∩ R2 )(a, b) , ¬(R1 ∪ R2 )(a, b) and



 , R1 (a, c) , ¬R1 (a, d) (◦) , R2 (c, b) (¬◦) , ¬R2 (d, b) , ,  (R1 ◦ R2 )(a, b)  ¬(R1 ◦ R2 )(a, b) , R1 (a, d) , ¬R1 (a, c) (∗) , R2 (d, b) (¬∗) , ¬R2 (c, b) , (R1 ∗ R2 )(a, b) , ¬(R1 ∗ R2 )(a, b)

where c is a new constant and d is a constant.



218

7 Role R-Calculus for Binary-Valued DL =t

t Definition 7.2.6 A 1/2-multisequent  is provable in N1/2 , denoted by 1/2 , if there is a sequence {δ1 , ..., δn } of 1/2-multisequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-multisequents by t one of the deduction rules in N1/2 .

Theorem 7.2.7 (Soundness and completeness theorem) For any 1/2-multisequent , =t =t 1/2  iff |=1/2 . 

7.2.4 R-Calculus St1/2 Given a 1/2-multisequent  and a statement R(a.b), a reduction |R(a.b) ⇒ =t [R  (a.b)] is St1/2 -valid, denoted by |=1/2 |R(a.b) ⇒ [R  (a.b)], if 



R (a.b) =

t -valid R(a.b) if [R(a.b)] is N1/2 λ otherwise.

That is, ER  (a  , b ) = R(a, b)( f1/2 R  (a  , b ), ¬R  (a  , b )) |R(a, b) ⇒ [R(a, b)] AR  (a  , b ) = R(a, b)( f1/2 R  (a  , b ) or  t1/2 ¬R  (a  , b )) (0) |R(a.b) ⇒ 

(−)

Because  f1/2 R  (a  , b ) is not recursively enumerable, we consider the (−)-case. Let R(a, b) ∈ . R-calculus St1/2 consists of the following axioms and deduction rules: • Axioms: − =t− [r (a, b)] ∩ ¬   = ∅ (A1/2 ) |r (a, b) ⇒ [r (a, b)] − ¬=t−  ∩ ¬ [¬r (a, b)]  = ∅ (A1/2 ) |¬r (a, b) ⇒ [¬r (a, b)] where  is a set of literal statements, and r (a, b) is atomic. • Deduction rules: (¬2− ) 

(∩− )

 ↑ R(a, b) ⇒ [R(a, b)]  ↑ ¬2 R(a, b) ⇒ [¬2 R(a, b)]

 ↑ R1 (a, b) ⇒ [R1 (a, b)]  ↑ R2 (a, b) ⇒ [R2 (a, b)]  ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]

7.2 1/2-Multisequents

219



(∪− ) 

(¬∩− )

and

 ↑ R1 (a, b) ⇒ [R1 (a, b)] [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]

 ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] [¬R1 (a, b)] ↑ ¬R2 (a, b) ⇒ [¬R1 (a, b), ¬R2 (a, b)]  ↑ ¬(R1 ∩ R2 )(a, b) ⇒ [¬(R1 ∩ R2 )(a, b)]   ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] (¬∩− )  ↑ ¬R2 (a, b) ⇒ [¬R2 (a, b)]  ↑ ¬(R1 ∪ R2 )(a, b) ⇒ [¬(R1 ∪ R2 )(a, b)]



 ↑ R1 (a, d) ⇒ [R1 (a, d)]  ↑ R2 (d, b) ⇒ [R2 (d, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, c) ⇒ [¬R1 (a, c)] (¬◦− ) [¬R1 (a, c)] ↑ ¬R2 (c, b) ⇒ [¬R1 (a, c), ¬R2 (c, b)]   ↑ ¬(R1 ◦ R2 )(a, b) ⇒ [¬(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, c) ⇒ [¬R1 (a, c)] (∗− ) [¬R1 (a, c)] ↑ ¬R2 (c, b) ⇒ [¬R1 (a, c), ¬R2 (c, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ R1 (a, d) ⇒ [R1 (a, d)] − (¬∗ )  ↑ R2 (d, b) ⇒ [R2 (d, b)]  ↑ ¬(R1 ∗ R2 )(a, b) ⇒ [¬(R1 ∗ R2 )(a, b)] (◦− )

Definition 7.2.8 A 1/2-reduction δ =  ↑ R(a, b) ⇒ , R  (a, b) is provable in =t St1/2 , denoted by 1/2 δ, if there is a sequence {δ1 , ..., δn } of 1/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/3-reductions by one of the deduction rules in St1/2 . Theorem 7.2.9 (Soundness and completeness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ , R(a, b), =t =t 1/2 δ iff |=1/2 δ.  Theorem 7.2.10 (Soundness and incompleteness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ , =t =t 1/2 δ implies |=1/2 δ; and

=t

=t

|=1/2 δ may not imply 1/2 δ.



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7 Role R-Calculus for Binary-Valued DL

7.3 1/2-Co-multisequents 1/2

1/2

A 1/2-co-multisequent  is Lt -valid, denoted by |=f , if there is an interpretation I such that for each statement R(a, b) ∈ , I (R(a, b)) = t. 1/2 1/2 A 1/2-co-multisequent  is Kt -valid, denoted by |=t , if there is an interpretation I such that for each statement R(a, b) ∈ , I (R(a, b)) = t.

1/2

7.3.1 Incomplete Deduction System Lt 1/2

Deduction system Lt consists of the following axiom and deduction rules: • Axiom: − 1/2  ∩ ¬  = ∅ (At )  where  is a set of atomic role statements. • Deduction rules: , R(a, b) (¬2 ) 2 , ¬ R(a, b)  , R1 (a, b) , R1 (a, b) (∩) , R2 (a, b) (∪) , R2 (a, b) ,(R1 ∩ R2 )(a, b) ,(R1 ∪ R2 )(a, b) , ¬R1 (a, b) , ¬R1 (a, b) (¬∩) , ¬R2 (a, b) (¬∪) , ¬R2 (a, b) , ¬(R1 ∩ R2 )(a, b) , ¬(R1 ∪ R2 )(a, b) and



 , R1 (a, c) , ¬R1 (a, d) (◦) , R2 (c, b) (¬◦) , ¬R2 (d, b) , ,  (R1 ◦ R2 )(a, b)  ¬(R1 ◦ R2 )(a, b) , R1 (a, d) , ¬R1 (a, c) (∗) , R2 (d, b) (¬∗) , ¬R2 (c, b) , (R1 ∗ R2 )(a, b) , ¬(R1 ∗ R2 )(a, b)

where c is a new constant and d is a constant. 1/2

1/2

Definition 7.3.1 A 1/2-co-multisequent  is provable in Lf , denoted by f , if there is a sequence {δ1 , ..., δn } of 1/2-co-multisequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-co-multisequents 1/2 by one of the deduction rules in Lf .

7.3 1/2-Co-multisequents

221

Theorem 7.3.2 (Soundness and incompleteness theorem) For any 1/2-comultisequent , =t  implies |==t ; and |==t  may not imply =t . 

1/2

7.3.2 R-Calculus Qt

Let  =   , R(a, b). R-calculus Qt consists of the following axioms and deduction rules: • Axioms: 1/2 ¬r (a, b) ∈  (At− ) |r (a, b) ⇒ [r (a, b)] r (a, b) ∈  1/2 (A¬t− ) |¬r (a, b) ⇒ [¬r (a, b)] 1/2

where  is a set of literal statements, and r (a, b) is atomic. • Deduction rules:  ↑ R(a, b) ⇒ [R(a, b)] (¬2− ) 2 2   ↑ ¬ R(a, b) ⇒ [¬ R(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∩− )  ↑ R2 (a, b) ⇒ [R2 (a, b)]   ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∪− ) [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]  ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] − (¬∩ ) [¬R1 (a, b)] ↑ ¬R2 (a, b) ⇒ [¬R1 (a, b), ¬R2 (a, b)]   ↑ ¬(R1 ∩ R2 )(a, b) ⇒ [¬(R1 ∩ R2 )(a, b)]  ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] (¬∩− )  ↑ ¬R2 (a, b) ⇒ [¬R2 (a, b)]  ↑ ¬(R1 ∪ R2 )(a, b) ⇒ [¬(R1 ∪ R2 )(a, b)] and

222

7 Role R-Calculus for Binary-Valued DL



 ↑ R1 (a, d) ⇒ [R1 (a, d)]  ↑ R2 (d, b) ⇒ [R2 (d, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, c) ⇒ [¬R1 (a, c)] (¬◦− ) [¬R1 (a, c)] ↑ ¬R2 (c, b) ⇒ [¬R1 (a, c), ¬R2 (c, b)]   ↑ ¬(R1 ◦ R2 )(a, b) ⇒ [¬(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, c) ⇒ [¬R1 (a, c)] (∗− ) [¬R1 (a, c)] ↑ ¬R2 (c, b) ⇒ [¬R1 (a, c), ¬R2 (c, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ R1 (a, d) ⇒ [R1 (a, d)] (¬∗− )  ↑ R2 (d, b) ⇒ [R2 (d, b)]  ↑ ¬(R1 ∗ R2 )(a, b) ⇒ [¬(R1 ∗ R2 )(a, b)]

(◦− )

Definition 7.3.3 A 1/2-reduction δ =  ↑ R(a, b) ⇒ [R  (a, b)] is provable in 1/2 1/2 Qt , denoted by =t δ, if there is a sequence {δ1 , ..., δn } of 1/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the pre1/2 vious 2/3-reductions by one of the deduction rules in Qt . Theorem 7.3.4 (Soundness and completeness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ [R(a, b)], 1/2 1/2 =t δ iff |==t δ.  Theorem 7.3.5 (Soundness and incompleteness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ , 1/2 1/2 =t δ implies |==t δ; and

1/2

1/2

|==t δ may not imply =t δ.  1/2

7.3.3 Incomplete Deduction System Kt 1/2

Deduction system Kt consists of the following axiom and deduction rules: • Axiom: − 1/2  ∩ ¬  = ∅ (At )  where  is a set of atomic role statements.

7.3 1/2-Co-multisequents

223

• Deduction rules: , R(a, b) (¬2 ) 2 , ¬ R(a, b)  , R1 (a, b) , R1 (a, b) (∩) , R2 (a, b) (∪) , R2 (a, b) ,(R1 ∩ R2 )(a, b) ,(R1 ∪ R2 )(a, b) , ¬R1 (a, b) , ¬R1 (a, b) (¬∩) , ¬R2 (a, b) (¬∪) , ¬R2 (a, b) , ¬(R1 ∩ R2 )(a, b) , ¬(R1 ∪ R2 )(a, b) and



 , R1 (a, d) , ¬R1 (a, c) (d, b) , R , ¬R2 (c, b) (◦) (¬◦) 2 , ,  (R1 ◦ R2 )(a, b)  ¬(R1 ◦ R2 )(a, b) , R1 (a, c) , ¬R1 (a, d) (∗) , R2 (c, b) (¬∗) , ¬R2 (d, b) , (R1 ∗ R2 )(a, b) , ¬(R1 ∗ R2 )(a, b)

where c is a new constant and d is a constant. 1/2

1/2

Definition 7.3.6 A 1/2-co-multisequent  is provable in Kt , denoted by t , if there is a sequence {δ1 , ..., δn } of 1/2-co-multisequents such that δn = , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/2-co-multisequents 1/2 by one of the deduction rules in Kt . Theorem 7.3.7 (Soundness and incompleteness theorem) For any 1/2-comultisequent , 1/2 1/2 t  implies |=t , and

1/2

1/2

|=t  may not imply t .



1/2

7.3.4 R-Calculus Pt

Let  =   , R(a, b). R-calculus Pt consists of the following axioms and deduction rules: • Axioms: 1/2 ¬r (a, b) ∈  (At− ) |r (a, b) ⇒ [r (a, b)] r (a, b) ∈  1/2 (A¬t− ) |¬r (a, b) ⇒ [¬r (a, b)] 1/2

where  is a set of literal statements, and r (a, b) ∈  is atomic.

224

7 Role R-Calculus for Binary-Valued DL

• Deduction rules:  ↑ R(a, b) ⇒ [R(a, b)] (¬2− ) 2 2   ↑ ¬ R(a, b) ⇒ [¬ R(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∩− ) [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]   ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∪− )  ↑ R2 (a, b) ⇒ [R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]  ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] (¬∩− )  ↑ ¬R2 (a, b) ⇒ [¬R2 (a, b)]   ↑ ¬(R1 ∩ R2 )(a, b) ⇒ [¬(R1 ∩ R2 )(a, b)]  ↑ ¬R1 (a, b) ⇒ [¬R1 (a, b)] (¬∪− ) [¬R1 (a, b)] ↑ ¬R2 (a, b) ⇒ [¬R1 (a, b), ¬R2 (a, b)]  ↑ ¬(R1 ∪ R2 )(a, b) ⇒ [¬(R1 ∩ R2 )(a, b)] and



 ↑ R1 (a, c) ⇒ [R1 (a, c)] [R1 (a, c)] ↑ R2 (c, b) ⇒ [R1 (a, c), R2 (c, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, d) ⇒ [¬R1 (a, d)] (¬◦− )  ↑ ¬R2 (d, b) ⇒ [¬R2 (d, b)]   ↑ ¬(R1 ◦ R2 )(a, b) ⇒ [¬(R1 ◦ R2 )(a, b)]  ↑ R1 (a, d) ⇒ [R1 (a, d)] (∗− )  ↑ R2 (d, b) ⇒ [R2 (d, b)]  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]  ↑ ¬R1 (a, c) ⇒ [¬R1 (a, c)] (¬∗− ) [¬R1 (a, c)] ↑ ¬R2 (c, b) ⇒ [¬R1 (a, c), ¬R2 (c, b)]  ↑ ¬(R1 ∗ R2 )(a, b) ⇒ [¬(R1 ∗ R2 )(a, b)]

(◦− )

Definition 7.3.8 A 1/2-reduction δ =  ↑ R(a, b) ⇒ [R  (a, b)] is provable in 1/2 1/2 Pt , denoted by t δ, if there is a sequence {δ1 , ..., δn } of 1/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the pre1/2 vious 2/3-reductions by one of the deduction rules in Pt . Theorem 7.3.9 (Soundness and completeness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ [R(a, b)], 1/2 1/2 t δ iff |=t δ.  Theorem 7.3.10 (Soundness and incompleteness theorem) For any 1/2-reduction δ = |R(a, b) ⇒ , 1/2 1/2 t δ implies |=t δ;

7.4 2/2-Multisequents

and

225

1/2

1/2

|=t δ may not imply t δ. 

7.4 2/2-Multisequents = A 2/2-multisequent | is M2/2 -valid, denoted by |== 2/2 |, if for any interpretation I, either for some R(a, b) ∈ , I (R(a, b)) = t, or for some Q(a, b) ∈ , I (Q(a, b)) = f. = Lemma 7.4.1 Let ,  be a set of atoms. | is M2/2 -valid if and only if  ∩  = ∅.

 = 7.4.1 Deduction System M2/2 = Deduction system M2/2 consists of the following deduction rules and axiom: • Axiom:  ∩  = ∅ , (A= 2/2 ) |

where ,  are sets of atoms. • Deduction rules: |, R(a, b) , Q(a, b)| (¬ Q ) , ¬R(a, b)| |¬Q(a, b),    , R1 (a, b)| |Q 1 (a, b),  (∩ R ) , R2 (a, b)| (∩ Q ) |Q 2 (a, b),  , |(Q 1 ∩ Q 2 )(a, b),   (R1 ∩ R2 )(a, b)|  , R1 (a, b)| |Q 1 (a, b),  (∪ R ) , R2 (a, b)| (∪ Q ) |Q 2 (a, b),  , (R1 ∪ R2 )(a, b)| |(Q 1 ∪ Q 2 )(a, b),  (¬ R )

and



 , R1 (a, d)| |R1 (a, c),  (◦ L ) , R2 (d, b)| (∗ R ) |R2 (c, b),  , |(R1 ∗ R2 )(a, b),   (R1 ◦ R2 )(a, b)|  , R1 (a, c)| |R1 (a, d)| (◦ L ) , R2 (c, b)| (◦ R ) |R2 (d, b)| , (R1 ◦ R2 )(a, b)| |(R1 ◦ R2 )(a, b), 

= Definition 7.4.2 A 2/2-multisequent | is provable in M2/2 , denoted by = 2/2 |, if there is a sequence {δ1 , ..., δn } of 2/2-multisequents such that δn = |, and for

226

7 Role R-Calculus for Binary-Valued DL

each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-multisequents by one of the = . deduction rules in M2/2 Theorem 7.4.3 (Soundness and completeness theorem) For any 2/2-multisequent |, = = 2/2 | iff |=2/2 |.  = 7.4.2 R-Calculi R2/2

Let R(a, b) ∈  and Q(a  , b ) ∈ . A reduction | ↑ (R(a, b), Q(a  , b )) ⇒ =     -valid, denoted by |==   | is R2/2 2/2 | ↑ (R(a, b), Q(a , b )) ⇒  | , if 

= [R(a, b)] if [R(a, b)]| is M2/2 -valid  otherwise;  = [Q(a  , b )] if   |[Q(a  , b )] is M2/2 -valid   =  otherwise.

 =

= R-calculus R2/2 consists of the following deduction rules and axioms: • Axioms: r (a, b) ∈ / (A=L− 2/2 ) | ↑ r (a, b) ⇒ [r (a, b)]| t (a, b) ∈ / (A=R− 2/2 )   | ↑ t (a, b) ⇒   |[t (a, b)],

where | is atomic, and r (a, b) ∈ , t (a, b) ∈ . • Deduction rules: | ↑ 2 R(a, b) ⇒ |[R(a, b) | ↑ ¬R(a, b) ⇒ [¬R(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∩ L− ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪ L− ) | ↑ R2 (a, b) ⇒ [R2 (a, b)]| | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]|  | ↑ R1 (a, d) ⇒ [R1 (a, d)]| (◦ L− ) [R1 (a, d)]| ↑ R2 (d, b) ⇒ [R1 (a, d), R2 (d, b)]| | ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]|  | ↑ R1 (a, c) ⇒ [R1 (a, c)]| (∗ L− ) | ↑ R2 (c, b) ⇒ [R2 (c, b)]| | ↑ (R1 ∗ R2 )(a, b) ⇒ [(R1 ∗ R2 )(a, b)]| (¬ L− )

7.4 2/2-Multisequents

and

227

  | ↑ 1 Q(a, b) ⇒   [Q(a, b)]|   | ↑ ¬Q(a, b) ⇒   |[¬Q(a, b)    | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] R− (∩ )   | ↑ Q 2 (a, b) ⇒   |[Q 2 (a, b)]   ↑ (Q 1 ∩ Q 2 )(a, b) ⇒   |[(Q 1 ∩ Q 2 )(a, b)]  |   | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] R− (∪ )   [Q 1 (a, b)]| ↑ Q 2 (a, b) ⇒   |[Q 1 (a, b), Q 2 (a, b)]  ↑ (Q 1 ∪ Q 2 )(a, b) ⇒   |[(Q 1 ∪ Q 2 )(a, b)]  |   | ↑ Q 1 (a, c) ⇒   |[Q 1 (a, c)] R− (◦ )   | ↑ Q 2 (c, b) ⇒   |[Q 2 (c, b)]   ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]  |   | ↑ Q 1 (a, d) ⇒   |[Q 1 (a, d)] R− (∗ )   |[Q 1 (a, d) ↑ Q 2 (d, b) ⇒   |[Q 1 (a, d), Q 2 (d, b)]   | ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]

(¬ R− )

Definition 7.4.4 A 2/2-reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | is prov= , denoted by = able in R2/2 2/2 δ, if there is a sequence {δ1 , ..., δn } of 2/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the = . previous 2/2-reductions by one of the deduction rules in R2/2 Theorem 7.4.5 (Soundness and completeness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | such that either   =  or  = , = = 2/2 δ iff |=2/2 δ.

 Theorem 7.4.6 (Soundness and incompleteness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ |, = = 2/2 δ implies |=2/2 δ,

and

= |== 2/2 δ may not imply 2/2 δ.

 = 7.4.3 Deduction System N2/2 =

= A 2/2-multisequent | is N2/2 -valid, denoted by |=2/2 |, if for any interpretation I, either for some R(a, b) ∈ , I (R(a, b)) = t, or for some Q(a, b) ∈ , I (Q(a, b)) = f.

228

7 Role R-Calculus for Binary-Valued DL

= Lemma 7.4.7 Let ,  be a set of atoms. | is N2/2 -valid if and only if  ∩  = ∅.

= Deduction system N2/2 consists of the following deduction rules and axiom: • Axiom:  ∩  = ∅ = (A2/2 ) |,



where ,  are sets of atoms. • Deduction rules: |, R(a, b) , Q(a, b)| (¬ Q ) , ¬R(a, b)| |¬Q(a, b),    , R1 (a, b)| |Q 1 (a, b),  (∩ R ) , R2 (a, b)| (∩ Q ) |Q 2 (a, b),  , |(Q 1 ∩ Q 2 )(a, b),   (R1 ∩ R2 )(a, b)|  , R1 (a, b)| |Q 1 (a, b),  (∪ R ) , R2 (a, b)| (∪ Q ) |Q 2 (a, b),  , (R1 ∪ R2 )(a, b)| |(Q 1 ∪ Q 2 )(a, b),  (¬ R )

and



 , R1 (a, c)| |, Q 1 (a, d) (◦ R ) , R2 (c, b)| (◦ Q ) |, Q 2 (d, b) , |, (Q 1 ◦ Q 2 )(a, b)  (R1 ◦ R2 )(a, b)|  , R1 (a, d)| |, Q 1 (a, c) (∗ R ) , R2 (d, b)| (∗ R ) |, Q 2 (c, b) , (R1 ∗ R2 )(a, b)| |, (Q 1 ∗ Q 2 )(a, b) =

= Definition 7.4.8 A 2/2-multisequent | is provable in N2/2 , denoted by 2/2 |, if there is a sequence {δ1 , ..., δn } of 2/2-multisequents such that δn = |, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-multisequents by one of the = . deduction rules in N2/2

Theorem 7.4.9 (Soundness and completeness theorem) For any 2/2-multisequent |, = = 2/2 | iff |=2/2 |. 

7.4 2/2-Multisequents

229

7.4.4 R-Calculi S= 2/2 Let R(a, b) ∈ , Q(a  , b ) ∈ . A reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ =   | is S= 2/2 -valid, denoted by |=2/2 δ, if 

= [R(a, b)] if [R(a, b)]| is N2/2 -valid  otherwise;  = [Q(a  , b )] if   |[Q(a  , b )] is N2/2 -valid   =  otherwise. 

 =

R-calculus S= 2/2 consists of the following deduction rules and axioms: • Axioms: = R+ [r (a, b)] ∩   = ∅ (A2/2 ) | ↑ r (a, b) ⇒ [r (a, b)]|    = Q+  ∩ [t (a , b )]  = ∅ (A2/2 )    | ↑ t (a , b ) ⇒   |[t (a  , b )], where | is atomic, and r (a, b) ∈ , t (a  , b ) ∈ . • Deduction rules consists of two parts L R and R Q : ◦ LR : | ↑ 2 R(a, b) ⇒ |[R(a, b)] | ↑ ¬R(a, b) ⇒ [¬R(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∩ R− ) | ↑ R2 (a, b) ⇒ [R2 (a, b)]| | ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪ R− ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]| | ↑ R1 (a, d) ⇒ [R1 (a, d)]| (◦ R− ) | ↑ R2 (d, b) ⇒ [R2 (d, b)]| |  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]| | ↑ R1 (a, c) ⇒ [R1 (a, c)]| (∗ R− ) [R1 (a, c)]| ↑ R2 (c, b) ⇒ [R1 (a, c), R2 (c, b)]| | ↑ (R1 ∗ R2 )(a, b) ⇒ [(R1 ∗ R2 )(a, b)]|

(¬ R− )

◦ QQ :

230

7 Role R-Calculus for Binary-Valued DL

  | ↑ 1 Q(a, b) ⇒   [Q(a, b)]|   | ↑ ¬Q(a, b) ⇒   |[¬Q(a, b)    | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] Q− (∩ )   [Q 1 (a, b)]| ↑ Q 2 (a, b) ⇒   |[Q 1 (a, b), Q 2 (a, b)]   ↑ (Q 1 ∩ Q 2 )(a, b) ⇒   |[(Q 1 ∩ Q 2 )(a, b)]  |   | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] Q− (∪ )   | ↑ Q 2 (a, b) ⇒   |[Q 2 (a, b)]  ↑ (Q 1 ∪ Q 2 )(a, b) ⇒   |[(Q 1 ∪ Q 2 )(a, b)]  |   | ↑ Q 1 (a, c) ⇒   |[Q 1 (a, c)] Q− (◦ )   [Q 1 (a, c)]| ↑ Q 2 (c, b) ⇒   |[Q 1 (a, c), Q 2 (c, b)]   ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]  |   | ↑ Q 1 (a, d) ⇒   |[Q 1 (a, d)] Q− (∗ )   | ↑ Q 2 (d, b) ⇒   |[Q 2 (d, b)]   | ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]

(¬ Q− )

Definition 7.4.10 A 2/2-reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | is prov= able in S= 2/2 , denoted by 2/2 δ, if there is a sequence {δ1 , ..., δn } of 2/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/2-reductions by one of the deduction rules in S= 2/2 . Theorem 7.4.11 (Soundness and completeness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | such that either   =  or  = , =

=

2/2 δ iff |=2/2 δ.  Theorem 7.4.12 (Soundness and incompleteness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ |, =

=

2/2 δ implies |=2/2 δ, and

=

=

|=2/2 δ may not imply 2/2 δ. 

7.5 2/2-Co-multisequents 2/2

2/2 A 2/2-co-multisequent | is L= -valid, denoted by |== |, if there is an interpretation I such that for each R(a, b) ∈ , I (R(a, b)) = f, and for each Q(a, b) ∈ , I (Q(a, b)) = t. 2/2 -valid if and only if  ∩  = ∅. Lemma 7.5.1 Let ,  be a set of atoms. | is L=



7.5 2/2-Co-multisequents

231

2/2

7.5.1 Incomplete Deduction System L=

2/2

2/2 A 2/2-co-multisequent | is L= -valid, denoted by |== |, if there is an interpretation I such that for each R(a, b) ∈ , I (R(a, b)) = t, and for each Q(a, b) ∈ , I (Q(a, b)) = f. 2/2 Lemma 7.5.2 Let ,  be a set of atoms. | is L= -valid if and only if  ∩  = ∅.

2/2 Deduction system L= consists of the following deduction rules and axiom: • Axiom: 2/2  ∩  = ∅ , (A= ) |



where ,  are sets of atoms. • Deduction rules: |, R(a, b) , Q(a, b)| (¬ Q ) , ¬R(a, b)| |¬Q(a, b),    , R1 (a, b)| |Q 1 (a, b),  (∩ R ) , R2 (a, b)| (∩ Q ) |Q 2 (a, b),  , |(Q 1 ∩ Q 2 )(a, b),   (R1 ∩ R2 )(a, b)|  , R1 (a, b)| |Q 1 (a, b),  (∪ R ) , R2 (a, b)| (∪ Q ) |Q 2 (a, b),  , (R1 ∪ R2 )(a, b)| |(Q 1 ∪ Q 2 )(a, b),  (¬ R )

and



 , R1 (a, c)| |, Q 1 (a, d) (◦ R ) , R2 (c, b)| (◦ Q ) |, Q 2 (d, b) , |, (Q 1 ◦ Q 2 )(a, b)  (R1 ◦ R2 )(a, b)|  , R1 (a, d)| |, Q 1 (a, c) (∗ R ) , R2 (d, b)| (∗ R ) |, Q 2 (c, b) , (R1 ∗ R2 )(a, b)| |, (Q 1 ∗ Q 2 )(a, b) 2/2

2/2 Definition 7.5.3 A 2/2-co-multisequent | is provable in L= , denoted by = |, if there is a sequence {δ1 , ..., δn } of 2/2-co-multisequents such that δn = |, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-co-multisequents by one 2/2 . of the deduction rules in L=

Theorem 7.5.4 (Soundness and incompleteness theorem) For any 2/2-co-multisequent |, 2/2 2/2 = | implies |== |, and

2/2

2/2

|== | may not imply = |.



232

7 Role R-Calculus for Binary-Valued DL

2/2

7.5.2 R-Calculi Q=

Let  =   , R(a, b) and  =  , Q(a  , b ). A reduction δ = | ↑ (R(a, b), 2/2 Q(a  , b )) ⇒   | is Q2/2 = -valid, denoted by |== δ, if 

2/2 -valid [R(a, b)] if [R(a, b)]| is L=  otherwise;  2/2 -valid [Q(a  , b )] if   |[Q(a  , b )] is L=   =  otherwise. 

 =

R-calculus Q2/2 = consists of the following deduction rules and axioms: • Axioms: r (a, b) ∈  2/2 (A= R− ) | ↑ r (a, b) ⇒ [r (a, b)]| q(a  , b ) ∈   2/2 (A= Q− )   | ↑ q(a  , b ) ⇒   |[q(a  , b )] where | is atomic, and r (a, b) ∈ , q(a  , b ) ∈ . • Deduction rules consists of two parts L R and R Q : ◦ LR : | ↑ 2 R(a, b) ⇒ |[R(a, b)] | ↑ ¬R(a, b) ⇒ [¬R(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∩ R− ) | ↑ R2 (a, b) ⇒ [R2 (a, b)]| | ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪ R− ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]| | ↑ R1 (a, d) ⇒ [R1 (a, d)]| (◦ R− ) | ↑ R2 (d, b) ⇒ [R2 (d, b)]| |  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]| | ↑ R1 (a, c) ⇒ [R1 (a, c)]| R− (∗ ) [R1 (a, c)]| ↑ R2 (c, b) ⇒ [R1 (a, c), R2 (c, b)]| | ↑ (R1 ∗ R2 )(a, b) ⇒ [(R1 ∗ R2 )(a, b)]|

(¬ R− )

◦ QQ :

7.5 2/2-Co-multisequents

233

  | ↑ 1 Q(a, b) ⇒   [Q(a, b)]|   | ↑ ¬Q(a, b) ⇒   |[¬Q(a, b)    | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] Q− (∩ )   [Q 1 (a, b)]| ↑ Q 2 (a, b) ⇒   |[Q 1 (a, b), Q 2 (a, b)]   ↑ (Q 1 ∩ Q 2 )(a, b) ⇒   |[(Q 1 ∩ Q 2 )(a, b)]  |   | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] Q− (∪ )   | ↑ Q 2 (a, b) ⇒   |[Q 2 (a, b)]  ↑ (Q 1 ∪ Q 2 )(a, b) ⇒   |[(Q 1 ∪ Q 2 )(a, b)]  |   | ↑ Q 1 (a, c) ⇒   |[Q 1 (a, c)] Q− (◦ )   [Q 1 (a, c)]| ↑ Q 2 (c, b) ⇒   |[Q 1 (a, c), Q 2 (c, b)]   ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]  |   | ↑ Q 1 (a, d) ⇒   |[Q 1 (a, d)] Q− (∗ )   | ↑ Q 2 (d, b) ⇒   |[Q 2 (d, b)]   | ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]

(¬ Q− )

Definition 7.5.5 A 2/2-reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | is prov2/2 able in Q2/2 = , denoted by = δ, if there is a sequence {δ1 , ..., δn } of 2/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/2-reductions by one of the deduction rules in Q2/2 = . Theorem 7.5.6 (Soundness and completeness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | such that either   =  or  = , 2/2

2/2

= δ iff |== δ.  Theorem 7.5.7 (Soundness and incompleteness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ |, 2/2

2/2

= δ implies |== δ, and

2/2

2/2

|== δ may not imply = δ. 

2/2

7.5.3 Incomplete Deduction System K=

2/2 A 2/2-co-multisequent | is K= -valid, denoted by |=2/2 = |, if there is an interpretation I such that for each R(a, b) ∈ , I (R(a, b)) = t, and for each Q(a, b) ∈ , I (Q(a, b)) = f. 2/2 consists of the following deduction rules and axiom: Deduction system K=

234

7 Role R-Calculus for Binary-Valued DL

• Axiom: (A2/2 = )

∩=∅ , |

where ,  are sets of atoms. • Deduction rules: |, R(a, b) , Q(a, b)| (¬ Q ) , ¬R(a, b)| |¬Q(a, b),    , R1 (a, b)| |Q 1 (a, b),  (∩ R ) , R2 (a, b)| (∩ Q ) |Q 2 (a, b),  , |(Q 1 ∩ Q 2 )(a, b),   (R1 ∩ R2 )(a, b)|  , R1 (a, b)| |Q 1 (a, b),  (∪ R ) , R2 (a, b)| (∪ Q ) |Q 2 (a, b),  , (R1 ∪ R2 )(a, b)| |(Q 1 ∪ Q 2 )(a, b),  (¬ R )

and



 , R1 (a, d)| |R1 (a, c),  (◦ L ) , R2 (d, b)| (∗ R ) |R2 (c, b),  , |(R1 ∗ R2 )(a, b),   (R1 ◦ R2 )(a, b)|  , R1 (a, c)| |R1 (a, d)| L R (c, b)| , R (◦ ) (◦ ) |R2 (d, b)| 2 , (R1 ◦ R2 )(a, b)| |(R1 ◦ R2 )(a, b), 

2/2 , denoted by = Definition 7.5.8 A 2/2-multisequent | is provable in K= 2/2 |, if there is a sequence {δ1 , ..., δn } of 2/2-multisequents such that δn = |, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/2-multisequents by one of the 2/2 . deduction rules in K=

Theorem 7.5.9 (Soundness and incompleteness theorem) For any 2/2-multisequent |, 2/2 2/2 = | implies |== |, and 2/2 |=2/2 = | may not imply = |.

 2/2

7.5.4 R-Calculi P=

Let  =   , R(a, b) and  =  , Q(a  , b ). A reduction | ↑ (R(a, b), 2/2   -valid, denoted by |=2/2 Q(a  , b )) ⇒   | is P= = | ↑ (R(a, b), Q(a , b )) ⇒    | , if

7.5 2/2-Co-multisequents



2/2 -valid [R(a, b)] if [R(a, b)]| is K=  otherwise;  2/2 -valid [Q(a  , b )] if   |[Q(a  , b )] is K=  =  otherwise.

 =

2/2 consists of the following deduction rules and axioms: R-calculus P= • Axioms: r (a, b) ∈  2/2 (A=R− ) | ↑ r (a, b) ⇒ [r (a, b)]| t (a, b) ∈  2/2 (A=Q− )   | ↑ t (a, b) ⇒   |[t (a, b)],

where | is atomic, and r (a, b) ∈ , t (a, b) ∈ . • Deduction rules: | ↑ 2 R(a, b) ⇒ |[R(a, b) | ↑ ¬R(a, b) ⇒ [¬R(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∩ L− ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| |  ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]| | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪ L− ) | ↑ R2 (a, b) ⇒ [R2 (a, b)]| | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]| | ↑ R1 (a, d) ⇒ [R1 (a, d)]| (◦ L− ) [R1 (a, d)]| ↑ R2 (d, b) ⇒ [R1 (a, d), R2 (d, b)]| |  ↑ (R1 ◦ R2 )(a, b) ⇒ [(R1 ◦ R2 )(a, b)]| | ↑ R1 (a, c) ⇒ [R1 (a, c)]| (∗ L− ) | ↑ R2 (c, b) ⇒ [R2 (c, b)]| | ↑ (R1 ∗ R2 )(a, b) ⇒ [(R1 ∗ R2 )(a, b)]| (¬ L− )

and

  | ↑ 1 Q(a, b) ⇒   [Q(a, b)]|   | ↑ ¬Q(a, b) ⇒   |[¬Q(a, b)    | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] R− (∩ )   | ↑ Q 2 (a, b) ⇒   |[Q 2 (a, b)]   ↑ (Q 1 ∩ Q 2 )(a, b) ⇒   |[(Q 1 ∩ Q 2 )(a, b)]  |   | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] (∪ R− )   [Q 1 (a, b)]| ↑ Q 2 (a, b) ⇒   |[Q 1 (a, b), Q 2 (a, b)]  ↑ (Q 1 ∪ Q 2 )(a, b) ⇒   |[(Q 1 ∪ Q 2 )(a, b)]  |   | ↑ Q 1 (a, c) ⇒   |[Q 1 (a, c)] (◦ R− )   | ↑ Q 2 (c, b) ⇒   |[Q 2 (c, b)]   ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)]  |   | ↑ Q 1 (a, d) ⇒   |[Q 1 (a, d)] R− (∗ )   |[Q 1 (a, d) ↑ Q 2 (d, b) ⇒   |[Q 1 (a, d), Q 2 (d, b)]   | ↑ (Q 1 ◦ Q 2 )(a, b) ⇒   |[(Q 1 ◦ Q 2 )(a, b)] (¬ R− )

235

236

7 Role R-Calculus for Binary-Valued DL

Definition 7.5.10 A 2/2-reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | is prov2/2 , denoted by 2/2 able in P= = δ, if there is a sequence {δ1 , ..., δn } of 2/2-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the 2/2 previous 2/2-reductions by one of the deduction rules in P= . Theorem 7.5.11 (Soundness and completeness theorem) For any reduction δ =   | ↑ (R(a, b), Q(a  , b )) ⇒   | such that either   =  or  = , 2/2 2/2 = δ iff |== δ.

 Theorem 7.5.12 (Soundness and incompleteness theorem) For any reduction δ =   | ↑ (R(a, b), Q(a  , b )) ⇒ |, 2/2 2/2 = δ implies |== δ,

and 2/2 |=2/2 = δ may not imply = δ.



7.6 Conclusions Let Tt and Tf be sets of deduction rules for truth and false, respectively. Then t M1/2 = Tt + (At1/2 ), t N1/2 = Tf + (Af1/2 ), 1/2 1/2 Lt = Tf + (Af ), 1/2 1/2 Kt = Tt + (At ).

Let Rt and Rf be sets of deduction rules for truth and false in R-calculi, respectively. Then t R1/2 = Rt + (St1/2 ), t S1/2 = Rf + (Sf1/2 ), 1/2 1/2 Qt = Rf + (Sf ), 1/2 1/2 Pt = Rt + (St ), 1/2

1/2

1/2

1/2

t where (St1/2 ), (Sf1/2 ), (Sf ), (St ) are sets of axioms in R1/2 , St1/2 , Qt , Pt , respectively. Similar for 2/2-multisequents. Let Gt and Gf be sets of deduction rules for truth and false, respectively. Then

7.6 Conclusions

237 t M2/2 = Gt + (At2/2 ), t N2/2 = Gf + (Af2/2 ), 2/2 2/2 Lt = Gf + (Af ), 2/2 2/2 Kt = Gt + (At ).

Let Rt and Rf be sets of deduction rules for truth and false in R-calculi, respectively. Then t R2/2 = Rt + (Rt2/2 ), t S2/2 = Rf + (Rf2/2 ), 2/2 2/2 Qt = Rf + (Rf ), 2/2 2/2 Pt = Rt + (Rt ), 2/2

2/2

2/2

2/2

t where (Tt2/2 ), (Tf2/2 ), (Tf ), (Tt ) are sets of axioms in R2/2 , St2/2 , Qt , Pt , respectively. Notice that in R-calculus | ↑ (R(a, b), Q(a  , b )) ⇒   | , there may be an injury. Assume that there is a stage s0 such that

t1/2 | ↑ Q(a  , b ) ⇒s0 |[Q(a  , b )]; and at a later stage s1 > s0 such that t= |[Q(a  , b )] ↑ R(a, b) ⇒s1 [R(a, b)]|[Q(a  , b )]. There is a possibility that |=t= [R(a, b)]| ↑ Q(a  , b ) ⇒ [R(a, b)]|. So we use R-calculus for reduction [R(a, b)]| ↑ Q(a  , b ) ⇒ [R(a, b)]| and there are two cases: there is a stage s2 > s1 such that either t= [R(a, b)]| ↑ Q(a  , b ) ⇒s2 [R(a, b)]|[Q(a  , b )] or

t= [R(a, b)]| ↑ Q(a  , b ) ⇒s2 [R(a, b)]|.

In this case, eliminating Q(a  , b ) out of  at stage s0 + 1 is injured by eliminating R(a, b) out of  at stage s1 + 1. Li and Sui (2017), Rogers (1987), Soare (1987).

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7 Role R-Calculus for Binary-Valued DL

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Logic 50, 510–530 (1985) Avron, A.: Gentzen-type systems, resolution and tableaux. J. Autom. Reason. 10, 265–281 (1993) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation. Applications. Cambridge University Press, Cambridge, UK (2003) Baader, F., Horrocks, I., Sattler, U.: Chapter 3 description logics. In: van Harmelen, F., Lifschitz, V., Porter, B. (eds.), Handbook of Knowledge Representation. Elsevier (2007) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Fensel, D., van Harmelen, F., Horrocks, I., McGuinness, D., Patel-Schneider, P.F.: OIL: an ontology infrastructure for the semantic web. IEEE Intell. Syst. 16, 38–45 (2001) Gärdenfors, P., Rott, H.: Belief revision. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming. Epistemic and Temporal Reasoning, vol. 4, pp. 35–132. Oxford Science Pub (1995) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The R-calculus and the finite injury priority method. J. Comput. 12, 127–134 (2017) Rogers, H.: Theory of Recursive Functions and Effective Computability. The MIT Press (1987) Soare, R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Functions and Computably Generated Sets. Springer (1987) Takeuti, G.: Proof theory. In: Barwise, J. (ed.), Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, NL (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (eds.), Handbook of Philosophical Logic, vol. 2, (2nd edn), pp. 249–295. Kluwer, Dordrecht (2001) Zach, R.: Proof theory of finite-valued logics, Tech. Report TUW-E185.2-Z.1-93

Chapter 8

Role R-Calculus for Post Three-Valued DL

t t t t Ri/3 Si/3 Mi/3 N i/3 i/3 i/3 i/3 i/3 Lt Kt Qt Pt

In this chapter, we consider two kinds of validity: t t N1/3 :  ∈ N1/3 ↔ AI ER(a, b) ∈ (I (R(a, b)) = t) 1/3 1/3 Kt :  ∈ Kt ↔ EI AR(a, b) ∈ (I (R(a, b)) = t). 1/3

t N1/3 is complementary to Kt , respectively. =t t N1/3 |=1/3 1/3 Kt |=1/3 t =t

t For each validity, there is a sound and complete deduction system 1/3 for N1/3 validity Avron (1991), Baader et al. (2003), Bochvar (1938), Post (1920); and the 1/3 1/3 deduction system t for Kt -validity is sound and incomplete. There are two kinds of R-calculi Alchourron et al. (1985), Fermé and Hansson (2011), Ginsberg (1987), Li and R-calculus (2007):

 ↑ R(a, b) ⇒  − {R  (a, b)] ∈ St1/3  R(a, b)  ↔ R (a, b) = λ 1/3  ↑ R(a, b) ⇒ , R  (a, b) ∈ Pt  R(a, b) ↔ R  (a, b) = λ

=t

if |=1/3  − {R(a, b)] otherwise 1/3

if |=t , R(a, b) otherwise

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_8

239

240

8 Role R-Calculus for Post Three-Valued DL

and  =t

 =t

1/3  ↑ R(a, b) ⇒ [R(a, b)] iff |=1/3  ↑ R(a, b) ⇒ [R(a, b)]  =t

 =t

1/3  ↑ R(a, b) ⇒  implies |=1/3  ↑ R(a, b) ⇒ 

 =t  =t |=1/3  ↑ R(a, b) ⇒  may not imply 1/3  ↑ R(a, b) ⇒  1/3 1/3 t , R(a, b) ↑ R(a, b) ⇒  iff |=t , R(a, b) ↑ R(a, b) ⇒  1/3 1/3 t , R(a, b) ↑ R(a, b) ⇒ , R(a, b) implies |=t , R(a, b) ↑ R(a, b) ⇒ , R(a, b) 1/3 1/3 |=t , R(a, b) ↑ R(a, b) ⇒ , R(a, b) may not imply t , R(a, b) ↑ R(a, b) ⇒ , R(a, b).

We consider the following deduction systems and R-calculi: t tm = Deduction system N1/3 , N2/3 , N3/3 1/3 3/3 Kt , K= t tm = R-calculus S1/3 , S2/3 , S3/3 1/3 3/3 Pt , P=

and

t N1/3 , ∼ , ∼2  t N2/3 |, ∼  = N3/3 ||.

8.1 Post Three-Valued DL with Role Constructors The logical language of Post three-valued DL with role constructors is that of Post three-valued DL added role constructors as follows: • atomic roles: S0 , S1 , ...; • role constructors: ∼, ∩, ∪, ∗ , ◦ . Notice that ∗, ◦ are binary role constructors in the last chapter and ∗ , ◦ are unary role constructors in this chapter. Roles are defined inductively as follows: R ::= S| ∼ R|R1 ∩ R2 |R1 ∪ R2 |R ∗ |R ◦ , where S is an atomic role. Role statements are of form R(a, b), where R is a role and a, b are constants. Given a model M = (U, I ), we define interpretation of roles R as functions from U 2 to L3 such that for any x, y ∈ U,

8.1 Post Three-Valued DL with Role Constructors

⎧ ⎪ ⎪ I (S)(x, y) ⎪ ⎪ f ∼ (I (R1 ))(x, y) ⎪ ⎪ ⎨ I (R1 )(x, y)&I (R2 )(x, y) I (R)(x, y) = I (R1 )(x, y) or I (R2 )(x, y) ⎪ ⎪ ⎪ ⎪ I (R1 )∗ (x, y) ⎪ ⎪ ⎩ I (R1 )◦ (x, y)

241

if if if if if if

R R R R R R

=S =∼ R1 = R1 ∩ R2 = R1 ∪ R2 = R1∗ = R1◦

where (x, y) ∈ I (R)∗ if there are x1 , ..., xn such that (x, x1 ), ..., (xn−1 , xn ), (xn , y) ∈ I (R); and (x, y) ∈ I (R)◦ if for any x1 , ..., xn , there is an i ≤ n such that (xi , xi+1 ) ∈ I (R). Hence, we have the following equivalences: R ∗ (a, b) iff ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )) (∼ R ∗ )(a, b) iff ∀c1 , ..., cn ∃i ≤ n(∼ R(ci , ci+1 )∨ ∼2 R(ci , ci+1 )) (∼2 R ∗ )(a, b) iff ∀c1 , ..., cn ∃i ≤ n(∼2 R(ci , ci+1 )) R ◦ (a, b) iff ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )) (∼ R ◦ )(a, b) iff ∀c1 , ..., cn ∃i ≤ n(∼ R(ci , ci+1 ) ∨ R(ci , ci+1 )) (∼2 R ◦ )(a, b) iff ∃d1 , ..., dn ∀i ≤ n(∼2 R(di , di+1 )). Lemma 8.1.1 For I (R(a, b)) = t, we have the following equivalences: (R1 ∩ R2 )(a, b) ≡ R1 (x, y)∧R2 (x, y) ∼ (R1 ∩ R2 )(a, b) ≡ (R1 (a, b)∧ ∼ R2 (a, b))∨(∼ R1 (a, b)∧R2 (a, b)) ∨(∼ R1 (a, b)∧ ∼ R2 (a, b)) ∼2 (R1 ∩ R2 )(a, b) ≡∼2 R1 (a, b)∨ ∼2 R2 (a, b) (R1 ∪ R2 )(a, b) ≡ R1 (a, b)∨R2 (a, b) ∼ (R1 ∪ R2 )(a, b) ≡ (∼ R1 (a, b)∧ ∼ R2 (a, b))∨(∼ R1 (a, b)∧ ∼2 R2 (a, b)) ∨(∼2 R1 (a, b)∧ ∼ R2 (a, b)) ∼2 (R1 ∪ R2 )(a, b) ≡∼2 R1 (a, b)∧ ∼2 R2 (a, b) and R ∗ (a, b) ≡ ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )) (∼ R ∗ )(a, b) ≡ ∀c1 , ..., cn ∃i ≤ n(∼ R(ci , ci+1 )∨ ∼2 R(ci , ci+1 )) (∼2 R ∗ )(a, b) ≡ ∀c1 , ..., cn ∃i ≤ n(∼2 R(ci , ci+1 )) R ◦ (a, b) ≡ ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )) (∼ R ◦ )(a, b) ≡ ∀c1 , ..., cn ∃i ≤ n(∼ R(ci , ci+1 ) ∨ R(ci , ci+1 )) (∼2 R ◦ )(a, b) ≡ ∃d1 , ..., dn ∀i ≤ n(∼2 R(di , di+1 )). 

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8 Role R-Calculus for Post Three-Valued DL

Lemma 8.1.2 For I (R(a, b)) = t, we have the following equivalences: (R1 ∩ R2 )(a, b) ≡ R1 (x, y)∨R2 (x, y) ∼ (R1 ∩ R2 )(a, b) ≡ (R1 (a, b)∨ ∼ R2 (a, b))∧(∼ R1 (a, b)∨R2 (a, b)) ∧(∼ R1 (a, b)∨ ∼ R2 (a, b)) ∼2 (R1 ∩ R2 )(a, b) ≡∼2 R1 (a, b)∧ ∼2 R2 (a, b) (R1 ∪ R2 )(a, b) ≡ R1 (a, b)∧R2 (a, b) ∼ (R1 ∪ R2 )(a, b) ≡ (∼ R1 (a, b)∨ ∼ R2 (a, b))∧(∼ R1 (a, b)∨ ∼2 R2 (a, b)) ∧(∼2 R1 (a, b)∨ ∼ R2 (a, b)) ∼ (R1 ∪ R2 )(a, b) ≡∼2 R1 (a, b)∨ ∼2 R2 (a, b) 2

and R ∗ (a, b) ≡ ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )) (∼ R ∗ )(a, b) ≡ ∃d1 , ..., dn ∀i ≤ n(∼ R(di , di+1 )∧ ∼2 R(di , di+1 )) (∼2 R ∗ )(a, b) ≡ ∃d1 , ..., dn ∀i ≤ n(∼2 R(di , di+1 )) R ◦ (a, b) ≡ ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )) (∼ R ◦ )(a, b) ≡ ∃d1 , ..., dn ∀i ≤ n(∼ R(di , di+1 )∧R(di , di+1 )) (∼2 R ◦ )(a, b) ≡ ∀c1 , ..., cn ∃i ≤ n(∼2 R(ci , ci+1 )). 

8.2 1/3-Multisequents =t

t A 1/3-multisequent  is N1/3 -valid, denoted by |=1/3 , if for any interpretation I, there is a statement R(a, b) ∈  such that I (R(a, b)) = t.

8.2.1 Deduction System Nt1/3 t Deduction system N1/3 consists of the following axiom and deduction rules: • Axiom: ⎧ ⎨ ∩ ∼−  = ∅ ∩ ∼−2  = ∅ =t (A1/3 ) ⎩ − ∼ ∩ ∼−2  = ∅ 

where  is a set of literal role statements. • Deduction rules:

8.2 1/3-Multisequents

243

, R(a, b) (∼3 ) 3 , ∼ R(a, b)  , R1 (a, b) , R1 (a, b) (a, b) , R , R2 (a, b) (∩) (∪) 2 , ⎡ (R1 ∩ R2 )(a, b) , ⎡ (R1 ∪ R2 )(a, b) , ∼ R1 (a, b) , ∼ R1 (a, b) ⎢ , ∼ R2 (a, b) ⎢ , ∼ R2 (a, b) ⎢ ⎢ ⎢ , R1 (a, b) ⎢ , ∼2 R1 (a, b) ⎢ ⎢ ⎢ (∼ ∩) ⎢  , ∼ R2 (a, b) (∼ ∪) ⎢ ⎢  , ∼ R2 (a, b) ⎣ , ∼ R1 (a, b) ⎣ , ∼ R1 (a, b) , R2 (a, b) , ∼2 R2 (a, b) , ,  ∼ (R21 ∩ R2 )(a, b)  ∼ (R21 ∪ R2 )(a, b) , ∼ R1 (a, b) , ∼ R1 (a, b) (∼2 ∩) , ∼2 R2 (a, b) (∼2 ∪) , ∼2 R2 (a, b) , ∼2 (R1 ∩ R2 )(a, b) , ∼2 (R1 ∪ R2 )(a, b) and

⎡

, ∼ R(a, d1 ) ⎢ , ∼2 R(a, d1 ) ⎢ ⎢ , ∼ R(d1 , d2 ) ⎢ ⎢ , ∼2 R(d1 , d2 ) (∼ ∗) ⎢ ⎢··· ⎢ ⎣ , ∼ R(cn , b) , ∼2 R(cn , b) ∗ ⎡ ⎡ , (∼ R )(a, b) 2 , ∼ R(a, d1 ) , R(a, d1 ) ⎢ , ∼2 R(d1 , d2 ) ⎢ , R(d1 , d2 ) ⎢ ⎢ (∼2 ∗) ⎣ · · · (◦) ⎣ · · · , ∼2 R(dn , b) , R(dn , b) 2 ∗ , (∼ R )(a, b) , R ◦ (a, b) ⎡ , R(a, d1 ) ⎢ , ∼ R(a, d1 ) ⎧ ⎢ , ∼2 R(a, c1 ) ⎪ ⎢ , R(d1 , d2 ) ⎪ ⎨ ⎢ , ∼2 R(c1 , c2 ) ⎢ , ∼ R(d1 , d2 ) (∼2 ◦) ⎪ (∼ ◦) ⎢ ⎪··· ⎢··· ⎩ ⎢ , ∼2 R(cn , b) ⎣ , R(dn , b) , (∼2 R ◦ )(a, b) , ∼ R(dn , b) , (∼ R ◦ )(a, b) ⎧ , R(a, c1 ) ⎪ ⎪ ⎨ , R(c1 , c2 ) ··· (∗) ⎪ ⎪ ⎩ , R(cn , b) , R ∗ (a, b)

where ci is a new constant and di is a constant. =t

t Definition 8.2.1 A 1/3-multisequent  is provable in N1/3 , denoted by 1/3 , if there is a sequence {1 , ..., n } of 1/3-multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/3-multisequents by t . one of the deduction rules in N1/3

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8 Role R-Calculus for Post Three-Valued DL

Theorem 8.2.2 (Soundness and completeness theorem) For any 1/3-multisequent , =t =t 1/3  iff |=1/3 . 

8.2.2 R-Calculus St1/3 Given a 1/3-multisequent  and a statement R(a, b) ∈ , a reduction  ↑ R(a, b) ⇒ [R  (a, b)] is St1/3 -valid, denoted by |=t1/3  ↑ R(a, b) ⇒ [R  (a, b)], if R  (a, b) =



t -valid R(a, b) if [R(a, b)] is N1/3 λ otherwise.

R-calculus St1/3 consists of the following axioms and deduction rules: • Axioms: ⎧ ⎨ [r (a, b)]∩ ∼−  = ∅ [r (a, b)]∩ ∼−2  = ∅ =t− (A1/3 ) ⎩ − ∼ ∩ ∼−2  = ∅ ⎧↑ r (a, b) ⇒ [r (a, b)] ⎨ ∩ ∼− [∼ r (a, b)] = ∅ ∩ ∼−2  = ∅ ∼=t− (A1/3 ) ⎩ − ∼ [∼ r (a, b)]∩ ∼−2  = ∅ ⎧↑∼ r (a, b) ⇒ [∼ r (a, b)] ⎨ ∩ ∼−  = ∅ ∩ ∼−2 [∼2 r (a, b)] = ∅ ∼2 =t− (A1/3 ) ⎩ − ∼ ∩ ∼−2 [∼2 r (a, b)] = ∅  ↑∼2 r (a, b) ⇒ [∼2 r (a, b)] where  is a set of literals, and r (a, b) is an atomic role statement. • Deduction rules:  ↑ R(a, b) ⇒ [R(a, b)] (∼3− ) 3 R(a, b) ⇒ [∼3 R(a, b)]  ↑∼   ↑ R1 (a, b) ⇒ [R1 (a, b)] (∩− )  ↑ R2 (a, b) ⇒ [R2 (a, b)]   ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∪− ) [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]

8.2 1/3-Multisequents

245

and ⎧  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎨  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∗− ) ⎪ (◦− ) ⎪··· ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ R ∗ (a, b) ⇒ [R ∗ (a, b)] and

⎡

 ↑ R(a, d1 ) ⇒ [R(a, d1 )] ⎢  ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )] ⎢ ⎣···  ↑ R(dn , b) ⇒ [R(dn , b)]  ↑ R ◦ (a, b) ⇒ [R ◦ (a, b)]

⎡

 ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎢  ↑∼ R2 (a, b) ⇒ [∼ R2 (a, b)] ⎢ ⎢ [Z 1 ] ↑ R1 (a, b) ⇒ [Z 1 , R1 (a, b)] ⎢ (∼ ∩− ) ⎢ ⎢  [Z 1 ] ↑∼ R2 (a, b) ⇒ [Z 1 , ∼ R2 (a, b)] ⎣ [Z 2 ] ↑∼ R1 (a, b) ⇒ [Z 2 , ∼ R1 (a, b)] [Z 2 ] ↑ R2 (a, b) ⇒ [Z 2 , R2 (a, b)]  ↑∼ (R1 ∩ R2 )(a, b) ⇒ [∼ (R1 ∩ R2 )(a, b)]

where Z 1 =∼ R1 (a, b)∨ ∼ R2 (a, b), Z 2 = R1 (a, b)∨ ∼ R2 (a, b), and ⎡

 ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎢  ↑∼ R2 (a, b) ⇒ [∼ R2 (a, b)] ⎢ ⎢ [Z 1 ] ↑∼2 R1 (a, b) ⇒ [Z 1 , ∼2 R1 (a, b)] ⎢ (∼ ∪− ) ⎢ ⎢  [Z 1 ] ↑∼ R2 (a, b) ⇒ [Z 1 , ∼ R2 (a, b)] ⎣ [Z 3 ] ↑∼ R1 (a, b) ⇒ [Z 3 , ∼ R1 (a, b)] [Z 3 ] ↑∼2 R2 (a, b) ⇒ [Z 3 , ∼2 R2 (a, b)]  ↑∼ (R1 ∪ R2 )(a, b) ⇒ [∼ (R1 ∪ R2 )(a, b)] where Z 3 =∼ R1 (a, b)∨ ∼2 R2 (a, b), and ⎡

 ↑∼ R(a, d1 ) ⇒ [∼ R(a, d1 )] ⎢  ↑∼2 R(a, d1 ) ⇒ [∼2 R(a, d1 )] ⎢ ⎢  ↑∼ R(d1 , d2 ) ⇒ [∼ R(d1 , d2 )] ⎢ 2 2 ⎢ − ⎢  ↑∼ R(d1 , d2 ) ⇒ [∼ R(d1 , d2 )] (∼ ∗ ) ⎢ ⎢··· ⎣  ↑∼ R(dn , b) ⇒ [∼ R(dn , b)]  ↑∼2 R(dn , b) ⇒ [∼2 R(dn , b)]  ↑ (∼ R ∗ )(a, b) ⇒ [(∼ R ∗ )(a, b)] ⎡  ↑∼ R(a, d1 ) ⇒ [∼ R(a, d1 )] ⎢  ↑ R(a, d1 ) ⇒ [R(a, d1 )] ⎢ ⎢  ↑∼ R(d1 , d2 ) ⇒ [∼ R(d1 , d2 )] ⎢ ⎢ − ⎢  ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )] (∼ ◦ ) ⎢ ⎢··· ⎣  ↑∼ R(dn , b) ⇒ [∼ R(dn , b)]  ↑ R(dn , b) ⇒ [R(dn , b)]  ↑ (∼ R ◦ )(a, b) ⇒ [(∼ R ◦ )(a, b)]

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8 Role R-Calculus for Post Three-Valued DL

and 

 ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] (∼ ∩− ) [∼2 R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼2 R2 (a, b)] 2 2  1 ∩ R2 )(a, b) ⇒ [∼ (R1 ∩ R2 )(a, b)]  ↑∼ (R 2 2  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] (∼2 ∪− )  ↑∼2 R2 (a, b) ⇒ [∼2 R2 (a, b)]  ↑∼2 (R1 ∪ R2 )(a, b) ⇒ [∼2 (R1 ∪ R2 )(a, b)] 2

and

⎡

 ↑∼2 R(a, d1 ) ⇒ [∼2 R(a, d1 )] ⎢  ↑∼2 R(d1 , d2 ) ⇒ [∼2 R(d1 , d2 )] ⎢ (∼2 ∗− ) ⎣ · · ·  ↑∼2 R(dn , b) ⇒ [∼2 R(dn , b)]  ↑ (∼2 R ∗ )(a, b) ⇒ [(∼2 R ∗ )(a, b)] ⎧  ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )] ⎪ ⎪ ⎨  ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )] 2 − ··· (∼ ◦ ) ⎪ ⎪ ⎩  ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)]  ↑ (∼2 R ◦ )(a, b) ⇒ [(∼2 R ◦ )(a, b)]

Definition 8.2.3 Given a 1/3-multisequent  and a statement R(a, b) ∈ , a reduc=t tion  ↑ R(a, b) ⇒   is provable in St1/3 , denoted by 1/3  ↑ R(a, b) ⇒   , if there is a sequence {δ1 , ..., δn } of reductions such that δn =  ↑ R(a, b) ⇒   , and for each 1 ≤ i ≤ n, δi is deduced from the previous reductions by one of the deduction rules in St1/3 . Theorem 8.2.4 (Soundness and completeness theorem) For any R-statement R(a, b) and any reduction δ =  ↑ R(a, b) ⇒ , R(a, b), =t

=t

1/3 δ iff |=1/3 δ.  Theorem 8.2.5 (Soundness and incompleteness theorem) For any R-statement R(a, b) and any reduction δ =  ↑ R(a, b) ⇒ , =t

=t

1/3 δ implies |=1/3 δ,

8.2 1/3-Multisequents

and

247

=t

=t

|=1/3 δ may not imply 1/3 δ. 

1/3

8.2.3 Incomplete Deduction System Kt 1/3

1/3

A 1/3-multisequent  is Kt -valid, denoted by |=t , if there is an interpretation I such that for any statement R(a, b) ∈ , I (R(a, b)) = t. 1/3 Deduction system Kt consists of the following axiom and deduction rules: • Axiom: ⎡ ∩ ∼−  = ∅ ⎣ ∩ ∼−2  = ∅ 1/3 (At ) ∼− ∩ ∼−2  = ∅  where  is a set of literal role statements. • Deduction rules: , R(a, b) (∼3 ) 3 ,  ∼ R(a, b)  , R1 (a, b) , R1 (a, b) (∩) , R2 (a, b) (∪) , R2 (a, b) , ⎧ (R1 ∩ R2 )(a, b) , ⎧ (R1 ∪ R2 )(a, b) , ∼ R , ∼ R1 (a, b) (a, b) ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a, b) , ∼ R , ∼ R2 (a, b) ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎨ , R1 (a, b) , ∼2 R1 (a, b) (∼ ∩) ⎪ (∼ ∪) ⎪ ⎪  , ∼ R2 (a, b) ⎪  , ∼ R2 (a, b) ⎪ ⎪ ⎪ ⎪ , ∼ R , ∼ R1 (a, b) (a, b) ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , R2 (a, b) , ∼2 R2 (a, b) , ,  ∼ (R21 ∩ R2 )(a, b)  ∼ (R21 ∪ R2 )(a, b) , ∼ R1 (a, b) , ∼ R1 (a, b) (∼2 ∩) , ∼2 R2 (a, b) (∼2 ∪) , ∼2 R2 (a, b) , ∼2 (R1 ∩ R2 )(a, b) , ∼2 (R1 ∪ R2 )(a, b)

248

and

8 Role R-Calculus for Post Three-Valued DL

⎧ , ∼ R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎡ , ∼2 R(a, c1 ) ⎪ ⎪ , R(a, d1 ) ⎪ ⎪ ⎨ , ∼ R(c1 , c2 ) ⎢ , R(d1 , d2 ) ⎢ , ∼2 R(c1 , c2 ) (∗) ⎣ · · · (∼ ∗) ⎪ ⎪ ··· ⎪ ⎪ , R(dn , b) ⎪ ⎪ , ∼ R(cn , b) ⎪ ⎪ , R ∗ (a, b) ⎩ , ∼2 R(cn , b) ∗ ⎧ ⎧ , (∼ R )(a, b) 2 , ∼ R(a, c1 ) ⎪ ⎪ , R(a, c1 ) ⎪ ⎪ ⎨ ⎨ , ∼2 R(c1 , c2 ) , R(c1 , c2 ) ··· (∼2 ∗) ⎪ (◦) ⎪ ⎪··· ⎪ ⎩ ⎩ , ∼2 R(cn , b) , R(cn , b) 2 ∗ , (∼ R )(a, b) , R ◦ (a, b) ⎧ , R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎡ , ∼ R(a, c1 ) ⎪ ⎪ , ∼2 R(a, d1 ) ⎪ ⎪ ⎨ , R(c1 , c2 ) ⎢ , ∼2 R(d1 , d2 ) ⎢ , ∼ R(c1 , c2 ) 2 (∼ ◦) ⎪ (∼ ◦) ⎣ · · · ⎪ ⎪··· ⎪ , ∼2 R(dn , b) ⎪ ⎪ ⎪ , R(cn , b) ⎪ , (∼2 R ◦ )(a, b) ⎩ , ∼ R(cn , b) , (∼ R ◦ )(a, b)

where ci is a new constant and di is a constant. 1/3

1/3

Definition 8.2.6 A 1/3-multisequent  is provable in Kt , denoted by t , if there is a sequence {1 , ..., n } of 1/3-multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/3-multisequents by 1/3 one of the deduction rules in Kt . Theorem 8.2.7 (Soundness and incompleteness theorem) For any 1/3-multisequent , 1/3

1/3

t  implies |=t , and

1/3

1/3

|=t  may not imply t .  1/3

8.2.4 R-Calculus Pt

Given a 1/3-multisequent  and a statement R(a, b), a reduction |R(a, b) ⇒ 1/3 1/3 , R  (a, b) is Pt -valid, denoted by |=t |R(a, b) ⇒ , R  (a, b), if

8.2 1/3-Multisequents

249

R  (a, b) =



1/3

R(a, b) if , R(a, b) is Kt -valid λ otherwise.

Let   = (R(a, b)). R-calculus Pt consists of the following axioms and deduction rules: • Axioms:  r (a, b) ∈∼−   1/3 (At− ) r (a, b) ∈∼−2    ↑ r (a, b) ⇒   [r (a, b)] r (a, b) ∈   1/3 (At∼− ) r (a, b) ∈∼−2    ↑∼ r (a, b) ⇒   [∼ r (a, b)] r (a, b) ∈   1/3 (At∼2 − ) r (a, b) ∈∼−     ↑∼2 r (a, b) ⇒   [∼2 r (a, b)] 1/3

where   is a set of literals. • Deduction rules:   ↑ R(a, b) ⇒   [R(a, b)] (∼3− )  3  [∼3 R(a, b)]   ↑∼ R(a, b) ⇒   ↑ R1 (a, b) ⇒   [R1 (a, b)] (∩− )   [R1 (a, b)] ↑ R2 (a, b) ⇒   [R1 (a, b), R2 (a, b)]   (R1 ∩ R2 )(a, b) ⇒   [(R1 ∩ R2 )(a, b)]  ↑   ↑ R1 (a, b) ⇒   [R1 (a, b)] (∪− )   ↑ R2 (a, b) ⇒   [R2 (a, b)]   ↑ (R1 ∪ R2 )(a, b) ⇒   [(R1 ∪ R2 )(a, b)] and

⎡

  ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢   ↑ R(d1 , d2 ) ⇒   [R(d1 , d2 )] ⎢ (∗− ) ⎣ · · ·   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ R ∗ (a, b) ⇒   [R ∗ (a, b)] ⎧   ↑ R(a, c1 ) ⇒   [R(a, c1 )] ⎪ ⎪ ⎨   ↑ R(c1 , c2 ) ⇒   [R(c1 , c2 )] − ··· (◦ ) ⎪ ⎪ ⎩   ↑ R(cn , b) ⇒   [R(cn , b)]   ↑ R ◦ (a, b) ⇒   [R ◦ (a, b)]

250

8 Role R-Calculus for Post Three-Valued DL

and ⎧    ↑∼ R1 (a, b) ⇒   [∼ R1 (a, b)] ⎪ ⎪ ⎪ ⎪   [∼ R1 (a, b)] ↑∼ R2 (a, b) ⇒   [∼ R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪ ⎨    ↑ R1 (a, b) ⇒   [R1 (a, b)] (∼ ∩− ) ⎪ ⎪    [R1 (a, b)] ↑∼ R2 (a, b) ⇒   [R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪   ↑∼ R1 (a, b) ⇒   [∼ R1 (a, b)] ⎪ ⎪ ⎩   [∼ R1 (a, b)] ↑ R2 (a, b) ⇒   [∼ R1 (a, b), R2 (a, b)]   ↑∼ (R1 ∩ R2 )(a, b) ⇒   [∼ (R1 ∩ R2 )(a, b)] and ⎧    ↑∼ R1 (a, b) ⇒   [∼ R1 (a, b)] ⎪ ⎪ ⎪ ⎪   [∼ R1 (a, b)] ↑∼ R2 (a, b) ⇒   [∼ R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪ ⎨    ↑∼2 R1 (a, b) ⇒   [∼2 R1 (a, b)]  2  2 − (∼ ∪ ) ⎪ 2 (a, b) ⇒  [∼ R1 (a, b), ∼ R2 (a, b)] ⎪    [∼ R1 (a, b)] ↑∼ R ⎪  ⎪  ↑∼ R1 (a, b) ⇒  [∼ R1 (a, b)] ⎪ ⎪ ⎩   [∼ R1 (a, b)] ↑∼2 R2 (a, b) ⇒   [∼ R1 (a, b), ∼2 R2 (a, b)]   ↑∼ (R1 ∪ R2 )(a, b) ⇒   [∼ (R1 ∪ R2 )(a, b)] and

⎧   ↑∼ R(a, c1 ) ⇒   [∼ R(a, c1 )] ⎪ ⎪ ⎪  ⎪  ↑∼2 R(a, c1 ) ⇒   [∼2 R(a, c1 )] ⎪ ⎪ ⎪  ⎪ ⎨  ↑∼ R(c1 , c2 ) ⇒   [∼ R(c1 , c2 )]   ↑∼2 R(c1 , c2 ) ⇒   [∼2 R(c1 , c2 )] (∼ ∗− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪   ↑∼ R(cn , b) ⇒   [∼ R(cn , b)] ⎪ ⎪ ⎩   ↑∼2 R(cn , b) ⇒   [∼2 R(cn , b)]   ↑ (∼ R ∗ )(a, b) ⇒   [(∼ R ∗ )(a, b)] ⎧   ↑∼ R(a, c1 ) ⇒   [∼ R(a, c1 )] ⎪ ⎪ ⎪ ⎪   ↑ R(a, c1 ) ⇒   [R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨   ↑∼ R(c1 , c2 ) ⇒   [∼ R(c1 , c2 )]   ↑ R(c1 , c2 ) ⇒   [R(c1 , c2 )] (∼ ◦− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪   ↑∼ R(cn , b) ⇒   [∼ R(cn , b)] ⎪ ⎪ ⎩   ↑ R(cn , b) ⇒   [R(cn , b)]   ↑ (∼ R ◦ )(a, b) ⇒   [(∼ R ◦ )(a, b)]

8.2 1/3-Multisequents

251

and    ↑∼2 R1 (a, b) ⇒   [∼2 R1 (a, b)] (∼2 ∩− )   ↑∼2 R2 (a, b) ⇒   [∼2 R2 (a, b)]   ↑∼2 (R1 ∩ R2 )(a, b) ⇒   [∼2 (R1 ∩ R2 )(a, b)]    ↑∼2 R1 (a, b) ⇒   [∼2 R1 (a, b)] 2 − (∼ ∪ )   [∼2 R1 (a, b)] ↑∼2 R2 (a, b) ⇒   [∼2 R1 (a, b), ∼2 R2 (a, b)]   ↑∼2 (R1 ∪ R2 )(a, b) ⇒   [∼2 (R1 ∪ R2 )(a, b)]

and

⎧   ↑∼2 R(a, c1 ) ⇒   [∼2 R(a, c1 )] ⎪ ⎪ ⎨   ↑∼2 R(c1 , c2 ) ⇒   [∼2 R(c1 , c2 )] 2 − ··· (∼ ∗ ) ⎪ ⎪ ⎩   ↑∼2 R(cn , b) ⇒   [∼2 R(cn , b)]   ↑ (∼2 R ∗ )(a, b) ⇒   [(∼2 R ∗ )(a, b)] ⎡   ↑∼2 R(a, d1 ) ⇒   [∼2 R(a, d1 )] ⎢   ↑∼2 R(d1 , d2 ) ⇒   [∼2 R(d1 , d2 )] ⎢ (∼2 ◦− ) ⎣ · · ·   ↑∼2 R(dn , b) ⇒   [∼2 R(dn , b)]   ↑ (∼2 R ◦ )(a, b) ⇒   [(∼2 R ◦ )(a, b)]

where ci is a new constant and di is a constant. Definition 8.2.8 Given a 1/3-multisequent   and a statement R(a, b) ∈   , a 1/3 reduction   |R(a, b) ⇒   is provable in Pt , denoted by t   |R(a, b) ⇒   , 1/3

if there is a sequence {δ1 , ..., δn } of reductions such that δn =   |R(a, b) ⇒   , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by 1/3 one of the deduction rules in Pt . Theorem 8.2.9 (Soundness and completeness theorem) For any R-statement R(a, b) ∈   and any reduction δ =   |R(a, b) ⇒ , 1/3

1/3

t δ iff |=t δ.  Theorem 8.2.10 (Soundness and incompleteness theorem) For any R-statement R(a, b) ∈   and any reduction δ =   |R(a, b) ⇒   , 1/3

1/3

t δ implies |=t δ, and

1/3

1/3

|=t δ may not imply t δ. 

252

8 Role R-Calculus for Post Three-Valued DL

8.3 2/3-Multisequents tm A 2/3-multisequent | is N2/3 -valid, denoted by |=tm 2/3 |, if for any interpretation I, either for some R(a, b) ∈ , I (R(a, b)) = t, or for some Q(a, b) ∈ , I (Q(a, b)) = m. 2/3 2/3 A 2/3-multisequent | is Ktm -valid, denoted by |=tm |, if there is an interpretation I such that for each R(a, b) ∈ , I (R(a, b)) = t, and for each Q(a, b) ∈ , I (Q(a, b)) = m.

tm 8.3.1 Deduction System N2/3 tm Deduction system N2/3 consists of the following deduction rules and axiom: • Axiom: ⎧ ⎨  ∩  = ∅ ∩ ∼−  = ∅ (Atm 2/3 ) ⎩ ∩ ∼−   = ∅ |,

where  is a set of role atoms, and  is a set of role literals. • Deduction rules: , Q(a, b)| (∼2,Q ) 2 |, ⎡  ∼ Q(a, b) |, Q 1 (a, b) ⎢ |, Q 2 (a, b) ⎢  ⎢ , Q 1 (a, b)| , R1 (a, b)| ⎢ R Q (∩ ) , R2 (a, b)| (∩ ) ⎢ ⎢  |, Q 2 (a, b) ⎣ |, Q 1 (a, b) , (R1 ∩ R2 )(a, b)| , Q 2 (a, b)| |(Q ⎡  1 ∩ Q 2 )(a, b),  |, Q 1 (a, b) ⎢ |, Q 2 (a, b) ⎢  ⎢ |, ∼ Q 1 (a, b) , R1 (a, b)| ⎢ R Q (∪ ) , R2 (a, b)| (∪ ) ⎢ ⎢  |, Q 2 (a, b) ⎣ |, Q 1 (a, b) , (R1 ∪ R2 )(a, b)| |, ∼ Q 2 (a, b) |(Q   1 ∪ Q 2 )(a, b),  |, ∼ Q 1 (a, b) |, ∼ Q 1 (a, b) (∼ ∩ Q ) |, ∼ Q 2 (a, b) (∼ ∪ Q ) |, ∼ Q 2 (a, b) |, ∼ (Q 1 ∩ Q 2 )(a, b) |, ∼ (Q 1 ∪ Q 2 )(a, b) (∼ L )

|, R(a, b) , ∼ R(a, b)|

8.3 2/3-Multisequents

and

253

⎡

|, Q(a, d1 ) ⎢ |, ∼ Q(a, d1 ) ⎢ ⎢ |, Q(d1 , d2 ) ⎢ ⎢ Q ⎢ |, ∼ Q(d1 , d2 ) (∗ ) ⎢ ⎢··· ⎣ |, Q(dn , b) |, ∼ Q(dn , b) |, Q ∗ (a, b) ⎡ ⎡ |, ∼ Q(a, d1 ) , R(a, d1 )| ⎢ |, ∼ Q(d1 , d2 ) ⎢ , R(d1 , d2 )| ⎢ ⎢ (∼ ∗ Q ) ⎣ · · · (◦ R ) ⎣ · · · |, ∼ Q(dn , b) , R(dn , b)| ∗ |, (∼ R )(a, b) , R ◦ (a, b)| ⎡ , Q(a, d1 )| ⎢ |, Q(a, d1 ) ⎧ ⎢ |, ∼ Q(a, c1 ) ⎪ ⎢ , Q(d1 , d2 )| ⎪ ⎨ ⎢ |, ∼ Q(c1 , c2 ) ⎢ |, Q(d1 , d2 ) Q · ·· (◦ Q ) ⎢ ) (∼ ◦ ⎪ ⎢··· ⎪ ⎩ ⎢ |, ∼ Q(cn , b) ⎣ , Q(dn , b)| |, (∼ Q ◦ )(a, b) |, Q(dn , b) |, Q ◦ (a, b)

⎧ , R(a, c1 )| ⎪ ⎪ ⎨ , R(c1 , c2 )| ··· (∗ R ) ⎪ ⎪ ⎩ , R(cn , b)| , R ∗ (a, b)|

where ci is a new constant and di is a constant. =tm

tm Definition 8.3.1 A 2/3-multisequent | is provable in N2/3 , denoted by 2/3 |, if there is a sequence {δ1 , ..., δn } of 2/3-multisequents such that δn = |, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/3-multisequents by one of tm . the deduction rules in N2/3

Theorem 8.3.2 (Soundness and completeness theorem) For any 2/3-multisequent |, =tm =tm |=2/3 | iff 2/3 |. 

8.3.2 R-Calculus Stm 2/3 Let R(a, b) ∈  and Q(a  , b ) ∈ . A reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ =tm [R  (a, b)]|[Q  (a  , b )] is Stm 2/3 -valid, denoted by |=2/3 δ, if 

tm -valid R(a, b) if [R(a, b)]| is N2/3 λ otherwise;  tm -valid Q(a  , b ) if   |[Q(a  , b )] is N2/3    Q (a , b ) = λ otherwise.

R  (a, b) =

254

8 Role R-Calculus for Post Three-Valued DL

R-calculus Stm 2/3 consists of the following deduction rules and axioms: • Axioms: ⎧ ⎨ [r (a, b)] ∩  = ∅ [r (a, b)]∩ ∼−  = ∅ R−tm (A2/3 ) ⎩ ∩ ∼−  = ∅ | ⎧ ↑ r (a, b) ⇒ [r (a, b)]| ⎨  ∩ [q(a, b)] = ∅ ∩ ∼−  = ∅ Q−tm (A2/3 ) ⎩ [q(a, b)]∩ ∼−  = ∅ | ⎧ ↑ q(a, b) ⇒ |[q(a, b)] ⎨  ∩  = ∅ ∩ ∼− [∼ q(a, b)] = ∅ ∼Q−tm (A2/3 ) ⎩ ∩ ∼− [∼ q(a, b)] = ∅ | ↑∼ q(a, b) ⇒ |[∼ q(a, b)] where ,  are sets of role literals, and q(a, b) is a role atom. • Deduction rules: | ↑ 2 R(a, b) ⇒ |[R(a, b)] | ↑∼ R(a, b) ⇒ [∼ R(a, b)]| | ↑ l Q(a, b) ⇒ [Q(a, b)]| 2Q (∼− ) 2 2 | ↑∼ Q(a, b) ⇒ |[∼ Q(a, b)] | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∩−R ) | ↑ R2 (a, b) ⇒ [R2 (a, b)]| | ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪−R ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)| (∼−R )

and

⎧ | ↑ R(a, c1 ) ⇒ [R(a, c1 )]| ⎪ ⎪ ⎨ | ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )]| · ·· (∗ R− ) ⎪ ⎪ ⎩ | ↑ R(cn , b) ⇒ [R(cn , b)]| | ↑ R ∗ (a, b) ⇒ [R ∗ (a, b)]| ⎡ | ↑ R(a, d1 ) ⇒ [R(a, d1 )]| ⎢ | ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )]| ⎢ (◦ R− ) ⎣ · · · | ↑ R(dn , b) ⇒ [R(dn , b)]| | ↑ R ◦ (a, b) ⇒ [R ◦ (a, b)]|

8.3 2/3-Multisequents

and

255

⎡

| ↑ Q 1 (a, b) ⇒ |[Q 1 (a, b)] ⎢ ||Q 2 (a, b) ⇒ |[Q 2 (a, b)] ⎢ ⎢ |[Y1 ] ↑ 1 Q 1 (a, b) ⇒ [Q 1 (a, b)]|[Y1 ] ⎢ Q ⎢ (∩− ) ⎢  |[Y1 ]|Q 2 (a, b) ⇒ |[Y1 , Q 2 (a, b)] ⎣ |[Y2 ] ↑ Q 1 (a, b) ⇒ |[Y2 , Q 1 (a, b)] |[Y2 ]|1 Q 2 (a, b) ⇒ [Q 2 (a, b)]|[Y2 ] | ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ |[(Q 1 ∩ Q 2 )(a, b)]

where Y1 = Q 1 (a, b) ∨ Q 2 (a, b), Y2 = Y1 , ∼2 Q 1 (a, b) ∨ Q 2 (a, b), and ⎡

| ↑ Q 1 (a, b) ⇒ |[Q 1 (a, b)] ⎢ | ↑ Q 2 (a, b) ⇒ |[Q 2 (a, b)] ⎢ ⎢ |[Y1 ] ↑∼ Q 1 (a, b) ⇒ |[Y1 , ∼ Q 1 (a, b)] ⎢ Q ⎢ (∪− ) ⎢  |[Y1 ] ↑ Q 2 (a, b) ⇒ |[Y1 , Q 2 (a, b)] ⎣ |[Y3 ] ↑ Q 1 (a, b) ⇒ |[Y3 , Q 1 (a, b)] |[Y3 ] ↑∼ Q 2 (a, b) ⇒ |[Y3 , ∼ Q 2 (a, b)] | ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ |[(Q 1 ∪ Q 2 )(a, b)] where Y3 = Y1 , ∼ Q 1 (a, b) ∨ Q 2 (a, b), and 

| ↑∼ Q 1 (a, b) ⇒ |[∼ Q 1 (a, b)] |[∼ Q 1 (a, b)]| ∼ Q 2 (a, b) ⇒ |[∼ Q 1 (a, b), ∼ Q 2 (a, b)] |  ↑∼ (Q 1 ∩ Q 2 )(a, b) ⇒ |[∼ (Q 1 ∩ Q 2 )(a, b)] | ↑∼ Q 1 (a, b) ⇒ |[∼ Q 1 (a, b)] (∼ ∪−Q ) || ∼ Q 2 (a, b) ⇒ |[∼ Q 2 (a, b)] | ↑∼ (Q 1 ∪ Q 2 )(a, b) ⇒ |[∼ (Q 1 ∪ Q 2 )(a, b)]

(∼ ∩−Q )

and

⎡

| ↑∼ Q(a, d1 ) ⇒ |[∼ Q(a, d1 )] ⎢ | ↑∼2 Q(a, d1 ) ⇒ |[∼2 Q(a, d1 )] ⎢ ⎢ | ↑∼ Q(d1 , d2 ) ⇒ |[∼ Q(d1 , d2 )] ⎢ 2 2 ⎢ Q− ⎢ | ↑∼ Q(d1 , d2 ) ⇒ |[∼ Q(d1 , d2 )] (∗ ) ⎢ · · · ⎢ ⎣ | ↑∼ Q(dn , b) ⇒ |[∼ Q(dn , b)] | ↑∼2 Q(dn , b) ⇒ |[∼2 Q(dn , b)] | ↑ Q ◦ (a, b) ⇒ |[Q ◦ (a, b)]

256

and

and

8 Role R-Calculus for Post Three-Valued DL

⎡

| ↑ Q(a, c1 ) ⇒ |[Q(a, c1 )] ⎢ | ↑∼ Q(a, c1 ) ⇒ |[∼ Q(a, c1 )] ⎢ ⎢ | ↑ Q(c1 , c2 ) ⇒ |[Q(c1 , c2 )] ⎢ ⎢ Q− ⎢ | ↑∼ Q(c1 , c2 ) ⇒ |[∼ Q(c1 , c2 )] (◦ ) ⎢ ⎢··· ⎣ | ↑ Q(cn , b) ⇒ |[Q(cn , b)] | ↑∼ Q(cn , b) ⇒ |[∼ Q(cn , b)] | ↑ Q ◦ (a, b) ⇒ |[Q ◦ (a, b)] ⎡

| ↑∼ Q(a, d1 ) ⇒ |[∼ Q(a, d1 )] ⎢ | ↑∼ Q(d1 , d2 ) ⇒ |[∼ Q(d1 , d2 )] ⎢ (∼ ∗ Q− ) ⎣ · · · | ↑∼ Q(dn , b) ⇒ |[∼ Q(dn , b)] ∗ ∗ | ⎧ ↑ (∼ Q )(a, b) ⇒ |[(∼ Q )(a, b)] | ↑∼ Q(a, c1 ) ⇒ |[∼ Q(a, c1 )] ⎪ ⎪ ⎨ | ↑∼ Q(c1 , c2 ) ⇒ |[∼ Q(c1 , c2 )] ··· (∼ ◦ Q− ) ⎪ ⎪ ⎩ | ↑∼ Q(cn , b) ⇒ |[∼ Q(cn , b)] | ↑ (∼ Q ◦ )(a, b) ⇒ |[(∼ Q ◦ )(a, b)]

where ci is a new constant and di is a constant. Definition 8.3.3 A 2/3-reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒   | is prov=tm δ, if there is a sequence {δ1 , ..., δn } of reductions able in Stm 1/3 , denoted by  such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in Stm 1/3 . Theorem 8.3.4 (Soundness and completeness theorem) Let R(a, b) ∈  and Q(a  , b ) ∈ . For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ , R  (a, b)|, Q  (a  , b ) such that either R  (a, b) = λ or Q  (a  , b ) = λ, =tm

=tm

2/3 δ iff |=2/3 δ.  Theorem 8.3.5 (Soundness and incompleteness theorem) Let R(a, b) ∈  and Q(a  , b ) ∈ . For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ |, =tm

=tm

2/3 δ implies |=2/3 δ, and

=tm

=tm

|=2/3 δ may not imply 2/3 δ. 

8.3 2/3-Multisequents

257

2/3

8.3.3 Incomplete Deduction System Ktm 2/3

Deduction system Ktm consists of the following deduction rules and axiom: • Axiom: ⎡ ∩=∅ − 2/3 ⎣ ∩ ∼  = ∅ (Atm ) ∩ ∼−  = ∅ |, where  is a set of atoms, and  is a set of literals. • Deduction rules: , Q(a, b)| (∼2,Q ) ∼2 Q(a, b) ⎧|,  |, Q 1 (a, b) ⎪ ⎪ ⎪ ⎪ |, Q ⎪ 2 (a, b) ⎪  ⎨ , Q 1 (a, b)| , R1 (a, b)| (∩ R ) , R2 (a, b)| (∩ Q ) ⎪ ⎪  |, Q 2 (a, b) ⎪ ⎪ |, Q 1 (a, b) , (R1 ∩ R2 )(a, b)| ⎪ ⎪ ⎩ , Q 2 (a, b)| |(Q ⎧  1 ∩ Q 2 )(a, b),  |, Q 1 (a, b) ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b) ⎪ ⎪  ⎨ |, ∼ Q 1 (a, b) , R1 (a, b)| (∪ R ) , R2 (a, b)| (∪ Q ) ⎪ ⎪  |, Q 2 (a, b) ⎪ ⎪ |, Q 1 (a, b) , (R1 ∪ R2 )(a, b)| ⎪ ⎪ ⎩ |, ∼ Q 2 (a, b) |(Q   1 ∪ Q 2 )(a, b),  |, ∼ Q 1 (a, b) |, ∼ Q 1 (a, b) (∼ ∩ Q ) |, ∼ Q 2 (a, b) (∼ ∪ Q ) |, ∼ Q 2 (a, b) |, ∼ (Q 1 ∩ Q 2 )(a, b) |, ∼ (Q 1 ∪ Q 2 )(a, b) (∼ L )

|, R(a, b) , ∼ R(a, b)|

258

and

8 Role R-Calculus for Post Three-Valued DL

⎧ |, Q(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎡ |, ∼ Q(a, c1 ) ⎪ ⎪ , R(a, d1 )| ⎪ ⎪ ⎨ |, Q(c1 , c2 ) ⎢ , R(d1 , d2 )| ⎢ |, ∼ Q(c1 , c2 ) (∗ R ) ⎣ · · · (∗ Q ) ⎪ ⎪ · ·· ⎪ ⎪ , R(dn , b)| ⎪ ⎪ |, Q(cn , b) ⎪ ⎪ , R ∗ (a, b)| ⎩ |, ∼ Q(cn , b) |, Q ∗ (a, b) ⎧ ⎧ |, ∼ Q(a, c1 ) ⎪ ⎪ , R(a, c1 )| ⎪ ⎪ ⎨ ⎨ |, ∼ Q(c1 , c2 ) , R(c1 , c2 )| ··· (∼ ∗ Q ) ⎪ (◦ R ) ⎪ ⎪··· ⎪ ⎩ ⎩ |, ∼ Q(cn , b) , R(cn , b)| ∗ |, (∼ R )(a, b) , R ◦ (a, b)| ⎧ , Q(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎡ |, Q(a, c1 ) ⎪ ⎪ |, ∼ Q(a, d1 ) ⎪ ⎪ ⎨ , Q(c1 , c2 )| ⎢ |, ∼ Q(d1 , d2 ) ⎢ |, Q(c1 , c2 ) (◦ Q ) ⎪ (∼ ◦ Q ) ⎣ · · · ⎪ ⎪··· ⎪ |, ∼ Q(dn , b) ⎪ ⎪ ⎪ , Q(cn , b)| ⎪ |, (∼ Q ◦ )(a, b) ⎩ |, Q(cn , b) |, Q ◦ (a, b)

where ci is a new constant and di is a constant. 2/3

=tm

Definition 8.3.6 A 2/3-multisequent | is provable in Ktm , denoted by 2/3 |, if there is a sequence {δ1 , ..., δn } of 2/3-multisequents such that δn = |, and for each 1 ≤ i ≤ n, δi is deduced from the previous 2/3-multisequents by one of 2/3 the deduction rules in Ktm . Theorem 8.3.7 (Soundness and incompleteness theorem) For any 2/3-multisequent |, 2/3 2/3 tm | implies |=tm |, and

2/3

2/3

|=tm | may not imply tm |.  2/3

8.3.4 R-Calculus Ptm

A reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ , R  (a, b)|, Q  (a  , b ) is Ptm 2/3 valid, denoted by |=tm δ, if 2/3

8.3 2/3-Multisequents

259



2/3

, R(a, b) if , R(a, b)| is Ktm -valid  otherwise;  2/3   , Q(a , b ) if   |, Q(a  , b ) is Ktm -valid  =  otherwise.

 =

Let   = (R(a, b)) and  = (Q(a  , b )). R-calculus Ptm consists of the following deduction rules and axioms: • Axioms:  r (a, b) ∈  1/3 (A R−tm ) ∼ r (a, b) ∈     ⇒   [r (a, b)]|  |  ↑ r (a, b)  q(a , b ) ∈  1/3 (A Q−tm ) ∼ q(a  , b ) ∈   | ↑ q(a  , b ) ⇒   | [q(a  , b )] q(a  , b ) ∈   1/3  q(a , b ) ∈  (A∼Q−tm )    | ↑∼ q(a  , b ) ⇒   | [∼ q(a  , b )] 2/3

where   ,  are sets of role literals, and r (a, b), q(a  , b ) are role atoms. • Deduction rules:   | ↑ 2 R(a, b) ⇒   | [R(a, b)]   | ↑∼ R(a, b) ⇒   [∼ R(a, b)]|   | ↑ l Q(a, b) ⇒   [Q(a, b)]| 2Q (∼− )   2   2 b)]  |  ↑∼ Q(a, b) ⇒  | [∼ Q(a,  | ↑ R1 (a, b) ⇒  [R1 (a, b)]| (∩−R )   [R1 (a, b)]| ↑ R2 (a, b) ⇒   [R1 (a, b), R2 (a, b)]|    ↑ (R1 ∩ R2 )(a, b) ⇒   [(R1 ∩ R2 )(a, b)]|  |   | ↑ R1 (a, b) ⇒   [R1 (a, b)]| R (∪− )   | ↑ R2 (a, b) ⇒   [R2 (a, b)]|   | ↑ (R1 ∪ R2 )(a, b) ⇒   [(R1 ∪ R2 )(a, b)| (∼−R )

and

⎡

  | ↑ R(a, d1 ) ⇒   [R(a, d1 )]| ⎢   | ↑ R(d1 , d2 ) ⇒   [R(d1 , d2 )]| ⎢ R− ⎣ · · · (∗ )   | ↑ R(dn , b) ⇒   [R(dn , b)]|   | ↑ R ∗ (a, b) ⇒   [R ∗ (a, b)]| ⎧ ⎪   | ↑ R(a, c1 ) ⇒   [R(a, c1 )]| ⎪ ⎨    | ↑ R(c1 , c2 ) ⇒   [R(c1 , c2 )]| ··· (◦ R− ) ⎪ ⎪ ⎩    | ↑ R(cn , b) ⇒   [R(cn , b)]|   | ↑ R ◦ (a, b) ⇒   [R ◦ (a, b)]|

260

8 Role R-Calculus for Post Three-Valued DL

and ⎧     | ↑ Q 1 (a, b) ⇒   | [Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪   | [Q 1 (a, b)]|Q 2 (a, b) ⇒   | [Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨    1  | ↑ Q 1 (a, b) ⇒   [Q 1 (a, b)]| Q  (∩− ) ⎪ ⎪   [Q 1 (a, b)]| |Q 2 (a, b) ⇒   [Q 1 (a, b)]| [Q 2 (a, b)] ⎪ ⎪   | ↑ Q 1 (a, b) ⇒   | [Q 1 (a, b)] ⎪ ⎪ ⎩   | [Q 1 (a, b)]|1 Q 2 (a, b) ⇒   [Q 2 (a, b)]| [Q 1 (a, b)]   | ↑ (Q 1 ∩ Q 2 )(a, b) ⇒   | [(Q 1 ∩ Q 2 )(a, b)] and ⎧     | ↑ Q 1 (a, b) ⇒   | [Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪   | [Q 1 (a, b)] ↑ Q 2 (a, b) ⇒   | [Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨     | ↑∼ Q 1 (a, b) ⇒   | [∼ Q 1 (a, b)] Q     (∪− ) ⎪ ⎪    | [∼ Q 1 (a, b)] ↑ Q2 (a, b) ⇒  | [∼ Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪  | ↑ Q 1 (a, b) ⇒  | [Q 1 (a, b)] ⎪ ⎪ ⎩   | [Q 1 (a, b)] ↑∼ Q 2 (a, b) ⇒   | [Q 1 (a, b), ∼ Q 2 (a, b)]   | ↑ (Q 1 ∪ Q 2 )(a, b) ⇒   | [(Q 1 ∪ Q 2 )(a, b)] and 

  | ↑∼ Q 1 (a, b) ⇒   | [∼ Q 1 (a, b)]   | | ∼ Q 2 (a, b) ⇒   | [∼ Q 2 (a, b)] (∼    ↑∼ (Q 1 ∩ Q 2 )(a, b) ⇒   | [∼ (Q 1 ∩ Q 2 )(a, b)]  |    | ↑∼ Q 1 (a, b) ⇒   | [∼ Q 1 (a, b)] Q (∼ ∪− )   | [∼ Q 1 (a, b)]| ∼ Q 2 (a, b) ⇒   | [∼ Q 1 (a, b), ∼ Q 2 (a, b)]   | ↑∼ (Q 1 ∪ Q 2 )(a, b) ⇒   | [∼ (Q 1 ∪ Q 2 )(a, b)] ∩−Q )

and

⎧    | ⎪ ⎪ ⎪ ⎪   | ⎪ ⎪ ⎪ ⎪ ⎨   |   | (∗ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪   | ⎪ ⎪ ⎩    |   | ↑

↑∼ Q(a, c1 ) ⇒   | [∼ Q(a, c1 )] ↑∼2 Q(a, c1 ) ⇒   | [∼2 Q(a, c1 )] ↑∼ Q(c1 , c2 ) ⇒   | [∼ Q(c1 , c2 )] ↑∼2 Q(c1 , c2 ) ⇒   | [∼2 Q(c1 , c2 )] ↑∼ Q(cn , b) ⇒   | [∼ Q(cn , b)] ↑∼2 Q(cn , b) ⇒   | [∼2 Q(cn , b)] Q ◦ (a, b) ⇒   | [Q ◦ (a, b)]

8.3 2/3-Multisequents

and

and

261

⎧    | ⎪ ⎪ ⎪ ⎪   | ⎪ ⎪ ⎪ ⎪ ⎨   |   | (◦ Q− ) ⎪ ⎪ · ·· ⎪ ⎪ ⎪  ⎪ |  ⎪ ⎪ ⎩    |   | ↑

↑ Q(a, c1 ) ⇒   | [Q(a, c1 )] ↑∼ Q(a, c1 ) ⇒   | [∼ Q(a, c1 )] ↑ Q(c1 , c2 ) ⇒   | [Q(c1 , c2 )] ↑∼ Q(c1 , c2 ) ⇒   | [∼ Q(c1 , c2 )] ↑ Q(cn , b) ⇒   | [Q(cn , b)] ↑∼ Q(cn , b) ⇒   | [∼ Q(cn , b)] Q ◦ (a, b) ⇒   | [Q ◦ (a, b)]

⎧    | ↑∼ Q(a, c1 ) ⇒   | [∼ Q(a, c1 )] ⎪ ⎪ ⎨    | ↑∼ Q(c1 , c2 ) ⇒   | [∼ Q(c1 , c2 )] Q− ··· (∼ ∗ ) ⎪ ⎪ ⎩    | ↑∼ Q(cn , b) ⇒   | [∼ Q(cn , b)]   | ↑ (∼ Q ∗ )(a, b) ⇒   | [(∼ Q ∗ )(a, b)] ⎡   | ↑∼ Q(a, d1 ) ⇒   | [∼ Q(a, d1 )] ⎢   | ↑∼ Q(d1 , d2 ) ⇒   | [∼ Q(d1 , d2 )] ⎢ (∼ ◦ Q− ) ⎣ · · ·   | ↑∼ Q(dn , b) ⇒   | [∼ Q(dn , b)]   | ↑ (∼ Q ◦ )(a, b) ⇒   | [(∼ Q ◦ )(a, b)]

where ci is a new constant and di is a constant. Definition 8.3.8 A 2/3-reduction δ =   | ↑ (R(a, b), Q(a  , b )) ⇒   | is prov2/3 2/3 able in Ptm , denoted by tm δ, if there is a sequence {δ1 , ..., δn } of reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the pre2/3 vious reductions by one of the deduction rules in Ptm . Theorem 8.3.9 (Soundness and completeness theorem) For any reduction δ =   | ↑ (R(a, b), Q(a  , b )) ⇒ , R  (a, b)|, Q  (a  , b ) such that one of R  (a, b), Q  (a  , b ) is not the empty string, 2/3

2/3

tm δ iff |=tm δ.  Theorem 8.3.10 (Soundness and incompleteness theorem) For any reduction δ =   | ↑ (R(a, b), Q(a  , b )) ⇒ |, 2/3

2/3

tm δ implies |=tm δ, and

2/3

2/3

|=tm δ may not imply tm δ. 

262

8 Role R-Calculus for Post Three-Valued DL

8.4 3/3-Multisequents Let , ,  be sets of statements. A multisequent δ is of form ||. We say that = = -valid, denoted by |=3/3 ||, if for any interpretation I, δ is N3/3 ⎧ ⎨ for some statement R(a, b) ∈ , I (R(a, b)) = t for some statement Q(a, b) ∈ , I (Q(a, b)) = m ⎩ for some statement C(a) ∈ , I (C(a)) = f. 3/3 A multisequent δ = || is K= -valid, denoted by |=3/3 = ||, if there is an interpretation I such that

⎡

for each statement R(a, b) ∈ , I (R(a, b)) = t ⎣ for each statement Q(a, b) ∈ , I (Q(a, b)) = m for each statement C(a) ∈ , I (C(a)) = f.

= 8.4.1 Deduction System N3/3 = Deduction system N3/3 consists of the following axiom and deduction rules: • Axiom: ⎧ ⎨  ∩  = ∅  ∩  = ∅ = (A3/3 ) ⎩  ∩  = ∅ ||,

where , ,  are sets of atoms. • Deduction rules: |, R(a, b)| ||, Q(a, b) (∼ Q ) , ∼ R(a, b)|| |, ∼ Q(a, b)| , P(a, b)|| (∼ P ) ||, ∼ P(a, b)

(∼ R )

8.4 3/3-Multisequents

and

and

263

⎡

|Q 1 (a, b), | ⎢ |Q 2 (a, b), | ⎢   ⎢ , Q 1 (a, b)|| , R1 (a, b)|| ⎢ (∩ R ) , R2 (a, b)|| (∩ Q ) ⎢ ⎢  |Q 2 (a, b), | ⎣ |Q 1 (a, b), | , (R1 ∩ R2 )(a, b)|| , Q 2 (a, b)||  |(Q 1 ∩ Q 2 )(a, b), | ||P1 (a, b),  , R1 (a, b)|| (∩ P ) ||P2 (a, b),  (∪ R ) , R2 (a, b)|| ||(P , (R1 ∪ R2 )(a, b)|| 1 ∩ P2 )(a, b),  ⎡ |Q 1 (a, b), | ⎢ |Q 2 (a, b), | ⎢  ⎢ |Q 1 (a, b), | ||P1 (a, b),  ⎢ P (a, b) ||, Q ||P2 (a, b),  ) (∪ (∪ Q ) ⎢ 2 ⎢ ⎣ ||, Q 1 (a, b) ||(P1 ∪ P2 )(a, b),  |Q 2 (a, b), | |, (Q 1 ∪ Q 2 )(a, b)| ⎡

|, Q(a, d1 )| ⎢ ||, Q(a, d1 ) ⎢ ⎢ |, Q(d1 , d2 )| ⎢ ⎢ ||, Q(d1 , d2 ) (∗ Q ) ⎢ ⎢··· ⎢ ⎣ |, Q(dn , b)| ||, Q(dn , b) |, Q ∗ (a, b)| ⎡ ⎡ ||, P(a, d1 ) , R(a, d1 )|| ⎢ ||, P(d1 , d2 ) ⎢ , R(d1 , d2 )|| ⎢ ⎢ (∗ P ) ⎣ · · · (◦ R ) ⎣ · · · ||, P(dn , b) , R(dn , b)|| ∗ ||, P (a, b) , R ◦ (a, b)|| ⎡ , Q(a, d1 )|| ⎢ |, Q(a, d1 )| ⎧ ⎢ ||, P(a, c1 ) ⎪ ⎢ , Q(d1 , d2 )|| ⎪ ⎨ ⎢ ||, P(c1 , c2 ) ⎢ Q ⎢ |, Q(d1 , d2 )| P · (◦ ) ⎢ (◦ ) ⎪ ⎪ ·· ⎩ ⎢··· ||, P(cn , b) ⎣ , Q(dn , b)|| ||, P ◦ (a, b) |, Q(dn , b)| |, Q ◦ (a, b)|

⎧ , R(a, c1 )|| ⎪ ⎪ ⎨ , R(c1 , c2 )|| ··· (∗ R ) ⎪ ⎪ ⎩ , R(cn , b)|| , R ∗ (a, b)||

where ci is a new constant and di is a constant.

264

8 Role R-Calculus for Post Three-Valued DL

= Definition 8.4.1 A 3/3-multisequent δ = || is provable in N3/3 , denoted by = 3/3 δ, if there is a sequence {δ1 , ..., δn } of 3/3-multisequents such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 3/3= multisequents by one of the deduction rules in N3/3 .

Theorem 8.4.2 (Soundness and completeness theorem) For any 3/3-multisequent ||, = = 3/3 || if and only if |=3/3 ||. 

8.4.2 R-Calculus S= 3/3 Let R(a, b) ∈ , Q(a, b) ∈  and P(a, b) ∈ . A reduction δ = || ↑ = (R(a, b), Q(a, b), P(a, b)) ⇒   | |  is S= 3/3 -valid, denoted by |=3/3 δ, if  ⎧ = R(a, b) if |=3/3 [R(a, b)]|| ⎪  ⎪ R (a, b) = ⎪ ⎪ otherwise ⎪ ⎪ λ ⎪ ⎨ = Q(a, b) if |=3/32 [R  (a, b)]|[Q(a, b)]| Q  (a, b) = ⎪ λ otherwise ⎪ ⎪  ⎪ = ⎪ ⎪ P(a, b) if |=3/3 [R  (a, b)]|[Q  (a, b)]|[P(a, b)] ⎪ ⎩ P  (a, b) = λ otherwise. Let X = || and X(R(a, b)) = , R(a, b)|| X[R(a, b)] =  − {R(a, b)}|| X(Q(a, b)) = |, Q(a, b)| X[Q(a, b)] = | − {Q(a, b)}| X(P(a, b)) = ||, P(a, b) X[P(a, b)] = || − {P(a, b)}.

Let R(a, b) ∈ , Q(a, b) ∈  and P(a, b) ∈ . R-calculus S= 3/3 consists of the following axioms and deduction rules: • Axioms: ⎧ ⎨ [r (a, b)] ∩   = ∅

[r (a, b)] ∩   = ∅ R−= (t3/3 ) ⎩  ∩  = ∅

|| ↑ r (a, b) ⇒ [r (a, b)]|| ⎧ ⎨  ∩ [q(a  , b )]  = ∅  ∩  = ∅ Q−= (t3/3 ) ⎩ [q(a  , b )] ∩   = ∅ || ↑ q(a  , b ) ⇒ |[q(a  , b )]| ⎧ ⎨  ∩  = ∅  ∩ [ p(a  , b )]  = ∅ P−= (t3/3 ) ⎩  ∩ [ p(a  , b )]  = ∅ || ↑ p(a  , b ) ⇒ ||[ p(a  , b )]

8.4 3/3-Multisequents

265

where , ,  are sets of role atoms, and r (a, b) ∈ , q(a  , b ) ∈ , p(a  , b ) ∈  are role atoms. • Deduction rules: X ↑ 2 R(a, b) ⇒ X[2 R(a, b)] X ↑∼ R(a, b) ⇒ X[∼ R(a, b)] 3 3 Q− X ↑ Q(a, b) ⇒ X[ Q(a, b)] (∼ ) X ↑∼ Q(a, b) ⇒ X[∼ Q(a, b)] 1 1 P− X ↑ P(a, b) ⇒ X[ P(a, b)] (∼ ) X ↑∼ P(a, b) ⇒ X[∼ P(a, b)] (∼ R− )

and



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X ↑ R2 (a, b) ⇒ X[R2 (a, b)] X ↑ (R1 ∩ R2 )(a, b) ⇒ X[(R1 ∩ R2 )(a, b)] ⎡ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎢ X ↑ Q 2 (a, b) ⇒ X[Q 2 (a, b)] ⎢ ⎢ X[Z 1 ] ↑ 1 Q 1 (a, b) ⇒ X[Z 1 , 1 Q 1 (a, b)] ⎢ Q− ⎢ X[Z ] ↑ Q (a, b) ⇒ X[Z , Q (a, b)] (∩ ) ⎢  1 2 1 2 ⎣ X[Z 2 ] ↑ Q 1 (a, b) ⇒ X[Z 2 , Q 1 (a, b)] X[Z 2 ] ↑ 1 Q 2 (a, b) ⇒ X[Z 2 , 1 Q 2 (a, b)] X ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ X[(Q 1 ∩ Q 2 )(a, b)] X ↑ P1 (a, b) ⇒ X[P1 (a, b)] (∩ P− ) X[P1 (a, b)] ↑ P2 (a, b) ⇒ X[P1 (a, b), P2 (a, b)] X ↑ (P1 ∩ P2 )(a, b) ⇒ X[(P1 ∩ P2 )(a, b)]

(∩ R− )

where Z 1 = Q 1 (a, b) ∨ Q 2 (a, b), Z 2 = Z 1 , ∼2 Q 1 (a, b) ∨ Q 2 (a, b), and 

X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X[R1 (a, b)] ↑ R2 (a, b) ⇒ X[R1 (a, b), R2 (a, b)] X ↑ (R1 ∪ R2 )(a, b) ⇒ X[(R1 ∪ R2 )(a, b)] ⎡ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎢ X ↑ Q 2 (a, b) ⇒ X[Q 2 (a, b)] ⎢ ⎢ X[Z 1 ] ↑ 3 Q 1 (a, b) ⇒ X[Z 1 , 3 Q 1 (a, b)] ⎢ Q− ⎢ X[Z ] ↑ Q (a, b) ⇒ X[Z , Q (a, b)] (∪ ) ⎢  1 2 1 2 ⎣ X[Z 3 ] ↑ Q 1 (a, b) ⇒ X[Z 3 , Q 1 (a, b)] X[Z 3 ] ↑ 3 Q 2 (a, b) ⇒ X[Z 3 , 3 Q 2 (a, b)] X ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ X[(Q 1 ∪ Q 2 )(a, b)]  X ↑ P1 (a, b) ⇒ X[P1 (a, b)] (∪ P− ) X ↑ P2 (a, b) ⇒ X[P2 (a, b)] X ↑ (P1 ∪ P2 )(a, b) ⇒ X[(P1 ∪ P2 )(a, b)]

(∪ R− )

where Z 3 = Z 1 , ∼ Q 1 (a, b) ∨ Q 2 (a, b), and

266

8 Role R-Calculus for Post Three-Valued DL

⎧ ⎪ ⎪ X ↑ R(a, c1 ) ⇒ X[R(a, c1 )] ⎨ X ↑ R(c1 , c2 ) ⇒ X[R(c1 , c2 )] · ·· (∗ R− ) ⎪ ⎪ ⎩ X ↑ R(cn , b) ⇒ X[R(cn , b)] ∗ ∗ X ⎡ ↑ R (a, b) ⇒ X[R (a, b)] X ↑ Q(a, d1 ) ⇒ X[Q(a, d1 )] ⎢ X ↑ 3 Q(a, d1 ) ⇒ X[3 Q(a, d1 )] ⎢ ⎢ X ↑ Q(d1 , d2 ) ⇒ X[Q(d1 , d2 )] ⎢ ⎢ X ↑ 3 Q(d1 , d2 ) ⇒ X[3 Q(d1 , d2 )] (∗ Q− ) ⎢ ⎢··· ⎢ ⎣ X ↑ Q(dn , b) ⇒ X[Q(dn , b)] X ↑ 3 Q(dn , b) ⇒ X[3 Q(dn , b)] X ↑ Q ∗ (a, b) ⇒ X[Q ∗ (a, b)] ⎡ X ↑ P(a, d1 ) ⇒ X[P(a, d1 )] ⎢ X ↑ P(d1 , d2 ) ⇒ X[P(d1 , d2 )] ⎢ (∗ P− ) ⎣ · · · X ↑ P(dn , b) ⇒ X[P(dn , b)] X ↑ P ∗ (a, b) ⇒ X[P ∗ (a, b)] and

⎡

X ↑ R(a, d1 ) ⇒ X[R(a, d1 )] ⎢ X ↑ R(d1 , d2 ) ⇒ X[R(d1 , d2 )] ⎢ (◦ R− ) ⎣ · · · X ↑ R(dn , b) ⇒ X[R(dn , b)] ◦ ◦ X b)] ⎡ ↑ R 1(a, b) ⇒ X[R (a, X ↑ Q(a, d1 ) ⇒ X[1 Q(a, d1 )] ⎢ X ↑ Q(a, d1 ) ⇒ X[Q(a, d1 )] ⎢ ⎢ X ↑ 1 Q(d1 , d2 ) ⇒ X[1 Q(d1 , d2 )] ⎢ ⎢ Q− ⎢ X ↑ Q(d1 , d2 ) ⇒ X[Q(d1 , d2 )] (◦ ) ⎢ ⎢··· 1 ⎣ X ↑ Q(dn , b) ⇒ X[1 Q(dn , b)] X ↑ Q(dn , b) ⇒ X[Q(dn , b)] ◦ ◦ X ⎧ ↑ Q (a, b) ⇒ X[Q (a, b)] X ↑ P(a, c1 ) ⇒ X[P(a, c1 )] ⎪ ⎪ ⎨ X ↑ P(c1 , c2 ) ⇒ X[P(c1 , c2 )] · ·· (◦ P− ) ⎪ ⎪ ⎩ X ↑ P(cn , b) ⇒ X[P(cn , b)] X ↑ P ◦ (a, b) ⇒ X[P ◦ (a, b)]

where ci is a new constant and di is a constant. Definition 8.4.3 A 3/3-reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒   | | 

8.4 3/3-Multisequents

267 =

is provable in S= 3/3 , denoted by 3/3 δ, if there is a sequence {δ1 , ..., δn } of 3/3reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in S= 3/3 . Theorem 8.4.4 (Soundness and completeness theorem) Let R(a, b) ∈ , Q(a  , b ) ∈  and P(a  , b ) ∈ . For any reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒   | |  such that one of R  (a, b), Q  (a  , b ), P  (a  , b ) is not the empty string, =

=

3/3 δ iff |=3/3 δ.  Theorem 8.4.5 (Soundness and incompleteness theorem) Let R(a, b) ∈ , Q(a  , b ) ∈  and P(a  , b ) ∈ . For any reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒ ||, =

=

3/3 δ implies |=3/3 δ, and

=

=

|=3/3 δ may not imply |=3/3 δ. 

3/3

8.4.3 Incomplete Deduction System K=

3/3 Deduction system K= consists of the following axiom and deduction rules: • Axiom: ⎡ ∩=∅ ⎣ ∩ =∅ (A3/3 = ) ∩ =∅ ||,

where , ,  are sets of atoms. • Deduction rules: |, R(a, b)| ||, Q(a, b) (∼ Q ) , ∼ R(a, b)|| |, ∼ Q(a, b)| , P(a, b)|| (∼ P ) ||, ∼ P(a, b)

(∼ R )

268

8 Role R-Calculus for Post Three-Valued DL

and

⎧ |Q 1 (a, b), | ⎪ ⎪ ⎪ ⎪ |Q 2 (a, b), | ⎪ ⎪  ⎨ , R1 (a, b)|| , Q 1 (a, b)|| (∩ R ) , R2 (a, b)|| (∩ Q ) ⎪ ⎪  |Q 2 (a, b), | ⎪ ⎪ |Q 1 (a, b), | , (R1 ∩ R2 )(a, b)|| ⎪ ⎪ ⎩ , Q 2 (a, b)|| |(Q 1 ∩ Q 2 )(a, b), |   ||P1 (a, b),  , R1 (a, b)|| (∩ P ) ||P2 (a, b),  (∪ R ) , R2 (a, b)|| ||(P , (R1 ∪ R2 )(a, b)|| 1 ∩ P2 )(a, b),  ⎧ |, Q (a, b)| ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪  |, Q 2 (a, b)|  ⎨ |, Q 1 (a, b)| ||, P1 (a, b) P (a, b) ||, Q ||, P2 (a, b) ) (∪ (∪ Q ) ⎪ 2 ⎪ ⎪ ⎪ (a, b) ||, Q ||, (P1 ∪ P2 )(a, b) ⎪ 1 ⎪ ⎩ |, Q 2 (a, b)| |, (Q 1 ∪ Q 2 )(a, b)|

and

⎧ ⎪ ⎪ |, Q(a, c1 )| ⎪ ⎪ ⎡ ||, Q(a, c1 ) ⎪ ⎪ , R(a, d1 )|| ⎪ ⎪ |, Q(c1 , c2 )| ⎨ ⎢ , R(d1 , d2 )|| ⎢ ||, Q(c1 , c2 ) R ⎣··· Q (∗ ) (∗ ) ⎪ ⎪ ··· ⎪ ⎪ , R(dn , b)|| ⎪ ⎪ |, Q(cn , b)| ⎪ ⎪ , R ∗ (a, b)|| ⎩ ||, Q(cn , b) |, Q ∗ (a, b)| ⎧ ⎧ ||, P(a, c1 ) , R(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ||, P(c1 , c2 ) , R(c1 , c2 )|| R · · · ··· ) (∗ P ) ⎪ (◦ ⎪ ⎪ ⎪ ⎩ ⎩ ||, P(cn , b) , R(cn , b)|| ∗ ||, P (a, b) , R ◦ (a, b)|| ⎧ , Q(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ⎡ |, Q(a, c1 )| ⎪ ⎪ ||, P(a, d1 ) ⎪ ⎪ ⎨ , Q(c1 , c2 )|| ⎢ ||, P(d1 , d2 ) ⎢ |, Q(c1 , c2 )| (◦ Q ) ⎪ (◦ P ) ⎣ · · · ⎪ ··· ⎪ ⎪ ||, P(dn , b) ⎪ ⎪ , Q(cn , b)|| ⎪ ⎪ ||, (P ◦ )(a, b) ⎩ |, Q(cn , b)| |, Q ◦ (a, b)|

where ci is a new constant and di is a constant.

8.4 3/3-Multisequents

269

3/3 Definition 8.4.6 A 3/3-multisequent δ = || is provable in K= , denoted by 3/3 = δ, if there is a sequence {δ1 , ..., δn } of 3/3-multisequents such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 3/33/3 multisequents by one of the deduction rules in K= .

Theorem 8.4.7 (Soundness and incompleteness theorem) For any 3/3-multisequent ||, 3/3 3/3 = || implies |== ||, and 3/3 |=3/3 = || may not imply = ||.

 3/3

8.4.4 R-Calculus P=

3/3 A reduction δ = || ↑ (R(a, b), Q(a, b), P(a, b)) ⇒   | |  is P= -valid, 3/3 denoted by |== δ, if



R(a, b) if |=3/3 = , R(a, b)|| λ otherwise   Q(a, b) if |=3/3  = , R (a, b)|, Q(a, b)| Q (a, b) = otherwise λ   P(a, b) if |=3/3 = , R (a, b)|, Q (a, b)|, P(a, b) P  (a, b) = λ otherwise. 

R (a, b) =

Let X = , R(a, b)|, Q(a  , b )|, P(a  , b ), and X[R(a, b)] =  − {R(a, b)}|| X[Q(a, b)] = | − {Q(a, b)}| X[P(a, b)] = || − {P(a, b)}. 3/3 consists of the following axioms and deduction rules: R-calculus P= • Axioms:  r (a, b) ∈  3/3 (A R−= ) r (a, b) ∈  || ↑ r (a, b) ⇒ [r (a, b)]||  q(a  , b ) ∈  3/3 (A Q−= ) q(a  , b ) ∈  || ↑ q(a  , b ) ⇒ |[q(a  , b )]|  p(a  , b ) ∈  3/3 (A P−= ) p(a  , b ) ∈  || ↑ p(a  , b ) ⇒ ||[ p(a  , b )]

270

8 Role R-Calculus for Post Three-Valued DL

where , ,  are sets of role atoms, and r (a, b), q(a  , b ), p(a  , b ) are role atoms. • Deduction rules: X ↑ 2 R(a, b) ⇒ X[2 R(a, b)] X ↑∼ R(a, b) ⇒ X[∼ R(a, b)] 3 3 Q− X ↑ Q(a, b) ⇒ X[ Q(a, b)] (∼ ) X ↑∼ Q(a, b) ⇒ X[∼ Q(a, b)] 1 1 P− X ↑ P(a, b) ⇒ X[ P(a, b)] (∼ ) X ↑∼ P(a, b) ⇒ X[∼ P(a, b)] (∼ R− )

and



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X[R1 (a, b)] ↑ R2 (a, b) ⇒ X[R1 (a, b), R2 (a, b)] X ⎧↑  (R1 ∩ R2 )(a, b) ⇒ X[(R1 ∩ R2 )(a, b)] X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪ X[Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨ X ↑ 1 Q 1 (a, b) ⇒ X[1 Q 1 (a, b)] 1 1 (∩ Q− ) ⎪ ⎪  X[ Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[ Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎩ X[Q 1 (a, b)] ↑ 1 Q 2 (a, b) ⇒ X[Q 1 (a, b), 1 Q 2 (a, b)] X ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ X[(Q 1 ∩ Q 2 )(a, b)]  X ↑ P1 (a, b) ⇒ X[P1 (a, b)] (∩ P− ) X ↑ P2 (a, b) ⇒ X[P2 (a, b)] X ↑ (P1 ∩ P2 )(a, b) ⇒ X[(P1 ∩ P2 )(a, b)]

(∩ R− )

and



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X ↑ R2 (a, b) ⇒ X[R2 (a, b)] X ⎧↑  (R1 ∪ R2 )(a, b) ⇒ X[(R1 ∪ R2 )(a, b)] ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪  X[Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎨ X ↑ 3 Q 1 (a, b) ⇒ X[3 Q 1 (a, b)] (∪ Q− ) ⎪ ⎪  X[3 Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[3 Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎩ X[Q 1 (a, b)] ↑ 3 Q 2 (a, b) ⇒ X[Q 1 (a, b), 3 Q 2 (a, b)] X ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ X[(Q 1 ∪ Q 2 )(a, b)] X ↑ P1 (a, b) ⇒ X[P1 (a, b)] P− (∪ ) X[P1 (a, b)] ↑ P2 (a, b) ⇒ X[P1 (a, b), P2 (a, b)] X ↑ (P1 ∪ P2 )(a, b) ⇒ X[(P1 ∪ P2 )(a, b)] (∪ R− )

8.4 3/3-Multisequents

and

and

271

⎡

X ↑ R(a, d1 ) ⇒ X[R(a, d1 )] ⎢ X ↑ R(d1 , d2 ) ⇒ X[R(d1 , d2 )] ⎢ (∗ R− ) ⎣ · · · X ↑ R(dn , b) ⇒ X[R(dn , b)] ∗ ∗ X ⎧ ↑ R (a, b) ⇒ X[R (a, b)] ⎪ X ↑ Q(a, c1 ) ⇒ X[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 3 Q(a, c1 ) ⇒ X[3 Q(a, c1 )] ⎪ ⎪ ⎪ ⎨ X ↑ Q(c1 , c2 ) ⇒ X[Q(c1 , c2 )] X ↑ 3 Q(c1 , c2 ) ⇒ X[3 Q(c1 , c2 )] Q− (∗ ) ⎪ ⎪ ⎪··· ⎪ ⎪ ⎪ ⎪ X ↑ Q(cn , b) ⇒ X[Q(cn , b)] ⎪ ⎩ X ↑ 3 Q(cn , b) ⇒ X[3 Q(cn , b)] X ↑ Q ∗ (a, b) ⇒ X[Q ∗ (a, b)] ⎧ X ↑ P(a, c1 ) ⇒ X[P(a, c1 )] ⎪ ⎪ ⎨ X ↑ P(c1 , c2 ) ⇒ X[P(c1 , c2 )] · ·· (∗ P− ) ⎪ ⎪ ⎩ X ↑ P(cn , b) ⇒ X[P(cn , b)] X ↑ P ∗ (a, b) ⇒ X[P ∗ (a, b)] ⎧ X ↑ R(a, c1 ) ⇒ X[R(a, c1 )] ⎪ ⎪ ⎨ X ↑ R(c1 , c2 ) ⇒ X[R(c1 , c2 )] · ·· (◦ R− ) ⎪ ⎪ ⎩ X ↑ R(cn , b) ⇒ X[R(cn , b)] ◦ ◦ X b)] ⎧ ↑ R 1(a, b) ⇒ X[R (a, 1 X ↑ Q(a, c ) ⇒ X[ Q(a, c1 )] ⎪ 1 ⎪ ⎪ ⎪ ) ⇒ X[Q(a, c )] X ↑ Q(a, c ⎪ 1 1 ⎪ ⎪ ⎪ ⎨ X ↑ 1 Q(c1 , c2 ) ⇒ X[1 Q(c1 , c2 )] X ↑ Q(c1 , c2 ) ⇒ X[Q(c1 , c2 )] (◦ Q− ) ⎪ ⎪ · ·· ⎪ ⎪ ⎪ ⎪ X ↑ 1 Q(cn , b) ⇒ X[1 Q(cn , b)] ⎪ ⎪ ⎩ X ↑ Q(cn , b) ⇒ X[Q(cn , b)] ◦ ◦ X ⎡ ↑ Q (a, b) ⇒ X[Q (a, b)] X ↑ P(a, d1 ) ⇒ X[P(a, d1 )] ⎢ X ↑ P(d1 , d2 ) ⇒ X[P(d1 , d2 )] ⎢ (◦ P− ) ⎣ · · · X ↑ P(dn , b) ⇒ X[P(dn , b)] X ↑ P ◦ (a, b) ⇒ X[P ◦ (a, b)]

where ci is a new constant and di is a constant. Definition 8.4.8 A 3/3-reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒   | | 

272

8 Role R-Calculus for Post Three-Valued DL

3/3 is provable in P= , denoted by 3/3 = δ, if there is a sequence {δ1 , ..., δn } of 3/3reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced 3/3 . from the previous reductions by one of the deduction rules in P=

Theorem 8.4.9 (Soundness and completeness theorem) For any reduction δ = X ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒   | |  such that one of R  (a, b), Q  (a  , b ), P  (a  , b ) is not the empty string, 3/3 3/3 = δ iff |== δ.

 Theorem 8.4.10 (Soundness and incompleteness theorem) For any reduction δ = X ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒ ||, 3/3 3/3 = δ implies |== δ,

and 3/3 |=3/3 = δ may not imply = δ.



8.5 Conclusions In R-calculus || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒   | |  there may be several injuries. Assume that there are two levels of injuries: • (first-level injuries) there is a stage s0 such that     = 3/3 || ↑ P(a , b ) ⇒s0 ||[P(a , b )];

and at a later stage s1 > s0 such that     = 3/3 || ↑ Q(a , b ) ⇒s1 |[Q(a , b )]|;

and at a later stage s2 > s1 such that = 3/3 || ↑ R(a, b) ⇒s2 [R(a, b)]||. • (second-level injuries) there is a stage t0 such that     = 3/3 [R(a, b)]|| ↑ P(a , b ) ⇒t0 [R(a, b)]||[P(a , b )];

References

273

and at a later stage t1 > t0 such that     = 3/3 [R(a, b)]|| ↑ Q(a , b ) ⇒t1 [R(a, b)]|[Q(a , b )]|;

and at a later stage t2 > t1 such that         = 3/3 [R(a, b)]|[Q(a , b )]| ↑ P(a , b ) ⇒t2 [R(a, b)]|[Q(a , b )]|[P(a , b )].

Then, there are two injures at first-level: eliminating P(a  , b ) from  at stage s0 + 1 may be injured by eliminating Q(a  , b ) from  at stage s1 + 1; and eliminating Q(a  , b ) from  at stage s1 + 1 may be injured by eliminating R(a, b) from  at stage s2 + 1; and there is one injury at second-level: eliminating P(a  , b ) from  at stage t0 + 1 may be injured by eliminating Q(a  , b ) from  at stage t1 + 1. Friedberg (1957), Li and Sui (2017), Muchnik (1956), Soare (1987).

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symbolic Logic 50, 510–530 (1985) Avron, A.: Natural 3-valued logics: Characterization and proof theory. J. of Symbolic Logic 56, 276–294 (1991) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation. Applications. Cambridge University Press, Cambridge, UK (2003) Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Logic 2, 87–112 (1938) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Log. 40, 295–331 (2011) Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957) Gärdenfors, P., Rott, H.: Belief revision. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 4, pp. 35–132. Oxford Science Pub, Epistemic and Temporal Reasoning (1995) Ginsberg, M.L. (ed.): Readings in Nonmonotonic Reasoning. Morgan Kaufmann, San Francisco (1987) Li, W.: R-calculus: an inference system for belief revision. The Computer J. 50, 378–390 (2007) Li, W., Sui, Y.: The R-calculus and the finite injury priority method. J. Comput. 12, 127–134 (2017) Muchnik, A.A.: On the separability of recursively enumerable sets (in Russian). Dokl. Akad. Nauk SSSR, N.S. 109, 29–32 (1956) Post, E.L.: Determination of all closed systems of truth tables. Bulletin American Mathematical Society 26, 437 (1920) Post, E.L.: Introduction to a general theory of elementary propositions. American Journal Mathematics 43, 163–185 (1921) Soare, R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Functions and Computably Generated Sets. Springer (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F. (Eds.), Handbook of Philosophical Logic, 2nd edn., vol. 2, pp. 249-295. Dordrecht, Kluwer (2001) Zach, R.: Proof theory of finite-valued logics, Technical Report TUW-E185.2-Z.1-93

Chapter 9

Role R-Calculus for B22 -Valued DL

t t t t Ri/2 Mi/2 2 Ni/22 2 Si/22 i/22

i/22

Lt

Kt

i/22

Qt

i/22

Pt

In this chapter, we consider two kinds of validity: t t M1/2 2 :  ∈ M1/22 ↔ AI E A ∈ (I (A) = t) 1/22

Lt

1/22

:  ∈ Kt

↔ EI AA ∈ (I (A) = t).

1/22

t M1/2 , respectively. 2 is complementary to Lt t |=t1/22 M1/2 2 1/22

1/22

|==t

Lt

For each validity, there is a Arieli and Avron (1996) sound and complete deduction system (Arieli and Avron 1996, 2000; Belnap 1977; Bochvar 1938; Font 1997) 1/22 t t1/22 for M1/2 2 -validity; and if  =t is recursive then there is a sound and complete 1/22

1/22

deduction system =t for Lt -validity (Ponse and van der Zwaag 2006; Pynko 1995; Urquhart 2001; Zach 2023). =t 1/22 1/22

=t

iff

|=t1/22 1/22

|==t

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_9

275

9 Role R-Calculus for B22 -Valued DL

276

1/22

1/22

If t1/22 is not recursive then the deduction system =t for Lt -validity is sound and incomplete. We consider two kinds of R-calculi (Alchourrón et al. 1985; Baader et al. 2003; Fermé and Hansson 2011; Li 2007; Li and Sui 2017): 

 ↑ A ⇒ [A ] ∈

t R1/2 2

 ↑ A ⇒ , A ∈

1/22 Qt





↔A =  ↔ A =

A if |=t1/22 [A] λ otherwise 1/22

A if |==t , A λ otherwise

and prove that t

 ↑ R(a, b) ⇒ [R(a, b)] iff |=t 2  ↑ R(a, b) ⇒ [R(a, b)] 1/22 1/2 t  2  ↑ R(a, b) ⇒  implies |=t 2  ↑ R(a, b) ⇒  1/2 1/2 |=t 2  ↑ R(a, b) ⇒  may not imply t 2  ↑ R(a, b) ⇒  1/2 1/2 1/22 1/22 =t , R(a, b) ↑ R(a, b) ⇒ [R(a, b)] iff |==t , R(a, b) ↑ R(a, b) ⇒ [R(a, b)] 1/22 1/22 =t , R(a, b) ↑ R(a, b) ⇒  implies |==t  ↑ R(a, b) ⇒  1/22 1/22 |==t , R(a, b) ↑ R(a, b) ⇒  may not imply =t  ↑ R(a, b) ⇒ .

We consider the following deduction systems and R-calculi (Alchourrón et al. 1985; Fermé and Hansson 2011; Li 2007): t t t⊥ = Deduction system M1/2 2 , M2/22 , M3/22 , M4/22 1/22

Lt , R-calculus

2

t t t⊥ = R1/2 2 , R2/22 , R3/22 , R4/22 1/22

Qt , and t M1/2 2 t M2/2 2 = M3/2 2 = M4/2 2

4/2 L=

, ∼1 , ∼2 ,  , ∼2 , | , || |||.

4/2 P=

2

9.1 B22 -Valued DL with Role Constructors

277

9.1 B22 -Valued DL with Role Constructors Let U be a universe and B22 = {t, , ⊥, f], where t  ⊥ f ∪ t  ⊥ f

f ∼1 f ∼2 f   ⊥ f t f ⊥ f t  ⊥  t t⊥ f tt t t t t  t t ⊥⊥ t⊥ f

∩ t⊥f t t⊥f  ff ⊥⊥ f ⊥f f f f ff → t ⊥f t t t tt  t t ⊥ ⊥ tt f f ⊥t

The logical language of B22 -valued description logic with role constructors contains the following symbols: • atomic roles: S0 , S1 , ...; • role constructors: ∼1 , ∼2 , , ∩, ∪, ∗ , ◦ . Concepts are defined inductively as follows: R ::= S| ∼1 R| ∼2 R|  R|R1 ∩ R2 |R1 ∪ R2 |R ∗ |R ◦ , where S is an atomic role, and R is a role. A model M is a pair (U, I ), where U is a non-empty set, and I is an interpretation such that ◦ for any atomic role S, I (S) : U 2 → B22 . Given an atomic role S, we define roles S, ∼1 S, ∼2 S, S as follows: for any x ∈ U, S(x, y) ∼1 S(x, y) ∼2 S(x, y) S(x, y) t  ⊥ f t f ⊥  f t  ⊥ ⊥  t f The interpretation R I of a role R is a function from U 2 to B22 such that for any x, y ∈ U, ⎧ I (S)(x, y) ⎪ ⎪ ⎨ f ∗ (R I )(x, y) I R (x, y) = min{R1I (x, y), R2I (x, y)] ⎪ ⎪ ⎩ max{R1I (x, y), R2I (x, y)] where ∗ ∈ {∼1 , ∼2 , ].

if if if if

R R R R

=S = ∗R1 = R1 ∩ R2 = R1 ∪ R2

9 Role R-Calculus for B22 -Valued DL

278

About (R(a, b)) I = t, we have the following equivalences: (R1 ∩ R2 )(x, y) ∼ = ∼1 (R1 ∩ R2 )(x, y) ∼ =

R1 (x, y)∧R2 (x, y) (R1 (x, y)∧ ∼1 R2 (x, y))∨(∼1 R1 (x, y)∧R2 (x, y))

∼2 (R1 ∩ R2 )(x, y) ∼ =

∨(∼1 R1 (x, y)∧ ∼1 R2 (x, y)) (R1 (x, y)∧ ∼2 R2 (x, y))∨(∼2 R1 (x, y)∧R2 (x, y))

(R1 ∩ R2 )(x, y) ∼ =

∨(∼2 R1 (x, y)∧ ∼2 R2 (x, y)) R1 (x, y)∨  R2 (x, y)∨(∼1 R1 (x, y)∧ ∼2 R2 (x, y))

(R1 ∪ R2 )(x, y) ∼ = ∼1 (R1 ∪ R2 )(x, y) ∼ =

∨(∼2 R1 (x, y)∧ ∼1 R2 (x, y)); R1 (x, y)∨R2 (x, y)∨(∼1 R1 (x, y)∧ ∼2 R2 (x, y)) ∨(∼2 R1 (x, y)∧ ∼1 R2 (x, y)) (∼1 R1 (x, y)∧ ∼1 R2 (x, y))∨(∼1 R1 (x, y)∧  R2 (x, y)) ∨(R1 (x, y)∧ ∼1 R2 (x, y))

∼2 (R1 ∪ R2 )(x, y) ∼ = (∼2 R1 (x, y)∧ ∼2 R2 (x, y))∨(∼2 R1 (x, y)∧  R2 (x, y)) ∨(R1 (x, y)∧ ∼2 R2 (x, y)) ∼ (R1 ∪ R2 )(x, y) = R1 (x, y)∧  R2 (x, y),

and

R ∗ (x, y) iff ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )) (∼1 R ∗ )(x, y) iff ∀c1 , ..., cn ∃i ≤ n(∼1 R(ci , ci+1 ) ∨ R(ci , ci+1 )) (∼2 R ∗ )(x, y) iff ∀c1 , ..., cn ∃i ≤ n(∼2 R(ci , ci+1 ) ∨ R(ci , ci+1 )) (R ∗ )(x, y) iff ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )) R ◦ (x, y) iff ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )) (∼1 R ◦ )(x, y) iff ∀c1 , ..., cn ∃i ≤ n(∼1 R(ci , ci+1 ) ∨ R(ci , ci+1 )) (∼2 R ◦ )(x, y) iff ∀c1 , ..., cn ∃i ≤ n(∼2 R(ci , ci+1 ) ∨ R(ci , ci+1 )) (R ◦ )(x, y) iff ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )),

where ∼1 (R1 ∩ R2 )(x, y) ∼ = (R1 (x, y)∧ ∼1 R2 (x, y))∨(∼1 R1 (x, y)∧R2 (x, y)) ∨(∼1 R1 (x, y)∧ ∼1 R2 (x, y)) means that for any interpretation I, I (∼1 (R1 ∩ R2 )) (x, y) = t iff either (1) I (R1 (x, y)) = t and I (∼1 R2 (x, y)) = t, or (2) I (∼1 R1 (x, y)) = t and I (R2 (x, y)) = t, or (3) I (∼1 R1 (x, y)) = t and I (∼1 R2 (x, y)) = t. About (R(a, b)) I = t, we have the following equivalences: (R1 ∩ R2 )(x, y) ∼ = ∼1 (R1 ∩ R2 )(x, y) ∼ =

R1 (x, y)∨R2 (x, y) (R1 (x, y)∨ ∼1 R2 (x, y))∧(∼1 R1 (x, y)∨R2 (x, y))

∼2 (R1 ∩ R2 )(x, y) ∼ =

∧(∼1 R1 (x, y)∨ ∼1 R2 (x, y)) (R1 (x, y)∨ ∼2 R2 (x, y))∧(∼2 R1 (x, y)∨R2 (x, y)) ∧(∼2 R1 (x, y)∨ ∼2 R2 (x, y))

9.2 1/22 -Multisequents

279

(R1 ∩ R2 )(x, y) ∼ =

R1 (x, y)∧  R2 (x, y)∧(∼1 R1 (x, y)∨ ∼2 R2 (x, y))

(R1 ∪ R2 )(x, y) ∼ =

∧(∼2 R1 (x, y)∨ ∼1 R2 (x, y)); R1 (x, y)∧R2 (x, y)∧(∼1 R1 (x, y)∨ ∼2 R2 (x, y)) ∧(∼2 R1 (x, y)∨ ∼1 R2 (x, y))

∼1 (R1 ∪ R2 )(x, y) ∼ =

(∼1 R1 (x, y)∨ ∼1 R2 (x, y))∧(∼1 R1 (x, y)∨  R2 (x, y)) ∧(R1 (x, y)∨ ∼1 R2 (x, y))

∼2 (R1 ∪ R2 )(x, y) ∼ = (∼2 R1 (x, y)∨ ∼2 R2 (x, y))∧(∼2 R1 (x, y)∨  R2 (x, y)) ∧(R1 (x, y)∨ ∼2 R2 (x, y)) ∼ (R1 ∪ R2 )(x, y) = R1 (x, y)∨  R2 (x, y),

and R ∗ (x, y) iff ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )) (∼1 R ∗ )(x, y) iff ∃d1 , ..., dn ∀i ≤ n(∼1 R(di , di+1 )∧  R(di , di+1 )) (∼2 R ∗ )(x, y) iff ∃d1 , ..., dn ∀i ≤ n(∼2 R(di , di+1 )∧  R(di , di+1 )) (R ∗ )(x, y) iff ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )) R ◦ (x, y) iff ∃d1 , ..., dn ∀i ≤ n(R(di , di+1 )) (∼1 R ◦ )(x, y) iff ∃d1 , ..., dn ∀i ≤ n(∼1 R(di , di+1 )∧R(di , di+1 )) (∼2 R ◦ )(x, y) iff ∃d1 , ..., dn ∀i ≤ n(∼2 R(di , di+1 )∧R(di , di+1 )) (R ◦ )(x, y) iff ∀c1 , ..., cn ∃i ≤ n(R(ci , ci+1 )).

9.2 1/22 -Multisequents t t A 1/22 -multisequent  is M1/2 2 -valid, denoted by |=1/22 , if for any interpretation I, there is a statement R(a, b) ∈  such that I (R(a, b)) = t.

t 9.2.1 Deduction System M1/2 2 t Deduction system M1/2 2 consists of the following axiom and deduction rules: • Axiom: − − ∩ ∼− 1 ∩ ∼2  ∩    = ∅ (At1/22 ) ,

where  is a set of literals. • Deduction rules: (∼21 )

, R(a, b) , R(a, b) , R(a, b) (∼22 ) (2 ) 2 2 , 2 R(a, b) , ∼1 R(a, b) , ∼2 R(a, b)

280

9 Role R-Calculus for B22 -Valued DL

and

⎧ , ∼1 R1 (a, b) ⎪ ⎪ ⎪ ⎪ , ∼1 R2 (a, b) ⎪ ⎪  ⎨ , R1 (a, b) , R1 (a, b) (∩) , R2 (a, b) (∼1 ∩) ⎪ ⎪  , ∼1 R2 (a, b) ⎪ ⎪ , ∼1 R1 (a, b) , (R1 ∩ R2 )(a, b) ⎪ ⎪ ⎩ , R2 (a, b) ⎧ ⎧, ∼1 (R1 ∩ R2 )(a, b)) , ∼2 R1 (a, b) ⎪ ⎪ , R1 (a, b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R (a, b) , ∼ , R2 (a, b) ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ ⎨ , R1 (a, b) , ∼1 R1 (a, b) R (a, b) , ∼ (∼2 ∩) ⎪ (∩) ⎪ 2 2 ⎪ ⎪   , ∼2 R2 (a, b) ⎪ ⎪ ⎪ ⎪ , ∼2 R1 (a, b) ⎪ ⎪ , ∼2 R1 (a, b) ⎪ ⎪ ⎩ ⎩ , R2 (a, b) , ∼1 R2 (a, b) , ∼2 (R1 ∩ R2 )(a, b)) , (R1 ∩ R2 )(a, b)

and

⎧ ⎧ , ∼1 R1 (a, b) , R1 (a, b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a, b) , R , ∼1 R2 (a, b) ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎨ , ∼1 R1 (a, b) , ∼1 R1 (a, b) (∪) ⎪ (∼1 ∪) ⎪ ⎪ ⎪  , ∼2 R2 (a, b)  , R2 (a, b) ⎪ ⎪ ⎪ ⎪ R (a, b) , ∼ , R1 (a, b) ⎪ ⎪ 2 1 ⎪ ⎪ ⎩ ⎩ , ∼1 R2 (a, b) , ∼1 R2 (a, b) , (R , ∼1 (R1 ∪ R2 )(a, b) 1 ∪ R2 )(a, b) ⎧ , ∼ R (a, b) ⎪ 2 1 ⎪ ⎪ ⎪ ⎪ ⎪  , ∼2 R2 (a, b)  ⎨ , ∼2 R1 (a, b) , R1 (a, b) (a, b) , R , R2 (a, b) (∼2 ∪) ⎪ (∪) 2 ⎪ ⎪ ⎪ (a, b) , R , (R ⎪ 1 1 ∪ R2 )(a, b) ⎪ ⎩ , ∼2 R2 (a, b) , ∼2 (R1 ∪ R2 )(a, b)

• Deduction rules for quantifier constructors: ⎧ , ∼1 R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎡ , R(a, c1 ) ⎪ ⎪ , R(a, d1 ) ⎪ ⎪ ⎨ , ∼1 R(c1 , c2 ) ⎢ , R(d1 , d2 ) ⎢ , R(c1 , c2 ) (∗) ⎣ · · · (∼1 ∗) ⎪ ⎪ ⎪··· ⎪ , R(dn , b) ⎪ ⎪ , ∼1 R(cn , b) ⎪ ⎪ , R ∗ (a, b) ⎩ , R(cn , b) , (∼1 R ∗ )(a, b)

9.2 1/22 -Multisequents

281

⎧ ⎪ ⎪ , ∼2 R(a, c1 ) ⎪ ⎪ ⎧ , R(a, c1 ) ⎪ ⎪ , R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎨ , ∼2 R(c1 , c2 ) ⎨ , R(c1 , c2 ) , R(c1 , c2 ) ··· (∼2 ∗) ⎪ (∗) ⎪ ⎪ ⎪ ··· ⎪ ⎩ ⎪ , R(cn , b) ⎪ ⎪ , ∼2 R(cn , b) ⎪ ⎪ , (R ∗ )(a, b) ⎩ , R(cn , b) , (∼2 R ∗ )(a, b) where c is a new constant (does not occur in ), and d is a constant, and ⎧ , R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎧ , ∼1 R(a, c1 ) ⎪ ⎪ , R(a, c1 ) ⎪ ⎪ ⎪ ⎪ , ⎨ R(c1 , c2 ) ⎨ , R(c1 , c2 ) , ∼1 R(c1 , c2 ) ··· (◦) ⎪ (∼1 ◦) ⎪ ⎪ ⎪ · ·· ⎪ ⎩ ⎪ , R(cn , b) ⎪ ⎪ , ⎪ R(cn , b) ⎪ , R ◦ (a, b) ⎩ , ∼1 R(cn , b) , (∼1 R ◦ )(a, b)) ⎧ , R(a, c ) ⎪ 1 ⎪ ⎪ ⎪ ⎡ , ∼2 R(a, c1 ) ⎪ ⎪ , R(a, d1 ) ⎪ ⎪ , c ) , R(c ⎨ 1 2 ⎢ , R(d1 , d2 ) ⎢ , ∼2 R(c1 , c2 ) (∼2 ◦) ⎪ (◦) ⎣ · · · ⎪ · · · ⎪ ⎪ , R(dn , b) ⎪ ⎪ , R(cn , b) ⎪ ⎪ , (R ◦ )(a, b) ⎩ , ∼2 R(cn , b) , (∼2 R ◦ )(a, b)) where ci is a new constant and di is a constant. t t Definition 9.2.1 A 1/22 -multisequent  is provable in M1/2 2 , denoted by 1/22 , if 2 there is a sequence {1 , ..., n } of 1/2 -multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/22 -multisequents t by one of the deduction rules in M1/2 2.

Theorem 9.2.2 (Soundness and completeness theorem) For any 1/22 -multisequent , t1/22  if and only if |=t1/22 . 

t 9.2.2 R-Calculus R1/2 2

Given a 1/22 -multisequent  and a statement R(a, b) ∈ , a reduction  ↑ R(a, b) t t  ⇒ [R  (a, b)] is R1/2 2 -valid, denoted by |=1/22  ↑ R(a, b) ⇒ [R (a, b)], if

9 Role R-Calculus for B22 -Valued DL

282

R  (a, b) =



t R(a, b) if [R(a, b)] is M1/2 2 -valid λ otherwise.

t R-calculus R1/2 2 consists of the following axioms and deduction rules: • Axioms: − − [r (a, b)]∩ ∼− 1 ∩ ∼2  ∩    = ∅  ↑ r (a, b) ⇒ [r (a, b)] − − − ∼1 Rt− ∩ ∼1 [∼1 r (a, b)]∩ ∼2  ∩    = ∅ (A1/22 )  ↑∼1 r (a, b) ⇒ [∼1 r (a, b)] − − − ∼2 Rt− ∩ ∼1 ∩ ∼2 [∼2 r (a, b)] ∩    = ∅ (A1/22 )  ↑∼2 r (a, b) ⇒ (∼2 r (a, b)) − − − Rt− ∩ ∼1 ∩ ∼2  ∩  [r (a, b)]  = ∅ (A1/22 )  ↑ r (a, b) ⇒ (r (a, b))

Rt− (A1/2 2 )

where  is a set of literals, and r (a, b) is an R-atom. • Deduction rules:  ↑ R(a, b) ⇒ [R(a, b)]  ↑∼21 R(a, b) ⇒ [∼21 R(a, b)]  ↑ R(a, b) ⇒ [R(a, b)] (∼22− )  ↑∼22 R(a, b) ⇒ [∼22 R(a, b)]  ↑ R(a, b) ⇒ [R(a, b)] (2− )  ↑ 2 R(a, b) ⇒ [2 R(a, b)]

(∼21− )

and 

 ↑ R1 (a, b) ⇒ [R1 (a, b)] [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]  ⎧ ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] ⎪ ⎪ ⎪ ⎪  ↑ R2 (a, b) ⇒ [R2 (a, b)] ⎪ ⎪ ⎨  ↑∼1 R1 (a, b) ⇒ [∼1 R1 (a, b)] (∪− ) ⎪ ⎪  [∼1 R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼1 R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪  ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪ ⎩ [∼2 R1 (a, b)] ↑∼1 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼1 R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]

(∩− )

and

9.2 1/22 -Multisequents

283

⎡

 ↑ R(a, d1 ) ⇒ [R(a, d1 )] ⎢  ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )] ⎢ (∗− ) ⎣ · · ·  ↑ R(dn , b) ⇒ [R(dn , b)] ∗ ∗  ⎧ ↑ R (a, b) ⇒ [R (a, b)]  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎨  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] · ·· (◦− ) ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ R ◦ (a, b) ⇒ [R ◦ (a, b)] and ⎧  ↑∼1 R1 (a, b) ⇒ [∼1 R1 (a, b)] ⎪ ⎪ ⎪ ⎪ [∼1 R1 (a, b)] ↑∼1 R2 (a, b) ⇒ [∼1 R1 (a, b), ∼1 R2 (a, b)] ⎪ ⎪ ⎨  ↑ R1 (a, b) ⇒ [R1 (a, b)] [R (∼1 ∩− ) ⎪ 1 (a, b)] ↑∼1 R2 (a, b) ⇒ [R1 (a, b), ∼1 R2 (a, b)] ⎪  ⎪ ⎪  ↑∼ ⎪ 1 R1 (a, b) ⇒ [∼1 R1 (a, b)] ⎪ ⎩ [∼1 R1 (a, b)] ↑ R2 (a, b) ⇒ [∼1 R1 (a, b), R2 (a, b)]  ↑∼1 (R1 ∩ R2 )(a, b) ⇒ [∼1 (R1 ∩ R2 )(a, b)] ⎧  ↑∼1 R1 (a, b) ⇒ [∼1 R1 (a, b)] ⎪ ⎪ ⎪ ⎪ [∼1 R1 (a, b)] ↑∼1 R2 (a, b) ⇒ [∼1 R1 (a, b), ∼1 R2 (a, b)] ⎪ ⎪ ⎨  ↑ R1 (a, b) ⇒ [R1 (a, b)] [R (∼1 ∪− ) ⎪ 1 (a, b)] ↑∼1 R2 (a, b) ⇒ [R1 (a, b), ∼1 R2 (a, b)] ⎪  ⎪ ⎪ R1 (a, b) ⇒ [∼1 R1 (a, b)]  ↑∼ ⎪ 1 ⎪ ⎩ [∼1 R1 (a, b)] ↑ R2 (a, b) ⇒ [∼1 R1 (a, b), R2 (a, b)]  ↑∼1 (R1 ∪ R2 )(a, b) ⇒ [∼1 (R1 ∪ R2 )(a, b)] and

⎧  ↑∼1 R(a, c1 ) ⇒ [∼1 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨  ↑∼1 R(c1 , c2 ) ⇒ [∼1 R(c1 , c2 )]  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∼1 ∗− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼1 R(cn , b) ⇒ [∼1 R(cn , b)] ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ (∼1 R ∗ )(a, b) ⇒ [(∼1 R ∗ )(a, b)] ⎧  ↑∼1 R(a, c1 ) ⇒ [∼1 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨  ↑∼1 R(c1 , c2 ) ⇒ [∼1 R(c1 , c2 )]  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∼1 ◦− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼1 R(cn , b) ⇒ [∼1 R(cn , b)] ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ (∼1 R ◦ )(a, b) ⇒ [(∼1 R ◦ )(a, b)]

9 Role R-Calculus for B22 -Valued DL

284

and ⎧  ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪ ⎪ ⎪ [∼2 R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪ ⎨  ↑ R1 (a, b) ⇒ [R1 (a, b)] [R (∼2 ∩− ) ⎪ ⎪ 1 (a, b)] ↑∼2 R2 (a, b) ⇒ [R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪  ↑∼ ⎪ 2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎩ [∼2 R1 (a, b)] ↑ R2 (a, b) ⇒ [∼ R1 (a, b), R2 (a, b)]  ↑∼2 (R1 ∩ R2 )(a, b) ⇒ [∼2 (R1 ∩ R2 )(a, b)] ⎧ ⎪  ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪ ⎪ ⎪  [∼2 R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎨  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∼2 ∪− ) ⎪ ⎪  [R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪ ⎪  ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎩ [∼2 R1 (a, b)] ↑ R2 (a, b) ⇒ [∼2 R1 (a, b), R2 (a, b)]  ↑∼2 (R1 ∪ R2 )(a, b) ⇒ [∼2 (R1 ∪ R2 )(a, b)] and

⎧ ⎪ ⎪  ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )] ⎪ ⎪  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨  ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )]  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∼2 ∗− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)] ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ (∼2 R ∗ )(a, b) ⇒ [(∼2 R ∗ )(a, b)] ⎧  ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨  ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )]  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∼2 ◦− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)] ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ (∼2 R ◦ )(a, b) ⇒ [(∼2 R ◦ )(a, b)]

and ⎧  ↑ R1 (a, b) ⇒ [R1 (a, b)] ⎪ ⎪ ⎪ ⎪  ↑ R2 (a, b) ⇒ [R2 (a, b)] ⎪ ⎪ ⎨  ↑∼1 R1 (a, b) ⇒ [∼1 R1 (a, b)] (∩− ) ⎪ ⎪  [∼1 R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼1 R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪  ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪ ⎩ [∼2 R1 (a, b)] ↑∼1 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼1 R2 (a, b)]  ↑∼2 (R1 ∩ R2 )(a, b)) ⇒ [∼2 (R1 ∩ R2 )(a, b)]

9.2 1/22 -Multisequents

285



(∪− ) and

 ↑ R1 (a, b) ⇒ [R1 (a, b)] [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]

⎧  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎨  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] ··· (∗− ) ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)] ∗ ∗  ⎡ ↑ (R )(a, b) ⇒ [(R )(a, b)]  ↑ R(a, d1 ) ⇒ [R(a, d1 )] ⎢  ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )] ⎢ (◦− ) ⎣ · · ·  ↑ R(dn , b) ⇒ [R(dn , b)]  ↑ (R ◦ )(a, b) ⇒ [(R ◦ )(a, b)]

where ci is a new constant and di is a constant. Definition 9.2.3 Given a 1/22 -multisequent  and a statement R(a, b) ∈ , a 1/22 t t reduction  ↑ R(a, b) ⇒   is provable in R1/2 2 , denoted by 1/22  ↑ R(a, b) ⇒   , if there is a sequence {δ1 , ..., δn } of 1/22 -reductions such that δn =  ↑ R(a, b) ⇒   , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/22 t reductions by one of the deduction rules in R1/2 2. Theorem 9.2.4 (Soundness and completeness theorem) For any reduction δ =  ↑ R(a, b) ⇒ [R(a, b)], t1/22  ↑ R(a, b) ⇒ [R(a, b)] iff |=t1/22  ↑ R(a, b) ⇒ [R(a, b)].  Theorem 9.2.5 (Soundness and incompleteness theorem) For any reduction δ =  ↑ R(a, b) ⇒ [R(a, b)], t1/22  ↑ R(a, b) ⇒  implies |=t1/22  ↑ R(a, b) ⇒ , and

|=t1/22  ↑ R(a, b) ⇒  may not imply t1/22  ↑ R(a, b) ⇒ . 

9 Role R-Calculus for B22 -Valued DL

286

1/22

9.2.3 Incomplete Deduction System Lt 1/22

1/22

A 1/22 -multisequent  is Lt -valid, denoted by |==t , if there is an interpretation I such that for each statement R(a, b) ∈ , I (R(a, b)) = t. 1/22 Deduction system Lt consists of the following axiom and deduction rules: • Axiom: − 1 − 1/22 ∩ ∼1 ∩ ∼2  ∩   = ∅ (At ) , where  is a set of literals. • Deduction rules for logical constructors: (∼21 ) and

, R(a, b) , R(a, b) , R(a, b) (∼22 ) (2 ) 2 2 , 2 R(a, b) , ∼1 R(a, b) , ∼2 R(a, b) ⎡

, ∼1 R1 (a, b) ⎢ , ∼1 R2 (a, b) ⎢  ⎢ , R1 (a, b) , R1 (a, b) ⎢ (∩) , R2 (a, b) (∼1 ∩) ⎢ ⎢  , ∼1 R2 (a, b) ⎣ , ∼1 R1 (a, b) , (R1 ∩ R2 )(a, b) , R2 (a, b) , ∼ 1 (R1 ∩ R2 )(a, b)) ⎡ ⎡ , ∼2 R1 (a, b) , R1 (a, b) ⎢ , ∼2 R2 (a, b) ⎢ , R2 (a, b) ⎢ ⎢ ⎢ , R1 (a, b) ⎢ , ∼1 R1 (a, b) ⎢ ⎢ ⎢ (∼2 ∩) ⎢  , ∼2 R2 (a, b) (∩) ⎢ ⎢  , ∼2 R2 (a, b) ⎣ , ∼2 R1 (a, b) ⎣ , ∼2 R1 (a, b) , R2 (a, b) , ∼1 R2 (a, b) , ∼2 (R1 ∩ R2 )(a, b)) , (R1 ∩ R2 )(a, b)

and ⎡ , ∼1 R1 (a, b) , R1 (a, b) ⎢ , ∼1 R2 (a, b) ⎢ , R2 (a, b) ⎢ ⎢ ⎢ , ∼1 R1 (a, b) ⎢ , ∼1 R1 (a, b) ⎢ ⎢ ⎢ (∪) ⎢  , ∼2 R2 (a, b) (∼1 ∪) ⎢ ⎢  , R2 (a, b) ⎣ , R1 (a, b) ⎣ , ∼2 R1 (a, b) , ∼1 R2 (a, b) , ∼1 R2 (a, b) , (R1 ∪ R2 )(a, b) , ∼1 (R1 ∪ R2 )(a, b) ⎡

9.2 1/22 -Multisequents

287

⎡

, ∼2 R1 (a, b) ⎢ , ∼2 R2 (a, b) ⎢  ⎢ , ∼2 R1 (a, b) , R1 (a, b) ⎢ (a, b) , R , R2 (a, b) (∼2 ∪) ⎢ (∪) 2 ⎢ ⎣ , R1 (a, b) , (R1 ∪ R2 )(a, b) , ∼2 R2 (a, b) , ∼2 (R1 ∪ R2 )(a, b) • Deduction rules for quantifier constructors: ⎡

⎧ , R(a, c1 ) ⎪ ⎪ ⎨ , R(c1 , c2 ) ··· (∗) ⎪ ⎪ ⎩ , R(cn , b) , R ∗ (a, b) ⎡

, ∼1 R(a, d1 ) ⎢ , R(a, d1 ) ⎢ ⎢ , ∼1 R(d1 , d2 ) ⎢ ⎢ , R(d1 , d2 ) (∼1 ∗) ⎢ ⎢··· ⎢ ⎣ , ∼1 R(dn , b) , R(dn , b) , (∼1 R ∗ )(a, b)

, ∼2 R(a, d1 ) ⎢ , R(a, d1 ) ⎡ ⎢ , R(a, d1 ) ⎢ , ∼2 R(d1 , d2 ) ⎢ ⎢ , R(d1 , d2 ) ⎢ , R(d1 , d2 ) ⎢ (∼2 ∗) ⎢ (∗) ⎣ · · · ⎢··· ⎢ , R(dn , b) ⎣ , ∼2 R(dn , b) , (R ∗ )(a, b) , R(dn , b) , (∼2 R ∗ )(a, b) ⎡

and ⎡

, R(a, d1 ) ⎢ , R(d1 , d2 ) ⎢ (◦) ⎣ · · · , R(dn , b) , R ◦ (a, b) ⎡

, R(a, d1 ) ⎢ , ∼1 R(a, d1 ) ⎢ ⎢ , R(d1 , d2 ) ⎢ ⎢ , ∼1 R(d1 , d2 ) (∼1 ◦) ⎢ ⎢··· ⎢ ⎣ , R(dn , b) , ∼1 R(dn , b) , (∼1 R ◦ )(a, b))

, R(a, d1 ) ⎢ , ∼2 R(a, d1 ) ⎧ ⎢ , R(a, c1 ) ⎪ ⎢ , R(d1 , d2 ) ⎪ ⎨ ⎢ , R(c1 , c2 ) ⎢ , ∼2 R(d1 , d2 ) ··· (∼2 ◦) ⎢ (◦) ⎪ ⎢··· ⎪ ⎩ ⎢ , R(cn , b) ⎣ , R(dn , b) , (R ◦ )(a, b) , ∼2 R(dn , b) , (∼2 R ◦ )(a, b)) where ci is a new constant and di is a constant.

9 Role R-Calculus for B22 -Valued DL

288

1/22

1/22

Definition 9.2.6 A 1/22 -multisequent  is provable in Lt , denoted by =t , if there is a sequence {1 , ..., n } of 1/22 -multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/22 -multisequents 1/22 by one of the deduction rules in Lt . Theorem 9.2.7 (Soundness and incompleteness theorem) For any 1/22 multisequent , 1/22 1/22 =t  implies |==t , and

1/22

1/22

|==t  may not imply =t . 

1/22

9.2.4 R-Calculus Qt

Given a 1/22 -multisequent  and a statement R(a, b), a reduction  ↑ R(a, b) ⇒ 1/22 1/22 , R  (a, b) is Qt -valid, denoted by |==t  ↑ R(a, b) ⇒ , R  (a, b), if 

R (a, b) =



1/22

R(a, b) if , R(a, b) is Lt -valid λ otherwise. 1/22

Let   = , R(a, b). R-calculus Qt consists of the following axioms and deduction rules: • Axioms: ⎡ ∼1 r (a, b) ∈   ⎣ ∼2 r (a, b) ∈   1/22 (A Rt+ ) r (a, b) ∈     ⎡↑ r (a, b) ⇒   [r (a, b)] r (a, b) ∈   ⎣ ∼2 r (a, b) ∈   1/22 (A∼1 Rt+ ) r (a, b) ∈      ⎡ ↑∼ r (a, b) ⇒  [∼ r (a, b)] ∼1 r (a, b) ∈  ⎣ r (a, b) ∈   1/22 (A∼2 Rt+ ) r (a, b) ∈     ↑∼2 r (a, b) ⇒   [∼2 r (a, b)] ⎡ ∼1 r (a, b) ∈   ⎣ ∼2 r (a, b) ∈   1/22 (ARt+ ) r (a, b) ∈     ↑ r (a, b) ⇒   [r (a, b)] where   is a set of literals, and r (a, b) is an R-atom.

9.2 1/22 -Multisequents

• Deduction rules:   ↑ R(a, b) ⇒   [R(a, b)]   ↑∼21 R(a, b) ⇒   [∼21 R(a, b)]   ↑ R(a, b) ⇒   [R(a, b)] (∼22− )   ↑∼22 R(a, b) ⇒   [∼22 R(a, b)]   ↑ R(a, b) ⇒   [R(a, b)] (2− )   ↑ 2 R(a, b) ⇒   [2 R(a, b)]

(∼21− )

and    ↑ R1 (a, b) ⇒   [R1 (a, b)] (∩− )   ↑ R2 (a, b) ⇒   [R2 (a, b)]   ↑ (R1 ∩ R2 )(a, b) ⇒   [(R1 ∩ R2 )(a, b)] ⎡   ↑ R1 (a, b) ⇒   [R1 (a, b)] ⎢   [R1 (a, b)] ↑ R2 (a, b) ⇒   [R1 (a, b), R2 (a, b)] ⎢   ⎢  [R1 (a, b), R2 (a, b)] ↑∼1 R1 (a, b) ⇒   [R1 (a, b), R2 (a, b), ∼1 R1 (a, b)] ⎢   (∪− ) ⎢ ⎢   [R1 (a, b), R2 (a, b)] ↑∼2 R2 (a, b) ⇒  [R1 (a, b), R2 (a, b), ∼2 R2 (a, b)] ⎣   [Z 1 ] ↑∼2 R1 (a, b) ⇒   [Z 1 , ∼2 R1 (a, b)]   [Z 1 ] ↑∼1 R2 (a, b) ⇒   [Z 1 , ∼1 R2 (a, b)]   ↑ (R1 ∪ R2 )(a, b) ⇒   [(R1 ∪ R2 )(a, b)]

where Z 1 = R1 (a, b), R2 (a, b), ∼1 R1 (a, b)∨ ∼2 R2 (a, b), and ⎧   ↑ R(a, c1 ) ⇒   [R(a, c1 )] ⎪ ⎪ ⎨   ↑ R(c1 , c2 ) ⇒   [R(c1 , c2 )] − · (∗ ) ⎪ ⎪ ·· ⎩   ↑ R(cn , b) ⇒   [R(cn , b)]   ↑ R ∗ (a, b) ⇒   [R ∗ (a, b)] ⎡   ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢   ↑ R(d1 , d2 ) ⇒   [R(d1 , d2 )] ⎢ (◦− ) ⎣ · · ·   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ R ◦ (a, b) ⇒   [R ◦ (a, b)] and ⎡

  ↑∼1 R1 (a, b) ⇒   [∼1 R1 (a, b)] ⎢   ↑∼1 R2 (a, b) ⇒   [∼1 R2 (a, b)] ⎢   ⎢  [Z 2 ] ↑ R1 (a, b) ⇒   [Z 2 , R1 (a, b)] ⎢   (∼1 ∩− ) ⎢ ⎢    [Z 2 ] ↑∼1 R2 (a, b) ⇒   [Z 2 , ∼1 R2 (a, b)] ⎣  [Z 3 ] ↑∼1 R1 (a, b) ⇒  [Z 3 , ∼1 R1 (a, b)]   [Z 3 ] ↑ R2 (a, b) ⇒   [Z 3 , R2 (a, b)]   ↑∼1 (R1 ∩ R2 )(a, b) ⇒   [∼1 (R1 ∩ R2 )(a, b)]

289

9 Role R-Calculus for B22 -Valued DL

290

⎡

  ↑∼1 R1 (a, b) ⇒   [∼1 R1 (a, b)] ⎢   ↑∼1 R2 (a, b) ⇒   [∼1 R2 (a, b)] ⎢   ⎢  [Z 2 ] ↑ R1 (a, b) ⇒   [Z 2 , R1 (a, b)] ⎢   (∼1 ∪− ) ⎢ ⎢    [Z 2 ] ↑∼1 R2 (a, b) ⇒   [Z 2 , ∼1 R2 (a, b)] ⎣  [Z 4 ] ↑∼1 R1 (a, b) ⇒  [Z 4 , ∼1 R1 (a, b)]   [Z 4 ] ↑ R2 (a, b) ⇒   [Z 4 , R2 (a, b)]   ↑∼1 (R1 ∪ R2 )(a, b) ⇒   [∼1 (R1 ∪ R2 )(a, b)] Z 2 =∼1 R1 (a, b)∨ ∼1 R2 (a, b), where Z 3 = Z 2 , R1 (a, b)∨ ∼1 R2 (a, b), and Z 4 = Z 2 , R1 (a, b)∨ ∼1 R2 (a, b), ⎡

  ↑∼1 R(a, d1 ) ⇒   [∼1 R(a, d1 )] ⎢   ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢  ⎢  ↑∼1 R(d1 , d2 ) ⇒   [∼1 R(d1 , d2 )] ⎢   ⎢ − ⎢  ↑ R(d1 , d2 ) ⇒  [R(d1 , d2 )] (∼1 ∗ ) ⎢ ⎢ · ·· ⎣  ↑∼1 R(dn , b) ⇒   [∼1 R(dn , b)]   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ (∼1 R ∗ )(a, b) ⇒   [(∼1 R ∗ )(a, b)] ⎡   ↑∼1 R(a, d1 ) ⇒   [∼1 R(a, d1 )] ⎢   ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢  ⎢  ↑∼1 R(d1 , d2 ) ⇒   [∼1 R(d1 , d2 )] ⎢   ⎢ − ⎢  ↑ R(d1 , d2 ) ⇒  [R(d1 , d2 )] (∼1 ◦ ) ⎢ ⎢ · ·· ⎣  ↑∼1 R(dn , b) ⇒   [∼1 R(dn , b)]   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ (∼1 R ◦ )(a, b) ⇒   [(∼1 R ◦ )(a, b)] and

⎡

  ↑∼2 R1 (a, b) ⇒   [∼2 R1 (a, b)] ⎢   ↑∼2 R2 (a, b) ⇒   [∼2 R2 (a, b)] ⎢   ⎢  [Z 5 ] ↑ R1 (a, b) ⇒   [Z 5 , R1 (a, b)] ⎢   (∼2 ∩− ) ⎢ ⎢    [Z 5 ] ↑∼2 R2 (a, b) ⇒   [Z 5 , ∼2 R2 (a, b)] ⎣  [Z 6 ] ↑∼2 R1 (a, b) ⇒  [Z 6 , ∼2 R1 (a, b)]   [Z 6 ] ↑ R2 (a, b) ⇒   [Z 6 , R2 (a, b)]    ⎡ ↑∼ 2 (R1 ∩ R2 )(a, b) ⇒  [∼2 (R1 ∩ R2 )(a, b)]  ↑∼2 R1 (a, b) ⇒  [∼2 R1 (a, b)] ⎢   ↑∼2 R2 (a, b) ⇒   [∼2 R2 (a, b)] ⎢   ⎢  [Z 5 ] ↑ R1 (a, b) ⇒   [Z 5 , R1 (a, b)] ⎢   (∼2 ∪− ) ⎢ ⎢    [Z 5 ] ↑∼2 R2 (a, b) ⇒   [Z 5 , ∼2 R2 (a, b)] ⎣  [Z 7 ] ↑∼2 R1 (a, b) ⇒  [Z 7 , ∼2 R1 (a, b)]   [Z 7 ] ↑ R2 (a, b) ⇒   [Z 7 , R2 (a, b)]   ↑∼2 (R1 ∪ R2 )(a, b) ⇒   [∼2 (R1 ∪ R2 )(a, b)]

9.2 1/22 -Multisequents

291

Z 5 =∼2 R1 (a, b)∨ ∼2 R2 (a, b), where Z 6 = Z 5 , R1 (a, b)∨ ∼2 R2 (a, b), and Z 7 = Z 6 , R1 (a, b)∨ ∼2 R2 (a, b), ⎡

  ↑∼2 R(a, d1 ) ⇒   [∼2 R(a, d1 )] ⎢   ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢  ⎢  ↑∼2 R(d1 , d2 ) ⇒   [∼2 R(d1 , d2 )] ⎢   ⎢ − ⎢  ↑ R(d1 , d2 ) ⇒  [R(d1 , d2 )] (∼2 ∗ ) ⎢ ⎢ · ·· ⎣  ↑∼2 R(dn , b) ⇒   [∼2 R(dn , b)]   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ (∼2 R ∗ )(a, b) ⇒   [(∼2 R ∗ )(a, b)] ⎡   ↑∼2 R(a, d1 ) ⇒   [∼2 R(a, d1 )] ⎢   ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢  ⎢  ↑∼2 R(d1 , d2 ) ⇒   [∼2 R(d1 , d2 )] ⎢   ⎢ − ⎢  ↑ R(d1 , d2 ) ⇒  [R(d1 , d2 )] (∼2 ◦ ) ⎢ ⎢ · ·· ⎣  ↑∼2 R(dn , b) ⇒   [∼2 R(dn , b)]   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ (∼2 R ◦ )(a, b) ⇒   [(∼2 R ◦ )(a, b)] and ⎡

  ↑ R1 (a, b) ⇒   [R1 (a, b)] ⎢   [R1 (a, b)] ↑ R2 (a, b) ⇒   [R1 (a, b), R2 (a, b)] ⎢   ⎢  [Z 8 ] ↑∼1 R1 (a, b) ⇒   [Z 8 , ∼1 R1 (a, b)] ⎢   (∩− ) ⎢ ⎢    [Z 8 ] ↑∼2 R2 (a, b) ⇒   [Z 8 , ∼2 R2 (a, b)] ⎣  [Z 9 ] ↑∼2 R1 (a, b) ⇒  [Z 9 , ∼2 R1 (a, b)]   [Z 9 ] ↑∼1 R2 (a, b) ⇒   [Z 9 , ∼1 R2 (a, b)]    2 (R1 ∩ R2 )(a, b)) ⇒  [∼2 (R1 ∩ R2 )(a, b)]  ↑∼    ↑ R1 (a, b) ⇒  [R1 (a, b)] (∪− )   ↑ R2 (a, b) ⇒   [R2 (a, b)]   ↑ (R1 ∪ R2 )(a, b) ⇒   [(R1 ∪ R2 )(a, b)] where Z 8 = R1 (a, b), R2 (a, b); Z 9 = Z 8 , ∼1 R1 (a, b)∨ ∼2 R2 (a, b), and ⎡

  ↑ R(a, d1 ) ⇒   [R(a, d1 )] ⎢   ↑ R(d1 , d2 ) ⇒   [R(d1 , d2 )] ⎢ (∗− ) ⎣ · · ·   ↑ R(dn , b) ⇒   [R(dn , b)]   ↑ (R ∗ )(a, b) ⇒   [(R ∗ )(a, b)]

9 Role R-Calculus for B22 -Valued DL

292

⎧   ⎪ ⎪   ↑ R(a, c1 ) ⇒  [R(a, c1 )] ⎨  ↑ R(c1 , c2 ) ⇒  [R(c1 , c2 )] ··· (◦− ) ⎪ ⎪ ⎩   ↑ R(cn , b) ⇒   [R(cn , b)]   ↑ (R ◦ )(a, b) ⇒   [(R ◦ )(a, b)] where ci is a new constant and di is a constant. Definition 9.2.8 Given a 1/22 -multisequent  and a statement R(a, b), a 1/22 1/22 1/22 reduction  ↑ R(a, b) ⇒ , R  (a, b) is provable in Qt , denoted by =t  ↑ R(a, b) ⇒ , R  (a, b), if there is a sequence {δ1 , ..., δn } of 1/22 -reductions such that δn =  ↑ R(a, b) ⇒ , R  (a, b), and for each 1 ≤ i ≤ n, δi is either an axiom 1/22 or deduced from the previous 1/22 -reductions by one of the deduction rules in Qt . Theorem 9.2.9 (Soundness and completeness theorem) For any reduction δ =  ↑ R(a, b) ⇒ , R(a, b), t1/22 δ iff |=t1/22 δ.  Theorem 9.2.10 (Soundness and incompleteness theorem) For any reduction δ =  ↑ R(a, b) ⇒ , t1/22 δ implies |=t1/22 δ, and

t1/22 δ may not imply |=t1/22 δ. 

9.3 2/22 -Multisequents t t A 2/22 -multisequent | is M2/2 2 -valid, denoted by |=2/22 |, if for any interpretation I, (i) I (R(a, b)) = t for some R(a, b) ∈ , and (ii) I (Q(a, b)) =  for some Q(a, b) ∈ . We have the following equivalences:

, ∼1 R(a, b)| ≡ |, R(a, b) |, ∼1 Q(a, b) ≡ , Q(a, b)| |, ∼2 Q(a, b) ≡ , Q(a, b)| |, Q(a, b) ≡ , ∼2 Q(a, b)|.

9.3 2/22 -Multisequents

293

t 9.3.1 Deduction System M2/2 2 t Deduction system M2/2 2 contains the following axiom and deduction rules. • Axiom: −  ∩ ∩ ∼− 2  ∩   = ∅ (At 2/22 ) |

where  is a set of literals and  is a set of atoms. • Deduction rules for unary connectives: |, R(a, b) , Q(a, b)| (∼1Q ) , ∼1 R(a, b)| |, ∼1 Q(a, b) Q , Q(a, b)| 2R , R(a, b)| (∼2 ) (∼2 ) , ∼22 R(a, b)| |, ∼2 Q(a, b) 2R , R(a, b)| Q , ∼2 Q(a, b)| ( ) ( ) , 22 R(a, b)| |, Q(a, b) (∼1R )

• Deduction rules for logical connectives: ⎧ |, Q 1 (a, b) ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b) ⎪ ⎪  ⎨ , R1 (a, b)| , Q 1 (a, b)| (∩ R ) , R2 (a, b)| (∩ Q ) ⎪ ⎪  |, Q 2 (a, b) ⎪ ⎪ |, Q 1 (a, b) , (R1 ∩ R2 )(a, b)| ⎪ ⎪ ⎩ , Q 2 (a, b)| |, ⎧ ⎧ (Q 1 ∩ Q 2 )(a, b) , R1 (a, b)| , ∼ R (a, b)| ⎪ ⎪ 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R (a, b)| , ∼ , R2 (a, b)| ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ ⎨ , R1 (a, b)| , ∼2 R1 (a, b)| R R R (a, b)| , ∼ (∼2 ∩ ) ⎪ (∩ ) ⎪ ⎪  |, R2 (a, b) 2 2 ⎪  ⎪ ⎪ ⎪ ⎪ R (a, b)| , ∼ ⎪ ⎪ |, R1 (a, b) 2 1 ⎪ ⎪ ⎩ ⎩ , R2 (a, b)| , ∼2 R2 (a, b)| , ∼2 (R1 ∩ R2 )(a, b)| , (R1 ∩ R2 )(a, b)| and ⎧ , R1 (a, b)| ⎪ ⎪ ⎪ ⎪ , R2 (a, b)| ⎪ ⎪ ⎨ |, R1 (a, b) Q , (∪ R ) ⎪ ⎪  ∼2 R2 (a, b) ⇒  (∪ ) ⎪ ⎪ , ∼2 R1 (a, b)| ⎪ ⎪ ⎩ |, R2 (a, b) , (R1 ∪ R2 )(a, b)|

⎧ |, Q 1 (a, b) ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b) ⎪ ⎪ ⎨ , Q 1 (a, b)| ⎪ ⎪  |, Q 2 (a, b) ⎪ ⎪ |, Q 1 (a, b) ⎪ ⎪ ⎩ , Q 2 (a, b)| |, (Q 1 ∪ Q 2 )(a, b)

9 Role R-Calculus for B22 -Valued DL

294

⎧ ⎪ ⎪ , ∼2 R1 (a, b)| ⎪ ⎪ , ∼2 R2 (a, b)| ⎪ ⎪  ⎨ , R1 (a, b)| , R1 (a, b)| R R (a, b)| , ∼ , R2 (a, b)| (∼2 ∪ R ) ⎪ ) (∪ 2 2 ⎪ ⎪ ⎪ R (a, b)| , ∼ , (R1 ∪ R2 )(a, b)| ⎪ 2 1 ⎪ ⎩ , R2 (a, b)| , ∼2 (R1 ∪ R2 )(a, b)| • Deduction rules for quantifier constructors: ⎡

, R(a, d)| ⎢ , R(d1 , d2 )| ⎢ (∗ R ) ⎣ · · · , R(dn , b)| , R ∗ (a, b)|

⎧ |, Q(a, c1 ) ⎪ ⎪ ⎪ ⎪ , Q(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎨ |, Q(c1 , c2 ) , Q(c1 , c2 )| (∗ Q ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ |, Q(cn , b) ⎪ ⎪ ⎩ , Q(cn , b)| |, (Q ∗ )(a, b)

⎧ , R(a, c1 )| ⎪ ⎪ ⎪ ⎪ , ∼2 R(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎨ , R(c1 , c2 )| , ∼2 R(c1 , c2 )| (∼2 ∗ R ) ⎪ (∗ R ) ⎪ ··· ⎪ ⎪ ⎪ ⎪ , R(cn , b)| ⎪ ⎪ ⎩ , ∼2 R(cn , b)| , (∼2 R ∗ )(a, b)| and ⎧ , R(a, c1 )| ⎪ ⎪ ⎨ , R(c1 , c2 )| ··· (◦ R ) ⎪ ⎪ ⎩ , R(cn , b)| , R ◦ (a, b)|

⎧ , R(a, c1 )| ⎪ ⎪ ⎨ , R(c1 , c2 )| ··· ⎪ ⎪ ⎩ , R(cn , b)| , (R ∗ )(a, b)|

⎧ ⎪ ⎪ |, Q(a, c1 ) ⎪ ⎪ , Q(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎨ |, Q(c1 , c2 ) , Q(c1 , c2 )| (◦ Q ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ |, Q(cn , b) ⎪ ⎪ ⎩ , Q(cn , b)| |, Q ◦ (a, b)

⎧ ⎪ ⎪ , ∼2 R(a, c1 )| ⎪ ⎪ , R(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎨ , ∼2 R(c1 , c2 )| , R(c1 , c2 )| (∼2 ◦ R ) ⎪ (◦ R ) ⎪ · · · ⎪ ⎪ ⎪ ⎪ , ∼2 R(cn , b)| ⎪ ⎪ ⎩ , R(cn , b)| , (∼2 R ◦ )(a, b)| where di is a constant and ci is a new constant.

⎡

, R(a, d)| ⎢ , R(d1 , d2 )| ⎢ ⎣··· , R(dn , b)| , (R ◦ )(a, b)|

9.3 2/22 -Multisequents

295

t t Definition 9.3.1 A 2/22 -multisequent | is provable in M2/2 2 , denoted by 2/22 |, if there is a sequence {1 |1 , ..., n |n } of 2/22 -multisequents such that n |n = |, and for each 1 ≤ i ≤ n, i |i is either an axiom or deduced from t the previous 2/22 -multisequents by one of the deduction rules in M2/2 2.

Theorem 9.3.2 (Soundness and completeness theorem) For any 2/22 -multisequent |, t |=t 2/22 | iff 2/22 |. 

t 9.3.2 R-Calculus R2/2 2

Given a 2/4-multisequent | and two statements R(a, b) ∈  and Q(a  , b ) ∈ , a 2/22 -reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )] t t is R1/2 2 -valid, denoted by |=2/22 δ, if 

t R(a, b) if [R(a, b)]| is M2/2 2 -valid λ otherwise.  t Q(a  , b ) if [R  (a, b)]|[Q(a  , b )] is M2/2 2 -valid    Q (a , b ) = λ otherwise.

R  (a, b) =

t R-calculus R2/2 2 consists of the following deduction rules and axioms: • Axioms: − [r (a, b)] ∩ ∩ ∼− 2  ∩   = ∅ | ↑ r (a, b) ⇒ [r (a, b)]| −   − Qt−  ∩ [q(a , b )]∩ ∼2  ∩    = ∅ (t2/22 )    | ↑ q(a , b ) ⇒ |[q(a , b )] −  ∩ ∩ ∼− 2 Rt− 2 [∼2 r (a, b)] ∩    = ∅ ) (t∼ 2 2/2 | ↑∼2 r (a, b) ⇒ [∼2 r (a, b)]| −  ∩ ∩ ∼− 2  ∩  [r (a, b)]  = ∅ ) (tRt− 2 2/2 | ↑ r (a, b) ⇒ [r (a, b)]|

Rt− ) (t2/2 2

where  is literal and  is atomic.

9 Role R-Calculus for B22 -Valued DL

296

• Deduction rules: | ↑ 2 R(a, b) ⇒ |[R(a, b)] | ↑∼1 R(a, b) ⇒ [∼1 R(a, b)]| | ↑ R(a, b) ⇒ |[R(a, b)] 2R− (∼2 ) | ↑∼22 R(a, b) ⇒ [∼22 R(a, b)]| | ↑ R(a, b) ⇒ [R(a, b)]| (2R− ) | ↑ 2 R(a, b) ⇒ [2 R(a, b)]| | ↑ 1 Q(a, b) ⇒ [Q(a, b)]| (∼1Q− ) | ↑∼1 Q(a, b) ⇒ |[∼1 Q(a, b)]

(∼1R− )

and 

| ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∩ ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ⎧ ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]| ⎪ | ↑ R1 (a, b) ⇒ [R1 (a, b)]| ⎪ ⎪ ⎪ | ↑ R2 (a, b) ⇒ [R2 (a, b)]| ⎪ ⎪ ⎨ | ↑ 2 R1 (a, b) ⇒ |[R1 (a, b)] R− (∪ ) ⎪ ⎪  |[R1 (a, b) ↑∼2 R2 (a, b) ⇒ [∼2 R2 (a, b)]|[R1 (a, b)] ⎪ ⎪ ⎪ | ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)]| ⎪ ⎩ [∼2 R1 (a, b)]| ↑ 2 A2 (a, b) ⇒ [∼2 R1 (a, b)]|[R2 (a, b)] | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]| R−

and ⎧ | ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)]| ⎪ ⎪ ⎪ ⎪ [∼2 R1 (a, b)]| ↑∼2 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼2 R2 (a, b)]| ⎪ ⎪  ⎨ | ↑ R1 (a, b) ⇒ [R1 (a, b)]| R− (∼2 ∩ ) ⎪ ⎪  [R1 (a, b)]| ↑∼2 R2 (a, b) ⇒ [R1 (a, b), ∼2 R2 (a, b)]| ⎪ ⎪ ⎪ | ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)]| ⎪ ⎩ [∼2 R1 (a, b)]| ↑ R2 (a, b) ⇒ [∼2 R1 (a, b), R2 (a, b)]| | ↑∼2 (R1 ∩ R2 )(a, b) ⇒ [∼2 (R1 ∩ R2 )(a, b)]|

and ⎧ | ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)]| ⎪ ⎪ ⎪ ⎪ [∼ ⎪ 2 R1 (a, b)]| ↑∼2 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼2 R2 (a, b)]| ⎪ ⎨ | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∼2 ∪ R− ) ⎪ ⎪  [R1 (a, b)]| ↑∼2 R2 (a, b) ⇒ [R1 (a, b), ∼2 R2 (a, b)]| ⎪ ⎪ | ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)]| ⎪ ⎪ ⎩ [∼2 R1 (a, b)]| ↑ R2 (a, b) ⇒ [∼2 R1 (a, b), R2 (a, b)]| | ↑∼2 (R1 ∪ R2 )(a, b) ⇒ [∼2 (R1 ∪ R2 )(a, b)]|

9.3 2/22 -Multisequents

297

and ⎧ | ↑ R1 (a, b) ⇒ [R1 (a, b)]| ⎪ ⎪ ⎪ ⎪ | ↑ R2 (a, b) ⇒ [R2 (a, b)]| ⎪ ⎪ ⎨  | ↑∼ R (a, b) ⇒ [∼ R (a, b)]| 2

1

2

1

(∩ R− ) ⎪ ⎪  [∼2 R1 (a, b)]| ↑ 2 R2 (a, b) ⇒ [∼2 R1 (a, b)|[R2 (a, b)] ⎪ ⎪ ⎪ | ↑ 2 R1 (a, b) ⇒ |[R1 (a, b)] ⎪ ⎩ |[R1 (a, b) ↑∼2 R2 (a, b) ⇒ [∼2 R2 (a, b)]|[R1 (a, b)] | ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]|  | ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪ R− ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]|

and

and

⎡

| ↑ R(a, d1 ) ⇒ [R(a, d1 )]| ⎢ | ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )]| ⎢ (∗ R− ) ⎣ · · · | ↑ R(dn , b) ⇒ [R(dn , b)]| ∗ ∗ | ⎧ ↑ R (a, b) ⇒ [R (a, b)]| | ↑ R(a, c1 ) ⇒ [R(a, c1 )]| ⎪ ⎪ ⎨ | ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )]| · ·· (◦ R− ) ⎪ ⎪ ⎩ | ↑ R(cn , b) ⇒ [R(cn , b)]| | ↑ R ◦ (a, b) ⇒ [R ◦ (a, b)]| ⎧ | ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )]| ⎪ ⎪ ⎪ ⎪ | ↑ R(a, c1 ) ⇒ [R(a, c1 )]| ⎪ ⎪ ⎪ ⎪ ⎨ | ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )]| | ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )]| (∼2 ∗ R− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ | ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)]| ⎪ ⎪ ⎩ | ↑ R(cn , b) ⇒ [R(cn , b)]| ∗ ∗ ⎧| ↑ (∼2 R )(a, b) ⇒ [(∼2 R )(a, b)]| | ↑ R(a, c1 ) ⇒ [R(a, c1 )]| ⎪ ⎪ ⎨ | ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )]| ··· (∗ R− ) ⎪ ⎪ ⎩ | ↑ R(cn , b) ⇒ [R(cn , b)]| | ↑ (R ∗ )(a, b) ⇒ [(R ∗ )(a, b)]|

298

and

9 Role R-Calculus for B22 -Valued DL

⎧ | ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )]| ⎪ ⎪ ⎪ ⎪ | ↑ 1 R(a, c1 ) ⇒ [R(a, c1 )]| ⎪ ⎪ ⎪ ⎪ ⎨ | ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )]| | ↑ 1 R(c1 , c2 ) ⇒ [R(c1 , c2 )]| R− (∼2 ◦ ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ | ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)]| ⎪ ⎪ ⎩ | ↑ 1 R(cn , b) ⇒ [R(cn , b)]| ◦ ◦ ⎡| ↑ (∼2 R )(a, b) ⇒ [(∼2 R )(a, b)]| | ↑ R(a, d1 ) ⇒ [R(a, d1 )]| ⎢ | ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )]| ⎢ (◦ R− ) ⎣ · · · | ↑ R(dn , b) ⇒ [R(dn , b)]| | ↑ (R ◦ )(a, b) ⇒ [(R ◦ )(a, b)]| ⎧   | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪   |[Q 1 (a, b) ↑ Q 2 (a, b) ⇒   |[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨   | ↑ 1 Q 1 (a, b) ⇒   [Q 1 (a, b)]|  Q− b) ⇒   [Q 1 (a, b)]|[Q 2 (a, b)] (∩ ) ⎪ ⎪    [Q 1 (a, b)]| ↑ Q 2 (a, ⎪  ⎪  | ↑ Q 1 (a, b) ⇒  |[Q 1 (a, b)] ⎪ ⎪ ⎩   |[Q 1 (a, b) ↑ 1 Q 2 (a, b) ⇒   [Q 2 (a, b)]|[Q 1 (a, b)]   (Q 1 ∩ Q 2 )(a, b) ⇒   |[(Q 1 ∩ Q 2 )(a, b)] ⎧ |  ↑   | ↑ Q 1 (a, b) ⇒   |[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪   |[Q 1 (a, b) ↑ Q 2 (a, b) ⇒   |[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨   | ↑ 1  Q 1 (a, b) ⇒   [Q 1 (a, b)]|   Q− (∪ ) ⎪ ⎪    [Q 1 (a, b)]| ↑ Q2 (a, b) ⇒  [Q 1 (a, b)]|[Q 2 (a, b)] ⎪ ⎪  | ↑ Q 1 (a, b) ⇒  |[Q 1 (a, b)] ⎪ ⎪ ⎩   |[Q 1 (a, b) ↑ 1  Q 2 (a, b) ⇒   [Q 2 (a, b)]|[Q 1 (a, b)]   | ↑ (Q 1 ∪ Q 2 )(a, b) ⇒   |[(Q 1 ∪ Q 2 )(a, b)]

and ⎧   | ↑ Q(a, c1 ) ⇒ |[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪   | ↑ 1  Q(a, c1 ) ⇒   [Q(a, c1 )]| ⎪ ⎪ ⎪ ⎪ ⎨   | ↑ Q(c1 , c2 ) ⇒ |[Q(c1 , c2 )]   | ↑ 1  Q(c1 , c2 ) ⇒   [Q(c1 , c2 )]| (∗ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪   | ↑ Q(cn , b) ⇒ |[Q(cn , b)] ⎪ ⎪ ⎩   | ↑ 1  Q(cn , b) ⇒   [Q(cn , b)]|   | ↑ Q ∗ (a, b) ⇒   |[Q ∗ (a, b)]

9.4 3/22 -Multisequents

299

⎧  c1 )] ⎪ ⎪   | ↑ 1Q(a, c1 ) ⇒ |[Q(a, ⎪  ⎪ | ↑ Q(a, c ) ⇒  [Q(a, c )]|  ⎪ 1 1 ⎪ ⎪ ⎪ ⎨   | ↑ Q(c1 , c2 ) ⇒ |[Q(c1 , c2 )]   | ↑ 1 Q(c1 , c2 ) ⇒   [Q(c1 , c2 )]| (◦ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪   | ↑ Q(cn , b) ⇒ |[Q(cn , b)] ⎪ ⎪ ⎩   | ↑ 1 Q(cn , b) ⇒   [Q(cn , b)]|   | ↑ Q ◦ (a, b) ⇒   |[Q ◦ (a, b)] where ci is a new constant and di is a constant. Definition 9.3.3 A 2/22 -reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ [R  (a, b)]| t t [Q  (a  , b )] is provable in R2/2 2 , denoted by 2/22 δ, if there is a sequence 2 {δ1 , ..., δn } of 2/2 -reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/4-reductions by one of the deduction rules t . in R2/4 Theorem 9.3.4 (Soundness and completeness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )] such that one of R  (a, b), Q  (a  , b ) is not the empty string, t t 2/22 δ iff |=2/22 δ.

 Theorem 9.3.5 (Soundness and incompleteness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ |, t t 2/22 δ implies |=2/22 δ,

and

t |=t 2/22 δ may not imply 2/22 δ.



9.4 3/22 -Multisequents t⊥ t A 3/22 -multisequent || is M3/2 2 -valid, denoted by |=3/22 ||, if for any interpretation I, either I (R(a, b)) = t for some R(a, b) ∈ , I (Q(a, b)) =  for some Q(a, b) ∈ , or I (P(a, b)) =⊥ for some P(a, b) ∈ . We have the following equivalences:

9 Role R-Calculus for B22 -Valued DL

300

, ∼1 R(a, b)|| ≡ |, R(a, b)| |, ∼1 Q(a, b)| ≡ , Q(a, b)|| |, Q(a, b)| ≡ ||, Q(a, b) ||, ∼2 P(a, b) ≡ , P(a, b)||

, ∼2 R(a, b)|| ≡ ||, R(a, b) |, ∼2 Q(a, b)| ≡ , Q(a, b)|| ||, ∼1 P(a, b) ≡ , P(a, b)|| ||, P(a, b) ≡ |, P(a, b)|.

t⊥ 9.4.1 Deduction System M3/2 2 t⊥ Deduction system M3/2 2 contains the following axiom and deduction rules: • Axiom:  ∩  ∩  ∩ −  = ∅ (At⊥ 3/22 ) ||

where  is literal and ,  are atomic. • Deduction rules for unary logical connectives: |, R(a, b)| , ∼1 R(a, b)|| , P(a, b)|| (∼1P ) ||, ∼1 P(a, b) , Q(a, b)|| Q (∼2 ) |, ∼2 Q(a, b)| , R(a, b)|| 2R ( ) , 2 R(a, b)|| |, P(a, b)| ( P ) ||, P(a, b) (∼1R )

, Q(a, b)|| |, ∼1 Q(a, b)| ||, R(a, b) (∼2R ) , ∼2 R(a, b)|| , P(a, b)|| P (∼2 ) ||, ∼2 P(a, b) ||, Q(a, b) Q ( ) |, Q(a, b)| (∼1Q )

• Deduction rules for binary logical connective ∩: ⎧ |, Q 1 (a, b)| ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b)| ⎪ ⎪  ⎨ , Q 1 (a, b)|| , R1 (a, b)|| |, Q 2 (a, b)| (∩ R ) , R2 (a, b)|| (∩ Q ) ⎪ ⎪  ⎪ ⎪ |, Q 1 (a, b)| , (R1 ∩ R2 )(a, b)|| ⎪ ⎪ ⎩ , Q 2 (a, b)|| |, ⎧ ⎧ (Q 1 ∩ Q 2 )(a, b)| , R1 (a, b)|| ||, P (a, b) ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a, b) ||, P , R2 (a, b)|| ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎨ , P1 (a, b)|| |, R1 (a, b)| R (a, b) ||, P ) (∩ P ) ⎪ (∩ ⎪ 2 ⎪ ⎪   ||, R2 (a, b) ⎪ ⎪ ⎪ ⎪ (a, b) ||, P ||, R1 (a, b) ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , P2 (a, b)|| |, R2 (a, b)| ||, (P1 ∩ P2 )(a, b) , (R1 ∩ R2 )(a, b)||

9.4 3/22 -Multisequents

301

• Deduction rules for binary logical connective ∪: ⎧ ⎧ , R1 (a, b)|| |, Q 1 (a, b)| ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a, b)|| , R |, Q 2 (a, b)| ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎨ ||, R1 (a, b) , Q 1 (a, b)|| Q (a, b)| |, R (∪ R ) ⎪ ) (∪ ⎪ 2 ⎪ ⎪   |, Q 2 (a, b)| ⎪ ⎪ ⎪ ⎪ (a, b)| |, R |, Q 1 (a, b)| ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎩ ||, R2 (a, b) , Q 2 (a, b)|| , |, (Q 1 ∪ Q 2 )(a, b)| ⎧ (R1 ∪ R2 )(a, b)|| ||, P (a, b) ⎪ 1 ⎪ ⎪ ⎪ ||, P2 (a, b) ⎪ ⎪  ⎨ , P1 (a, b)|| , R1 (a, b)|| R (a, b) ||, P , R2 (a, b)|| ) (∪ P ) ⎪ (∪ 2 ⎪  ⎪ ⎪ (a, b) ||, P , (R1 ∪ R2 )(a, b)|| ⎪ 1 ⎪ ⎩ , P2 (a, b)|| ||, (P1 ∪ P2 )(a, b) • Deduction rules for quantifier ∀: ⎡

, R(a, d1 )|| ⎢ , R(d1 , d2 )|| ⎢ (∗ R ) ⎣ · · · , R(dn , b)|| , R ∗ (a, b)||

⎧ ⎪ ⎪ |, Q(a, c1 )| ⎪ ⎪ ⎪ ⎪ , Q(a, c1 )|| ⎪ ⎪ ⎨ |, Q(c1 , c2 )| , Q(c2 , c2 )|| Q (∗ ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ |, Q(cn , b)| ⎪ ⎪ ⎩ , Q(cn , b)|| |, Q ∗ (a, b)|

⎧ ⎪ ⎪ ||, P(a, c1 ) ⎪ ⎪ , P(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ⎨ ||, P(c1 , c2 ) , P(c1 , c2 )|| (∗ P ) ⎪ (∗ R ) ⎪ · · · ⎪ ⎪ ⎪ ⎪ ||, P(cn , b) ⎪ ⎪ ⎩ , P(cn , b)|| ||, P ∗ (a, b) and

⎧ , R(a, c1 )|| ⎪ ⎪ ⎨ , R(c1 , c2 )|| ··· ⎪ ⎪ ⎩ , R(cn , b)|| , (R ∗ )(a, b)||

9 Role R-Calculus for B22 -Valued DL

302

⎧ ⎪ ⎪ |, Q(a, c1 )| ⎪ ⎪ ⎧ , Q(a, c1 )|| ⎪ ⎪ , R(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ⎨ |, Q(c1 , c2 )| ⎨ , R(c1 , c2 )|| , Q(c1 , c2 )|| ··· (◦ R ) ⎪ (◦ Q ) ⎪ ⎪ ⎪ ··· ⎪ ⎩ ⎪ , R(cn , b)|| ⎪ ⎪ |, Q(cn , b)| ⎪ ◦ ⎪ , R (a, b)|| ⎩ , Q(cn , b)|| |, Q ◦ (a, b)| ⎧ ⎪ ⎪ ||, P(a, c1 ) ⎪ ⎪ ⎡ , P(a, c1 )|| ⎪ ⎪ , R(a, d1 )|| ⎪ ⎪ ⎨ ||, P(c1 , c2 ) ⎢ , R(d1 , d2 )|| ⎢ , P(c1 , c2 )|| (◦ P ) ⎪ (◦ R ) ⎣ · · · ⎪ ··· ⎪ ⎪ , R(dn , b)|| ⎪ ⎪ ||, P(cn , b) ⎪ ⎪ , (R ◦ )(a, b)|| ⎩ , P(cn , b)|| ||, P ◦ (a, b) where di is a constant and ci is a new constant. t⊥ Definition 9.4.1 A 3/22 -multisequent || is provable in M3/2 2 , denoted by 2 t⊥ || if there is a sequence { | | , ...,  | | } of 3/2 -multisequents 1 1 1 n n n 3/22 such that n |n |n = ||, and for each 1 ≤ i ≤ n, i |i |i is either an axiom or deduced from the previous 3/22 -multisequents by one of the deduction rules in t⊥ M3/2 2 .

Theorem 9.4.2 (Soundness and completeness theorem) For any 3/22 -multisequent ||, t⊥ |=t⊥ 3/22 || iff 3/22 ||. 

t⊥ 9.4.2 R-Calculus R3/2 2

Let R(a  , b ) ∈ , Q(a  , b ) ∈  and P(a  , b ) ∈ . A reduction δ = || ↑ (R(a  , b ), Q(a  , b ), P(a  , b )) ⇒ [R  (a  , b )]|[Q  (a  , b )]|[P  (a  , b )] t⊥ t⊥ is R3/2 2 -valid, denoted by |=3/22 δ, if

9.4 3/22 -Multisequents

303



  R(a  , b ) if |=t⊥ 3/22 [R(a , b )]||  λ   otherwise      Q(a , b ) if |=t⊥ 3/22 [R (a , b )]|[Q(a , b )]| Q  (a  , b ) = λ   otherwise         P(a , b ) if |=t⊥ 3/22 [R (a , b )]|[Q (a , b )]|[P(a , b )] P  (a  , b ) = λ otherwise.

R  (a  , b ) =

Given a statement triple X = (R(a, b), Q(a  , b ), P(a  , b )), let X = || and X[R(a, b)] =  − {R(a, b)}|| X[Q(a, b)] = | − {Q(a, b)}| X[P(a, b)] = || − {P(a, b)};

X[1 R(a, b)] =  − {R(a, b)}|| X[2 R(a, b)] = | − {R(a, b)}| X[3 R(a, b)] = || − {R(a, b)}.

t⊥ R-calculus R3/2 2 consists of the following axioms and deduction rules: • Axioms:

[r (a, b)] ∩  ∩  ∩ −  = ∅ || ↑ r (a, b) ⇒ [r (a, b)]||   − Qt⊥−  ∩ [q(a , b )] ∩  ∩    = ∅ ) (A3/2 2   || ↑ q(a , b ) ⇒ |[q(a  , b )]|   − Pt⊥−  ∩  ∩ [ p(a , b )] ∩    = ∅ ) (A3/2 2   || ↑ p(a , b ) ⇒ ||[ p(a  , b )]  ∩  ∩  ∩ − [r (a, b)] = ∅ ) (ARt⊥− 2 3/2 || ↑ r (a, b) ⇒ ||[r (a, b)]

Rt⊥− ) (A3/2 2

where ,  are sets of atoms,  is a set of literals, and r (a, b) ∈ , q(a  , b ) ∈ , p(a  , b ) ∈  are atoms. • Deduction rules: (∼1R− )

X ↑ 2 R(a, b) ⇒ X[2 R(a, b)] X ↑∼1 R(a, b) ⇒ X[∼1 R(a, b)]

(∼1Q− )

X ↑ 1 Q(a, b) ⇒ X[1 Q(a, b)] X ↑∼1 Q(a, b) ⇒ X[∼1 Q(a, b)]

(∼1P− )

X ↑ 1  P(a, b) ⇒ X[1  P(a, b)] X ↑∼1 P(a, b) ⇒ X[∼1 P(a, b)]

(∼2R− ) (∼2Q− )

X ↑ 3 R(a, b) ⇒ X[3 R(a, b)] X ↑∼2 R(a, b) ⇒ X[∼2 R(a, b)]

X ↑ 1  Q(a, b) ⇒ X[1  Q(a, b)] X ↑∼2 Q(a, b) ⇒ X[∼2 Q(a, b)]

(∼2P− )

X ↑ 1 P(a, b) ⇒ X[1 P(a, b)] X ↑∼2 P(a, b) ⇒ X[∼2 P(a, b)]

9 Role R-Calculus for B22 -Valued DL

304

(2R− )

and

X ↑ R(a, b) ⇒ X[R(a, b)] X ↑ 2 R(a, b) ⇒ X[2 R(a, b)]

( Q− )

X ↑ 3 Q(a, b) ⇒ X[3 Q(a, b)] X ↑ Q(a, b) ⇒ X[Q(a, b)]

( P− )

X ↑ 2 P(a, b) ⇒ X[2 P(a, b)] X ↑ P(a, b) ⇒ X[P(a, b)]



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X[R1 (a, b)] ↑ R2 (a, b) ⇒ X[R1 (a, b), R2 (a, b)] X ⎧↑  (R1 ∩ R2 )(a, b) ⇒ X[(R1 ∩ R2 )(a, b)] X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪ X[Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨ X ↑ 1 Q 1 (a, b) ⇒ X[1 Q 1 (a, b)] 1 1 (∩ Q− ) ⎪ ⎪  X[ Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[ Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎩ X[Q 1 (a, b)] ↑ 1 Q 2 (a, b) ⇒ X[Q 1 (a, b), 1 Q 2 (a, b)] X ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ X[(Q 1 ∩ Q 2 )(a, b)]

(∩ R− )

and ⎧ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎪ ⎪ X[P1 (a, b)] ↑ P2 (a, b) ⇒ X[P1 (a, b), P2 (a, b)] ⎪ ⎪ ⎨ X ↑ 1 P1 (a, b) ⇒ X[1 P1 (a, b)] P− (∩ ) ⎪ ⎪  X[1 P1 (a, b)] ↑ P2 (a, b) ⇒ X[1 P1 (a, b), P2 (a, b)] ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎩ X[P1 (a, b)] ↑ 1 P2 (a, b) ⇒ X[P1 (a, b), 1 P2 (a, b)] X⎧ ↑ (P1 ∩ P2 )(a, b) ⇒ X[(P1 ∩ P2 )(a, b)] ⎪ X ↑ R1 (a, b) ⇒ X[R1 (a, b)] ⎪ ⎪ ⎪ ↑ R2 (a, b) ⇒ X[R2 (a, b)] ⎪X ⎪ ⎨ X ↑ 2 R1 (a, b) ⇒ X[2 R1 (a, b)] (∩ R− ) ⎪ ⎪  X[2 R1 (a, b)] ↑ 3 R2 (a, b) ⇒ X[2 R1 (a, b), 3 R2 (a, b)] ⎪ ⎪ ⎪ X ↑ 3 R1 (a, b) ⇒ X[3 R1 (a, b)] ⎪ ⎩ X[3 R1 (a, b)] ↑ 2 R2 (a, b) ⇒ X[3 R1 (a, b), 2 R2 (a, b)] X ↑ (R1 ∩ R2 )(a, b) ⇒ X[(R1 ∩ R2 )(a, b)] and ⎧ ⎪ ⎪ X ↑ R1 (a, b) ⇒ X[R1 (a, b)] ⎪ ⎪ ⎪ ⎪X  ↑ R32 (a, b) ⇒ X[R2 3(a, b)] ⎨ X ↑ R1 (a, b) ⇒ X[ R1 (a, b)] 3 R1 (a, b)] ↑ 2 R2 (a, b) ⇒ X[3 R1 (a, b), 2 R2 (a, b)] X[ (∪ R− ) ⎪ ⎪ ⎪ ⎪ X ↑ 2 R1 (a, b) ⇒ X[2 R1 (a, b)] ⎪ ⎪ ⎩ X[2 R1 (a, b)] ↑ 3 R2 (a, b) ⇒ X[2 R1 (a, b), 3 R2 (a, b)] X ↑ (R1 ∪ R2 )(a, b) ⇒ X[(R1 ∪ R2 )(a, b)]

9.4 3/22 -Multisequents

305

⎧ ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ X[Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨ X ↑ 1  Q 1 (a, b) ⇒ X[1  Q 1 (a, b)] Q− 1 1 (∪ ) ⎪ ⎪  X[  Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[  Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎩ X[Q 1 (a, b)] ↑ 1  Q 2 (a, b) ⇒ X[Q 1 (a, b), 1  Q 2 (a, b)] X ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ X[(Q 1 ∪ Q 2 )(a, b)] and ⎧ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎪ ⎪ X[P1 (a, b)] ↑ P2 (a, b) ⇒ X[P1 (a, b), P2 (a, b)] ⎪ ⎪ ⎨ X ↑ 1  P1 (a, b) ⇒ X[1  P1 (a, b)] 1 1 P− (∪ ) ⎪ ⎪  X[  P1 (a, b)] ↑ P2 (a, b) ⇒ X[  P1 (a, b), P2 (a, b)] ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎩ X[P1 (a, b)] ↑ 1  P2 (a, b) ⇒ X[P1 (a, b), 1  P2 (a, b)] X↑ (P1 ∪ P2 )(a, b) ⇒ X[(P1 ∪ P2 )(a, b)] X ↑ R1 (a, b) ⇒ X[R1 (a, b)] (∪ R− ) X ↑ R2 (a, b) ⇒ X[R2 (a, b)] X ↑ (R1 ∪ R2 )(a, b) ⇒ X[(R1 ∪ R2 )(a, b)] and

and

⎡

X ↑ R(a, d1 ) ⇒ X[R(a, d1 )] ⎢ X ↑ R(d1 , d2 ) ⇒ X[R(d1 , d2 )] ⎢ (∗ R− ) ⎣ · · · X ↑ R(dn , b) ⇒ X[R(dn , b)] ∗ ∗ X ⎧ ↑ R (a, b) ⇒ X[R (a, b)] X ↑ Q(a, c1 ) ⇒ X[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 1  Q(a, c1 ) ⇒ X[1  Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ X ↑ Q(c1 , c2 ) ⇒ X[Q(c1 , c2 )] X ↑ 1  Q(c1 , c2 ) ⇒ X[1  Q(c1 , c2 )] (∗ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ X ↑ Q(cn , b) ⇒ X[Q(cn , b)] ⎪ ⎪ ⎩ X ↑ 1  Q(cn , b) ⇒ X[1  Q(cn , b)] X ↑ Q ∗ (a, b) ⇒ X[Q ∗ (a, b)]

306

9 Role R-Calculus for B22 -Valued DL

⎧ ⎪ ⎪ X ↑ 1P(a, c1 ) ⇒ X[P(a,1 c1 )] ⎪ ⎪ X ↑  P(a, c1 ) ⇒ X[  P(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ X ↑ P(c1 , c2 ) ⇒ X[P(c1 , c2 )] X ↑ 1  P(c1 , c2 ) ⇒ X[1  P(c1 , c2 )] (∗ P− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ X ↑ P(cn , b) ⇒ X[P(cn , b)] ⎪ ⎪ ⎩ X ↑ 1  P(cn , b) ⇒ X[1  P(cn , b)] X⎧ ↑ P ∗ (a, b) ⇒ X[P ∗ (a, b)] X ↑ R(a, c1 ) ⇒ X[R(a, c1 )] ⎪ ⎪ ⎨ X ↑ R(c1 , c2 ) ⇒ X[R(c1 , c2 )] · ·· (∗ R− ) ⎪ ⎪ ⎩ X ↑ R(cn , b) ⇒ X[R(cn , b)] X ↑ (R ∗ )(a, b) ⇒ X[(R ∗ )(a, b)] and

⎧ X ↑ R(a, c1 ) ⇒ X[R(a, c1 )] ⎪ ⎪ ⎨ X ↑ R(c1 , c2 ) ⇒ X[R(c1 , c2 )] · ·· (◦ R− ) ⎪ ⎪ ⎩ X ↑ R(cn , b) ⇒ X[R(cn , b)] ◦ X ⎧ ↑ R (a, b) ⇒ X X ↑ Q(a, c1 ) ⇒ X[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 1 Q(a, c1 ) ⇒ X[1 Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ X ↑ Q(c1 , c2 ) ⇒ X[Q(c1 , c2 )] X ↑ 1 Q(c1 , c2 ) ⇒ X[1 Q(c1 , c2 )] (◦ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ X ↑ Q(cn , b) ⇒ X[Q(cn , b)] ⎪ ⎪ ⎩ X ↑ 1 Q(cn , b) ⇒ X[1 Q(cn , b)] X ↑ Q ◦ (a, b) ⇒ X[Q ◦ (a, b)]

and

⎧ X ↑ P(a, c1 ) ⇒ X[P(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 1 P(a, c1 ) ⇒ X[1 P(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ X ↑ P(c1 , c2 ) ⇒ X[P(c1 , c2 )] X ↑ 1 P(c1 , c2 ) ⇒ X[1 P(c1 , c2 )] (◦ P− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ X ↑ P(cn , b) ⇒ X[P(cn , b)] ⎪ ⎪ ⎩ X ↑ 1 P(cn , b) ⇒ X[1 P(cn , b)] X⎡↑ P ◦ (a, b) ⇒ X[P ◦ (a, b)] X ↑ R(a, d1 ) ⇒ X[R(a, d1 )] ⎢ X ↑ R(d1 , d2 ) ⇒ X[R(d1 , d2 )] ⎢ (◦ R− ) ⎣ · · · X ↑ R(dn , b) ⇒ X[R(dn , b)] X ↑ (R ◦ )(a, b) ⇒ X[(R ◦ )(a, b)]

9.5 4/22 -Multisequents

307

Definition 9.4.3 Given a 3/22 -multisequent || and statements R(a, b) ∈ , Q(a  , b ) ∈  and P(a  , b ) ∈ , a 3/22 -reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )] is provable in t⊥ t⊥ 2 R3/2 2 , denoted by 3/22 δ, if there is a sequence {δ1 , ..., δn } of 3/2 -reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the t⊥ previous reductions by one of the deduction rules in R3/2 2 . Theorem 9.4.4 (Soundness and completeness theorem) For any multisequent X = ||, any triple (R(a, b), Q(a  , b ), P(a  , b )) of statements, and a δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒ [R  (a, b)]| 3/22 -reduction      [Q (a , b )]|[P (a , b )] such that one of R  (a, b), Q  (a  , b ), P  (a  , b ) is not the empty string, t⊥ t⊥ 3/22 δ iff |=3/22 δ.  Theorem 9.4.5 (Soundness and incompleteness theorem) For any multisequent X = ||, any triple (R(a, b), Q(a  , b ), P(a  , b )) of statements, and a 3/22 reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒ ||, t⊥ t⊥ 3/22 δ implies |=3/22 δ

and

t⊥ |=t⊥ 3/22 δ may not imply 3/22 δ.



9.5 4/22 -Multisequents A 4/22 -multisequent ||| is M= -valid, denoted by |= |||, if for any interpretation I, ⎧ I (R(a, b)) = t for some R(a, b) ∈ , ⎪ ⎪ ⎨ I (Q(a, b)) =  for some Q(a, b) ∈ , I (P(a, b)) =⊥ for some P(a, b) ∈ , ⎪ ⎪ ⎩ I (O(a, b)) = f for some O(a, b) ∈ .

9 Role R-Calculus for B22 -Valued DL

308

= 9.5.1 Deduction System M4/2 2 = Deduction system M4/2 2 contains the following axiom and deduction rules: • Axiom:  ∩  ∩  ∩  = ∅ (A= 4/22 ) |||

where , , ,  are sets of atoms. • Deduction rules for unary logical connective ¬: (∼1R ) (∼1P ) (∼2R ) (∼2P ) ( R ) ( P )

|, R(a, b)|| , ∼1 R(a, b)||| |||, P(a, b) ||, ∼1 P(a, b)| ||, R(a, b)| , ∼2 R(a, b)||| , P(a, b)||| ||, ∼2 P(a, b)| |||, R(a, b) , R(a, b)||| |, P(a, b)|| ||, P(a, b)|

(∼1Q ) (∼1O ) (∼2Q ) (∼2O ) ( Q ) ( O )

, Q(a, b)||| |, ∼1 Q(a, b)|| ||, O(a, b)| |||, ∼1 O(a, b) |||, Q(a, b) |, ∼2 Q(a, b)|| |, O(a, b)|| |||, ∼2 O(a, b) ||, Q(a, b)| |, Q(a, b)|| , O(a, b)||| |||, O(a, b)

• Deduction rules for binary logical connective ∩: ⎧ |, Q 1 (a, b)|| ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b)|| ⎪ ⎪  ⎨ , Q 1 (a, b)||| , R1 (a, b)||| (∩ R ) , R2 (a, b)||| (∩ Q ) ⎪ ⎪  |, Q 2 (a, b)|| ⎪ ⎪ |, Q 1 (a, b)|| , (R1 ∩ R2 )(a, b)||| ⎪ ⎪ ⎩ , Q 2 (a, b)||| |, (Q 1 ∩ Q 2 )(a, b)|| ⎧ ⎧ |||, O1 (a, b) ||, P (a, b)| ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a, b)| ||, P |||, O2 (a, b) ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎨ , P1 (a, b)||| |, O1 (a, b)|| (∩ P ) ⎪ (∩ O ) ⎪ ⎪ ⎪  ||, O2 (a, b),   ||, P2 (a, b)| ⎪ ⎪ ⎪ ⎪ (a, b)| ||, P ||, O1 (a, b)| ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎩ , P2 (a, b)||| |, O2 (a, b)|| ||, (P1 ∩ P2 )(a, b)| |||, (O1 ∩ O2 )(a, b)

9.5 4/22 -Multisequents

• Deduction rules for binary logical connective ∪: ⎧ ⎧ , R1 (a, b)||| |, Q 1 (a, b)|| ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a, b)||| , R |, Q 2 (a, b)|| ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎨ ||, R1 (a, b)| |||, O1 (a, b) (∪ R ) ⎪ (∪ Q ) ⎪ ⎪ ⎪  |, R2 (a, b)||  |, Q 2 (a, b)|| ⎪ ⎪ ⎪ ⎪ (a, b)|| |, R |, Q 1 (a, b)|| ⎪ ⎪ 1 ⎪ ⎪ ⎩ ⎩ ||, R2 (a, b)| |||, O2 (a, b) , |, (Q 1 ∪ Q 2 )(a, b)|| ⎧ (R1 ∪ R2 )(a, b)||| ||, P (a, b)| ⎪ 1 ⎪ ⎪ ⎪ ||, P2 (a, b)| ⎪ ⎪  ⎨ |||, P1 (a, b) |||, O1 (a, b) O (a, b)| ||, P |||, O2 (a, b) ) (∪ P ) ⎪ (∪ 2 ⎪  ⎪ ⎪ (a, b)| ||, P |||, (O1 ∪ O2 )(a, b) ⎪ 1 ⎪ ⎩ |||, P2 (a, b) ||, (P1 ∪ P2 )(a, b)| • Deduction rules for quantifier ∀: ⎧ ⎪ ⎪ |, Q(a, c1 )|| ⎪ ⎪ ⎡ ⎪ ⎪ |||, Q(a, c1 ) , R(a, d1 )||| ⎪ ⎪ ⎨ |, Q(c1 , c2 )|| ⎢ , R(d1 , d2 )||| ⎢ |||, Q(c1 , c2 ) R ⎣··· Q (∗ ) (∗ ) ⎪ ⎪ · ·· ⎪ ⎪ , R(dn , b)||| ⎪ ⎪ |, Q(cn , b)|| ⎪ ∗ ⎪ , R (a, b)||| ⎩ |||, Q(cn , b) |, Q ∗ (a, b)|| ⎧ ⎪ ⎪ ||, P(a, c1 )| ⎪ ⎪ ⎧ |||, P(a, c1 ) ⎪ ⎪ |||, O(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎨ ||, P(c1 , c2 )| ⎨ |||, O(c1 , c2 ) |||, P(c1 , c2 ) O · ·· (∗ P ) ⎪ (∗ ) ⎪ ⎪ ⎪ ··· ⎪ ⎩ ⎪ |||, O(cn , b) ⎪ ⎪ ⎪ ⎪ ||, P(cn , b)| |||, O ∗ (a, b) ⎩ |||, P(cn , b) ||, P ∗ (a, b)|

309

9 Role R-Calculus for B22 -Valued DL

310

and

⎧ |, Q(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ⎧ , Q(a, c1 )||| ⎪ ⎪ , R(a, c1 )||| ⎪ ⎪ ⎪ ⎪ ⎨ |, Q(c1 , c2 )|| ⎨ , R(c1 , c2 )||| , Q(c1 , c2 )||| ··· (◦ R ) ⎪ (◦ Q ) ⎪ ⎪ ⎪ ··· ⎪ ⎩ ⎪ , R(cn , b)||| ⎪ ⎪ |, Q(cn , b)|| ⎪ ⎪ , R ◦ (a, b)||| ⎩ , Q(cn , b)||| |, Q ◦ (a, b)|| ⎧ ||, P(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎡ ⎪ , P(a, c1 )||| ⎪ |||, O(a, d1 ) ⎪ ⎪ ⎨ ||, P(c1 , c2 )| ⎢ |||, O(d1 , d2 ) ⎢ , P(c1 , c2 )||| (◦ O ) ⎣ · · · (◦ P ) ⎪ ⎪ ⎪··· ⎪ |||, O(dn , b) ⎪ ⎪ ⎪ ||, P(cn , b)| ⎪ |||, O ◦ (a, b) ⎩ , P(cn , b)||| ||, P ◦ (a, b)|

where di is a constant and ci is a new constant. = Definition 9.5.1 A 4/22 -multisequent ||| is provable in M4/2 2 , denoted by = 4/22 |||, if there is a sequence

{1 |1 |1 |1 , ..., n |n |n |n } of 4/22 -multisequents such that n |n |n |n = |||, and for each 1 ≤ i ≤ n, i |i |i |i is either an axiom or deduced from the previous 4/22 -multisequents = by one of the deduction rules in M4/2 2. Theorem 9.5.2 (Soundness and completeness theorem) For any 4/22 -multisequent |||, = |== 4/22 ||| iff 4/22 |||. 

= 9.5.2 R-Calculus R4/2 2

Given statements R(a, b) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈ , O(a  , b ) ∈ , a 4/22 -reduction δ = ||| ↑ (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )]|[O  (a  , b )] = = is R4/2 2 -valid, denoted by |=4/22 δ, if

9.5 4/22 -Multisequents

311



[R(a, b)]||| R(a, b) if |== 4/22 λ otherwise  Q(a  , b ) if |== [R  (a, b)]|[Q(a  , b )]||    4/22 Q (a , b ) = λ otherwise    [R  (a, b)]|[Q  (a  , b )]|[P(a  , b )]| P(a , b ) if |== 4/22 P  (a  , b ) = λ otherwise   , b ) if |== O(a [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )]|[O(a  , b )] 4/22 O  (a  , b ) = λ otherwise. R  (a, b) =

Let Y = ||| and Y[R(a, b)] = [R(a, b)]||| Y[Q(a, b)] = |[Q(a, b)]|| Y[P(a, b)] = ||[P(a, b)]| Y[O(a, b)] = |||[O(a, b)]

Y[1 R(a, b)] = [R(a, b)]||| Y[2 R(a, b)] = |[R(a, b)]|| Y[3 R(a, b)] = ||[R(a, b)]| Y[4 R(a, b)] = |||[R(a, b)]

= R-calculus R4/2 2 consists of the following axioms and deduction rules: • Axioms:

[r (a, b)] ∩  ∩  ∩  = ∅ ||||r (a, b) ⇒ [r (a, b)]|||   Q=−  ∩ [q(a , b )] ∩  ∩   = ∅ (A4/2 2 ) ||||q(a  , b ) ⇒ |[q(a  , b )]||  ∩  ∩ [ p(a  , b )] ∩  = ∅ P=− (A4/2 2 ) |||| p(a  , b ) ⇒ ||[ p(a  , b )]|   O=−  ∩  ∩  ∩ [o(a , b )]  = ∅ (A4/2 2 )   ||||o(a , b ) ⇒ |||[o(a  , b )] R=− (A4/2 2 )

where , , ,  are sets of atoms, and r (a, b), q(a  , b ), p(a  , b ), o(a  , b ) are atoms. • Deduction rules: (∼1R− ) (∼1P− ) (∼2R− ) (∼2P− ) ( R− ) ( P− )

Y ↑ 2 R(a, b) ⇒ Y[2 R(a, b)] Y ↑∼1 R(a, b) ⇒ Y[∼1 R(a, b)] Y ↑ 4 P(a, b) ⇒ Y[4 P(a, b)] Y ↑∼1 P(a, b) ⇒ Y[∼1 P(a, b)] Y ↑ 3 R(a, b) ⇒ Y[3 R(a, b)] Y ↑∼2 R(a, b) ⇒ Y[∼2 R(a, b)] Y ↑ 1 P(a, b) ⇒ Y[1 P(a, b)] Y ↑∼2 P(a, b) ⇒ Y[∼2 P(a, b)] Y ↑ 4 R(a, b) ⇒ Y[4 R(a, b)] Y ↑ R(a, b) ⇒ Y[R(a, b)] Y ↑ 2 P(a, b) ⇒ Y[2 P(a, b)] Y ↑ P(a, b) ⇒ Y[P(a, b)]

Q−

(∼1

)

(∼1O− ) Q−

(∼2

)

(∼2O− ) ( Q− ) ( O− )

Y ↑ 1 Q(a, b) ⇒ Y[1 Q(a, b)] Y ↑∼1 Q(a, b) ⇒ Y[∼1 Q(a, b)] Y ↑ 3 O(a, b) ⇒ Y[3 O(a, b)] Y ↑∼1 O(a, b) ⇒ Y[∼1 O(a, b)] Y ↑ 4 Q(a, b) ⇒ Y[4 Q(a, b)] Y ↑∼2 Q(a, b) ⇒ Y[∼2 Q(a, b)] Y ↑ 2 O(a, b) ⇒ Y[2 O(a, b)] Y ↑∼2 O(a, b) ⇒ Y[∼2 O(a, b)] Y ↑ 3 Q(a, b) ⇒ Y[3 Q(a, b)] Y ↑ Q(a, b) ⇒ Y[Q(a, b)] Y ↑ 1 O(a, b) ⇒ Y[1 O(a, b)] Y ↑ O(a, b) ⇒ Y[O(a, b)]

9 Role R-Calculus for B22 -Valued DL

312

and



Y ↑ R1 (a, b) ⇒ Y[R1 (a, b)] Y[R1 (a, b)] ↑ R2 (a, b) ⇒ Y[R1 (a, b), R2 (a, b)] Y ⎧↑  (R1 ∩ R2 )(a, b) ⇒ Y[(R1 ∩ R2 )(a, b)] Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  Y[Q 11 (a, b)] ↑ Q 2 (a,1b) ⇒ Y[Q 1 (a, b), Q 2 (a, b)] ⎨ Y ↑ Q 1 (a, b) ⇒ Y[ Q 1 (a, b)] (∩ Q− ) ⎪ ⎪  Y[1 Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ Y[1 Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎪ Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎪ ⎩ Y[Q 1 (a, b)] ↑ 1 Q 2 (a, b) ⇒ Y[Q 1 (a, b), 1 Q 2 (a, b)] Y ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ Y[(Q 1 ∩ Q 2 )(a, b)]

(∩ R− )

and

⎧ Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎪ ⎪ ⎪ ⎪ Y[P1 (a, b)] ↑ P2 (a, b) ⇒ Y[P1 (a, b), P2 (a, b)] ⎪ ⎪ ⎨ Y ↑ 1 P1 (a, b) ⇒ Y[1 P1 (a, b)] 1 1 (∩ P− ) ⎪ ⎪  Y[ P1 (a, b)] ↑ P2 (a, b) ⇒ Y[ P1 (a, b), P2 (a, b)] ⎪ ⎪ Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎪ ⎪ ⎩ Y[P1 (a, b)] ↑ 1 P2 (a, b) ⇒ Y[P1 (a, b), 1 P2 (a, b)] Y ⎧ ↑ (P1 ∩ P2 )(a, b) ⇒ Y[(P1 ∩ P2 )(a, b)] Y ↑ O1 (a, b) ⇒ Y[O1 (a, b)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Y  ↑ O22 (a, b) ⇒ Y[O22(a, b)] ⎨ Y ↑ O1 (a, b) ⇒ Y[ O1 (a, b)] 2 O1 (a, b)] ↑ 3 O2 (a, b) ⇒ Y[2 O1 (a, b), 3 O2 (a, b)] Y[ (∩ O− ) ⎪ ⎪ ⎪ ⎪ Y ↑ 3 O1 (a, b) ⇒ Y[3 O1 (a, b)] ⎪ ⎪ ⎩ Y[3 O1 (a, b)] ↑ 2 O2 (a, b) ⇒ Y[3 O1 (a, b), 2 O2 (a, b)] Y ↑ (O1 ∩ O2 )(a, b) ⇒ Y[(O1 ∩ O2 )(a, b)]

and

⎧ Y ↑ R1 (a, b) ⇒ Y[R1 (a, b)] ⎪ ⎪ ⎪ ⎪ Y ↑ R2 (a, b) ⇒ Y[R2 (a, b)] ⎪ ⎪ ⎨ Y ↑ 3 R1 (a, b) ⇒ Y[3 R1 (a, b)] 3 2 R− b) ⇒ Y[3 R1 (a, b), 2 R2 (a, b)] (∪ ) ⎪ ⎪  Y[ R21 (a, b)] ↑ R2 (a, ⎪ 2 ⎪ Y ↑ R1 (a, b) ⇒ Y[ R1 (a, b)] ⎪ ⎪ ⎩ Y[2 R1 (a, b)] ↑ 3 R2 (a, b) ⇒ Y[2 R1 (a, b), 3 R2 (a, b)] Y ⎧↑  (R1 ∪ R2 )(a, b) ⇒ Y[(R1 ∪ R2 )(a, b)] Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪ Y[Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ Y[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ ⎨ Y ↑ 4 Q 1 (a, b) ⇒ Y[4 Q 1 (a, b)] 4 4 (∪ Q− ) ⎪ ⎪  Y[ Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ Y[ Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎪ ⎪ ⎩ Y[Q 1 (a, b)] ↑ 4 Q 2 (a, b) ⇒ Y[Q 1 (a, b), 4 Q 2 (a, b)] Y ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ Y[(Q 1 ∪ Q 2 )(a, b)]

and

9.5 4/22 -Multisequents

313

⎧ ⎪ ⎪ Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎪ ⎪ Y[P1 (a, b)] ↑ P2 (a, b) ⇒ Y[P1 (a, b), P2 (a, b)] ⎪ ⎪ ⎨ Y ↑ 4 P1 (a, b) ⇒ Y[4 P1 (a, b)] 4 4 P− (∪ ) ⎪ ⎪  Y[ P1 (a, b)] ↑ P2 (a, b) ⇒ Y[ P1 (a, b), P2 (a, b)] ⎪ ⎪ Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎪ ⎪ ⎩ Y[P1 (a, b)] ↑ 4 P2 (a, b) ⇒ Y[P1 (a, b), 4 P2 (a, b)] Y  ↑ (P1 ∪ P2 )(a, b) ⇒ Y[(P1 ∪ P2 )(a, b)] Y ↑ O1 (a, b) ⇒ Y[O1 (a, b)] (∪ O− ) Y[O1 (a, b)] ↑ O2 (a, b) ⇒ Y[O1 (a, b), O2 (a, b)] Y ↑ (O1 ∪ O2 )(a, b) ⇒ Y[(O1 ∪ O2 )(a, b)] and

and

and

⎡

Y ↑ R(a, d1 ) ⇒ Y[R(a, d1 )] ⎢ Y ↑ R(d1 , d2 ) ⇒ Y[R(d1 , d2 )] ⎢ (∗ R− ) ⎣ · · · Y ↑ R(dn , b) ⇒ Y[R(dn , b)] ∗ ∗ Y ⎧ ↑ [R (a, b)] ⇒ Y[R (a, b)] Y ↑ R(a, c1 ) ⇒ Y[R(a, c1 )] ⎪ ⎪ ⎨ Y ↑ R(c1 , c2 ) ⇒ Y[R(c1 , c2 )] · ·· (◦ R− ) ⎪ ⎪ ⎩ Y ↑ R(cn , b) ⇒ Y[R(cn , b)] Y ↑ R ◦ (a, b) ⇒ Y[R ◦ (a, b)] ⎧ Y ↑ Q(a, c1 ) ⇒ Y[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ Y ↑ 4 Q(a, c1 ) ⇒ Y[4 Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ Y ↑ Q(c1 , c2 ) ⇒ Y[Q(c1 , c2 )] Y ↑ 4 Q(c1 , c2 ) ⇒ Y[4 Q(c1 , c2 )] (∗ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ Y ↑ Q(cn , b) ⇒ Y[Q(cn , b)] ⎪ ⎪ ⎩ Y ↑ 4 Q(cn , b) ⇒ Y[4 Q(cn , b)] Y ↑ [Q ∗ (a, b)] ⇒ Y[Q ∗ (a, b)] ⎧ Y ↑ Q(a, c1 ) ⇒ Y[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎪ Y ↑ 1 Q(a, c1 ) ⇒ Y[1 Q(a, c1 )] ⎪ ⎪ ⎪ ⎨ Y ↑ Q(c1 , c2 ) ⇒ Y[Q(c1 , c2 )] Y ↑ 1 Q(c1 , c2 ) ⇒ Y[1 Q(c1 , c2 )] (◦ Q− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ Y ↑ Q(cn , b) ⇒ Y[Q(cn , b)] ⎪ ⎪ ⎩ Y ↑ 1 Q(cn , b) ⇒ Y[1 Q(cn , b)] Y ↑ [Q ◦ (a, b)] ⇒ Y[Q ◦ (a, b)]

9 Role R-Calculus for B22 -Valued DL

314

⎧ c1 )] ⎪ ⎪ Y ↑ 4P(a, c1 ) ⇒ Y[P(a, ⎪ 4 ⎪ P(a, c ) ⇒ Y[ P(a, c1 )] Y ↑ ⎪ 1 ⎪ ⎪ ⎪ ⎨ Y ↑ P(c1 , c2 ) ⇒ Y[P(c1 , c2 )] Y ↑ 4 P(c1 , c2 ) ⇒ Y[4 P(c1 , c2 )] (∗ P− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ Y ↑ P(cn , b) ⇒ Y[P(cn , b)] ⎪ ⎪ ⎩ Y ↑ 4 P(cn , b) ⇒ Y[4 P(cn , b)] Y ↑ [P ∗ (a, b)] ⇒ Y[P ∗ (a, b)] ⎧ Y ↑ P(a, c1 ) ⇒ Y[P(a, c1 )] ⎪ ⎪ ⎪ ⎪ Y ↑ 1 P(a, c1 ) ⇒ Y[1 P(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ Y ↑ P(c1 , c2 ) ⇒ Y[P(c1 , c2 )] Y ↑ 1 P(c1 , c2 ) ⇒ Y[1 P(c1 , c2 )] P− (◦ ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ Y ↑ P(cn , b) ⇒ Y[P(cn , b)] ⎪ ⎪ ⎩ Y ↑ 1 P(cn , b) ⇒ Y[1 P(cn , b)] Y ↑ [P ◦ (a, b)] ⇒ Y[P ◦ (a, b)] and

⎧ Y ↑ O(a, c1 ) ⇒ Y[O(a, c1 )] ⎪ ⎪ ⎨ Y ↑ O(c1 , c2 ) ⇒ Y[O(c1 , c2 )] · ·· (∗ O− ) ⎪ ⎪ ⎩ Y ↑ O(cn , b) ⇒ Y[O(cn , b)] Y ↑ [O ∗ (a, b)] ⇒ Y[O ∗ (a, b)] ⎡ Y ↑ O(a, d1 ) ⇒ Y[O(a, d1 )] ⎢ Y ↑ O(d1 , d2 ) ⇒ Y[O(d1 , d2 )] ⎢ (◦ Q− ) ⎣ · · · Y ↑ O(dn , b) ⇒ Y[O(dn , b)] Y ↑ O ◦ (a, b) ⇒ Y[O ◦ (a, b)]

Definition 9.5.3 Let Y = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )), Y =   [R(a, b)]| [Q(a  , b )]|  [P(a  , b )]| [O(a  , b )]. = = A 4/22 -reduction δ = Y ↑ Y ⇒ Y is provable in R4/2 2 , denoted by 4/22 δ, if there is 2 a sequence {δ1 , ..., δn } of 4/2 -reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 4/22 -reductions by one of the = deduction rules in R4/2 2.

Theorem 9.5.4 (Soundness and completeness theorem) For any 4/22 -reduction δ = Y ↑ Y ⇒ Y such that one of R  (a, b), Q  (a  , b ), P  (a  , b ), O  (a  , b ) is not the empty string, = = 4/22 δ iff |=4/22 δ. 

9.5 4/22 -Multisequents

315

Theorem 9.5.5 (Soundness and incompleteness theorem) For any 4/22 -reduction δ = Y ↑ Y ⇒ Y, = = 4/22 δ implies |=4/22 δ, and

= |== 4/22 δ may not imply |=4/22 δ.



4/22

9.5.3 Incomplete Deduction System L=

4/22

2

4/2 A 4/22 -multisequent ||| is L= -valid, denoted by |== is an interpretation I such that

|||, if there

⎡

I (R(a, b)) = t for each R(a, b) ∈ , ⎢ I (Q(a, b)) =  for each Q(a, b) ∈ , ⎢ ⎣ I (P(a, b)) =⊥ for each P(a, b) ∈ , I (O(a, b)) = f for each O(a, b) ∈ . 2

4/2 Deduction system L= contains the following axiom and deduction rules: • Axiom: 4/22  ∩  ∩  ∩  = ∅ (A= ) |||

where , , ,  are sets of atoms. • Deduction rules for unary logical connective ¬: (∼1R ) (∼1P ) (∼2R ) (∼2P ) ( R ) ( P )

|, R(a, b)|| , ∼1 R(a, b)||| |||, P(a, b) ||, ∼1 P(a, b)| ||, R(a, b)| , ∼2 R(a, b)||| , P(a, b)||| ||, ∼2 P(a, b)| |||, R(a, b) , R(a, b)||| |, P(a, b)|| ||, P(a, b)|

(∼1Q ) (∼1O ) (∼2Q ) (∼2O ) ( Q ) ( O )

, Q(a, b)||| |, ∼1 Q(a, b)|| ||, O(a, b)| |||, ∼1 O(a, b) |||, Q(a, b) |, ∼2 Q(a, b)|| |, O(a, b)|| |||, ∼2 O(a, b) ||, Q(a, b)| |, Q(a, b)|| , O(a, b)||| |||, O(a, b)

9 Role R-Calculus for B22 -Valued DL

316

• Deduction rules for binary logical connective ∩: ⎡

|, Q 1 (a, b)|| ⎢ |, Q 2 (a, b)|| ⎢  ⎢ , Q 1 (a, b)||| , R1 (a, b)||| ⎢ R Q (∩ ) , R2 (a, b)||| (∩ ) ⎢ ⎢  |, Q 2 (a, b)|| ⎣ |, Q 1 (a, b)|| , (R1 ∩ R2 )(a, b)||| , Q 2 (a, b)||| |, (Q 1 ∩ Q 2 )(a, b)|| ⎡ ⎡ |||, O1 (a, b) ||, P1 (a, b)| ⎢ ||, P2 (a, b)| ⎢ |||, O2 (a, b) ⎢ ⎢ ⎢ , P1 (a, b)||| ⎢ |, O1 (a, b)|| ⎢ ⎢ P ⎢ ||, P (a, b)| O ⎢ ||, O (a, b)| (∩ ) ⎢  (∩ ) ⎢  2 2 ⎣ ||, P1 (a, b)| ⎣ ||, O1 (a, b)| , P2 (a, b)||| |, O2 (a, b)|| ||, (P1 ∩ P2 )(a, b)| |||, (O1 ∩ O2 )(a, b) • Deduction rules for binary logical connective ∪: ⎡ |, Q 1 (a, b)|| , R1 (a, b)||| ⎢ , R2 (a, b)||| ⎢ |, Q 2 (a, b)|| ⎢ ⎢ ⎢ ||, R1 (a, b)| ⎢ |||, O1 (a, b) ⎢ ⎢ R ⎢ |, R (a, b)|| Q ⎢ |, Q (a, b)|| (∪ ) ⎢  (∪ ) ⎢  2 2 ⎣ |, R1 (a, b)|| ⎣ |, Q 1 (a, b)|| ||, R2 (a, b)| |||, O2 (a, b) , (R ∪ R )(a, b)||| |, (Q 1 ∪ Q 2 )(a, b)|| 1 2 ⎡ ||, P1 (a, b)| ⎢ ||, P2 (a, b)| ⎢  ⎢ |||, P1 (a, b) |||, O1 (a, b) ⎢ O (a, b)| ||, P |||, O2 (a, b) ) (∪ (∪ P ) ⎢ 2 ⎢ ⎣ ||, P1 (a, b)| |||(O1 ∪ O2 )(a, b),  |||, P2 (a, b) ||, (P1 ∪ P2 )(a, b)| ⎡

• Deduction rules for quantifier ∀: ⎡

⎧ , R(a, c1 )||| ⎪ ⎪ ⎨ , R(c1 , c2 )||| ··· (∗ R ) ⎪ (∗ Q ) ⎪ ⎩ , R(cn , b)||| , R ∗ (a, b)|||

|, Q(a, d1 )|| ⎢ |||, Q(a, d1 ) ⎢ ⎢ |, Q(d1 , d2 )|| ⎢ ⎢ |||, Q(d1 , d2 ) ⎢ ⎢··· ⎢ ⎣ |, Q(dn , b)|| |||, Q(dn , b) |, Q ∗ (a, b)||

9.5 4/22 -Multisequents

317

⎡

||, P(a, d1 )| ⎢ |||, P(a, d1 ) ⎢ ⎢ ||, P(d1 , d2 )| ⎢ ⎢ |||, P(d1 , d2 ) O (∗ P ) ⎢ (∗ ) ⎢··· ⎢ ⎣ ||, P(dn , b)| |||, P(dn , b) ||, P ∗ (a, b)| and

⎡

|||, O(a, d1 ) ⎢ |||, O(d1 , d2 ) ⎢ ⎣··· |||, O(dn , b) |||, O ∗ (a, b)

⎡

|, Q(a, d1 )|| ⎢ , Q(a, d1 )||| ⎢ , R(a, d1 )||| ⎢ |, Q(d1 , d2 )|| ⎢ ⎢ , R(d1 , d2 )||| ⎢ ⎢ R ⎣··· Q ⎢ , Q(d1 , d2 )||| (◦ ) (◦ ) ⎢ ⎢··· , R(dn , b)||| ⎣ |, Q(dn , b)|| , R ◦ (a, b)||| , Q(dn , b)||| |, Q ◦ (a, b)|| ⎡ ||, P(a, d1 )| ⎢ , P(a, d1 )||| ⎧ ⎢ |||, O(a, c1 ) ⎪ ⎢ ||, P(d1 , d2 )| ⎪ ⎨ ⎢ |||, O(c1 , c2 ) ⎢ P ⎢ , P(d1 , d2 )||| O (◦ ) ⎢ (◦ ) ⎪ ⎪··· ⎩ ⎢··· |||, O(cn , b) ⎣ ||, P(dn , b)| |||, O ◦ (a, b) , P(dn , b)||| ||, P ◦ (a, b)| ⎡

where di is a constant and ci is a new constant. 2

4/2 Definition 9.5.6 A 4/22 -multisequent ||| is provable in L= , denoted by 4/22 = |||, if there is a sequence {1 |1 |1 |1 , ..., n |n |n |n } of 4/22 multisequents such that n |n |n |n = |||, and for each 1 ≤ i ≤ n, i | i |i |i is either an axiom or deduced from the previous 4/22 -multisequents by 4/22 . one of the deduction rules in L=

Theorem 9.5.7 (Soundness and incompleteness theorem) For any 4/22 -multisequent |||, 4/22 4/22 = ||| implies |== |||, and

4/22

|==

4/22

||| may not imply =

|||. 

9 Role R-Calculus for B22 -Valued DL

318

4/22

9.5.4 R-Calculus Q= A 4/22 -reduction

δ = ||| ↑ (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ⇒ , R  (a, b)|, Q  (a  , b )|, P  (a  , b )|, O  (a  , b ) 4/22

2

is Q4/2 = -valid, denoted by |==  R  (a, b)

=

δ, if

4/22

R(a, b) if |== , R(a, b)||| λ otherwise  4/22

Q(a  , b ) if |== , R  (a, b)|, Q(a  , b )|| otherwise λ

Q  (a  , b ) = P  (a  , b ) =

O  (a  , b ) =

4/22

P(a  , b ) if |== , R  (a, b)|, Q  (a  , b )|, P(a  , b )| λ otherwise  4/22

O(a  , b ) if |== , R  (a, b)|, Q  (a  , b )|, P  (a  , b )|, O(a  , b ) λ otherwise.

Let Y = , R(a, b)|, Q(a  , b )|, P(a  , b )|, O(a  , b ) and Y[R(a, b)] = [R(a, b)]||| Y[Q(a, b)] = |[Q(a, b)]|| Y[P(a, b)] = ||[P(a, b)]| Y[O(a, b)] = |||[O(a, b)]

Y[1 R(a, b)] = [R(a, b)]||| Y[2 R(a, b)] = |[R(a, b)]|| Y[3 R(a, b)] = ||[R(a, b)]| Y[4 R(a, b)] = |||[R(a, b)]

2

R-calculus Q4/2 consists of the following axioms and deduction rules: = • Axioms: r (a, b) ∈  ∩  ∩  ||| ↑ r (a, b) ⇒ [r (a, b)]||| q(a  , b ) ∈  ∩  ∩  4/22 (A Q=− ) ||| ↑ q(a  , b ) ⇒ |[q(a  , b )]|| 2 p(a  , b ) ∈  ∩  ∩  = ∅ 4/2 (A P=− ) ||| ↑ p(a  , b ) ⇒ ||[ p(a  , b )]| 2 o(a  , b ) ∈  ∩  ∩  4/2 (A O=− ) ||| ↑ o(a  , b ) ⇒ |||[o(a  , b )] 4/22

(A R=− )

where , , ,  are sets of atoms, and r (a, b), q(a  , b ), p(a  , b ), o(a  , b ) are atoms.

9.5 4/22 -Multisequents

319

• Deduction rules: (∼1R− ) (∼1P− ) (∼2R− ) (∼2P− ) ( R− ) ( P− )

and

Y ↑ 2 R(a, b) ⇒ Y[2 R(a, b)] Y ↑∼1 R(a, b) ⇒ Y[∼1 R(a, b)] Y ↑ 4 P(a, b) ⇒ Y[4 P(a, b)] Y ↑∼1 P(a, b) ⇒ Y[∼1 P(a, b)] Y ↑ 3 R(a, b) ⇒ Y[3 R(a, b)] Y ↑∼2 R(a, b) ⇒ Y[∼2 R(a, b)] Y ↑ 1 P(a, b) ⇒ Y[1 P(a, b)] Y ↑∼2 P(a, b) ⇒ Y[∼2 P(a, b)] Y ↑ 4 R(a, b) ⇒ Y[4 R(a, b)] Y ↑ R(a, b) ⇒ Y[R(a, b)] Y ↑ 2 P(a, b) ⇒ Y[2 P(a, b)] Y ↑ P(a, b) ⇒ Y[P(a, b)]

Q−

(∼1

)

(∼1O− ) Q−

(∼2

)

(∼2O− ) ( Q− ) ( O− )

Y ↑ 1 Q(a, b) ⇒ Y[1 Q(a, b)] Y ↑∼1 Q(a, b) ⇒ Y[∼1 Q(a, b)] Y ↑ 3 O(a, b) ⇒ Y[3 O(a, b)] Y ↑∼1 O(a, b) ⇒ Y[∼1 O(a, b)] Y ↑ 4 Q(a, b) ⇒ Y[4 Q(a, b)] Y ↑∼2 Q(a, b) ⇒ Y[∼2 Q(a, b)] Y ↑ 2 O(a, b) ⇒ Y[2 O(a, b)] Y ↑∼2 O(a, b) ⇒ Y[∼2 O(a, b)] Y ↑ 3 Q(a, b) ⇒ Y[3 Q(a, b)] Y ↑ Q(a, b) ⇒ Y[Q(a, b)] Y ↑ 1 O(a, b) ⇒ Y[1 O(a, b)] Y ↑ O(a, b) ⇒ Y[O(a, b)]



Y ↑ R1 (a, b) ⇒ Y[R1 (a, b)] Y ↑ R2 (a, b) ⇒ Y[R2 (a, b)] Y ↑ (R1 ∩ R2 )(a, b) ⇒ Y[(R1 ∩ R2 )(a, b)] ⎡ Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎢ Y ↑ Q 2 (a, b) ⇒ Y[Q 2 (a, b)] ⎢ ⎢ Y[Z 1 ] ↑ 1 Q 1 (a, b) ⇒ Y[Z 1 , 1 Q 1 (a, b)] ⎢ Q− ⎢ Y[Z ] ↑ Q (a, b) ⇒ Y[Z , Q (a, b)] (∩ ) ⎢  1 2 1 2 ⎣ Y[Z 2 ] ↑ Q 1 (a, b) ⇒ Y[Z 2 , Q 1 (a, b)] Y[Z 2 ] ↑ 1 Q 2 (a, b) ⇒ Y[Z 2 , 1 Q 2 (a, b)] Y ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ Y[(Q 1 ∩ Q 2 )(a, b)]

(∩ R− )

where Z 1 = Q 1 (a, b) ∨ Q 2 (a, b), Z 2 = Z 1 , ∼1 Q 1 (a, b) ∨ Q 2 (a, b), and ⎡

Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎢ Y ↑ P2 (a, b) ⇒ Y[P2 (a, b)] ⎢ ⎢ Y[Z 3 ] ↑ 1 P1 (a, b) ⇒ Y[Z 3 , 1 P1 (a, b)] ⎢ (∩ P− ) ⎢ ⎢  Y[Z 3 ] ↑ P2 (a, b) ⇒ Y[Z 3 , P2 (a, b)] ⎣ Y[Z 4 ] ↑ P1 (a, b) ⇒ Y[Z 4 , P1 (a, b)] Y[Z 4 ] ↑ 1 P2 (a, b) ⇒ Y[Z 4 , 1 P2 (a, b)] Y ⎡ ↑ (P1 ∩ P2 )(a, b) ⇒ Y[(P1 ∩ P2 )(a, b)] Y ↑ O1 (a, b) ⇒ Y[O1 (a, b)] ⎢ Y[O1 (a, b)] ↑ O2 (a, b) ⇒ Y[O1 (a, b), O2 (a, b)] ⎢ ⎢ Y ↑ 2 O1 (a, b) ⇒ Y[Z 5 , 2 O1 (a, b)] ⎢ 3 3 (∩ O− ) ⎢ ⎢  Y[Z 5 ] ↑ 3 O2 (a, b) ⇒ Y[Z 5 , 3 O2 (a, b)] ⎣ Y[Z 6 ] ↑ O1 (a, b) ⇒ Y[Z 6 , O1 (a, b)] Y[Z 6 ] ↑ 2 O2 (a, b) ⇒ Y[Z 6 , 2 O2 (a, b)] Y ↑ (O1 ∩ O2 )(a, b) ⇒ Y[(O1 ∩ O2 )(a, b)]

9 Role R-Calculus for B22 -Valued DL

320

Z3 Z where 4 Z5 Z6

= = = =

P1 (a, b) ∨ P2 (a, b), Z 3 , ∼2 P1 (a, b) ∨ P2 (a, b), and O1 (a, b) ∨ O2 (a, b), Z 5 , ∼2 O1 (a, b)∨ ∼1 O2 (a, b), ⎡

Y ↑ R1 (a, b) ⇒ Y[R1 (a, b)] ⎢ Y[R1 (a, b)] ↑ R2 (a, b) ⇒ Y[R1 (a, b), R2 (a, b)] ⎢ ⎢ Y[Y1 ] ↑ 3 R1 (a, b) ⇒ Y[Y1 , 3 R1 (a, b)] ⎢ R− ⎢ Y[Y ] ↑ 2 R (a, b) ⇒ Y[Y , 2 R (a, b)] (∪ ) ⎢  1 2 1 2 ⎣ Y[Y2 ] ↑ 2 R1 (a, b) ⇒ Y[Y2 , 2 R1 (a, b)] Y[Y2 ] ↑ 3 R2 (a, b) ⇒ Y[Y2 , 3 R2 (a, b)] Y ↑ (R1 ∪ R2 )(a, b) ⇒ Y[(R1 ∪ R2 )(a, b)] ⎡ Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎢ Y ↑ Q 2 (a, b) ⇒ Y[Q 2 (a, b)] ⎢ ⎢ Y[Y3 ] ↑ 4 Q 1 (a, b) ⇒ Y[Y3 , 4 Q 1 (a, b)] ⎢ Q− ⎢ Y[Y ] ↑ Q (a, b) ⇒ Y[Y , Q (a, b)] (∪ ) ⎢  3 2 3 2 ⎣ Y[Y4 ] ↑ Q 1 (a, b) ⇒ Y[Y4 , Q 1 (a, b)] Y[Y4 ] ↑ 4 Q 2 (a, b) ⇒ Y[Y4 , 4 Q 2 (a, b)] Y ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ Y[(Q 1 ∪ Q 2 )(a, b)] Y1 Y where 2 Y3 Y4

= R1 (a, b), R2 (a, b); = Y1 , ∼2 R1 (a, b)∨ ∼1 R2 (a, b), and = Q 1 (a, b), Q 2 (a, b); = Y3 , ∼2 Q 1 (a, b) ∨ Q 2 (a, b), ⎡

Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎢ Y ↑ P2 (a, b) ⇒ Y[P2 (a, b)] ⎢ ⎢ Y[Y5 ] ↑ 4 P1 (a, b) ⇒ Y[Y5 , 4 P1 (a, b)] ⎢ P− ⎢ Y[Y ] ↑ P (a, b) ⇒ Y[Y , P (a, b)] (∪ ) ⎢  5 2 5 2 ⎣ Y[Y6 ] ↑ P1 (a, b) ⇒ Y[Y6 , P1 (a, b)] Y[Y6 ] ↑ 4 P2 (a, b) ⇒ Y[Y6 , 4 P2 (a, b)] Y  ↑ (P1 ∪ P2 )(a, b) ⇒ Y[(P1 ∪ P2 )(a, b)] Y ↑ O1 (a, b) ⇒ Y[O1 (a, b)] (∪ O− ) Y ↑ O2 (a, b) ⇒ Y[O2 (a, b)] Y ↑ (O1 ∪ O2 )(a, b) ⇒ Y[(O1 ∪ O2 )(a, b)] where Y5 = P1 (a, b) ∨ P2 (a, b), Y6 = Y5 , ∼1 P1 (a, b) ∨ P2 (a, b), and ⎧ Y ↑ R(a, c1 ) ⇒ Y[R(a, c1 )] ⎪ ⎪ ⎨ Y ↑ R(c1 , c2 ) ⇒ Y[R(c1 , c2 )] · ·· (∗ R− ) ⎪ ⎪ ⎩ Y ↑ R(cn , b) ⇒ Y[R(cn , b)] Y ↑ [R ∗ (a, b)] ⇒ Y[R ∗ (a, b)]

9.5 4/22 -Multisequents

321

⎡

Y ↑ R(a, d1 ) ⇒ Y[R(a, d1 )] ⎢ Y ↑ R(d1 , d2 ) ⇒ Y[R(d1 , d2 )] ⎢ (◦ R− ) ⎣ · · · Y ↑ R(dn , b) ⇒ Y[R(dn , b)] Y ↑ R ◦ (a, b) ⇒ Y[R ◦ (a, b)] and

⎡

Y ↑ Q(a, d1 ) ⇒ Y[Q(a, d1 )] ⎢ Y ↑ 4 Q(a, d1 ) ⇒ Y[4 Q(a, d1 )] ⎢ ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[Q(d1 , d2 )] ⎢ 4 4 ⎢ Q− ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[ Q(d1 , d2 )] (∗ ) ⎢ ⎢··· ⎣ Y ↑ Q(dn , b) ⇒ Y[Q(dn , b)] Y ↑ 4 Q(dn , b) ⇒ Y[4 Q(dn , b)] Y ↑ [Q ∗ (a, b)] ⇒ Y[Q ∗ (a, b)] ⎡ Y ↑ Q(a, d1 ) ⇒ Y[Q(a, d1 )] ⎢ Y ↑ 1 Q(a, d1 ) ⇒ Y[1 Q(a, d1 )] ⎢ ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[Q(d1 , d2 )] ⎢ 1 1 ⎢ Q− ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[ Q(d1 , d2 )] (◦ ) ⎢ ⎢··· ⎣ Y ↑ Q(dn , b) ⇒ Y[Q(dn , b)] Y ↑ 1 Q(dn , b) ⇒ Y[4 Q(dn , b)] Y ↑ [Q ◦ (a, b)] ⇒ Y[Q ◦ (a, b)]

and ⎡

Y ↑ P(a, d1 ) ⇒ Y[P(a, d1 )] ⎢ Y ↑ 4 P(a, d1 ) ⇒ Y[4 P(a, d1 )] ⎢ ⎢ Y ↑ P(d1 , d2 ) ⇒ Y[P(d1 , d2 )] ⎢ 4 4 ⎢ P− ⎢ Y ↑ P(d1 , d2 ) ⇒ Y[ P(d1 , d2 )] (∗ ) ⎢ ⎢··· ⎣ Y ↑ P(dn , b) ⇒ Y[P(dn , b)] Y ↑ 4 P(dn , b) ⇒ Y[4 P(dn , b)] Y ↑ [P ∗ (a, b)] ⇒ Y[P ∗ (a, b)] ⎡ Y ↑ P(a, d1 ) ⇒ Y[P(a, d1 )] ⎢ Y ↑ 1 P(a, d1 ) ⇒ Y[1 P(a, d1 )] ⎢ ⎢ Y ↑ P(d1 , d2 ) ⇒ Y[P(d1 , d2 )] ⎢ 1 1 ⎢ P− ⎢ Y ↑ P(d1 , d2 ) ⇒ Y[ P(d1 , d2 )] (◦ ) ⎢ ⎢··· ⎣ Y ↑ P(dn , b) ⇒ Y[P(dn , b)] Y ↑ 1 P(dn , b) ⇒ Y[1 P(dn , b)] Y ↑ [P ◦ (a, b)] ⇒ Y[P ◦ (a, b)] and

9 Role R-Calculus for B22 -Valued DL

322

⎡

Y ↑ O(a, d1 ) ⇒ Y[O(a, d1 )] ⎢ Y ↑ O(d1 , d2 ) ⇒ Y[O(d1 , d2 )] ⎢ (∗ O− ) ⎣ · · · Y ↑ O(dn , b) ⇒ Y[O(dn , b)] ∗ ∗ Y ⎧ ↑ [O (a, b)] ⇒ Y[O (a, b)] Y ↑ O(a, c1 ) ⇒ Y[O(a, c1 )] ⎪ ⎪ ⎨ Y ↑ O(c1 , c2 ) ⇒ Y[O(c1 , c2 )] · ·· (◦ Q− ) ⎪ ⎪ ⎩ Y ↑ O(cn , b) ⇒ Y[O(cn , b)] Y ↑ O ◦ (a, b) ⇒ Y[O ◦ (a, b)] where ci is a new constant and di is a constant. Definition 9.5.8 A 4/22 -reduction δ = Y ↑ Y ⇒ Y is provable in Q4/2 = , denoted 4/22 2 by = δ, if there is a sequence {δ1 , ..., δn } of 4/2 -reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 4/22 2 reductions by one of the deduction rules in Q4/2 = . 2

Theorem 9.5.9 (Soundness and completeness theorem) For any 4/22 -reduction δ = Y ↑ Y ⇒ Y such that one of R  (a, b), Q  (a  , b ), P  (a  , b ), O  (a  , b ) is not the empty string, 4/22 4/22 = δ iff |== δ.  Theorem 9.5.10 (Soundness and incompleteness theorem) For any 4/22 -reduction δ = Y ↑ Y ⇒ Y, 4/22 4/22 = δ implies |== δ, and

4/22

|==

4/22

δ may not imply =

δ. 

9.6 Conclusions In R-calculus ||| ↑ (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ⇒   | |  | there may be several injuries. Assume that ◦ three first-level injuries: there is a stage s0 such that     = 4/22 ||| ↑ O(a , b ) ⇒s0 |||[O(a , b )];

9.6 Conclusions

323

and at a later stage s1 > s0 such that     = 4/22 ||| ↑ P(a , b ) ⇒s1 ||[P(a , b )]|;

and at a later stage s2 > s1 such that     = 4/22 ||| ↑ Q(a , b ) ⇒s2 |[Q(a , b )]||;

and at a later stage s3 > s2 such that = 4/22 ||| ↑ R(a, b) ⇒s3 [R(a, b)]|||; ◦ two second-level injuries: there is a stage t0 > s3 such that     = 4/22 [R(a, b)]||| ↑ O(a , b ) ⇒t0 [R(a, b)]|||[O(a , b )];

and at a later stage t1 > t0 such that     = 4/22 [R(a, b)]||| ↑ P(a , b ) ⇒t1 [R(a, b)]||[P(a , b )]|;

and at a later stage t2 > t1 such that     = 4/22 [R(a, b)]||| ↑ Q(a , b ) ⇒t2 [R(a, b)]|[Q(a , b )]||;

◦ one third-level injury: there is a stage u 0 > t2 such that     = 4/22 [R(a, b)]|[Q(a , b )]|| ↑ O(a , b )     ⇒u 0 [R(a, b)]|[Q(a , b )]||[O(a , b )];

and at a later stage u 1 > u 0 such that     = 4/22 [R(a, b)]|[Q(a , b )]|| ↑ P(a , b )     ⇒u 1 [R(a, b)]|[Q(a , b )]|[P(a , b )]|;

and at a later stage u 2 > u 1 such that       = 4/22 [R(a, b)]|[Q(a , b )]|[P(a , b )]| ↑ O(a , b )       ⇒u 2 [R(a, b)]|[Q(a , b )]|[P(a , b )]|[O(a , b )].

Then, for first-level injury (Friedberg 1957; Li and Sui 2017; Muchnik 1956; Soare 1987), eliminating O(a  , b ) from  at stage s0 + 1 may be injured by eliminating P(a  , b ) from  at stage s1 + 1; eliminating P(a  , b ) from  at stage s1 + 1 may be injured by eliminating Q(a  , b ) from  at stage s2 + 1 and eliminating Q(a  , b ) from  at stage s2 + 1 may be injured by eliminating R(a, b) from  at stage s3 + 1

324

9 Role R-Calculus for B22 -Valued DL

for second-level injury, eliminating O(a  , b ) from  at stage t0 + 1 may be injured by eliminating P(a  , b ) from  at stage t1 + 1; and eliminating P(a  , b ) from  at stage t1 + 1 may be injured by eliminating Q(a  , b ) from  at stage t2 + 1 for third-level injury, eliminating O(a  , b ) from  at stage u 0 + 1 may be injured by eliminating P(a  , b ) from  at stage u 1 + 1.

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symbolic Logic 50, 510–530 (1985) Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inform. 5, 25–63 (1996) Arieli, O., Avron, A.: Bilattices and paraconsistency. Front. Paraconsist. Logic, Stud. Logic Comput. 8, 11–27 (2000) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation Applications. Cambridge University Press, Cambridge (2003) Belnap, N.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.), Modern Uses of Multiplevalued Logic, pp. 8–37. D. Reidel (1977) Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Logic 2, 87–112 (1938) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Log. 40, 295–331 (2011) Font, J.M.: Belnap’s four-valued logic and De Morgan lattices. Logic J. I.G.P.L. 5, 413–440 (1997) Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957) Ginsberg, M.L.: Multi-valued logics: a uniform approach to reasoning in artificial intelligence. Comput. Intell. 4, 256–316 (1988) Gottwald, S.: A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9. Baldock: Research Studies Press Ltd. (2001) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Li, W., Sui, Y.: The R-calculus and the finite injury priority method. J. Comput. 12, 127–134 (2017) Muchnik, A.A.: On the separability of recursively enumerable sets (in Russian). Dokl. Akad. Nauk SSSR, N.S. 109, 29–32 (1956) Ponse, A., van der Zwaag, M.B.: A generalization of ACP using Belnap’s logic. Electron. Notes Theor. Comput. Sci. 162, 287–293 (2006) Pynko, A.P.: Characterizing Belnap’s logic via De Morgan’s laws. Math. Log. Q. 41(4), 442–454 (1995) Soare, R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Functions and Computably Generated Sets. Springer (1987) Urquhart, A.: Basic many-valued logic. In: Gabbay, D., Guenthner, F., (eds.), Handbook of Philosophical Logic, vol. 2, 2ed edn., pp. 249–295. Kluwer, Dordrecht (2001) Zach, R.: Proof theory of finite-valued logics, Technical Report TUW-E185.2-Z.1-93

Chapter 10

Role R-Calculus for Post L4 -Valued DL

t t t t Mi/4 Ni/4 Ri/4 Si/4 i/4 i/4 i/4 i/4 Lt Kt Qt Pt

In this chapter, we consider two kinds of validity (Arieli and Avron 1996, 2000; Baader et al. 2003; Belnap 1977; Bochvar 1938; Ginsberg 1988; Gottwald 2001; Zach 2023): t t :  ∈ M1/4 ↔ AI E A ∈ (I (A) = t) M1/4 t t N1/4 :  ∈ N1/4 ↔ AI E A ∈ (I (A) = t). t t is contrary to N1/4 , respectively. M1/4 t t N1/4 M1/4 =t |=t1/4 |=1/4 =t

For each validity, there are sound and complete deduction systems t1/4 , 1/4 for respectively.

t t , N1/4 -validity, M1/4

|=t1/4 =t 1/4 =t iff =t 1/4 |=1/4 We consider two kinds of R-calculi (Alchourrón et al. 1985; Fermé and Hansson 2011; Font1997; Li 2007; Muchnik 1956; Ponse and van der Zwaag 2006; Pynko1995): let R(a, b) ∈ ,

© Science Press 2024 W. Li and Y. Sui, R-Calculus, V: Description Logics, Perspectives in Formal Induction, Revision and Evolution, https://doi.org/10.1007/978-981-99-6460-4_10

325

326

10 Role R-Calculus for Post L4 -Valued DL t  ↑ R(a, b) ⇒ [R(a, b)] ∈R1/4 R(a, b) ↔ R  (a, b) = λ t  ↑ R(a, b) ⇒ [R(a, b )] ∈ N1/4  R(a, b) ↔ R  (a, b) = λ

if |=t1/4 [R(a, b)] otherwise =t

if |=1/4  − {R(a, b)} otherwise

and prove that t1/4  ↑ R(a, b) ⇒ [R(a, b)] iff |=t1/4  ↑ R(a, b) ⇒ [R(a, b)] t1/4  ↑ R(a, b) ⇒  implies |=t1/4  ↑ R(a, b) ⇒  |=t1/4  ↑ R(a, b) ⇒  may not imply t1/4  ↑ R(a, b) ⇒  =t

=t

1/4 , R(a, b) ↑ R(a, b) ⇒ [R(a, b)] iff |=1/4 , R(a, b) ↑ R(a, b) ⇒ [R(a, b)] =t

=t

1/4 , R(a, b) ↑ R(a, b) ⇒  implies |=1/4  ↑ R(a, b) ⇒ 

=t |=1/4

=t

, R(a, b) ↑ R(a, b) ⇒  may not imply 1/4  ↑ R(a, b) ⇒ .

We consider the following deduction systems and R-calculi: t t t⊥ = Deduction system M1/4 , M2/4 , M3/4 , M4/4 t = N1/4 , N4/4 t t t⊥ = R-calculus R1/4 , R2/4 , R3/4 , R4/4 t = S1/4 , S4/4

and

t M1/4 t M2/4 = M3/4 = M4/4

, ∼ , ∼2 , ∼3  |, ∼ , ∼2  . ||, ∼  |||

10.1 Post L4 -Valued DL Let U be a universe and L4 = {t, , ⊥, f}, where

10.1 Post L4 -Valued DL

327

t  ⊥ f ∪ t  ⊥ f

f ∼ f ∼2 f ∼3  ⊥ f ⊥ f t f t  t  ⊥ t⊥ f tt t t t t⊥⊥ t⊥ f

∩ t ⊥f t t ⊥f ⊥f ⊥⊥⊥⊥f f f f ff → t ⊥f t t t tt  t tt ⊥ ⊥tt f f ⊥t

The logical language of Post L4 -valued DL with role constructors contains the following symbols: • atomic roles: S0 , S1 , . . . ; • role constructors: ∼, ∩, ∪, ∗ , ◦ . Concepts are defined inductively as follows: R ::= S| ∼ R|R1 ∩ R2 |R1 ∪ R2 |R ∗ |R ◦ , where S is an atomic role, and R is a role. A model M is a pair (U, I ), where U is a non-empty set, and I is an interpretation such that for any atomic role S, I (S) : U 2 → L4 . Given an atomic role S, we define roles S, ∼ S, ∼2 S, ∼3 S as follows: for any x ∈ U, S(x, y) ∼ S(x, y) ∼2 S(x, y) ∼3 S(x, y) t  ⊥ f t f ⊥  f t  ⊥ ⊥  t f The interpretation R I of a role R is a function from U 2 to L4 such that for any x, y ∈ U, ⎧ y) ⎪ ⎪ I (S)(x, ⎨ f ∗ (R I )(x, y) I R (x, y) = min{R1I (x, y), R2I (x, y)} ⎪ ⎪ ⎩ max{R1I (x, y), R2I (x, y)}

if if if if

R R R R

=S = ∗R1 = R1 ∩ R2 = R1 ∪ R2

where ∗ ∈ {∼, ∼2 , ∼3 }. About (R(a, b)) I = t, we have the following equivalences:

328

10 Role R-Calculus for Post L4 -Valued DL

(R1 ∩ R2 )(x, y) ∼ = R1 (x, y)∧R2 (x, y) ∼ (R1 ∩ R2 )(x, y) ∼ = (R1 (x, y)∧ ∼ R2 (x, y))∨(∼ R1 (x, y)∧R2 (x, y)) ∨(∼ R1 (x, y)∧ ∼ R2 (x, y)) 2 ∼ ∼ (R1 ∩ R2 )(x, y) = (R1 (x, y)∧ ∼2 R2 (x, y))∨(∼2 R1 (x, y)∧R2 (x, y)) ∨(∼ R1 (x, y)∧ ∼2 R2 (x, y))∨(∼2 R1 (x, y)∧ ∼ R2 (x, y)) ∨(∼2 R1 (x, y)∧ ∼2 R2 (x, y)) ∼3 (R1 ∩ R2 )(x, y) ∼ = ∼3 R1 (x, y)∨ ∼3 R2 (x, y); (R1 ∪ R2 )(x, y) ∼ = R1 (x, y)∨R2 (x, y) ∼ (R1 ∪ R2 )(x, y) ∼ = (∼ R1 (x, y)∧ ∼ R2 (x, y))∨(∼ R1 (x, y)∧ ∼2 R2 (x, y)) ∨(∼ R1 (x, y)∧ ∼3 R2 (x, y))∨(∼2 R1 (x, y)∧ ∼ R2 (x, y)) ∨(∼3 R1 (x, y)∧ ∼ R2 (x, y)) ∼2 (R1 ∪ R2 )(x, y) ∼ = (∼2 R1 (x, y)∧ ∼2 R2 (x, y))∨(∼2 R1 (x, y)∧ ∼3 R2 (x, y)) ∨(∼3 R1 (x, y)∧ ∼2 R2 (x, y)) ∼3 (R1 ∪ R2 )(x, y) ∼ = ∼3 R1 (x, y)∧ ∼3 R2 (x, y),

and R ∗ (x, y) iff ∃d1 , . . . , dn ∀i ≤ n(R(di , di+1 )) (∼ R ∗ )(x, y) iff ∀c1 , . . . , cn ∃i ≤ n(∼ R(ci , ci+1 )∨ ∼2 R(ci , ci+1 )∨ ∼3 R(ci , ci+1 )) (∼2 R ∗ )(x, y) iff ∀c1 , . . . , cn ∃i ≤ n(∼2 R(ci , ci+1 )∨ ∼3 R(ci , ci+1 )) (∼3 R ∗ )(x, y) iff ∀c1 , . . . , cn ∃i ≤ n(∼3 R(ci , ci+1 )) R ◦ (x, y) iff ∀c1 , . . . , cn ∃i ≤ n(R(ci , ci+1 )) (∼ R ◦ )(x, y) iff ∀c1 , . . . , cn ∃i ≤ n(∼ R(ci , ci+1 ) ∨ R(ci , ci+1 )) (∼2 R ◦ )(x, y) iff ∀c1 , . . . , cn ∃i ≤ n(∼2 R(ci , ci+1 )∨ ∼ R(ci , ci+1 ) ∨ R(ci , ci+1 )) (∼3 R ◦ )(x, y) iff ∃d1 , . . . , dn ∀i ≤ n(∼3 R(di , di+1 )),

where ∼ (R1 ∩ R2 )(x, y) ∼ = (R1 (x, y)∧ ∼ R2 (x, y))∨(∼ R1 (x, y)∧R2 (x, y))∨(∼ R1 (x, y)∧ ∼ R2 (x, y)) means that for any interpretation I, I (∼ (R1 ∩ R2 ))(x, y) = t iff either (1) I (R1 (x, y)) = t and I (∼ R2 (x, y)) = t, or (2) I (∼ R1 (x, y)) = t and I (R2 (x, y)) = t, or (3) I (∼ R1 (x, y)) = t and I (∼ R2 (x, y)) = t. About the inequality, we have the following equivalences:

10.2 1/4-Multisequents

329

(R1 ∩ R2 )(x, y) ∼ = R1 (x, y)∨R2 (x, y) ∼ (R1 ∩ R2 )(x, y) ∼ = (R1 (x, y)∨ ∼ R2 (x, y))∧(∼ R1 (x, y)∨R2 (x, y)) ∧(∼ R1 (x, y)∨ ∼ R2 (x, y)) 2 ∼ ∼ (R1 ∩ R2 )(x, y) = (R1 (x, y)∨ ∼2 R2 (x, y))∧(∼ R1 (x, y)∨R2 (x, y)) ∨(∼2 R1 (x, y)∨ ∼2 R2 (x, y))∧(∼2 R1 (x, y)∨ ∼ R2 (x, y)) ∧(∼2 R1 (x, y)∨R2 (x, y)) ∼3 (R1 ∩ R2 )(x, y) ∼ = ∼3 R1 (x, y)∨ ∼3 R2 (x, y); (R1 ∪ R2 )(x, y) ∼ = R1 (x, y)∧R2 (x, y) ∼ (R1 ∪ R2 )(x, y) ∼ = (∼ R1 (x, y)∨ ∼ R2 (x, y))∧(∼ R1 (x, y)∨ ∼2 R2 (x, y)) ∧(∼ R1 (x, y)∨ ∼3 R2 (x, y))∧(∼2 R1 (x, y)∨ ∼ R2 (x, y)) ∧(∼3 R1 (x, y)∨ ∼ R2 (x, y)) ∼2 (R1 ∪ R2 )(x, y) ∼ = (∼2 R1 (x, y)∨ ∼2 R2 (x, y))∧(∼2 R1 (x, y)∨ ∼3 R2 (x, y)) ∧(∼3 R1 (x, y)∨ ∼2 R2 (x, y)) ∼3 (R1 ∪ R2 )(x, y) ∼ = ∼3 R1 (x, y)∧ ∼3 R2 (x, y), where ∼ (R1 ∩ R2 )(x, y) ∼ = (R1 (x, y)∨ ∼ R2 (x, y))∧(∼ R1 (x, y)∨R2 (x, y))∧(∼ R1 (x, y)∨ ∼ R2 (x, y)) means that for any interpretation I, I (∼ (R1 ∩ R2 )(x, y) = t iff (1) either I (R1 (x, y)) = t or I (∼ R2 (x, y)) = t, (2) either I (∼ R1 (x, y)) = t or I (R2 (x, y)) = t, and (3) either I (∼ R1 (x, y)) = t or I (∼ R2 (x, y)) = t, and R ∗ (x, y) iff ∀c1 , . . . , cn ∃i ≤ n(R(ci , ci+1 )) (∼ R ∗ )(x, y) iff ∃d1 , . . . , dn ∀i ≤ n(∼ R(di , di+1 )∧ ∼2 R(di , di+1 )∧ ∼3 R(di , di+1 )) (∼2 R ∗ )(x, y) iff ∃d1 , . . . , dn ∀i ≤ n(∼2 R(di , di+1 )∧ ∼3 R(di , di+1 )) (∼3 R ∗ )(x, y) iff ∃d1 , . . . , dn ∀i ≤ n(∼3 R(di , di+1 )) R ◦ (x, y) iff ∃d1 , . . . , dn ∀i ≤ n(R(di , di+1 )) (∼ R ◦ )(x, y) iff ∃d1 , . . . , dn ∀i ≤ n(∼ R(di , di+1 ) ∧ R(di , di+1 )) (∼2 R ◦ )(x, y) iff ∃d1 , . . . , dn ∀i ≤ n(∼2 R(di , di+1 )∧ ∼ R(di , di+1 ) ∧ R(di , di+1 )) (∼3 R ◦ )(x, y) iff ∀c1 , . . . , cn ∃i ≤ n(∼3 R(ci , ci+1 )).

10.2 1/4-Multisequents t A 1/4-multisequent  is M1/4 -valid, denoted by |=t1/4 , if for any interpretation I, there is a statement R(a, b) ∈  such that I (R(a, b)) = t.

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t 10.2.1 Deduction System M1/4 t Deduction system M1/4 consists of the following axiom and deduction rules: • Axiom: ∩ ∼− ∩ ∼−2 ∩ ∼−3  = ∅ (At1/4 ) ,

where  is a set of literals. • Deduction rules: (∼4 )

, R(a, b) , ∼4 R(a, b)



(∩)

, R1 (a, b) , R2 (a, b) , (R1 ∩ R2 )(a, b)

⎧ , ∼ R1 (a, b) ⎪ ⎪ ⎪ ⎪ , ∼ R2 (a, b) ⎪ ⎪ ⎨ , R1 (a, b) , (∼ ∩) ⎪ ⎪  ∼ R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎩ , R2 (a, b) , ∼ (R1 ∩ R2 )(a, b))

⎧ , ∼2 R1 (a, b) ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪  , ∼ R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎪ ⎪ , ∼2 R2 (a, b) ⎪ ⎪  ⎨ , R1 (a, b) , ∼3 R1 (a, b) 2 2 3 (∼ ∩) ⎪ (∼ ∩) , ∼3 R2 (a, b) ⎪  , ∼2 R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) , ∼3 (R1 ∩ R2 )(a, b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  , ∼2R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎩ , R2 (a, b) , ∼2 (R1 ∩ R2 )(a, b)) and

10.2 1/4-Multisequents



(∪)

, R1 (a, b) , R2 (a, b) , (R1 ∪ R2 )(a, b)

331

⎧ ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎪ ⎪  , ∼ R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎪ ⎪ , ∼2 R2 (a, b) ⎪ ⎪ ⎨ , ∼ R1 (a, b) 3 , (∼ ∪) ⎪ ⎪  ∼3 R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎪ ⎪ , ⎪ ⎪  ∼2R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) ⎪ ⎪ ⎩ , ∼ R2 (a, b) , ∼ (R1 ∪ R2 )(a, b)

⎧ 2 ⎪ ⎪ , ∼2 R1 (a, b) ⎪ ⎪ , ∼ R2 (a, b) ⎪ ⎪  ⎨ , ∼3 R1 (a, b) , ∼2 R1 (a, b) 3 2 3 (∼ ∪) ⎪ (∼ ∪) , ∼3 R2 (a, b) ⎪  , ∼3 R2 (a, b) ⎪ ⎪ , ∼ R1 (a, b) , ∼3 (R1 ∪ R2 )(a, b) ⎪ ⎪ ⎩ 2 , ∼ R2 (a, b) , ∼2 (R1 ∪ R2 )(a, b) • Deduction rules for quantifier constructors:

⎡

, R(a, d1 ) ⎢ , R(d1 , d2 ) ⎢ (∗) ⎣ · · · , R(dn , b) , R ∗ (a, b)

⎧ , ∼ R(a, c1 ) ⎪ ⎪ ⎪ ⎪ , ∼2 R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎪ , ∼3 R(a, c1 ) ⎪ ⎪ ⎪ , ∼ R(c1 , c2 ) ⎪ ⎪ ⎨ , ∼2 R(c1 , c2 ) (∼ ∗) ⎪ ⎪ , ∼3 R(c1 , c2 ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ , ∼ R(cn , b) ⎪ ⎪ ⎪ ⎪ , ∼2 R(cn , b) ⎪ ⎪ ⎩ , ∼3 R(cn , b) , (∼ R ∗ )(a, b)

⎧ , ∼2 R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎧ , ∼3 R(a, c1 ) ⎪ ⎪ , ∼3 R(a, c1 ) ⎪ ⎪ 2 ⎪ ⎪ R(c , c ) , ∼ ⎨ ⎨ 1 2 , ∼3 R(c1 , c2 ) , ∼3 R(c1 , c2 ) 2 3 ··· (∼ ∗) ⎪ (∼ ∗) ⎪ ⎪ ⎪ ⎪··· ⎩ ⎪ , ∼3 R(cn , b) ⎪ ⎪ ⎪ , ∼2 R(cn , b) ⎪ , (∼3 R ∗ )(a, b) ⎩ , ∼3 R(cn , b) , (∼2 R ∗ )(a, b) where c is a new constant (does not occur in ), and d is a constant, and

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10 Role R-Calculus for Post L4 -Valued DL

⎧ , R(a, c1 ) ⎪ ⎪ ⎨ , R(c1 , c2 ) ··· (◦) ⎪ ⎪ ⎩ , R(cn , b) , R ◦ (a, b)

⎧ ⎪ ⎪ , R(a, c1 ) ⎪ ⎪ , ∼ R(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎨ , R(c1 , c2 ) , ∼ R(c1 , c2 ) (∼ ◦) ⎪ ⎪ · ·· ⎪ ⎪ ⎪ ⎪ , R(cn , b) ⎪ ⎪ ⎩ , ∼ R(cn , b) , (∼ R ◦ )(a, b))

⎧ ⎪ ⎪ , R(a, c1 ) ⎪ ⎪ , ∼ R(a, c1 ) ⎪ ⎪ ⎪ ⎪ , ∼2 R(a, c1 ) ⎪ ⎪ ⎡ ⎪ ⎪ , ∼3 R(a, d1 ) , R(c1 , c2 ) ⎪ ⎪ ⎨ 3 ⎢ , ∼ R(c1 , c2 ) ⎢ , ∼ R(d1 , d2 ) 2 2 3 ⎣ , ∼ R(c1 , c2 ) (∼ ◦) · · · (∼ ◦) ⎪ ⎪ ⎪ ⎪ ··· , ∼3 R(dn , b) ⎪ ⎪ ⎪ ⎪ , R(cn , b) , (∼3 R ◦ )(a, b) ⎪ ⎪ ⎪ ⎪ , ∼ R(cn , b) ⎪ ⎪ ⎩ , ∼2 R(cn , b) , (∼2 R ◦ )(a, b)) where ci is a new constant and di is a constant. t Definition 10.2.1 A 1/4-multisequent  is provable in M1/4 , denoted by t1/4 , if there is a sequence {1 , . . . , n } of 1/4-multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/4-multisequents by t . one of the deduction rules in M1/4

Theorem 10.2.2 (Soundness and completeness theorem) For any 1/4-multisequent , 1/4 1/4 t  iff |=t . 

t 10.2.2 R-Calculus R1/4

Given a 1/4-multisequent  and a statement R(a, b) ∈ , a reduction  ↑ R(a, b) ⇒ t -valid, denoted by |=t1/4  ↑ R(a, b) ⇒ [R  (a, b)], if [R  (a, b)] is R1/4 R  (a, b) =



t -valid R(a, b) if [R(a, b)] is M1/4 λ otherwise.

t Let R(a, b) ∈ . R-calculus R1/4 consists of the following axioms and deduction rules:

10.2 1/4-Multisequents

333

• Axioms: [r (a, b)]∩ ∼− ∩ ∼−2 ∩ ∼−3  = ∅  ↑ r (a, b) ⇒ [r (a, b)] − −2 ∩ ∼−3  = ∅ ∼Rt− ∩ ∼ [∼ r (a, b)]∩ ∼ (A1/4 )  ↑∼ r (a, b) ⇒ [∼ r (a, b)] − −2 [∼2 r (a, b)]∩ ∼−3  = ∅ ∼2 Rt− ∩ ∼ ∩ ∼ (A1/4 ) 2  ↑∼ r (a, b) ⇒ [∼2 r (a, b)] − −2 3 ∩ ∼−3 [∼3 r (a, b)] = ∅ Rt− ∩ ∼ ∩ ∼ ) (A∼ 3 1/4  ↑∼ r (a, b) ⇒ [∼3 r (a, b)]

Rt− ) (A1/4

where  is a set of literals, and r (a, b) is a role atom. • Deduction rules:  ↑ R(a, b) ⇒ [R(a, b)] 4 R(a, b) ⇒ [∼4 R(a, b)]  ↑∼   ↑ R1 (a, b) ⇒ [R1 (a, b)] (∩− ) [R1 (a, b)] ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]   ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]  ↑ R1 (a, b) ⇒ [R1 (a, b)] (∪− )  ↑ R2 (a, b) ⇒ [R2 (a, b)]  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]

(∼4− )

and

and

⎡

 ↑ R(a, d1 ) ⇒ [R(a, d1 )] ⎢  ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )] ⎢ (∗− ) ⎣ · · ·  ↑ R(dn , b) ⇒ [R(dn , b)] ∗ ∗  ⎧ ↑ R (a, b) ⇒ [R (a, b)]  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎨  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] · ·· (◦− ) ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ R ◦ (a, b) ⇒ [R ◦ (a, b)]

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⎧ ⎪ ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ [∼ R1 (a, b)] ↑∼ R2 (a, b) ⇒ [∼ R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪ ⎨  ↑ R1 (a, b) ⇒ [R1 (a, b)] [R (∼ ∩− ) ⎪ 1 (a, b)] ↑∼ R2 (a, b) ⇒ [R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪ ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎩ [∼ R1 (a, b)] ↑ R2 (a, b) ⇒ [∼ R1 (a, b), R2 (a, b)]  ↑∼ (R1 ∩ R2 )(a, b) ⇒ [∼ (R1 ∩ R2 )(a, b)] ⎧  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  [∼ R2 1 (a, b)] ↑∼ R2 (a,2 b) ⇒ [∼ R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎪ 2 ⎪ R1 (a, b)] ↑∼ R2 (a, b) ⇒ [∼2 R1 (a, b), ∼ R2 (a, b)] [∼ ⎪ ⎪ ⎨ 3  ↑∼ R1 (a, b) ⇒ [∼3 R1 (a, b)] 3 3 (∼ ∪− ) ⎪ ⎪  [∼ R1 (a, b)] ↑∼ R2 (a, b) ⇒ [∼ R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎪ ⎪ [∼ R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼ R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪ ⎪ ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎩ [∼ R1 (a, b)] ↑∼3 R2 (a, b) ⇒ [∼ R1 (a, b), ∼3 R2 (a, b)]  ↑∼ (R1 ∪ R2 )(a, b) ⇒ [∼ (R1 ∪ R2 )(a, b)] and

and

⎧  ↑∼ R(a, c1 ) ⇒ [∼ R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑∼3 R(a, c1 ) ⇒ [∼3 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑∼ R(c1 , c2 ) ⇒ [∼ R(c1 , c2 )] ⎪ ⎪ ⎨  ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )]  ↑∼3 R(c1 , c2 ) ⇒ [∼3 R(c1 , c2 )] (∼ ∗− ) ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼ R(cn , b) ⇒ [∼ R(cn , b)] ⎪ ⎪ ⎪ ⎪  ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)] ⎪ ⎪ ⎩  ↑∼3 R(cn , b) ⇒ [∼3 R(cn , b)]  ↑ (∼ R ∗ )(a, b) ⇒ [(∼ R ∗ )(a, b)] ⎧  ↑∼ R(a, c1 ) ⇒ [∼ R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨  ↑∼ R(c1 , c2 ) ⇒ [∼ R(c1 , c2 )]  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∼ ◦− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼ R(cn , b) ⇒ [∼ R(cn , b)] ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ (∼ R ◦ )(a, b) ⇒ [(∼ R ◦ )(a, b)]

10.2 1/4-Multisequents

⎧ 2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪  ↑∼ ⎪ 2 2 2 2 ⎪ ⎪ ⎪  [∼ R1 (a, b)] ↑∼ R2 (a, b) ⇒ [∼ R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪  ↑ R1 (a, b) ⇒ [R1 (a, b)] ⎪ ⎪ ⎪ 2 2 ⎪ [R ⎪ 1 (a, b)] ↑∼ R2 (a, b) ⇒ [R1 (a, b), ∼ R2 (a, b)] ⎪ ⎨  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] 2 b) ⇒ [∼ R1 (a, b), ∼2 R2 (a, b)] (∼2 ∩− ) ⎪ ⎪  [∼ R2 1 (a, b)] ↑∼ R2 (a, ⎪ 2 ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎪ 2 ⎪ b) ⇒ [∼2 R1 (a, b), ∼ R2 (a, b)] ⎪ ⎪  [∼ 2R1 (a, b)] ↑∼ R2 (a, ⎪ 2 ⎪  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎪ ⎪ ⎩ [∼ R1 (a, b)] ↑ R2 (a, b) ⇒ [∼2 R1 (a, b), R2 (a, b)]  ↑∼2 (R1 ∩ R2 )(a, b) ⇒ [∼2 (R1 ∩ R2 )(a, b)] ⎧  ↑∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪ ⎪ 2 2 ⎪ b) ⇒ [∼2 R1 (a, b), ∼2 R2 (a, b)] ⎪ ⎪  [∼ 3R1 (a, b)] ↑∼ R2 (a, ⎨ 3  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] 3 R1 (a, b)] ↑∼2 R2 (a, b) ⇒ [∼3 R1 (a, b), ∼2 R2 (a, b)] [∼ (∼2 ∪− ) ⎪ ⎪ ⎪ 2 ⎪  ↑∼ R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎪ ⎪ ⎩ [∼2 R1 (a, b)] ↑∼3 R2 (a, b) ⇒ [∼2 R1 (a, b), ∼3 R2 (a, b)]  ↑∼2 (R1 ∪ R2 )(a, b) ⇒ [∼2 (R1 ∪ R2 )(a, b)] and

and

⎧  ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑∼3 R(a, c1 ) ⇒ [∼3 R(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨  ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )]  ↑∼3 R(c1 , c2 ) ⇒ [∼3 R(c1 , c2 )] (∼2 ∗− ) ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪  ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)] ⎪ ⎪ ⎩  ↑∼3 R(cn , b) ⇒ [∼3 R(cn , b)]  ↑ (∼2 R ∗ )(a, b) ⇒ [(∼2 R ∗ )(a, b)] ⎧  ↑∼2 R(a, c1 ) ⇒ [∼2 R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑∼ R(a, c1 ) ⇒ [∼ R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑ R(a, c1 ) ⇒ [R(a, c1 )] ⎪ ⎪ ⎪ ⎪  ↑∼2 R(c1 , c2 ) ⇒ [∼2 R(c1 , c2 )] ⎪ ⎪ ⎨  ↑∼ R(c1 , c2 ) ⇒ [∼ R(c1 , c2 )]  ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )] (∼2 ◦− ) ⎪ ⎪ ⎪ ⎪ · · · ⎪ ⎪ ⎪ ⎪  ↑∼2 R(cn , b) ⇒ [∼2 R(cn , b)] ⎪ ⎪ ⎪ ⎪  ↑∼ R(cn , b) ⇒ [∼ R(cn , b)] ⎪ ⎪ ⎩  ↑ R(cn , b) ⇒ [R(cn , b)]  ↑ (∼2 R ◦ )(a, b) ⇒ [(∼2 R ◦ )(a, b)]

335

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10 Role R-Calculus for Post L4 -Valued DL



 ↑∼3 R1 (a, b) ⇒ [∼3 R1 (a, b)] (∼ ∩− )  ↑∼3 R2 (a, b) ⇒ [∼3 R2 (a, b)] 2 2  1 ∩ R2 )(a, b)) ⇒ [∼ (R1 ∩ R2 )(a, b)]  ↑∼ (R 3 3  ↑∼ R1 (a, b) ⇒ [∼ R1 (a, b)] (∼3 ∪− ) [∼3 R1 (a, b)] ↑∼3 R2 (a, b) ⇒ [∼3 R1 (a, b), ∼3 R2 (a, b)]  ↑∼3 (R1 ∪ R2 )(a, b) ⇒ [∼3 (R1 ∪ R2 )(a, b)] 3

and

⎧  ↑∼3 R(a, c1 ) ⇒ [∼3 R(a, c1 )] ⎪ ⎪ ⎨  ↑∼3 R(c1 , c2 ) ⇒ [∼3 R(c1 , c2 )] 3 − ··· (∼ ∗ ) ⎪ ⎪ ⎩  ↑∼3 R(cn , b) ⇒ [∼3 R(cn , b)]  ↑ (∼3 R ∗ )(a, b) ⇒ [(∼3 R ∗ )(a, b)] ⎡  ↑∼3 R(a, d1 ) ⇒ [∼3 R(a, d1 )] ⎢  ↑∼3 R(d1 , d2 ) ⇒ [∼3 R(d1 , d2 )] ⎢ 3 − ⎣··· (∼ ◦ )  ↑∼3 R(dn , b) ⇒ [∼3 R(dn , b)]  ↑ (∼3 R ◦ )(a, b) ⇒ [(∼3 R ◦ )(a, b)]

where ci is a new constant and di is a constant. Definition 10.2.3 Given a 1/4-multisequent  and a statement R(a, b) ∈ , a 1/4t reduction  ↑ R(a, b) ⇒   is provable in R1/4 , denoted by t1/4  ↑ R(a, b) ⇒   , if there is a sequence {δ1 , . . . , δn } of 1/4-reductions such that δn =  ↑ R(a, b) ⇒   , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/4t . reductions by one of the deduction rules in R1/4 Theorem 10.2.4 (Soundness and completeness theorem) For any reduction δ =  ↑ R(a, b) ⇒ [R(a, b)], t1/4 δ iff |=t1/4 δ.  Theorem 10.2.5 (Soundness and incompleteness theorem) For any reduction δ =  ↑ R(a, b) ⇒ , t1/4 δ implies |=t1/4 δ, and

|=t1/4 δ may not imply t1/4 δ. 

10.2 1/4-Multisequents

337

t 10.2.3 Deduction System N1/4 =t

t A 1/4-multisequent  is N1/4 -valid, denoted by |=1/4 , if for any interpretation I, for some statement R(a, b) ∈ , I (R(a, b)) = t. t consists of the following axiom and deduction rules: Deduction system N1/4 • Axiom: ⎧ ∩ ∼−  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎪ ⎨ ∩ ∼−3  = ∅ =t ∼− ∩ ∼−2  = ∅ (A1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− ∩ ∼−3  = ∅ ⎪ ⎪ ⎩ −2 ∼ ∩ ∼−3  = ∅ 

where  is a set of literals. • Deduction rules for logical constructors: (∼4 )

, R(a, b) , ∼4 R(a, b)



(∩)

, R1 (a, b) , R2 (a, b) , (R1 ∩ R2 )(a, b) ⎡

⎡

, ∼ R1 (a, b) ⎢ , ∼ R2 (a, b) ⎢ ⎢ , R1 (a, b) ⎢ (∼ ∩) ⎢ ⎢  , ∼ R2 (a, b) ⎣ , ∼ R1 (a, b) , R2 (a, b) , ∼ (R1 ∩ R2 )(a, b))

, ∼2 R1 (a, b) ⎢ , ∼2 R2 (a, b) ⎢ ⎢ , ∼ R1 (a, b) ⎢ ⎢ , ∼2 R2 (a, b) ⎢  ⎢ , R1 (a, b) , ∼3 R1 (a, b) ⎢ 2 2 3 ⎢ (∼ ∩) ⎢  , ∼ R2 (a, b) (∼ ∩) , ∼3 R2 (a, b) ⎢ , ∼2 R1 (a, b) , ∼3 (R1 ∩ R2 )(a, b) ⎢ ⎢ , R2 (a, b) ⎢ ⎣ , ∼2 R1 (a, b) , ∼ R2 (a, b) , ∼2 (R1 ∩ R2 )(a, b)) and

338

10 Role R-Calculus for Post L4 -Valued DL

⎡



(∪)

, R1 (a, b) , R2 (a, b) , (R1 ∪ R2 )(a, b)

⎡

, ∼ R1 (a, b) ⎢ , ∼ R2 (a, b) ⎢ ⎢ , ∼ R1 (a, b) ⎢ ⎢ , ∼2 R2 (a, b) ⎢ ⎢ , ∼ R1 (a, b) ⎢ 3 (∼ ∪) ⎢ ⎢  , ∼2 R2 (a, b) ⎢ , ∼ R1 (a, b) ⎢ ⎢ , ∼ R2 (a, b) ⎢ ⎣ , ∼3 R1 (a, b) , ∼ R2 (a, b) , ∼ (R1 ∪ R2 )(a, b)

, ∼2 R1 (a, b) ⎢ , ∼2 R2 (a, b) ⎢  ⎢ , ∼2 R1 (a, b) , ∼3 R1 (a, b) ⎢ 3 2 3 ⎢ (∼ ∪) ⎢  , ∼ R2 (a, b) (∼ ∪) , ∼3 R2 (a, b) ⎣ , ∼3 R1 (a, b) , ∼3 (R1 ∪ R2 )(a, b) 2 , ∼ R2 (a, b) , ∼2 (R1 ∪ R2 )(a, b) • Deduction rules for quantifier constructors: ⎡

⎧ , R(a, c1 ) ⎪ ⎪ ⎨ , R(c1 , c2 ) ··· (∗) ⎪ ⎪ ⎩ , R(cn , b) , R ∗ (a, b)

⎡

, ∼ R(a, d1 ) ⎢ , ∼2 R(a, d1 ) ⎢ ⎢ , ∼3 R(a, d1 ) ⎢ ⎢ , ∼ R(d1 , d2 ) ⎢ ⎢ , ∼2 R(d1 , d2 ) ⎢ 3 (∼ ∗) ⎢ ⎢ , ∼ R(d1 , d2 ) ⎢··· ⎢ ⎢ , ∼ R(dn , b) ⎢ ⎣ , ∼2 R(dn , b) , ∼3 R(dn , b) , (∼ R ∗ )(a, b)

, ∼2 R(a, d1 ) ⎢ , ∼3 R(a, d1 ) ⎡ ⎢ , ∼3 R(a, d1 ) ⎢ , ∼2 R(d1 , d2 ) 3 ⎢ ⎢ ⎢ , ∼3 R(d1 , d2 ) ⎢ , ∼ R(d1 , d2 ) 2 3 ⎢ ⎣ (∼ ∗) ⎢ (∼ ∗) · · · ⎢··· 2 , ∼3 R(dn , b) ⎣ , ∼ R(dn , b) , (∼3 R ∗ )(a, b) , ∼3 R(dn , b) , (∼2 R ∗ )(a, b) and

10.2 1/4-Multisequents

339

⎡

⎡

, R(a, d1 ) ⎢ , R(d1 , d2 ) ⎢ (◦) ⎣ · · · , R(dn , b) , R ◦ (a, b) ⎡

, R(a, d1 ) ⎢ , ∼ R(a, d1 ) ⎢ ⎢ , R(d1 , d2 ) ⎢ ⎢ , ∼ R(d1 , d2 ) (∼ ◦) ⎢ ⎢··· ⎢ ⎣ , R(dn , b) , ∼ R(dn , b) , (∼ R ◦ )(a, b))

, R(a, d1 ) ⎢ , ∼ R(a, d1 ) ⎢ ⎢ , ∼2 R(a, d1 ) ⎢ ⎧ ⎢ , R(d1 , d2 ) , ∼3 R(a, c1 ) ⎪ ⎢ ⎪ ⎨ ⎢ , ∼ R(d1 , d2 ) , ∼3 R(c1 , c2 ) ⎢ 2 2 3 ⎢ (∼ ◦) ⎢ , ∼ R(d1 , d2 ) (∼ ◦) ⎪ ⎪··· 3 ⎩ ⎢··· , ∼ R(cn , b) ⎢ ⎢ , R(dn , b) , (∼3 R ◦ )(a, b) ⎢ ⎣ , ∼ R(dn , b) , ∼2 R(dn , b) , (∼2 R ◦ )(a, b)) where c is a new constant, and d is a constant. t Definition 10.2.6 A 1/4-multisequent  is provable in N1/4 , denoted by t1/4 , if there is a sequence {1 , . . . , n } of 1/4-multisequents such that n = , and for each 1 ≤ i ≤ n, i is either an axiom or deduced from the previous 1/4-multisequents by t . one of the deduction rules in N1/4

Theorem 10.2.7 (Soundness and completeness theorem) For any 1/4-multisequent , t1/4  iff |=t1/4 . 

10.2.4 R-Calculus St1/4 Given a 1/4-multisequent  and a statement R(a, b) ∈ , a reduction  ↑ R(a, b) ⇒ =t , R  (a, b) is St1/4 -valid, denoted by |=1/4  ↑ R(a, b) ⇒ , R  (a, b), if R  (a, b) =



t -valid R(a, b) if [R(a, b)] is N1/4 λ otherwise.

R-calculus St1/4 consists of the following axioms and deduction rules:

340

• Axioms:

10 Role R-Calculus for Post L4 -Valued DL

⎧ [r (a, b)]∩ ∼−  = ∅ ⎪ ⎪ ⎪ ⎪ [r (a, b)]∩ ∼−2  = ∅ ⎪ ⎪ ⎨ [r (a, b)]∩ ∼−3  = ∅ Rt− (A1/4 ) ⎪ ⎪ ∼− ∩ ∼−2  = ∅ ⎪ ⎪ ⎪ ∼− ∩ ∼−3  = ∅ ⎪ ⎩ −2 ∼ ∩ ∼−3  = ∅ ⎧↑ r (a, b) ⇒ [r (a, b)] ⎪ ∩ ∼− [∼ r (a, b)] = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎨ ∩ ∼−3  = ∅ ∼Rt− (A1/4 ) ⎪ ⎪ ∼− [∼ r (a, b)]∩ ∼−2  = ∅ ⎪ ⎪ ⎪ ∼− [∼ r (a, b)]∩ ∼−3  = ∅ ⎪ ⎩ −2 ∼ ∩ ∼−3  = ∅ ⎧↑∼ r (a, b) ⇒ [∼ r (a, b)] ∩ ∼−  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−2 [∼2 r (a, b)] = ∅ ⎪ ⎪ ⎨ ∩ ∼−3  = ∅ ∼2 Rt− (A1/4 ) ⎪ ⎪ ∼− ∩ ∼−2 [∼2 r (a, b)] = ∅ ⎪ ⎪ ⎪ ∼− ∩ ∼−3  = ∅ ⎪ ⎩ −2 ∼ [∼2 r (a, b)]∩ ∼−3  = ∅  ↑∼2 r (a, b) ⇒ [∼2 r (a, b)] ⎧ ∩ ∼−  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎪ ⎨ ∩ ∼−3 [∼3 r (a, b)] = ∅ ∼3 Rt− ∼− ∩ ∼−2  = ∅ (A1/4 ) ⎪ ⎪ ⎪ ⎪ ∼− ∩ ∼−3 [∼3 r (a, b)] = ∅ ⎪ ⎪ ⎩ −2 ∼ ∩ ∼−3 [∼3 r (a, b)] = ∅  ↑∼3 r (a, b) ⇒ [∼3 r (a, b)]

where  is a set of literals, and r (a, b) is an R-atom. • Deduction rules: |R(a, b) ⇒ [R(a, b)] (∼4− ) 4 R(a, b) ⇒ [∼4 R(a, b)] | ∼  |R1 (a, b) ⇒ [R1 (a, b)] (∩− ) |R2 (a, b) ⇒ [R2 (a, b)] |(R  1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)] |R1 (a, b) ⇒ [R1 (a, b)] (∪− ) [R1 (a, b)]|R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)] |(R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)] and

10.2 1/4-Multisequents

341

⎧ ⎪ ⎪ |R(a, c1 ) ⇒ [R(a, c1 )] ⎨ |R(c1 , c2 ) ⇒ [R(c1 , c2 )] · ·· (∗− ) ⎪ (◦− ) ⎪ ⎩ |R(cn , b) ⇒ [R(cn , b)] |R ∗ (a, b) ⇒ [R ∗ (a, b)] and

|R(a, d1 ) ⇒ [R(a, d1 )] ⎢ |R(d1 , d2 ) ⇒ [R(d1 , d2 )] ⎢ ⎣··· |R(dn , b) ⇒ [R(dn , b)] |R ◦ (a, b) ⇒ [R ◦ (a, b)]

⎡

| ∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎢ | ∼ R2 (a, b) ⇒ [∼ R2 (a, b)] ⎢ ⎢ [Z 1 ]|R1 (a, b) ⇒ [Z 1 , R1 (a, b)] ⎢ − ⎢ [Z ]| ∼ R (a, b) ⇒ [Z , ∼ R (a, b)] (∼ ∩ ) ⎢  1 2 1 2 ⎣ [Z 2 ]| ∼ R1 (a, b) ⇒ [Z 2 , ∼ R1 (a, b)] [Z 2 ]|R2 (a, b) ⇒ [Z 2 , R2 (a, b)] | ∼ ⎡  (R1 ∩ R2 )(a, b) ⇒ [∼ (R1 ∩ R2 )(a, b)] | ∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎢ | ∼ R2 (a, b) ⇒ [∼ R2 (a, b)] ⎢ ⎢ [Z 1 ]| ∼2 R1 (a, b) ⇒ [Z 1 , ∼2 R1 (a, b)] ⎢ ⎢ [Z 1 ]| ∼ R2 (a, b) ⇒ [Z 1 , ∼ R2 (a, b)] ⎢ ⎢ [Z 3 ]| ∼3 R1 (a, b) ⇒ [Z 3 , ∼3 R1 (a, b)] ⎢ − ⎢ [Z ]| ∼ R (a, b) ⇒ [Z , ∼ R (a, b)] (∼ ∪ ) ⎢  3 2 3 2 ⎢ [Z 4 ]| ∼ R1 (a, b) ⇒ [Z 4 , ∼ R1 (a, b)] ⎢ ⎢ [Z 4 ]| ∼2 R2 (a, b) ⇒ [Z 4 , ∼2 R2 (a, b)] ⎢ ⎣ [Z 5 ]| ∼ R1 (a, b) ⇒ [Z 5 , ∼ R1 (a, b)] [Z 5 ]| ∼3 R2 (a, b) ⇒ [Z 5 , ∼3 R2 (a, b)] | ∼ (R1 ∪ R2 )(a, b) ⇒ [∼ (R1 ∪ R2 )(a, b)]

where Z1 Z2 Z3 Z4 Z5 and

⎡

=∼ R1 (a, b)∨ ∼ R2 (a, b) = Z 1 , R1 (a, b)∨ ∼ R2 (a, b)] = Z 1 , ∼2 R1 (a, b)∨ ∼ R2 (a, b)] = Z 3 , ∼3 R1 (a, b)∨ ∼ R2 (a, b)] = Z 4 , ∼ R1 (a, b)∨ ∼2 R2 (a, b)]

342

10 Role R-Calculus for Post L4 -Valued DL

⎡

| ∼ R(a, d1 ) ⇒ [∼ R(a, d1 )] ⎢ | ∼2 R(a, d1 ) ⇒ [∼2 R(a, d1 )] ⎢ ⎢ | ∼3 R(a, d1 ) ⇒ [∼3 R(a, d1 )] ⎢ ⎢ | ∼ R(d1 , d2 ) ⇒ [∼ R(d1 , d2 )] ⎢ ⎢ | ∼2 R(d1 , d2 ) ⇒ [∼2 R(d1 , d2 )] ⎢ − ⎢ | ∼3 R(d , d ) ⇒ [∼3 R(d , d )] (∼ ∗ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ | ∼ R(dn , b) ⇒ [∼ R(dn , b)] ⎢ ⎣ | ∼2 R(dn , b) ⇒ [∼2 R(dn , b)] | ∼3 R(dn , b) ⇒ [∼3 R(dn , b)] |(∼ R ∗ )(a, b) ⇒ [(∼ R ∗ )(a, b)] ⎡ | ∼ R(a, d1 ) ⇒ [∼ R(a, d1 )] ⎢ |R(a, d1 ) ⇒ [R(a, d1 )] ⎢ ⎢ | ∼ R(d1 , d2 ) ⇒ [∼ R(d1 , d2 )] ⎢ ⎢ |R(d1 , d2 ) ⇒ [R(d1 , d2 )] (∼ ◦− ) ⎢ ⎢··· ⎢ ⎣ | ∼ R(dn , b) ⇒ [∼ R(dn , b)] |R(dn , b) ⇒ [R(dn , b)] |(∼ R ◦ )(a, b) ⇒ [(∼ R ◦ )(a, b)] and

where

⎡

| ∼2 R1 (a, b) ⇒ [∼2 R1 (a, b)] ⎢ | ∼2 R2 (a, b) ⇒ [∼2 R2 (a, b)] ⎢ ⎢ [Z 6 ]| ∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎢ ⎢ [Z 6 ]| ∼2 R2 (a, b) ⇒ [Z 6 , ∼2 R2 (a, b)] ⎢ ⎢ [Z 7 ]|R1 (a, b) ⇒ [Z 7 , R1 (a, b)] ⎢ 2 − ⎢ [Z ]| ∼2 R (a, b) ⇒ [Z , ∼2 R (a, b)] (∼ ∩ ) ⎢  7 2 7 2 ⎢ [Z 8 ]| ∼2 R1 (a, b) ⇒ [Z 8 , ∼2 R1 (a, b)] ⎢ ⎢ [Z 8 ]| ∼ R2 (a, b) ⇒ [Z 8 , ∼ R2 (a, b)] ⎢ ⎣ [Z 9 ]| ∼2 R1 (a, b) ⇒ [Z 9 , ∼2 R1 (a, b)] [Z 9 ]|R2 (a, b) ⇒ [Z 9 , R2 (a, b)] 2 2 | ∼ (R1 ∩ R2 )(a, b)] ⎡  (R21 ∩ R2 )(a, b) ⇒ [∼ 2 | ∼ R1 (a, b) ⇒ [∼ R1 (a, b)] ⎢ | ∼2 R2 (a, b) ⇒ [∼2 R2 (a, b)] ⎢ ⎢ [Z 6 ]| ∼3 R1 (a, b) ⇒ [Z 6 , ∼3 R1 (a, b)] ⎢ 2 − ⎢ [Z ]| ∼2 R (a, b) ⇒ [Z , ∼2 R (a, b)] (∼ ∪ ) ⎢  6 2 6 2 ⎣ [Z 10 ]| ∼2 R1 (a, b) ⇒ [Z 10 , ∼2 R1 (a, b)] [Z 10 ]| ∼3 R2 (a, b) ⇒ [Z 10 , ∼3 R2 (a, b)] | ∼2 (R1 ∪ R2 )(a, b) ⇒ [∼2 (R1 ∪ R2 )(a, b)]

10.2 1/4-Multisequents

343

Z 6 =∼2 R1 (a, b)∨ ∼2 R2 (a, b) Z 7 = Z 6 , ∼ R1 (a, b)∨ ∼2 R2 (a, b) Z 8 = Z 7 , R1 (a, b)∨ ∼2 R2 (a, b) Z 9 = Z 8 , ∼2 R1 (a, b)∨ ∼ R2 (a, b) Z 10 = Z 6 , ∼3 R1 (a, b)∨ ∼2 R2 (a, b), ⎡

and

| ∼2 R(a, d1 ) ⇒ [∼2 R(a, d1 )] ⎢ | ∼3 R(a, d1 ) ⇒ [∼3 R(a, d1 )] ⎢ ⎢ | ∼2 R(d1 , d2 ) ⇒ [∼2 R(d1 , d2 )] ⎢ ⎢ | ∼3 R(d1 , d2 ) ⇒ [∼3 R(d1 , d2 )] (∼2 ∗− ) ⎢ ⎢··· ⎢ ⎣ | ∼2 R(dn , b) ⇒ [∼2 R(dn , b)] | ∼3 R(dn , b) ⇒ [∼3 R(dn , b)] |(∼2 R ∗ )(a, b) ⇒ [(∼2 R ∗ )(a, b)] ⎡ | ∼2 R(a, d1 ) ⇒ [∼2 R(a, d1 )] ⎢ | ∼ R(a, d1 ) ⇒ [∼ R(a, d1 )] ⎢ ⎢ |R(a, d1 ) ⇒ [R(a, d1 )] ⎢ ⎢ | ∼2 R(d1 , d2 ) ⇒ [∼2 R(d1 , d2 )] ⎢ ⎢ | ∼ R(d1 , d2 ) ⇒ [∼ R(d1 , d2 )] ⎢ 2 − ⎢ |R(d , d ) ⇒ [R(d , d )] (∼ ◦ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ | ∼2 R(dn , b) ⇒ [∼2 R(dn , b)] ⎢ ⎣ | ∼ R(dn , b) ⇒ [∼ R(dn , b)] |R(dn , b) ⇒ [R(dn , b)] |(∼2 R ◦ )(a, b) ⇒ [(∼2 R ◦ )(a, b)]

and 

| ∼3 R1 (a, b) ⇒ [∼3 R1 (a, b)] (∼ ∩ ) [∼3 R1 (a, b)]| ∼3 R2 (a, b) ⇒ [∼3 R1 (a, b), ∼3 R2 (a, b)] 2 2 | 1 ∩ R2 )(a, b)) ⇒ [∼ (R1 ∩ R2 )(a, b)]  ∼ (R 3 3 | ∼ R1 (a, b) ⇒ [∼ R1 (a, b)] (∼3 ∪− ) | ∼3 R2 (a, b) ⇒ [∼3 R2 (a, b)] | ∼3 (R1 ∪ R2 )(a, b) ⇒ [∼3 (R1 ∪ R2 )(a, b)] 3

and

−

344

10 Role R-Calculus for Post L4 -Valued DL

⎡

| ∼3 R(a, d1 ) ⇒ [∼3 R(a, d1 )] ⎢ | ∼3 R(d1 , d2 ) ⇒ [∼3 R(d1 , d2 )] ⎢ (∼3 ∗− ) ⎣ · · · | ∼3 R(dn , b) ⇒ [∼3 R(dn , b)] |(∼3 R ∗ )(a, b) ⇒ [(∼3 R ∗ )(a, b)] ⎧ | ∼3 R(a, c1 ) ⇒ [∼3 R(a, c1 )] ⎪ ⎪ ⎨ | ∼3 R(c1 , c2 ) ⇒ [∼3 R(c1 , c2 )] 3 − ··· (∼ ◦ ) ⎪ ⎪ ⎩ | ∼3 R(cn , b) ⇒ [∼3 R(cn , b)] |(∼3 R ◦ )(a, b) ⇒ [(∼3 R ◦ )(a, b)] where ci is a new constant and di is a constant. Definition 10.2.8 Given a 1/4-multisequent  and a statement R(a, b) ∈ , a 1/4=t reduction  ↑ R(a, b) ⇒   is provable in St1/4 , denoted by 1/4  ↑ R(a, b) ⇒   , if there is a sequence {δ1 , . . . , δn } of 1/4-reductions such that δn =  ↑ R(a, b) ⇒   , and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 1/4reductions by one of the deduction rules in St1/4 . Theorem 10.2.9 (Soundness and completeness theorem) For any reduction |R (a, b) ⇒ [R(a, b)], t1/4 |R(a, b) ⇒ [R  (a, b)] iff |=t1/4 |R(a, b) ⇒ [R  (a, b)].  Theorem 10.2.10 (Soundness and incompleteness theorem) For any reduction |R(a, b) ⇒ , t1/4 |R(a, b) ⇒  implies |=t1/4 |R(a, b) ⇒ , and

|=t1/4 |R(a, b) ⇒  may not imply t1/4 |R(a, b) ⇒ . 

10.3 2/4-Multisequents =t

t A 2/4-multisequent | is N2/4 -valid, denoted by |=2/4 |, if for any interpretation I, either I (R(a, b)) = t for some R(a, b) ∈ , or I (Q(a, b)) =  for some Q(a, b) ∈ . We have the following equivalences:

, ∼ A| ≡ |, A |, ∼3 B ≡ , B| 2 , ∼ A| ≡ |, ∼ A , ∼3 A| ≡ |, ∼2 A.

10.3 2/4-Multisequents

345

t 10.3.1 Deduction System N2/4 t Deduction system N2/4 contains the following axiom and deduction rules. • Axiom: ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎪ ⎨ ∩ ∼−2  = ∅ (At ) ⎪ ∩ ∼−  = ∅ 2/4 ⎪ ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎩ − ∼ ∩ ∼−2  = ∅ |

where  is a set of atoms and  is a set of literals. • Deduction rules for unary connectives: |, R(a, b) , Q(a, b)| (∼3Q ) , ∼ R(a, b)| |, ∼3 Q(a, b) |, ∼2 R(a, b) |, ∼2 R(a, b) (∼3R ) (∼2 R) 2 , ∼ R(a, b)| , ∼3 R(a, b)|

(∼ R )

• Deduction rules for logical connectives: ⎡



(∩ R )

, R1 (a, b)| , R2 (a, b)| , (R1 ∩ R2 )(a, b)| ⎡

|, Q 1 (a, b) ⎢ |, Q 2 (a, b) ⎢ ⎢ , Q 1 (a, b)| ⎢ Q ⎢ |, Q (a, b) (∩ ) ⎢  2 ⎣ |, Q 1 (a, b) , Q 2 (a, b)| |, (Q 1 ∩ Q 2 )(a, b)

|, ∼ Q 1 (a, b) ⎢ |, ∼ Q 2 (a, b) ⎢ ⎢ , Q 1 (a, b)| ⎢ ⎢ |, ∼ Q 2 (a, b) ⎢  ⎢ |, Q 1 (a, b) |, ∼2 Q 1 (a, b) ⎢ Q ⎢ |, ∼ Q (a, b) 2 Q (∼ ∩ ) ⎢  (∼ ∩ ) |, ∼2 Q 2 (a, b) 2 ⎢ |, ∼ Q 1 (a, b) |, ∼2 (Q 1 ∩ Q 2 )(a, b) ⎢ ⎢ |, Q 2 (a, b) ⎢ ⎣ |, ∼ Q 1 (a, b) , Q 2 (a, b)| |, ∼ (Q 1 ∩ Q 2 )(a, b) and

346

10 Role R-Calculus for Post L4 -Valued DL

⎡



(∪ R )

, R1 (a, b)| , R2 (a, b)| , (R1 ∪ R2 )(a, b)|

⎡

|, Q 1 (a, b) ⎢ |, Q 2 (a, b) ⎢ ⎢ |, ∼ Q 1 (a, b) ⎢ ⎢ |, Q 2 (a, b) ⎢ ⎢ |, ∼2 Q 1 (a, b) ⎢ Q ⎢ |, Q (a, b) (∪ ) ⎢  2 ⎢ |, Q 1 (a, b) ⎢ ⎢ |, ∼ Q 2 (a, b) ⎢ ⎣ |, Q 1 (a, b) |, ∼2 Q 2 (a, b) |, (Q 1 ∪ Q 2 )(a, b)

|, ∼ Q 1 (a, b) ⎢ |, ∼ Q 2 (a, b) ⎢  ⎢ |, ∼ Q 1 (a, b) |, ∼2 Q 1 (a, b) ⎢ Q ⎢ |, ∼2 Q (a, b) 2 Q (∼ ∪ ) ⎢  (∼ ∪ ) |, ∼2 Q 2 (a, b) 2 ⎣ |, ∼2 Q 1 (a, b) , ∼2 (Q 1 ∪ Q 2 )(a, b)| |, ∼ Q 2 (a, b) , ∼ (Q 1 ∪ Q 2 )(a, b)| • Deduction rules for quantifier constructors: ⎡

⎧ , R(a, c)| ⎪ ⎪ ⎨ , R(c1 , c2 )| R · ·· (∗ ) ⎪ ⎪ ⎩ , R(cn , b)| , R ∗ (a, b)|

⎡

|, Q(a, d1 ) ⎢ |, ∼ Q(a, d1 ) ⎢ ⎢ |, ∼2 Q(a, d1 ) ⎢ ⎢ |, Q(d1 , d2 ) ⎢ ⎢ |, ∼ Q(d1 , d2 ) ⎢ Q ⎢ |, ∼2 Q(d , d ) (∗ ) ⎢ 1 2 ⎢··· ⎢ ⎢ |, Q(dn , b) ⎢ ⎣ |, ∼ Q(dn , b) |, ∼2 Q(dn , b) |, Q ∗ (a, b)

|, ∼ Q(a, d1 ) ⎢ |, ∼2 Q(a, d1 ) ⎢ ⎢ |, ∼ Q(d1 , d2 ) ⎢ 2 ⎢ Q ⎢ |, ∼ Q(d1 , d2 ) (∼ ∗ ) ⎢ (∼2 ∗ Q ) · · · ⎢ ⎣ |, ∼ Q(dn , b) |, ∼2 Q(dn , b) |, (∼ Q ∗ )(a, b) and

⎡

|, ∼2 Q(a, d1 ) ⎢ |, ∼2 Q(d1 , d2 ) ⎢ ⎣··· |, ∼2 Q(dn , b) |, (∼2 Q ∗ )(a, b)

10.3 2/4-Multisequents

347

⎡

⎡

, R(a, d1 )| ⎢ , R(d1 , d2 )| ⎢ (◦ R ) ⎣ · · · , R(dn , b)| , R ◦ (a, b)| ⎡

|, Q(a, d1 ) ⎢ , Q(a, d1 )| ⎢ ⎢ |, Q(d1 , d2 ) ⎢ ⎢ , Q(d1 , d2 )| (◦ Q ) ⎢ ⎢··· ⎢ ⎣ |, Q(dn , b) , Q(dn , b)| |, Q ◦ (a, b)

, Q(a, d1 )| ⎢ |, Q(a, d1 ) ⎢ ⎢ |, ∼ Q(a, d1 ) ⎢ ⎢ , Q(d1 , d2 )| ⎢ ⎢ |, Q(d1 , d2 ) ⎢ Q ⎢ |, ∼ Q(d , d ) (∼ ◦ ) ⎢ (∼2 ◦ Q ) 1 2 ⎢··· ⎢ ⎢ , Q(dn , b)| ⎢ ⎣ |, Q(dn , b) |, ∼ Q(dn , b) |, (∼ Q ◦ )(a, b)

⎧ 2 ⎪ ⎪ |, ∼2 Q(a, c1 ) ⎨ |, ∼ Q(c1 , c2 ) ··· ⎪ ⎪ ⎩ |, ∼2 Q(cn , b) |, (∼2 Q ◦ )(a, b)

where di is a constant and ci is a new constant. =t

t Definition 10.3.1 A 2/4-multisequent | is proof in N2/4 , denoted by 2/4 |, if there is a sequence {1 |1 , . . . , n |n } of 2/4-multisequents such that n |n = |, and for each 1 ≤ i ≤ n, i |i is either an axiom or deduced from the previous t . 2/4-multisequents by one of the deduction rules in N2/4

Theorem 10.3.2 (Soundness and completeness theorem) For any 2/4-multisequent |, =t =t |=2/4 | iff 2/4 |. 

10.3.2 R-Calculus St 2/4 Given a 2/4-multisequent | and two statements R(a, b) ∈  and Q(a  , b ) ∈ , using | to revise (R(a, b), Q(a  , b )) and obtaining [R  (a, b)]|[Q  (a  , b )], denoted by a reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )], =t

is St 2/4 -valid, denoted by |=2/4 δ, if

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10 Role R-Calculus for Post L4 -Valued DL



t R(a, b) if [R(a, b)]| is N2/4 -valid λ otherwise.  t -valid Q(a  , b ) if [R  (a, b)]|[Q(a  , b )] is N2/4    Q (a , b ) = λ otherwise.

R  (a, b) =

R-calculus St 2/4 consists of the following deduction rules and axioms: • Axioms: ⎧ [r (a, b)] ∩  = ∅ ⎪ ⎪ ⎪ ⎪ [r (a, b)]∩ ∼−  = ∅ ⎪ ⎪ ⎨ [r (a, b)]∩ ∼−2  = ∅ Rt− (A2/4 ) ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎩ − ∼ ∩ ∼−2  = ∅ | ⎧ ↑ r (a, b) ⇒ [r (a, b)]|  ∩ [q(a , b )] = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎪ ⎨ ∩ ∼−2  = ∅ Qt− [q(a  , b )]∩ ∼−  = ∅ (A2/4 ) ⎪ ⎪ ⎪ ⎪ ⎪ [q(a  , b )]∩ ∼−2  = ∅ ⎪ ⎩ − ∼ ∩ ∼−2  = ∅     | ⎧ ↑ q(a , b ) ⇒ |[q(a , b )]  ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼− [∼ q(a  , b )] = ∅ ⎪ ⎪ ⎨ ∩ ∼−2  = ∅ ∼Qt− ∩ ∼− [∼ q(a  , b )] = ∅ ) ⎪ (A2/4 ⎪ ⎪ ⎪ ∩ ∼−2  = ∅ ⎪ ⎪ ⎩ − ∼ [∼ q(a  , b )]∩ ∼−2  = ∅ 2 2 | ⎧ ↑∼ r (a, b) ⇒ [∼ r (a, b)]|  ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎪ ⎨ ∩ ∼−2 [∼2 q(a  , b )] = ∅ ∼2 Qt− ∩ ∼−  = ∅ ) ⎪ (A2/4 ⎪ ⎪ ⎪ ∩ ∼−2 [∼2 q(a  , b )] = ∅ ⎪ ⎪ ⎩ − ∼ ∩ ∼−2 [∼2 q(a  , b )] = ∅ | ↑∼2 q(a  , b ) ⇒ |[∼2 q(a  , b )] where , r (a, b), q(a  , b ) is a set of role atoms and  is a set of role literals.

10.3 2/4-Multisequents

• Deduction rules: | ↑ 2 R(a, b) ⇒ |[R(a, b)] | ↑∼ R(a, b) ⇒ [∼ R(a, b)]| 2 2R− | ↑ ∼ R(a, b) ⇒ |[∼ R(a, b)] (∼ ) | ↑∼2 R(a, b) ⇒ [∼2 R(a, b)]| | ↑ 2 ∼2 R(a, b) ⇒ |[∼2 R(a, b)] (∼3R− ) | ↑∼3 R(a, b) ⇒ [∼3 R(a, b)]| | ↑ 1 Q(a, b) ⇒ [Q(a, b)]| (∼3Q− ) | ↑∼3 Q(a, b) ⇒ |[∼3 Q(a, b)]

(∼ R− )

and



| ↑ R1 (a, b) ⇒ [R1 (a, b)]| | ↑ R2 (a, b) ⇒ [R2 (a, b)]| | ⎡  ↑ (R1 ∩ R2 )(a, b) ⇒ [(R1 ∩ R2 )(a, b)]| | ↑ Q 1 (a, b) ⇒ |[Q 1 (a, b)] ⎢ | ↑ Q 2 (a, b) ⇒ |[Q 2 (a, b)] ⎢ ⎢ [Z 1 ]| ↑ 1 Q 1 (a, b) ⇒ [Z 1 , Q 1 (a, b)]| ⎢ Q− ⎢ [Z ]| ↑ Q (a, b) ⇒ [Z , Q (a, b)]|[Q (a, b)] (∩ ) ⎢  1 2 1 1 2 ⎣ [Z 2 ]| ↑ Q 1 (a, b) ⇒ [Z 2 ]|[Q 1 (a, b)] [Z 2 ]| ↑ 1 Q 2 (a, b) ⇒ [Z 2 , Q 2 (a, b)]| | ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ [(Q 1 ∩ Q 2 )(a, b)]|

(∩ R− )

where

Z 1 = Q 1 (a, b) ∨ Q 2 (a, b)] and Z 2 = Z 1 , ∼3 Q 1 (a, b) ∨ Q 2 (a, b), ⎡

| ↑∼ Q 1 (a, b) ⇒ |[∼ Q 1 (a, b)] ⎢ | ↑∼ Q 2 (a, b) ⇒ |[∼ Q 2 (a, b)] ⎢ ⎢ |[Z 3 ] ↑ Q 1 (a, b) ⇒ |[Z 3 , Q 1 (a, b)] ⎢ ⎢ |[Z 3 ] ↑∼ Q 2 (a, b) ⇒ |[Z 3 , ∼ Q 2 (a, b)] ⎢ ⎢ |[Z 4 ] ↑ 1 Q 1 (a, b) ⇒ [Q 1 (a, b)]|[Z 4 ] ⎢ Q− ⎢ (∼ ∩ ) ⎢  |[Z 4 ] ↑∼ Q 2 (a, b) ⇒ |[Z 4 , ∼ Q 2 (a, b)] ⎢ |[Z 5 ] ↑∼ Q 1 (a, b) ⇒ |[Z 5 , ∼ Q 1 (a, b)] ⎢ ⎢ |[Z 5 ] ↑ Q 2 (a, b) ⇒ |[Z 5 , Q 2 (a, b)] ⎢ ⎣ |[Z 6 ] ↑∼ Q 1 (a, b) ⇒ |[Z 6 , ∼ Q 1 (a, b)] |[Z 6 ] ↑ 1 Q 2 (a, b) ⇒ [Q 2 (a, b)]|[Z 5 ] | ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ [(Q 1 ∩ Q 2 )(a, b)]| Z3 Z4 where Z5 Z6

=∼ Q 1 (a, b)∨ ∼ Q 2 (a, b) = Z 3 , Q 1 (a, b)∨ ∼ Q 2 (a, b) and = Z 4 , ∼3 Q 1 (a, b)∨ ∼ Q 2 (a, b) = Z 5 , ∼ Q 1 (a, b) ∨ Q 2 (a, b),

349

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10 Role R-Calculus for Post L4 -Valued DL



(∼2 ∩ Q− )

| ↑∼2 Q 1 (a, b) ⇒ |[∼2 Q 1 (a, b)] |[∼2 Q 1 (a, b)] ↑∼2 Q 2 (a, b) ⇒ |[∼2 Q 1 (a, b), ∼2 Q 2 (a, b)] | ↑∼2 (Q 1 ∩ Q 2 )(a, b) ⇒ |[∼2 (Q 1 ∪ Q 2 )(a, b)]



and

| ↑ R1 (a, b) ⇒ [R1 (a, b)]| (∪ ) [R1 (a, b)]| ↑ R2 (a, b) ⇒ [R1 (a, b), R2 (a, b)]| | ⎡  ↑ (R1 ∪ R2 )(a, b) ⇒ [(R1 ∪ R2 )(a, b)]| | ↑ Q 1 (a, b) ⇒ |[Q 1 (a, b)] ⎢ | ↑ Q 2 (a, b) ⇒ |[Q 2 (a, b)] ⎢ ⎢ |[Y1 ] ↑∼ Q 1 (a, b) ⇒ |[Y1 , ∼ Q 1 (a, b)] ⎢ ⎢ |[Y1 ] ↑ Q 2 (a, b) ⇒ |[Y1 , Q 2 (a, b)] ⎢ ⎢ |[Y2 ] ↑∼2 Q 1 (a, b) ⇒ |[Y2 , ∼2 Q 1 (a, b)] ⎢ Q− ⎢ |[Y ] ↑ Q (a, b) ⇒ |[Y , Q (a, b)] (∪ ) ⎢  2 2 2 2 ⎢ |[Y3 ] ↑ Q 1 (a, b) ⇒ |[Y3 , Q 1 (a, b)] ⎢ ⎢ |[Y3 ] ↑∼ Q 2 (a, b) ⇒ |[Y3 , ∼ Q 2 (a, b)] ⎢ ⎣ |[Y4 ] ↑ Q 1 (a, b) ⇒ |[Y4 , Q 1 (a, b)] |[Y4 ] ↑∼2 Q 2 (a, b) ⇒ |[Y4 , ∼2 Q 2 (a, b)] | ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ |[(Q 1 ∪ Q 2 )(a, b) R−

Y1 Y2 where Y3 Y4

= Q 1 (a, b) ∨ Q 2 (a, b) = Y1 , ∼ Q 1 (a, b) ∨ Q 2 (a, b) and = Y2 , ∼2 Q 1 (a, b) ∨ Q 2 (a, b) = Y3 , Q 1 (a, b)∨ ∼ Q 2 (a, b), ⎡

| ↑∼ Q 1 (a, b) ⇒ |[∼ Q 1 (a, b)] ⎢ | ↑∼ Q 2 (a, b) ⇒ |[∼ Q 2 (a, b)] ⎢ ⎢ |[Y5 ] ↑∼ Q 1 (a, b) ⇒ |[Y5 , ∼ Q 1 (a, b)] ⎢ Q− ⎢ |[Y ] ↑∼2 Q (a, b) ⇒ |[Y , ∼2 Q (a, b)] (∼ ∪ ) ⎢  5 2 5 2 ⎣ |[Y6 ] ↑∼2 Q 1 (a, b) ⇒ |[Y6 , ∼2 Q 1 (a, b)] |[Y6 ] ↑∼ Q 2 (a, b) ⇒ |[Y6 , ∼ Q 2 (a, b)] | ↑∼ (Q 1 ∪ Q 2 )(a, b) ⇒ |[∼ (Q 1 ∪ Q 2 )(a, b) where

Y5 =∼ Q 1 (a, b)∨ ∼ Q 2 (a, b) and Y6 = Y5 , ∼ Q 1 (a, b)∨ ∼2 Q 2 (a, b) 

(∼ ∪ 2

and

Q−

| ↑∼2 Q 1 (a, b) ⇒ |[∼2 Q 1 (a, b)] ) | ↑∼2 Q 2 (a, b) ⇒ |[∼2 Q 2 (a, b)] | ↑∼2 (Q 1 ∪ Q 2 )(a, b) ⇒ |[∼2 (Q 1 ∪ Q 2 )(a, b)

10.3 2/4-Multisequents

351

⎧ ⎪ ⎪ | ↑ R(a, c1 ) ⇒ [R(a, c1 )]| ⎨ | ↑ R(c1 , c2 ) ⇒ [R(c1 , c2 )]| · ·· (∗ R− ) ⎪ ⎪ ⎩ | ↑ R(cn , b) ⇒ [R(cn , b)]| ∗ ∗ | ⎡ ↑ R (a, b) ⇒ [R (a, b)]| | ↑ Q(a, d1 ) ⇒ |[Q(a, d1 )] ⎢ | ↑∼ Q(a, d1 ) ⇒ |[∼ Q(a, d1 )] ⎢ ⎢ | ↑∼2 Q(a, d1 ) ⇒ |[∼2 Q(a, d1 )] ⎢ ⎢ | ↑ Q(d1 , d2 ) ⇒ |[Q(d1 , d2 )] ⎢ ⎢ | ↑∼ Q(d1 , d2 ) ⇒ |[∼ Q(d1 , d2 )] ⎢ Q− ⎢ | ↑∼2 Q(d , d ) ⇒ |[∼2 Q(d , d )] (∗ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ | ↑ Q(dn , b) ⇒ |[Q(dn , b)] ⎢ ⎣ | ↑∼ Q(dn , b) ⇒ |[∼ Q(dn , b)] | ↑∼2 Q(dn , b) ⇒ |[∼2 Q(dn , b)] | ↑ Q ∗ (a, b) ⇒ |[Q ∗ (a, b)] and

and

⎡

| ↑∼ Q(a, d1 ) ⇒ |[∼ Q(a, d1 )] ⎢ | ↑∼2 Q(a, d1 ) ⇒ |[∼2 Q(a, d1 )] ⎢ ⎢ | ↑∼ Q(d1 , d2 ) ⇒ |[∼ Q(d1 , d2 )] ⎢ 2 2 ⎢ Q− ⎢ | ↑∼ Q(d1 , d2 ) ⇒ |[∼ Q(d1 , d2 )] (∼ ∗ ) ⎢ ⎢··· ⎣ | ↑∼ Q(dn , b) ⇒ |[∼ Q(dn , b)] | ↑∼2 Q(dn , b) ⇒ |[∼2 Q(dn , b)] ∗ ∗ | ⎡ ↑ (∼ Q2 )(a, b) ⇒ |[∼ Q2 (a, b)] | ↑∼ Q(a, d1 ) ⇒ |[∼ Q(a, d1 )] ⎢ | ↑∼2 Q(d1 , d2 ) ⇒ |[∼2 Q(d1 , d2 )] ⎢ (∼2 ∗ Q− ) ⎣ · · · | ↑∼2 Q(dn , b) ⇒ |[∼2 Q(dn , b)] | ↑ (∼2 Q ∗ )(a, b) ⇒ |[∼2 Q ∗ (a, b)]

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10 Role R-Calculus for Post L4 -Valued DL

⎡

| ↑ R(a, d1 ) ⇒ [R(a, d1 )]| ⎢ | ↑ R(d1 , d2 ) ⇒ [R(d1 , d2 )]| ⎢ (◦ R− ) ⎣ · · · | ↑ R(dn , b) ⇒ [R(dn , b)]| ◦ ◦ | ⎡ ↑ R (a, b) ⇒ [R (a, b)]| | ↑ Q(a, d1 ) ⇒ |[Q(a, d1 )] ⎢ | ↑ 1 Q(a, d1 ) ⇒ [Q(a, d1 )]| ⎢ ⎢ | ↑ Q(d1 , d2 ) ⇒ |[Q(d1 , d2 )] ⎢ ⎢ | ↑ 1 Q(d1 , d2 ) ⇒ [Q(d1 , d2 )]| (◦ Q− ) ⎢ ⎢··· ⎢ ⎣ | ↑ Q(dn , b) ⇒ |[Q(dn , b)] | ↑ 1 Q(dn , b) ⇒ [Q(dn , b)]| | ↑ Q ◦ (a, b) ⇒ |[Q ◦ (a, b)] and

⎡

| ↑ 1 Q(a, d1 ) ⇒ [Q(a, d1 )]| ⎢ | ↑ Q(a, d1 ) ⇒ |[Q(a, d1 )] ⎢ ⎢ | ↑∼ Q(a, d1 ) ⇒ |[∼ Q(a, d1 )] ⎢ ⎢ | ↑ 1 Q(d1 , d2 ) ⇒ [Q(d1 , d2 )]| ⎢ ⎢ | ↑ Q(d1 , d2 ) ⇒ |[Q(d1 , d2 )] ⎢ Q− ⎢ | ↑∼ Q(d , d ) ⇒ |[∼ Q(d , d )] (∼ ◦ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ | ↑ 1 Q(dn , b) ⇒ [Q(dn , b)]| ⎢ ⎣ | ↑ Q(dn , b) ⇒ |[Q(dn , b)] | ↑∼ Q(dn , b) ⇒ |[∼ Q(dn , b)] ◦ ◦ | ⎧ ↑ (∼ Q2 )(a, b) ⇒ |[∼ Q2 (a, b)] | ↑∼ Q(a, c1 ) ⇒ |[∼ Q(a, c1 )] ⎪ ⎪ ⎨ | ↑∼2 Q(c1 , c2 ) ⇒ |[∼2 Q(c1 , c2 )] 2 Q− ··· (∼ ◦ ) ⎪ ⎪ ⎩ | ↑∼2 Q(cn , b) ⇒ |[∼2 Q(cn , b)] | ↑ (∼2 Q ◦ )(a, b) ⇒ |[∼2 Q ◦ (a, b)]

where ci is a new constant and di is a constant. Definition 10.3.3 A 2/4-reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ [R  (a, b)]| =t [Q  (a  , b )] is provable in Qt 2/4 , denoted by 2/4 δ, if there is a sequence {δ1 , . . . , δn } of 2/4-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 2/4-reductions by one of the deduction rules in Qt 2/4 . Theorem 10.3.4 (Soundness and completeness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ , R  (a, b)|, Q  (a  , b ) such that one of R  (a, b), Q  (a  , b ) is not the empty string, =t =t 2/4 δ iff |=2/4 δ. 

10.4 3/4-Multisequents

353

Theorem 10.3.5 (Soundness and incompleteness theorem) For any reduction δ = | ↑ (R(a, b), Q(a  , b )) ⇒ |, =t

=t

2/4 δ implies |=2/4 δ, and

=t

=t

|=2/4 δ may not imply 2/4 δ. 

10.4 3/4-Multisequents =t⊥

t⊥ A 3/4-multisequent || is N3/4 -valid, denoted by |=3/4 ||, if for any interpretation I, either I (R(a, b)) = t for some R(a, b) ∈ , I (Q(a, b)) =  for some Q(a, b) ∈ , or I (P(a, b)) =⊥ for some P(a, b) ∈ . We have the following equivalences:

, ∼ R(a, b)|| ≡ |, R(a, b)| |, ∼ Q(a, b)| ≡ ||, Q(a, b) ||, ∼2 P(a, b) ≡ , P(a, b)||.

t⊥ 10.4.1 Deduction System N3/4 t⊥ Deduction system N3/4 contains the following axiom and deduction rules: • Axiom: ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎨ ∩ ∼−  = ∅ t⊥  (A3/4 ) ⎪ ⎪ ∩  = ∅ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎩ ∩ ∼−  = ∅ ||

where , ,  are sets of literal statements. • Deduction rules for unary logical connectives: (∼ R )

|, R(a, b)| ||, Q(a, b) , P(a, b)|| (∼ Q ) (∼2P ) , ∼ R(a, b)|| |, ∼ Q(a, b)| ||, ∼2 P(a, b)

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10 Role R-Calculus for Post L4 -Valued DL

• Deduction rules for binary logical connective ∩: ⎡

|, Q 1 (a, b)| ⎢ |, Q 2 (a, b)| ⎢  ⎢ , Q 1 (a, b)|| , R1 (a, b)|| ⎢ R Q (∩ ) , R2 (a, b)|| (∩ ) ⎢ ⎢  |, Q 2 (a, b)| ⎣ |, Q 1 (a, b)| , (R1 ∩ R2 )(a, b)|| , Q 2 (a, b)|| |, (Q 1 ∩ Q 2 )(a, b)| ⎡ ||, P1 (a, b) ⎢ ||, P2 (a, b) ⎢ ⎢ |, P1 (a, b)| ⎢ ⎢ ||, P2 (a, b) ⎢  ⎢ , P1 (a, b)|| ||, ∼ P1 (a, b) ⎢ P (a, b) ||, P ||, ∼ P2 (a, b) (∩ P ) ⎢ (∼ ∩ ) 2 ⎢ ⎢ ||, P1 (a, b) ||, ∼ (P1 ∩ P2 )(a, b) ⎢ ⎢ , P2 (a, b)|| ⎢ ⎣ ||, P1 (a, b) |, P2 (a, b)| ||, (P1 ∩ P2 )(a, b) • Deduction rules for binary logical connective ∪: ⎡

|, Q 1 (a, b)| ⎢ |, Q 2 (a, b)| ⎢ ⎢ ||, Q 1 (a, b) ⎢ ⎢ |, Q 2 (a, b)| ⎢  ⎢ ||, ∼ Q 1 (a, b) , R1 (a, b)|| ⎢ R Q (∪ ) , R2 (a, b)|| (∪ ) ⎢ ⎢  |, Q 2 (a, b)| ⎢ |, Q 1 (a, b)| , (R1 ∪ R2 )(a, b)|| ⎢ ⎢ ||, ∼ Q 2 (a, b) ⎢ ⎣ |, Q 1 (a, b)| ||, Q 2 (a, b) |, (Q 1 ∪ Q 2 )(a, b)| ⎡ ||, P1 (a, b) ⎢ ||, P2 (a, b) ⎢  ⎢ ||, ∼ P1 (a, b) ||, ∼ P1 (a, b) ⎢ P (a, b) ||, P ||, ∼ P2 (a, b) ) (∼ ∪ (∪ P ) ⎢ 2 ⎢ ⎣ ||, P1 (a, b) ||, ∼ (P1 ∪ P2 )(a, b) ||, ∼ P2 (a, b) ||, (P1 ∪ P2 )(a, b)

10.4 3/4-Multisequents

355

• Deduction rules for constructors ∗ and ◦: ⎡ ⎧ , R(a, c1 )|| ⎪ ⎪ ⎨ , R(c1 , c2 )|| R ··· (∗ ) ⎪ ⎪ ⎩ , R(cn , b)|| , R ∗ (a, b)||

⎡

|, Q(a, d1 )| ⎢ ||, Q(a, d1 ) ⎢ ⎢ ||, ∼ Q(a, d1 ) ⎢ ⎢ |, Q(d1 , d2 )| ⎢ ⎢ ||, Q(d2 , d2 ) ⎢ Q ⎢ ||, ∼ Q(d , d ) (∗ ) ⎢ 2 2 ⎢··· ⎢ ⎢ |, Q(dn , b)| ⎢ ⎣ ||, Q(dn , b) ||, ∼ Q(dn , b) |, Q ∗ (a, b)|

||, P(a, d1 ) ⎢ ||, ∼ P(a, d1 ) ⎢ ⎢ ||, P(d1 , d2 ) ⎢ ⎢ P ⎢ ||, ∼ P(d1 , d2 ) (∗ ) ⎢ (∼ ∗ P ) ⎢··· ⎣ ||, P(dn , b) ||, ∼ P(dn , b) ||, P ∗ (a, b) and

⎡

⎡

||, ∼ P(a, d1 ) ⎢ ||, ∼ P(d1 , d2 ) ⎢ ⎣··· ||, ∼ P(dn , b) ||, (∼ P ∗ )(a, b)

|, Q(a, d1 )| ⎢ , Q(a, d1 )|| ⎢ , R(a, d1 )|| ⎢ |, Q(d1 , d2 )| ⎢ ⎢ , R(d1 , d2 )|| ⎢ , Q(d1 , d2 )|| ⎢ (◦ R ) ⎣ · · · (◦ Q ) ⎢ ⎢··· ⎢ , R(dn , b)|| ⎣ |, Q(dn , b)| , R ◦ (a, b)|| , Q(dn , b)|| |, Q ◦ (a, b)| ⎡ ||, P(a, d1 ) ⎢ |, P(a, d1 )| ⎢ ⎢ , P(a, d1 )|| ⎢ ⎧ ⎢ ||, P(d1 , d2 ) ||, ∼ P(a, c1 ) ⎪ ⎢ ⎪ ⎨ ⎢ |, P(d1 , d2 )| ||, ∼ P(c1 , c2 ) ⎢ P , d )|| , P(d ··· (◦ P ) ⎢ (∼ ◦ ) ⎪ 1 2 ⎢ ⎪ ⎩ ⎢··· ||, ∼ P(cn , b) ⎢ ⎢ ||, P(dn , b) ||, (∼ P ◦ )(a, b) ⎢ ⎣ |, P(dn , b)| , P(dn , b)|| ||, P ◦ (a, b) ⎡

where di is a constant and ci is a new constant.

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10 Role R-Calculus for Post L4 -Valued DL

t⊥ Definition 10.4.1 A 3/4-multisequent || is provable in N3/4 , denoted by =t⊥ 3/4 || if there is a sequence {1 |1 |1 , . . . , n |n |n } of 3/4-multisequents such that n |n |n = ||, and for each 1 ≤ i ≤ n, i |i |i is either an axiom or deduced from the previous 3/4-multisequents by one of the deduction rules in t⊥ . N3/4

Theorem 10.4.2 (Soundness and completeness theorem) For any 3/4-multisequent ||, =t⊥ =t⊥ |=3/4 || iff 3/4 ||. 

10.4.2 R-Calculus St⊥ 3/4 Given statements R(a  , b ) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈ , a reduction δ = || ↑ (R(a  , b ), Q(a  , b ), P(a  , b )) ⇒ [R  (a  , b )]|[Q  (a  , b )]|[P  (a  , b )] =t⊥

is St⊥ 3/4 -valid, denoted by |=3/4 

δ, if

=t⊥

R(a  , b ) if |=3/4 [R(a  , b )]|| λ otherwise  =t⊥   , b ) if |=3/4 [R  (a  , b )]|[Q(a  , b )]| Q(a Q  (a  , b ) = λ otherwise  =t⊥   P(a , b ) if |=3/4 [R  (a  , b )]|[Q  (a  , b )]|[P(a  , b )] P  (a  , b ) = λ otherwise. R  (a  , b ) =

Given any statements R(a, b) ∈ , Q(a  , b ) ∈ , and P(a  , b ) ∈ , let X = || and X[R(a, b)] =  − {R(a, b)}|| X[Q(a, b)] = | − {Q(a, b)}| X[P(a, b)] = || − {P(a, b)};

X[1 R(a, b)] =  − {R(a, b)}|| X[2 R(a, b)] = | − {R(a, b)}| X[3 R(a, b)] = || − {R(a, b)}.

R-calculus St⊥ 3/4 consists of the following axioms and deduction rules:

10.4 3/4-Multisequents

357

• Axioms: ⎧ [r (a, b)] ∩  = ∅ ⎪ ⎪ ⎪ ⎪ [r (a, b)] ∩  = ∅ ⎪ ⎪ ⎨ [r (a, b)]∩ ∼−  = ∅ Rt⊥−  ∩  = ∅ (A3/4 ) ⎪ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎪ ⎩ ∩ ∼−  = ∅ || ↑ r (a, b) ⇒ [r (a, b)]|| ⎧ ⎪  ∩ [q(a  , b )] = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎨ ∩ ∼−  = ∅ Qt⊥− ) ⎪ (A3/4 ⎪ [q(a  , b )] ∩  = ∅ ⎪ ⎪ ⎪ [q(a  , b )]∩ ∼−  = ∅ ⎪ ⎩ ∩ ∼−  = ∅ || ↑ q(a  , b ) ⇒ |[q(a  , b )]| ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ⎪  ∩ [ p(a  , b )] = ∅ ⎪ ⎨ ∩ ∼−  = ∅ Pt⊥− ) ⎪ (A3/4 ⎪  ∩ [ p(a  , b )] = ∅ ⎪ ⎪ ⎪ ∩ ∼−  = ∅ ⎪ ⎩ [ p(a  , b )]∩ ∼−  = ∅ || ↑ p(a  , b ) ⇒ ||[ p(a  , b )] ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎨ ∩ ∼− [∼ p(a  , b )] = ∅ ∼Pt⊥−  ∩  = ∅ )⎪ (A3/4 ⎪ ⎪ ⎪ ∩ ∼− [∼ p(a  , b )] = ∅ ⎪ ⎪ ⎩ ∩ ∼− [∼ p(a  , b )] = ∅ || ↑∼ p(a  , b ) ⇒ ||[∼ p(a  , b )] where ,  are sets of role atoms,  is a set of role literals, and r (a, b), q(a  , b ), p(a  , b ) are role atoms. • Deduction rules: X ↑ 2 R(a, b) ⇒ X[2 R(a, b)] X ↑∼ R(a, b) ⇒ X[∼ R(a, b)] X ↑ 3 Q(a, b) ⇒ X[3 Q(a, b)] (∼ Q− ) X ↑∼ Q(a, b) ⇒ X[∼ Q(a, b)] X ↑ 1 P(a, b) ⇒ X[1 P(a, b)] (∼2P− ) X ↑∼2 P(a, b) ⇒ X[∼2 P(a, b)]

(∼ R− )

and

358

10 Role R-Calculus for Post L4 -Valued DL



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X ↑ R2 (a, b) ⇒ X[R2 (a, b)] X ↑ (R1 ∩ R2 )(a, b) ⇒ X[(R1 ∩ R2 )(a, b)] ⎡ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎢ X ↑ Q 2 (a, b) ⇒ X[Q 2 (a, b)] ⎢ ⎢ X[Z 1 ] ↑ 1 Q 1 (a, b) ⇒ X[Z 1 , 1 Q 1 (a, b)] ⎢ Q− ⎢ X[Z ] ↑ Q (a, b) ⇒ X[Z , Q (a, b)] (∩ ) ⎢  1 2 1 2 ⎣ X[Z 2 ] ↑ Q 1 (a, b) ⇒ X[Z 2 , Q 1 (a, b)] X[Z 2 ] ↑ 1 Q 2 (a, b) ⇒ X[Z 2 , 1 Q 2 (a, b)] X ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ X[(Q 1 ∩ Q 2 )(a, b)]

(∩ R− )

where

Z 1 = Q 1 (a, b) ∨ Q 2 (a, b) and Z 2 = Z 1 , 1 Q 1 (a, b) ∨ Q 2 (a, b), ⎡

X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎢ X ↑ P2 (a, b) ⇒ X[P2 (a, b)] ⎢ ⎢ X[Y1 ] ↑ 1 P1 (a, b) ⇒ X[Y1 , 1 P1 (a, b)] ⎢ ⎢ X[Y1 ] ↑ P2 (a, b) ⇒ X[Y1 , P2 (a, b)] ⎢ ⎢ X[Y2 ] ↑ 2 P1 (a, b) ⇒ X[Y2 , 2 P1 (a, b)] ⎢ (∩ P− ) ⎢ ⎢  X[Y2 ] ↑ P2 (a, b) ⇒ X[Y2 , P2 (a, b)] ⎢ X[Y3 ] ↑ P1 (a, b) ⇒ X[Y3 , P1 (a, b)] ⎢ ⎢ X[Y3 ] ↑ 1 P2 (a, b) ⇒ X[Y3 , 1 P2 (a, b)] ⎢ ⎣ X[Y4 ] ↑ P1 (a, b) ⇒ X[Y4 , P1 (a, b)] X[Y4 ] ↑ 2 P2 (a, b) ⇒ X[Y4 , 2 P2 (a, b)] X↑  (P1 ∩ P2 )(a, b) ⇒ X[(P1 ∩ P2 )(a, b)] X ↑∼ P1 (a, b) ⇒ X[∼ P1 (a, b)] (∼ ∩ P− ) X[∼ P1 (a, b)] ↑∼ P2 (a, b) ⇒ X[∼ P1 (a, b), ∼ P2 (a, b)] X ↑∼ (P1 ∩ P2 )(a, b) ⇒ X[∼ (P1 ∩ P2 )(a, b)] Y1 Y where 2 Y3 Y4

= P1 (a, b) ∨ P2 (a, b) = Y1 , 1 P1 (a, b) ∨ P2 (a, b) and = Y2 , 2 P1 (a, b) ∨ P2 (a, b) = Y2 , P1 (a, b) ∨ 1 P2 (a, b),

10.4 3/4-Multisequents



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X[R1 (a, b)] ↑ R2 (a, b) ⇒ X[R1 (a, b), R2 (a, b)] X ↑ (R1 ∪ R2 )(a, b) ⇒ X[(R1 ∪ R2 )(a, b)] ⎡ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎢ X ↑ Q 2 (a, b) ⇒ X[Q 2 (a, b)] ⎢ ⎢ X[X 1 ] ↑∼ Q 1 (a, b) ⇒ X[X 1 , ∼ Q 1 (a, b)] ⎢ ⎢ X[X 1 ] ↑ Q 2 (a, b) ⇒ X[X 1 , Q 2 (a, b)] ⎢ ⎢ X[X 2 ] ↑∼2 Q 1 (a, b) ⇒ X[X 2 , ∼2 Q 1 (a, b)] ⎢ Q− ⎢ X[X ] ↑ Q (a, b) ⇒ X[X , Q (a, b)] (∪ ) ⎢  2 2 2 2 ⎢ X[X 3 ] ↑ Q 1 (a, b) ⇒ X[X 3 , Q 1 (a, b)] ⎢ ⎢ X[X 3 ] ↑∼ Q 2 (a, b) ⇒ X[X 3 , ∼ Q 2 (a, b)] ⎢ ⎣ X[X 4 ] ↑ Q 1 (a, b) ⇒ X[X 4 , Q 1 (a, b)] X[X 4 ] ↑∼2 Q 2 (a, b) ⇒ X[X 4 , ∼2 Q 2 (a, b)] X ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ X[(Q 1 ∪ Q 2 )(a, b)]

(∪ R− )

X1 X2 where X3 X4

= = = =

Q 1 (a, b) ∨ Q 2 (a, b) X 1 , ∼ Q 1 (a, b) ∨ Q 2 (a, b) and X 2 , ∼2 Q 1 (a, b) ∨ Q 2 (a, b) X 3 , Q 1 (a, b)∨ ∼ Q 2 (a, b), ⎡

X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎢ X ↑ P2 (a, b) ⇒ X[P2 (a, b)] ⎢ ⎢ X[Y1 ] ↑∼ P1 (a, b) ⇒ X[Y1 , ∼ P1 (a, b)] ⎢ (∪ P− ) ⎢ ⎢  X[Y1 ] ↑ P2 (a, b) ⇒ X[Y1 , P2 (a, b)] ⎣ X[Y5 ] ↑ P1 (a, b) ⇒ X[Y5 , P1 (a, b)] X[Y5 ] ↑∼ P2 (a, b) ⇒ X[Y5 , ∼ P2 (a, b)] X↑  (P1 ∪ P2 )(a, b) ⇒ X[(P1 ∪ P2 )(a, b)] X ↑∼ P1 (a, b) ⇒ X[∼ P1 (a, b)] (∼ ∪ P− ) X[∼ P1 (a, b)] ↑∼ P2 (a, b) ⇒ X[∼ P1 (a, b), ∼ P2 (a, b)] X ↑∼ (P1 ∪ P2 )(a, b) ⇒ X[∼ (P1 ∪ P2 )(a, b)] where Y5 = Y1 , ∼ P1 (a, b) ∨ P2 (a, b), and

359

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10 Role R-Calculus for Post L4 -Valued DL

⎧ ⎪ ⎪ X ↑ R(a, c1 ) ⇒ X[R(a, c1 )] ⎨ X ↑ R(c1 , c2 ) ⇒ X[R(c1 , c2 )] · ·· (∗ R− ) ⎪ ⎪ ⎩ X ↑ R(cn , b) ⇒ X[R(cn , b)] ∗ ∗ X ⎡ ↑ R (a, b) ⇒ X[R (a, b)] X ↑ Q(a, d1 ) ⇒ X[Q(a, d1 )] ⎢ X ↑ 3 Q(a, d1 ) ⇒ X[3 Q(a, d1 )] ⎢ ⎢ X ↑ 3 ∼ Q(a, d1 ) ⇒ X[3 ∼ Q(a, d1 )] ⎢ ⎢ X ↑ Q(d1 , d2 ) ⇒ X[Q(d1 , d2 )] ⎢ ⎢ X ↑ 3 Q(d1 , d2 ) ⇒ X[3 Q(d1 , d2 )] ⎢ Q− ⎢ X ↑ 3 ∼ Q(d , d ) ⇒ X[3 ∼ Q(d , d )] (∗ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ X ↑ Q(dn , b) ⇒ X[Q(dn , b)] ⎢ ⎣ X ↑ 3 Q(dn , b) ⇒ X[3 Q(dn , b)] X ↑ 3 ∼ Q(dn , b) ⇒ X[3 ∼ Q(dn , b)] X ↑ Q ∗ (a, b) ⇒ X[Q ∗ (a, b)] and

and

⎡

X ↑ P(a, d1 ) ⇒ X[P(a, d1 )] ⎢ X ↑∼ P(a, d1 ) ⇒ X[∼ P(a, d1 )] ⎢ ⎢ X ↑ P(d1 , d2 ) ⇒ X[P(d1 , d2 )] ⎢ ⎢ P− ⎢ X ↑∼ P(d1 , d2 ) ⇒ X[∼ P(d1 , d2 )] (∗ ) ⎢ ⎢··· ⎣ X ↑ P(dn , b) ⇒ X[P(dn , b)] X ↑∼ P(dn , b) ⇒ X[∼ P(dn , b)] ∗ ∗ X↑ ⎡ P (a, b) ⇒ X[P (a, b)] X ↑∼ P(a, d1 ) ⇒ X[∼ P(a, d1 )] ⎢ X ↑∼ P(d1 , d2 ) ⇒ X[∼ P(d1 , d2 )] ⎢ (∼ ∗ P− ) ⎣ · · · X ↑∼ P(dn , b) ⇒ X[∼ P(dn , b)] X ↑ (∼ P ∗ )(a, b) ⇒ X[(∼ P ∗ )(a, b)]

10.4 3/4-Multisequents

361

⎡

X ↑ R(a, d1 ) ⇒ X[R(a, d1 )] ⎢ X ↑ R(d1 , d2 ) ⇒ X[R(d1 , d2 )] ⎢ (◦ R− ) ⎣ · · · X ↑ R(dn , b) ⇒ X[R(dn , b)] ◦ X ⎡ ↑ R (a, b) ⇒ X X ↑ Q(a, d1 ) ⇒ X[Q(a, d1 )] ⎢ X ↑ 1 Q(a, d1 ) ⇒ X[1 Q(a, d1 )] ⎢ ⎢ X ↑ Q(d1 , d2 ) ⇒ X[Q(d1 , d2 )] ⎢ ⎢ X ↑ 1 Q(d1 , d2 ) ⇒ X[1 Q(d1 , d2 )] (◦ Q− ) ⎢ ⎢··· ⎢ ⎣ X ↑ Q(dn , b) ⇒ X[Q(dn , b)] X ↑ 1 Q(dn , b) ⇒ X[1 Q(dn , b)] X ↑ Q ◦ (a, b) ⇒ X[Q ◦ (a, b)] and

⎡

X ↑ P(a, d1 ) ⇒ X[P(a, d1 )] ⎢ X ↑ 2 P(a, d1 ) ⇒ X[2 P(a, d1 )] ⎢ ⎢ X ↑ 1 P(a, d1 ) ⇒ X[1 P(a, d1 )] ⎢ ⎢ X ↑ P(d1 , d2 ) ⇒ X[P(d1 , d2 )] ⎢ ⎢ X ↑ 2 P(d1 , d2 ) ⇒ X[2 P(d1 , d2 )] ⎢ P− ⎢ X ↑ 1 P(d , d ) ⇒ X[1 P(d , d )] (◦ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ X ↑ P(dn , b) ⇒ X[P(dn , b)] ⎢ ⎣ X ↑ 2 P(dn , b) ⇒ X[2 P(dn , b)] X ↑ 1 P(dn , b) ⇒ X[1 P(dn , b)] ◦ ◦ X↑ ⎧ P (a, b) ⇒ X[P (a, b)] X ↑∼ P(a, c1 ) ⇒ X[∼ P(a, c1 )] ⎪ ⎪ ⎨ X ↑∼ P(c1 , c2 ) ⇒ X[∼ P(c1 , c2 )] ··· (∼ ◦ P− ) ⎪ ⎪ ⎩ X ↑∼ P(cn , b) ⇒ X[∼ P(cn , b)] X ↑ (∼ P ◦ )(a, b) ⇒ X[(∼ P ◦ )(a, b)]

where ci is a new constant and di is a constant. Definition 10.4.3 Given a 3/4-multisequent || and statements R(a, b) ∈ , Q(a  , b ) ∈  and P(a  , b ) ∈ , a 3/4-reduction δ = || ↑ (R(a, b), Q(a  , b ), P(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )] is provable in =t⊥ St⊥ δ, if there is a sequence {δ1 , . . . , δn } of 3/4-reductions 3/4 , denoted by 3/4 such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous reductions by one of the deduction rules in St⊥ 3/4 . Theorem 10.4.4 (Soundness and completeness theorem) For any multisequent X = || and a triple X = (R(a, b), Q(a  , b ), P(a  , b )) of statements such that one of R  (a, b), Q  (a  , b ), P  (a  , b ) is not the empty string,

362

10 Role R-Calculus for Post L4 -Valued DL =t⊥

3/4

=t⊥

X ↑ X ⇒ X iff |=3/4

X ↑ X ⇒ X . 

Theorem 10.4.5 (Soundness and incompleteness theorem) For any multisequent X = || and a triple X = (R(a, b), Q(a  , b ), P(a  , b )) of statements, =t⊥

3/4 and

=t⊥

|=3/4

=t⊥

X ↑ X ⇒ X implies |=3/4

X ↑ X ⇒ X,

=t⊥

X ↑ X ⇒ X may not imply 3/4

X ↑ X ⇒ X. 

10.5 4/4-Multisequents = A 4/4-multisequent ||| is M4/4 -valid, denoted by |= |||, if for any interpretation I,

⎧ ⎪ ⎪ I (R(a, b)) = t for some R(a, b) ∈ , ⎨ I (Q(a, b)) =  for some Q(a, b) ∈ , I (P(a, b)) =⊥ for some P(a, b) ∈ , ⎪ ⎪ ⎩ I (O(a, b)) = f for some O(a, b) ∈ .

= 10.5.1 Deduction System M4/4 = Deduction system M4/4 contains the following axiom and deduction rules: • Axiom:  ∩  ∩  ∩  = ∅ (A= 4/4 ) |||

where , , ,  are sets of atoms. • Deduction rules for unary logical connective ¬: |, R(a, b)|| ||, Q(a, b)| (∼ Q ) , ∼ R(a, b)||| |, ∼ Q(a, b)|| |||, P(a, b) , O(a, b)||| (∼ O ) (∼ P ) ||, ∼ P(a, b)| |||, ∼ O(a, b)

(∼ R )

10.5 4/4-Multisequents

• Deduction rules for binary logical connective ∩: ⎧ |, Q 1 (a, b)|| ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b)|| ⎪ ⎪  ⎨ , Q 1 (a, b)||| , R1 (a, b)||| (∩ R ) , R2 (a, b)||| (∩ Q ) ⎪ ⎪  |, Q 2 (a, b)|| ⎪ ⎪ |, Q 1 (a, b)|| , (R1 ∩ R2 )(a, b)||| ⎪ ⎪ ⎩ , Q 2 (a, b)||| |, (Q 1 ∩ Q 2 )(a, b)|| ⎧ ||, P (a, b)| ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪  ||, P2 (a, b)| ⎪ ⎪ , P1 (a, b)||| ⎪ ⎪ ⎪ ⎪ ||, P2 (a, b)| ⎪ ⎪  ⎨ |, P1 (a, b)|| |||, O1 (a, b) O (a, b)| ||, P |||, O2 (a, b) (∩ P ) ⎪ (∩ ) 2 ⎪  ⎪ ⎪ (a, b)| ||, P |||, (O1 ∩ O2 )(a, b) ⎪ 1 ⎪ ⎪ ⎪ (a, b)||| , P ⎪ 2 ⎪  ⎪ ⎪ ||, P1 (a, b)| ⎪ ⎪ ⎩ |, P2 (a, b)|| ||, (P1 ∩ P2 )(a, b)| • Deduction rules for binary logical connective ∪: ⎧ |, Q 1 (a, b)|| ⎪ ⎪ ⎪ ⎪ |, Q 2 (a, b)|| ⎪ ⎪  ⎪ ⎪ ||, Q 1 (a, b)| ⎪ ⎪ ⎪ ⎪ (a, b)|| |, Q ⎪ 2 ⎪  ⎨ |||, Q 1 (a, b) , R1 (a, b)||| (∪ R ) , R2 (a, b)||| (∪ Q ) ⎪ ⎪  |, Q 2 (a, b)|| ⎪ ⎪ |, Q 1 (a, b)|| , (R1 ∪ R2 )(a, b)||| ⎪ ⎪ ⎪ ⎪ ||, Q 2 (a, b)| ⎪ ⎪ ⎪ ⎪ (a, b)|| |, Q ⎪ 1 ⎪ ⎩ |||, Q 2 (a, b) |, (Q 1 ∪ Q 2 )(a, b)|| ⎧ ||, P (a, b)| ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪  ||, P2 (a, b)|  ⎨ |||, O1 (a, b) |||, P1 (a, b) O (a, b)| ||, P |||, O2 (a, b) ) (∪ (∪ P ) ⎪ 2 ⎪ ⎪ ⎪ (a, b)| ||, P |||, (O1 ∪ O2 )(a, b) ⎪ 1 ⎪ ⎩ |||, P2 (a, b) ||, (P1 ∪ P2 )(a, b)|

363

364

10 Role R-Calculus for Post L4 -Valued DL

• Deduction rules for quantifier ∀: ⎧ |, Q(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ||, Q(a, c1 )| ⎪ ⎪ ⎪ ⎪ |||, Q(a, c1 ) ⎪ ⎪ ⎡ ⎪ ⎪ |, Q(c1 , c2 )|| , R(a, d1 )||| ⎪ ⎪ ⎨ ⎢ , R(d1 , d2 )||| ||, Q(c1 , c2 )| ⎢ |||, Q(c1 , c2 ) (∗ R ) ⎣ · · · (∗ Q ) ⎪ ⎪ ⎪ ⎪ · · · , R(dn , b)||| ⎪ ⎪ ⎪ ⎪ |, Q(cn , b)|| , R ∗ (a, b)||| ⎪ ⎪ ⎪ ⎪ ||, Q(cn , b)| ⎪ ⎪ ⎩ |||, Q(cn , b) |, Q ∗ (a, b)|| ⎧ ||, P(a, c1 )| ⎪ ⎪ ⎪ ⎪ ⎧ |||, P(a, c1 ) ⎪ ⎪ |||, O(a, c1 ) ⎪ ⎪ ⎪ ⎪ ⎨ ||, P(c1 , c2 )| ⎨ |||, O(c1 , c2 ) |||, P(c1 , c2 ) ··· (∗ P ) ⎪ (∗ O ) ⎪ ⎪ ⎪ ··· ⎪ ⎩ ⎪ |||, O(cn , b) ⎪ ⎪ , b)| ||, P(c ⎪ n ⎪ |||, O ∗ (a, b) ⎩ |||, P(cn , b) ||, P ∗ (a, b)| and

⎧ |, Q(a, c1 )|| ⎪ ⎪ ⎪ ⎪ ⎧ , Q(a, c1 )||| ⎪ ⎪ , R(a, c1 )||| ⎪ ⎪ ⎪ ⎪ ⎨ |, Q(c1 , c2 )|| ⎨ , R(c1 , c2 )||| , Q(c1 , c2 )||| ··· (◦ R ) ⎪ (◦ Q ) ⎪ ⎪ ⎪ ··· ⎪ ⎩ ⎪ , R(cn , b)||| ⎪ ⎪ |, Q(cn , b)|| ⎪ ⎪ , R ◦ (a, b)||| ⎩ , Q(cn , b)||| |, Q ◦ (a, b)|| ⎧ ||, P(a, c1 )| ⎪ ⎪ ⎪ ⎪ |, P(a, c1 )|| ⎪ ⎪ ⎪ ⎪ , P(a, c1 )||| ⎪ ⎪ ⎡ ⎪ ⎪ ||, P(c1 , c2 )| |||, O(a, d1 ) ⎪ ⎪ ⎨ ⎢ |||, O(d1 , d2 ) |, P(c1 , c2 )|| ⎢ , P(c1 , c2 )||| (◦ O ) ⎣ · · · (◦ P ) ⎪ ⎪ ⎪ ⎪ ··· |||, O(dn , b) ⎪ ⎪ ⎪ ⎪ , b)| ||, P(c |||, O ◦ (a, b) ⎪ n ⎪ ⎪ ⎪ |, P(cn , b)|| ⎪ ⎪ ⎩ , P(cn , b)||| ||, P ◦ (a, b)|

where di is a constant and ci is a new constant.

10.5 4/4-Multisequents

365

= Definition 10.5.1 A 4/4-multisequent ||| is provable in M4/4 , denoted by = 4/4 |||, if there is a sequence

{1 |1 |1 |1 , . . . , n |n |n |n } of 4/4-multisequents such that n |n |n |n = |||, and for each 1 ≤ i ≤ n, i |i |i |i is either an axiom or deduced from the previous 4/4-multisequents = . by one of the deduction rules in M4/4 Theorem 10.5.2 (Soundness and completeness theorem) For any 4/4-multisequent |||, = |== 4/4 ||| iff 4/4 |||. 

= 10.5.2 R-Calculus R4/4

Let R(a  , b ) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈ , O(a  , b ) ∈ . A reduction δ = ||| ↑ (R(a  , b ), Q(a  , b ), P(a  , b ), O(a  , b )) ⇒ [R  (a  , b )]| [Q =   (a , b )]|[P  (a  , b )]|[O(a  , b )] is R4/4 -valid, denoted by |== 4/4 δ, if  R  (a  , b ) =



  R(a  , b ) if |== 4/4 [R(a , b )]||| λ otherwise

     Q(a  , b ) if |== 4/4 [R (a , b )]|[Q(a , b )]|| λ otherwise          P(a  , b ) if |== 4/4 [R (a , b )]|[Q (a , b )]|[P(a , b )]| P  (a  , b ) = λ otherwise            Q(a  , b ) if |== 4/4 [R (a , b )]|[Q (a , b )]|[P(a , b )]|[O(a , b )] Q  (a  , b ) = λ otherwise.

Q  (a  , b ) =

Given any statements R(a, b) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈  and O(a  , b ) ∈ , let X = ||| and X[R(a, b)] = [R(a, b)]||| X[Q(a, b)] = |[Q(a, b)]|| X[P(a, b)] = ||[P(a, b)]| X[O(a, b)] = |||[O(a, b)];

X[1 R(a, b)] = [R(a, b)]||| X[2 R(a, b)] = |[R(a, b)]|| X[3 R(a, b)] = ||[R(a, b)]| X[4 R(a, b)] = |||[R(a, b)].

= R-calculus R4/4 consists of the following axioms and deduction rules:

366

10 Role R-Calculus for Post L4 -Valued DL

• Axioms: [r (a, b)] ∩  ∩  ∩  = ∅ ||||r (a, b) ⇒ [r (a, b)]|||   Q=−  ∩ [q(a , b )] ∩  ∩   = ∅ ) (A4/4 ||||q(a  , b ) ⇒ |[q(a  , b )]||   P=−  ∩  ∩ [ p(a , b )] ∩   = ∅ ) (A4/4   |||| p(a , b ) ⇒ ||[ p(a  , b )]|   O=−  ∩  ∩  ∩ [o(a , b )]  = ∅ ) (A4/4   ||||o(a , b ) ⇒ |||[o(a  , b )]

R=− ) (A4/4

where , , ,  are sets of atoms, and r (a, b), q(a  , b ), p(a  , b ), o(a  , b ) are atoms. • Deduction rules: X ↑ 2 R(a, b) ⇒ X[2 R(a, b)] X ↑∼ R(a, b) ⇒ X[∼ R(a, b)] X ↑ 3 Q(a, b) ⇒ X[3 Q(a, b)] (∼ Q− ) X ↑∼ Q(a, b) ⇒ X[∼ Q(a, b)] X ↑ 4 P(a, b) ⇒ X[4 P(a, b)] (∼ P− ) X ↑∼ P(a, b) ⇒ X[∼ P(a, b)] 1 1 O− X ↑ O(a, b) ⇒ X[ O(a, b)] (∼ ) X ↑∼ O(a, b) ⇒ X[∼ O(a, b)] (∼ R− )

and



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X[R1 (a, b)] ↑ R2 (a, b) ⇒ X[R1 (a, b), R2 (a, b)] X ⎧↑  (R1 ∩ R2 )(a, b) ⇒ X[(R1 ∩ R2 )(a, b)] X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  X[Q 11 (a, b)] ↑ Q 2 (a,1b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎨ X ↑ Q 1 (a, b) ⇒ X[ Q 1 (a, b)] 1 1 (∩ Q− ) ⎪ ⎪  X[ Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[ Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎩ X[Q 1 (a, b)] ↑ 1 Q 2 (a, b) ⇒ X[Q 1 (a, b), 1 Q 2 (a, b)] X ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ X[(Q 1 ∩ Q 2 )(a, b)]

(∩ R− )

and

10.5 4/4-Multisequents

⎧ ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎪ ⎪  X[P11(a, b)] ↑ P2 (a, 1b) ⇒ X[P1 (a, b), P2 (a, b)] ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[ P1 (a, b)] ⎪ ⎪ ⎪ ⎪ X[1 P1 (a, b)] ↑ P2 (a, b) ⇒ X[1 P1 (a, b), P2 (a, b)] ⎪ ⎪ ⎨ X ↑ 2 P1 (a, b) ⇒ X[2 P1 (a, b)] 2 2 P− (∩ ) ⎪ ⎪  X[ P1 (a, b)] ↑ P2 (a, b) ⇒ X[ P1 (a, b), P2 (a, b)] ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪  X[P1 (a, b)] ↑ P2 (a, b) ⇒ X[P1 (a, b), P2 (a, b)] ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎩ X[P1 (a, b)] ↑ 2 P2 (a, b) ⇒ X[P1 (a, b), 2 P2 (a, b)] X  ↑ (P1 ∩ P2 )(a, b) ⇒ X[(P1 ∩ P2 )(a, b)] X ↑ O1 (a, b) ⇒ X[O1 (a, b)] (∩ O− ) X ↑ O2 (a, b) ⇒ X[O2 (a, b)] X ↑ (O1 ∩ O2 )(a, b) ⇒ X[(O1 ∩ O2 )(a, b)] and



X ↑ R1 (a, b) ⇒ X[R1 (a, b)] X ↑ R2 (a, b) ⇒ X[R2 (a, b)] X ⎧↑  (R1 ∪ R2 )(a, b) ⇒ X[(R1 ∪ R2 )(a, b)] X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  X[Q 31 (a, b)] ↑ Q 2 (a,3b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[ Q 1 (a, b)] ⎪ ⎪ ⎪ 3 ⎪ Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[3 Q 1 (a, b), Q 2 (a, b)] X[ ⎪ ⎪ ⎨ X ↑ 4 Q 1 (a, b) ⇒ X[4 Q 1 (a, b)] 4 4 Q− (∪ ) ⎪ ⎪  X[ Q 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[ Q 1 (a, b), Q 2 (a, b)] ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎪ 3 3 ⎪ X[Q ⎪ 1 (a, b)] ↑ Q 2 (a, b) ⇒ X[Q 1 (a, b), Q 2 (a, b)] ⎪  ⎪ ⎪ X ↑ Q 1 (a, b) ⇒ X[Q 1 (a, b)] ⎪ ⎪ ⎩ X[Q 1 (a, b)] ↑ 4 Q 2 (a, b) ⇒ X[Q 1 (a, b), 4 Q 2 (a, b)] X ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ X[(Q 1 ∪ Q 2 )(a, b)] (∪ R− )

and

and

⎧ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎪ ⎪ X[P1 (a, b)] ↑ P2 (a, b) ⇒ X[P1 (a, b), P2 (a, b)] ⎪ ⎪ ⎨ X ↑ 4 P1 (a, b) ⇒ X[4 P1 (a, b)] 4 4 P− (∪ ) ⎪ ⎪  X[ P1 (a, b)] ↑ P2 (a, b) ⇒ X[ P1 (a, b), P2 (a, b)] ⎪ ⎪ X ↑ P1 (a, b) ⇒ X[P1 (a, b)] ⎪ ⎪ ⎩ X[P1 (a, b)] ↑ 4 P2 (a, b) ⇒ X[P1 (a, b), 4 P2 (a, b)] X  ↑ (P1 ∪ P2 )(a, b) ⇒ X[(P1 ∪ P2 )(a, b)] X ↑ O1 (a, b) ⇒ X[O1 (a, b)] (∪ O− ) X ↑ O2 (a, b) ⇒ X[O2 (a, b)] X ↑ (O1 ∪ O2 )(a, b) ⇒ X[(O1 ∪ O2 )(a, b)]

367

368

10 Role R-Calculus for Post L4 -Valued DL

⎡

X ↑ R(a, d1 ) ⇒ X[R(a, d1 )] ⎢ X ↑ R(d1 , d2 ) ⇒ X[R(d1 , d2 )] ⎢ (∗ R− ) ⎣ · · · X ↑ R(dn , b) ⇒ X[R(dn , b)] ∗ ∗ X ⎧ ↑ R (a, b) ⇒ X[R (a, b)] X ↑ Q(a, c1 ) ⇒ X[Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 3 Q(a, c1 ) ⇒ X[3 Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 4 Q(a, c1 ) ⇒ X[4 Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ Q(c1 , c2 ) ⇒ X[Q(c1 , c2 )] ⎪ ⎪ ⎨ X ↑ 3 Q(c1 , c2 ) ⇒ X[3 Q(c1 , c2 )] X ↑ 4 Q(c1 , c2 ) ⇒ X[4 Q(c1 , c2 )] (∗ Q− ) ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ X ↑ Q(cn , b) ⇒ X[Q(cn , b)] ⎪ ⎪ ⎪ ⎪ X ↑ 3 Q(cn , b) ⇒ X[3 Q(cn , b)] ⎪ ⎪ ⎩ X ↑ 4 Q(cn , b) ⇒ X[4 Q(cn , b)] X ↑ Q ∗ (a, b) ⇒ X[Q ∗ (a, b)] and

(∗ P− )

(∗ O− )

and (◦ R− )

(◦ Q− )

⎧ X ↑ P(a, c1 ) ⇒ X[P(a, c1 )] ⎪ ⎪ ⎪ X ↑ 4 P(a, c ) ⇒ X[4 P(a, c )] ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎨ X ↑ P(c1 , c2 ) ⇒ X[P(c1 , c2 )] X ↑ 4 P(c1 , c2 ) ⇒ X[4 P(c1 , c2 )] ⎪ ⎪ ⎪ · ·· ⎪ ⎪ ⎪ ⎪ X ↑ P(cn , b) ⇒ X[P(cn , b)] ⎪ ⎩ X ↑ 4 P(cn , b) ⇒ X[4 P(cn , b)] ∗ ∗ X ⎧ ↑ P (a, b) ⇒ X[P (a, b)] X ↑ O(a, c ) ⇒ X[O(a, c1 )] ⎪ 1 ⎪ ⎨ X ↑ O(c1 , c2 ) ⇒ X[O(c1 , c2 )] ··· ⎪ ⎪ ⎩ X ↑ O(cn , b) ⇒ X[O(cn , b)] X ↑ O ∗ (a, b) ⇒ X[O ∗ (a, b)] ⎧ X ↑ R(a, c1 ) ⇒ X[R(a, c1 )] ⎪ ⎪ ⎨ X ↑ R(c1 , c2 ) ⇒ X[R(c1 , c2 )] ··· ⎪ ⎪ ⎩ X ↑ R(cn , b) ⇒ X[R(cn , b)] X ↑ R ◦ (a, b) ⇒ X ⎧ ⎪ X ↑ Q(a, c1 ) ⇒ X[Q(a, c1 )] ⎪ ⎪ ⎪ X ↑ 1 Q(a, c1 ) ⇒ X[1 Q(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎨ X ↑ Q(c1 , c2 ) ⇒ X[Q(c1 , c2 )] X ↑ 1 Q(c1 , c2 ) ⇒ X[1 Q(c1 , c2 )] ⎪ ⎪ ⎪ · ·· ⎪ ⎪ ⎪ ⎪ X ⎪ ↑ Q(cn , b) ⇒ X[Q(cn , b)] ⎩ X ↑ 1 Q(cn , b) ⇒ X[1 Q(cn , b)] X ↑ Q ◦ (a, b) ⇒ X[Q ◦ (a, b)]

10.5 4/4-Multisequents

369

and

(◦ P− )

(◦ O− )

⎧ X ↑ P(a, c1 ) ⇒ X[P(a, c1 )] ⎪ ⎪ ⎪ ⎪ ⎪ X ↑ 2 P(a, c1 ) ⇒ X[2 P(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ 1 P(a, c1 ) ⇒ X[1 P(a, c1 )] ⎪ ⎪ ⎪ ⎪ X ↑ P(c1 , c2 ) ⇒ X[P(c1 , c2 )] ⎪ ⎪ ⎨ X ↑ 2 P(c1 , c2 ) ⇒ X[2 P(c1 , c2 )] ⎪ X ↑ 1 P(c1 , c2 ) ⇒ X[1 P(c1 , c2 )] ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎪ ⎪ X ↑ P(cn , b) ⇒ X[P(cn , b)] ⎪ ⎪ ⎪ ⎪ X ↑ 2 P(cn , b) ⇒ X[2 P(cn , b)] ⎪ ⎪ ⎩ X ↑ 1 P(cn , b) ⇒ X[1 P(cn , b)] ◦ ◦ X ⎡ ↑ P (a, b) ⇒ X[P (a, b)] X ↑ O(a, d1 ) ⇒ X[O(a, d1 )] ⎢ X ↑ O(d1 , d2 ) ⇒ X[O(d1 , d2 )] ⎢ ⎣··· X ↑ O(dn , b) ⇒ X[O(dn , b)] X ↑ O ◦ (a, b) ⇒ X[O ◦ (a, b)]

where ci is a new constant and di is a constant. Definition 10.5.3 Given a 4/4-multisequent ||| and statements R(a, b) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈ , O(a  , b ) ∈ , a 4/4-reduction δ = ||| ↑ (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )]|[O  (a  , b )] = is provable in R4/4 , denoted by = 4/4 δ, if there is a sequence {δ1 , . . . , δn } of 4/4reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced = . from the previous reductions by one of the deduction rules in R4/4

Theorem 10.5.4 (Soundness and completeness theorem) For any multisequent X = ||| and a quadruple X = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) of statements such that one of R  (a, b), Q  (a  , b ), P  (a  , b ), O  (a  , b ) is not the empty string,  =  = 4/4 X ↑ X ⇒ X iff |=4/4 X ↑ X ⇒ X .  Theorem 10.5.5 (Soundness and incompleteness theorem) For any multisequent X = ||| and a quadruple X = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) of statements, = = 4/4 X ↑ X ⇒ X implies |=4/4 X ↑ X ⇒ X, and

= |== 4/4 X ↑ X ⇒ X may not imply 4/4 X ↑ X ⇒ X.



370

10 Role R-Calculus for Post L4 -Valued DL

= 10.5.3 Deduction System N4/4 =

= A 4/4-multisequent ||| is N4/4 -valid, denoted by |=4/4 |||, if for any interpretation I, either

⎧ ⎪ ⎪ I (R(a, b))  = t for some R(a, b) ∈ , ⎨ I (Q(a, b))  =  for some Q(a, b) ∈ , I (P(a, b))  =⊥ for some P(a, b) ∈ , ⎪ ⎪ ⎩ I (O(a, b))  = f for some O(a, b) ∈ . = Deduction system N4/4 contains the following axiom and deduction rules: • Axiom: ⎧

 ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎨  ∩  = ∅ = ⎪  ∩  = ∅ (A4/4 ) ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎩  ∩  = ∅ |||

where , , ,  are sets of role atoms. • Deduction rules for unary logical connective ¬: |, R(a, b)|| ||, Q(a, b)| (∼ Q ) , ∼ R(a, b)||| |, ∼ Q(a, b)|| |||, P(a, b) , O(a, b)||| (∼ P ) (∼ O ) ||, ∼ P(a, b)| |||, ∼ O(a, b)

(∼ R )

• Deduction rules for binary logical connective ∩: ⎡

|, Q 1 (a, b)|| ⎢ |, Q 2 (a, b)|| ⎢ ⎢ , Q 1 (a, b)||| , R1 (a, b)||| ⎢ (∩ R ) , R2 (a, b)||| (∩ Q ) ⎢ ⎢  |, Q 2 (a, b)|| ⎣ |, Q 1 (a, b)|| , (R1 ∩ R2 )(a, b)||| , Q 2 (a, b)||| |, (Q 1 ∩ Q 2 )(a, b)||  ⎡ ||, P1 (a, b)| ⎢ ||, P2 (a, b)| ⎢ ⎢ , P1 (a, b)||| ⎢ ⎢ ||, P2 (a, b)| ⎢  ⎢ |, P1 (a, b)|| |||, O1 (a, b) ⎢ P O ⎢ (∩ ) ⎢  ||, P2 (a, b)| (∩ ) |||, O2 (a, b) ⎢ ||, P1 (a, b)| |||, (O1 ∩ O2 )(a, b) ⎢ ⎢ , P (a, b)||| 2 ⎢ ⎣ ||, P (a, b)| 1 |, P2 (a, b)|| ||, (P1 ∩ P2 )(a, b)| 

10.5 4/4-Multisequents

371

• Deduction rules for binary logical connective ∪: ⎡

|, Q 1 (a, b)|| ⎢ |, Q 2 (a, b)|| ⎢ ⎢ ||, Q 1 (a, b)| ⎢ ⎢ |, Q 2 (a, b)|| ⎢  ⎢ |||, Q 1 (a, b) , R1 (a, b)||| ⎢ R Q (∪ ) , R2 (a, b)||| (∪ ) ⎢ ⎢  |, Q 2 (a, b)|| ⎢ |, Q 1 (a, b)|| , (R1 ∪ R2 )(a, b)||| ⎢ ⎢ ||, Q 2 (a, b)| ⎢ ⎣ |, Q 1 (a, b)|| |||, Q 2 (a, b) |, (Q 1 ∪ Q 2 )(a, b)|| ⎡ ||, P1 (a, b)| ⎢ ||, P2 (a, b)| ⎢  ⎢ |||, P1 (a, b) |||, O1 (a, b) ⎢ O (a, b)| ||, P |||, O2 (a, b) ) (∪ P ) ⎢ (∪ 2 ⎢ ⎣ ||, P1 (a, b)| |||, (O1 ∪ O2 )(a, b) |||, P2 (a, b) ||, (P1 ∪ P2 )(a, b)| • Deduction rules for quantifier ∀: ⎡

|, Q(a, d1 )|| ⎢ ||, Q(a, d1 )| ⎢ ⎢ |||, Q(a, d1 ) ⎢ ⎧ ⎢ |, Q(d1 , d2 )|| , R(a, c1 )||| ⎪ ⎢ ⎪ ⎨ ⎢ ||, Q(d1 , d2 )| , R(c1 , c2 )||| ⎢ R Q ··· (∗ ) ⎪ (∗ ) ⎢ ⎢ |||, Q(d1 , d2 ) ⎪ ⎩ ⎢··· , R(cn , b)||| ⎢ ∗ ⎢ |, Q(dn , b)|| , R (a, b)||| ⎢ ⎣ ||, Q(dn , b)| |||, Q(dn , b) |, Q ∗ (a, b)|| ⎡ ||, P(a, d1 )| ⎢ |||, P(a, d1 ) ⎡ ⎢ |||, O(a, d1 ) ⎢ ||, P(d1 , d2 )| ⎢ ⎢ |||, O(d1 , d2 ) ⎢ |||, P(d1 , d2 ) ⎢ (∗ P ) ⎢ (∗ O ) ⎣ · · · ⎢··· ⎢ |||, O(dn , b) ⎣ ||, P(dn , b)| |||, O ∗ (a, b) |||, P(dn , b) ||, P ∗ (a, b)|

372

and

10 Role R-Calculus for Post L4 -Valued DL

⎡

|, Q(a, d1 )|| ⎢ , Q(a, d1 )||| ⎢ , R(a, d1 )||| ⎢ |, Q(d1 , d2 )|| ⎢ ⎢ , R(d1 , d2 )||| ⎢ ⎢ R ⎣··· Q ⎢ , Q(d1 , d2 )||| (◦ ) (◦ ) ⎢ ⎢··· , R(dn , b)||| ⎣ |, Q(dn , b)|| , R ◦ (a, b)||| , Q(dn , b)||| |, Q ◦ (a, b)|| ⎡ ||, P(a, d1 )| ⎢ |, P(a, d1 )|| ⎢ ⎢ , P(a, d1 )||| ⎢ ⎧ ⎢ ||, P(d1 , d2 )| ⎪ |||, O(a, c1 ) ⎢ ⎪ ⎨ ⎢ |, P(d1 , d2 )|| |||, O(c1 , c2 ) ⎢ O (◦ P ) ⎢ ⎪··· ⎢ , P(d1 , d2 )||| (◦ ) ⎪ ⎩ ⎢··· |||, O(cn , b) ⎢ ⎢ ||, P(dn , b)| |||, O ◦ (a, b) ⎢ ⎣ |, P(dn , b)|| , P(dn , b)||| ||, P ◦ (a, b)| ⎡

where di is a constant and ci is a new constant. = Definition 10.5.6 A 4/4-multisequent ||| is provable in N4/4 , denoted by = 4/4 |||, if there is a sequence {1 |1 |1 |1 , . . . , n |n |n |n } of 4/4multisequents such that n |n |n |n = |||, and for each 1 ≤ i ≤ n, i |i |i |i is either an axiom or deduced from the previous 4/4-multisequents = . by one of the deduction rules in N4/4

Theorem 10.5.7 (Soundness and completeness theorem) For any 4/4-multisequent |||, = = |=4/4 ||| iff 4/4 |||. 

10.5 4/4-Multisequents

373

10.5.4 R-Calculus S= 4/4 Let

R(a  , b ) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈ , O(a  , b ) ∈ .

A 4/4-reduction δ = ||| ↑ (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ⇒ [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )]|[O  (a  , b )] is S= 4/4 -valid, denoted by = |=4/4 δ, if 

=

R(a, b) if |=4/4 [R(a, b)]||| λ otherwise  =  Q(a , b ) if |=4/4 [R  (a, b)]|[Q(a  , b )]|| Q  (a  , b ) = otherwise λ R  (a, b) =

P  (a  , b ) = O  (a  , b ) =

=

P(a  , b ) if |=4/4 [R  (a, b)]|[Q  (a  , b )]|[P(a  , b )]| λ otherwise  =

O(a  , b ) if |=4/4 [R  (a, b)]|[Q  (a  , b )]|[P  (a  , b )]|[O(a  , b )] λ otherwise.

Let R(a, b) ∈ , Q(a  , b ) ∈ , P(a  , b ) ∈ , O(a  , b ) ∈ , Y = ||| and Y[R(a, b)] = [R(a, b)]||| Y[Q(a, b)] = |[Q(a, b)]|| Y[P(a, b)] = ||[P(a, b)]| Y[O(a, b)] = |||[O(a, b)]

Y[1 R(a, b)] = [R(a, b)]||| Y[2 R(a, b)] = |[R(a, b)]|| Y[3 R(a, b)] = ||[R(a, b)]| Y[4 R(a, b)] = |||[R(a, b)]

R-calculus S= 4/4 consists of the following axioms and deduction rules:

374

10 Role R-Calculus for Post L4 -Valued DL

• Axioms:

R=−

(A4/4 )

Q=−

(A4/4 )

P=−

(A4/4 )

O=−

(A4/4 )

⎧ [r (a, b)] ∩  = ∅ ⎪ ⎪ ⎪ ⎪ [r (a, b)] ∩  = ∅ ⎪ ⎪ ⎨ [r (a, b)] ∩  = ∅  ∩  = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎩  ∩  = ∅ ||| ↑ r (a, b) ⇒ [r (a, b)]||| ⎧ ⎪  ∩ [q(a  , b )] = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎨  ∩  = ∅ ⎪ [q(a  , b )] ∩  = ∅ ⎪ ⎪ ⎪ ⎪ [q(a  , b )] ∩  = ∅ ⎪ ⎩  ∩  = ∅ ||| ↑ q(a  , b ) ⇒ |[q(a  , b )]|| ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪ ⎪  ∩ [ p(a  , b )] = ∅ ⎪ ⎨  ∩  = ∅ ⎪  ∩ [ p(a  , b )] = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎩ [ p(a  , b )] ∩  = ∅ ||| ↑ p(a  , b ) ⇒ ||[ p(a  , b )]| ⎧  ∩  = ∅ ⎪ ⎪ ⎪ ⎪  ∩  = ∅ ⎪ ⎪ ⎨  ∩ [o(a  , b )] = ∅  ∩  = ∅ ⎪ ⎪ ⎪ ⎪  ∩ [o(a  , b )] = ∅ ⎪ ⎪ ⎩  ∩ [o(a  , b )] = ∅ ||| ↑ o(a  , b ) ⇒ |||[o(a  , b )]

where , , ,  are sets of atoms, and r (a, b), q(a  , b ), p(a  , b ), o(a  , b ) are atoms. • Deduction rules: Y ↑ 2 R(a, b) ⇒ Y[2 R(a, b)] (∼ Q− ) Y ↑∼ R(a, b) ⇒ Y[∼ R(a, b)] Y ↑ 4 P(a, b) ⇒ Y[4 P(a, b)] (∼ O− ) (∼ P− ) Y ↑∼ P(a, b) ⇒ Y[∼ P(a, b)] (∼ R− )

and

Y ↑ 3 Q(a, b) ⇒ Y[3 Q(a, b)] Y ↑∼ Q(a, b) ⇒ Y[∼ Q(a, b)] Y ↑ 1 O(a, b) ⇒ Y[1 O(a, b)] Y ↑∼ O(a, b) ⇒ Y[∼ O(a, b)]

10.5 4/4-Multisequents

375



Y ↑ R1 (a, b) ⇒ Y[R1 (a, b)] Y ↑ R2 (a, b) ⇒ Y[R2 (a, b)] Y ↑ (R1 ∩ R2 )(a, b) ⇒ Y[(R1 ∩ R2 )(a, b)] ⎡ Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎢ Y ↑ Q 2 (a, b) ⇒ Y[Q 2 (a, b)] ⎢ ⎢ Y[Z 1 ] ↑ 1 Q 1 (a, b) ⇒ Y[Z 1 , 1 Q 1 (a, b)] ⎢ Q− ⎢ Y[Z ] ↑ Q (a, b) ⇒ Y[Z , Q (a, b)] (∩ ) ⎢  1 2 1 2 ⎣ Y[Z 2 ] ↑ Q 1 (a, b) ⇒ Y[Z 2 , Q 1 (a, b)] Y[Z 2 ] ↑ 1 Q 2 (a, b) ⇒ Y[Z 2 , 1 Q 2 (a, b)] Y ↑ (Q 1 ∩ Q 2 )(a, b) ⇒ Y[(Q 1 ∩ Q 2 )(a, b)]

(∩ R− )

where Z 1 = Q 1 (a, b) ∨ Q 2 (a, b), Z 2 = Z 1 , 1 Q 1 (a, b) ∨ Q 2 (a, b), and ⎡

Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎢ Y ↑ P2 (a, b) ⇒ Y[P2 (a, b)] ⎢ ⎢ Y[Y1 ] ↑ 1 P1 (a, b) ⇒ Y[Y1 , 1 P1 (a, b)] ⎢ ⎢ Y[Y1 ] ↑ P2 (a, b) ⇒ Y[Y1 , P2 (a, b)] ⎢ ⎢ Y[Y2 ] ↑ 2 P1 (a, b) ⇒ Y[Y2 , 2 P1 (a, b)] ⎢ P− ⎢ Y[Y ] ↑ P (a, b) ⇒ Y[Y , P (a, b)] (∩ ) ⎢  2 2 2 2 ⎢ Y[Y3 ] ↑ P1 (a, b) ⇒ Y[Y3 , P1 (a, b)] ⎢ ⎢ Y[Y3 ] ↑ 2 P2 (a, b) ⇒ Y[Y3 , 2 P2 (a, b)] ⎢ ⎣ Y[Y4 ] ↑ P1 (a, b) ⇒ Y[Y4 , P1 (a, b)] Y[Y4 ] ↑ 1 P2 (a, b) ⇒ Y[Y4 , 1 P2 (a, b)] Y  ↑ (P1 ∩ P2 )(a, b) ⇒ Y[(P1 ∩ P2 )(a, b)] Y ↑ O1 (a, b) ⇒ Y[O1 (a, b)] (∩ O− ) Y[O1 (a, b)] ↑ O2 (a, b) ⇒ Y[O1 (a, b), O2 (a, b)] Y ↑ (O1 ∩ O2 )(a, b) ⇒ Y[(O1 ∩ O2 )(a, b)] Y1 Y2 where Y3 Y4

= P1 (a, b) ∨ P2 (a, b) = Y1 , 1 P1 (a, b) ∨ P2 (a, b) and = Y2 , 2 P1 (a, b) ∨ P2 (a, b) 2 = Y3 , P1 (a, b) ∨ P2 (a, b),

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10 Role R-Calculus for Post L4 -Valued DL



Y ↑ R1 (a, b) ⇒ Y[R1 (a, b)] Y[R1 (a, b)] ↑ R2 (a, b) ⇒ Y[R1 (a, b), R2 (a, b)] Y ↑ (R1 ∪ R2 )(a, b) ⇒ Y[(R1 ∪ R2 )(a, b)] ⎡ Y ↑ Q 1 (a, b) ⇒ Y[Q 1 (a, b)] ⎢ Y ↑ Q 2 (a, b) ⇒ Y[Q 2 (a, b)] ⎢ ⎢ Y[X 1 ] ↑ 3 Q 1 (a, b) ⇒ Y[X 1 , 3 Q 1 (a, b)] ⎢ ⎢ Y[X 1 ] ↑ Q 2 (a, b) ⇒ Y[X 1 , Q 2 (a, b)] ⎢ ⎢ Y[X 2 ] ↑ 4 Q 1 (a, b) ⇒ Y[X 2 , 4 Q 1 (a, b)] ⎢ Q− ⎢ Y[X ] ↑ Q (a, b) ⇒ Y[X , Q (a, b)] (∪ ) ⎢  2 2 2 2 ⎢ Y[X 3 ] ↑ Q 1 (a, b) ⇒ Y[X 3 , Q 1 (a, b)] ⎢ ⎢ Y[X 3 ] ↑ 4 Q 2 (a, b) ⇒ Y[X 3 , 4 Q 2 (a, b)] ⎢ ⎣ Y[X 4 ] ↑ Q 1 (a, b) ⇒ Y[X 4 , Q 1 (a, b)] Y[X 4 ] ↑ 3 Q 2 (a, b) ⇒ Y[X 4 , 3 Q 2 (a, b)] Y ↑ (Q 1 ∪ Q 2 )(a, b) ⇒ Y[(Q 1 ∪ Q 2 )(a, b)]

(∪ R− )

X1 X2 where X3 X4

= = = =

Q 1 (a, b) ∨ Q 2 (a, b) X 1 , 3 Q 1 (a, b) ∨ Q 2 (a, b) and X 2 , 4 Q 1 (a, b) ∨ Q 2 (a, b) 4 X 3 , Q 1 (a, b) ∨ Q 2 (a, b), ⎡

Y ↑ P1 (a, b) ⇒ Y[P1 (a, b)] ⎢ Y ↑ P2 (a, b) ⇒ Y[P2 (a, b)] ⎢ ⎢ Y[Y1 ] ↑ 4 P1 (a, b) ⇒ Y[Y1 , 4 P1 (a, b)] ⎢ P− ⎢ Y[Y ] ↑ P (a, b) ⇒ Y[Y , P (a, b)] (∪ ) ⎢  1 2 1 2 ⎣ Y[Y5 ] ↑ P1 (a, b) ⇒ Y[Y5 , P1 (a, b)] Y[Y5 ] ↑ 4 P2 (a, b) ⇒ Y[Y5 , 4 P2 (a, b)] Y  ↑ (P1 ∪ P2 )(a, b) ⇒ Y[(P1 ∪ P2 )(a, b)] Y ↑ O1 (a, b) ⇒ Y[O1 (a, b)] (∪ O− ) Y ↑ O2 (a, b) ⇒ Y[O2 (a, b)] Y ↑ (O1 ∪ O2 )(a, b) ⇒ Y[(O1 ∪ O2 )(a, b)] where Y5 = Y1 , 4 P1 (a, b) ∨ P2 (a, b), and ⎧ Y ↑ R(a, c1 ) ⇒ Y[R(a, c1 )] ⎪ ⎪ ⎨ Y ↑ R(c1 , c2 ) ⇒ Y[R(c1 , c2 )] · (∗ R− ) ⎪ ⎪ ·· ⎩ Y ↑ R(cn , b) ⇒ Y[R(cn , b)] Y ↑ [R ∗ (a, b)] ⇒ Y[R ∗ (a, b)] ⎡ Y ↑ R(a, d1 ) ⇒ Y[R(a, d1 )] ⎢ Y ↑ R(d1 , d2 ) ⇒ Y[R(d1 , d2 )] ⎢ (◦ R− ) ⎣ · · · Y ↑ R(dn , b) ⇒ Y[R(dn , b)] Y ↑ R ◦ (a, b) ⇒ Y[R ◦ (a, b)]

10.5 4/4-Multisequents

and

and

and

377

⎡

Y ↑ Q(a, d1 ) ⇒ Y[Q(a, d1 )] ⎢ Y ↑ 3 Q(a, d1 ) ⇒ Y[3 Q(a, d1 )] ⎢ ⎢ Y ↑ 4 Q(a, d1 ) ⇒ Y[4 Q(a, d1 )] ⎢ ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[Q(d1 , d2 )] ⎢ ⎢ Y ↑ 3 Q(d1 , d2 ) ⇒ Y[3 Q(d1 , d2 )] ⎢ Q− ⎢ Y ↑ 4 Q(d , d ) ⇒ Y[4 Q(d , d )] (∗ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ Y ↑ Q(dn , b) ⇒ Y[Q(dn , b)] ⎢ ⎣ Y ↑ 3 Q(dn , b) ⇒ Y[3 Q(dn , b)] Y ↑ 4 Q(dn , b) ⇒ Y[4 Q(dn , b)] Y ↑ [Q ∗ (a, b)] ⇒ Y[Q ∗ (a, b)] ⎡ Y ↑ Q(a, d1 ) ⇒ Y[Q(a, d1 )] ⎢ Y ↑ 1 Q(a, d1 ) ⇒ Y[1 Q(a, d1 )] ⎢ ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[Q(d1 , d2 )] ⎢ 1 1 ⎢ Q− ⎢ Y ↑ Q(d1 , d2 ) ⇒ Y[ Q(d1 , d2 )] (◦ ) ⎢ · · · ⎢ ⎣ Y ↑ Q(dn , b) ⇒ Y[Q(dn , b)] Y ↑ 1 Q(dn , b) ⇒ Y[1 Q(dn , b)] Y ↑ [Q ◦ (a, b)] ⇒ Y[Q ◦ (a, b)] ⎡

Y ↑ P(a, d1 ) ⇒ Y[P(a, d1 )] ⎢ Y ↑ 4 P(a, d1 ) ⇒ Y[4 P(a, d1 )] ⎢ ⎢ Y ↑ P(d1 , d2 ) ⇒ Y[P(d1 , d2 )] ⎢ ⎢ Y ↑ 4 P(d1 , d2 ) ⇒ Y[4 P(d1 , d2 )] (∗ P− ) ⎢ ⎢··· ⎢ ⎣ Y ↑ P(dn , b) ⇒ Y[P(dn , b)] Y ↑ 4 P(dn , b) ⇒ Y[4 P(dn , b)] Y ↑ [P ∗ (a, b)] ⇒ Y[P ∗ (a, b)] ⎡ Y ↑ P(a, d1 ) ⇒ Y[P(a, d1 )] ⎢ Y ↑ 2 P(a, d1 ) ⇒ Y[2 P(a, d1 )] ⎢ ⎢ Y ↑ 1 P(a, d1 ) ⇒ Y[1 P(a, d1 )] ⎢ ⎢ Y ↑ P(d1 , d2 ) ⇒ Y[P(d1 , d2 )] ⎢ ⎢ Y ↑ 2 P(d1 , d2 ) ⇒ Y[2 P(d1 , d2 )] ⎢ P− ⎢ Y ↑ 1 P(d , d ) ⇒ Y[1 P(d , d )] (◦ ) ⎢ 1 2 1 2 ⎢··· ⎢ ⎢ Y ↑ P(dn , b) ⇒ Y[P(dn , b)] ⎢ ⎣ Y ↑ 2 P(dn , b) ⇒ Y[2 P(dn , b)] Y ↑ 1 P(dn , b) ⇒ Y[1 P(dn , b)] Y ↑ [P ◦ (a, b)] ⇒ Y[P ◦ (a, b)]

378

10 Role R-Calculus for Post L4 -Valued DL

⎡

Y ↑ O(a, d1 ) ⇒ Y[O(a, d1 )] ⎢ Y ↑ O(d1 , d2 ) ⇒ Y[O(d1 , d2 )] ⎢ (∗ O− ) ⎣ · · · Y ↑ O(dn , b) ⇒ Y[O(dn , b)] ∗ ∗ Y ⎧ ↑ [O (a, b)] ⇒ Y[O (a, b)] Y ↑ O(a, c1 ) ⇒ Y[O(a, c1 )] ⎪ ⎪ ⎨ Y ↑ O(c1 , c2 ) ⇒ Y[O(c1 , c2 )] · ·· (◦ O− ) ⎪ ⎪ ⎩ Y ↑ O(cn , b) ⇒ Y[O(cn , b)] Y ↑ (∃R.O)(a, b) ⇒ Y[O ◦ (a, b)] where ci is a new constant and di is a constant. Definition 10.5.8 A 4/4-reduction δ = Y ↑ Y ⇒ Y is provable in S= 4/4 , denoted by = 4/4 δ, if there is a sequence {δ1 , . . . , δn } of 4/4-reductions such that δn = δ, and for each 1 ≤ i ≤ n, δi is either an axiom or deduced from the previous 4/4-reductions by one of the deduction rules in S= 4/4 . Theorem 10.5.9 (Soundness and completeness theorem) For any multisequent Y = ||| and quadruple Y = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ∈ Y of statements, = = 4/4 Y ↑ Y ⇒ Y iff |=4/4 Y ↑ Y ⇒ Y , if Y = Y.



Theorem 10.5.10 (Soundness and incompleteness theorem) For any multisequent Y = ||| and quadruple Y = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b )) ∈ Y of statements, =

=

4/4 Y ↑ Y ⇒ Y implies |=4/4 Y ↑ Y ⇒ Y, and

=

=

|=4/4 Y ↑ Y ⇒ Y may not imply 4/4 Y ↑ Y ⇒ Y,  Theorem 10.5.11 (Limit soundness and completeness theorem) For any multisequent Y = ||| and quadruple Y = (R(a, b), Q(a  , b ), P(a  , b ), O(a  , b ) of statements such that one of R  (a, b), Q  (a  , b ), P  (a  , b ), O  (a  , b ) is not the empty string, =

=

|=4/4 Y ↑ Y ⇒ Y iff lim 4/4 Y ↑ Y ⇒ Y = Y ↑ Y ⇒ Y . s→∞



References

379

10.6 Conclusions The same injuries occur in R-calculus for L4 -valued description logics.

References Alchourrón, C.E., Gärdenfors, P., Makinson, D.: On the logic of theory change: partial meet contraction and revision functions. J. Symb. Logic 50, 510–530 (1985) Arieli, O., Avron, A.: Reasoning with logical bilattices. J. Logic Lang. Inf. 5, 25–63 (1996) Arieli, O., Avron, A.: Bilattices and paraconsistency. Front. Paraconsistent Logic Studies Logic Comput. 8, 11–27 (2000) Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F.: The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, Cambridge, UK (2003) Belnap, N.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.), Modern Uses of Multiplevalued Logic, pp. 8–37. D. Reidel (1977) Bochvar, D.A.: On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. Hist. Philos. Logic 2, 87–112 (1938) Fermé, E., Hansson, S.O.: AGM 25 years, twenty-five years of research in belief change. J. Philos. Logic 40, 295–331 (2011) Font, J.M.: Belnap’s four-valued logic and De Morgan lattices. Logic J. I.G.P.L. 5, 413–440 (1997) Ginsberg, M.L.: Multi-valued logics: a uniform approach to reasoning in artificial intelligence. Comput. Intell. 4, 256–316 (1988) Gottwald, S.: A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol. 9. Research Studies Press Ltd., Baldock (2001) Li, W.: R-calculus: an inference system for belief revision. Comput. J. 50, 378–390 (2007) Muchnik, A.A.: On the separability of recursively enumerable sets (in Russian). Dokl. Akad. Nauk SSSR, N.S. 109, 29–32 (1956) Ponse, A., van der Zwaag, M.B.: A generalization of ACP using Belnap’s logic. Electron. Notes Theor. Comput. Sci. 162, 287–293 (2006) Pynko, A.P.: Characterizing Belnap’s logic via De Morgan’s laws. Math. Logic Q. 41(4), 442–454 (1995) Zach, R.: Proof theory of finite-valued logics. Technical Report TUW-E185.2-Z.1-93

Appendix: Finite Injury Priority Method

Let {e} be the computable function computed by the eth Turing machine under a coding. If {e} with input x halts then we say {e}(x) converges, denoted by {e}(x) ↓; otherwise, diverges, denoted by {e}(x) ↑ . Let {e}s (x) be the approximate computation of {e}(x) at stage s. Let {e} A be the function computed by the eth Turing machine with oracle A under a coding, and {e}sAs (x) its approximation at stage s. Let u(A, e, x, s) be the use function of {e} A (x), the maximal number of A used in computing {e}sAs (x). The recursive sets are decidable sets. The recursively enumerable sets are the domain of recursive functions (equivalently, the ranges of the total recursive functions). Notation: We = dom({e}), the eth recursively enumerable set. The halting problem K 0 = {e : {e}(e) ↓} is undecidable. Let K = {(e, x) : {e}(x) ↓}. Then, K is undecidable. The Turing degree deg(A) of a set A is the equivalence class of Turing reductions. Let 0 = deg(∅) 0 = deg(K ). Because both 0 and 0 are recursively enumerable, the recursively enumerable version of Post’s problem is Post’s problem II. Whether there exists a recursively enumerable degree a between 0 and 0 , i.e., 0 se then by induction on t ≥ s, r (e, t) = r (e, s) and {e}tBt (x) = {e}sBs (x) = As (x) = 0 for all t ≥ s, so Bs  r (e, s) = B  r (e, s), and hence {e} B (x) is defined and x is  enumerated in A so that {e} B (x) = 0 = 1 = A(x). Lemma A.4 For every e, requirement Pe is met. Proof By the last lemma, choose s such that ∀t ≥ s∀e ≤ i(r (e, t) = r (e)). Choose s  ≥ s such that no R j , j < 2e, receives attention after stage s  .

384

Appendix: Finite Injury Priority Method

Choose t > s  such that ∃x({e} At (x) = 0 = Bt (x)&∀i ≤ 2e(r (i) < x)). Now Pi receives attention at stage t + 1, and x is enumerated into Bt+1 . Hence, Pe is satisfied by the end of stage t + 1. 

References Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957) Muchnik, A.A.: On the separability of recursively enumerable sets (in Russian). Dokl. Akad. Nauk SSSR, N.S. 109, 29–32 (1956) Rogers, H.: Theory of Recursive Functions and Effective Computability. The MIT Press (1987) Soare, R.I.: Recursively Enumerable Sets and Degrees, a Study of Computable Functions and Computably Generated Sets. Springer (1987)