Introduction to Logic and Logical Discourse 9789811626890

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Satya Sundar Sethy

Introduction to Logic and Logical Discourse

Introduction to Logic and Logical Discourse

Satya Sundar Sethy

Introduction to Logic and Logical Discourse

Satya Sundar Sethy Department of Humanities and Social Sciences Indian Institute of Technology Madras Chennai, Tamil Nadu, India

ISBN 978-981-16-2688-3 ISBN 978-981-16-2689-0 (eBook) https://doi.org/10.1007/978-981-16-2689-0 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to my mother Sukanti Sethy

Preface

This manuscript is an outcome of teaching a ‘Logic’ course to the Philosophy students, MA Integrated students, Economics students, and Undergraduate Engineering students for about two decades in various higher education institutions of India. This manuscript aims to help students to develop their creative and critical minds while searching truth of worldly affairs. It discusses deductive logic, inductive logic, symbolic logic, predicate logic, basic sets, and laws of algebra. It consists of six parts, and these parts together embrace 17 chapters. Each part is designed and developed keeping in mind the target audience and methods to arouse student’s interest to learn course content through self-study. Logic course teachers’ pedagogical approaches for teaching the logic course content are also taken into consideration while explaining exercises in the chapters. Thus, the manuscript is taken utmost care and concern to develop as well as analyse the concepts and elucidate the symbolical arguments. The aim of the book to assist and guide students in learning logic course contents in an easy, thorough, and comfortable way is hence achieved. Part I, entitled Logic and Language, consists of six chapters. This part discusses definition, nature, the scope of logic, classification of logical propositions, the square of the opposition of propositions, the laws of thought, and logical paradoxes. It deals with the basic concepts of logic and language that are used in logical discourses. It elucidates the concept of ‘term’, ‘word’, ‘proposition’, and ‘sentence’. Part II, entitled Immediate and Mediate Inference, comprises three chapters. This part discusses the immediate and mediate logic and valid syllogism of mediate logic (syllogism). It analyses the pure and mixed syllogism. It elaborates the notion of ‘formal and informal fallacies’ as well. Part III, named Symbolic Logic, consists of two chapters. This part discusses logical connectives, truth functions, truth table method, conjunctive normal formula, and truth tree method with exercises. Part IV, entitled Predicate Logic, has one chapter, and it discusses universal and existential quantifiers and their applications in symbolising ordinary language sentences. It enunciates the procedures to verify the validity and invalidity of arguments and translation of logical propositions to predicate logic form with exercises. Part V, entitled Basic Sets and Laws of Algebra, consists of two chapters that discuss a variety of sets and the operation of sets among other concepts. It deals vii

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with the notion of basic sets and their operations, laws of algebra, and some proofs. The last Part VI is named Induction. This part consists of three chapters, and these chapters discuss the types of induction, J. S. Mill’s inductive methods, science, and hypothesis in detail. While discussing the notion of induction, it critically analyses the concepts of causality and universality and Mill’s inductive methods. Further, it discusses the notion of hypothesis and probability. It highlights the significance of logic and the application of logic in human beings’ everyday lives. Summarily, this book may be considered as a unique contribution to the logic subject for the benefit of logic teachers and students across the globe. Chennai, India

Satya Sundar Sethy

Acknowledgements

It is a pleasure to record my debts of gratitude. I am grateful to Prof. P. R. Bhat (IIT Bombay, Mumbai), Prof. N. N. Chakraborty (Ravindra Bharati University, West Bengal), late Prof. K. Srinivas (Pondicherry University, Pondicherry), Dr. Laxminarayan Lenka (University of Hyderabad, Telangana), Prof. Gopal Sahu (University of Allahabad, Uttar Pradesh), Dr. Venusha Tinyi (University of Hyderabad, Telangana), and Dr. A. V. Ravishankar Sarma (IIT Kanpur, Uttar Pradesh) for reviewing the chapters and providing valuable suggestions in finalising the manuscript. I am thankful to my colleagues hailing from various engineering departments and the Department of Humanities and Social Sciences of Indian Institute of Technology Madras, Chennai, for their encouragement to complete the manuscript on time. I express my heartfelt thanks to Dean (Administration), Head of Department of Humanities and Social Sciences, Indian Institute of Technology Madras, Chennai, for granting me the sabbatical leave to complete the book-writing project. I am indebted to Prof. L. Udaya Kumar, Acharya Nagarjuna University, Andhra Pradesh, for his constant encouragement to complete the manuscript on time and interest to see the book in print form. I am grateful to Prof. Prasant Kumar Panda, Department of Economics, Central University of Tamil Nadu, Thiruvarur, for inviting me to engage ‘Logic’ classes for their department students. My praise and appreciation go to Dr. Suresh M. (Postdoctoral Fellow) and Mr. Nishant Kumar (Research Scholar), Department of Humanities and Social Sciences of IIT Madras, Chennai, for preparing the ‘References’ of this manuscript. I appreciate and acknowledge Mr. Nishant Kumar’s efforts in making a few diagrams of the chapters. My sincere thanks to Ms. Satvinder Kaur (Senior Editor, Humanities and Social Sciences, Springer, India) and Mrs. Jayanthi Krishnamoorthi (Project Coordinator, Springer Book Production) for their help in replying to my queries and reminding me of all the deadlines of the book project a week before. I thank B.Tech. students, Dual-Degree students, and MA Integrated students of IIT Madras to whom I taught the ‘Introduction to Logic’ course for the last ten years and ix

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received feedback from them from time to time. Their feedback on course contents motivated me to put new examples and arguments in this manuscript. I thank the Central University of Tamil Nadu, Thiruvarur, students to whom I taught the ‘Logic’ course for three years and received feedback from time to time to develop certain new perspectives and arguments in this manuscript. I owe my sincere thanks to Mrs. Sasmita Sethy (Assistant Professor of Sanskrit, Government of Odisha) and Mrs. Sanjukta Sethy (Assistant Professor of Economics, Government of Odisha) for their constant encouragement to complete this manuscript on time. I acknowledge Mr. D. Sasikumar’s and Ms. P. V. Suguna’s (Technical staff, Department of Humanities and Social Sciences, IIT Madras) computer-related help throughout the book-writing project. My sincere thanks to the Centre for Continuing Education (CCE), IIT Madras, Chennai, for financially supporting me partially to develop, design, and write this manuscript. I am indebted to my uncle, Mr. Dayanidhi Sethy, Associate Professor of Philosophy, Fakir Mohan College, Odisha, India, for his blessings, constant moral support, and academic encouragement. He is the source of inspiration for my academic and personal achievements and successes. Chennai, India April 2021

Satya Sundar Sethy

Contents

Part I

Logic and Language

1

Definition, Nature, and Scope of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 What is Logic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definition of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Methods to Acquire Valid Knowledge . . . . . . . . . . . . . . . . . . . . . . 1.4 Logic, Language, and Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Ideas Versus Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Thinking, Asserting, and Judging . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Usage of Logic in Everyday Life . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Form and Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Formal Logic and Material Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Deductive Logic and Inductive Logic . . . . . . . . . . . . . . . . . . . . . .

3 3 6 8 10 11 13 13 14 16 17

2

Language, Logic, and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Interrelation between Logic and Language . . . . . . . . . . . . . . . . . . 2.2 Syntax, Semantics, and Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Natural Language, Metalanguage, Logically Perfect Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Functions of Language: Directive, Informative, and Expressive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Formation of Abstract and Concrete Concepts: The Role of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 24

Classification of Logical Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 What is a Word? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Types of Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Words Versus Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Types of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Sentence Versus Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Propositions and Theories of Truth . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Types of Logical Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 32 32 34 36 37 40

3

26 28

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3.8

Formal and Material Proposition: Their Truth-Value Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identifying Formal and Material Arguments . . . . . . . . . . . . . . . . . Categorical Propositions: A, E, I, O . . . . . . . . . . . . . . . . . . . . . . . . Distribution of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 46 48 50

4

Square of Opposition of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Sub-alternation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Sub-contrary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Contradictory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Truth and Validity of SOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Modern Interpretations of SOP . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54 54 54 55 56 58

5

Fundamental Principles of Logic (The Laws of Thought) . . . . . . . . . . 5.1 The Law of Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Law of Excluded Middle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Law of Non-contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Law of Sufficient Reason . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 62 64 65 68

6

Logical Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 What is a Paradox? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Classification of the Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Differences among Paradox, Ambiguity, and Vagueness . . . . . . 6.4 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Liar’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Barber’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Zeno’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 The Paradox of the Stone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Grelling’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72 73 75 76 76 77 79 79

3.9 3.10 3.11

Part II 7

8

Immediate and Mediate Inference

Immediate Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Obversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Contraposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Converting Sentences to Categorical Propositions for Immediate Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 84 87 90 92

Mediate Inference (Syllogism) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1 Figures of Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2 Moods of Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3 Rules of Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.4 Determining the Valid Moods of Syllogism . . . . . . . . . . . . . . . . . 106 8.5 Valid Moods of the First Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.6 Valid Moods of the Second Figure . . . . . . . . . . . . . . . . . . . . . . . . . 110

Contents

8.7 8.8 8.9 8.10 9

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Valid Moods of the Third Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . Valid Moods of the Fourth Figure . . . . . . . . . . . . . . . . . . . . . . . . . . Formal Fallacy Versus Material Fallacy . . . . . . . . . . . . . . . . . . . . . Informal Fallacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 115 119 120

Pure and Mixed Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Pure Categorical Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Dictum de Omni et Nullo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Enthymemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Sorites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Pure Hypothetical Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Pure Disjunctive Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Hypothetical Categorical Syllogism . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Disjunctive Categorical Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Forms of Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 126 127 127 130 132 134 135 137 139 140

Part III Symbolic Logic 10 Symbolic Logic-01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Birth of Symbolic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Ideograms Versus Phonograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Propositional Variables and Logical Connectives . . . . . . . . . . . . . 10.4 Translation of Logical Propositions into Symbolic Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Truth Functions and Truth Table Method . . . . . . . . . . . . . . . . . . . 10.6 The Conjunctive Function and the Disjunctive Function . . . . . . . 10.7 Implicative Function and Equivalence Function . . . . . . . . . . . . . . 10.8 Dagger Function and Stroke Function . . . . . . . . . . . . . . . . . . . . . . 10.9 Truth-Value: Tautology, Contradiction, and Contingent . . . . . . .

145 145 147 147 159 163 166 167 168 169

11 Symbolic Logic-02 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Indirect Method of Truth Table Decision . . . . . . . . . . . . . . . . . . . . 11.2 Beth Tree (Truth Tree) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Propositional Derivation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Disjunctive Normal Form (DNF) . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conjunctive Normal Form (CNF) . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Proving the Validity of Arguments . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 179 187 191 193 196

Part IV Predicate Logic 12 Predicate Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 What is Predicate Logic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Universal Quantifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Existential Quantifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Atomic Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Opposition of Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 206 207 207 209

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12.6 12.7 Part V

Translation of Logical Propositions to Predicate Logic . . . . . . . . 210 Proving Validity of Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Basic Sets and Laws of Algebra

13 Basic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction to Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Forms of a Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Equality of Sets and the Null Set . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Subset, Proper Subset, and Superset . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Universal Set, Power Set, and Comparability of Sets . . . . . . . . . 13.7 A Set of Sets, and Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . 13.8 Disjoint Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Line Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.10 Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219 219 220 220 221 223 224 225 226 227 229

14 Basic Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Union and Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Relative Complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Absolute Complement and Double Complement . . . . . . . . . . . . . 14.4 Some Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Laws of the Algebra of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 231 233 234 236 237 238

Part VI

Induction

15 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 What is Induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Types of Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Scientific and Unscientific Induction . . . . . . . . . . . . . . . . . . . . . . . 15.4 Induction by Analogy and Enumerative Induction . . . . . . . . . . . . 15.5 Induction by Simple Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Induction by Complete Enumeration . . . . . . . . . . . . . . . . . . . . . . . 15.7 David Hume and the Problem of Induction . . . . . . . . . . . . . . . . . . 15.8 Induction and Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 243 247 248 251 254 256 257 258

16 J. S. Mill’s Inductive Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 J. S. Mill’s Proposal on Inductive Methods . . . . . . . . . . . . . . . . . . 16.2 The Method of Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Method of Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 The Joint Method of Agreement and Difference . . . . . . . . . . . . . 16.5 The Method of Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 The Method of Concomitant Variations . . . . . . . . . . . . . . . . . . . . .

261 261 262 266 267 268 269

Contents

17 Science and Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 What is a Hypothesis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Definition, Nature, and Importance of ‘Hypothesis’ . . . . . . . . . . 17.3 Sources of a Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Types of Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Verification of a Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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271 271 272 275 277 279

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About the Author

Satya Sundar Sethy is an Associate Professor of Philosophy in the Department of Humanities and Social Sciences, Indian Institute of Technology Madras, Chennai, India. He has published several papers in international and national journals, contributed book chapters to the edited books. He is credited with an authored book Meaning and Language(2016), and two edited books Higher Education and Professional Ethics: Roles and Responsibilities of Teachers(2018) and Contemporary Ethical Issues in Engineering (2015).His current research interests include Analytic Philosophy, Logic, Engineering Ethics, Professional Ethics in Higher Education, Indian Philosophy, Consciousness Studies, Assessment and Evaluation in Higher Education settings.

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Part I

Logic and Language

Chapter 1

Definition, Nature, and Scope of Logic

This chapter introduces logic subject to students and highlights its importance and usage in their everyday lives and various academic disciplines. It explains the valid sources to acquire correct and true knowledge about worldly affairs. It elucidates the interrelation between logic and language and the correlation among thinking, asserting, and judging. It illustrates the difference between ideas and thoughts. Further, it explains the concept of formal logic and material logic, and deductive logic and inductive logic. Succinctly speaking, this chapter aims to discuss the definition, nature, and scope of logic. Before entering into the discussion of various themes of logic, let us discuss what logic is.

1.1 What is Logic? In our everyday lives, we hear people saying that ‘speak logically, as we do not understand what you are saying!’, ‘illogical statements are not worthy of pondering on’, etc. People also use the following expressions in their conversation: ‘This movie is boring and frustrating because there is no logic in the story’, ‘Mr. X often makes illogical statements, hence do not heed him seriously’, ‘it is illogical to say that a person can cross a 250 m wide river in 20 s by sailing a country boat’, ‘it is logical to assert that anyone born on this earth will share his or her death after some time’, ‘it is logical to accept that water quenches human beings’ thirst’, etc. So, the word ‘logic’ is familiar to most of us, as we use it in our everyday conversation with others. Logic, in this sense, is related to our day-to-day social lives. Logic is used to differentiate true beliefs from blind beliefs, rational beliefs from emotional beliefs, and so on. It guides human beings to search for truth. Scientists, non-scientists, and common people use the word ‘logic’ in their day-to-day activities for various purposes. Upon applying logic to a given situation or context, we can find out what correct information is and what incorrect information is, which are considered to be true facts and which are considered to be false facts, etc. These © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_1

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are a few usages of logic in the social lives of human beings. The basic question still remains, ‘what is logic?’. In order to answer this question, we need to learn the nature, scope, and definition of logic. The word ‘logic’ is derived from the Greek word ‘logike’, which means ‘forms of thinking’. Thinking is an act of the mind by means of which human beings obtain knowledge about worldly affairs. But all sorts of thinking, such as imagining, hallucinating, and remembering, are not parts of logic. Here, the word ‘thinking’ refers to the expression ‘reasoning’. Logic deals with the science of reasoning. The science of reasoning differentiates correct reasoning from incorrect reasoning. However, it is not true that those who have not studied logic most often use incorrect reasoning and those who have studied logic use only correct reasoning to obtain true knowledge about worldly affairs. But it is true that like a trained musician who performs better than an untrained musician, a logic student who has learned the logical tools and methods can mostly apply correct reasoning to obtain true knowledge of worldly affairs than persons who have not studied the subject ‘logic’. Correct reasoning guides to find out the cause of an effect and the relation between evidence and the conclusion of an argument. An argument consists of a few pieces of evidence followed by a conclusion. In the logical discourse, we consider the evidence as premises that help in inferring a conclusion in the argument. The conclusion of an argument is indicated through the following words, but not limited to these alone— hence, so, therefore, etc. An argument, thus, necessarily consists of premises and a conclusion. Premises alone cannot be treated as an argument. Further, a conclusion alone without premises cannot be treated as an argument. The examples below are considered as arguments. Example 1 If there is sun, then there is light. There is a sun. Therefore, there is light. Example 2 All men are mortal. Gandhi is a man. Therefore, Gandhi is mortal. Example 3 All logic students are wise beings. Miku is a logic student. Therefore, Miku is a wise being. It is noted that in an argument if the premises are true, the conclusion drawn from the premises must be true. Further, if the conclusion of an argument is false, then one of the premises must be false. There will not be a case where the conclusion of an argument is true, and one of the premises is false. Refer to the premises and conclusion of the arguments given below.

1.1 What is Logic?

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Example 1 All birds have two legs. (Premise-1) All horses have two legs. (Premise-2) Therefore, all horses are birds. (Conclusion) In this argument, the conclusion is false because premise-2 is false. Example-2 All cows are white. (Premise-1) All swans are white. (Premise-2) Therefore, all swans are cows. (Conclusion) In this argument, the conclusion is false because premise-1 is false. Example-3 Some logic students are cricket players. (Premise-1) Some football players are logic students. (Premise-2) Therefore, some football players are cricket players. (Conclusion) In this argument, the conclusion is true because the premises are regarded as true. Example-4 Some dancers are not tall persons. (Premise-1) Some History students are dancers. (Premise-2) Therefore, some history students are not tall persons. (Conclusion) In this argument, the conclusion is true because the premises are regarded as true. Logic deals with arguments, and each argument consists of a few premises. An argument is either valid or invalid, whereas a premise is either true or false. An argument can be arranged in the following ways: (i) Premises are mentioned in the beginning, and conclusion is derived from the premises, (ii) Conclusion is asserted in the beginning and justifications to the conclusion are mentioned thereafter in the form of premises. Consider the example below. Rina was not well yesterday. (Premise) The doctor advised her to take a rest in the hostel. (Premise) Hence, Rina could not come to logic class yesterday. (Conclusion) This argument can be rearranged as Rina did not come to logic class yesterday because the doctor advised her to take a rest in the hostel, as she was not well yesterday. There are many subjects in the academic arena, such as physics, biology, political science, philosophy, history, mathematics, mechanical engineering, computer science, and management sciences. Each subject deals with many information, ideas, concepts, arguments, and findings. Logic is an integral part of each and every subject. It is an indispensable component of all the subjects of study. It is so because logic is used to communicate subject contents, analyse concepts, formulate arguments, present research findings to others, etc. It guides in finding out whether an idea is correct or not, whether an argument is valid or not, so on, and so forth. Logic is an indispensable part of all scientific enquiries. In this sense, ‘logic’ is embedded in every subject. Although logic and logical analyses are found in every subject of

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study, yet it is true that ‘logic’ as an independent subject of study does not include any subject in its scope. It is regarded as a science subject because it studies its subject contents (topics) in a systematic and scientific manner as a science subject does. In this sense, logic is a method as well as a subject of study. The aim of logic is to assist in obtaining the correct information and true knowledge about worldly affairs. Logic, as the science of reasoning, helps in deriving ‘unknown phenomenon’ from known facts or ‘observed phenomena’. The observed phenomena are regarded as premises or evidence, and the ‘unknown phenomenon’ is regarded as the conclusion. For example, ‘Crow X is black’ is perceived in India. ‘Crow Y is black’ is perceived in India. ‘Crow Z is black’ is perceived in India. Therefore, all crows are black in India. In this argument, three premises are observed phenomena, and the conclusion is treated as unknown phenomena.

1.2 Definition of Logic Many definitions of logic are found in logical discourses. We shall mention some of them by highlighting their merits and lacunas. Then, we shall find out which definition fits well to explain ‘logic’ as a subject of study. Mill (1882), in his work A System of Logic, defines ‘Logic is the science of the operations of the understanding which are subservient to the estimation of evidence: both the process itself of advancing from known truths to unknown, and all other intellectual operations in so far as auxiliary to this’ (p. 18). This definition signifies that logic is a science that examines the premises (evidence) from which a conclusion is derived. This exercise is meant for determining the truth of the premises. In addition to this exercise, logic deals with other intellectual operations that include classification, assertion, naming, etc. These tasks are considered theoretical aspects of an argument. Thus, logic includes both theoretical and practical aspects of an argument. In Mill’s (1882) words, logic comprises the science of reasoning as well as an art founded on that science (p. 10). To simplify, logic is both science and art. It is a science because it deals with laws and principles of valid arguments. It is an art because it follows the laws religiously to achieve the goals, and in the process of achieving the goals, it identifies the errors in the arguments. This definition sounds convincing and logical as it incorporates all the required features of the formation and evaluation of arguments. According to Hamilton (1860), ‘Logic is the science of the necessary forms of thought’ (p. 17). This definition explains that logic deals with the necessary forms of thought without clearly stating what does the expression ‘necessary forms of thought’ mean: Is it an abstract thought or a concrete thought or something else altogether? Further, it ignores the theoretical aspect of logic and highlights only the practical aspect of logic. Hence, this definition is not inclusive in nature. In addition to these shortcomings, this definition confines the subject logic to formal logic only. It does not include material logic. But in reality, logic encompasses both formal and material logic.

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In Arnauld’s view, logic is the science of understanding in the pursuit of truth.1 (Woods, 2012). This definition highlights the theoretical aspect of logic and ignores its practical side. It explains about ‘truth’ but does not mention whether it is about formal truth or material truth or both. This definition highlights the significance of logic to derive a conclusion from the premises of an argument, but it ignores an important fact. That is, logic is also used as a method to find out the truth of the premises. Hence, this definition is not free from lacunas. Logic, as a science, deals with the principles and criteria of validity of inference and demonstration: the science of the formal principles of reasoning (Gabbay et al., 2005, p. 1; Webster’s Dictionary, 1987). This definition highlights the scientific aspect of logic and avoids the theoretical aspect of logic. It deals with formal logic and ignores material logic. It mentions the validity of an argument without mentioning the methods to determine the truth of the premises of an argument. Hence, this definition is not a satisfactory one. According to Simpson (2000), ‘Logic is the science of formal principles of reasoning or correct inference’ (p. 2). This definition, while highlighting the notion of formal logic, does not mention material logic. It delineates the scientific aspect of logic and ignores the significance of the theoretical aspect of logic. It gives importance to correct inference only and does not state anything about incorrect inference. In reality, inferences refer not only to correct inferences but also to incorrect inferences. Hence, this definition is not inclusive in its explanation. Gensler (2017; 3rd Ed) defines, ‘Logic is the systematic study of the forms of inference, i.e. the relations that lead to the acceptance of one proposition (the conclusion) on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.’2 (p. 1). This definition sounds appropriate and satisfactory. It is inclusive in its scope and explanation. The reason is this definition suggests the relation between premises and the conclusion of an argument. It incorporates both the material and formal arguments. It delineates about theoretical and practical aspects of a logical argument. From the above definitions, it may be asserted that logic assists in accumulating correct information and true knowledge about worldly affairs. To obtain true knowledge about worldly affairs, we need to adopt the correct method. If the method(s) are correct and justifiable, then the knowledge obtained through these method(s) is considered valid knowledge. Logic guides us to use the appropriate and correct method(s) to acquire correct information and true knowledge (valid knowledge) on the objects of the world and the concepts of worldly affairs.

1 Woods,

J. (2012). A history of the fallacies in western logic, In Logic: A history of its central concepts (pp. 513–610), Eds. D.M. Gabbay, F.J.Pelletier, J. Woods. USA: Elsevier Publication. 2 Gensler, Harry J. (2017). Chapter 1: Introduction. Introduction to logic (3rd ed.), New York: Routledge. p. 1.

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1.3 Methods to Acquire Valid Knowledge Human beings are knowledge seekers. They search for new knowledge, and in the process, they uphold correct information and reject incorrect information about worldly affairs. They apply scientific reasoning (i.e. logic) to find out the correct information about objects, facts, events, and concepts of the world. To possess true knowledge of an object, a subject (a person), an object, and a method are indubitably required. Further, the subject must have a desire to acquire knowledge about the object by applying the valid method. For example, Ms. Seema (a subject) desires to know about a red ink pen (an object) that lies on her study table. She perceives (a method) the pen and acquires knowledge about it. In this regard, logicians, scientists, and philosophers propose that there are six valid methods (pramanas) through which we can obtain true knowledge about worldly affairs. These are perception (pratyaksha), inference (anumana), comparison (upamana), verbal testimony (sabda), postulation (arthapatti), and non-perception (anupalabdhi). Let us discuss each valid method with a suitable example. Perception (Pratyaksha) Perception is the direct and immediate method to cognise the objects of the world. In this case, the cognition of an object occurs due to the interaction between the object and the sense organs. Perception includes any sense organ’s (ear, tongue, etc.) contact with an object, and it is not confined to only visual sense organ’s (eyes) contact with the object. There are two types of perception: indeterminate perception and determinate perception. In the case of indeterminate perception, a person cannot determine the features of an object, such as colour, shape, and size. As a result, he/she is not able to explain the object precisely and clearly. Unlike indeterminate perception, in the case of determinate perception, a person recognises most of the features of an object and thus cognises what the object is. Further, the person gives a name to that object for its recognition. In this sense, perception gives human beings an immediate knowledge about an object. An example, by seeing a table, a person acquires knowledge of that table. Perceptual knowledge is valid when a cogniser (person) perceives the object with its possible features. Inference (Anumana) Inference is an independent and valid method that is used to acquire valid knowledge about worldly affairs. It assists in obtaining new knowledge about an object that is not perceivable to the cogniser (person). In inference, an object is inferred to be present in a particular case because it has been invariably perceived to be present in all such similar cases in the past. Thus in inference, the cognition of an object is based on a person’s prior knowledge about it. For example, a person sees smoke on a hill from a distance. By seeing smoke, he/she relates his/her previous knowledge of smoke with fire, that is, wherever there is smoke, there is fire, such as in the kitchen, in a lamp, and in candlelight. From his/her previous knowledge about fire smoke invariable, unconditional, and universal relation, the person claims that since the hill is smoky,

1.3 Methods to Acquire Valid Knowledge

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there must be fire. An example of inference is mentioned below for reference and discernment. Whatever is smoky is fiery. This hill is smoky. Therefore, this hill is fiery. Comparison (Upamana) Comparison is a valid-independent method to acquire valid knowledge of objects of the world. This method helps in obtaining new knowledge of an unknown object by comparing it with a known object. In this method, knowledge of an unknown object is acquired by comparing it with similar kind of known objects of the world. It is a source of knowledge of the relation between a word and its denotation (what the word refers to). For example, a person does not know what a ‘squirrel’ is. She/he is told by a forester that it is a small mammal like a country rat, but it has a long furry tail and stripes on its whole body. After some period of time, when the person sees such a mammal in a forest that matches with the forester’s descriptions about the squirrel and his/her knowledge about the country rat, she/he compares the country rat with the new perceived mammal and finally cognises it as a squirrel. In the case of comparison, there are four steps involved in acquiring new knowledge of an object. First, a person (cogniser) listens to authoritative statements about a word (i.e. squirrel) that denotes an object of certain descriptions. Second, at a later period of time, the person finds a new object that matches with the authority’s descriptions and has a resemblance with his/her previous knowledge of a known object of a similar type. Third, the person compares the unknown object with the known object by considering the authority’s descriptions about the unknown object. Finally, the person obtains new knowledge about the unknown object. Verbal testimony (Sabda) Knowledge gained through verbal testimony is also regarded as valid knowledge. But words and sentences uttered by any person cannot be treated as verbal testimonies because all verbal expressions do not necessarily help in accumulating valid knowledge. Instructive assertion of a reliable person is only considered as verbal testimony. A reliable person may be a risi, a mlechha, an arya, or anybody for that matter, who has expertise in a certain matter and is willing to communicate his/her experience of it. For example, a person wanted to cross a river, but she did not know the depth of the river. In this case, she asked a fisherman who is fishing there, ‘What is the depth of the river? Can I cross the river?’ Since the fisherman is a local person who has knowledge about the depth of the river and desires to communicate his experience with the person, the words of the fisherman are to be considered as verbal testimonies. In this case, the fisherman replied that she could cross the river easily, and for that, she needed to walk towards her right in every step after stepping into the river. By considering the fisherman’s instructions, the lady crossed the river and gained new knowledge about the river.

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Postulation (Arthapatti) Postulation is a unique method to acquire valid knowledge about worldly affairs. This method resolves a conflict between two facts. It suggests a presumption that solves the conflict between two facts. Postulation is an assumption of an unperceived fact that reconciles two inconsistent perceived facts. For example, Devadatta is a bulky man who fasts in the daytime. In this proposition, a cogniser finds two contradictory facts. First, Devadatta is a person who is very fat. Second, Devadatta is not eating in the daytime. In order to resolve this conflict (i.e. fasting in the daytime and being a bulky man), the cogniser postulates the existence of a third fact, that is, Devadatta must be eating in the night. In this way, the cogniser gains a piece of new knowledge about Devadatta’s bulky body. Non-perception (Anupalabdhi) Non-perception is also an independent method that assists in obtaining new knowledge about worldly affairs. It provides an immediate knowledge of the non-existence of an object in a particular place and at a given time. It suggests that an object does not exist in a particular place and at a given time, but it may exist elsewhere. To perceive the non-existence of an object in a given situation, a cogniser uses the non-perception method. Thus, this method is unique in its usage and application to acquire new knowledge about non-existent objects of the world in a designated place and time. An example, ‘There is no laptop on the study table’. Here, the cogniser does not perceive the laptop through his/her sense organs. But the cogniser gains new knowledge of the absence (non-existence) of a laptop on the table at a given time. This knowledge is accumulated due to the non-perception of the laptop in a designated place and at a given time. Perceiving the absence (non-existence) of an object at a designated place and time is regarded as a non-perception of the object. There is a polemic about the above-mentioned six valid methods. Some logicians and philosophers say that these six valid methods are considered as the validindependent methods to acquire valid knowledge about worldly affairs, and a few others state that out of these six valid methods, some of the valid methods can be reduced to either inference or a combination of perception and inference. We are not discussing this issue as it is outside the scope of this chapter. However, it is to be noted that the Nyaya School proposed the first four valid methods (pramanas), and Mimamsa School adds the last two valid methods to the list to acquire valid knowledge about worldly affairs.

1.4 Logic, Language, and Reasoning Logic assists in applying the valid method(s) to obtain true knowledge about worldly affairs. It validates the knowledge about an object and finds out the truth of it. It is concerned with formulating arguments and determining the validity of the arguments. An argument consists of propositions. A proposition comprises words of a language

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system. In this sense, logic is linked to a language system. Logic is communicated in language. Logic brings clarity to the sentences that are communicated to others. Logic and language, in this sense, are related to each other. Logic is not ‘reasoning’, but it is the science and art of reasoning. It guides how to apply reasoning to a fact or an event to find out its truth. By using logic in our thinking processes, we bring coherence in our thoughts, analyse them, and present them to others in a meaningful manner. Logic also helps us to comprehend the lexical and intended meaning of sentences written on a piece of paper and uttered by a speaker. Logic is used as a vehicle of thought in a language system. A thought is communicated to others when it is formulated with logic and conformed to logical norms. A few random thoughts without complying with logical norms do not bring coherence among themselves in a language system. For example, Rama is a good boy. Grass is green. Ocean water is salty. These three propositions convey three different thoughts, but these thoughts are not related to each other. In other words, there is no relation of consistency found among these propositions. Hence, all the propositions mentioned in a passage as a whole do not communicate any meaning in a language system. A logical thought has many constituents. The reasoning is one of them. Reasoning helps to guide our thinking about an issue or an event in a sequential and logical manner. It helps in producing justified, rational thoughts in a language system. It guides the usage of words and propositions in our thinking processes to produce valid thoughts. It assists in formulating valid arguments in a language system. Thus, logic is treated as the science of thought in a language system. It is to be noted here that thought cannot be expressed without language. So, language is the vehicle of our thoughts. In this way, logic, language, and reasoning are interrelated to one another.

1.5 Ideas Versus Thoughts Logic helps in producing valid thoughts in a language system. According to Frege (1956), a thought is invariably and logically related to propositions. As a result, we are able to determine whether the proposition is true or false. In this regard, a question may arise, ‘Are thoughts the same as ideas?’ Frege brings the distinction between ‘thought’ and ‘idea’ in the logical discourses. In his view, ideas are subjective, whereas thoughts are objective and eternal (Frege, 1956, p. 299). His arguments are as follows: (i)

(ii)

Ideas cannot be seen, touched, smelled, and tasted. It is the impression of an observer stored in his/her mind. More precisely, ideas cannot be presented to our sense organs. For example, I saw a beautiful landscape. I have a visual impression of that landscape. I have it, but I do not see it. An idea, which someone has, belongs to the content of his or her consciousness. The contents might be in the form of sensations, feelings, moods, inclinations, wishes, etc. For example, a person is very much fond of apple. Once, he went to a fruit vendor to purchase an apple. He saw that the fruit vendor

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(iii)

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1 Definition, Nature, and Scope of Logic

arranged different varieties of fruits, including apple, in his rickshaw. The person who went to purchase apple may see different fruits arranged systematically in the vendor’s rickshaw, but his content of consciousness (alertness) is on the apple. So the idea of apple exits for him. Thus, it is stated that without a bearer, an idea cannot exist. In a similar fashion, an experience is not possible without an experiencer. Things of the outer world are independent of their bearer. However, ideas need a bearer. Ideas are subjective. Since ideas are subjective, they differ from person to person. Even though a few persons recognise an object, ideas about the object may differ from person to person. It is so because the approach to cognise the object may differ from one person to another. For example, some people see an animal named ‘Bucephalus’ as a horse. The idea of that horse may differ from person to person, as a zoologist’s idea of that animal may differ from the idea of a poet, the idea of a poet may differ from the idea of a horse rider, and so on. Hence, each idea needs a bearer. Without a bearer, an idea cannot exist. Two individuals cannot have the same idea on a particular object because every idea needs only one bearer, and ideas are subjective. So, no two men can have the same idea. An example, I am having toothache and feeling its pain. No other person can share my pain, but someone may have sympathy for me. So here, my pain belongs to me, and his sympathy belongs to him. Hence, he does not have my pain, and I do not have his sympathy. Thus, we assert that each idea has one and only one bearer. If there will be more than one bearer of an idea, then an idea will be regarded as objective and universal; but it is not the case.

If an idea needs a bearer, then a question erupts, who is the bearer of my idea as ‘myself’? Frege answers that ‘I am’ the idea of ‘myself’. Here, I have an idea of myself, but I am not identical to this idea.3 The content of my consciousness is ‘my idea’, which could be sharply distinguished from what is an object of my thought on ‘myself’. It will be clear if we explain this with the help of an analogy. In a strict and logical sense, I can see others as in the form of objects, but I cannot see myself as an object. Frege asserts that I may see myself by gathering the impressions that I could get from the mirror. Thus, ideas need a bearer, and without a bearer, ideas have no existence. It is just like if there is no ruler, there are no subjects. According to Frege, ideas are the power of thought and immediate objects of apperception. Further, ideas correspond to the apprehension of thought. In thinking, we do not produce thoughts, but we apprehend them. When there is an idea, it is an idea of someone. It belongs to someone and is known to someone. On the contrary, a thought is transmitted from one person to another person, one generation to another. A thought is communicated to others. Thus, ideas are subjective and personal, whereas thoughts are objective and eternal. Frege expresses that ‘By a thought, I understand not the subjective performance of thinking but its objective content which is capable 3 For

more details, see Frege, G. (1977). Logical Investigation. Edited by P.T. Geach, Oxford: Basil Blackwell Publication, p. 22.

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of being the common property of several thinkers’ (Frege, 1952, p. 62). He states that ‘every sentence has a thought because the thought is in itself immaterial clothes found in the material garment of a sentence and thereby becomes comprehensible to us.’ (Frege, 1956, p. 292).

1.6 Thinking, Asserting, and Judging Logic deals with our thinking, asserting, and judging of facts or events of the world. Gottlob Frege (1848–1925), who is regarded as the father of modern philosophical logic and analytic philosophy, distinguishes functions of thinking, asserting, and judging in the logical discourse. He writes apprehension of a thought is called thinking. Recognition of truth of a thought is treated as judgement, and manifestation of a judgement in a language system is regarded as an assertion.4 He enunciates that consider an indicative sentence where our main concern is to know whether we can apprehend the thought from the indicative sentence or not. This is about the function of thinking. Then, we can ask if we apprehend the thought of the sentence, is it possible for us to investigate the fact of what the sentence states about? This is about the functions of judging the fact. After carrying out appropriate investigations, if we are able to judge whether the sentence is true or false, it is considered as the function of asserting the fact. Combining the functions of thinking, judging, and asserting a fact, we can claim that an indicative sentence is necessarily either true or false.

1.7 Usage of Logic in Everyday Life It is true that we use logic in sentences to communicate our thoughts to others. We also use logic to find out the truth of a fact or an event. Further, we use logic to argue on an issue coherently, cogently, and convincingly. In this sense, logic plays a pivotal role in our day-to-day affairs. Like language, logic is also associated with our everyday lives. A few examples are mentioned from our day-to-day affairs to attest to the claim. By seeing a rainbow from her windowpane, a person infers that it is raining and sun rays are falling on the earth. Assume that a person went for a morning walk and found that roads are dirty with dried leaves, potholes are filled with mud water, and roads are wet. From these phenomena, she infers that perhaps yesterday night, there was heavy wind and rain. Take another example: a student finds that the price of a particular model of mobile phone is increased suddenly in the market. She applies logic to this situation to find out the possible reasons for the escalation of mobile phone price. She concludes that perhaps the supply of that particular model of mobile phone is not adequate in the market. Consider one more example: a person 4 Frege,

G. (1977). Logical investigations. Edited by P.T. Geach, Oxford: Basil Blackwell Publication, p. 7.

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sees that a big fish is floating in the river. He uses logic and infers that perhaps the fish is dead. By applying logic to the school education system, parents infer that if their kids are passing in the 8th standard examinations, they are eligible to take admission in the 9th standard. People, by applying logic to the day and night timings, state that when the sun sets in the west, evening approaches. It is logical to assert that when a water bottle is filled with water, more water cannot be poured into it. Further, by applying logic, we know that leap year comes once in four years. So, it would be illogical to think that each and every year should be treated as a leap year. Consider another example; a person sees a big wooden log. By applying logic to this context, she concludes that it would be so heavy that she cannot lift it on her own. Another example: a person sees a bird’s feathers, and by applying logic to the context, she concludes that feathers are so light and she can lift those in her hands. In this way, human beings use logic in their daily lives. It would not be an exaggeration to state that logic is associated with human lives from the cradle to the grave.

1.8 Form and Matter Logic deals with the form and matter of an object of worldly affairs. Each object is made of a certain matter, and it has a unique ‘form’. Human beings identify an object with a name due to its unique form. For example, a table has a form; as a result, it is identified as a table and not as a chair, a desk, a kitchen slab, etc. If the ‘form’ of an object changes, then the name of the object also changes. ‘Form’ is similar to the concept of an object. An object exists in a given space and time. Anything that occupies space and time does not exist permanently and thereby eternally. ‘Form’ of an object does not occupy space and time, hence exists eternally. It exists beyond the space–time continuum. An example, a laptop, as an object, exists at a given space and time, but the ‘form’ of a laptop exists eternally bereft of the destruction of many old laptops and the creation of many new laptops. A laptop is a member of the concept ‘laptop’ where all varieties of laptop exist. The concept of a laptop is conceived as a form of a laptop. ‘Form’ and ‘matter’ of an object are associated with each other. It is so because there is no such thing as matter without form and form without matter. If there is no matter, then there is no form. According to Aristotle, ‘form’ is not to be understood as ‘shape’. The reason is ‘form’ is a concept of an object that represents a class. For example, a person can draw a big square, a small square, a medium square, etc. These are shapes of the square, but not the ‘form’ of the square. The form of the square is one and only one. It does not change along with the shapes of the square. Even though the shape of the square changes from time to time and place to place, the form of the square will remain unchanged and unmodified. Logic deals with arguments, and each argument has a form and matter. The content of an argument is regarded as ‘matter’, and the way contents are arranged in an argument is treated as ‘form’. According to Russell (1963), ‘form’ is not a constituent

1.8 Form and Matter

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of an argument but the way constituents are arranged in an argument. He argues that even non-existent objects cannot be thought of as bereft of matter and form. So, there is no formless matter and matterless form. A matter can have different forms (e.g. ‘wood’ is the matter, and from wood, we can make a wooden chair, wooden table, wooden desk, etc., as different forms.), and a form can be conceived through different matters (e.g. the form of a chair can be conceived through many matters, such as wood, plastic, iron, and other materials). In the context of logic, a form of an argument can be conceived through a variety of matters (contents). For example, No M is P. All S is M. Therefore, No S is P. In this argument (i.e. a form of argument), M, P, and S are the matters (contents) of the argument. We can put different matters in this argument, let us say, B, C, N, instead of M, P, S to formulate a similar argument. By putting different matters in place of M, P, S, we can have many arguments, but the form of the argument remains the same (i.e. the way premises and conclusion are arranged in an order of this argument). To be precise, although matters of the arguments can differ from one to another, yet all the arguments share a common ‘form’. Further, in the context of logic, matter of an argument can be arranged in different ways; hence, we can have different forms of argument. To put it simply, content (matter) of an argument can be used to formulate different forms of argument. For example, P, M, and S are contents of an argument. From P, M, and S contents, we can formulate the following arguments that differ from each other in their forms. Argument-1 All M is P. All S is M. Therefore, all S is P. Argument-2 No P is M. All S is M. Therefore, No S is P. Argument-3 All M is P. All M is S. Therefore, Some S is P. Contents of argument-1, argument-2, and argument-3 are arranged uniquely. Hence, each argument differs from other arguments based on its ‘form’, even though the contents of all the three arguments remain the same.

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1.9 Formal Logic and Material Logic Formal logic concerns the ‘form’ of an argument, whereas material logic deals with the ‘form’ and ‘content’ of an argument. Formal logic aims at formal truth, whereas material logic aims at both formal and material truth. Formal logic is related to the form of propositions (premises and conclusion) of an argument. It focuses on the structure of an argument and does not concern the verification of the truth of the propositions. It considers propositions of the argument as true without verification. But, it applies the logical rules to the arguments to find out the validity of the arguments. In the formal logic, content (matter) of an argument does not play a significant role to derive a valid conclusion of the argument. Rather, the conclusion of the argument is derived from the premises by applying logical rules and without committing any logical fallacy. A few examples of formal logic are as follows: Example-1 All men are mortal. Sita is a man. Therefore, Sita is mortal. This argument is valid even though the second proposition (premise) is not true. Example-2 All men are honest beings. All men are biped creatures. Therefore, Some biped creatures are honest beings. This argument is valid even though the first proposition is not true. Example-3 Some men are capable of seeing. All blind persons are men. Therefore, Some blind persons are capable of seeing. This argument is valid even though the second proposition is not true. In contrast to formal logic, material logic is concerned with both formal truth and material truth of an argument. Material logic verifies the truth of the propositions of an argument. It finds out whether the contents of the propositions exist in the empirical world or not. Further, it finds out whether the contents of the propositions refer to the objects and facts of the phenomenal world or not. If the content of a proposition corresponds to worldly affairs, and it is found in the phenomenal world, the proposition is judged as true. But if the content of a proposition does not correspond to worldly affairs, the proposition is treated as false. Material logic aims for truth as a matter of fact. In material logic, if propositions of an argument correspond to the actual facts of the world, then the conclusion derived from the propositions also corresponds to the actual fact of the world. However, to derive the true conclusion from true premises in

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material logic, we need to apply the logical rules to the argument. Thus, it is asserted that in material logic, if the propositions are true, the conclusion must be true; and if one of the propositions is false, then the conclusion must be false. Material logic, in this sense, is considered as an applied logic and the perfect form of logic. An example of material logic is as follows: All crows are black. X is a Crow. Therefore, X is black. From the above analyses, it is asserted that an argument that satisfies material logic norms has a wider scope than the argument that satisfies the norms of formal logic. However, both material logic and formal logic play vital roles in formulating correct arguments and finding out the validity of the arguments in a language system.

1.10 Deductive Logic and Inductive Logic Logic is broadly divided into two kinds based on the ‘nature’ and ‘form’ of arguments. These are deductive logic and inductive logic. In deductive logic, the conclusion is derived from the premises in accordance with logical rules. In this case, the conclusion of an argument necessarily follows from the premises. It supports formal logic both in principle and practice. In contrast to deductive logic, in inductive logic, the conclusion of an argument is inferred from a few observed instances (verified facts) that are presented in the form of the premise of an argument. In this case, the conclusion of an argument is always associated with a probability and not with certainty. However, inductive logic supports material logic both in principle and practice, as its premises correspond to the existing facts and events of the phenomenal world. In deductive logic, the conclusion is inherently embedded in the premises. The quantity stated in the conclusion already exists in the premises of a valid deductive argument. Thus, the quantity of the conclusion cannot be greater than the sum total of the premises. It is either equal to or less than the premises of an argument. Unlike deductive logic, in inductive logic, quantity of the conclusion can be greater than or equal to the premises. An example of deductive logic is as follows: All swans are white. X is a Swan. Therefore, X is white. In this argument, the conclusion states that X is a swan, and its colour is white. The quantity (i.e. a swan) of the conclusion already exists in the first premise (i.e. all swans are white), as it states that all swans are white in colour. So in this example, the conclusion already exists implicitly in the first premise of the argument. More

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about deductive logic and the application of rules to determine valid arguments of deductive logic are found in Chaps. 7–9 of Part II of this manuscript. An example of inductive logic is as follows: Crow L is black. Crow M is black. Crow N is black. ––––––––– {Inductive leap} ––––––––– Therefore, perhaps all crows are black. In this argument, the quantity expressed in the conclusion is greater than the quantity expressed in the premises put together. The notion of ‘inductive leap’ suggests that a few more observed instances (premises) of similar type could have been added to the existing premises to strengthen the conclusion. The details about inductive logic are found in Chaps. 15 and 16 of Part VI of this book. In deductive logic, since a conclusion implicitly exists in the premises, the conclusion cannot be true unless the premises are true. This implies that if the conclusion is false, then one of the premises must be false, and if the premises are true, then the conclusion following from the premises is necessarily true. Further, if the premises are true and the conclusion is true, then the argument is treated as valid, as premises support the conclusion of the argument. A deductive argument is invalid when its premises do not support the conclusion. Thus, a deductive argument is necessarily either valid or invalid. Unlike deductive arguments, inductive arguments are not judged as either valid or invalid. Rather, they are judged as stronger or weaker, good or bad, convincing or unconvincing. The reason is, in an inductive argument, there may be a case that all the premises are true, but the conclusion is false, yet the falsity of the conclusion does not contradict the truth of the premises. Conclusion inferred from the limited number of observed instances (premises) is considered tentative or provisional, as some more observed instances (premises) could have been added to the given premises to make a difference to the conclusion. Concerning the above-mentioned inductive example, the conclusion states that ‘perhaps all crows are black’. This proposition does not guarantee that there will not be a case where a crow’s colour would be not-black in the future. In future, if there would be a case where a crow’s colour is not-black, then the conclusion inferred in the inductive argument will not be considered true. So, the conclusion of an inductive argument is not to be considered as a true fact with certainty and surety, as it is associated with a probable situation. The strength of the conclusion of inductive arguments varies in degree. It is so because in some cases, premises (evidence) are stronger, and on some occasions, premises are weaker. This phenomenon suggests that the conclusion of inductive arguments is sometimes highly probable (stronger) and sometimes less probable (weaker).

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In deductive arguments, the premises are taken for granted as true. They are not empirically verified for the determination of their truth. The conclusion derived from the premises is based on a set of logical rules. Any violation of these logical rules will encounter the fallacy of the argument. In inductive arguments, premises are empirically verified as true, and the conclusion is inferred from the verified true premises (evidence). Due to the involvement of ‘inductive leap’ in inductive arguments, the truth of the conclusion varies in degree, but not in kind. However, in the case of deductive arguments, the truth of the conclusion does not associate with ‘inductive leap’, and thereby it does not vary in degree, but maybe in kind. Thus, deductive arguments are judged as either valid or invalid. Both deductive logic and inductive logic are used as methods to differentiate true fact from false fact, correct reasoning from incorrect reasoning, correct information from incorrect information, true beliefs from blind beliefs, and so on. They complement and supplement each other in the sense that the conclusion of an inductive argument may serve as the premise of deductive logic. Logicians, social scientists, basic scientists, applied scientists, and others use logic (deductive logic and inductive logic) as scientific methods for their scientific inquiries, as they are interested in evidence (premises) that need to support the conclusion (findings) of their research studies.

Chapter 2

Language, Logic, and Concepts

In the previous chapter, we mentioned definitions of logic and discussed the relation between logic, language, and reasoning. We elucidated the origin, nature, and scope of logic. Further, we explained the difference between ‘idea’ and ‘thought’, ‘form’ and ‘matter’, ‘formal logic’, and ‘material logic’. We illustrated the usage of logic in everyday life. In addition to these topics, we elucidated ‘deductive logic’ and ‘inductive logic’. In continuation of the previous chapter, in this chapter, we will discuss the relation between logic and language. We will illustrate the concept of semantics, syntax, and grammar. Further, we will enunciate types of language found in a language system. In addition to these topics, we will discuss the functions of language and the role of logic in formulating abstract and concrete concepts.

2.1 Interrelation between Logic and Language Logic and language are inherently and implicitly related to each other. It is so because the subject matter of logic cannot be explained without language, and language bereft of logic does not convey meaning in a context. Language is the vehicle of our thoughts, as thinking is not possible without language. Even in soliloquy, a person cannot speak to himself or herself without a language. Human beings use language to communicate their thoughts to others. Logic assists in producing rational and valid thoughts. The study of logic is necessary and important because it improves human beings’ rational and critical thinking about worldly affairs. Rational and critical thinking are used to recognise, analyse, and evaluate arguments of a language system. According to Harman (1973), ‘Language makes thought possible. Learning a language is not just learning a new way to put our thoughts into words; it is also learning a new way to think’ (pp. 84–85). Wittgenstein (1989–1951), an analytic philosopher, states that ‘the limits of my language mean the limits of my world’. ‘The limits of my world’ are comprehended as his limited thoughts about the phenomenal world.

© Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_2

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Language helps us to think about a fact or an event, but logic helps us to arrange our thoughts about a fact or an event in a sequence. The sequence of thoughts about a fact assists in comprehending the fact concisely, precisely, and clearly. In an argument, the sequence of thoughts is presented from premises through conclusion. Logic helps in deriving the conclusion from the premises and finding out the validity and invalidity of the argument. In this sense, logic and language are interrelated to each other.

2.2 Syntax, Semantics, and Grammar A language has two components, namely syntax and semantics. The mechanism of production of a language is called syntax, and when meaning is assigned to the basic unit of a language, it is called semantics. ‘Grammar’ is a part of the syntax of a language. It guides both speakers and listeners about time (i.e. present, past, future) that is expressed in the speaker’s utterances and sentences. In the linguistic system, a letter plays an important role in formulating a meaningful word, and a word plays a significant role in formulating a meaningful sentence. The mechanism of producing words and sentences of a language is called syntax. Syntax deals with the structure of a word and a sentence. Syntax is treated as either correct or incorrect. A correct combination of a few letters produces a correct and meaningful word. Further, a correct concatenation of a few words produces a correct and meaningful sentence. For example, if we write ‘khahdfli’, it does not convey any meaning because no such word (combination of letters) ‘khahdfli’ exists in the English language. Thus, it is not the correct combination of letters. But if we write ‘khadi’, it conveys meaning, that is, an Indian homespun cotton cloth. It is so because each letter in the word ‘khadi’ plays a vital role in formulating the word ‘khadi’. Thus, it is affirmed that a correct syntax is associated with meaning but an incorrect syntax does not convey any meaning. Bare words, sounds, and signs do not convey any meaning on their own. We assign meaning to a word, a sign, and a sound to convey our thoughts to others. When we assign meaning to a sign, or a word, or a sound, or a sentence, it is called semantics. To put it simply, a bare ‘word’ is a sign that does not convey a meaning on its own. We, human beings, assign meaning to a word; as a result, it conveys a meaning. We assign meaning to a word based on certain conventions. In this context, we may say that a bare sign is not meaningful, but when we assign meaning to a sign, it becomes meaningful and thereby treated as a symbol. So, each symbol has a meaning. A symbol conveys a designated meaning in the linguistic system and in the logical discourse. For example, the signs ‘1’ and ‘one’ do not convey any meaning on their own. We assign meaning to it that ‘1’ is a numerical representation of a single object, and ‘one’ is an English word that stands for an object or a concept. Human beings assign meaning to different signs and sounds in their daily lives to communicate their thoughts to others. Assigning meaning to a sign, a word or a sound is called semantics. The semantics of a word or a sentence is judged as either true or false and not correct or incorrect.

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In the language discourse, each sentence is formulated with a grammatical structure. ‘Grammar’ is part of the syntactical structure of a sentence. It not only guides the speaker but also guides the listeners to know the time that the sentence is stating about. A sentence can be formulated to express facts of the present time, past events, and future events. So, the notion of ‘time’ plays a vital role in comprehending the written sentences and utterances of a speaker precisely and clearly. If a sentence is not grammatically correct, then the thought expressed in the sentence will not be communicated to the hearers intently and aptly. Rather, hearers may understand the meaning of the sentence differently that may not be intended by the speaker when uttered. For example, Mr. X utters, ‘Miku was a runner’. Here, Mr. X means that in the past, Miku was participating in running competitions, and he was a runner. But this sentence shall not be understood as ‘Miku is a runner even in the present time’. Another example, Mr. Y utters that ‘Ritu is a singer’. In this sentence, Mr. Y communicates his thought to the listeners that Ritu sings many songs and she is continuing her singing career at the present time. Take another example, Mr. Z says, ‘Sita will not score good grade in her examination’. In this sentence, Mr. Z speaks about Sita’s future that she will not be able to score a good grade in her upcoming examination. So, in a linguistic system, grammar is a part of syntax that plays an important role in obtaining the meaning of the sentences correctly uttered by the speakers. Indian logicians (the Naiyayikas) belonging to the Nyaya School argue that a sentence is a string of words that are collectively able to convey a specific meaning. But any kind of combination of words may not be able to convey a meaning. They propose that a sentence should satisfy the following four conditions to be treated as a meaningful sentence. (i) (ii) (iii) (iv)

Expectancy Compatibility Proximity Intention

Expectancy Expectancy is to be met to obtain the meaning of a sentence. It signifies ‘mutual implication’. Here, the constituent words that compose a sentence must be interrelated in such a manner that it conveys a specific sense. A mere combination of unrelated words obviously cannot constitute a sentence. For example, horses fly fine elephants excellent mosquitoes. Although each of these words conveys its meaning, they together are unable to convey any meaning. Thus, it is necessary that a sentence must consist of a string of related words. Compatibility This condition expresses that the constituent words of a sentence should possess ‘fitness’ to convey a particular meaning. They should not contradict each other. For example, ‘A triangle is a plane figure consisting of four angles’, ‘Mortal beings never

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die’. These sentences are contradicting themselves. They do not fulfil the compatibility criterion to be regarded as meaningful sentences. Therefore, the compatibility criterion should be satisfied to obtain the meaning of a sentence. Proximity The third condition ‘proximity’ states that the constituent words of a sentence must be uttered or written in sequential order, but not in a long gap. For example, if we utter a sentence, ‘Please……..give……..me……..a…….…piece………of……..paper’, the meaning of the sentence cannot be grasped by the hearer. It is so because the words are uttered with long gaps. On the other hand, if we utter the sentence, ‘Please give me a piece of paper’, it will convey a meaning to the hearer and hence intelligible. Here, the words are uttered in a suitable manner and in sequential order. Intention The fourth or the last condition expresses that if a word in a sentence possesses more than one meaning, then the meaning of the sentence should be derived by considering its use in the context, i.e. how a word is used in the sentence and in which context. This condition arises if a word is used ambiguously in a sentence. For example, the word ‘saindhava’ means ‘salt’ as well as ‘horse’. While eating, if a speaker utters, ‘please bring saindhava for me’, obviously it will be understood as salt and not a horse. In this condition, the intention of the speaker is to be taken into account while deriving the meaning of the proposition. Another example is ‘The ladder is crying. Here the word ‘crying’ has multiple meanings such as tears of eyes and breaking down. But, in this context, it will be understood as the ladder is going to break down as more than the required load has been put on it. Thus, the meaning of a sentence is also derived on the basis of its contextual use.

2.3 Natural Language, Metalanguage, Logically Perfect Language Language describes objects and concepts of the world. A word used in a language refers to either an object or a concept of the world. There are innumerable objects existing in the world, and each object displays some kind of structure. To describe the structure, we need an appropriate language that is termed as ‘first-order language’ or ‘ordinary language’ or ‘natural language’. It assists us in expressing objects and facts of the world. We use this language to share our thoughts, desires, cravings, passions, etc., with others. Thus, it is also known as object language. Logically speaking, the language in which we shall be characterising the language of logic is known as the ordinary language (Singh and Goswami, 1998, p. 1), and an extended version of ordinary language is known as metalanguage. Metalanguage cannot be explained on its own. It needs ordinary language for its elucidation.

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Etymologically speaking, the language through which we speak or express an object of the phenomenal world is called object language. A language through which we formulate statements of object language is known as metalanguage. In metalanguage, sentences embrace semantic and non-semantic components. The nonsemantic component furnishes an expressive potential that is as rich as that of the object language. Thus, a correspondence relation is found between the nonsemantic component of the metalanguage and the object language, while no such correspondence is sought in the semantic part of the metalanguage. Metalanguage satisfies the following requirements. i. ii.

It must allow for all expressions of the object language to be explicable in it. It must contain semantic concepts in relation to the object language that cannot be formulated in the object language.

For example, “‘Grass is green’ is true if and only if the grass is green”. In this sentence, the subject part within single quotation marks is treated as metalanguage because it is just a concatenation of words. It does not convey any meaning. It expresses a meaning when the predicate part affirms the facts by examining their availability in the empirical world. The predicate part grass is green, which is free from quote conveys that there are objects called grass existing in the empirical world, and their colour is green. This is called the object language as it deals with objects of the world. But the subject part of the sentence enclosed in single quotes is treated as metalanguage because it does not convey any meaning on its own. It needs the help of the object language for its explanation. The third variety of language is known as ‘logically perfect language’. It is a unique language that is free from all sorts of logical errors that we often encounter in ordinary language usage. Its nature and structure are exhibited. It is grammatically correct, referent assigner, and meaning conveyor. It conveys meaning about an object or a concept of worldly affairs. Sentences belonging to this language offer both sense (meaning) and reference. Frege (1848–1925), an analytic philosopher, has propounded this language which he calls Begriffsschrift. According to Frege, natural language (ordinary language) is not a well-defined and perfect language because defects lie in the structure of the sentences. For example, ‘Practising austerity is a good habit’. In this sentence, how do we refer to the words ‘practice’ and ‘austerity’, and further, what will we refer to when we say that something is a good habit? In this way, ambiguity arises in natural language. In order to overcome these shortcomings, especially relating to reference failure, Frege suggests that we need a formal and logically perfect language, a language where every sentence is endowed with both ‘sense’ and ‘reference’. As a result, each and every sentence can be judged as either true or false.

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2.4 Functions of Language: Directive, Informative, and Expressive We use language for various purposes such as describing a fact or an event, sharing emotions with others, requesting others to do a task, and so on. Language, in this sense, is used as a tool or an instrument to communicate the thoughts of our everyday lives. A language is not only considered as a tool to communicate our thoughts but also considered as a phenomenon of our lives. It is so because a language has many functions. For example, a word can be used in multiple ways to convey different meanings in different contexts. A sentence can also be used in various ways to convey varieties of meanings in different contexts. In this regard, Wittgenstein (1958), an analytic philosopher, in his work on Philosophical Investigations, states that using language is like playing games, which he calls ‘language games’. Language is a complex phenomenon as it has many functions in a linguistic discourse. But for our present purpose, we shall explain only three functions of a language; these are directive, informative, and expressive. Directive function Human beings use language to request or ask others to perform a task. For example, X requests someone to shut the windows, Y directs someone to put the file on the table, Z orders someone to complete the office work by today’s evening, L requests someone to open the door, M requests the audience for their kind attention to her seminar presentation, etc. To be precise, a person can direct someone either to perform an action or refrain from doing an action. These kinds of expressions (sentences) perform the directive function of a language. Also note that, asking questions to someone also comes under the directive function of a language. The reason is, questions are asked to get answers, maybe a probable one. For example, what is the time now? do we have logic class tomorrow? can I take leave tomorrow?, etc. These sentences are considered as directive functions of the language. Since these sentences do not describe any object of the fact of the phenomenal world, they cannot be judged as either true or false. Informative function When a sentence describes a fact, or an event, or an object of the phenomenal world, it is treated as an informative function of a language. In this case, a sentence informs about an object or a fact, and the object or the fact exists in the phenomenal world. In this function of a language, a sentence must have a referent in the empirical world. If the referent is found in the empirical world, then the sentence is judged as true, and if the referent is not found in the empirical world, then the sentence is treated as false. Thus, all informative sentences are judged as either true or false, as they describe the state of affairs of the world. The informative function of a language assists in communicating one’s beliefs, thoughts, and opinions to others. For example, snow is white; the grass is green; a table is a solid object; Sita carries six apples in a basket; five cows are grazing the grass in a green field, etc.

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Informative sentences cannot be judged as not-true and not-false at a time. Further, these sentences cannot be both true and false at the same time. But, sentences serving informative functions are necessarily judged as either true or false. These sentences help in formulating arguments in a linguistic system. Logicians are interested in finding out the truth and falsity of a sentence while formulating arguments in a linguistic system. Thus, logicians are more interested in the informative function of a language. Expressive function Human beings use words and expressions of a language to express their emotions, feelings, desires, and so on to others. Words and expressions used to convey a person’s emotions, feelings, and so on are not to be judged as either true or false, because they do not describe an object or an event or a fact of the world. They only communicate the passions, desires, and feelings of a person, not the information about worldly affairs. The expressive function of a language expresses a person’s joy, pain, pleasure, happiness, etc. For example, ‘Wow! It is wonderful!’, ‘Bravo!’, ‘Great achievement!’ ‘Sweet dreams!’, ‘Best wishes!’, etc. Expressions conforming to the expressive function of a language are treated as non-logical because they do not state any information about anything. Instead, they communicate feelings to the listeners or hearers. Consider a poet who describes a person’s desire for heavenly life. In this case, the poetic descriptions of the person’s desire are part of the poet’s emotions and imaginations. The poetic descriptions do not correspond to the objects or facts of the world. Hence, the expressions used to describe a person’s desire for a heavenly life cannot be judged as either true or false. However, these descriptions communicate meaning to the readers and listeners. In this way, the expressive function of a language does not aim at establishing the truth or falsity of the expressions; rather shares the feelings and emotions of a person with others. The expressive function of a language is rarely used in logic and logical discourse, as it does not contribute towards formulating logical arguments. The directive, informative, and expressive functions of a language are not mutually exclusive. It is so because, on some occasions, it is found that these three functions of a language are overlapping with each other. There may be a case where an expression has both informative and expressive functions. For example, ‘Wow! He owns a luxury car in a lottery!’, ‘Wow! It is a beautiful snow-capped mountain!’. Similarly, an expression can satisfy both expressive and directive functions of a language. Further, a statement may conform to both informative and directive functions of a language. Even though these three functions of a language overlap with each other, yet the function of an expression would be identified based on the ‘context’ and its ‘use’ in the linguistic system. Thus, ‘context’ and ‘use’ of an expression play vital roles in identifying the function of a language in the linguistic discourse.

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2.5 Formation of Abstract and Concrete Concepts: The Role of Logic Logic and language are intrinsically related to each other. Language assists in producing thoughts, while logic evaluates these thoughts and finds out their validities. Logic is used to formulate thoughts or concepts of a linguistic system. The application of logic to formulate concepts or thoughts of worldly affairs is so rudimentary that one cannot dissociate logic from a concept or thought and vice versa. Logic not only helps in formulating concepts but also brings clarity to concepts while communicating to others. Two types of concepts are found in logic and logical discourse. These are the abstract concept and concrete concept. Abstract concept An abstract concept does not have a referent in the empirical world. But it communicates meaning to the listeners and readers of the concept. For example, man, tree, triangle, book, etc., are regarded as abstract concepts. Consider the word ‘book’ from the examples. The word ‘book’ does not refer to any particular book, but it means all the books belonging to all subjects of the academic disciplines. Even though there are many subjects and many books existing in the academic fields, the notion of ‘book’ represents something that is essential to all the books. It does not describe a book in particular as such; rather, it describes a commonality that is found in all the books belonging to all academic disciplines. The abstract concept represents a class where all members of the class must have at least one essential feature and that feature necessarily defines a member of the class. Abstract concepts are eternal as they exist timelessly. For example, the word ‘man’ is an abstract concept. It represents all the male persons on this earth. The notion of ‘man’ will remain as it is even though many males will be born and will die on this earth. Similarly, the word ‘tree’ is an abstract concept. The notion of ‘tree’ will remain unaltered even though many trees would get destroyed, and many would grow in a forest. Logic helps in formulating abstract concepts of the world. It eliminates accidental features (that changes from time to time and place to place) of an object and identifies the essential features (that do not change with time) of the object to formulate the abstract concept of that object. By applying logic, we can comprehend the meaning of a word that represents the name of a class and identify a member belonging to the class. Concrete concept Unlike abstract concepts, concrete concepts have referents. Concrete concepts refer to the existence of objects and facts of the world. The referents are empirically verifiable to affirm their existence in the phenomenal world. For example, the present president of India, a banyan tree in front of the Humanities and Social Sciences department of the Indian Institute of Technology Madras (IITM), an analytic philosophy book in the third bookshelf of IITM central library, etc., consider the expression ‘the prime minister of India in the year 2020’. This expression refers to a designated

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individual, not anyone else; that is honourable Sri Narendra Damodardas Modi. Here, the expression ‘the prime minister of India in the year 2020’ represents a concrete concept of an individual. Logic helps in determining the meaning of the sentences by finding out referents of the objects and individuals mentioned in the sentences. Since a concrete concept expresses an object or an individual that is verifiable and referable in the empirical world, the sentence through which the concrete concept is expressed is judged as either true or false. Logicians deal with sentences that are judged as either true or false, as they are interested in formulating and evaluating arguments in the linguistic discourse. The abstract and concrete concepts can be explained through the ‘type-token’ distinction. An object that exists in the phenomenal world and has a referent is called the ‘token’ of that object. The word ‘type’ refers to a ‘class’. It represents a class name where many individuals are members of the class, having at least one common and essential feature among them. It is not verifiable and referable. The word ‘type’ signifies an abstract concept of worldly affairs. In short, a concrete concept is uttered in ‘token’ form and represents an existent object, whereas an abstract concept signifies a ‘type’ that represents a class where many individuals are members of it. The distinction between concrete and abstract concepts is placed below for reference, clarity, and discernment. Abstract concept Concrete concept Man

Mr. Jawaharlal Nehru, Mr. Vallabhbhai Jhaverbhai Patel, Mr. Subhas Chandra Bose, Mr. Mahatma Gandhi, etc.

Tree

A neem tree, a banyan tree, a mango tree, etc.

Book

A deductive logic book, an inductive logic book, a symbolic logic book, etc.

Pen

A red dot pen, a green ink pen, a blue ink pen, etc.

Chapter 3

Classification of Logical Propositions

In the previous chapter, we discussed the relation between logic and language. We analysed the syntax, semantics, and grammar of a sentence. We explained the types of language. Further, we illustrated the functions of language and the role of logic in formulating abstract and concrete concepts. In continuation of the previous chapter, in this chapter, we will discuss types of words, kinds of terms, the difference between word and term, the difference between sentence and proposition, types of logical propositions, determination of the truth-value of a proposition, and the distribution of terms in categorical propositions. Let us start with a query: What is a word?

3.1 What is a Word? Logic, as a subject of study, is concerned with arguments. Each argument consists of propositions, and a proposition comprises mainly three constituents: subject, predicate, and a copula (helping verb). The subject, a predicate, and copula of a proposition are ‘words’ of a language. A word consists of a letter or more than one letter that conveys a specific meaning. For example, a, an, the, are, of, tree, tall, table, etc., are words. A word is considered a basic unit of a language. In this regard, a question may arise, whether discussion of ‘word’ comes under the purview of logic? The answer is ‘affirmative’ because logic is primarily concerned with reasoning. Reasoning assists in finding out whether a proposition is true or false in an argument. Since a ‘word’ is an essential component of a proposition, the discussion of ‘word’ is warranted in the logic subject.

© Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_3

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3.2 Types of Words Words are of three types: categorematic, syncategorematic, and acategorematic. A ‘categorematic’ word is one that is used in the subject and predicate part of a proposition. It conveys meaning independent of other words of the proposition. For example, John is a swimmer. In this proposition, ‘John’ and ‘swimmer’ are the categorematic words. Categorematic words are recognised as nouns, pronouns (excluding relative pronouns), adjectives, and participles of the English language. A ‘syncategorematic’ word does not convey any meaning on its own. It needs the support of other terms of a proposition to convey the meaning. In the example ‘John is a swimmer’, the expressions ‘is’ and ‘a’ are regarded as syncategorematic words. Articles, conjunctions, prepositions, and adverbs of the English language are known as syncategorematic words. Examples of syncategorematic words are, of, on, a, was, were, are, had, will, ought, etc. Sometimes, it is noticed that the word ‘the’ is pre-owned in the subject of a proposition. A question arises: Can we treat ‘the’ as a categorematic word? Logicians claim that as long as words are used as substantives (conjunctions, adverbs) and not as articles of propositions, we treat them as a categorematic word. For example, but, only, the, etc., are considered as a categorematic word as long as they convey meaning independent of other words of a proposition. Consider an example, ‘Only the eleven selected Indian cricket players can play for India in the international one-day cricket match’. Here the word ‘only’ is categorematic while ‘the’ is syncategorematic. Another example, ‘Ramesh is a mathematician, but he claims that his knowledge about the philosophical subject is sound’. Here the word ‘but’ is used as a conjunction. Therefore, it is treated as a syncategorematic word. Besides these two types of words, there is another type of word existing in the language discourse, known as the ‘acategorematic’ word. ‘Acategorematic’ words are exclamatory. For example, oh! ah!, Ouch!, wow!, etc. These words can neither be used as subject nor predicate of a proposition. Further, these words cannot be used as a conjunction of the propositions. Rather, they are used as the interjection of propositions. They convey meaning based on the context where they are used. Without context, these words do not convey any meaning.

3.3 Words Versus Terms For logicians, ‘words’ are not the same as ‘terms’. Only categorematic words are to be regarded as ‘terms’. It is so because a term is used either as a subject or a predicate of a proposition. It does not seek any assistance from other words of a proposition to convey meaning and for its subsistence. In the example, John is a swimmer, ‘John’ and ‘swimmer’ are considered as terms, while ‘john’, ‘is’, ‘a’, ‘swimmer’ are regarded as words. Thus, logicians suggest that all terms are words, but all words are not terms. In this regard, a question may arise: What about proper names?

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Are proper names ‘terms’? Proper names, such as cow, horse, John, Rama, Sita, Jim, and Jack, are terms since they are used in the subject and/or predicate part of a proposition. Proper names are both connotative and non-connotative. They are connotative when we attribute qualities to them and non-connotative when we do not attribute qualities to them. Irrespective of the connotative or non-connotative nature of proper names, they necessarily refer to individuals or entities of the world. For example, Vishnu, Sita, Rama, and Hari are proper names and belong to the connotative category, as we attribute Hindu deities’ qualities to them. In contrast, Smith and Jack are proper names that belong to the non-connotative category, as we do not attribute any deity’s or entity’s quality to them. According to Frege (1848–1925), proper names are the terms that have both ‘sense’ and ‘reference’. Sense for him is the ‘mode of presentation’, and reference is the ‘mode of applicability’ (referent). He admits that there are cases where two proper names have two different senses, but both possess one referent. An example, ‘The morning star’ and ‘The evening star’ have two different meanings (senses), but they refer to one object (referent), i.e. ‘Venus’. The detailed discussion about proper names is found in Frege’s On Sense and Reference. For John Searle (1958), a proper name has a reference because it is associated with a cluster of descriptions.1 For example, to know the meaning of the proper name ‘Aristotle’, we need to know a cluster of propositions associated with Aristotle, such as the pupil of Plato, the teacher of Alexander the Great, and the author of Nichomachean Ethics. Kripke (1980) disagrees with Searle’s view on the proper name. In his opinion, a proper name is a rigid designator that has only one referent. The referent must be an object or an individual and that should exist in every possible world. Further, he enunciates that most of the definite descriptions are non-rigid designators2 , as they misguide us to identify the exact referent. For example, ‘A tree is having green leaves of four to six metres long’ does not necessarily refer to a coconut tree. Natural kind terms Unlike proper names that refer to individual objects, natural kind terms pick out ‘kind’. For example, tiger, gold, water, etc., are natural kind terms, and these terms are known as universal terms. When we say ‘tiger’, it does not refer to an individual tiger; instead, it refers to all the tigers of the world irrespective of their colours, height, and other differences. Putnam and Kripke opined that these terms are found in the natural language3 . John Locke4 , an empiricist, said that natural kind terms signify ideas of speakers’ and listeners’ minds. For example, when a speaker utters 1 Searle,

J. (1958). Proper names. Mind, 67, pp. 166–173. S. (2002). Beyond rigidity. Oxford: OUP. Also see, Kripke, S. (1980) (2nd Edition). Naming and necessity. Oxford: Blackwell Lectures I & II. 3 Putnam, H. (1973). Meaning and reference. Journal of Philosophy, 70, pp. 699–711. Also, see Kripke, S. (1980). (2nd edition). Naming and necessity. Oxford: Blackwell. 4 Locke, J. (1975). An essay concerning human understanding. Edited by P.Niddith. Oxford: Oxford University Press. 2 Soames,

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the word ‘gold’, it creates a mental image in the listeners’ minds, that is, a yellow body is having a certain weight, malleable, fusible, and solid. So, the speaker shares the idea of ‘gold’ with listeners.

3.4 Types of Terms The following types of terms are found in the logical discourse. (i) (ii) (iii) (iv) (v)

Simple and composite terms Assertive and negative terms Absolute and relative terms Abstract and concrete terms Universal and particular terms

Simple and composite terms A simple term is one that is devoid of a combination of words. It is a single and independent unity dissociated from other words. It conveys a specific meaning independent of other words, for example, horse, student, table, book, etc. A composite term, on the other hand, consists of a few words. All these words are put together to convey a specific meaning. In other words, a composite term is a collection of words used to logically and scientifically convey a meaning. For example, a leather industry, an intelligent lady, a smart student, etc., in the case of ‘an intelligent lady’, three words are combined to convey the meaning of the term. Assertive (affirmative) and negative terms A term that conveys meaning affirmatively is viewed as an assertive term. Assertive terms are known as declarative terms and popularly called ‘positive terms’. For example, Jil is a good student. This sentence consists of three terms (i.e. Jil, good, and student) and five words (i.e. Jil, is, a, good, and student). Each term asserts something positively about the state of affairs of the world. Thus, Jil, good, and student are assertive terms. Consider another example: Harish is an insincere student. In this sentence, ‘insincere’ is not an assertive term as it negates sincerity. In short, assertive (affirmative) terms affirm the presence of attributes in them. In contrast to assertive terms, negative terms deny the presence of qualities in them. Whether a term is said to be assertive or negative, it is based on its semantics. Merely ‘form’ of a term does not convey whether it is a negative or an assertive term. The terms, such as immortal, unintelligent, incompetent, and unspeakable, are considered as negative terms. In these terms, we find a negative prefix. But there are a few terms such as miserable and foolish are also treated as negative terms, even though there is no negative prefix attached to these terms.

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Absolute and relative terms An absolute term is the name of the class that does not refer to an individual member of that class. For example, when we say ‘tree’, it means all trees of the world and not a single tree. Here, ‘tree’ is an absolute term. Other examples of absolute terms are cow, chair, human being, flower, etc. Relative terms do not convey any meaning of their own. They convey meaning about something. The expression ‘something’ may stand for an object, concept, event, fact, action, quality, relation, etc. Thus, a relative term is a name that denotes either an object or a concept of the state of affairs of the world. For example, the meaning of the term ‘student’ cannot be comprehended without relating it to the concept of ‘teacher’, ‘learning’, etc. Thus, the term ‘student’ is a relative term, and also it is a positive term. In the case of a negative relative term, the meaning of the term is obtained by negatively relating it to an object or a concept of worldly affairs. For example, ‘widow’, the meaning of the term is understood with the death of a married woman’s husband. Here, ‘death (not alive)’ is a negative term, and ‘widow’ is a relative term. Abstract and concrete terms Human beings in their everyday lives experience numerous objects of the phenomenal world. Some objects are verifiable, empirically observable, and thereby referable. Their existence asserts that they occupy a certain space at a given time by virtue of their form and size. These objects are expressed through concrete terms. So, concrete terms refer to tangible, intangible, perishable, and durable objects. For example, a tree, a duster, a chair, etc., unlike concrete terms, abstract terms represent concepts of worldly affairs. They exist in the form of ideas. An idea does not have a referent but has meaning. It acquires meaning because some of its constituents are related to objects of the world, and some of its constituents are construed through mental activities (reasoning). Abstract terms like virtue, happiness, honesty, truth, and colour do not have any referent but have meaning. Consider an example, the term ‘virtue’. We understand the meaning of the term ‘virtue’. But we cannot refer to anything by pointing out our fingers as ‘virtue’, as we do in the case of a chair, a table, etc. Universal and particular terms A universal term is one that holds all members of a class as a unity. It expresses the commonness or essence of members (individual objects) belonging to a class. For example, when we say ‘cow’, it stands for black, non-black, white, non-white, shorttail, long-tail cows, etc. It does not refer to an individual cow. Rather, it indicates ‘cowness’ (the essential quality) that exists in each cow irrespective of their shapes, sizes, and other differences. Universalness has many classes. The division of class takes place upon the consideration of how big the scope of the term is. For example, ‘man’ is a universal term as it embraces all men. But if we say ‘human being’, then it includes ‘man’, ‘woman’, ‘child’, etc., in it. So, here ‘human being’ is a bigger universal term than ‘man’. Further, if we consider ‘mammal’, it includes ‘human being’, ‘man’, ‘reptile’, etc., in it. If we enlarge further and broaden the universal

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term, we can say ‘creature’ wherein the universal terms mammal, human being, man, etc., are part of it. If we examine all these universal terms, we find the differences lie among them in degree, but not in kind. Particular terms, on the other hand, are known as singular terms. A particular term denotes an object. It refers to an object that is empirically verifiable and referable. For example, a green pen, a brown chair, etc., particular terms are also treated as concrete terms.

3.5 Sentence Versus Proposition Words and terms are the constituents of a sentence. These constituents are the building blocks of a sentence. They are considered essential components of a sentence. Without these components, formulating a sentence is an impossible phenomenon. Propositions do also possess these constituents. So a question arises: Are propositions the same as sentences? Before answering this question, let us explain ‘what is a sentence?’, ‘what is a proposition?’, and find out the similarities and differences between ‘sentence’ and ‘proposition’. Sentence In the English language, a proposition often refers to a sentence in logic. But in the logical discourse, there are differences found between ‘sentence’ and ‘proposition’. A sentence consists of two parts: subject and predicate. But it must not have a ‘copula’ (helping verb). For example, Ram sings. This is a sentence, and it does not have a copula (e.g. is, are, were, should, may, had, must, etc.). Another feature of ‘sentence’ is it necessarily affirms or denies a fact, but to do so, it does not require a copula. The sentence ‘Sita swims’ is an affirmative sentence, whereas ‘Gita declines a job offer’ is a negative sentence. Proposition Propositions possess three components invariably: subject, predicate, and copula. The subject and predicate of a proposition are considered as the categorematic words, and copula is a syncategorematic word. The predicate term states something about the subject term necessarily. The subject term is one about which something is stated. Copula does indicate the status of the proposition, i.e. whether the proposition is affirmative or negative. For example, New Delhi is the capital of India. Here, the copula ‘is’ is making a statement of the relation between subject and predicate in an affirmative way. Thus, the proposition is judged as an affirmative proposition. Two important questions revolve around the word ‘copula’. First, does ‘copula’ belong to the present tense or in any tense? Second, what is the nature of ‘copula’—is it affirmative or negative? Regarding the first question, logicians claim that copula belongs to the present tense because it expresses the relation between predicate and subject only when the sentence is presented before us either in spoken or written form. Thus, it is not free

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from the present time. We assign meaning to a sentence when it is presented to us. Thus, copula should be confined to the present tense only. Refuting this claim, J. S. Mill (1806–1873) vehemently expressed that copula should belong to past, present, and future tense. The reason is that even though sentences and propositions are communicated to us in spoken or written form, we assign meaning to a sentence based on its copula. If the copula is in the past tense, the meaning we assign to the sentence is about past time. In a similar vein, present tense and future tense copulas follow their respective time. Concerning the second query, ‘Is copula affirmative or negative?’ it is unanimously agreed that the nature of copula is both affirmative and negative depending on the context. A copula cannot convey its meaning on its own. It conveys meaning when it is used in a proposition. It states a relation between the subject and predicate of a proposition. For example, the sky is blue. In this proposition, the copula ‘is’ is stating a relation between sky and blue colour affirmatively. Here, the copula ‘is’ is considered affirmative. There are certain situations where we use the syncategorematic word ‘is’ as ‘to be’. Here, ‘is’ implies ‘to exist’. Depending on the usage of ‘is’ in a sentence or a proposition, we can find out whether ‘is’ is a substantive verb or a predicate. For example, ‘Truth is’, this sentence says ‘Truth exists’. In this case, ‘is’ is a predicate. But if we say ‘Evil is’ and it means ‘Evil is existing’, then in this case ‘is’ is treated as a ‘copula’. Thus, copula should not be confused with the predicate of a sentence. Sentences are of various types, such as interrogative, assertive, and exclamatory. But propositions are only declarative. It is so because a proposition is composed of three elements: subject, predicate, and a copula, and a sentence is not necessarily composed of these three elements. Only in some cases, we find sentences of having these three elements. Thus, the scope and nature of ‘sentence’ are wider than the proposition. In this context, it is important to note that all propositions are sentences, but all sentences are not propositions.

3.6 Propositions and Theories of Truth According to Wittgenstein, an analytic philosopher, a proposition is a picture of reality (Tractatus, 4.021). It either asserts or denies object, fact, and/or event of the world. The reality of the world is a conglomeration of both existent and non-existent objects, facts, and events. Thus, a picture invariably depicts the reality of the world. Since a proposition confirms the association of subject and predicate, it conveys a specific meaning. The meaning is not an empty or void entity. It represents things and/or facts of worldly affairs. If the thing (object) and/or fact what the proposition is stating about are found in the phenomenal world, the proposition is judged as true. But if the object and/or fact is not found in the phenomenal world, then the proposition is judged as false. Thus, a proposition is necessarily either true (T) or false (F).

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Concerning the determination of the truth-value of propositions, there are three theories found in the logical discourse. In this context, I wish to place an excerpt from Rene Descartes’s writing. According to Descartes,5 ‘I have never had any doubts about truth, because it seems a notion so transcendentally clear that nobody can be ignorant of it… the word ‘truth’, in the strict sense, denotes the conformity of thought with its object’ (Letter to Mersenne: 16 October 1639, The Philosophical Writings of Descartes, Vol. 3). The three theories of truth are correspondence, coherence, and pragmatic theory of truth. The correspondence theory This theory suggests that a proposition expresses a fact or an event of worldly affairs. The truth of a fact or an event depends on the relationship between a belief and the fact or an event of the world. When we possess a belief, it is about either the physical or mental (psychological) world. And, the belief is made true or false not by relating to other beliefs but by corresponding to something available in the empirical or mental world. Let us take two distinct examples to clarify the notion of the correspondence theory of truth. Example-1: Anu believes that her blue pen is in her study table drawer. Her belief is true only when the blue pen is available in her study table drawer. In a sense, we can refer to the blue pen (object) in the drawer. But if she believes that there is a pigeon in her study table drawer and in reality there is no pigeon in the drawer, then her belief turns out to be false. It is so because her belief does not correspond to the ‘pigeon’ in the drawer. Example-2: Susama believes that she had a toothache yesterday. Her belief is made true if she had a toothache yesterday indeed. This example is about the mental world, as pain realised in one’s mind, whereas the prior example (Example-1) was about the physical world. The coherence theory Hegel (1770–1831) and Bradley (1846–1924) among others have proposed the coherence theory of truth. This theory suggests that the truth of a fact is ascertained through the examination of the relationship among beliefs (judgements). A fact is considered true when it conforms to more than one belief those are related to each other. Thus, the coherence theory of truth accepts the degrees of truth that can be more or less true, but certainly cannot be untrue. No belief is absolutely true because we can never attain a complete and absolute coherent system. Thus, one can assert that some beliefs (judgments) are truer than others by virtue of being close to the ideal. For example, on a fine morning, John while reading India’s national newspapers found news that ‘Indian Institute of Technology (IIT) is the best engineering institution in India’. Although different journalists wrote about this information in their ways in different newspapers, yet John understands the truth of the fact. Here, the truth of the fact is ascertained by virtue of the coherence among many newspaper columns about IITs. Thus, the coherence theory of truth expresses that a statement is true if it is logically consistent with other statements 5 Descartes

is a rationalist philosopher.

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and that is held to be true. Further, a statement is false if it is inconsistent with other statements that are held to be true. This theory of truth has some lacunas. Such as, it is hard to believe that a belief (judgement) is more or less true. A question arises: How can a belief ‘New Delhi is the capital of India’ be more or less true? It can, at best, be either true or false. This criticism is one of the reasons for the emergence of the pragmatic theory of truth. The pragmatic theory William James (1842–1910) is believed to be the propounder of the pragmatic theory of truth. Richard Rorty (1931–2007) is considered to be a contemporary adherent to this theory. The term ‘pragmatic’ means practical or useful. William James defines the pragmatic theory of truth in terms of the usefulness of a belief. He thinks that useful beliefs are true, and useless beliefs are false. Thus, the pragmatic theory of truth suggests that beliefs are true if they are useful and false if they are not useful. Accordingly, pragmatists express that ‘the pragmatic truth’ stands for beliefs that ‘work’. Further, they claim that we cannot attain the absolute truth, and therefore, we must be content with what works. Pragmatists believe that true beliefs must be consistent with each other and argue that if these are not consistent with each other, then they will not work. Some criticisms are made against this theory. Such as, it is quite conceivable that a belief might work well but still not considered true. On the other hand, a belief may work badly, but it is still considered true. For example, if an intoxicated person believes that consuming alcohol for the next time would kill him, he would be greatly benefited, but this does not make his belief true. It is so because every time he consumes alcohol thinks about the next time. Further, there are occasions where we find that what works for an individual may not work for another. Again, what works well for an individual in a particular context may not work in another context. If these situations prevail, then how can we assign truth to a belief that just works in a given situation? In the logical discourse, a proposition is judged as true when it satisfies the correspondence theory of truth. In other words, a proposition is true when it corresponds to either objects or concepts of the world.6 For example, ‘A cow is a quadruped animal’. The content of the proposition corresponds to worldly affairs, and therefore, it is a true proposition, but if someone says, ‘A cow is a flying animal’, it is a false fact as it does not correspond to worldly affairs. Hence, it is treated as a false proposition. Another example, ‘That tree is tall’. This proposition is judged as true provided what it refers to must be a tree, and it is indeed tall. But if someone says, ‘A tree gives milk’; this proposition is treated as a false proposition, as it does not correspond to worldly affairs. As a subject of study, logic deals with various kinds of propositions that are termed logical propositions. Logical propositions are divided into different kinds based on their structure, relation, quality, quantity, and so on. The detailed analyses of types of logical propositions are as follows. 6 The

detailed analyses are available in Frege’s paper ‘On Sense and Reference’.

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3.7 Types of Logical Propositions (a)

STRUCTURE Based on structure, propositions are divided into two kinds: simple proposition and compound proposition. A simple proposition is not combined with any other proposition and expresses a state of affairs of the world. It is also known as an atomic or elementary proposition. It is atomic because it cannot be divided further into an atomic proposition. It is elementary because it is the basic proposition. We can add propositions to this type of proposition, but certainly, we cannot divide this sort of proposition into further atomic propositions. For example, tiger is an animal, dog is an animal, human beings are rational beings, etc., are the atomic propositions. Compound propositions, on the other hand, are a combination of more than one simple proposition. For example, Davis is a researcher and a piano player. This sentence can be divided into two atomic (simple) propositions. These are Davis is a researcher, and Davis is a piano player. Compound propositions are further divided into two kinds.

(i) (ii)

Remotive Copulative

In the case of a remotive proposition, two negative propositions are combined and form a compound proposition, whereas copulative propositions are consisting of more than one affirmative simple proposition. Example of a remotive proposition, Jack is neither a swimmer nor a cricket player. This proposition is divided into two negative simple propositions. That is, Jack is not a swimmer and Jack is not a cricket player. Example of a copulative proposition: Anu is a schoolteacher and a singer. This proposition is divided into two affirmative simple propositions. That is, Anu is a schoolteacher, and Anu is a singer. (b)

RELATION Based on ‘relation’, propositions are divided into two kinds: categorical and conditional. In the case of categorical propositions, the predicate does not attribute any condition to the subject but it states something either affirmatively or negatively about the subject. In short, the relation between the predicate and the subject of a proposition takes place without any condition. For example, all men are mortal, no man is perfect, some cows are white, some flowers are not red, etc. In the first and third proposition, the predicate does not attribute any condition to the subject, but it states something about the subject. In the case of the second and fourth propositions, the predicate negates the subject without any condition.

In conditional propositions, the relation between subject and predicate is made under certain conditions. For example, if I were a bird, then I would fly to my desired island every weekend. Another example, if there is sun, then there is light. In both examples, the consequent is related to the antecedent under certain conditions.

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Conditional propositions are of two types: hypothetical and disjunctive. The hypothetical proposition consists of two parts: antecedent and consequent, and it is conveyed with an expression ‘if...then’. In a hypothetical proposition, the consequent is associated with antecedent under certain conditions. For example, if there is heavy and continuous rainfall, then there will be a flood. The disjunctive proposition is expressed through ‘either…or’ expression. A proposition that is formulated and expressed through ‘either…or’ expression invariably affirms one alternative while rejecting the other. It excludes the middle ground of the two alternatives. In other words, a disjunctive proposition does not hold any middle ground between the two alternatives. For example, either he is tall or short. Another example, she is rich or poor. In the first example, we find two simple propositions; he is tall and he is short. These two simple propositions are construed with ‘either…or’ expression. So, by principle, if he is tall, he cannot be short at the same time, and if he is short, he cannot be considered as tall. Thus, either of the alternatives needs to be affirmed. It is, therefore, stated that all disjunctive propositions are affirmative propositions. A proposition that is formulated with ‘neither…nor’ expression cannot be treated as a disjunctive proposition because both the alternatives are getting rejected in the proposition. Thus, it violates the disjunctive proposition conditions. At best, this sort of proposition can be placed under the ‘remotive’ proposition. Hypothetical propositions are either affirmative or negative. But how should we determine which hypothetical proposition is affirmative and which one is negative? Logicians stated that the ‘consequent’ of the hypothetical proposition determines whether the proposition is affirmative or negative. It is so because the antecedent of a hypothetical proposition only contains the condition, but not the statement of relation. Thus, if the consequent is affirmative, the hypothetical proposition is affirmative. And if the consequent is negative, the hypothetical proposition is judged as a negative proposition. Let us consider the below hypothetical propositions to find out whether they are affirmative and negative hypothetical propositions. (i) (ii) (iii) (iv)

If X is Y, Z is L. If X is not Y, Z is L. If X is Y, Z is not L. If X is not Y, Z is not L.

In these examples, ‘X’ and ‘Z’ are subject; ‘Y’ and ‘L’ are the predicate of the propositions. Here, each hypothetical proposition is unique because of its structure. The first and second propositions are regarded as hypothetical affirmative propositions because their consequents are affirmative propositions. The third and fourth propositions are judged as hypothetical negative propositions for the reason that their consequents are negative propositions. (iii)

QUALITY Based on quality, propositions are divided into two kinds: affirmative and negative. An affirmative proposition is one whose predicate affirms the subject. A negative proposition is one whose predicate denies the subject. For example, ‘Vivek is a good student’ is an affirmative proposition. ‘Doli is not a dancer’

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is a negative proposition. In the case of an affirmative proposition, the copula must be affirmative, and in the case of a negative proposition, it must be negative. Negative propositions should not be confused with negative terms. ‘Some students are not wise’ is a negative proposition, whereas ‘some students are unwise’ is an affirmative proposition with a negative predicate. QUANTITY In consideration of ‘quantity’, propositions are divided into particular and universal. A particular proposition is one where the predicate either affirms or denies a part of the subject, for example, some dogs are black, some men are not tall, etc. The ‘symbols’ used to express particular propositions in logic are some, some…not, a few, a little, more than one, less than the whole, etc. The word ‘some’ in logic implies at least one but it does not refer to the whole, for example, ‘Some students are swimmers’. This proposition states that at least one student is a swimmer. About ‘number’, the word ‘some’ suggests any number between 1 and the highest possible digit. Let us say some flowers are kept in the basket. This proposition states that at least one flower is kept in the basket. Here, the concept of ‘more than one’ becomes subjective, as it might be 10 for you, 12 for me, and so on.

In the natural language, when we say, ‘some flowers are kept in the basket’, it conveys that there are lso flowers kept outside the basket, i.e. ‘All flowers are not kept in the basket’. But in the logical discourse, it does not say so. It only states about the flowers that are kept in the basket. From these analyses, it is asserted that ‘some’ in the logical discourse necessarily implies ‘at least one’ element. A universal proposition, on the other hand, is one where the predicate either affirms or denies the whole of the subject. For example, ‘All men are mortal’, in this proposition, the predicate affirms the whole of the subject. It says if x is a man, then x is mortal. Another example is ‘Birds are not quadruped animals’. In this proposition, the predicate negates the subject wholly. This proposition can be expressed alternatively as ‘no birds are quadruped animals’. In this case, the predicate negates each bird that we include in the subject. The ‘symbols’ used to communicate universal propositions are all, every, none, each one, no, etc. The quantity of a proposition, whether universal or particular, is determined by the quantity of its subject term. If all the entities of the subject term of a proposition have been taken into consideration either for affirmation or denial, then the proposition is judged as a universal proposition. In contrast to this, if some of the entities of the subject term of a proposition are taken into consideration either for affirmation or denial, the proposition is treated as a particular proposition. In the case of a hypothetical proposition, the quantity of the proposition is determined by its antecedent and not by its consequent. A hypothetical proposition is universal when the consequent without any exception follows the antecedent. For example, if they will come to my home, I will take them to a movie. A hypothetical proposition is judged as ‘particular’ when consequent follows some elements of the antecedent. For example, if some students will perform well in the examination, they will score good grades.

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(e)

(f) (g) (h)

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MODALITY The concept of ‘modality’ is associated with the notion of ‘probability’. The modality of a proposition means the probability of the proposition, where the predicate either affirms or denies the subject. Based on ‘modality’, propositions are classified into three types. Necessary proposition Assertoric proposition Problematic proposition

In the case of necessary propositions, the predicate is associated with the subject inseparably, universally, and indubitably. For example, B must be C. Another example, three angles of a triangle are equal to two right angles. Concerning assertoric propositions, predicate affirms the subject without any condition. In this case, the proposition states something about the phenomenal world and what it states about is found in the phenomenal world. Further, the object found in the phenomenal world is verifiable through sense observation data, examples, swans are white, dogs are barking animals, etc. In the case of problematic propositions, the relation between predicate and subject is probable. In some situations, the predicate may affirm the subject, but not necessarily, for example, it may rain tomorrow, she may not go to the market today, etc. (f)

IMPORT Based on ‘import’, propositions are divided into two kinds: analytic and synthetic. Analytic propositions are those in which predicate repeats in the subject. The predicate does not convey anything new about the subject. These propositions are necessarily true propositions because to find out their truthvalues we need not search for anything in the empirical world. They are true, necessarily and timelessly. They are true by definition. For example, cows are cows, bachelors are bachelors, temples are temples, the morning star is a morning star, etc., these propositions were true in the past, true at present, and will be true in the future as well.

Synthetic propositions, on the other hand, are those where the predicate states something about the subject either partly or wholly, and either affirmatively or negatively. Synthetic propositions invariably state something about worldly affairs. To find out what a synthetic proposition states about, we need to verify it through our sensory experiences. If the fact that the proposition is stating about is found in the empirical world, the proposition is judged as true, and if it is not found in the empirical world, then the proposition is judged as false. So, synthetic propositions are either true or false. From this analysis, it is deduced that the truth-value of a synthetic proposition is contingent. Examples of synthetic propositions are some cows are white, all birds are biped creatures, etc. According to Gottlob Frege (1848–1925), there are five differences found between analytic and synthetic propositions. These are

44

(i) (ii) (iii)

(iv)

(v)

(vi)

3 Classification of Logical Propositions

The truth-values of analytic propositions are tautology, whereas truth-values of synthetic propositions are contingent. Analytic propositions do not require any verification for establishing their truth-values, whereas synthetic propositions do. To find out the meaning of an analytic proposition, we need not search anything in the worldly affairs, whereas in the case of the synthetic proposition, we need to do so. The truth of an analytic proposition is not context-specific, as it is independent of place, person, and time, whereas the determination of the truth-value of a synthetic proposition depends on an individual, a place, and a given time. Hence, it is context-specific. In analytic propositions, the predicate repeats in the subject, whereas in synthetic propositions, the predicate does not repeat in the subject. Instead, it states something about the subject. EXPERIENCE Based on ‘experience’, propositions are divided into ‘a priori’ and ‘a posteriori’. In the case of an ‘a priori’ proposition, the meaning of the proposition is obtained independently of our experiences. We human beings (subjects/cognisers) do not need any verification and experiment to verify the truth-value of the a priori propositions. For example, ‘Two plus two is equal to four’. Here, we assign meaning to the proposition without even looking at which two and which four objects or concepts the sentence is referring to. Here, we gain knowledge of the proposition without gaining any experience of the phenomenal world. Logicians claimed that a priori knowledge exists in human minds, and it comes to human beings along with their birth.

‘A posteriori’ propositions are those in which we assign meaning to the propositions based on our experience about facts or events that the propositions state about. In short, the meaning of an ‘a posteriori’ proposition is obtained through our experience. The truth-value of an ‘a posteriori’ proposition is determined based on our experience. To put it in different words, truth-values of a posteriori propositions are determined when we verify meanings of the propositions corresponding to the objects and facts of the world. For example, the lily flower is white. This proposition will be judged as true when we will find a flower named lily and its colour is white. Otherwise, this proposition will be judged as false. To know the flower lily and its colour, we need to use our sense organs to verify the object. Since a person’s experience determines truth-values of ‘a posteriori’ propositions, truth-values of the ‘a posteriori’ propositions have resulted in contingency (i.e. either true or false). We have already discussed ‘analytic’, ‘synthetic’, ‘a priori’, and ‘a posteriori’ propositions. Now, we consider the permutation and combination of these four types of propositions and analyse their logical structures. In consideration of the permutation and combination of these four types of propositions, we get the following four varieties of propositions. (i) (ii)

Analytic a priori (e.g. bachelors are bachelors) Analytic a posteriori (e.g. bachelors are unmarried men)

3.7 Types of Logical Propositions

(iii) (iv)

45

Synthetic a priori (e.g. three plus three is equal to six) Synthetic a posteriori (e.g. bachelors of arts are organising a picnic party)

Immanuel Kant (1724–1804) discussed these four varieties of propositions. According to him, analytic a priori, analytic a posteriori, and synthetic a posteriori propositions are found in the logical discourse. But it is a challenge to find out a proposition that is synthetic and a priori at the same time in the logical discourse. So, he was more interested in explaining the ‘synthetic a priori’ proposition. He argued that ‘synthetic a priori’ propositions are too found in the logical discourse. Consider an example, three plus three is equal to six. By considering this example, he argued that the concept ‘number’ is an a priori concept. It is so because a numerical digit’s existence necessarily presupposes the existence of the concept of ‘number’. Further, the predicate states something about the subject affirmatively. This feature justifies the proposition to be treated as a synthetic proposition. However, numbers in the propositions are not verifiable through our sense experiences, as ‘number’ is an a priori concept. Thus, this proposition is regarded as a ‘synthetic a priori’ proposition.

3.8 Formal and Material Proposition: Their Truth-Value Determination In the logical discourse, each proposition is judged as either true or false. Now, let us discuss when we can consider a proposition is materially true, and formally true? Before identifying the truth-values of the material and formal propositions, let us briefly discuss what it means to say a proposition is material or formal. Further, are material propositions the same as formal propositions? The concept of ‘matter’ and ‘form’ has been discussed since Aristotle. Without entering into Aristotle’s theory of causation, we can summarise that matter is the basic ingredient for all creations and/or productions. It is used to produce something. It is a verifiable and referable entity. It has existence. The verification of a matter is possible either through observation or experience. ‘Form’, on the other hand, is an idea of a human mind. It is a concept about something. It has no referent. The following example can clarify the notion of ‘matter’ and ‘form’. A lump of clay is the matter which is verifiable because it has existence. But the shape and size of a pot, which is to be conceived by a potter and made accordingly out of clay, are called ‘form’, that is not verifiable. It remains as a concept in the potter’s mind. But the actual ‘pot’, which is a product of matter (clay), is verifiable. In the case of logical propositions, the proposition itself is the form, whereas the contents of the proposition are its matter. We can use the contents of a proposition to formulate another proposition. Thus, the contents of a proposition are the matter, and the proposition itself is the form. A proposition is judged as true or false based on its contents what it states. If the contents are found in the empirical world, then the proposition is judged as true, and if the contents are not found in the empirical world, then the proposition is judged as false.

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3 Classification of Logical Propositions

A proposition is materially true when it states something about worldly affairs and what it states about are available in the phenomenal world. For example, a cow is a quadruped animal. This is a materially true proposition since cows are found in the empirical world, and they possess four legs. In contrast to it, if the contents of a proposition do not correspond to worldly affairs, then the proposition is judged as materially false. For example, Golden Mountain exists in the southern part of Europe. A proposition is judged as formally false when the predicate says something about the subject but does not make any connection with it. For example, horses are flying in the air. This proposition is formally false and thereby materially not true. A proposition is judged as formally true when the predicate states something about the subject, and a connection exists between the subject and predicate. For example, some logic students are badminton players. This proposition is formally true and might also be materially true. From the above analyses, we get four types of propositions. These are (i) (ii) (iii) (iv)

A formally false proposition (e.g. dusters are crying.) A formally true proposition (e.g. cows are grazing the grass.) A materially true position (e.g. crows are black.) A materially false proposition (e.g. human beings are non-emotional creatures.)

If we do permutation and combination of these four possible propositions, we will get the following propositions. (i) (ii) (iii) (iv)

A formally false and materially false proposition A formally true and materially true proposition A formally true and materially false proposition A formally false and materially true proposition

The fourth variety of propositions (i.e. formally false and materially true proposition) does not exist in the logical discourse because there will not be a case where a formally false proposition can have contents that are true materially.

3.9 Identifying Formal and Material Arguments Propositions are the basic constituents of an argument. An argument is either valid or invalid. A valid argument is one where the premises (propositions) support the conclusion. An invalid argument is one where the conclusion is not supported by the premises. A valid material argument must consist of true propositions and a true conclusion. If one of the propositions (premises) is judged as false, then the argument cannot be judged as a materially valid argument. Example of a materially valid argument: All men are mortal.

3.9 Identifying Formal and Material Arguments

47

Gandhi is a man. Therefore, Gandhi is mortal. Example of a materially invalid argument: All crows are black. All pigs are black. Therefore, all pigs are crows. An argument is formally valid when its structure conforms to the argument norms, even though its contents do not correspond to the facts or events of the empirical world. For example, I touched the table. The table touched the floor. Therefore, I touched the floor. In the case of a formally invalid argument, one of the premises must not conform to the structure and norms of the argument. For example, palm trees are tall. Human beings are rational animals. In this case, no conclusion can be drawn. It is so because there is no common term found between two propositions (premises), which could establish the link between two propositions and help to draw a conclusion. This is a violation of argument norms known as ‘fallacy of four terms’. All the rules, norms, and structures of arguments are discussed in Part II of this manuscript. It is to be noted here that it is not necessary that a valid argument must have a true conclusion, and an invalid argument must have a false conclusion. The reason is there are arguments where conclusions are treated as false, but the arguments are judged as valid, and there are arguments where conclusions are treated as true, but the arguments are judged as invalid. Thus, it is stated that in an argument, if the conclusion is false, and one of the premises is false, then the argument is valid. Further, if the conclusion is true and if the premises are true, then the argument is regarded as valid. In our everyday lives, we come across many arguments. But all these arguments are not formulated through premises to a conclusion in a sequence. Seldom it is noticed that arguments have their conclusion first, and then certain premises justify the conclusion. For example, a student said, I could not submit the assignment on time because I was hospitalised for the last five days due to a road accident. This argument can be rearranged in the following way by considering its premises first and then derive a conclusion from the premises. The argument would appear as; I had a road accident. Due to the road accident, I was hospitalised for five days. Hence, I could not submit the assignment on time. In this regard, logicians claim that whether arguments are construed from premises to conclusion form or from conclusion to premises form, the content of the argument would remain the same. That is, an argument consists of a minimum of one premise and a conclusion. If the premises of an argument support the conclusion, the argument is treated as valid. And, if the premises of an argument do not support the conclusion, then the argument is judged as invalid.

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3 Classification of Logical Propositions

3.10 Categorical Propositions: A, E, I, O In the classification of logical propositions, based on ‘quantity’, we find ‘universal’ and ‘particular’ propositions, and based on ‘quality’, we find ‘affirmative’ and ‘negative’ propositions. QUANTITY QUALITY Universal Affirmative Particular Negative But if we combine these propositions through permutation and combination, we get four possible categorical propositions. These are (i) (ii) (iii) (iv)

Universal affirmative Universal negative Particular affirmative Particular negative

A universal affirmative proposition is one in which the predicate affirms the whole of the subject. In the case of a universal negative proposition, the predicate denies the whole of the subject. In the case of a particular affirmative proposition, the predicate affirms the part of the subject, whereas, in the case of a particular negative proposition, the predicate negates part of the subject. The word ‘affirmative’ derives from the Latin word ‘AffIrmo’. Thus, logicians conventionally represent universal affirmative propositions with the sign ‘A’ and particular affirmative proposition with ‘I’. Similarly, the word ‘negative’ is derived from the Latin word ‘nEgO’. Logicians, thus, assigned the ‘E’ sign to universal negative propositions and ‘O’ sign to particular negative propositions. In short, the signs A, E, I, and O represent the following categorical proposition, respectively, such as universal affirmative (UA) proposition, universal negative (UN) proposition, particular affirmative (PA) proposition, and particular negative (PN) proposition. To identify the nature and type of a categorical proposition, logicians have assigned a unique symbol to each categorical proposition. A universal affirmative proposition is represented with the symbol ‘all’. A universal negative proposition is identified with the symbol ‘no’. A particular affirmative proposition is represented with the symbol ‘some’, and a particular negative proposition is represented with the symbol ‘some…not’ (Table 3.1). Table 3.1 Details of categorical propositions

Categorical proposition

Abbreviation

Sign

Symbol

Universal affirmative

UA

A

All

Universal negative

UN

E

No

Particular affirmative

PA

I

Some

Particular negative

PN

O

Some…not

3.10 Categorical Propositions: A, E, I, O

49

Let us formulate categorical propositions by adapting each unique symbol. UA: All men are mortal. UN: No man is perfect. PA: Some cows are white. PN: Some students are not cricket players. These four propositions are judged as categorical because in all these cases, predicate either affirms or denies the whole or part of the subject without any condition. A question arises, there are many propositions we use in our everyday lives which are not of the above kind. So, can we translate them into either A, or E, or I, or O categorical proposition form? Logicians answer that we need to observe the structure of the proposition and then find out the meaning of it. Accordingly, we can translate the proposition into a categorical proposition without compromising the semantics of it. The translation is viable only in the case of declarative sentences, but not in other types of sentences, such as interrogative sentence and exclamatory sentence. The following examples will clarify this notion. Example-1: Rose is beautiful. This proposition is not stating about a particular rose. Rather, it expresses all the roses of the past, present, and future, and further, these are beautiful. Thus, this proposition can be translated as ‘All roses are beautiful’. Thus, the proposition ‘Rose is beautiful’ is judged as a universal affirmative (A) proposition. Other examples belonging to this category are (i) (ii) (iii) (iv) (v)

The kindergarten teachers are older than students. Water quenches thirst. Honesty is desirable. Yellow flowers are beautiful, etc.

Example-2: Crows are not white. The structure of the proposition does not hold any symbol that conforms to either A, E, I, or O proposition. But in consideration of its semantics, we find the predicate denies the whole of the subject. So, we can translate this proposition into ‘No crows are white’. Thus, this proposition is judged as a universal negative (E) proposition. Other examples belonging to this category are (i) (ii) (iii) (iv)

Dead persons do not cry. Morality is not stupidity. Emotion is not the same as a reason, etc.

Examples 3: A few logic students are cricket players.

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3 Classification of Logical Propositions

This proposition does not correspond to the categorical propositions directly. But if we examine its structure and meaning, we find the predicate states something about the subject affirmatively, and it affirms only a part of the subject. Thus, it fits into a particular affirmative (I) proposition. Thus, it can be translated into ‘Some logic students are cricket players’. Other examples belonging to this category are (i) (ii) (iii) (iv)

At least five flowers in the basket look red. Sita seldom reads her WhatsApp messages. A few varieties of red apples are kept in the refrigerator, etc.

Example-4: A little water cannot save his life. This proposition does not outrightly represent any of the categorical propositions. It is so because it does not bear any of the symbols assigned to categorical propositions. But if we consider its semantics, we find the predicate denies the part of the subject, which conforms to a particular negative (O) proposition. Other examples are (i) (ii) (iii) (iv)

A few unnecessary stitches in his paint do not enhance his stardom. A beautiful and cute lady does not mean she is free from ego and attitude problems. Speaking over the phone to Harish on a few occasions does not guarantee his friendship, etc.

Even though categorical propositions (A, E, I, O) are unique because of their structure and semantics, yet one commonality is found among them, that is, they have the subject, predicate, and a copula. Since all the categorical propositions have the subject, predicate, and a copula, a question arises, how to find out which term (either subject or predicate) is distributed in which categorical proposition? Before analysing the distribution of terms, we should know what do logicians mean by ‘distribution of a term’?

3.11 Distribution of Terms The expression ‘distribution’ is used in the logic subject in a technical sense. It expresses that when the subject term or the predicate term of a proposition either includes or excludes all the members of a class, it is regarded as distributed. To put it simply, a term is said to be distributed when it either excludes or includes all the members of its class. In contrast, a term is said to be undistributed when it does not include or exclude all the members of a class. Further, it may include or exclude some members of a class. Concerning the above explanations of the distribution of a term, let us examine all the categorical propositions (A, E, I, O) and find out the status of the terms used in them.

3.11 Distribution of Terms

(i)

(ii)

(iii)

(iv)

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Universal Affirmative Proposition: A Consider an example of a universal affirmative proposition, ‘All cats are quadruped animals’. In this proposition, we find subject term (hereafter, S), i.e. cats, and predicate term (hereafter, P), i.e. quadruped animals. The predicate affirms the whole of the subject. It suggests the subject term includes all the members of its class. To simplify further, ‘S’ refers to all cats of the phenomenal world. Hence, S is distributed. But the predicate part is undistributed as it refers to some quadruped animals only. It signifies that besides cats, there are other creatures considered as quadruped animals, such as cow, horse, dog, and tiger. Thus, in an ‘A’ proposition, the subject part is distributed, and the predicate part is undistributed. Universal Negative Proposition: E Take an example of a universal negative proposition, ‘No cows are biped animals’. In this proposition, the predicate denies the whole of the subject. By implication, the predicate excludes the subject. So, S is excluded from P and vice versa. Therefore, in S, all the members of the cow are excluded from all the members of P, i.e. biped animals. Hence, all the members of ‘cows’ and ‘biped animals’ are included in their respective classes. Therefore, in the ‘E’ proposition, both subject and predicate terms are distributed. Particular Affirmative Proposition: I Consider an example of a particular affirmative proposition, ‘Some students are hockey players’. In this proposition, the predicate affirms part of the subject. As per the definition of distribution of a term, since the subject of the proposition refers to only some members of the ‘student’ class, it is undistributed. Again, the predicate of the proposition does not include all the members of hockey players. It is so because, besides students, there are other hockey players found in society. So, it is undistributed. Therefore in the ‘I’ proposition, both S and P are undistributed. Particular Negative Proposition: O Take an example of a particular negative proposition, ‘Some students are not swimmers’. In this proposition, the predicate denies the part of the subject. When the predicate denies part of the subject, it does not consider all the subject term members, i.e. students. But the predicate (i.e. swimmers) term includes all the members of its class, as it states that each swimmer belongs to the predicate term. So, the predicate term is distributed. But concerning the subject term, since it includes only a few members of its class, it is undistributed. It refers to only some students and does not say anything about all the students. Therefore, in the ‘O’ proposition, the subject term is not distributed, but the predicate term is distributed.

The flow chart summarises the distribution of terms of the categorical propositions (Fig. 3.1; Table 3.2).

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3 Classification of Logical Propositions Categorical Propositions

Quality

Quantity

Universal Proposition

A

E

Particular Proposition

I

Af irmative Proposition

O

A

I

Negative Proposition

E

O

Common Feature

Common Feature

Common Feature

Common Feature

Subject term is distributed

Subject term is undistributed

Predicate term is undistributed

Predicate term is distributed

Fig 3.1 Distribution of terms of the categorical propositions Table 3.2 Details of categorical propositions and distribution of terms

Categorical proposition

Symbol

Distributed term(s)

Undistributed term(s)

Universal affirmative (UA)

All

Subject

Predicate

Universal negative (UN)

No

Subject and predicate

XXXX

Particular affirmative (PA)

Some

XXXX

Subject and predicate

Predicate

Subject

Particular Some…not negative (PN)

Chapter 4

Square of Opposition of Propositions

In the previous chapter, we discussed the types of logical propositions, formal and material propositions, formal and material arguments, and theories of truth. We also analysed categorical propositions (A, E, I, O) and the distribution of terms. In continuation of the previous chapter, in this chapter, we will explain the relationship between categorical propositions. The relationships of categorical propositions are construed through ‘opposition of propositions’. The traditional logic (Aristotelian logic) has described the opposition of propositions through a square diagram. Hence, the opposition of propositions is popularly known as square of opposition of propositions (SOP). We will discuss the four forms of the opposition of propositions. Further, we will illustrate the truth and validity of SOP. In the end, we will elucidate the differences and commonalities between the traditional SOP and the modern SOP. By ‘opposition of propositions’, it means the relationship between two propositions of having the same subject and predicate terms but differs in quantity, or quality, or in both quantity and quality. There are four forms of the opposition of propositions found in the traditional logic. These are

Fig. 4.1 Square of Opposition of Propositions

© Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_4

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4 Square of Opposition of Propositions

(i) (ii) (iii) (iv)

Sub-alternation Contrary Sub-contrary Contradictory (Fig. 4.1)

4.1 Sub-alternation Sub-alternation is a relationship that exists between two categorical propositions having the same subject term, predicate term, and quality, but differing in quantity. It is a relationship between a universal proposition and its corresponding particular proposition and vice versa, that is, between A and I propositions, and E and O propositions. In these opposite pairs, the universal proposition is called ‘sub-alternant’, and the particular proposition is called ‘sub-alternate’. Example-1 A: All men are mortal (sub-alternant). I: Some men are mortal (sub-alternate). Example-2 E: No crows are yellow (sub-alternant). O: Some crows are not yellow (sub-alternate).

4.2 Contrary Contrary opposition is a relation between two universal propositions having the same subject and predicate terms but differing in quality. That is, A proposition opposes the E proposition, and E proposition, in turn, opposes the A proposition by contrary relation. Example-1 A: All swans are white. E: No swans are white.

4.3 Sub-contrary Sub-contrary opposition is a relation between two particular propositions having the same subject and predicate terms, but differing in quality. That is, I proposition and its corresponding O proposition are related to each other through sub-contrary relation. Example-1 I: Some students are swimmers. O: Some students are not swimmers.

4.4 Contradictory

55

4.4 Contradictory This is the perfect form of opposition of propositions. It is so because it is the relation between two propositions having the same subject and predicate terms, but differing in quality and quantity. There are two pairs of propositions opposed to each other by contradictory relation. The two pairs are A and O and E and I. Example-1 A: All men are mortal. O: Some men are not mortal. Example-2 E: No crows are yellow. I: Some crows are yellow. The SOP is viewed from broad and narrow perspectives. From the broad perspective, four oppositions of propositions are considered. These are sub-alternation, contrary, sub-contrary, and contradictory. And, from the narrow perspective, only two oppositions of propositions, contrary and contradictory, are considered. It is so because, in the case of contrary opposition, A and E oppose each other by quality. Since quality changes, all the qualities that are affirmed in A proposition get negated in E proposition, and all the qualities that are negated in E proposition get affirmed in A proposition. Concerning the contradictory opposition of propositions, the propositions are having the same subject and predicate terms, but oppose each other based on quality and quantity. This sort of opposition is unique in a SOP. It is treated as the perfect form of opposition of propositions. In sub-alternation opposition, propositions differ in quantity only. It implies that the opposition is not an opposition in a true sense of opposition. To explain, let us say, A stands for the proposition ‘All swans are white’, and I stands for the proposition ‘Some swans are white’. Even though A opposes I and I opposes A, what is true in I is already found true in A, as I is a component of A. Further, what is found true in A must be found true in I, as I is a component of A. So, the question of A opposing I and vice versa in a real sense does not arise. With regard to sub-contrary opposition, propositions differ in quality only. Even though the opposition is based on quality, it is not to be treated as the perfect form of opposition of propositions because the quantity of I and O propositions remains the same. For example, let us say, I proposition stands for ‘Some students are swimmers’, and O proposition stands for ‘Some students are not swimmers’. Here, with regard to both the propositions, it is found that swimmers are attributed to some students only affirmatively (in the case of I proposition) and negatively (in the case of O proposition), but not to all students. Hence, no clear opposition is found in subcontrary opposition of proposition.

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4 Square of Opposition of Propositions

4.5 Truth and Validity of SOP In the SOP diagram, the affirmative propositions are placed on the left side, the negative propositions are placed on the right side, the universal propositions are placed at the top, and the particular propositions are placed at the bottom. Each proposition is judged as either true or false by dint of the fact that each proposition describes something about worldly affairs. The methods to determine the truth-value of a proposition has been discussed in the previous chapter. In this chapter, we shall discuss the following: if a categorical proposition is judged as true, then what will be the truth-values of the other three categorical propositions that are opposing the proposition by some or other relations. Further, if a categorical proposition is judged as false, then what will be the truth-values of the other three categorical propositions that are opposing the proposition by some or other relations. Let us formulate A, E, I, O propositions having the subject term X and the predicate term Y. A: All X is Y. E: No X is Y. I: Some X is Y. O: Some X is not Y. Concerning the sub-alternation relation, two pairs of opposition of propositions are found. These are A and I, and E and O. In this case if a universal proposition (A or E) is true, then its opposite particular proposition (I or O) is true. If a particular proposition (I or O) is false, then its opposite universal proposition (A or E) is false. But, if a particular proposition (I or O) is true, then the truth-value of its opposite universal proposition (A or E) is undetermined. The term ‘undetermined’ is used here to express ‘uncertainty’, i.e. may be true or false. Similarly, if a universal proposition (A or E) is false, then the truth-value of its opposite particular proposition (I or O) is undetermined. The truth-value determination for sub-alternation opposition of propositions can be summed up in the following lines. The truth of the universal proposition implies the truth of its corresponding particular proposition, but not conversely. The falsity of the particular proposition implies the falsity of its corresponding universal proposition, but not conversely. With reference to the contrary opposition of propositions (A and E), if A is true, then E is false. If E is true, then A is false. But, if A is false, then the truth-value of E is undetermined. It means the truth-value of E may be false or true. Similarly, if E is false, then the truth-value of A is undetermined. However, both A and E propositions cannot be true together but can be false together. From these analyses, we can formulate the following rule to determine the truth-values of contrary opposition of propositions. That is, the truth of one implies the falsity of the other, but not conversely. About the sub-contrary opposition of propositions (I and O), if I is false, then O is true. If O is false, then I is true. But, if I is true, then the truth-value of O is undetermined. It means the truth-value of O may be true or false. Similarly, if O is

4.5 Truth and Validity of SOP

57

true, then the truth-value of I is undetermined. However, both I and O propositions cannot be false together, but can be true together. We can formulate the following rule to determine the truth-values of sub-contrary opposition of propositions, that is, the falsity of one implies the truth of the other, but not conversely. With regard to contradictory relation, two pairs of opposition of propositions are found. These are A and O, and E and I. In this case, if universal proposition (A or E) is true, then its corresponding particular proposition (O and I) is false. If universal proposition (A or E) is false, then its corresponding particular proposition (O and I) is true. Further, if particular proposition (I or O) is true, then its corresponding universal proposition (E and A) is false. If particular proposition (I or O) is false, then its corresponding universal proposition (E and A) is true. From these analyses, we can formulate the following rule to determine the truth-values of contradictory opposition of propositions. That is, the truth of a proposition implies the falsity of its corresponding proposition and vice versa. The SOP offers a logical ground to determine the truth-value of a categorical proposition in relation to another categorical proposition. It states that we can formulate arguments where there could be a premise and a conclusion, and the conclusion follows from the premise immediately. In this sense, the SOP is treated as a form of inference, more specifically termed as ‘immediate inference’. The reason is that the conclusion of an argument is inferred from a single premise. Table 4.1 mentions that if a categorical proposition is judged as true or false subsequently, what will be the truth-values of the remaining three categorical propositions that oppose the proposition either by sub-alternation, or contrary, or sub-contrary, or contradictory relation. Table 4.1 Truth-value of SOP Assume a proposition is either true or false

Truth-value of A proposition

Truth-value of E proposition

Truth-value of I proposition

Truth-value of O proposition

If A is T

T

F

T

F

If A is F

F

?

?

T

If E is T

F

T

F

T

If E is F

?

F

T

?

If I is T

?

F

T

?

If I is F

F

T

F

T

If O is T

F

?

?

T

If O is F

T

F

T

F

F = false, T = true, ? = undetermined

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4.6 Modern Interpretations of SOP Modern logicians criticised the traditional SOP based on ‘existential import’. A proposition is said to have ‘existential import’ when the truth of the proposition asserts the existence of members of the subject term. In their views, I and O propositions have existential import, whereas A and E propositions do not have existential import. That is, I and O propositions deal with ‘some’ which are verifiable, countable, and thereby referable, but A and E propositions deal with ‘all’, which are neither verifiable nor referable, and thus not able to assert their true existence. Further, they state that only contradictory opposition of proposition is considered as a valid form of opposition of proposition, and rest (i.e. contrary, sub-alternation, sub-contrary) are not the valid form of opposition of propositions. Concerning ‘existential import’, it is stated that when the subject term of A and E propositions has no member or entity, then A and E propositions are treated true propositions. Further, if the subject term of the A and E propositions has members or entities, then A and E propositions are judged false propositions. In contrast to this, when the subject term of I and O propositions has no member or entity, then I and O propositions are called false propositions. But if the subject term of I and O propositions has members or entities, then I and O propositions may become true propositions. By considering the modern logician’s views on SOP, let us discuss the contrary, sub-contrary, and sub-alternation relations of the square of opposition of propositions. Regarding the contrary relation, the traditional SOP states that both A and E cannot be true together, but can be false together. However, the modern logicians’ interpretations of SOP state that both A and E can be true together when the subject term of the A and E propositions has no members. Since the subject term does not have members or entities, A and E propositions do not have contrary relation to each other. For example, let us say, A proposition stands for ‘All ponds filled up with milk are milk ponds’, and E proposition stands for ‘No ponds filled up with milk are milk ponds’. Since there is no such pond filled up with milk exists on this earth, A and E propositions can be treated as true together. Regarding the sub-contrary relation between I and O propositions, the traditional SOP states that I and O propositions cannot both be false together, but can be true together. But modern logicians’ interpretations of SOP express that if the subject term of I and O propositions has no members or entities, then I and O propositions can be false together. For example, ‘A renowned deity drinking milk is found in India’ and ‘A renowned deity drinking milk is not found in India’ are false together because the subject part ‘A renowned deity drinks milk’ has no existence or has no members. So, the truth-value of I and O propositions can be false together. In the case of sub-alternation relation, the relationship between A and I propositions, and E and O propositions, the traditional SOP states that the truth of the universal proposition implies the truth of the corresponding particular proposition, but not conversely. However, if we consider the existential import of A and E propositions where the subject term of A and E propositions need not have any member

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or entity, then from the truth of A proposition, we cannot infer the truth of I proposition, and from the truth of E proposition, we cannot infer the truth of O proposition. The reason is when I and O propositions are existentially loaded, they cannot be inferred from the non-existentially loaded A and E propositions. In the deductive arguments, the conclusion is derived from the premises, so the entity or member found in conclusion must be found in the premises. With these arguments, modern logicians vehemently express that contrary, sub-contrary, and sub-alternation relations of the opposition of propositions are not to be treated as valid forms of the opposition of propositions. But contradictory relations between A and O propositions, E, and I propositions are considered as the perfect form of opposition of propositions.

Chapter 5

Fundamental Principles of Logic (The Laws of Thought)

In this chapter, we will discuss the fundamental principles (laws of thought) of logic: the law of identity, the law of excluded middle, the law of non-contradiction, and the law of sufficient reason. We will mention criticisms of the law of identity, the law of excluded middle, and the law of non-contradiction. Further, we will elucidate the differences and commonalities between the law of excluded middle and the law of non-contradiction. In the end, we will describe Leibnitz’s proposal on the law of sufficient reason. Every branch of knowledge relies on certain fundamental laws. Logic in this way has relied on some fundamental laws, and they are the ‘law of identity’, the ‘law of excluded middle’, and the ‘law of non-contradiction’. Aristotle has mentioned these three laws in his philosophical works. At a later period, Leibnitz added a new law to the existing three laws, the ‘law of sufficient reason’. These four laws are related to logical propositions, as logical propositions are judged as either true or false. A logical proposition cannot be both true and false, and at the same time, neither true nor false condition is also not possible. The truth-values of logical propositions are exclusive and exhaustive because they are the basis of our consistent and rational thinking. The objectives of the laws of thought are to formulate correct propositions in a linguistic system, express our thoughts correctly, and bring consistency in an argument. The laws of thought are employed to prove everything on this earth, but they cannot prove themselves. Since they are fundamental, nothing else can be used to prove these laws. It is affirmed that these laws are foundations for all proofs, and at the same time, they are beyond the scope of proof. Thus, we can say that fundamental principles form the very foundation of all proofs, but they cannot be the subject matter of proof. Thus, truth-values of the propositions of fundamental principles are always true, hence treated as a tautology. When something is called ‘fundamental’, it is regarded either as a basic or an elementary unit. The basic unit cannot be divided further. In connection with a proposition, Ludwig Wittgenstein, an analytic philosopher, in his Tractatus states that elementary propositions are the atomic propositions of a linguistic system because © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_5

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they cannot be divided further into more basic propositions. For example, ‘This is a tree’. We cannot divide this proposition into a further basic proposition. To know more about an elementary proposition, read Chap. 3 of this manuscript. A fundamental principle or a law by definition is a statement of universal truth. The truth of the universal statement applies to every single case of worldly affairs. If the truth of a statement applies to only a few cases, then it is not regarded as a statement of universal truth. In this context, we can state that a law or a principle is fundamental when it does not require any further proofs for its subsistence, as it is a basic proof on its own. Thus, fundamental principles are regarded as elementary principles. They are self-proved, self-evident, and intuitive. In this sense, fundamental principles are treated as axioms and postulates. Logic is a branch of philosophy that is based on certain fundamental principles like the ‘law of identity’, the ‘law of excluded middle’, the ‘law of non-contradiction’, and the ‘law of sufficient reason’. These fundamental principles assist in formulating true statements in a linguistic discourse. They also guide human beings to bring consistency and coherence in their thinking, communication, and arguments. Further, they help human civilization to conceive and express their thoughts correctly. Logic, with the help of these fundamental principles, tries to prove everything on this earth, but not the fundamental principles. It is like a normal human being who sees each and everything on this earth cannot see ‘herself’. Even though she sees herself in a mirror, she could see only her images, not herself. Depending on the elements used in manufacturing the mirror, her image may change. Aristotle has formulated the law of identity, the law of excluded middle, and the law of non-contradiction. At a later time, Leibnitz added another law to these three laws. That is the law of sufficient reason. These four laws are expressed in propositions that are discussed below.

5.1 The Law of Identity The law of identity states that if a proposition is true, then it is true, and if a proposition is false, then it is false. To put it symbolically, if X is true, then X is true. Further, if X is false, then X is false. For example, if ‘snow is white’ is a true proposition, then ‘snow is white’ is true in reality. If ‘Elephants are flying in the sky’ is a false proposition, then ‘elephants are flying in the sky’ is false in reality. The law of identity is expressed through many propositions in a linguistic system. It is so because a proposition is either true or false and can never be neither true nor false. In this sense, the truth-value of a proposition is exclusive and exhaustive. The law of identity is expressed in the following propositions. (i) (ii) (iii) (iv) (v)

Everything is equal to itself. Everything is identical to itself. A is A (e.g. river is a river). If everything is A, then it is A. Everything has its own nature.

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The law of identity has the formula ‘If X then X’. For example, if we consider an object has certain attributes, we always identify that object with these attributes. Once meaning is assigned to a term or a proposition, we understand the term or the proposition with that meaning. Take an example, if a logic student understands the word ‘classroom’ means the presence of a teacher and students in a room, where there is a blackboard, a projector, and a live lecture by a teacher then, he/she will understand the meaning of the classroom in this way only. So for him/her, the classroom means a conglomeration of teacher(s), students, a projector, a blackboard, and a live lecture by a teacher. In this sense, the law of identity is a necessary condition of our thinking. It is a prerequisite to assign meaning to the objects and concepts of the world. The law of identity is conveyed that a thing implies itself. Symbolically speaking, A → A. It reads as ‘A implies A’. The truth-value of the law of identity proposition has resulted in true all the time. Hence, it is regarded as a tautology. The law of identity is symbolically stated as follows:

A T F

T T

A T F

A: proposition, T: true, F: false,

: implication

To explain, A is a proposition. A may be true (T) or false (F). If A is T, then A is T. If A is false, then A is false. The truth-value of ‘A → A’ in two possible cases has resulted in true. Thus, it is a tautology. The truth-values of the law of identity propositions are proved as true by confirming to themselves. It says a thing implies itself. The law of identity expresses about the analytic proposition, i.e. A is A. It is so because the predicate part repeats in the subject part, and the truth-value of the proposition has always resulted true. It is true by definition. For example, if we say a river is a river, then it is necessarily a true proposition. If a person says, ‘If this is a table, then this is not a table’, it suggests that he/she has ignored the law of identity while making the statement. As a result, this sentence does not convey any meaning about worldly affairs. Further, it is regarded as a self-contradictory statement. Thus, we may assert that rejecting the law of identity while formulating propositions in a linguistic discourse results in self-contradicting, meaningless, and unintelligible statements. The law of identity has been criticised based on the doctrine of ‘momentariness’. The doctrine of momentariness states that nothing is permanent on this earth. Everything is in a state of constant flux. So, if something is ‘A’ at a given moment, then it must not be ‘A’ in the next moment. For example, a tadpole turns into a frog and will not remain as a tadpole anymore. In this case, we can assert that ‘A is ~ A (not A)’ does not always hold false. Further, the law of identity does not state anything about the progress of the universe. Rather it states that when there is A, then there is A, but in reality, when A turns into B, then it is B and not A. Logicians consider this criticism is trivial and misleading, as it is not based on the correct understanding of the law of identity. The law of identity shall not be

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understood in the absence of a space–time continuum. It is so because this law states that if A is a tadpole, then A is a tadpole in a given space (S1) and at a given time (T1). If A is a frog, then A is a frog at space (S1) and time (T1), nothing else. So, it is important to consider time and space while adopting the law of identity to formulate a proposition for communication and convey a thought in an argument.

5.2 The Law of Excluded Middle The ‘law of excluded middle’ asserts that a statement is either true or false, and there is no third alternative to it and neither true nor false condition is also not possible. If a proposition is true, then its negation must be false. If there is A, it cannot be not-A (Read; ‘not dash A’) at the same time and same place. For example, if a flower colour is white, it cannot be not-white at the same time and same place. So, there are only two alternatives found in case of a proposition: truth and falsity. The ‘law of excluded’ middle is presented in the following propositions. A is either P or not-P. Everything must be either P or not-P. Either X is a rich woman or a not-rich woman. Consider the first proposition ‘A is either P or not-P’. In this proposition, A and P are variables. The law of excluded middle while affirms one alternative of the proposition rejects the other and vice versa. This law excludes the middle position between P and not-P. It states that either P is true or false. If P is true, it cannot be false, and if P is false, it cannot be true. The truth-value of this proposition has, thus, resulted in tautology. It expresses that an object necessarily possesses either of the two contradictory qualities at a given time and space. Consider the third proposition; either X is a rich woman or a not-rich woman. If it is true that X is a rich woman at space (S1) and time (T1), then the other alternative that X is a not-rich woman at space (S1) and time (T1) must be false and vice versa. There would not be any such case where X would be treated as a rich and not-rich woman in a given time (T1) and at a given space (S1). The law of excluded middle is presented symbolically as P v ~ P. It is read as ‘P or not P’. The truth-value of the law of excluded middle proposition has resulted in true all the time. Hence, it is regarded as a tautology.

P: proposition, T: true, F: false, ~: negation, v: disjunction

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To explain, P is a proposition. P may be true (T) or false (F). If P is T, then ~ P is F. Further, if P is false, then ~ P is true. The truth-value of ‘P v ~ P’ in two possible cases has resulted in true. The reason is under disjunctive proposition case, if the truthvalue of a proposition is true, the truth-value of the negation of that proposition must be false and vice versa, thereby the truth-value under disjunctive logical connective results is true. Thus, it is a tautology. The ‘law of excluded middle’ is criticised on the ground that besides the truth and falsity of a proposition, we may find the third alternative of the proposition. For example, ‘A unicorn is either red or not-red’. If the unicorn is red is true, then the unicorn is not-red is false, and if the unicorn is not-red is true, then the unicorn is red is false. However, as unicorn does not exist in this universe, this proposition may be considered neither true nor false. This is the third alternative of the proposition. A few logicians and mathematicians are researching on multivalued logic by adopting ‘n-values’. It means the value of a proposition can be one, two, three, four, etc., at a given time and place. So, it is wrong to claim that a proposition can have only two values (true and false). Again, it may be argued that some individuals may not know about some propositions exhaustively. Hence, they cannot assign any truth-value to these propositions. Here, it is argued that it is wrong to accept that a proposition has necessarily two truth-values. However, these criticisms are not worth considering because these are based on certain propositions that are not meaningful, and in some cases, criticisms are made outside the scope of the law of excluded middle discussions. In the case of ‘Unicorn is either red or not-red’, it is considered as a pseudo-proposition. Hence, the law of excluded middle does not apply to this proposition. About researchers dealing with propositions that have n-values, it may be stated that the law of excluded middle deals with only two-valued logic. Thus, the question of the third alternative does not arise. For propositions, that are not known to some individuals comprehensively are regarded as meaningless propositions for them. Hence, criticism of not assigning any truth-values to these propositions is beyond the scope of the law of excluded middle.

5.3 The Law of Non-contradiction The ‘law of non-contradiction’ states that no proposition can be both true and false together. Symbolically speaking, ‘X cannot be both Y and not-Y’. In other words, nothing can be both Y and not-Y. A person cannot be male and not-male at the same time. If X is a male, then X cannot be a not-male. Further, if X is not a male, then X cannot be a male. It explains that Y and not-Y cannot be both true of one thing. If X has the quality Y, it cannot possess the contradictory quality not-Y at the same time and same place. In short, a proposition cannot be both true and false at the same time. If it were so, then the proposition would be considered illogical and insignificant. By not abiding by the law of non-contradiction, if we formulate the propositions in a linguistic system, the propositions would be treated as meaningless.

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Consider an example, ‘X is alive and not-alive’. If ‘X is alive’ is true, then ‘X is not-alive’ cannot be true. If ‘X is alive’ is false, then ‘X is not-alive’ cannot be false. In brief, X cannot be both alive and not-alive at the given time and place. According to Aristotle, this is a fundamental law because it assists in formulating meaningful propositions in the linguistic discourse. Leibniz (1646–1716), a rationalist philosopher, states that the law of non-contradiction is the foundation of mathematics. The reason is it is such a law that the entire language, logic, and nature rest on it. The law of non-contradiction can be presented in the following sentences. A cannot be both P and not-P. Nothing can be P and not-P at a time and place. Consider the proposition ‘A cannot be both P and not-P’. In this proposition, let us assume that ‘A’ is an object and ‘P’ is a quality of it. Please note, in the case of the law of non-contradiction, ‘A’ (i.e. the subject term of the proposition) cannot be considered as a general term. It must refer to an individual or an object or an entity. It refers to a single entity. The law of non-contradiction states that P and not-P are two contradictory qualities. These two qualities cannot subsist in object A at a given space and time. If A possesses P, it would not possess not-P, and if A possesses not-P, then it would not possesses P. Thus, we can conclude that both P and not-P are not true of one object. The law of non-contradiction is symbolically presented as ~ (P ∧ ~ P). It is read as negation of P and not P. The truth-value of the law of contradiction has resulted in true, and thereby it is a tautology. The truth-value of the law of non-contradiction is mentioned below.

P : proposition, T: true, F: false, ~: negation,

: conjunction

To explain, P is a proposition. P may be true (T) or false (F). If P is T, then ~ P is F. If P is false, then ~ P is true. The truth-values of P ∧ ~ P in two possible cases have resulted in false. The reason is in conjunction if truth-values of both the propositions (P and ~ P) are true, the truth-value of the conjunctive proposition would be true, otherwise false. Further, the truth-value of the negation of the conjunctive proposition will be true when the truth-value of the conjunctive proposition is false. In short, the negation of P and ~ P has resulted in true in the above two possible cases. There is a criticism charged against the law of non-contraction. That is, we human beings live on this earth and find many ‘contradictions’ in our everyday lives, and it is a fact of the world. For example, X likes both green and not-green apple at

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the same time and same place. Another example, private industries’ owners and their labour unions are always in conflict. They have contradictions on numerous issues. However, labourers work in private industries. In this case, even though we observe a contradiction between labourers and owners of the industries, it is not a true form of ‘negation’ or ‘denial’ to each other. Hence, the law of non-contradiction is not free from lacunas. Logicians responded to this criticism. Concerning ‘X likes both green apple and not-green apple’, logicians pointed out that it is not a law of non-contradiction statement. The law of non-contradiction statement claims that an object does not have contradictory qualities. It does not say that an individual may not like two contradictory objects at a time and place. Hence, the correct statement that would be formulated under the law of non-contradiction is, ‘X is a green apple and not-green apple’. If ‘X is a green apple’ is true, then ‘X is not-green apple’ is false and vice versa. Further, it is logically wrong to say that at a given time and space, X could like both green and not-green apple. If X likes green apple is true at a given place and time, then it would be logically wrong to state that X likes not-green apple is true at the same place and time. There may be a possibility that in the next moment and a different place, X may like not-green apple would be true. In this sense, the law of non-contradiction is an error-free fundamental principle of logic. The truth-value of the law of non-contradiction proposition has always resulted in true. There are some differences and commonalities found between the ‘law of noncontradiction’ and the ‘law of excluded middle’. In the case of the law of excluded middle, two alternatives of a proposition cannot be false of a thing at a time. For example, X is either rich or not-rich. If ‘X is rich’ is false, then ‘X is not-rich’ will not be false at the same time and same place. It implies that to negate one alternative of the proposition is to affirm other alternatives of the proposition. Concerning the law of non-contradiction, two contradictory properties of a thing cannot be true at a given time. If one is true, another must be false and vice versa. For example, X cannot be both rich and not-rich. If X is rich is true, then X is not-rich must be false at a given time. Further, if X is rich is false, then X is not-rich must be true. The law of non-contradiction and the law of excluded middle refer to ‘contradictory terms’, not ‘contrary terms’. Contrary terms cannot both be true but can be false of a thing. For example, milk cannot be in both white and black colour, but it can be in some other colour other than white and black. Here, white and black are contrary terms, not contradictory terms. At a given time and place, it is not possible to think milk is in both white and black colours. So, the black and white colours of milk cannot be true but can be false at a given time and space. In the case of contradictory terms, let us say, black and not-black, white and not-white cannot both be true and false at a given time and space. It is so because one of them must be true, and therefore, other alternatives must be false and vice versa. Besides alleging specific criticisms to each law of thought, few general criticisms are made against the three laws of thought. It is stated that these three laws are disappointing since they do not describe anything new about objects and their state of affairs with the world. For example, when we say, ‘the river is a river’ as a statement for the ‘law of identity’, we do not obtain any new knowledge about ‘river’. It is merely

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a repetition of the predicate in the subject. Again, if someone utters a statement, ‘This is a table and not-table’, the listeners will think that the speaker is unintelligible because he or she is contradicting himself or herself. Further, it is alleged that the ‘law of identity’ is implicitly embedded in the ‘law of non-contradiction’. This is because the latter law is merely a negative way of expressing the same fact asserted by the former law. Modern logicians refute these criticisms by stating that these criticisms are trivial and have no merits. They evoke that the laws of thought are naturally fundamental. These laws are the basis on which one can think about worldly affairs and assert something about worldly affairs. If we deny these laws, we would not be able to conceive our thoughts correctly, formulate meaningful propositions, bring coherence in our conversation and consistency in our arguments, etc. Thus, these laws are not only useful for the logic subject but also useful for language discourse. Logicians vehemently enunciate that affirmation and negation are essentially different from each other, and therefore, the ‘law of identity’ and the ‘law of non-contradiction’ are to be considered as two different laws and shall be maintained as two distinct laws. Plato emphasised the importance of the law of non-contradiction in Book IV of his Republic. Aristotle discussed these three laws in Books IV and XI of his Metaphysics.

5.4 The Law of Sufficient Reason The law of sufficient reason states that for every fact (F) or event (E), there must be sufficient reason(s) why F or E is the case. To put it symbolically, for every X, there is a Y such that Y is the sufficient reason for X. To explain, there is nothing that happens on this earth without a reason. On some occasions, human beings find a reason for a fact or an event, and in some cases, human beings cannot find out reasons for a fact or an event. It is so because human beings possess limited knowledge about worldly affairs. With their limited knowledge, they cannot find out all the possible reasons (i.e. necessary reasons and sufficient reasons) of a fact or an event of the phenomenal world. But the fact of the matter is that an event or a fact could not exist without a reason or reasons. On some occasions, we, human beings, with our limited knowledge, find out the necessary reasons for a fact but may not know the sufficient reasons for the fact. In these cases, we are not convinced about the necessary reasons alone that result in the occurrence of the fact. For example, a mosquito bite is the cause (necessary reason) of malaria. But merely mosquito bite will not cause malaria, as there are other medical conditions (sufficient reasons) involved to get affected with malaria disease. Another example, Z wants to reach a designated place on time. In the case of Z, she must travel on the train to reach the destination on time, provided the train departs from the platforms on time and maintains its schedule timing. Here, travelling on the train may be considered as the necessary reason to reach the destination on time, but travelling on the train cannot be considered as sufficient reasons to reach the destination on time. It is so because without sufficient reasons (i.e. the train departs

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from the platforms on time, maintains its schedule timing, etc.), Z cannot reach her destination on time. Thus, we may assert that a necessary condition must be present for an event to occur, but sufficient conditions would produce the event. A necessary condition alone would not be able to produce the event. Logically speaking, if A is necessary for B, then B cannot be true unless A is true. But if A is sufficient for B, then A being true always implies B is true, but A not being true does not often imply B is not true. Let us consider another example, X sees a logic textbook on her study table. In this sentence, the necessary reasons for seeing a logic textbook on the study table are the logic textbook must be placed near X, and there must be proper light to perceive the textbook. But a question may arise, are these essential reasons enough for X to see the logic textbook on her study table? Logicians argue that along with necessary reasons, there must be sufficient reasons to find out the truth-value of the proposition. The sufficient reasons are X must have the intention or desire to perceive the book, X must have good eyesight, X must have the capability to read the front page of the book to identify that it is a logic textbook or not, X must not be dreaming about the logic textbook on her study table, etc. Thus, sufficient reasons and necessary reasons are both required to find out the truth-value of the proposition. Leibnitz, a rationalist philosopher, states that the law of sufficient reason is inevitable to conceive valid thoughts and communicate those in a linguistic system. This fundamental principle is an essential supplement to the law of identity. The reason is if anything changes, there must be necessary and sufficient reasons for the change to occur. For example, a tadpole matures into a frog. In this case, there must be necessary and sufficient reasons for a tadpole to transform into a frog, and because of these reasons, we identify tadpoles as tadpoles and frogs as frogs but do not cognize tadpoles as frogs or frogs as tadpoles. The above four laws are essential and indispensable to express correct thoughts about worldly affairs. A question may arise, whether the ‘laws of thought’ are adequate for deducing other logical principles in the logical discourse, such as the ‘law of inference’, the ‘principle of tautology’, and the ‘principle of contingent’? Logicians have stated that the four laws of thought are not exhaustive to deduce other logical principles from them. However, the four laws of thought are prerequisite and necessary to govern correct and consistent thinking about worldly affairs.

Chapter 6

Logical Paradoxes

In this chapter, we will discuss the concept of paradox and types of paradox. We will elucidate the differences between paradox and ambiguity, paradox and vagueness, and ambiguity and vagueness. Further, we will illustrate Russell’s paradox, liar’s paradox, barber’s paradox, and Zeno’s paradox amid others. Let us begin by knowing ‘what is a paradox?’.

6.1 What is a Paradox? The term ‘paradox’ is derived from two words: ‘Para’ and ‘Doxa’. The word ‘Para’ means the contrary, while ‘Doxa’ means opinion. So, etymologically ‘paradox’ means contrary opinion. Here, the term ‘contrary’ means ‘opposite’. To be precise, a statement is regarded as a paradox when its connotation goes against generally accepted opinion about the definition of the statement. But in the logical discourse, the term ‘paradox’ has a more precise elucidation. A paradox consists of either two contrary or more than two contradictory statements, in which they are led by apparently sound arguments. An argument is treated as a sound argument when it does not seem to create any confusion when used in another context. It is only in a particular combination the paradox occurs and leads to confusion. According to Quine (1976), a paradox is just any conclusion that at first sounds absurd, but that has an argument to sustain it. In his words, ‘the argument that sustains a paradox may expose the absurdity of a buried premise or of some preconception previously reckoned as central to physical theory, to mathematics, or to the thinking process. Paradox has been the occasion for major reconstruction at the foundations of thought’ (p. 1). A paradoxical statement refers to the apparent contradiction. It means, a paradoxical statement seems to be true in one particular context, but it may create two contrary ideas in another context. For example, ‘This statement is false’. If we consider this statement is true, then the statement is false. But if we consider this statement is false, then the statement is true. To put in other words, if the speaker of © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_6

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this sentence is truthful, then the sentence is false, but if he/she claims that whatever he/she said is false, then the statement is not false, and thereby it is true. Thus, it is paradoxical.

6.2 Classification of the Paradoxes Paradoxes are not only found in the logic subject but also found in mathematics and language discourses. It is of two types: logical paradox and semantic paradox.1 Logical paradoxes are related to the mathematics and logic subjects concerning set theories, numbers, etc., whereas semantic paradoxes are related to the language discourse concerning thought, symbols, the postulation of a sentence, etc. (Ramsey, 1926). Both types of paradox comprise veridical paradox, falsidical paradox, and antinomies. A veridical paradox is one where claims made in a statement are considered true though these claims are concluded as unreasonable. A veridical paradox is a counter-intuitive solution to a problem statement. A falsidical paradox is based on fallacious reasoning. It makes false claims and creates a fallacy in the proof. A falsidical paradox statement not only appears as a false statement but also results in false. It leads to a contradictory and false conclusion through bad reasoning. An antinomy is treated as a paradox because even if we apply the correct reasoning to a statement, we will get only contradictory results. In other words, the application of correct reasoning to a statement has resulted in a false conclusion. This chapter aims to discuss some of the important paradoxes found in the logic, language, and mathematics discourses. Let us start with an example of a veridical paradox. A person after his/her fifth birthday celebration reaches his/her age of twenty-one. If it is so, then on which date he/she was born? Anyone celebrates his twenty-one birthday means he/she was born 20 years back. But in the problem statement, it is mentioned that at the age of twenty-one, the person is celebrating his/her fifth birthday. So, he/she might have born on a leap year, as the February month of every year does not have 29 days. Now, consider an example of a falsidical paradox, that is, ‘2 = 1’. This paradox belongs to the mathematical discourse. It is typically a comical proof of mathematics. A British mathematician, Augustus De Morgan (1806–1871), offered the demonstration for ‘2 = 1’. Let x is 1. Step 1 : x = 1 Step 2 : x 2 = 1 Step 3 : x 2 − 1 = x − 1 1 Horsten,

L. (2015). One hundred years of semantic paradox. Journal of Philosophical Logic, 44, 681–695. Also, see Ramsey, F. (1926). The foundations of mathematics. Proceedings of the London Mathematical Society, 25, 338–384.

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(x 2 − 1) (X − 1) = (X − 1) (X − 1) (X − 1) (x − 1)(x + 1) = Step 5 : (X − 1) (X − 1) Step 6 : x + 1 = 1

Step 4 :

Step 7 : 2 = 1 To explain step 1, we assumed that x is 1. In the second step, we argued that ‘12 is equal to 1’, that is, ‘1 × 1’ is equal to 1. In the third step, we added (− 1) to each side of the argument. In the fourth step, we expanded the argument stating that 12 –1 divided by 1–1 is equal to 1–1 is divided by 1–1. In fifth step, we simplified (a2 − b2 ) rule and mentioned (a − b) and (a + b). In the sixth step, after applying the division rule, we get (x + 1) is equal to 1. Hence, we concluded that (1 + 1) is equal to 1, i.e. ‘2 = 1’. The fallacy arises in the step 4, i.e. x − 1 = 0, as we cannot divide 0 by 0. Even if we try to divide 0 by 0, we will get the result 0, not 1. The crux of the falsidical paradox is that it has a fallacy in the argument itself. It is endowed with a false claim and a fallacy in the proof itself. An example of an antinomy, a person states that a paradox satisfies all the logical rules. By applying our reasoning, we know that a paradox does not conform to all the logical rules and thereby does not satisfy all the logical rules. As a consequence, this statement is resulted in self-contradictory. This is called the antinomy. Take another example, if we state that ‘“not true of self” is true of a thing if and only if the thing is not true of itself’, then it resulted in self-contradiction. It is so because the statement states that ‘“not true of self” is true of itself if and only if it is not true of itself’. Now consider an example of a semantic paradox, ‘A beautiful woman said women are ugly’. If we explain the semantics of the statement, it has resulted in contradictory conclusions, hence paradoxical. The reason is when we consider ‘women are ugly’, it necessarily includes the beautiful woman who made the statement about women. So, is it possible to exclude a beautiful woman when she has made a statement about women? If women are ugly is true, then ‘she is a beautiful woman’ is false, and if women are not ugly, then her statement is not true.

6.3 Differences among Paradox, Ambiguity, and Vagueness A paradox is a statement that has resulted in a logically unacceptable and selfcontradictory conclusion. Upon applying valid reasoning to a paradox statement, we find self-contradictory elements in it that may be interrelated among each other and endure over time. Ambiguity is a type of uncertainty of meaning in which several interpretations are plausible. It arises with the semantic aspect of a language. If a word or a sentence conveys more than one meaning, we regard it as an ambiguous word or an ambiguous sentence. For example, the word ‘bank’ has more than one meaning. It may mean a financial institution, a shoreline of a river, and a few. Hence,

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the term ‘bank’ is treated as an ambiguous word. Ambiguous words are overdetermined because ambiguity involves uncertainty about the mapping between levels of representation with different structural characteristics. There are many ambiguous expressions found in a linguistic system. For example, Madhu likes visiting her uncle’s house. In this expression, the listener may not know whether Madhu wishes to visit her maternal uncle or paternal uncle’s house. Consider another sentence; John is wearing a green colour t-shirt. Here, the listener may be wondering what kind of green colour t-shirt: mint leaf green colour, lime green colour, or something else. Concerning vagueness, it is stated that when the meaning of a term is not determined precisely, it is treated as a vague term. Vague terms do not convey exact and accurate meaning. It is so because these terms lack precision in a linguistic system. For example, the term ‘bald’ is vague because when exactly a person would be called bald remains unclear. The term ‘bald’ has many and multiple interpretations in a linguistic system, as it lacks precision and seeks more information to fix the vagueness of the term. Another example is the expression ‘light blue colour’. This expression is vague because it allows the listeners to wonder which shade of blue light would be regarded as light blue colour? According to Quine (1960), Lakoff (1970), and a few logicians, ambiguity differs from vagueness. Simpson (1970) states that the word ‘green’ is ambiguous and not vague. He justifies his claim in the following lines. When a speaker utters the word ‘green’, the hearer(s) may not understand green as intended by the speaker, as there may be a possibility that the speaker means ‘mint-green’ and the hearer understands it as ‘lime-green’. The reason is the colour green has hexadecimal colour codes. The word ‘green’ thus does not have one meaning, rather more than one meaning in a linguistic discourse. So, we can assert that a word or a sentence is ambiguous when it has more than one meaning or interpretation. Friedrich (2017) suggests two criteria to find out whether a word or a sentence is ambiguous or not. These are: (i) (ii)

A word or a sentence is ambiguous if and only if more than one interpretation of the word or the sentence exists, and a rational speaker can possibly use the word or the sentence appropriately to convey the intended meaning of the word or the sentence correctly.

For example, Miku is going to his aunt’s house. This sentence is meaningful. But it conveys more than one meaning because of the term ‘aunt’. The term ‘aunt’ refers to ‘sister of the father’ and ‘sister of the mother’. In this sentence, the hearer may not be able to know the speaker’s intention for the word ‘aunt’. To fix the ambiguity of the sentence, the speaker formulates the sentence as ‘Miku is going to his father’s sister house’. About vagueness, the hearer finds difficulty in assigning the correct and intended meaning to the words and sentences uttered by a speaker. The hearer also finds difficulty in interpreting the speaker’s utterances correctly. Vagueness arises due to a lack of accuracy in the meaning of the word and the sentence, where the word and sentence are used by the speaker in his/her utterances, do not convey any designated meaning. For example, if a speaker says, ‘She is going to bank’. In this sentence, the word

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‘bank’ is considered a vague word. The reason is ‘bank’ means riverbed, commercial organisation, etc. Further, if the hearer considers the bank means a ‘riverbed’ then which riverbed is not known to the hearer. Again, if a hearer considers a bank means a ‘commercial organisation’, then which type of commercial organisation, what is the name of the commercial organisation, etc., are not clearly known to the hearer. All these questions may arise in the hearer’s mind when he/she is trying to assign meaning to the sentence ‘She is going to bank’. Thus, the sentence ‘She is going to the bank’ is treated as a vague sentence, as it lacks precision. However, the vagueness of this sentence can be fixed with an accurate and precise meaning, i.e. ‘She is going to the State Bank of India located inside Indian Institute of Technology Madras, Chennai campus’. Some more examples of ambiguous sentences are given below. Please look at the underlined expressions. (i) (ii) (iii) (iv)

Smita desires to visit her cousins. The chickens are ready to eat. Sadhu purchased a red colour t-shirt. Miku is carrying notebooks in his school bag.

A few examples of vague sentences. Please look at the underlined expressions. (i) (ii) (iii) (iv)

Sadhu is a good person. Manu likes games. Smita went to the bank. Miku is the champion of an athletic game.

Now we will discuss some of the important paradoxes of logic and logical discourse.

6.4 Russell’s Paradox Russell’s paradox is a logical paradox. Professor Bertrand Russell (1872–1990), a logician-cum-mathematician, argued that any number of efforts we will make to formalise Georg Cantor’s naïve set theory would lead to self-contradiction. His argument on this account is known as Russell’s paradox. The naïve set theory explains that a set is a collection of distinct and distinguishable objects. For example, let R is a set consisting of horse, cow, goat, crow, snake, and bird. Each member of the set R is a distinct entity and differs from other entities. So, we can say ‘horse’ is a member of R, but ‘table’ is not a member of R. To write it symbolically, horse ∈ R and table ∈ / R. Russell argues that consider a set ‘S’ be the set of all possible sets of the world that are not members of themselves. If so, then a question arises, does ‘S’ a member of itself? There are two possible answers to this question. First, if ‘S’ is a member of itself, then ‘S’ is not a set of all possible sets of the world. Second, if ‘S’ is not

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a member of itself, then ‘S’ cannot be regarded as a set. So in both cases, we find contradictions. Hence, it is a paradox.

6.5 Liar’s Paradox Liar’s paradox is a semantic paradox. Epimenides advocates this paradox. The paradox of Epimenides is popularly known as Liar’s paradox. Consider the statement, ‘A teacher told all the teachers are liars’. If so, a question arises, was the statement spoken by the teacher true? On considering the teacher’s statement as true, we end up in contradiction. The reason is being a teacher, he is also a liar. Hence, whatever he said is also a lie. Thus, the statement is false. If we consider the teacher statement to be false, then he is not lying. So, the statement is true. Hence, it leads to self-contradiction. In either of the cases, answers to the question result in self-contradiction. Thus, it is paradoxical. In short, if the teacher tells the truth, then he is a liar. So the sentence is false. If the teacher has lied that ‘all teachers are liars’, then the sentence is regarded as true. In either of the cases, we lead to the paradox.

6.6 Barber’s Paradox The barber’s paradox is derived from Russell’s paradox. This paradox considers the following situation. In a village, there is a man who is a barber. This barber shaves all and only those men in the village who do not shave on their own. If so, a question arises does the barber shave on his own or not? No matter how we answer this question, it will lead to a paradox. The following three suppositions can be considered to answer the above question. First, if the barber shaves on his own, then as per the above-mentioned situation (i.e. the barber shaves only those who do not shave on their own), the barber is not shaving his beard. Thus, it has resulted in self-contradiction. Second, if we consider that the barber does not shave on his own, then another problem arises. That is, as per the above-mentioned situation (i.e. the barber shaves everyone those do not shave), if the barber does not shave himself, then he shaves. This has also resulted in self-contradiction. Third, if we consider the barber is a woman for argument’s sake, even then the conclusion leads to the paradox. The reason is, concerning the above-mentioned situation, we may say that either she shaves or does not shave. If she shaves on her own, then the barber does not shave one of the members of the village. If she does not shave on her own, then the barber shaves one of the members of the village. So the question ‘who shaves the barber’ remains unanswered.

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6.7 Zeno’s Paradox Zeno (494–435 BC) propounded a set of paradoxes, such as the arrow paradox, the Achilles and tortoise paradox, etc. These paradoxes are regarded as falsidical paradoxes. His objective was to establish an argument in favour of ‘motion is impossible’. Look at the arrow below and other details to comprehend ‘the arrow paradox’. L M

Y B

P X

K

−→ We name the arrow as ‘MK’. The middle point of the arrow is ‘Y’. We can name −→ the arrow until ‘Y’ as ‘MY’. The middle point between ‘M’ and ‘Y’ is ‘B’. So, until −→ B point, we can name the arrow as ‘MB’. Likewise, on the further division between −→ ‘M’ and ‘B’, we can have ‘ML’ as depicted in the arrow. If we go on finding out the middle point between ‘M’ and ‘L’ followed by finding middle points between any two identified points of the arrow, at one point of time, there will be no middle point between two points of the arrow as there would be no further space between two points and the last middle point between the two points of the arrow would be a static one. Zeno’s paradox states that when an arrow is moving from one place to another place by taking some time, how is it the case that at a given point of time and space, the arrow is static? Zeno argues that an arrow is moving on the one hand and it is static on the other hand, become contradictory to each other, which has resulted in a paradox. Zeno reiterates that at a given point of time and space, the arrow is motionless. It is so because if we break down the arrow into individual frames, we will notice that in each frame, the arrow is hovering in the air. In each frame, the arrow is motionless. At each point in time, the arrow is motionless. It is static. But when we put all the arrow frames together, it appears to move. The paradox arises when we say an arrow is moving, although it is motionless. Motion indeed occurs through space, not at a single point in space. Anything to move through space must start from one point to reach another point. So, upon consideration of each point, we find the arrow is motionless. Motion, too, takes time. If we consider a specific time, the arrow cannot be moving. If at every point of time the arrow is static, then how is it possible for the arrow to reach the target after moving from a bow? On the one hand, an arrow is moving to hit the target, and on the other hand, it is static. This has resulted in a paradox. Zeno, to prove his argument ‘motion is impossible’, has given another example of running competition between Achilles and a tortoise. He claims that if the tortoise were given a head start, the fastest runner (Achilles) could not be able to catch the

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Picture 6.1 Achilles and tortoise paradox

slowest runner (tortoise), and therefore the slowest runner will reach the destination before the fastest runner. Zeno argues that so long as a runner keeps running, however slowly, any fast runner can never overtake him. Look at the picture to comprehend Zeno’s paradox (Picture 6.1). Both Achilles and the tortoise start running to reach the destination ‘D’. To run from one place to another takes time. Even if the distance between the two locations is minimal, it takes time to reach the destination. If a runner’s running speed is very fast, even then, it takes time to reach the destination. At best, the fast runner may reach the destination in a very short time. So, the assertion is it is impossible to move from one place to another instantaneously. In the running competition between Achilles and a tortoise, Achilles is a fast runner, and the tortoise is a slow runner. So, it is presumed that Achilles will overtake the tortoise to win the running competition and reach the destination earlier than the tortoise. Zeno argues that if the tortoise is given a head start, then no matter how fast Achilles runs, he will never be able to overtake the tortoise to win the race. He explains Achilles needs to cross the distance between him and the tortoise. For that, Achilles needs to move from where he is now to where the tortoise is. If there is any distance between Achilles and the tortoise, then Achilles will take time to catch the tortoise. Tortoise is running and hoping to reach the destination earlier than its competitor. This suggests that when Achilles is trying to reach the tortoise place, the tortoise is moving away and covering a little more distance, which Achilles needs to cover, and for that, it will take time for him. The argument is, the time Achilles takes to run between where he is now to where the tortoise is, the tortoise will move on to a forward point. To cover the forward point, Achilles will take time, and by that time tortoise will move on to the next forward point to reach the destination. As stated in the above diagram, Achilles starts running from A1, and the tortoise starts running from T1 to reach the destination ‘D’. The time Achilles takes to reach point A2, which is similar to T1, the tortoise will move on to point T2. Then, Achilles will need to reach the point A3 that tortoise is now (T2). Achilles will have to run from point A2 to point A3. To run from A2 to A3, Achilles will take time. By the time Achilles will reach point A3, the tortoise will move on point T3. This process can be repeated at ad infinitum; as a result, Achilles would never be able to overtake the tortoise to win the race. This argument suggests that each time the pursuer reaches a sport where the pursued has been, the pursued has moved a bit beyond. The paradox

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arises because any infinite succession of intervals of time has to add up to all eternity (Quine, 1976, p. 3).

6.8 The Paradox of the Stone In our society, people are either theists or atheists. Those are theists who believe that God is the creator, sustainer, and destroyer of the universe. He is omniscient, omnipotent, eternal, and ubiquitous. Since God is an unseen force and omnipotent, He can do everything that a human being cannot do with his/her limited knowledge and small lifespan. The paradox arises when it is asked if there is nothing that God cannot do, can He create a stone that is so heavy that He cannot lift it? If God has created a stone that He cannot lift, then He is not omnipotent. If He cannot lift the stone, then there is something that He cannot do, that is lifting a heavy stone. On the other hand, if God cannot create a stone that He cannot lift, then God is not regarded as omnipotent, as He cannot create something that He cannot lift. In either of the cases, answering the question leads to a paradox known as the paradox of stone. For atheists, this paradox does not arise because, for them, there is no being named God who is omnipotent and omniscient and exists in the universe.

6.9 Grelling’s Paradox Grelling’s paradox is a semantic paradox. Kurt Grelling (1886–1942) proposed a paradox that deals with heterological adjectives that are not self-descriptive. To explain, in a linguistic system, we have two types of adjectives, autological and heterological. An autological adjective is one that describes itself. Autological adjectives are true of themselves because they are self-descriptive. For example, adjectives ‘English’ is ‘English’, ‘short’ is ‘short’, etc. A heterological adjective is not true of itself because it does not describe itself. For example, ‘long’ is not a long, ‘German’ is not German, ‘monosyllabic’ is not monosyllabic.2 So the question arises, is the adjective ‘heterological’ heterological? If we say that the adjective ‘heterological’ is heterological, then it must describe itself, and hence it is not heterological but autological, and if we are saying the adjective ‘heterological’ is not heterological, then it does not describe itself. So it is heterological. In other words, if we say the adjective ‘heterological’ is autological, then the adjective is true for itself. This implies the adjective ‘heterological’ is heterological. This paradox is indeed falsidical because this paradox leads to a contradictory and false conclusion through correct reasoning.

2 For details, see Quine, W.V. (1976). The ways of paradox and other essays. USA: Harvard University Press.

Part II

Immediate and Mediate Inference

Chapter 7

Immediate Inference

In this chapter, we will discuss the notion of inference and types of inference. We will illustrate immediate inference and kinds of immediate inference. Further, we will explain three kinds of immediate inference: conversion, obversion, and contraposition, with suitable examples. Inference consists of one or more than one proposition. In inference, we infer a conclusion from a proposition or a set of propositions. The proposition is termed as ‘premise’, and what we infer from the premise(s) is known as ‘conclusion’. The conclusion of inference is expressed through a proposition. It is to be noted here that in an inference, premise(s) is given to us, and we infer a conclusion from the premises. So conclusion, as a new proposition, is inferred from the premise(s). The inference is of two types: deductive inference and inductive inference. In the case of deductive inference, the conclusion cannot be more general than the premise(s). But in the case of inductive inference, the conclusion may be more general than the premises. In deductive inference, mostly we move from a universal proposition to a particular proposition, whereas in the case of inductive inference, we move from a particular proposition to a universal proposition often. An example of deductive inference, ‘All students are intelligent. Miku is a student. Therefore, Miku is intelligent’. An example of inductive inference, Crow X is black. Crow Y is black. Crow Z is black. One may add a few more premises to the existing premises from his/her experiences about the bird crow and its colour, such as Crow L is black, Crow N is black, etc. By considering these premises together, we can infer that ‘All crows are black’. Here, the conclusion ‘All crows are black’ is a universal proposition that is inferred from the given premises taking them together. A depiction of deductive inference and inductive inference are presented below for reference and discernment.

© Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_7

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Deductive Inference

Inductive Inference

Inductive leap

The deductive inference is further divided into immediate inference and mediate inference. The immediate inference is one where we infer the conclusion from one and only one premise. Mediate inference is one where we infer the conclusion from more than one premise by taking them together. In the case of mediate inference, we need a minimum of two premises to draw a conclusion. Mediate inference is known as ‘syllogism’. In a mediate inference, if the number of premises is more than two, then it is regarded as non-syllogistic. We will discuss ‘syllogism’ in the next chapter in detail. In this chapter, we will focus our discussions on immediate inference only. Although immediate inference and mediate inference differ in kinds, yet they have a commonality. That is the distribution of terms of categorical propositions.1 Violation of this rule results in an invalid inference. It may be stated here that a deductive inference is either valid or invalid. A valid deductive inference (mediate inference and immediate inference) is one where the conclusion is supported by the given premise(s) and the conclusion must not be greater (i.e. in quantity) than the premises. The immediate inference is of three kinds. These are conversion, obversion, and contraposition.

7.1 Conversion Conversion is a kind of immediate deductive inference, where a conclusion is drawn from a premise. The premise is named as ‘convertend’ and the conclusion is named as ‘converse’. The conversion has the following rules. These rules are to be followed to draw converse from the convertend. 1 Please

refer to Chap. 3 for the detailed analysis of the distribution of terms.

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85

Rule-1: The subject of the convertend becomes the predicate of the converse. Rule-2: The predicate of the convertend becomes the subject of the converse. Rule-3: The quality of the convertend and the converse remain unaltered. To explain, if the convertend is affirmative, the converse must be affirmative and if the convertend is negative, the converse must be negative. Rule-4: A term that is distributed in the converse must be distributed in the convertend to result in a valid immediate inference. We have four categorical propositions in the logical discourse. These are universal affirmative (A), universal negative (E), particular affirmative (I), and particular negative (O). Let us apply these four rules to the four categorical propositions and find out their converse, respectively. Conversion of universal affirmative (A) proposition The universal affirmative proposition is symbolically represented as ‘All S is P’, where S stands for the subject term and P stands for the predicate term. ‘All S is P’ is the convertend given to us. So a question arises, what would be the valid converse of ‘All S is P’. As per the rule-1, rule-2, and rule-3, we can have either ‘All P is S’ or ‘Some P is S’ converse. Upon applying rule-4 to ‘All P is S’ converse, we find P is distributed in the converse whereas it does not distribute in the convertend. Therefore, ‘All P is S’ is not a valid converse of ‘All S is P’ convertend. We are left with ‘Some P is S’ converse to verify whether it is a valid converse or not. In the case of ‘Some P is S’ converse, no term is distributed; hence, rule-4 does not apply to this converse. In other words, if no term is distributed in the converse, it is redundant to search for whether any term is getting distributed in the convertend or not. Thus, the conversion of A proposition is I proposition. That is, A: All S is P. (Convertend) I: Some P is S. (Converse) Take a concrete example of the conversion of A proposition. A: All cows are quadruped animals. (Convertend) I: Some quadruped animals are cows. (Converse) Conversion of universal negative (E) proposition The universal negative proposition is symbolically represented as ‘No S is P’, where S stands for the subject term and P stands for the predicate term. ‘No S is P’ is the convertend given to us. Now we will find out what would be the valid converse of ‘No S is P’. As per the rule-1, rule-2, and rule-3, we can have either ‘No P is S’ or ‘Some P is not S’ converse. Upon applying the rule-4 to ‘No P is S’ converse, we find that both P and S are distributed in the converse and they too are distributed in the convertend. Therefore, ‘No P is S’ is regarded as a valid converse of ‘No S is P’ convertend. We are left with the converse ‘Some P is not S’. In ‘Some P is not S’ converse, the term S is distributed in the converse and distributed in the convertend. Hence, it is also regarded as a valid converse. Logicians name the converse ‘Some

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P is not S’ as conversion per accidens (conversion by limitation). The reason is this converse is a weaker statement, although it is regarded as a valid converse. Logicians argue that in ‘Some P is not S’ converse, the term P is not distributed; only the term S is distributed. So, even though ‘Some P is not S’ converse does not violate any conversion rules, yet ‘No P is S’ is regarded as the appropriate converse for the convertend ‘No S is P’, as both P and S terms are distributed in the converse as well as the convertend. E: No S is P. (Convertend) E: No P is S. (Converse) Take a concrete example of the conversion of E proposition. E: No men are immortal beings. (Convertend) E: No immortal beings are men. (Converse) Conversion of particular affirmative (I) proposition The particular affirmative proposition is symbolically represented as ‘Some S is P’, where S stands for the subject term and P stands for the predicate term. The convertend ‘Some S is P’ is given to us. A question arises, what would be the valid converse of ‘Some S is P’. As per the rule-1, rule-2, and rule-3, we can have either ‘All P is S’ or ‘Some P is S’ converse. Upon applying the rule-4 to ‘All P is S’ converse, we find that P is distributed in the converse but it does not distribute in the convertend. Therefore, ‘All P is S’ is not a valid converse of ‘Some S is P’ convertend. We are left with ‘Some P is S’ converse to verify whether it is a valid converse of ‘Some S is P’ or not. In ‘Some P is S’ converse, no term is distributed, hence rule-4 does not apply to this converse. In other words, if no term is distributed in the converse, it is a redundant exercise to search for any term and its distribution in the convertend. Thus, the conversion of I proposition is I proposition, and it is a valid immediate inference. I: Some S is P. (Convertend) I: Some P is S. (Converse) Take a concrete example of the conversion of I proposition. I: Some students are swimmers. (Convertend) I: Some swimmers are students. (Converse) Conversion of particular negative (O) proposition The particular negative proposition is symbolically represented as ‘Some S is not P’, where S stands for the subject term and P stands for the predicate term. The convertend ‘Some S is not P’ is given to us. Now, we will find out what would be the valid converse of ‘Some S is not P’. As per the rule-1, rule-2, and rule-3, we can have either ‘No P is S’ or ‘Some P is not S’ converse. Upon applying the rule-4 to ‘No P is S’ converse, we find that both P and S terms are distributed in the converse but S term is not distributed in the convertend, whereas only P term is distributed in the

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convertend. Therefore, ‘No P is S’ cannot be treated as a valid converse of ‘Some S is not P’ convertend. We are left with ‘Some P is not S’ converse to verify whether it is a valid converse of ‘Some S is not P’ or not. In ‘Some P is not S’ converse, the term S is distributed, but it does not distribute in the convertend. So, it is a violation of rule-4. Hence, ‘Some P is not S’ cannot be regarded as a valid converse of ‘Some S is not P’ convertend. Thus, we submit that from the ‘O’ convertend, no valid converse is inferred. O: Some S is not P. (Convertend) No valid converse can be inferred. Take a concrete example of the conversion of O proposition. O: Some students are not swimmers. (Convertend) No valid converse can be inferred. Conversion Table Convertend

Valid converse

A: All S is P

I: Some P is S

E: No S is P

E: No P is S

I: Some S is P

I: Some P is S

O: Some S is not P

No valid converse

7.2 Obversion Obversion is a kind of immediate deductive inference, where we draw a conclusion from a premise. The premise is named as ‘obvertend’, and the conclusion is named as ‘obverse’. Obversion has the following rules. These rules are to be satisfied to draw an obverse from the obvertend. Rule-1: The subject of the obvertend and the obverse remains the same. Rule-2: The quantity of the obverse and obvertend will remain the same. That is, if the obvertend is universal, the obverse will be universal, and if the obvertend is particular, the obverse is particular. Rule-3: The quality of the obverse is the opposite of the quality of the obvertend. That is, if the obvertend is affirmative, the obverse would be negative and if the obvertend is negative, the obverse would be affirmative. Rule-4: The predicate of the obverse is the contradiction of the predicate of the obvertend. But the meaning of the obverse and obvertend will remain unaltered. The contradictory predicate of the obverse would be written as ‘non-P’. We are applying these four rules to the categorical propositions: universal affirmative (A), universal negative (E), particular affirmative (I), and particular negative

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(O), to find out their obverse, respectively. To put it differently, concerning obversion, we will find out what conclusion can be drawn from each of these categorical propositions. Obversion of universal affirmative (A) proposition The universal affirmative proposition is symbolically represented as ‘All S is P’, where S stands for the subject term and P stands for the predicate term. The obvertend ‘All S is P’ is given to us. Now, we will find out what would be the valid obverse of ‘All S is P’. Upon applying the obversion rules, E proposition ‘No S is non-P’ is inferred from A proposition ‘All S is P’. That is, A: All S is P. (Obvertend) E: No S is non-P. (Obverse) Take a concrete example of the obversion of A proposition. A: All cows are quadruped animals. (Obvertend) E: No cows are non-quadruped animals. (Obverse) In this argument, both premise and conclusion have the same quantity (i.e. universal) but differ in quality (i.e. obvertend is affirmative and the obverse is negative). The subject term of premise and conclusion remain the same. The predicate term of the conclusion contradicts the predicate term of the premise, but the meaning of the premise and the conclusion remain unaltered. The conclusion E proposition ‘No S is non-P’ is regarded as a negative proposition with a contradictory predicate. Obversion of universal negative (E) proposition The universal negative proposition is symbolically represented as ‘No S is P’, where S stands for the subject term and P stands for the predicate term. The obvertend ‘No S is P’ is given to us. Now let us find out what would be the valid obverse of ‘No S is P’. Upon applying the obversion rules, A proposition ‘All S is non-P’ is inferred from E proposition ‘No S is P’. That is, E: No S is P. (Obvertend) A: All S is non-P. (Obverse) Take a concrete example of the obversion of E proposition. E: No swans are black. (Obvertend) A: All swans are non-black. (Obverse) In this argument, both premise and conclusion have the same quantity (i.e. universal) but differ in quality (i.e. obvertend is negative and the obverse is affirmative). The subject term of premise and conclusion remain the same. The predicate term of the conclusion contradicts the predicate term of the premise, but the meaning of the premise and the conclusion remain unchanged. Here, the conclusion A proposition ‘All S is non-P’ is regarded as an affirmative proposition with the contradictory predicate.

7.2 Obversion

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Obversion of particular affirmative (I) proposition The particular affirmative proposition is symbolically represented as ‘Some S is P’, where S stands for the subject term and P stands for the predicate term. The obvertend ‘Some S is P’ is given to us. Now let us find out what would be the valid obverse of ‘Some S is P’. Upon applying the obversion rules, O proposition ‘Some S is not non-P’ is inferred from I proposition ‘Some S is P’. That is, I: Some S is P. (Obvertend) O: Some S is not non-P. (Obverse) Take a concrete example of the obversion of I proposition. I: Some men are intelligent beings. (Obvertend) O: Some men are not non-intelligent beings. (Obverse) In this argument, both premise and conclusion have the same quantity (i.e. particular) but differ in quality (i.e. obvertend is affirmative and the obverse is negative). The subject term of premise and conclusion remain the same. The predicate term of the conclusion contradicts the predicate term of the premise, but the meaning of the premise and the conclusion remain unaltered. Here, the conclusion O proposition ‘Some S is not non-P’ is considered as a negative proposition with a contradictory predicate. Obversion of particular negative (O) proposition The particular negative proposition is symbolically represented as ‘Some S is not P’, where S stands for the subject term and P stands for the predicate term. The obvertend ‘Some S is not P’ is given to us. Let us find out what would be the valid obverse of ‘Some S is not P’. Upon applying the obversion rules to O proposition ‘Some S is not P’, the I proposition ‘Some S is non-P’ is inferred. That is, O: Some S is not P. (Obvertend) I: Some S is non-P. (Obverse) Take a concrete example of the obversion of O proposition. O: Some women are not tall. (Obvertend) I: Some women are non-tall. (Obverse) In this argument, both premise and conclusion have the same quantity (i.e. particular) but differ in quality (i.e. obvertend is negative and the obverse is affirmative). The subject term of the premise and the conclusion remain the same. The predicate term of the conclusion contradicts the predicate term of the premise, but the meaning of the premise and conclusion remain unaltered. In this case, the conclusion I proposition ‘Some S is non-P’ is regarded as an affirmative proposition with a contradictory predicate.

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Obversion Table Obvertend

Valid obverse

A: All S is P

E: No S is non-P

E: No S is P

A: All S is non-P

I: Some S is P

O: Some S is not non-P

O: Some S is not P

I: Some S is non-P

7.3 Contraposition Contraposition is a kind of immediate deductive inference that can be reduced to a combination of obversion and conversion inferences. In contraposition, the conclusion is drawn from one of the categorical propositions A, E, I, O. To infer a contrapositive (i.e. conclusion) from one of the categorical proposition, we need to obvert the given premise, then convert the obverted premise, and finally obvert the converted premise. To do so, we have to apply obversion and conversion rules by turns so long as the conclusion does not satisfy the following conditions. The first condition states that the predicate of the conclusion is the contradiction of the subject of the premise. The second condition enunciates that the subject part of the conclusion is contradictory of the predicate part of the premise. The third and last condition expresses that quality of the premise and the conclusion remains unchanged. Contraposition of universal affirmative (A) proposition The universal affirmative proposition is symbolically represented as ‘All S is P’, where S stands for the subject term and P stands for the predicate term. The premise ‘All S is P’ is given to us. Now we will find out what would be the valid conclusion (i.e. contrapositive) of ‘All S is P’. A: All S is P. (Premise) E: No S is non-P. (Obverse) E: No non-P is S. (Converse) A: All non-P is non-S. (Obverse and the Conclusion) To explain, upon examination of the conclusion ‘All non-P is non-S’ (i.e. contrapositive), which is drawn from the premise A (i.e. All S is P), it is found that the conclusion satisfies contraposition’s above-mentioned three conditions. Hence, the contrapositive of A proposition is an A proposition. Take a concrete example of the contraposition of A proposition ‘All logic students are intelligent students’. The contrapositive of this proposition is ‘All non-intelligent students are non-logic students’.

7.3 Contraposition

91

Contraposition of universal negative (E) proposition The universal negative proposition is symbolically represented as ‘No S is P’, where S stands for the subject term and P stands for the predicate term. The premise ‘No S is P’ is given to us. Let us find out what would be the valid contrapositive of ‘No S is P’. E: No S is P. (Premise) A: All S is non-P. (Obverse) I: Some non-P is S. (Converse) O: Some non-P is not non-S. (Obverse and the Conclusion) To explain, about the perusal of the conclusion ‘Some non-p is not non-S’ (i.e. contrapositive), which is drawn from the premise E (i.e. No S is P), it is found that the conclusion satisfies the contraposition’s three conditions. Hence, the contrapositive of E proposition is an O proposition. Consider a concrete example of contraposition of E proposition ‘No poets are Indian international cricket players’. The contrapositive of this proposition is ‘Some non-Indian international cricket players are not non-poets’. Contraposition of particular affirmative (I) proposition The particular affirmative proposition is symbolically represented as ‘Some S is P’, where S stands for the subject term and P stands for the predicate term. The premise ‘Some S is P’ is given to us. Now let us find out what would be the valid contrapositive of ‘Some S is P’. I: Some S is P. (Premise) O: Some S is not non-P. (Obverse) No valid converse can be drawn from the ‘O’ premise. Hence, no valid contrapositive (i.e. conclusion) can be inferred from I premise. To explain, upon attempting to draw contrapositive of I proposition (i.e. Some S is P), it is found that no contrapositive can be drawn. The reason is we cannot draw a valid converse from the ‘O’ proposition (i.e. Some S is not non-P). Further, the premise I (i.e. Some S is P) does not satisfy three conditions of contraposition inference. Hence, from an I premise, no contrapositive can be inferred. Contraposition of particular negative (O) proposition The particular negative proposition is symbolically represented as ‘Some S is not P’, where S stands for the subject term and P stands for the predicate term. The premise ‘Some S is not P’ is given to us. Now we will find out what would be the valid contrapositive of ‘Some S is not P’. O: Some S is not P. (Premise) I: Some S is non-P. (Obverse) I: Some non-P is S. (Converse) O: Some non-P is not non-S. (Obverse and the Conclusion)

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To explain, upon examination of the contrapositive (i.e. Some non-P is not nonS), which is drawn from the premise O (i.e. Some S is not P), it is found that the conclusion (i.e. contrapositive) satisfies the three conditions of contraposition. Hence, the contrapositive of the O proposition is an O proposition. Consider a concrete example of contraposition of O proposition ‘Some politicians are not truth speakers’. The contrapositive of this proposition is ‘Some non-truth speakers are not nonpoliticians’. Contraposition Table Premise

Valid contrapositive (conclusion)

A: All S is P

A: All non-P is non-S

E: No S is P

O: Some non-P is not non-S

I: Some S is P

No valid contrapositive

O: Some S is not P

O: Some non-P is not non-S

The four categorical propositions (i.e. A, E, I, O) and the types of immediate inference (i.e. conversion, obversion, and contraposition) are presented below in Table 7.1 for the immediate reference and discernment. Table 7.1 depicts the converse, obverse, and contrapositive of A, E, I, O premise. Table 7.1 Premise and valid conclusion of immediate inference Premise

Converse

Obverse

Contrapositive

A: All S is P

I: Some P is s

E: No S is non-P

A: All non-P is non-S

E: No S is P

E: No P is S

A: All S is non-P

O: Some non-P is not non-S

I: Some S is P

I: Some P is S

O: Some S is not non-P No valid contrapositive

O: Some S is not P No valid converse I: Some S is non-P

O: Some non-P is not non-S

7.4 Converting Sentences to Categorical Propositions for Immediate Inference In our mundane lives, we use many sentences to communicate our ideas, thoughts, and feelings to others. Most of the sentences are not formulated in the standard logical form, and thereby they are not considered categorical propositions. Hence, to draw the conclusion from these sentences, we may result in chaos, confusion, and the debacle of the logical structure of the proposition. To arrest this lacuna, we need to translate these sentences to categorical propositions. This would enable us to draw valid immediate inferences from these sentences. While converting natural language sentences to categorical propositions, we need to adopt the following rules.

7.4 Converting Sentences to Categorical Propositions for …

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Rule-1: The subject and predicate terms of a categorical proposition must be the names of classes. They must be nouns and not adjectives. Rule-2: If the predicate term of a categorical proposition is not a noun, then it must be changed into a substantive noun. Rule-3: There must be a copula (i.e. is, are, is not, are not, was, was not, etc.) in the categorical proposition, which connects the subject term with the predicate term. Rule-4: While translating natural language sentences to categorical propositions, the meaning of the sentences shall not be altered. Rule-5: The quantity indicator (i.e. ‘All’ or ‘Some’) of a sentence should be fixed based on the context where it is used or uttered. Some natural language sentences are: (a) (b) (c) (d) (e) (f) (g)

A few logic students play chess. Most of the logic lessons are exciting. Some monkeys bite. Men are mortal. Men are cricketers. Whoever is a baby is cute. A handful of politicians are not honest. Translating the sentences of natural language to categorical propositions.

(a) (b) (c) (d) (e) (f) (g)

Some logic students are chess players. Some logic lessons are exciting lessons. Some biting animals are monkeys. All men are mortal. Some men are cricketers. All babies are cute living infants. Some politicians are not honest beings.

Now we will take two categorical propositions from the above-mentioned list to draw their immediate inferences (i.e. converse, obverse, and contrapositive). (e) (g)

Some men are cricketers. No politicians are honest beings.

These two propositions are regarded as I and E proposition, respectively. The I proposition is represented as ‘Some S is P’ and E proposition is represented as ‘No S is P’. Premise

Converse

Obverse

Contrapositive

I: Some men are cricketers

I: Some cricketers are men

I: Some men are not non-cricketers

No valid contrapositive can be drawn from the I premise

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Premise

Converse

Obverse

Contrapositive

E: No politicians are honest beings

E: No honest beings are politicians

A: All politicians are non-honest beings

O: Some non-honest beings are not non-politicians

Note for students: Write the immediate inferences (i.e. converse, obverse, and contrapositive) of the rest of the categorical propositions mentioned above. If you succeed in writing the correct immediate inferences, you should take pride in your effort in understanding the concept of immediate inference clearly and correctly.

Chapter 8

Mediate Inference (Syllogism)

In the previous chapter, we discussed the notion of immediate inference and types of immediate inference, such as conversion, obversion, and contraposition. We enumerated the rules for translating natural language sentences into standard logical forms (i.e. categorical propositions). In continuation of the previous chapter, in this chapter, we will discuss mediate inference (i.e. syllogism) in detail with suitable examples. We will analyse the moods and figures of syllogism, describe the rules for the syllogism, and apply the rules of syllogism to each mood by linking them to figures of syllogism to find out the valid moods and valid arguments of the syllogism. In the end, we will list out the valid moods of the syllogism corresponding to the four figures of syllogism. Unlike the immediate inference, in mediate inference, we consider two premises together to draw a conclusion. But the conclusion is not the total of two premises. A mediate inference is consisting of three propositions; the first and second propositions are regarded as ‘premises’, and the third proposition is treated as a ‘conclusion’. Mediate inference is popularly known as ‘syllogism’. In a syllogism, we use the standard logical forms of sentences, that are, categorical propositions (A, E, I, O). Out of the four categorical propositions, two propositions are taken as premises and the third proposition (i.e. conclusion) is also a categorical proposition. Since syllogism is a mediate and deductive inference, the conclusion of the syllogism can be as general as or less general than the premises. In a syllogism, the conclusion is judged as either true or false, and it all depends on the premises. If the premises are true, the conclusion is true, and if one of the premises is false, then the conclusion is false. Further, the truth-value of a premise (i.e. a premise is either true or false) is related to the concept of formal truth, not material truth. The formal truth of a premise is relied on the structure of the proposition, whereas the material truth of a premise is dependent on both structure and semantics of the proposition. For example, ‘All T is B’ is a categorical proposition of having T as the subject term and B as the predicate term. Both T and B are used as variables of the categorical proposition. This proposition is assumed as formally true but may not be materially true. This is because of the substitution of a subject term ‘tables’ for T and a predicate © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_8

95

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term ‘quadruped animals’ for B. The proposition ‘All tables are quadruped animals’ becomes a materially false proposition, as the semantics of the proposition does not have a referent in the empirical world. An example of mediate inference (syllogism): A: All ladies are beautiful women. (Premise I) A: All air hostesses are ladies. (Premise II) A: All air hostesses are beautiful women. (Conclusion)

8.1 Figures of Syllogism A syllogism consists of two premises and a conclusion. Each premise has two terms (subject term and predicate term), and the conclusion has two terms (subject term and predicate term). Therefore, a syllogism has six terms. Upon examination of these six terms, it is found that there are only three terms, and each term appears twice in a syllogism. Logicians named these three terms as the major term, minor term, and middle term. The subject term of the conclusion is called minor term, the predicate term of the conclusion is regarded as the major term, and a term that appears in the first and second premise but does not appear in conclusion is regarded as the middle term. The middle term establishes a link between two premises and assists to infer a conclusion in a syllogism. Further, a premise, which has both major term and middle term, is regarded as the major premise, and a premise that has both minor term and the middle term is regarded as the minor premise. Thus, the structure of a syllogism consists of a major premise, a minor premise, and a conclusion, wherein major term, minor term, and middle term appear twice in a syllogism. By considering these specifics and by referring to the above-mentioned syllogism example, it is identified that ‘air hostesses’ is the minor term (hereafter, we will use ‘S’ to indicate minor term), ‘beautiful women’ is the major term (hereafter, we will use ‘P’ to indicate major term), and ‘ladies’ is the middle term (hereafter, we will use ‘M’ to indicate the middle term). The premise I and premise II are regarded as major premise and minor premise, respectively. Symbolically, this syllogism is represented as: A: All M is P. (Major premise) {middle term and major term} A: All S is M. (Minor premise) {minor term and middle term} A: All S is P. (Conclusion) {minor term and major term} In a syllogism, the middle term appears in both the major and minor premises. The middle term can be positioned either in the subject or in the predicate of a major premise. It can also be placed either in the subject or in the predicate of the minor premise. Thus, the middle term placing can be done in four different ways in a syllogism. The figure of the syllogism is determined based on the position of the middle term (M) of a syllogism. Concerning the four positions of the middle term in a syllogism, we will have four figures of syllogism. The four figures are named as the first figure, the second figure, the third figure, and the fourth figure, respectively.

8.1 Figures of Syllogism

97

The first figure of syllogism In the first figure, the middle term is positioned as the subject term of the major premise and predicate term of the minor premise. The first figure is depicted as: M………P. (Major premise)

{Middle term and major term}

S………M. (Minor premise)

{Minor term and middle term}

S………P. (Conclusion)

{Minor term and major term}

The second figure of syllogism In the second figure, the middle term is positioned as the predicate term of the major premise and the minor premise. The second figure is depicted as: P……. M. (Major premise)

{Major term and middle term}

S………M. (Minor premise)

{Minor term and middle term}

S………P. (Conclusion)

{Minor term and major term}

The third figure of syllogism In the third figure, the middle term is positioned as the subject term of the major premise and the minor premise. The third figure is represented as: M……. P. (Major premise)

{Middle term and major term}

M……. S. (Minor premise)

{Middle term and minor term}

S………P. (Conclusion)

{Minor term and major term}

The fourth figure of syllogism In the fourth figure, the middle term is positioned as the predicate term of the major premise and the subject term of the minor premise. The fourth figure is portrayed as: P………M. (Major premise)

{Major term and middle term}

M………S. (Minor premise)

{Middle term and minor term}

S………P. (Conclusion)

{Minor term and major term}

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8.2 Moods of Syllogism A mood of a syllogism is the categorical proposition (A, E, I, O) it contains. Since every syllogism consists of three categorical propositions, it implies each (two premises and a conclusion) syllogism has a mood. A syllogism has a major premise, a minor premise, and a conclusion. Thus, the mood of a syllogism can be represented depending on the type of categorical proposition it contains. Consider the example below: A: All ladies are beautiful women. (Major premise) A: All air hostesses are ladies. (Minor premise) A: All air hostesses are beautiful women. (Conclusion) In this syllogism, major premise and minor premise are the universal affirmative propositions represented as ‘A’. The conclusion is also a universal affirmative proposition represented as ‘A’. Thus, the mood of the syllogism is ‘AAA’. The first A and the second A stand for major premise and minor premise, respectively, and the third A stands for the conclusion of the syllogism. As per the syllogism requirements, there must be two premises (i.e. categorical propositions) to draw a conclusion. Since there are four categorical propositions, a categorical proposition can be combined with another categorical proposition out of three categorical propositions to constitute two premises. This combination would take place in such a unique manner that two categorical propositions would not repeat further. The permutation and combination of the four categorical propositions would result in sixteen (4 × 4 = 16) possible moods. These are mentioned in Table 8.1. We can draw a conclusion from each possible mood. The conclusion may be either A, or E, or I, or O proposition. Hence, there are four possibilities for concluding a possible mood. Therefore, we have sixty-four possible moods (i.e. 16 × 4 = 64) along with four possible conclusions in the syllogism, for example, AAA, AAE, AAI, etc. Each possible mood with a conclusion (e.g. AAA) can be structured conforming to the four figures of syllogism. So in total, we have two hundred and fifty-six possible moods (i.e. 64 × 4 = 256) in syllogism. A question arises, are these 256 possible moods treated as valid syllogisms? To find out which mood is regarded as a valid mood and thereby a valid syllogism, we need to apply the syllogistic rules to each possible mood and verify their validities in the logical discourse. Table 8.1 Sixteen possible moods of syllogism

AA

EA

IA

OA

AE

EE

IE

OE

AI

EI

II

OI

AO

EO

IO

OO

8.3 Rules of Syllogism

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8.3 Rules of Syllogism Rule-1: A syllogism must contain three and only three terms. Rule-2: The middle term must be distributed at least once in the premises. Rule-3: A term that is distributed in the conclusion must be distributed in the premise. Rule-4: From two negative premises, no conclusion can be inferred. Rule-5: If one of the premises is negative, the conclusion must be negative. Rule-6: If two premises are universal, the conclusion must be universal. Rule-7: If two premises are particular, no conclusion can be inferred. Rule-8: From a particular major premise and a negative minor premise, no conclusion can be inferred. Elucidation of rule-1 Rule-1 states that a syllogism must have three and only three terms. That is, it should have the major, minor, and middle term. What if a syllogism violates this rule? Violation of this rule would result in a syllogism with less than three terms or more than three terms. Consider the former alternative, and assume that a syllogism has two terms. If it is so, then the structure of the syllogism does not satisfy the norms of a mediate deductive inference, as stated in the previous sections. If we consider the latter alternative (i.e. a syllogism has more than three terms), then there would not be a middle term that establishes a link between major premise and minor premise. As a result, a conclusion cannot be inferred from the premises by taking them together. It is a fallacy known as the fallacy of four terms. An example of the fallacy of four terms is as follows. Miku is a student. Mira is a teacher. Therefore, no conclusion follows. In this example, we have four terms: Mira, Miku, student, and teacher. But we do not have a middle term to establish a link between major premise and minor premise, and also assist in inferring the conclusion. Hence, no conclusion is inferred from the premises. In some cases, a syllogism is found to have three terms, but the meanings of the terms are ambiguous. If it were so, a syllogism would not have three terms indeed. It would have more than three terms. Here, we commit a fallacy known as the fallacy of equivocation. There are three kinds of the fallacy of equivocation, namely fallacy of ambiguous major, fallacy of ambiguous minor, and fallacy of ambiguous middle. In the case of fallacy of ambiguous major, the meaning of the major term of the conclusion is not the same as the meaning of the major term of the major premise. In the case of fallacy of ambiguous minor, the meaning of the minor term of the conclusion and the minor premise is not the same. About the fallacy of ambiguous middle, the meaning of the middle term of the major premise and the minor premise is

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not the same. An example of an ambiguous major, ambiguous minor, and ambiguous middle is given below. Fallacy of ambiguous major Mobile is essential to talk to others. (Major premise) Raju is not essential to talk to others. (Minor premise) Raju is not mobile. (Conclusion) In this syllogism, the major term is ‘mobile’ as it is positioned in the predicate term of the conclusion. The major term appears twice in the syllogism, the conclusion, and the major premise. But the meanings of the major term of the conclusion and the major premise are not the same. In conclusion, the term ‘mobile’ means moving freely, and in the major premise, it means a mobile phone. Hence, this syllogism has four terms instead of three terms. Due to the ambiguous major term, the conclusion does not support the given premises. We commit the fallacy of ambiguous major in this syllogism, and hence it is regarded as an invalid syllogism. Fallacy of ambiguous minor No man is made of paper. (Major premise) All pages are men. (Minor premise) No pages are made of paper. (Conclusion) In this syllogism, the minor term is ‘page’ as it is positioned in the subject term of the conclusion. The minor term appears twice in the syllogism, the conclusion, and the minor premise. But the meanings of the minor term in both places are not the same. In conclusion, the term ‘page’ means a paper of a book or a magazine, and in the minor premise, it means a male servant. Hence, this syllogism has four terms in reality. In this syllogism, the conclusion does not support the given premises. Therefore, it is an invalid syllogism, as we commit the fallacy of ambiguous minor. Fallacy of ambiguous middle This curry is hot. (Major premise) Sarmila is not hot. (Minor premise) Sarmila is not a curry. (Conclusion) In this syllogism, the middle term is ‘hot’ and it is found in the major and minor premises. The meanings of the middle term in both the premises are not the same. In the major premise, the term ‘hot’ means very spicy, and in the minor premise, it means not attractive and enticing. Hence, this syllogism has four terms in reality. In this syllogism, the middle term does not establish the link between major premise and minor premise, as we commit the fallacy of ambiguous middle. Thus, the conclusion of the syllogism does not support the given premises; as a result, the syllogism is regarded as an invalid syllogism. From the above analyses, it is asserted that a valid syllogism must not violate the rule-1. If it violates, then it encounters either fallacy of four terms or fallacy of equivocation.

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Explanation of rule-2 Rule-2 suggests that in a syllogism, the middle term must be distributed at least once in the premises. If the middle term is not distributed in one of the premises, it will not establish a link between the major premise and the minor premise. Further, if there is no link between the major premise and the minor premise of a syllogism, a valid conclusion cannot be inferred from two premises by taking them together. So, the syllogism would be treated as invalid. Violation of rule-2 commits the fallacy known as the fallacy of the undistributed middle. An example of the fallacy of the undistributed middle is mentioned below. Fallacy of the undistributed middle All men are mortal. (Major premise) All women are mortal. (Minor premise) All women are men. (Conclusion) This syllogism has three terms: the major term (i.e. men), the minor term (i.e. women), and the middle term (i.e. mortal). It corresponds to the second figure of the syllogism, as the middle term is positioned in the predicate part of both the major premise and minor premise. The major and the minor premises are A and A proposition, respectively. In an A proposition, only the subject part is distributed. Hence, the middle term is not distributed in any of the premises. Thus, the inferred conclusion from two premises is an erroneous one, as it commits the fallacy of the undistributed middle. Therefore, it is treated as an invalid syllogism. Elucidation of rule-3 Rule-3 enunciates that a term distributed in conclusion must be distributed in the premise because, in the deductive mediate inference, the conclusion cannot be more general than the premises. The possibility of a term distributed in conclusion but not distributed in the premise is ruled out. In other words, a term that is taken in conclusion for its entire denotation must be considered for its entire denotation in the premise. If it is not so, then the syllogism encounters either the fallacy of illicit major or the fallacy of illicit minor. The fallacy of illicit major explains that the major term of the conclusion is distributed, but it is not distributed in the major premise of a syllogism. Similarly, the fallacy of illicit minor conveys that minor term of the conclusion is distributed, but it is not distributed in the minor premise of a syllogism. An example of the fallacy of illicit major and the fallacy of illicit minor is given below. The fallacy of illicit major All queens are powerful. (Major premise) Some ladies are not queens. (Minor premise) Some ladies are not powerful. (Conclusion) This syllogism has three terms: the major term (i.e. powerful), the minor term (i.e. ladies), and the middle term (i.e. queens). It corresponds to the first figure of the

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syllogism. The major premise is A proposition, the minor premise is O proposition, and the conclusion is O proposition. The middle term is distributed in the major premise and the minor premise. In conclusion, the major term is distributed, and it is not distributed in the major premise. Therefore, it is regarded as an invalid syllogism, as it commits the fallacy of illicit major. The fallacy of illicit minor No philosophers are cricket players. (Major premise) All philosophers are swimmers. (Minor premise) No swimmers are cricket players. (Conclusion) This syllogism has three terms: the major term (i.e. cricket players), the minor term (i.e. swimmers), and the middle term (i.e. philosophers). It corresponds to the third figure of the syllogism. The major premise is E proposition, the minor premise is A proposition, and the conclusion is the E proposition. The middle term is distributed in the major premise and the minor premise. In conclusion, minor term is distributed, but it is not distributed in the minor premise. Therefore, it is regarded as an invalid syllogism, as we commit the fallacy of illicit minor. This rule of syllogism suggests that violating this rule invariably results in an invalid syllogism. In other words, a valid syllogism must conform to this rule without any exception. Explanation of rule-4 Rule-4 expresses that from two negative premises, no conclusion can be inferred. The reason is in negative categorical propositions (i.e. E and O), and the predicate part denies the subject part. As a result, there is no connection found between the subject term and the predicate term of the proposition. In a syllogism, if there is no connection found between the subject term and the predicate term of both the major premise and minor premise, the middle term is not able to connect with the major term in the major premise, and it is not able to connect with the minor term in the minor premise. As a consequence, the middle term is not able to establish a link between the major premise and the minor premise of a syllogism. Hence, no conclusion can be inferred from the two negative premises. Violation of this rule commits a fallacy known as the fallacy of exclusive premises. Examples of the fallacy of exclusive premises are mentioned below for information, reference, and discernment. Example-1: Fallacy of exclusive premises No men are quadruped animals. (Major premise) No quadruped animals are rational beings. (Minor premise) No rational beings are men. (Conclusion) This syllogism has three terms: the major term (i.e. men), the minor term (i.e. rational beings), and the middle term (i.e. quadruped animals). It corresponds to the fourth figure of the syllogism. The major premise is E proposition, the minor premise is E proposition, and the conclusion is the E proposition. The middle term is distributed in the major premise and the minor premise. In conclusion, the minor term

8.3 Rules of Syllogism

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and the major term are distributed and also distributed in the respective premises. Even then, the conclusion drawn from two premises is not valid, as the middle term does not link to the major term in the major premise and connects to the minor term in the minor premise. Hence, to draw a conclusion from two negative premises commits the fallacy known as the fallacy of exclusive premises. Example-2: Fallacy of exclusive premises No cows are dogs. (Major premise) Some cats are not cows. (Minor premise) Some cats are not dogs. (Conclusion) This syllogism has three terms: the major term (i.e. dogs), the minor term (i.e. cats), and the middle term (i.e. cows). It corresponds to the first figure of the syllogism. The major premise is E proposition, the minor premise is O proposition, and the conclusion is O proposition. The middle term is distributed in the major premise and in the minor premise. In conclusion, the major term is distributed and also distributed in the respective premise. Even then, the conclusion drawn from two premises is not treated as a valid conclusion, as the middle term does not link to the major term in the major premise and connects to the minor term in the minor premise. Hence, to draw a conclusion from two negative premises commits the fallacy known as the fallacy of exclusive premises. Explanation of rule-5 Rule-5 states that if one premise is negative, the conclusion must be negative because two premises cannot be negative. If two premises are negative, no conclusion can be drawn as per the justification and explanation given in rule-4. So, to draw a conclusion in a mediate deductive inference, if one premise is negative, the other premise must be affirmative. In the case of a negative premise, there is no connection between the subject term and the predicate term. But in the case of an affirmative premise, there is a connection between the subject and the predicate terms. Since there is no connection found between the subject term and the predicate term of a negative premise, the middle term does not establish a true link between the major premise and the minor premise. As a result, in conclusion, we can infer that there is no true link found between the major term and the minor term of the syllogism. So, the conclusion must be negative. The below example proves the claim made in rule-5. E: No logic students are potters. (Major premise) A: All badminton players are logic students. (Minor premise) E: No badminton players are potters. (Conclusion) This syllogism has three terms: the major term (i.e. potters), the minor term (i.e. badminton players), and the middle term (i.e. logic students). It corresponds to the first figure of the syllogism. The major premise is E proposition, the minor premise is A proposition, and the conclusion is E proposition. The middle term is distributed on the major premise only. In conclusion, both major and minor terms are distributed on the respective premises. In the case of a major premise, there is no link found between

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the subject and the predicate terms, but in the case of the minor premise, there is a link found between the subject and the predicate terms. Hence, in conclusion, we infer that there is no true connection existing between the major term and the minor term of the syllogism. If we infer an affirmative proposition in conclusion from one of the negative premises of the syllogism, the syllogism is judged as an invalid syllogism. This rule suggests that if both the premises are affirmative, the conclusion must be affirmative of a syllogism. Explanation of rule-6 Rule-6 enunciates that if two premises are universal, the conclusion must be universal. Modern logicians propose this rule by repudiating the traditional syllogism argument, which states that a particular conclusion can be inferred from two universal premises. The traditional syllogism argues that in a mediate deductive inference, the conclusion must not be more general than the premises. It implies that the conclusion may be a particular proposition, which could be drawn from universal premises. But modern logicians reject this argument. They argue that universal propositions (A and E) do not have existential import, whereas particular propositions (I and O) have existential import.1 Therefore, from two universal propositions, if we draw a particular proposition in conclusion, we will commit a fallacy known as ‘existential fallacy’. Thus, we need to infer a universal proposition in conclusion from two universal premises to make the syllogism a valid syllogism. This rule evokes that in the mediate deductive inference, the conclusion cannot possess something (information) that the premises do not have. It suggests that if one of the premises is particular, the conclusion must be a particular proposition of a syllogism with an exception to IE mood. The details about IE mood are mentioned in rule-8. An example of existential fallacy is given below. Existential fallacy No teachers are politicians. (Major premise) All politicians are liars. (Minor premise) Some liars are not teachers. (Conclusion) This syllogism has three terms: the major term (i.e. teachers), the minor term (i.e. liars), and the middle term (i.e. politicians). It corresponds to the fourth figure of the syllogism. The major premise is E proposition, the minor premise is A proposition, and the conclusion is O proposition. The middle term is distributed in the major premise and the minor premise. In conclusion, the major term is distributed and it is distributed in the major premise as well. Even then, the syllogism is not regarded as a valid syllogism because the conclusion possesses something (information) that the premises do not have. Further, the premises do not support the 1 In

Geroge Boole’s (1815–1864) view, a categorical proposition has existential import if and only if the truth of the categorical proposition requires a belief in the existence of members of the subject class. Thus, I and O propositions have existential import, whereas A and E propositions do not have existential import. I and O propositions assert that the classes designated by their subject terms are not empty.

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conclusion. Thus, whenever a particular proposition is drawn from two universal premises, the syllogism encounters the existential fallacy. Elucidation of rule-7 Rule-7 states that if two premises are particular, no conclusion can be inferred. The reason is we have four possible moods (i.e. II, IO, OI, and OO) of having two particular premises. By applying the rule-2 (i.e. the middle term must be distributed at least once in the premises), we can reject the II mood as a valid mood because in the I proposition neither the subject term nor the predicate term is distributed. Hence, the middle term is not distributed in any of the premises. If we attempt to infer a conclusion from II mood, we end up in a fallacy known as the fallacy of undistributed middle. By applying rule-4 (i.e. from two negative premises, no conclusion can be inferred), we can reject the OO mood, as it violates rule-4 to draw a valid conclusion. Now we will find out whether a valid conclusion can be drawn from IO and OI moods or not. Concerning IO mood, the major premise is I proposition and the minor premise is O proposition. In I proposition, neither the subject term nor the predicate term is distributed, but in the O proposition, the predicate term is distributed. Hence, to satisfy rule-2, we need to mention the middle term in the predicate part of the O proposition. Further, to satisfy rule-5, we have to infer either E or O proposition as a conclusion. Upon considering the possible conclusions of IO mood, we get the following moods: IOE and IOO. To infer E conclusion from IO mood has not resulted in a valid syllogism. The reason is, it is not satisfying rule-6 (i.e. if one of the premises is particular, the conclusion cannot be universal). About the IOO mood, the conclusion is O proposition, and thereby the major term is distributed in conclusion. However, the major term is not distributed in the major premise as I proposition is the major premise. So, the IOO mood encounters a fallacy known as the fallacy of illicit major. As a result, it is not regarded as a valid syllogism. About OI mood, the major premise is O proposition and the minor premise is I proposition. In O proposition, the predicate term is distributed, but in I proposition neither the subject term nor the predicate term is distributed. Hence, to satisfy the rule-2, we need to mention the middle term in the predicate part of the O proposition. Further, to satisfy rule-5, we have to draw either E or O proposition as a conclusion. Upon considering the possible conclusions of OI mood, we get the following moods: OIE and OIO. To infer E conclusion from OI mood has not resulted in a valid syllogism. The reason is it is not satisfying the rule-6 of syllogism; that is, if one of the premises is particular, the conclusion cannot be universal. Concerning OIO mood, the conclusion is O proposition, and thereby the major term is distributed in conclusion. However, the major term is not distributed in the major premise as it is positioned in the subject term of the major premise. So, OIO mood encounters a fallacy known as the fallacy of illicit major. From the above analyses and arguments, it is proved that if the major premise is particular and minor premise is particular, a conclusion cannot be inferred.

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Explanation of rule-8 Rule-8 states that from a particular major premise and a negative minor premise, no conclusion can be inferred. Let us verify whether we can draw a valid conclusion from a particular major premise and a negative minor premise or not. If we consider the major premise is particular and the minor premise is negative, then we have the following possible moods: IE, IO, OE, OO. By applying rule-4, we can reject OE and OO moods. By applying rule-7, we can reject the IO mood. So, we are left with an IE mood to test whether we can draw a valid conclusion from this mood or not. About IE mood, the major premise is I proposition and the minor premise is E proposition. In I proposition, neither the subject term nor the predicate term is distributed, but in E proposition, both subject term and predicate term are getting distributed. To satisfy rule-2 of syllogism, we need to put the middle term in either subject or predicate part of the E proposition. Therefore, the minor term of the E proposition is distributed. Further, to satisfy rule-5, we have to draw a negative conclusion from the IE mood. Again, by conforming to the rule-6 (i.e. if one of the premises is particular, the conclusion cannot be universal), we get the IEO mood. In the IEO mood, the conclusion is O proposition and thereby the major term is distributed in conclusion. However, the major term is not distributed in the major premise, as it is an I proposition. So, the IEO mood encounters a fallacy known as the fallacy of illicit major. From the above analyses and arguments, it is proved that if the major premise is particular and the minor premise is negative, no conclusion can be inferred in a syllogism.

8.4 Determining the Valid Moods of Syllogism Let us apply the eight syllogistic rules to the sixteen possible moods (refer to Table 8.1) and find out which moods are to be rejected outright as they violate the syllogistic rules and which moods are to be tested for their validities. By considering rule-4, we can reject EE, EO, OE, and OO moods, as no conclusion can be drawn from two negative premises. Upon consideration of rule-7, we can reject II, IO, OI, and OO moods, as no conclusion can be drawn from two particular premises. By upholding rule-8, we can discard IE mood, as from a particular major premise and negative minor premise, no valid conclusion can be inferred. After eliminating these eight moods, we are left with eight more possible moods (refer to Table 8.2) that are to be tested to determine their validities. We will test these eight moods by referring to four figures of syllogism one after another.

8.4 Determining the Valid Moods of Syllogism Table 8.2 Moods need to be tested

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AA

EA

IA

OA

AE

EE (Rejected by rule-4)

IE (Rejected by rule-8)

OE (Rejected by rule-4)

AI

EI

II (Rejected by rule-7)

OI (Rejected by rule-7)

AO

EO (Rejected by rule-4)

IO (Rejected by rule-7)

OO (Rejected by rule-4)

The following steps are to be followed to determine valid moods of four figures of syllogism. Step 1: A syllogism must have three terms. If not, it will commit a fallacy, either fallacy of four terms or fallacy of equivocation (i.e. ambiguous major, ambiguous minor, and ambiguous middle). Hence, the syllogism is not treated as a valid syllogism. Step 2: The middle term must distribute at least once in the premises of a syllogism. If it does not distribute in one of the premises, the syllogism encounters a fallacy known as the fallacy of undistributed middle. Hence, the syllogism is not regarded as a valid syllogism. Step 3: If a syllogism contains two negative premises, the syllogism encounters a fallacy termed as the fallacy of exclusive premises. In this case, the syllogism is regarded as an invalid syllogism. Step 4: If one of the premises is negative, a negative conclusion must be inferred to make the syllogism valid. Step 5: If two premises are particular propositions, no valid conclusion can be inferred in a syllogism. Step 6: If the major premise is a particular proposition and the minor premise is a negative proposition, we cannot draw a valid conclusion from these two premises. In this case, the syllogism is judged as an invalid syllogism. Step 7: If both the premises of a syllogism are universal propositions, we must infer a universal proposition in conclusion to make the syllogism valid. Step 8: If a term is distributed in conclusion and not distributed in the respective premise, it violates rule-3. In this case, we commit a fallacy, either fallacy of illicit major or fallacy of illicit minor. Hence, the syllogism is not regarded as a valid syllogism.

8.5 Valid Moods of the First Figure The first figure of syllogism is identified based on its middle term position on the premises. In the first figure, the middle term is placed in the subject part of the major

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premise and predicate part of the minor premise. By referring to the first figure of syllogism, we will test the non-rejected eight possible moods (refer to Table 8.2) to find out their validities in a sequential manner. This exercise aims to find out valid moods of the first figure of syllogism. Testing the mood AA A: All M is P. (Major premise) A: All S is M. (Minor premise) A: All S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. Both major premise and minor premise are affirmative; therefore, the conclusion must be affirmative. The premises are universal propositions, and thereby the conclusion must be a universal proposition. The categorical proposition A is the universal and affirmative proposition. Hence, the conclusion is the A proposition. The minor term is distributed in the conclusion, and it is distributed in the minor premise as well. Thus, we can infer A conclusion from AA mood. The mood AAA is, therefore, a valid syllogism, and it is named BARBARA. Testing the mood AE A: All M is P. (Major premise) E: No S is M. (Minor premise) E: No S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the major premise and the minor premise. Since the minor premise is negative, the conclusion must be negative. The premises are universal propositions, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal and negative proposition. Hence, the conclusion is the E proposition. In conclusion, both the major term and minor term are distributed. The minor term is distributed in the minor premise, but the major term is not distributed in the major premise. Thus, syllogism encounters a fallacy known as the fallacy of illicit major. Hence, we cannot infer E conclusion from the AE mood. The mood AEE is, therefore, not a valid syllogism. Testing the mood AI A: All M is P. (Major premise) I: Some S is M. (Minor premise) I: Some S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. Both major premise and minor premise are affirmative; therefore, the conclusion must be affirmative. The minor premise is particular, and thereby the conclusion must be particular. The categorical proposition I is a particular and affirmative proposition. Hence, the conclusion is the I proposition. Since no term is distributed in the conclusion, we need not search for whether major term and minor

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term are distributed in their respective premises or not. Thus, we can infer I conclusion from the AI mood. The mood AII is, therefore, a valid syllogism, and it is named DARII. Testing the mood AO A: All M is P. (Major premise) O: Some S is M. (Minor premise) O: Some S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the major premise. Since the minor premise is negative, the conclusion must be negative. The minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular and negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed, but it is not distributed in the major premise. Thus, this syllogism encounters a fallacy known as the fallacy of illicit major. Hence, it is wrong to infer O conclusion from the AO mood. The mood AOO is, therefore, not a valid syllogism. Testing the mood EA E: No M is P. (Major premise) A: All S is M. (Minor premise) E: No S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The major premise is a negative proposition; therefore, the conclusion must be a negative proposition. The major premise and the minor premise are universal propositions, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal and negative proposition. Hence, the conclusion is the E proposition. The minor term and major term of the conclusion are distributed, and they are distributed in the respective premises as well. Thus, we can draw E conclusion from the EA mood. The mood EAE is, therefore, a valid syllogism, and it is named CELARENT. Testing the mood EI E: No M is P. (Major premise) I: Some S is M. (Minor premise) O: Some S is not P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The major premise is negative; therefore, the conclusion must be negative. The minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is a particular and negative proposition. Hence, the conclusion is the O proposition. The major term is distributed in the conclusion, and it is distributed in the major premise as well. Thus, we can infer O conclusion from the EI mood. The mood EIO is, therefore, a valid syllogism, and it is named FERIO.

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Testing the mood IA I: Some M is P. (Major premise) A: All S is M. (Minor premise) No valid conclusion follows. In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the IA mood. This mood encounters a fallacy known as the fallacy of undistributed middle. Thus, IA mood is not a valid mood concerning the first figure of syllogism. Testing the mood OA O: Some M is not P. (Major premise) A: All S is M. (Minor premise) No valid conclusion follows. In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the OA mood. This mood encounters a fallacy termed as the fallacy of undistributed middle. Thus, OA mood is not a valid mood concerning the first figure of syllogism.

8.6 Valid Moods of the Second Figure The second figure of syllogism is identified based on its middle term position on the premises. In the second figure, the middle term is placed in the predicate part of both the major premise and the minor premise. By referring to the second figure of syllogism, we will test the non-rejected eight possible moods (refer to Table 8.2) to find out their validities. This exercise aims to find out valid moods of the second figure of syllogism. Testing the mood AA A: All P is M. (Major premise) A: All S is M. (Minor premise) No valid conclusion follows. In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the AA mood. The mood encounters a fallacy called the fallacy of the undistributed middle. Thus, AA mood is not a valid mood of the second figure of syllogism. Testing the mood AE A: All P is M. (Major premise) E: Some S is M. (Minor premise) E: No S is P. (Conclusion)

8.6 Valid Moods of the Second Figure

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This syllogism has three terms. The middle term is distributed in the minor premise. The minor premise is negative; therefore, the conclusion must be negative. The major premise and the minor premise are universal, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal and negative proposition. Hence, the conclusion is the E proposition. In conclusion, major term and minor term are distributed and they are getting distributed in their respective premises. Thus, we can infer E conclusion from the AE mood. The mood AEE is, therefore, a valid syllogism, and it is called CAMESTRES. Testing the mood AI A: All P is M. (Major premise) I: Some S is M. (Minor premise) No valid conclusion follows. In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the AI mood. This mood encounters a fallacy known as the fallacy of undistributed middle. Thus, AI mood is not a valid mood of the second figure of syllogism. Testing the mood AO A: All P is M. (Major premise) O: Some S is M. (Minor premise) O: Some S is not P. (Conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. The minor premise is negative; therefore, the conclusion must be negative. The minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is a particular and negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed and it is distributed in the respective premise. Thus, we can infer O conclusion from the AO mood. The mood AOO is, therefore, a valid syllogism, and it is called BAROKO. Testing the mood EA E: No P is M. (Major premise) A: All S is M. (Minor premise) E: No S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The major premise is negative; therefore, the conclusion must be negative. The major premise and the minor premise are universal, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal and negative proposition. Hence, the conclusion is the E proposition. In conclusion, major

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and minor terms are distributed and they are getting distributed in their respective premises as well. Thus, we can infer E conclusion from EA mood. The mood EAE is, therefore, a valid syllogism, and it is called CESARE. Testing the mood EI E: No P is M. (Major premise) I: Some S is M. (Minor premise) O: Some S is not P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The major premise is negative; therefore, the conclusion must be negative. The minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular and negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed and it is distributed in the major premise as well. Thus, we can infer O proposition from the EI mood. The mood EIO, therefore, is a valid syllogism known as FESTINO. Testing the mood IA I: Some P is M. (Major premise) A: All S is M. (Minor premise) No valid conclusion follows. In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the IA mood. This mood encounters a fallacy known as the fallacy of the undistributed middle. Thus, IA mood is not a valid mood of the second figure of syllogism. Testing the mood OA O: Some P is not M. (Major premise) A: All S is M. (Minor premise) O: Some S is not P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the major premise. Since the major premise is negative, the conclusion must be negative. The major premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular and negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed, but it is not distributed in the major premise. Thus, this syllogism encounters a fallacy known as the fallacy of illicit major. Therefore, we cannot infer O conclusion from OA mood. The mood OAO is not a valid syllogism.

8.7 Valid Moods of the Third Figure

113

8.7 Valid Moods of the Third Figure The third figure of syllogism is identified based on its middle term position on the premises. In the third figure, the middle term is placed on the subject side of both the major premise and the minor premise. By referring to the third figure of syllogism, we will test the non-rejected eight possible moods (refer to Table 8.2) to find out their validities. This exercise aims to find out valid moods of the third figure of syllogism. Testing the mood AA A: All M is P. (Major premise) A: All M is S. (Minor premise) A: All S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the major premise and the minor premise. Both the premises are affirmative; therefore, the conclusion must be affirmative. Further, both the premises are universal, and thereby the conclusion must be a universal proposition. The categorical proposition A is the universal and affirmative proposition. Hence, the conclusion is the A proposition. But in conclusion, minor term is distributed and it is not distributed in the minor premise. Hence, the syllogism encounters a fallacy known as the fallacy of illicit minor. Thus, we cannot infer A conclusion from AA mood. The mood AAA is, therefore, not a valid syllogism. Testing the mood AE A: All M is P. (Major premise) E: No M is S. (Minor premise) E: No S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the major premise and in the minor premise. The minor premise is negative; therefore, the conclusion must be negative. Both the premises are universal, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal and negative proposition. Hence, the conclusion is the E proposition. In conclusion, the major and minor terms are distributed. The minor term is also distributed in the minor premise, but the major term is not distributed in the major premise. Thus, this syllogism commits a fallacy known as the fallacy of illicit major. Therefore, we cannot infer E conclusion from the AE mood. The mood AEE is treated as an invalid syllogism. Testing the mood AI A: All M is P. (Major premise) I: Some M is S. (Minor premise) I: Some S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. Both the premises are affirmative; therefore, the conclusion must be affirmative. The minor premise is particular, and thereby the conclusion must be a particular

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proposition. The categorical proposition I is the particular and affirmative proposition. Hence, the conclusion is the I proposition. In conclusion, both the terms are not distributed; hence, we need not search for whether they are distributed in their respective premises or not. Thus, we can infer I conclusion from the AI mood. The mood AII is, therefore, a valid syllogism, and it is known as DATISI. Testing the mood AO A: All M is P. (Major premise) O: Some M is not S. (Minor premise) O: Some S is not P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The minor premise is negative; therefore, the conclusion must be negative. Further, the minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular and negative proposition. Hence, the conclusion is the O proposition. But in conclusion, the major term is distributed, but it is not distributed in the major premise. Thus, this syllogism encounters a fallacy known as the fallacy of illicit major. Therefore, we cannot infer O conclusion from AO mood. The mood AOO is not regarded as a valid syllogism. Testing the mood EA E: No M is P. (Major premise) A: All M is S. (Minor premise) E: No S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in both the premises. The major premise is negative; therefore, the conclusion must be negative. Further, both the premises are universal, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal and negative proposition. Hence, the conclusion is the E proposition. In conclusion, both the major term and the minor term are distributed. The major term is distributed in the major premise, but the minor term remains undistributed in the minor premise. Hence, the syllogism commits a fallacy known as the fallacy of illicit minor. Thus, we cannot infer E conclusion from the EA mood. The mood EAE is, therefore, not a valid syllogism of the third figure. Testing the mood EI E: No M is P. (Major premise) I: Some M is S. (Minor premise) O: Some S is not P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The major premise is negative; therefore, the conclusion must be negative. The minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular and negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is

8.7 Valid Moods of the Third Figure

115

distributed, and it is distributed in the respective premise as well. Thus, we can infer O conclusion from the EI mood. The mood EIO is, therefore, a valid syllogism, and it is known as FERISON. Testing the mood IA I: Some M is P. (Major premise) A: All M is S. (Minor premise) I: Some S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. Both the premises are affirmative; therefore, the conclusion must be affirmative. The major premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition I is the particular and affirmative proposition. Hence, the conclusion is the I proposition. In conclusion, both terms are not distributed; hence, we need not search for whether they are getting distributed in their respective premises or not. Thus, we can infer I conclusion from IA mood. The mood IAI is, therefore, a valid syllogism, and it is called DISAMIS. Testing the mood OA O: Some M is not P. (Major premise) A: All M is S. (Minor premise) O: Some S is not P. (Conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. The major premise is a negative proposition; therefore, the conclusion must be a negative proposition. The major premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular and negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed, and it is distributed in the major premise. Thus, we can infer O conclusion from the OA mood. The mood OAO is, therefore, a valid syllogism, and it is known as BOKARDO.

8.8 Valid Moods of the Fourth Figure The fourth figure of syllogism is identified based on its middle term position on the premises. In the fourth figure, the middle term is placed in the predicate part of the major premise and the subject part of the minor premise. By referring to the fourth figure of syllogism, we will test the non-rejected eight possible moods (refer to Table 8.2) to find out their validities. The objective of this exercise is to find out valid moods of the fourth figure of syllogism.

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Testing the mood AA A: All P is M. (Major premise) A: All M is S. (Minor premise) A: All S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. The major premise and minor premise are affirmative; therefore, the conclusion must be affirmative. Further, both the premises are universal, and thereby the conclusion must be a universal proposition. The categorical proposition A is the universal affirmative proposition. Hence, the conclusion is the A proposition. In conclusion, only the minor term is distributed, but it is not distributed in the minor premise. Thus, the syllogism encounters a fallacy known as the fallacy of illicit minor. Thus, we cannot infer A conclusion from AA mood. The mood AAA is, therefore, not a valid syllogism of the fourth figure. Testing the mood AE A: All P is M. (Major premise). E: No M is S. (Minor premise) E: No S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. The minor premise is negative; therefore, the conclusion must be negative. The major premise and the minor premise are universal, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal negative proposition. Hence, the conclusion is the E proposition. In conclusion, both major term and minor term are distributed, and they are distributed in their respective premises as well. Thus, we can infer E conclusion from the AE mood. The mood AEE is, therefore, a valid syllogism and is called CAMENES. Testing the mood AI A: All P is M. (Major premise) I: Some M is S. (Minor premise) No valid conclusion follows. In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the AI mood. This mood commits a fallacy known as the fallacy of undistributed middle. Thus, AI mood is not regarded as a valid mood of the fourth figure of syllogism. Testing the mood AO A: All P is M. (Major premise) O: Some M is not S. (Minor premise) No valid conclusion follows.

8.8 Valid Moods of the Fourth Figure

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In this mood, the middle term is not distributed in any of the premises. Hence, we cannot draw a conclusion from the AO mood. This mood encounters a fallacy known as the fallacy of undistributed middle. Thus, AO mood is not treated as a valid mood of the fourth figure of syllogism. Testing the mood EA E: No P is M. (Major premise) A: All M is S. (Minor premise) E: No S is P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in both the premises. The major premise is negative; therefore, the conclusion must be negative. The major premise and the minor premise are universal, and thereby the conclusion must be a universal proposition. The categorical proposition E is the universal negative proposition. Hence, the conclusion is the E proposition. In conclusion, both the major term and the minor term are distributed. The major term is distributed in the major premise, but the minor term is not distributed in the minor premise. Thus, the syllogism commits a fallacy known as the fallacy of illicit minor. Therefore, we cannot infer E conclusion from the EA mood. The mood EAE is not a valid syllogism of the fourth figure. Testing the mood EI E: No P is M. (Major premise) I: Some M is S. (Minor premise) O: Some S is not P. (Conclusion) This syllogism has three terms. The middle term is distributed in the major premise. The major premise is negative; therefore, the conclusion must be negative. The minor premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed and it is distributed in the respective premise as well. Thus, we can infer O conclusion from the EI mood. The mood EIO, therefore, is a valid syllogism known as FRESISON. Testing the mood IA I: Some P is M. (Major premise) A: All M is S. (Minor premise) I: Some S is P. (Conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. The major premise and minor premise are affirmative; therefore, the conclusion must be affirmative. The major premise is particular, and thereby the conclusion

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must be a particular proposition. The categorical proposition I is the particular affirmative proposition. Hence, the conclusion is the I proposition. In conclusion, neither the major term nor minor is distributed, and therefore we need not search for whether they are getting distributed in their respective premises or not. Thus, we can infer I conclusion from the IA mood. The mood IAI is, therefore, a valid syllogism, and it is named DIMARIS. Testing the mood OA O: Some P is not M. (Major premise) A: All M is S. (Minor premise) O: Some S is not P. (Not a valid conclusion) This syllogism has three terms. The middle term is distributed in the minor premise. The major premise is negative; therefore, the conclusion must be negative. Further, the major premise is particular, and thereby the conclusion must be a particular proposition. The categorical proposition O is the particular negative proposition. Hence, the conclusion is the O proposition. In conclusion, the major term is distributed, but it is not distributed in the major premise. Hence, this syllogism encounters a fallacy known as the fallacy of illicit major. Thus, we cannot infer O conclusion from OA mood. The mood OAO is, therefore, not a valid syllogism of the fourth figure. Let us summarise the valid moods of four figures of syllogism in Table 8.3. Table 8.3 Valid moods of four figures of syllogism

First figure

Second figure

Third figure

Fourth figure

Valid mood

Name of the valid syllogism

AAA

BARBARA

AII

DARII

EAE

CELARENT

EIO

FERIO

Valid mood

Name of the valid syllogism

AEE

CAMESTRES

AOO

BAROKO

EAE

CESARE

EIO

FESTINO

Valid mood

Name of the valid syllogism

AII

DATISI

IAI

DISAMIS

EIO

FERISON

OAO

BOKARDO

Valid mood

Name of the valid syllogism

AEE

CAMENES

EIO

FRESISON

IAI

DIMARIS

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There are fifteen valid moods found corresponding to four figures of syllogism, and each valid mood has resulted in a valid syllogism. We argued that if a syllogism violates a syllogistic rule then a valid conclusion cannot be drawn in the syllogism. Instead, the syllogism will commit one or other fallacy, such as the fallacy of four terms, illicit major, illicit minor, and the fallacy of the undistributed middle. We discussed these fallacies relating to the violation of syllogistic rules. These fallacies are regarded as formal fallacies, as syllogism deals with formal logic. Besides formal fallacies, material fallacies are also found in syllogism. Let us discuss ‘material fallacy’ and find out the commonalities and differences between formal fallacy and material fallacy.

8.9 Formal Fallacy Versus Material Fallacy A formal fallacy is dealt with the figure, mood, and rules of syllogism, whereas a material fallacy is concerned with the content of the propositions of a syllogism. If constituent propositions of a syllogism are judged as either true or false, the syllogism is treated as a material syllogism. The truth and falsity of a proposition depend on the existence of the content in the empirical world that the proposition signifies. If the content of a proposition is found in the empirical world, the proposition is judged as true, and if the content of a proposition is not found in the empirical world, the proposition is judged as false. A material syllogism concerns with the content of its constituent propositions. If the content of a proposition is judged as false, the proposition is regarded as materially false. So, the validity of a material syllogism lies in the contents of its constituent propositions. But in the case of a formal syllogism, constituent propositions are judged as either correct or incorrect. A proposition is treated as correct or incorrect based on the structure of the proposition. A formal syllogism is construed based on the figure and mood of syllogism that reveals the structure of the syllogism. So, the validity of a formal syllogism is based on the structure of the propositions and conformation to the syllogistic rules, but not on the contents of its constituent propositions. In brief, a material syllogism considers both structure and content of its constituent propositions, whereas formal syllogism considers the only structure of its constituent propositions to find out the validity of the syllogism. In the case of a material syllogism, if a premise is false, the syllogism is treated as an invalid or unsound syllogism, even though the structure of the syllogism is correct and it satisfies all the syllogistic rules. For example: A: All mosquitoes are singing a devotional song. (Major premise) I: Some mosquitoes are treated as dancing insects. (Minor premise) I: Therefore, some dancing insects are singing a devotional song. (Conclusion) This argument corresponds to the third figure of syllogism based on the position of its middle term. Upholding its structure, it is identified as a valid mood AII (i.e. DATISI) of the third figure of syllogism. This syllogism satisfies all the syllogistic rules. Thus, it is a valid argument. Concerning formal logic, this syllogism is valid, but it is not materially a valid syllogism. The reason is the contents of the major

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premise and minor premise do not correspond to the facts of the empirical world. Hence, the conclusion derived from these two premises is not treated as a valid conclusion. The major premise and minor premise of the syllogism are regarded as formally true propositions but not materially true propositions. They are formally true because they correspond to the standard logical form of categorical propositions, that is, A proposition and I proposition, respectively. Thus, it may be stated that a formal valid syllogism is not necessarily a material valid syllogism. But a material valid syllogism is necessarily regarded as a formal valid syllogism. A formal valid syllogism may not consist of materially true propositions, but a material valid syllogism must consist of materially true propositions. Thus, it is argued that a syllogism commits a material fallacy maybe regarded as a formal valid argument. In precise, material fallacy expresses that a mistake encountered in a syllogism is due to the non-existence content in the empirical world that a constituent proposition is stating about. An example of a formal valid argument with a material fallacy is as follows. A: All men are capable of seeing objects in the world. (Major premise) I: Some blind persons are men. (Minor premise) I: Some blind persons are capable of seeing objects in the world. (Conclusion) This argument refers to the first figure of syllogism based on its middle term position in the argument. It has three terms: the major term, the minor term, and the middle term. The middle term is distributed in the major premise. This argument satisfies all the syllogistic rules. It does not commit a formal fallacy. Hence, it is a valid syllogism named DARII. But if we consider the content of the major premise of the syllogism, we find that it is a materially false proposition. So, the conclusion derived from two premises is not valid. Thus, it is not to be regarded as a valid argument, as premises do not support the conclusion. So, this argument is formally valid but materially invalid, as it is materially fallacious.

8.10 Informal Fallacies In our daily lives, we commit many fallacies that are not directly related to the violation of formal logic syllogistic rules. These are called informal fallacies, and they are of two types. a. b.

Non-verbal fallacies of matter Verbal fallacies

The non-verbal fallacies of matter are known as ‘fallacies of relevance’. These fallacies arise due to the inappropriate and illogical ‘content’ (matter) of the argument. In this case, the premises of an argument become irrelevant to its conclusion. The premises appear to be psychologically relevant but not logically relevant to its conclusion. On examination of the premises’ relevance to the conclusion, we find that the argument is inadequate and incomplete.

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There are quite a few non-verbal fallacies of matter found in logical discourse. The reason is the premises of arguments can be judged irrelevant to the conclusions in many ways. We will discuss some of the significant non-verbal fallacies of matter for our present purpose and reference. These are: (a) (b) (c) (d) (e)

Argumentum ad hominem Complex question Argumentum ad populum (the appeal to emotion) Argumentum ad ignorantiam (the argument from ignorance) Hasty generalisation

Argumentum Ad Hominem The fallacy ‘argumentum ad hominem’ is committed when we consider the conveyer of the argument and on what circumstance the argument was stated. This fallacy source is traced back to the emotions, ideology, blind beliefs, personal attitudes, and political affiliation of a person. In this case, the premises’ merit is misplaced and sometimes not considered true because the conclusion of the argument is refuted, as it is stated by a person who has not a good reputation in society, little biased religious minds, and belongs to the opposite political party, etc. Consider an example. Mr. X makes an argument. Mr. Y examines and evaluates that argument and finds that Mr. X’s argument should not be treated as a sound argument on the following grounds. (i) (ii) (iii)

Mr. X has a lousy reputation in society. Mr. X belongs to a political party, and thereby his argument is biased in favouring the ideology of his political party. Mr. X conveyed his argument in a political debate platform (i.e. a given situation) which is questionable.

However, there is a possibility that Mr. X’s argument is sound and the reasons stated by Mr. Y are fallacious. Here, Mr. Y’s evaluation of Mr. X’s argument is based on his/her grounds and not on the strength of premises to the conclusion. This fallacy is noticed when two political opponents argue on a society’s policy-related issue or social problems. It is observed that when two opposite religious leaders argue on which spiritual practice is superior, they too commit the fallacy of argumentum ad hominem. Complex Question The fallacy ‘complex question’ is committed when a question is asked by considering a particular belief or an assumption as the basis for answering the question. For example, Mr. L asks Mr. P that ‘when did you stop smoking Ganja’? In this case, if Mr. P does not smoke Ganja, then he would be having difficulty dealing with this question. This question is regarded as complex because Mr. L beliefs and takes for granted that Mr. P smokes Ganja. It was observed that advocates in the court ask complex questions to a witness to confuse him/her, trap him/her, and make an implicit admission of his/her guilt before the honourable court. A complex question implicitly embraces many hidden

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questions. The best way to avoid the fallacy of complex questions is to refute all the presuppositions (beliefs and assumptions) concerning the complex question one after another. Argumentum Ad Populum (The Appeal to Emotion) The fallacy ‘argument ad populum’ arises when a person cleverly addresses a large gathering and tries to win over them by arousing their emotions. In some contexts, an issue is emotional, and on other occasions, ‘emotionally charged words’ are stated to win over the public. In this case, the premises are selectively chosen to use as a tool to manipulate public emotions. The speaker’s conclusion of the argument is accepted as true based on public emotions’ arousal and not on the logical grounds of the argument. For example, a politician defends himself/herself against money laundering accusations by stating his/her benevolent approach to the needy people, ethical life in the society, patriotism for the nation, fights for the public rights, etc. This fallacy is noticed in some of the advertisements also. For example, beauty cosmetic products are shown used by slim, fair, gorgeous, and smiling ladies. Some of the two-wheelers are related to thrilling of driving, couple romanticism, adventurism, etc. The objective of these advertisements is to manipulate and influence the public emotions. Although the public resists themselves from buying these products, seldom these advertisements tempt them to purchase these products. Argumentum Ad Ignorantiam (The Argument from Ignorance) In our daily lives, we have come across some facts and events that do not have evidence to support their existence and non-existence. Since no evidence is found to support their non-existence, their existence is taken for granted as true. Further, if no evidence supports their existence, their non-existence is taken for granted as true. The fallacy ‘argument from ignorance’ is committed when the lack of evidence of a fact or an event is used to justify an argument. The strength of an argument’s conclusion lies in our ignorance instead of knowing the fact or an event. This fallacy is explained in two ways.2 (i) (ii)

Arguing from the absence of disproof to the presence of proof Arguing from the absence of proof to the presence of disproof

Take an example: ‘Demon exists’ is true because no one has proved that ‘Demons do not exist’. Here, the argument about demons’ existence relies on an appeal to our ignorance instead of knowledge. Another example, a nastika (who does not believe in the existence of God) argues that since no one has proved with evidence that God (i.e. omnipotent and omniscient being) exists, we can claim that God does not exist. In this fallacy, we can establish the conclusion either in favour of or against the argument, as it lacks evidence for its support and justification. Consider a few more examples; when we ask a cosmetic shopkeeper about the quality of a product, he/she gives a standard answer, that is, ‘as of today, there is no 2 Schipper,

E.W., and Edward, S. (1960). A first course in modern logic. London: Routledge and Kegan Paul Publication.

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complaint received about the quality of the product’, and this answer becomes proof for accepting the quality of the cosmetic product. Another example is Mr. Y asks a carpenter about the durability and quality of a dining table. He replies that ‘for the last five years, no complaint has been made on this product’. This answer becomes proof for Mr. Y to believe and accept that the dining table is of good quality. Hasty Generalisation The fallacy ‘hasty generalisation’ expresses that it is a mistake in our judgement to infer a general conclusion that states about all the instances of a kind based on one or a few instances of that kind. To put it simply, it is fallacious to consider one or a few instances of a state of affairs and make a conclusion of all the instances of that state of affairs. In this case, the conclusion would not be logical and convincing to ascertain the truth and certainty of all the instances of the state of affairs. For example, Miss X has an unpleasant and disgusting experience with her boyfriend. She concludes that all boys are irritating and disgusting. In this case, it is fallacious to consider that all boys are annoying and awful, and further, they render unpleasant behaviour to their girlfriends. Another example is an auto-rickshaw driver with a Tamil Nadu licence plate cuts in a traffic violation. Here, a passenger concludes that all Tamil Nadu auto-rickshaw drivers are terrible drivers. In this case, the conclusion is an overgeneralisation of Tamil Nadu auto-rickshaw drivers. There might be the case that many auto-rickshaw drivers in Tamil Nadu obey traffic rules and drive carefully and skilfully. Consider another example: Mr. Y learned that his friend has consulted an Ayurvedic doctor for his skin disease and got cured. By considering this evidence alone, Mr. Y concludes that Ayurvedic doctors can cure all sorts of skin diseases. In this case, it is fallacious to consider that Ayurvedic doctors cure all kinds of skin diseases. Now, we will discuss some of the relevant and significant verbal fallacies that come under the informal fallacy category. These are: (a) (b) (c)

Fallacy of composition Fallacy of division Fallacy of accent

Verbal fallacies are known as ‘fallacies of ambiguity’. These fallacies arise due to the incorrect use of words in a linguistic system. These verbal fallacies are also named ‘fallacies in speech’ and ‘fallacies in diction’. This is because an argument will be expressed through language, and there can be multiple errors in expressing it. Fallacy of Composition The fallacy of composition arises when we argue what is true of a constituent that is necessarily true for an object, fact, and event. Logically speaking, in an argument, when a term in the premise is taken distributively, and in conclusion, it is treated collectively, we commit the fallacy of composition. For example, consider ‘Spain international football team’. Each player in the team may be an excellent player in his capacity and skills, but that does not mean the team as a whole is excellent. For

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a team to be excellent, it must possess certain qualities, such as organically united, the spirit of playing for the nation, and holding the team’s virtue; these may not be found in the team’s excellent individual players. Another example, it is fallacious to consider what is in the interest of individuals of a state must also be in the interest of the state. Fallacy of Division The fallacy of division is considered as an opposite to the fallacy of composition. The fallacy of division is committed when we argue what is predicated to the whole (i.e. an object, or a fact, or an event) can be predicated to its constituents. In this case, an argument is made from qualities of the whole to qualities of its constituent parts. To put it simply, what is true of the whole is wrongly considered as true of its parts. For example, it is a fallacy of division to argue that what is best for the state must necessarily benefit all individuals of the state. Fallacy of Accent The fallacy of accent is committed when the meaning of a proposition is altered by wrongly emphasising words or other elements, such as words in italics, bold letters, and underline the words. In this case, the accent of words in an argument makes the argument fallacious. That is, when a word should not have been emphasised but wrongly emphasised on an argument, the meaning of the argument changes, and thereby, the argument becomes fallacious. There are cases where the tone, accent, italics, underline, and bold letter words of a sentence change the meaning of the sentence entirely in an argument, and thereby the argument becomes fallacious. For example, we find the use of bold letters and large font size letters in the newspapers concerning selling a product to allure the readers’ attention to it. Most often, it is written, let us say, ‘up to 60% OFF’. Here, the expression ‘up to’ is written in an insignificant way. Since the emphasis is given on ‘60% OFF’, it is fallacious to understand that all the products have offer price ‘60% off’ of the shop. To avoid the fallacy of accent, one should read the words correctly or utter them properly in the context. Take another example; three sentences are written below as the same sentence. Still, due to the emphasis of certain words in each sentence, each sentence’s meaning is different from other sentences. Here, we commit the fallacy of accent. (i) (ii) (iii)

I do not want to talk to him. I do not want to talk to him. I do not want to talk to him.

The meaning of the first sentence is I have no interest in talking to him. The meaning of the second sentence is he is so irritating that he continuously talks nonsense without listening to anyone. The meaning of the third sentence is he cannot understand what I will tell him. Hence, it would be insignificant to talk to him. The above-mentioned informal fallacies are not exhaustive but essential and relevant for learners’ understanding and discernment.

Chapter 9

Pure and Mixed Syllogism

In the previous chapter, we discussed the four figures and possible moods of syllogism. We explained the rules of syllogism, and by applying the rules we tested all the possible moods of syllogism. We mentioned the names of valid moods about the four figures of syllogism. We discussed similarities and differences between formal fallacy and material fallacy. We elucidated some of the informal fallacies. In continuation of the previous chapter, in this chapter, we will discuss the nature and kinds of pure syllogism, the difference between pure syllogism and mixed syllogism, and types of mixed syllogism. We will describe ‘sorites’ and ‘enthymemes’ with suitable examples. Further, we will explain the ‘dilemma’ and forms of the dilemma with examples. The syllogism is of two kinds: pure syllogism and mixed syllogism. In the case of pure syllogism, all constituent propositions (major premise, minor premise, and conclusion) are of the same relation. For example, if all the propositions are categorical, the syllogism is regarded as a pure categorical syllogism. If all the propositions are hypothetical, the syllogism is regarded as a pure hypothetical syllogism. If all the propositions are disjunctive, the syllogism is regarded as a pure disjunctive syllogism. Thus, we have three kinds of pure syllogism: pure categorical syllogism, pure hypothetical syllogism, and pure disjunctive syllogism. Unlike pure syllogism, constituent propositions of a mixed syllogism are not of the same relation. The mixed syllogism is of three types: hypothetical categorical, disjunctive categorical, and dilemma. In the case of hypothetical categorical syllogism, the major premise is a hypothetical proposition; the minor premise and conclusion are categorical propositions. In the case of disjunctive categorical syllogism, the major premise is a disjunctive proposition; the minor premise and conclusion are categorical propositions. In the case of a dilemma, the major premise is a hypothetical compound proposition. The minor premise is a disjunctive proposition. The conclusion is either a categorical or disjunctive proposition. The drawing below portrays types of syllogism.

© Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_9

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Types of syllogism

9.1 Pure Categorical Syllogism A pure categorical syllogism accepts all the rules and norms of the formal syllogism, as discussed in the previous chapter. It is a mediate deductive inference. It consists of three categorical propositions, namely major premise, minor premise, and conclusion, and has three terms, namely major term, minor term, and middle term. An example of a pure categorical syllogism is placed below. A: All logic students are intelligent beings. (Major premise) I: Some cricketers are logic students. (Minor premise) I: Therefore, Some cricketers are intelligent beings. (Conclusion) This syllogism is valid because it corresponds to the first figure of syllogism and satisfies all the syllogistic rules. It is named DARII. In this syllogism, three terms are found, namely major term, minor term, and middle term. In the major premise, middle term (i.e. logic students) and major term (i.e. intelligent beings) agree with each other. In the minor premise, middle term and minor term (i.e. cricketers) agree with each other. In conclusion, both the major and minor terms agree with each other. Due to the agreement among three terms, the conclusion derived from two premises is judged as a true proposition in this syllogism. In a pure categorical syllogism, whatever is denied or affirmed of an entire class would be denied or affirmed of any part of it. In Aristotle’s words, it is known as Dictum de omni et nullo. This dictum applies to only the first figure of syllogism, as the first figure of syllogism is regarded as the perfect figure of syllogism (Patterson, 1993, p. 359). Except for the first figure, the rest of the figures of syllogism are not considered as perfect figures of syllogism. The reasons are, as stated by Aristotle, universal affirmative proposition ‘A’ is found in the conclusion of a valid syllogism (i.e. BARBARA) of the first figure alone, which does not find in other figures of valid syllogisms. Further, A, E, I, O propositions are found as a conclusion of valid moods

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of the first figure (i.e. AAA, EAE, AII, EIO), which do not find in other figures of syllogism.

9.2 Dictum de Omni et Nullo The Dictum de omni et nullo is translated into the English language as ‘the maxim of all and none’. Concerning Dictum de omni et nullo, Aristotle argues that in a pure categorical syllogism, for every proposition, what is predicated (or not predicated) of a group is predicated (or not predicated) of members contained in that.1 To put it simply, what is true of a certain class is true of members of that class, and what is not true of a certain class is not true of members of that class. For example, if white colour is true for all swans then white colour must be true for a few members of the ‘swan’ class. Further, if the yellow colour is not true for all crows, then yellow will not be true for a few members of the ‘crow’ class. An example of Dictum de Omni et Nullo No men are perfect beings. Gandhi is a man. Therefore, Gandhi is not a perfect being. In this argument, it is asserted that ‘men’ (i.e. a class) are not perfect beings. Gandhi is a man who belongs to the class of ‘men’, and thus Gandhi is not a perfect being. This argument states that whatever is not true for all men must not be true for Gandhi as an individual member belongs to the class of ‘men’. Another example of Dictum de omni et nullo Men are rational beings. Socrates is a man. Therefore, Socrates is a rational being. In this argument, it is asserted that ‘men’ (i.e. a class) are rational beings. Socrates is a man belonging to the class of ‘men’. Hence, Socrates is a rational being. This argument states that whatever is true for all men must be true for Socrates as an individual member belongs to the class of ‘men’.

9.3 Enthymemes In our daily lives, we formulate and share many arguments with the hearers in the form of mediate deductive inference (syllogism). While communicating the arguments to the hearers, we do not state all the propositions of the argument explicitly, as it is 1 Please

read Patterson, R. (1993). Aristotle’s perfect syllogisms, predictions, and the “Dictum de Omni”, Synthese, 96 (3), 359–378.

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believed that it will be redundant to state the well-known and trivial true propositions to the hearers of an argument. However, the speaker intends that the hearer will be able to insert the trivial true proposition in the argument appropriately, and the hearer will carry out this task by considering the intention of the speaker and the context of the argument. This is called the enthymeme. To put it simply, when an argument lacks one of its propositions, the unstated proposition (it may be a premise or a conclusion) is understood by the hearer based on the speaker’s intention and context of the argument, it is known as an enthymeme. This phenomenon of an argument arises when a speaker shares his/her partial or incomplete ideas with hearers and puts his/her views on something vehemently. The enthymemes are treated as rhetoric, and rhetoric is more powerful and vigorous when communicated through a language in comparison to an argument that is stated in detail with its constituent propositions. An example of an enthymeme All politicians are liars. Therefore, Shimi is a liar. This argument (syllogism) is incomplete because we have only two propositions: a premise and a conclusion. We need to justify the conclusion by adding a missing and tacit premise to the argument. Once we add the tacit premise to the argument, we can test the syllogism by applying all the syllogistic rules and find out whether the syllogism is regarded as a valid syllogism or not. The hearer easily adds the missing premise (tacit premise) to the argument from his/her knowledge and understanding about politicians and their public speeches. The complete argument (syllogism) would appear as: All politicians are liars. (Major premise) Shimi is a politician. (Minor premise) Therefore, Shimi is a liar. (Conclusion) The categorical proposition ‘Shimi is a politician’ is added to the syllogism as the minor premise to derive the conclusion. This is because, in a syllogism, the conclusion is derived from two premises by taking them together. This syllogism has A, I, I categorical propositions. It corresponds to the first figure of the syllogism based on the position of its middle term. It satisfies all the syllogistic rules. Hence, it is a valid syllogism of having AII valid mood named as DARII. Enthymemes are of three kinds: the first-order enthymeme, second-order enthymeme, and third-order enthymeme. These kinds of enthymemes are determined based on which unstated proposition is missing from the argument. In an argument, if the minor premise and the conclusion are mentioned, and the major premise is not stated, but it is implicitly understood, it is known as the first-order enthymeme. Similarly, in an argument, if the major premise and the conclusion are mentioned, and the minor premise is not stated, but it is implicitly understood, it is known as a second-order enthymeme. Further, in an argument, if the major and minor premises are mentioned, and the conclusion is not stated, but it is implicitly understood, it is called a third-order enthymeme.

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An example of the first-order enthymeme All native of India is a citizen of India. (The hearer adds this proposition later to the argument) Miku is a native of India. Therefore, Miku is a citizen of India. In this syllogism, the major premise was not stated by the speaker, but it is implicitly understood by the hearer as ‘All native of India is a citizen of India’. This syllogism corresponds to the first figure of syllogism based on its middle term position on the premises. It comprises of A, I, I categorical propositions. It does not violate any syllogistic rules. Hence, it is a valid syllogism of the first figure named DARII. An example of the second-order enthymeme All students are educated persons. All undergraduates are students. (The hearer adds this proposition later to the argument) Therefore, all undergraduates are educated persons. In this syllogism, the minor premise was not expressed by the speaker, but the hearer implicitly understood it by considering the intention of the speaker and context of the argument, as ‘All undergraduates are students’. This syllogism corresponds to the first figure of syllogism based on its middle term position on the premises. It comprises of A, A, A categorical propositions. It satisfies all the syllogistic rules. Hence, it is a valid syllogism of the first figure named BARBARA. An example of the third-order enthymeme No true Hindus are rigid religious-minded people. Some temple priests are rigid religious-minded people. Therefore, some temple priests are not true Hindus. (The hearer adds this proposition later to the argument) In this syllogism, the speaker did not state the conclusion, but the hearer understood it by considering the speaker’s intention and context of the argument, as ‘Some temple priests are not true Hindus’. This syllogism corresponds to the second figure of syllogism based on its middle term position on the premises. It comprises E, I, O categorical propositions. It satisfies all the syllogistic rules. Hence, it is a valid syllogism of the second figure named FESTINO. There are certain cases when a speaker shares two premises with the hearer, those are either negative categorical propositions (i.e. E and O) or particular categorical propositions (i.e. I and O), the hearer will not be able to derive any conclusion from these two premises. If the hearer attempts to derive the argument’s unstated conclusion, the argument would be regarded as an invalid syllogism, as it will not satisfy some of the syllogistic rules. Hence, from two negative categorical propositions and two particular categorical propositions, no conclusion (third-order enthymeme) can be inferred.

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9.4 Sorites A categorical syllogism consists of three propositions: two premises and one conclusion. The conclusion of a syllogism is drawn from two premises by considering them together. Unlike syllogism, a sorites requires more than three propositions to infer the conclusion. In short, a sorites consists of more than three propositions. The last proposition of the sorites is known as a conclusion, and the rest of the propositions have treated either premise(s) and/or conclusion(s) based on their positions in the argument. A sorites is not regarded as a syllogism, rather a chain of categorical syllogisms. In sorites, the whole argument is valid when the constituent syllogisms are treated as valid. The salient features of a sorites are as follows: it consists of more than three premises, and the last proposition is regarded as a conclusion. Each term in a constituent syllogism of the chain of categorical syllogisms appears two times only. Further, some of the premise(s) and/or conclusion(s) expressed enthymematically. The unstated premise and/or conclusion of the argument need to be added to the constituent syllogism of the chain of syllogisms to draw the desired conclusion. While testing the validity of a sorites, we need to consider the unstated or missing premise(s) and/or conclusion(s) of the whole argument. An example of sorites All birds are flying creatures. (Proposition-1) Some feathered species are birds. (Proposition-2) All feathered species are biped animals. In this example, three categorical propositions are given to us. Now, we need to find out whether these three categorical propositions are to be regarded as premise-1, premise-2, and conclusion or not. Upon examination, it is found that the last proposition is not linked to the other two propositions. Hence, it would not be considered as a conclusion of the argument. So, we need to add a proposition after the second proposition and before the last proposition. That is, ‘Some feathered species are flying creatures’. This proposition is inferred considering a syllogistic rule that states that each term must appear twice in a syllogism, and the middle term must establish a link between major term and minor term to draw a conclusion of an argument. So, the proposition ‘Some feathered species are flying creatures’ is treated as a third-order enthymeme. It is also treated as a conclusion when we consider proposition-1 and proposition-2 of the argument. After adding the proposition ‘Some feathered species are flying creatures’ to the argument, it is found that the last proposition is not to be considered as a conclusion of the sorites. The reasons are, we get a new term ‘biped animals’ in the predicate part of the proposition that does not appear twice in the argument. Further, in consideration of the propositions ‘Some feathered species are flying creatures’ and ‘All feathered species are biped animals’, we find a middle term (i.e. feathered species) that connects to ‘biped animals’ on the one hand, and ‘flying creatures’ on the other hand. As a result, it helps us to draw a conclusion, that is, ‘Some biped animals are flying creatures’. Now, if we consider the whole argument,

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we find the argument consists of five categorical propositions. The last proposition, ‘Some biped animals are flying creatures’, is regarded as a conclusion and the above propositions are regarded as premises. However, the proposition ‘Some feathered species are flying creatures’ is regarded as a premise as well as a conclusion. It is considered as a conclusion when we consider proposition-1 and proposition-2 of the argument. It is treated as a premise when we consider the proposition ‘All feathered species are biped animals’ is a premise, and ‘Some biped animals are flying creatures’ is a conclusion. In this case, the conclusion of the sorites is a third-order enthymeme, which was not stated explicitly in the given argument. This sorites consists of two syllogisms as follows: All birds are flying creatures. (Proposition-1) Some feathered species are birds. (Proposition-2) Some feathered species are flying creatures. (Proposition-3 and a conclusion) All feathered species are biped animals. (Proposition-4) Therefore, some biped animals are flying creatures. (Conclusion) Now, we need to test this sorites to find out whether it is valid sorites or not. Upon consideration of proposition-1, proposition-2, and proposition-3, we find that proposition-1 and proposition-2 are regarded as the major premise and minor premise, respectively, and proposition-3 is a valid conclusion. It is so because the conclusion is drawn from the two premises by taking them together without violating any syllogistic rules. The valid mood of the syllogism is AII, and it corresponds to the first figure of syllogism. This valid syllogism is named DARII. This syllogism is one of the constituents of the sorties. The reason is besides this syllogism, we find another syllogism when we consider proposition-3, proposition-4, and the conclusion (last proposition) of the sorites. In this syllogism, proposition-3 and proposition-4 are considered as the major premise and minor premise, respectively. The conclusion of the syllogism is inferred from proposition-3 and proposition-4 by taking them together. This syllogism corresponds to the third figure and satisfies all the syllogistic rules. The valid mood of the syllogism is IAI, and it is named DISAMIS. The proposition ‘Some biped animals are flying creatures’ is treated as the conclusion of the second syllogism and conclusion of the sorites as well. Another example of sorites All students are educated persons. All educated persons are swimmers. No quadruped animals are educated persons. Therefore, no quadruped animals are swimmers. In this example, four categorical propositions are given. Now, we need to find out whether the first three categorical propositions are to be regarded as premises and the fourth proposition is to be regarded as a conclusion of the sorites or not. Upon examination of these four propositions, it is found that the first proposition and second proposition are linked to each other, but they are not together linked to the third proposition. The reason is, in the third proposition, we find a new term ‘quadruped animals’ that does not find in either in the first proposition or in the second

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proposition. So, the third proposition ‘No quadruped animals are educated persons’ cannot be drawn from the first and second proposition by taking them together. Further, it is observed that the third proposition is linked to the fourth proposition (i.e. conclusion). So, a proposition is missing in the chain of syllogisms, which we need to identify. Once we add the unstated proposition to the chain of syllogisms, we will be able to test the sorites whether it is valid or not. If we consider the first and second propositions, we will draw the conclusion ‘All swimmers are students’. But this is not the third proposition as given in the argument. Further, this proposition is not linked to the given third proposition, ‘No quadruped animals are educated persons’. The reason is no middle term is found between ‘All swimmers are students’ and ‘No quadruped animals are educated persons’ propositions. Hence, we need to add a proposition, that is, ‘All swimmers are students’ before the first proposition of the given argument to draw the conclusion ‘All educated persons are swimmers’. The conclusion ‘All educated persons are swimmers’ is linked to the proposition ‘No quadruped animals are educated persons’ and thereby assists in drawing the conclusion ‘No quadruped animals are swimmers’. Here, ‘All swimmers are students’ is considered as the first-order enthymeme. The complete sorites is stated below. All swimmers are students. (Proposition-1) All students are educated persons. (Proposition-2) All educated persons are swimmers. (Proposition-3) No quadruped animals are educated persons. (Proposition-4) Therefore, no quadruped animals are swimmers. (Conclusion) This sorites is a chain of two syllogisms. The first syllogism consists of proposition-1, proposition-2, and proposition-3. The second syllogism consists of proposition-3, proposition-4, and the conclusion. The categorical proposition ‘All educated persons are swimmers’ is the conclusion of the first syllogism and the categorical proposition ‘No quadruped animals are swimmers’ is the conclusion of the second syllogism. Upon examination of this chain of syllogism, it is found that the first syllogism corresponds to the fourth figure of syllogism and it violates the syllogistic rules. It encounters a fallacy known as ‘illicit minor’. Since one of the constituents (first syllogism) of the sorites is not valid, the sorites is treated as invalid.

9.5 Pure Hypothetical Syllogism In a pure hypothetical syllogism, the constituent propositions must be hypothetical propositions. In other words, a syllogism is regarded as pure hypothetical when its major premise, minor premise, and conclusion are the hypothetical propositions. An example, If Miku studies logic then he will learn lessons. (Major premise) If he learns lessons then he will score a good grade. (Minor premise)

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Therefore, if Miku studies logic then he will score a good grade. (Conclusion) In this syllogism, all the constituent propositions are compound propositions because they have antecedent and consequent parts. Antecedent and consequent parts are combined with ‘if…then’ expression. In the major premise, ‘Miku studies logic’ is considered an antecedent, and ‘He will learn lessons’ is treated as a consequent. Both antecedent and consequent of a hypothetical proposition are treated as categorical propositions. With this logic, we can assert that a hypothetical proposition is a compound proposition of categorical propositions. The above-mentioned pure hypothetical syllogism can be symbolised as: If X then Y. (Major premise) If Y then Z. (Minor premise) Therefore, If X then Z. (Conclusion) In this syllogism, X, Y, and Z are used as propositional variables to represent categorical propositions. A pure hypothetical syllogism is judged as valid when it satisfies the following conditions. (i) (ii) (iii)

There must be one common categorical proposition that exists in the major premise and the minor premise. The common categorical proposition shall be antecedent of one premise and consequent of another premise. The conclusion must not have the common categorical proposition, but it must have an antecedent of one premise and a consequent of another premise.

Upon applying these three conditions to a pure hypothetical syllogism, we get two types of valid syllogism. These are: (a)

(b)

If X then Y. If Y then Z. Therefore, If X then Z. If Y then Z. If X then Y. Therefore, If X then Z.

Violation of any of these three conditions, the pure hypothetical syllogism would be judged as invalid. For example, (a)

(b)

(c)

If X then Y. If Y then Z. Therefore, If Z then X. If Y then Z. If X then Y. Therefore, If Z then X. If X then Y. If Z then Y. Therefore, If X then Z.

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If X then Y. If X then Z. Therefore, If Y then Z.

9.6 Pure Disjunctive Syllogism In a pure disjunctive syllogism, the major premise, the minor premise, and the conclusion are the disjunctive propositions. A disjunctive proposition consists of two categorical propositions combined with ‘either…or’ expression. For example, ‘Either Anu is an engineer or a dancer’. In this example, we find two categorical propositions, ‘Anu is an engineer’ and ‘Anu is a dancer’, and these propositions are combined. In the case of a disjunctive proposition, both the categorical propositions are affirmative propositions. It is so because we cannot formulate a disjunctive proposition by negating its two alternatives. Further, we cannot even negate one of its alternatives of a disjunctive proposition. If we attempt to negate one of the alternatives to a disjunctive proposition, we will commit an error, which is called the incorrect structure of the disjunctive proposition. For example, ‘Neither Sinu is a singer nor a dancer’. In this proposition, we cannot affirm any of its alternatives, as it rejects both of its alternatives. Hence, this proposition is not considered as a disjunctive proposition. Take another example, ‘Either Sinu is a swimmer or he is not a dancer’. In this proposition, two alternatives are not linked to each other. Hence, this proposition is also not regarded as a disjunctive proposition in the logical discourse. Since a disjunctive proposition embraces two affirmative propositions, the syllogistic rule about quality does not apply to the pure disjunctive syllogism. The pure disjunctive syllogisms are rarely found in logic and logical discourse. Hence, modern logicians do not give much importance to the pure disjunctive syllogism. For modern logicians, syllogism means only categorical syllogism, not any kind of argument that has merely two premises and a conclusion of many varieties. But for traditional logicians, a syllogism must consist of two premises and one conclusion irrespective of the structure of constituent propositions of the syllogism. An example of a pure disjunctive syllogism is placed below. Either Sita will marry John or she will marry Jack. (Major premise) Either Sita will marry Jack or she will marry Miku. (Minor premise) Therefore, Either Sita will marry Miku or she will marry John. (Conclusion) This syllogism can be represented symbolically as, Either X or Y. (Major premise) Either Y or Z. (Minor premise) Therefore, Either Z or X. (Conclusion) Please note, X, Y, and Z are the propositional variables, and they represent categorical propositions.

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A pure disjunctive syllogism is judged as valid when it satisfies the following conditions. (i) (ii) (iii)

There must be a common categorical proposition for the major premise and the minor premise. The common categorical proposition shall be one of the alternatives to the major premise and the minor premise. The conclusion must not have the common categorical proposition, but it should have an alternative proposition of the major premise and the minor premise.

Upon applying these three conditions to the pure disjunctive syllogism, we get two valid syllogisms that are mentioned below symbolically. (a)

(b)

Either X or Y. Either Y or Z. Therefore, Either Z or X. Either X or Y. Either Y or Z. Therefore, Either X or Z.

9.7 Hypothetical Categorical Syllogism A hypothetical categorical syllogism is a mixed syllogism in which the major premise is a hypothetical proposition. The minor premise and the conclusion are the categorical propositions. The antecedent of the major premise is the minor premise, and the consequent of the major premise is the conclusion of the syllogism. An example, If Smita comes to the piano class then she will learn the lessons. (Major premise) Smita comes to the piano class. (Minor premise) Therefore, She will learn the lessons. (Conclusion) This syllogism is symbolically represented as, If X then Y. X. Therefore, Y. Please note, the alphabet X and Y are the propositional variables, and they represent categorical propositions in the above-mentioned argument. A hypothetical categorical syllogism has two valid forms: modus ponens and modus tollens. Modus ponens is known as constructive syllogism, whereas modus tollens is known as a negative syllogism. Modus ponens is regarded as an affirmative mood. The reason is the word ponens is derived from the Latin word ponere, which means ‘to affirm’. Modus ponens has the following rules to validate an argument. (i)

Affirm the antecedent of the major premise in the minor premise, and

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Affirm the consequent of the major premise in conclusion.

Upon satisfying these two rules, we can have the following valid modus ponens, represented symbolically. (a)

If X then Y. X. Therefore, Y. If X then not Y. X. Therefore, not Y. If not X then Y. Not X. Therefore, Y. If not X then not Y. Not X. Therefore, not Y.

(b)

(c)

(d)

Violation of the modus ponens rules, we commit the fallacy known as ‘denying the antecedent’. For example, If X then Y. (Major premise) Not X. (Minor premise) Therefore, not Y. (Conclusion) This syllogism is fallacious because we deny the antecedent of the major premise in the minor premise and deny the consequent of the major premise in conclusion. If we reduce this syllogism to the pure categorical syllogism, we find the below syllogism. A: All cases of X are cases of Y. O: This is not a case of X. O: Therefore, this is not a case of Y. This syllogism is not valid because the major term is distributed in conclusion, but it is not distributed in the major premise. Hence, the syllogism commits a fallacy known as ‘illicit major’. Modus tollens is also a valid form of a hypothetical categorical syllogism. It is known as a negative syllogism, as it possesses a negative mood. The word tollens is derived from the Latin word tollere, which means ‘to negate or to deny’. Modus tollens has the following rules to validate an argument. (i) (ii)

Deny the consequent of the major premise in the minor premise, and Deny the antecedent of the major premise in conclusion.

Upon satisfying these rules, we can have the following valid modus tollens, as presented below symbolically.

9.7 Hypothetical Categorical Syllogism

(a)

(b)

(c)

(d)

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If X then Y. Not Y. Therefore, not X. If X then not Y. Y. Therefore, not X. If not X then Y. Not Y. Therefore, X. If not X then not Y. Y. Therefore, X.

Violation of the modus tollens rules, we commit the fallacy known as ‘affirming the consequent’. An example, If X then Y. (Major premise) Y. (Minor premise) Therefore, X. (Conclusion) This syllogism is fallacious because we affirm the consequent of the major premise in the minor premise and affirm the antecedent of the major premise in conclusion. If we reduce this syllogism to the pure categorical syllogism, we find the below syllogism. A: All cases of X are cases of Y. I: This is a case of Y. I: Therefore, this is a case of X. This is not a valid syllogism because the middle term is not distributed in any one of the premises. Hence, the syllogism commits the fallacy known as ‘undistributed middle’.

9.8 Disjunctive Categorical Syllogism A syllogism is treated as disjunctive categorical syllogism when its major premise is a disjunctive proposition. Its minor premise and the conclusion are categorical propositions. In this syllogism, the minor premise denies one of the major premises’ alternatives, and the conclusion affirms the other alternative of the major premise. Since all the constituent propositions are not of the same relation, the syllogism is regarded as a mixed syllogism. An example of the disjunctive categorical syllogism is placed below. Either Riku is a swimmer or a singer. (Major premise) Riku is not a swimmer. (Minor premise) Therefore, Riku is a singer. (Conclusion)

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This syllogism is symbolically represented as, Either X or Y. (Major premise) Not X. (Minor premise) Therefore, Y. (Conclusion) Please note, X and Y are used as propositional variables and they represent categorical propositions. In this syllogism, concerning the major premise, the truth of one alternative does not necessarily imply the falsehood of other alternatives. Rather, in some situations, both the alternatives can be considered as true. So, to find out the validity of the syllogism, we need to apply the disjunctive categorical syllogism rules to it. The rule says, to deny one of the alternatives of the major premise in the minor premise is to affirm the other alternative of the major premise in conclusion. The following valid syllogisms satisfy the disjunctive categorical syllogism rules. (a)

(b)

Either X or Y. Not X. Therefore, Y. Either X or Y. Not Y. Therefore, X.

Violation of the disjunctive categorical syllogism rules would result in the following invalid syllogisms. (a)

(b)

Either X or Y. X. Therefore, not Y. Either X or Y. Y. Therefore, not X.

But modern logicians claim that when two alternative categorical propositions of the major premise are mutually exclusive like contradictory propositions, we can affirm one alternative of the major premise in the minor premise and negate the other alternative of the major premise in conclusion. By doing that, we will not violate the disjunctive categorical syllogism rules. For example, Either Mita is a female or a male. (Major premise) Mita is a female. (Minor premise) Therefore, Mita is not a male. (Conclusion) In this syllogism, the truth of one alternative necessarily implies the falsity of the other alternative of the major premise. Hence, two alternatives mentioned in the major premise are mutually exclusive. But, when two categorical propositions of a disjunctive major premise are not mutually exclusive, we have to apply the disjunctive categorical syllogism rules to check the validity of the argument.

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9.9 Dilemma A dilemma is when human beings find it difficult to opt for an option between two or many alternatives. For example, a person wants to reach a destination in ten minutes. In this case, if she waits for a government bus, she may not reach the destination on time because a few government buses do not run on time. If she travels in a shared auto-rickshaw, she may not reach the destination on time because shared autorickshaws stop now and then get more passengers in their routes. At this juncture, she is in a dilemma whether to opt for a government bus or a shared auto-rickshaw to reach the destination. Another example, most of the parents in India are in a dilemma about whether to send their wards for engineering or medicine study who have passed out twelfth standard science examination with a high percentage of marks. A dilemma is a mixed syllogism in which the major premise is a hypothetical compound proposition, the minor premise is a disjunctive proposition, and the conclusion is either a categorical or a disjunctive proposition. In the minor premise, we need to either affirm the antecedent or deny the consequent of the major premise. A symbolical example of dilemma, If X then Y and if L then Y. (Major premise) Either X or L. (Minor premise) Therefore, Y. (Conclusion) Here, X, Y, and L are used as propositional variables, and they represent categorical propositions. The dilemma arises in this syllogism because of its compound hypothetical major premise. The reason is in the minor premise if we wish to affirm the antecedent, which propositions we have to take into consideration as an antecedent of the major premise. Is it ‘If X then Y’ or ‘X and L’? Further, if we wish to deny the consequent of the major premise in the minor premise, which propositions are to be taken into consideration of the major premise? Is it ‘If L then Y’ or ‘Y’ alone? The dilemma persists in the conclusion as well. That is about whether the conclusion is to be affirmed or denied. If we wish to affirm the propositions in conclusion then which propositions are to be affirmed, and if we wish to deny the propositions in conclusion then which propositions are to be denied? Further, whether the conclusion is to be categorical or disjunctive? A dilemma will be judged as constructive or destructive is based on the quality of its minor premise. In case of a constructive dilemma, the disjunctive minor premise alternatively affirms the antecedent of the compound hypothetical major premise. In case of a destructive dilemma, the disjunctive minor premise alternatively denies the consequent of the compound hypothetical major premise. A dilemma that will be treated as simple or complex is based on the conclusion of the dilemma. If the conclusion is a categorical proposition, the dilemma is regarded as a simple dilemma. But if the conclusion is a disjunctive proposition, then the dilemma is regarded as a complex dilemma. In consideration of a simple dilemma, complex dilemma,

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constructive dilemma, and destructive dilemma, we have four forms of dilemmas. These are, (i) (ii) (iii) (iv)

Simple constructive dilemma Complex constructive dilemma Simple destructive dilemma Complex destructive dilemma.

9.10 Forms of Dilemma In the below dilemma examples, X, Y, L, and M are used as propositional variables, and they represent categorical propositions. Simple constructive dilemma The symbolic representation of a simple constructive dilemma is, If X then Y and if L then Y. (Major premise) Either X or L. (Minor premise) Therefore, Y. (Conclusion) A concrete example of a simple constructive dilemma is, If a woman is guided by the opinion of others, then she will be criticised and if a woman acts according to her judgments, then she will be criticised. A woman acts either according to the opinion of others or according to her judgments. In any case, she will be criticised. Complex constructive dilemma The symbolic representation of a complex constructive dilemma is, If X then Y and if L then M. (Major premise) Either X or L. (Minor premise) Therefore, Either Y or M. (Conclusion) A concrete example of a complex constructive dilemma is, If Smita will sing a song, she will be revered and if Rita will play badminton, she will be acknowledged. Either Smita will sing a song or Rita will play badminton. Therefore, Either Smita will be revered or Rita will be acknowledged. Simple destructive dilemma The symbolic representation of a simple destructive dilemma is, If X then Y and if X then M. (Major premise)

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Either not Y or not M. (Minor premise) Therefore, not X. (Conclusion) A concrete example of a simple destructive dilemma is, If the sky is clear then there is a sun and if the sky is clear then there is a moon. Either there is no sun or there is no moon. Therefore, the sky is not clear. Complex destructive dilemma The symbolic representation of a complex destructive dilemma is, If X then Y and if L then M. (Major premise) Either not Y or not M. (Minor premise) Therefore, Either not X or not L. (Conclusion) A concrete example of a complex destructive dilemma is, If a student is obedient then he will obey orders and if he is intelligent then he will understand the lessons. Either he does not obey orders or he does not understand the lessons. Therefore, either he is not obedient or he is not intelligent.

Part III

Symbolic Logic

Chapter 10

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In Part II, we discussed the Aristotelian notion of syllogism in detail. We also discussed the difference between pure syllogism and mixed syllogism, types of mixed syllogism, forms of dilemma, and kinds of immediate inference, while Part III is devoted to ‘symbolic logic’. This chapter being the first chapter of Part III, we will discuss the need and significance of symbolic logic in the logical discourse. We will enumerate logical connectives relating to propositional variables. Further, we will translate logical propositions into symbolic propositions. We will also elucidate the truth function of a propositional variable and determine the truth-value of symbolic propositions through the truth table method.

10.1 Birth of Symbolic Logic In the previous chapters, we mentioned that logic deals with arguments. Arguments contain propositions, and a proposition consists of a few words as well as terms. Since arguments are formulated in natural language (ordinary language), words and terms used in the arguments are often associated with vagueness and ambiguities, and thereby, they mislead (misrepresent/distort) the meaning of the propositions of the arguments. Due to the misleading meaning of the propositions, the validity and invalidity of an argument would not be determined correctly. For example, ‘Logic students are not bad’. In this proposition, the term ‘bad’ is vague because it lacks precision. It may refer to the behaviour of logic students or refer to students’ performances in logic courses, or it may further refer to logic students’ reasoning ability. Thus, to derive the conclusion of the argument and judge the validity and invalidity of the argument correctly and easily, logicians proposed ‘symbolic logic’ in the logic subject. It is noticed that arguments formulated through natural language are too big, as it consists of a few compound and complex propositions. An argument involved with

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complexity often entangles with ambiguities and confusion. For example, the difference between the square of the two numbers is equal to the product of the sum and the difference of the two numbers. This expression is involved with complexity as it is formulated through natural language. It creates confusion for the readers unless they are well aware of elementary algebra, mathematical equations, and their interpretations. But to avoid confusion and bring  clarity to this expression, one can translate it into the symbolical form as x 2 − y 2 = (x + y)(x − y). In this symbolical expression, x and y are considered as variables, and any number can be assigned to these variables. In this way, to arrest the lacunas and avoid the inadequacies of the natural language of an argument, logicians attempted to bring reform in the classical logic and develop ‘symbolic logic’. In symbolic logic, each proposition is communicated through a propositional variable. So, it is easier to find out the validity or invalidity of an argument. According to Copi (1995), the special symbols of symbolic logic permit us to exhibit with greater clarity the logical structures of arguments that may be obscured by formulation in natural language (ordinary language) (p. 6). Aristotle introduced the notion of ‘variable’ to the logic subject in the fourth century. His works are limited to the formulation of the logical proposition through subject–predicate terms, find out the correct structure of the proposition of an argument, determine validity and invalidity of an argument by applying the syllogistic rules, etc. These works are regarded as the foundation of logic. Thus, Aristotelian logic is termed the classical logic. In the seventeenth century, G. W. Von Leibniz (1646–1716) proposed the development of classical logic. He suggested that ‘a universal calculus of reasoning could be devised which would provide an automatic method of solution for all problems and that could be expressed in the universal language’ (Basson & Connor, 2011, p. 4). Following Leibniz, George Boole (1815–1864), an English mathematiciancum-philosopher; Augustus De Morgan (1806–1871), an Indian born mathematician and logician; Charles Peirce (1839–1914), an American Philosopher have elaborated the topic ‘symbolic logic’ in detail and mentioned it as an independent subject of study. In the twentieth century, the German mathematician-cum-philosopher Gottlob Frege (1848–1925), the Italian mathematician Guiseppe Peano (1858–1935), the English mathematician-cum-philosopher Alfred North Whitehead (1861–1947), the British philosopher Bertrand Russell (1872–1970), the American philosopher C. I. Lewis (1883–1964), the Polish-American logician Alfred Tarski (1901–1983), and a few others have contributed their works on the ‘symbolic logic’ subject. This subject is now rich in its scope and important in its application on argument formulation and its validation. According to Agler (2013), ‘Symbolic logic is a branch of logic that represents how we ought to reason by using a formal language consisting of abstract symbols’. In Whitehead’s (1911) view, ‘by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain’ (p. 61). According to C. I. Lewis (1883–1964), symbolic logic has three essential components. These are ideograms, propositional variables, and logical constants (connectives).

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10.2 Ideograms Versus Phonograms Ideograms are those signs that are used to convey ideas or concepts about worldly affairs. These signs are not conveyed through sounds. They are soundless signs. For example, greater than sign ‘>’, subtraction sign ‘−’, etc. The greater than sign expresses that a value is greater than another value. The subtraction sign states that minus a number from another number. These signs do not have ambiguities and vagueness in their usage. Irrespective of human beings’ religion, age, gender, and other differences, these signs convey the same meaning to all. Ideograms are used in symbolic logic because they convey meaning without any ambiguities. Their meanings do not change along with time. The use of ideograms in symbolic logic assists in determining the truth-value of the propositions and validity of the arguments. Phonograms are those signs that are used to express objects and concepts of worldly affairs through sound or words. In other words, phonograms are the sounds that correspond to either objects or concepts of the phenomenal world. In natural language, phonograms are conveyed through sounds. For example, ‘question mark’, ‘subtraction’, ‘addition’, etc., are uttered in words in the natural language (i.e. ordinary language). Since natural langue is not free from ambiguities and vagueness, phonograms are not useful for symbolic logic.

10.3 Propositional Variables and Logical Connectives In the arithmetic, the small English alphabets x, y, z, etc., are used as variables of a mathematical statement and an equation. These variables represent a value or a number. But in the symbolic logic, x, y, z, etc., are used as variables to represent propositions. Hence, in the symbolic logic discourse, x, y, z, etc., are treated as propositional variables. For example, p is a propositional variable that may represent ‘Sky is blue’ or ‘Grass is green’ or ‘Hari is a tall boy’ or some other proposition. In the case of a compound proposition, more than one propositional variable is used to represent the proposition. Let us say, ‘Ram is a swimmer and Gopal is a cricket player’. In this proposition, two propositions are combined with the word ‘and’. So, we can assign a unique propositional variable to each constituent proposition. That is, p stands for ‘Ram is a swimmer’, and q stands for ‘Gopal is a cricket player’. So, we can write ‘p and q’. A proposition is necessarily judged as either true or false in the logical discourse. It cannot be judged as true and false at a time. Further, it is not possible to judge a proposition as neither true nor false at a given time. Thus, truth and falsity of a proposition are regarded as values of the proposition. If a proposition is true, then its negation will be false, and if a proposition is false, then its negation is considered true. So invariably, a proposition has a truth-value. A propositional variable stands for a variety of propositions in different contexts. So, it does not have any fixed meaning. Thus, the mere consideration of propositional

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variables would not suffice to translate natural language propositions to symbolic propositions. We need logical constants (connectives) besides propositional variables to translate natural language propositions to symbolic propositions. Logical connectives combine the propositional variables and put them together in a bracket as a whole to convey their meaning. The meanings of the logical connectives do not change from time to time, from one argument to another argument, and from one context to another context. Their meanings remain fixed all the time. Logical Connectives There are six logical connectives used in symbolic logic. In addition to that, ‘negation’ is used as a logical constant in the logical discourse. The ‘negation’ is treated as a unary operator because it is meant to negate a propositional variable in the symbolic logic and a proposition in the natural language. So, it is not regarded as a logical connective but as a truth-functional operator. The six logical connectives require a minimum of two propositional variables for their connection, and thereby, they are regarded as binary operators. Unary operator (logical constant)

~

Binary operator (logical connectives)

∧, ∨, →, ≡, ↓, |

The logical constant and logical connectives are (i) (ii) (iii) (iv) (v) (vi) (vii)

Negation Conjunction Disjunction Implication Equivalence Dagger Stroke

The logical constant and logical connectives have unique symbols and distinct connotations. The details are presented below. Name of the logical constant and logical connectives

Logical symbols

Connotation of the symbols

Negation

~

Not

Conjunction

∧

And

Disjunction

∨

Either or

Implication

→

If then

Equivalence

≡

If and only if

Dagger

↓

Neither nor

Stroke

|

Not both

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In symbolic logic, a propositional variable represents a simple or an atomic proposition. Let us say, ‘p’ is a propositional variable that represents an atomic proposition (simple proposition) ‘Snow is white’. We write the proposition in the symbolical form as ‘p’. If the proposition is negated, then we write ‘Snow is not white’, and its symbolical form is ‘~p’, which is read as ‘not p’. It is to be noted here that the logical constant ‘negation’ is always associated with a propositional variable only. For example, p—Snow is white. ~p—Snow is not white. ~~p—It is not true that snow is not white. The symbol ‘~’ is communicated through many expressions of natural languages, such as, it is not true, it is false, and it is not the case. Consider an example; let us consider p, r, t are propositional variables. These variables are presented symbolically for the following propositions. ~p—It is not true that the sun rises in the west. ~r—It is false that crows are white. ~t—It is not the case that beautiful women are immortal. The logical connective ‘conjunction’ is linked to two simple propositions and thereby connected with two propositional variables at a time. It is symbolised as ‘∧’ and read as ‘and’. For example, ‘m’ stands for ‘Miku is a good boy’, and ‘n’ stands for ‘Mini is a singer’. These two simple propositions are combined with the logical connective ‘conjunction’, and it is written as ‘Miku is a good boy and Mini is a singer’. This compound proposition is symbolised as ‘m ∧ n’. Some more examples are mentioned below. m—Miku is a good boy. n—Mini is a singer. m ∧ n—Miku is a good boy and Mini is a singer. ~m ∧ n—Miku is not a good boy and Mini is a singer. m ∧ ~n—Miku is a good boy and Mini is not a singer. ~m ∧ ~n—Miku is not a good boy and Mini is not a singer. The logical symbol ‘∧’ can be used for the following expressions of natural language, but not limited to these alone, ‘both’, ‘although’, ‘but’, ‘however’, ‘moreover’, ‘yet’, etc. Take an example, consider p, r, n, m are the propositional variables. These variables are presented symbolically for the following propositions. p—Rita is a swimmer. r—Rita is a short girl. n—Rita is sincere. m—Gita is a swimmer. p ∧ r—Rita is a swimmer but she is a short girl.

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p ∧ ~n—Rita is a swimmer although she is not sincere. m ∧ p—Both Gita and Rita are swimmers. ~n ∧ p—Rita is not sincere yet she is a swimmer. The logical connective ‘disjunction’ is linked to two propositional variables, and thus, it is associated with a compound proposition. It is symbolised as ‘∨’. For example, x stands for ‘Snow is white’, and y stands for ‘Grass is green’. If these two simple propositions are combined with the logical connective ‘disjunction’, it would become a compound proposition. This compound proposition is written as ‘Either snow is white or grass is green’. It is symbolically presented as ‘x ∨ y’. Some examples are mentioned below. x—Snow is white. y—Grass is green. x ∨ y—Either snow is white or grass is green. ~x ∨ y—Either snow is not white or grass is green. x ∨ ~y—Either snow is white or grass is not green. ~x ∨ ~y—Either snow is not white or grass is not green. A logical proposition formulated with an ‘if-then’ expression is called a hypothetical or implicative or conditional proposition. A hypothetical proposition consists of two parts, namely antecedent and consequent. The logical connective ‘implication’ connects with two parts (i.e. two propositional variables) and establishes their relation. ‘Implication’ is symbolised as ‘→’. Take an example, consider two simple propositions; ‘Anu will come to the playground’ and ‘Rani will return her class notes’. If we combine these two propositions with the ‘implication’ logical connective, the compound proposition would be written as ‘If Anu will come to the playground then Rani will return her class notes’. This hypothetical proposition is symbolised as ‘x → y’. Some more examples are as follows: x—Anu will come to the playground. y—Rani will return her class notes. x → y—If Anu will come to the playground then Rani will return her class notes. ~x → y—If Anu will not come to the playground then Rani will return her class notes. x → ~y—If Anu will come to the playground then Rani will not return her class notes. ~x → ~y—If Anu will not come to the playground then Rani will not return her class notes. The logical connective ‘equivalence’ is popularly known as ‘biconditional’. It relates to two atomic or simple propositions that are combined with an ‘if and only if’ expression. The expression ‘if and only if’ is symbolised as ‘≡’. For example, ‘p’ stands for ‘You will be promoted in the job’, and ‘q’ stands for ‘You will qualify the written test’. If these two simple propositions were combined with the logical connective ‘equivalence’, then it would become a compound proposition and would

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be written as ‘You will be promoted in the job if and only if you will qualify the written test’. This compound proposition is symbolically represented as ‘x ≡ y’. Some more examples are given below. p—You will be promoted in the job. q—You will qualify the written test. p ≡ q—You will be promoted in the job if and only if you will qualify the written test. ~p ≡ q—You will not be promoted in the job if and only if you will qualify the written test. p ≡ ~q—You will be promoted in the job if and only if you will not qualify the written test. ~p ≡ ~q—You will not be promoted in the job if and only if you will not qualify the written test. The logical connective ‘dagger’ is symbolised as ‘↓’. It connotes ‘neither nor’. It is understood as ‘joint denial’. It connects with two simple propositional variables. For example, ‘p’ stands for ‘He is honest’, and ‘q’ stands for ‘He is studious’. If these two simple propositions are combined with the logical connective ‘dagger’, then it would be written as ‘Neither he is honest nor he is studious’. This compound proposition is symbolised as ‘p ↓ q’. A few examples are placed below concerning dagger connective. p—He is honest. q—He is studious. p ↓ q—Neither he is honest nor he is studious. ~p ↓ q—Neither he is not honest nor he is studious. p ↓ ~q—Neither he is honest nor he is not studious. ~p ↓ ~q—Neither he is not honest nor he is not studious. The ‘stroke’ is a logical connective and a binary operator. It connects with two simple propositions and formulates a compound proposition. It is symbolised as ‘|’. It expresses about ‘alternative denial’ of the compound proposition. It connotes ‘not both’. For example, ‘p’ stands for ‘It is a chair’, and ‘q’ stands for ‘It is a table’. If these two simple propositions were combined with the logical connective ‘stroke’, then it would become a compound proposition and would be written as ‘It is not both a chair and a table’. It is symbolised as ‘p | q’. p—It is a chair. q—It is a table. p | q—It is not both a chair and a table. A symbolical argument consists of a few symbolical propositions. Each symbolical proposition consists of a few propositional variables and logical constant as well as logical connectives. If a compound symbolic proposition consists of many propositional variables and logical connectives along with logical constants, then we have to limit the scope of each proposition that is conjoined with another proposition. To do so, we need to use parentheses ‘( )’, braces ‘{ }’, and brackets ‘[ ]’ in a

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hierarchical order in the proposition. This would assist us to keep the symbolical propositions in order and clarify the link between two symbolical propositions in an argument. Description of symbolical propositions

Symbolic propositions

Indication to limit the scope of symbolic propositions

A propositional variable

p

Parenthesis is not required

A propositional variable with negation

~p

Parenthesis is not required

Two propositional variables with one logical connective

p∨q

Parenthesis is not required

Two propositional variables; one ~p → q is negated and one logical connective

Parenthesis is not required

Two propositional variables; two are negated and one logical connective

~p ∨ ~q

Parenthesis is not required

Compound proposition is negated

~(p ∧ q)

Parenthesis is required

Two propositional variables and (p ∨ q) → q two logical connectives

Parenthesis is required

Three propositional variables and two logical connectives

r ∨ (p → q)

Parenthesis is required

More than two propositional variables and logical connectives

{(p ∨ q) → ((q ∨ r) ∨ p)}

Parentheses and braces are required

More than three propositional variables and logical connectives

[{~(p ∧ s) → (q ∨ r)} ∨ p]

Parentheses, braces, and brackets are required

Symbolical propositions are formulated with propositional variables, logical constant, logical connectives with appropriate parentheses, braces, and brackets wherever needed. If a symbolic compound proposition consists of more than two variables and logical connectives, then we need to identify the main logical connective of the proposition. The main logical connective helps to identify the relation between two propositions. The main logical connective also guides in the translation of symbolic propositions to the logical proposition of natural language. By identifying the correct parentheses, braces, and brackets of a symbolic proposition, we can identify its main logical connective. The parentheses, braces, and brackets limit the scope of compound propositions of the symbolic proposition. The technique to identify the main logical connective of a symbolic proposition is elucidated below. Let us consider the symbolic proposition [{~(p ∧ s) → (q ∨ r)} ∨ p]. In this proposition, the right-most ‘∨’ has the greatest scope. It is treated as the main logical connective of the proposition because it contains other logical connectives and logical constant in its scope. The main logical connective ‘∨’ connects with two propositions

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‘{~(p ∧ s) → (q ∨ r)}’ and ‘p’. Next, the main logical connective of the proposition ‘{~(p ∧ s) → (q ∨ r)}’ is ‘→’. It is so because ‘~(p ∧ s)’ and ‘(q ∨ r)’ are in its scope. Next, in consideration of the ‘(q ∨ r)’ proposition, the main logical connective is ‘∨’, as it connects ‘q’ and ‘r’ propositions. In consideration of the ‘~(p ∧ s)’ proposition, it is the ‘~’ that negates the compound proposition ‘p ∧ s’. It is the least scope of the compound proposition. Further, it is the ‘∧’ that stands for the main logical connective of ‘p’ and ‘s’. Identifying the main logical connective of some of the symbolic propositions. Symbolic propositions

Main logical constant or connective

p ∧ (q ∨ r)

∧

~(p ∧ (q ∨ r))

~

~~r → s

→

{((p ≡ r) ∧ r) ∧ q}

∧

{(t ∧ ~q) ≡ ~t}

≡

[((k ≡ e) ∧ ~q) ∨ {(r ∨ e) ∧ (k ∧ q)}]

∨

Now, we are going to formulate symbolic arguments from symbolic propositions. Let us consider the following symbolic propositions. (i) (ii) (iii) (iv) (v) (vi)

{((p → q) ∧ p) → q} {((p → q) ∧ ~q) → ~p} {((p → q) ∧ ~p) → ~q} {(~(p ∧ q) ∧ p) → ~q} {((p ∨ q) ∧ ~p) → q} [{(p ∧ ~r) ∧ (~r ∨ q)} → ((r ∨ q) ∧ ~p)]

Consider the symbolic proposition ‘{((p → q) ∧ p) → q}’. In this proposition, the main logical connective is ‘→’. The main logical connective ‘implication’ connects with antecedent ‘((p → q) ∧ p)’ and consequent ‘q’ of the proposition. In the consequent, a propositional variable (simple proposition) ‘q’ is found. This consequent is derived from the antecedent ‘((p → q) ∧ p)’. Concerning the antecedent of the proposition, ‘∧’ is the main logical connective of the antecedent as per the parentheses. Therefore, the antecedent consists of two propositions, a compound proposition ‘(p → q)’ and a simple proposition ‘p’. The compound proposition and the simple proposition together conclude ‘q’, as the consequence of the proposition. Thus, we can say, ‘q’ follows from the antecedent ‘((p → q) ∧ p)’. While the symbolical proposition is translated into an argument form, we have to identify the main logical connective of the symbolical proposition. The main logical connective of the symbolic proposition is mostly ‘→’. It stands for the expression ‘ifthen’. The consequent part of the symbolical proposition is regarded as a conclusion, and the antecedent part(s) are considered as premises of the argument.

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In an argument, we write the premises in the form of propositions and draw a conclusion from the premises by taking them together. A symbolical argument can be formulated from the symbolical proposition ‘{((p → q) ∧ p) → q}’ in the following way. Symbolical proposition: {((p → q) ∧ p) → q}

(i)

Symbolic argument Premise 1: (p → q) Premise 2: p Conclusion: Therefore, q. (ii)

Symbolical proposition: {((p → q) ∧ ~q) → ~p}

Symbolic argument Premise 1: (p → q) Premise 2: ~q Conclusion: Therefore, ~p. (iii)

Symbolical proposition: {((p → q) ∧ ~p) → ~q}

Symbolic argument Premise 1: (p → q) Premise 2: ~p Conclusion: Therefore, ~q. (iv)

Symbolical proposition: {(~(p ∧ q) ∧ p) → ~q}

Symbolic argument Premise 1: ~(p ∧ q) Premise 2: p Conclusion: Therefore, ~q Symbolical proposition: {((p ∨ q) ∧ ~p) → q}

(v)

Symbolic argument Premise 1: (p ∨ q) Premise 2: ~p Conclusion: Therefore, q (vi)

Symbolical proposition: [{(p ∧ ~r) ∧ (~r ∨ q)} → ((r ∨ q) ∧ ~p)]

Symbolic argument Premise 1: (p ∧ ~r) Premise 2: (~r ∨ q) Conclusion: Therefore, (r ∨ q) ∧ ~p We can also convert a symbolic argument into a symbolic proposition. In other words, we can formulate a symbolic proposition by considering the premises and conclusion of a symbolic argument. Let us consider the below symbolic argument.

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Symbolic argument Premise 1: (p ∨ q) Premise 2: p Conclusion: Therefore q. This argument consists of two premises and one conclusion. Two premises together help in drawing the conclusion. The premises are required to be joined by ‘∧’ while converting these to a symbolic proposition. The premises are considered as an antecedent of the symbolic proposition because they together conclude the argument. The conclusion of the argument is treated as a consequent of the symbolic proposition. Since an argument consists of premises and a conclusion, and the conclusion is derived from the premises, the whole argument is symbolised as ‘if-then’ form. Thus, ‘if-then’ would be the main logical connective of the symbolic proposition. The logical connective ‘if-then’ is symbolised as ‘→’. We can thus formulate the following symbolic proposition from this argument. Symbolic proposition: {((p ∨ q) ∧ p) → q}. A few more examples are mentioned below, converting symbolical arguments to symbolical propositions. Premise 1: (p → q) Premise 2: ~q Conclusion: Therefore, ~p The symbolic proposition of this argument: {((p → q) ∧ ~q) → ~p} Premise 1: (p ∧ ~q) Premise 2: (p → q) Premise 3: (~q ∨ ~p) Conclusion: Therefore, (~q → p) The symbolic proposition of this argument: [{(p ∧ ~q) ∧ (p → q) ∧ (~q ∨ ~p)} → (~q → p)] Premise 1: ~(p ∧ q) Premise 2: p Conclusion: Therefore, ~q The symbolic proposition of this argument: {(~(p ∧ q) ∧ p) → ~q} Premise 1: (~p → q) Premise 2: (q → r) Premise 3: (~r → s) Conclusion: Therefore, ~(~p → s) The symbolic proposition of this argument: [{((~p → q) ∧ (q → r)) ∧ (~r → s)} → ~(~p → s)] Premise 1: (p → (~q → r)) Premise 2: (p ∧ ~r) Premise 3: q Conclusion: Therefore, (~q ∧ p)

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The symbolic proposition of this argument: [{((p → (~q → r)) ∧ (p ∧ ~r)) ∧ q} → (~q ∧ p)] Premise 1: ~p Premise 2: (q ∨ r) Premise 3: ~r Conclusion: Therefore, (~p ∧ r) The symbolic proposition of this argument: [{(~p ∧ (q ∨ r)) ∧ ~r} → (~p ∧ r)] There are cases where symbolic propositions consist of more than three propositional variables, a few logical connectives, and in addition to that many parentheses, braces, and brackets. Due to many and repetitive use of parentheses, braces, and brackets in a symbolic proposition, it would not be easy to discern every complex and compound proposition within the symbolic proposition and find out the main logical connective of each compound and complex proposition. To arrest these inconveniences and make the symbolic propositions more user-friendly, J. Lukasiewicz (1878–1956), a polish logician, developed four logical notations (N, A, K, C) that correspond to logical symbols ~, ∨, ∧, →, respectively. The logical notations should be written in English capital letters, and the propositional variables should be in the lower-case letters, such as p, q, r, s, for a symbolic proposition. The propositional variables are to be written immediately after the logical notations of a symbolic proposition. In the case of a compound symbolic proposition, where there are two propositional variables and that are connected with a logical connective, these two variables are to be written chronologically right next to the logical notation of the logical symbol. For example, Logical symbols

Logical notation

Symbolic propositions

Symbolic notations

~

N

~p

Np

∨

A

p∨q

Apq

∧

K

p∧q

Kpq

→

C

p→q

Cpq

By adopting logical notations, we can formulate compound and complex symbolic propositions easily and avoid the inconvenience of putting many parentheses, braces, and brackets on the symbolic proposition. The below examples show how to translate symbolical propositions to symbolic notation propositions through logical notations. (i) (ii) (iii) (iv) (v) (vi)

p ∧ (p ∧ q) p ∧ ~q (p → ~q) ∧ r (~p ∨ r) ∨ (q ∧ p) ~p ∧ (~q ∧ ~r) ~(p ∧ ~q) → (~q → ~r)

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Let us consider the symbolic proposition ‘p ∧ (p ∧ q)’. In this proposition, we find two propositional variables ‘p’ and ‘q’ and one logical connective ‘∧’. The main logical connective of the symbolic proposition is ‘∧’, as it connects with ‘p’ and ‘(p ∧ q)’. Further, ‘(p ∧ q)’ compound proposition has two variables ‘p’ and ‘q’ and a logical connective ‘∧’. As per the logical notation guidelines, we can translate ‘(p ∧ q)’ as Kpq, and thereafter, ‘p ∧ (p ∧ q)’ translated as KpKpq. The main logical notation of a symbolic proposition must be written at the beginning of the symbolic proposition. Symbolic proposition: p ∧ (p ∧ q) Step 1: p ∧ Kpq Step 2: KpKpq The symbolic proposition ‘p ∧ (p ∧ q)’ is translated as KpKpq. Symbolic proposition: p ∧ ~q Step 1: p ∧ Nq Step 2: KpNq The symbolic proposition ‘(p ∧ ~q)’ is translated as KpNq. Symbolic proposition: (p → ~q) ∧ r Step 1: (p → Nq) ∧ r Step 2: CpNq ∧ r Step 3: KCpNqr The symbolic proposition ‘(p → ~q) ∧ r’ is translated as KCpNqr. Symbolic proposition: (~p ∨ r) ∨ (q ∧ p) Step 1: (Np ∨ r) ∨ (q ∧ p) Step 2: (ANpr) ∨ (q ∧ p) Step 3: (ANpr) ∨ (Kqp) Step 4: AANprKqp The symbolic proposition ‘(~p ∨ r) ∨ (q ∧ p)’ is translated as AANprKqp. Symbolic proposition: ~p ∧ (~q ∧ ~r) Step 1: Np ∧ (Nq ∧ Nr) Step 2: Np ∧ KNqNr Step 3: KNpKNqNr The symbolic proposition ‘~p ∧ (~q ∧ ~r)’ is translated as KNpKNqNr. Symbolic proposition: ~(p ∧ ~q) → (~q → ~r) Step 1: ~(p ∧ Nq) → (Nq → Nr) Step 2: ~(KpNq) → (CNqNr) Step 3: (NKpNq) → (CNqNr) Step 4: CNKpNqCNqNr The symbolic proposition ‘~(p ∧ ~q) → (~q → ~r)’ is translated as CNKpNqCNqNr.

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Now, we can translate symbolic notation propositions to symbolic propositions by adopting the logical notation guidelines. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

CqKqp ACprr CCNqNpCpq NANpq ANpANrq ANpAqNr NAANpqr CNApKqpACNrqp.

Let us consider the symbolic notation proposition ‘CqKqp’. This proposition has two propositional variables ‘p’ and ‘q’ and two logical connectives ‘C’ and ‘K’. The logical notation ‘C’ would be considered as the main logical connective of the symbolic proposition, as it is placed at the beginning of the symbolic notation proposition. Since the logical notation ‘C’ stands for logical symbol ‘→’, it connects with two propositional variables, ‘q’ and ‘Kqp’. Further, Kqp is translated as ‘q ∧ p’. Thus, ‘CqKqp’ is translated as ‘(q → (q ∧ p))’. Symbolic notation proposition: CqKqp Step 1: Cq(q ∧ p) Step 2: (q → (q ∧ p)) Thus, CqKqp is translated as (q → (q ∧ p)). Symbolic notation proposition: ACprr Step 1: A(p → r)r Step 2: ((p → r) ∨ r) Thus, ACprr is translated as ((p → r) ∨ r). Symbolic notation proposition: CCNqNpCpq Step 1: CC ~ q ~ pCpq Step 2: C(~q → ~p)Cpq Step 3: C(~q → ~p)(p → q) Step 4: ((~q → ~p) → (p → q)) Thus, CCNqNpCpq is translated as ((~q → ~p) → (p → q)). Symbolic notation proposition: NANpq Step 1: NA ~ pq Step 2: N(~p ∨ q) Step 3: ~(~p ∨ q) Thus, NANpq is translated as ~(~p ∨ q). Symbolic notation proposition: ANpANrq Step 1: A ~ pA ~ rq Step 2: A ~ p(~r ∨ q) Step 3: (~p ∨ (~r ∨ q))

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Thus, ANpANrq is translated as (~p ∨ (~r ∨ q)). Symbolic notation proposition: ANpAqNr Step 1: A ~ pAq ~ r Step 2: A ~ p(q ∨ ~r) Step 3: (~p ∨ (q ∨ ~r)) Thus, ANpAqNr is translated as (~p ∨ (q ∨ ~r)). Symbolic notation proposition: NAANpqr Step 1: NAA ~ pqr Step 2: NA(~p ∨ q)r Step 3: N((~p ∨ q) ∨ r) Step 4: ~((~p ∨ q) ∨ r) Thus, NAANpqr is translated as ~((~p ∨ q) ∨ r). Symbolic notation proposition: CNApKqpACNrqp Step 1: CNApKqpAC ~ rqp Step 2: CNAp(q ∧ p)A(~r → q)p Step 3: CN (p ∨ (q ∧ p))A(~r → q)p Step 4: C{~(p ∨ (q ∧ p))}A(~r → q)p Step 5: C{~(p ∨ (q ∧ p))}((~r → q) ∨ p) Step 6: [{~(p ∨ (q ∧ p))} → ((~r → q) ∨ p)] Thus, CNApKqpACNrqp is translated as [{~(p ∨ (q ∧ p))} → ((~r → q) ∨ p)].

10.4 Translation of Logical Propositions into Symbolic Propositions In our mundane life, we use many logical propositions to convey our thoughts. For example, the sky is blue, the grass is green, snow is white, logic students are wise beings, etc. A question arises, can we translate these logical propositions to symbolic propositions? If we can do so, it would be easier for us to use the symbolic propositions in the formulation of arguments and finding out the validity or invalidity of the arguments easily and intelligibly. In this section, we will translate the logical propositions into symbolic propositions. Then, translate the symbolic propositions into logical propositions. Thereafter, we will translate the logical arguments into symbolical arguments and vice versa. Let ‘p’ stand for the proposition ‘He is tall’ and ‘q’ stand for the proposition ‘He is handsome’. Considering the two propositional variables for the logical propositions, let us translate the following sentential propositions to symbolic propositions. (i) (ii) (iii) (iv)

He is tall but he is not handsome. He is a handsome and tall person. It is false that he is short and handsome. He is either tall or handsome.

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He is a handsome but not tall person. It is not true that he is short or not handsome. Since he is tall, he must be handsome.

While translating these logical propositions to symbolic propositions, we must not change the semantics of the propositions. If we do so, then the whole symbolisation will go wrong, as the logical constant and logical connectives will get changed due to the change of meaning of the propositions. Hence, the meaning of the logical propositions and their translation of symbolic propositions will remain unaltered in case of a correct translation. The translation of the above-mentioned propositions to the symbolic propositions is mentioned below in sequential order. (i) (ii) (iii) (iv) (v) (vi) (vii)

p ∧ ~q q∧p ~(~p ∧ q) p∨q q ∧ ~p ~(~p ∨ ~q) p→q

Now, we will translate the symbolic propositions to logical propositions. Let ‘p’ stand for the proposition ‘It is a moonlit night’ and ‘q’ stand for the proposition ‘you are looking beautiful’. Considering these propositional variables, we are translating the following symbolic propositions to logical propositions. (i) (ii) (iii) (iv) (v) (vi)

p∧q ~q ∧ ~p ~(~p ∨ q) ~~p ((p ∧ ~q) → q) {~(p ∧ (~p ∨ q)) ∨ ~q}

While translating these symbolical propositions to logical propositions of natural language, we cannot afford to miss any logical constant and logical connective of the symbolic proposition. Further, we have to give utmost attention to the parentheses, braces, and brackets of the symbolic proposition while translating a compound and complex proposition to a logical proposition. The translation shall not change the meaning of the symbolic proposition. If it changes, then the meaning of logical propositions will be altered. The semantics of symbolic propositions and their translation of logical propositions must remain unchanged. The translation of the abovementioned symbolic propositions to logical propositions is mentioned below in a sequential manner. (i) (ii) (iii) (iv)

It is moonlit night and you are looking beautiful. You are not looking beautiful and it is not a moonlit night. It is false that either it is not a moonlit night or you are looking beautiful. It is false that it is not a moonlit night.

10.4 Translation of Logical Propositions into Symbolic Propositions

(v) (vi)

161

If it is moonlit night and you are not looking beautiful then you are looking beautiful. Either it is false that, it is moonlit night, and either it is not moonlit night or you are looking beautiful, or you are not looking beautiful.

Now, we will translate the below logical arguments into symbolic arguments by assigning a propositional variable to each sentence of the argument. A sentence ends with a full stop—it may be a simple sentence or a compound sentence. The terms ‘hence’, ‘thus’, ‘therefore’, etc., are associated with the conclusion of the argument. (a)

If I watch a cricket match on TV, then I cannot study for the end semester examination. If I do not study for the end semester examination, then tomorrow either I will not appear in the end semester examination or I will sit for the make-up examination. I will appear in the end semester examination tomorrow. Therefore, I will study for the end semester examination. Symbolic argument: I will watch a cricket match on TV—p I study for the end semester examination—q I will appear in the end semester examination—r I will sit for make-up examination—s 1st premise: (p → ~q) 2nd premise: (~q → (~r ∨ s)) 3rd premise: r Conclusion: q

(b)

If logic is difficult for you, then you have to attend all the classes. You must study lessons at your home. You do not study lessons at your home and do not attend the classes. Hence, logic is difficult for you. Symbolic argument: Logic is difficult for you—p You have to attend all the classes—q You must study lessons at your home—r 1st premise: p → q 2nd premise: r 3rd premise: ~r ∧ ~q Conclusion: p

(c)

Smita will get the job only if Rakesh does not appear in the job interview. Rakesh appeared in the job interview. Hence, Smita did not get the job. Symbolic argument: Smita will get the job—p Rakesh does appear in the job interview—q 1st premise: ~q → p 2nd premise: q

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Conclusion: ~p (d)

Either a disease is contagious or hereditary. Malaria is a contagious disease. Hence, it is not hereditary. Symbolic argument: A disease is contagious—p A disease is hereditary—q Malaria is a contagious disease—r Malaria is hereditary—s 1st premise: p ∨ q 2nd premise: r Conclusion: ~s.

(e)

If a student is interested in studies, then she will enjoy studies, and she will pay attention to the lessons. If a student does not pay attention to the lessons, then she will not learn the lessons. If she does not learn the lessons, then either she is not enjoying the studies or not paying attention to the lessons. She does not learn the lessons. Hence, she is not interested in studies. Symbolic argument: A student is interested in studies—p She will enjoy studies—q She will pay attention to the lessons—r She will learn the lessons—s She can do well in the examination—t 1st premise: p → (q ∧ r) 2nd premise: ~r → ~s 3rd premise: ~s → (~q ∨ ~r) 4th premise: ~s Conclusion: ~p

We will now translate symbolic arguments into logical arguments. While translating the symbolic arguments, we need to find out the main logical connective of the compound premises of the argument. Further, we need to notice the propositional variables used in the argument. In addition to these tasks, we need to consider that the ‘conclusion’ of the symbolic argument is the consequent and premises are antecedent of the logical argument. The conclusion of the symbolic argument shall be written by mentioning ‘therefore’, ‘hence’, ‘so’, etc., in the logical sentence. For example, Example-1 Let ‘p’ stand for ‘there is sun’ and ‘q’ stand for ‘there is light’. 1st premise: p → q 2nd premise: p Conclusion: q Logical argument: If there is sun there is light. There is sun. Therefore, there is light.

10.4 Translation of Logical Propositions into Symbolic Propositions

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Example-2 Let ‘p’ stand for ‘there is sun’ and ‘q’ stand for ‘there is light’. 1st premise: p → q 2nd premise: ~q Conclusion: ~p Logical argument: If there is sun there is light. There is no light. Therefore, there is no sun. Example-3 Let p—A teacher knew that the speed limit is thirty kilometres per hour. q—She would have been driving at seventy. r—She was caught by the traffic police. 1st premise: p → ~q 2nd premise: ~p ∧ r Conclusion: ~p ∨ r Logical argument: If a teacher knew that the speed limit is thirty kilometres per hour, then she would not have been driving at seventy. The teacher did not know that the speed limit is thirty kilometres per hour. She was driving at seventy kilometres per hour and was caught by the traffic police. Thus, either the teacher did not know that the speed limit is thirty kilometres per hour, or she was caught by the traffic police. As explained above, we can translate logical propositions to symbolic propositions and symbolic propositions to logical propositions. We can also translate logical arguments to symbolic arguments and symbolic arguments to logical arguments. Further, we can translate symbolic arguments to symbolic propositions. Logicians are also interested in determining the truth-value of a proposition and the validity and invalidity of an argument. Due to this interest, they search for the truth functions of a proposition.

10.5 Truth Functions and Truth Table Method A logical proposition cannot be neither true nor false. It cannot be true and false at the same time. Rather, it is either ‘true’ or ‘false’. So, truth and falsity are the two values of a logical proposition. These two values are known as ‘truth-values’ of a proposition. In symbolic logic, a proposition is linked to another proposition with a logical connective. Since each logical connective is unique due to its semantics and its functions, the truth-value of a compound proposition would result in unique truth-values.

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It is convenient for us to write the capital letter ‘T’ for truth and capital letter ‘F’ for the falsity of a proposition. If a proposition is T, then its negation is F, and if a proposition is F, then its negation is T. Thus, the truth function of a negative proposition is based on the truth-value of the proposition. For example, if ‘Grass is green’ is true, then ‘Grass is not green’ is false. Again, if ‘Grass is green’ is false, then ‘Grass is not green’ is true. When truth functions are presented in a table, it is called a ‘truth table’. Truth function of ‘Negation P’ p

~p

T

F

F

T

The negation of a proposition is the truth function of that proposition. Irrespective of the simple proposition or compound proposition, the negation of a proposition is contradictory to that proposition. One propositional variable has two truth-values, T and F. Truth-values of negation of a proposition are illustrated through the truth table method on the above. Here, the truth table consists of two rows and two columns as shown above. The two rows are exhaustive to mention the possible combination of the truth-values of a propositional variable. It is to be noted here that if there are two negations for a propositional variable, then the truth-value of the double negation of that proposition remains the same as the truth-value of the proposition. For example, if p is true, then ~~p is true, and if p is false, then ~~p is false. p

~p

~ ~p

T

F

T

F

T

F

In consideration of two propositional variables connecting with one logical connective, we can have combinations of four possible truth-values. These are TT, TF, FT, and FF. These truth-values are arranged in an order keeping in mind the equal and proper distribution of truth and false value between two propositional variables. If p and q are two propositional variables connecting with a logical connective, say ‘→’, then p would have TTFF values, and q would have TFTF values to determine the truth function of p → q. A truth table is drawn below for the illustration purpose without mentioning the truth function of p → q, which would be explained a little later in this chapter. p

q

T

T

T

F

p→q

(continued)

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165

(continued) p

q

F

T

F

F

p→q

In the case of three propositional variables of a symbolic proposition, let us say, ‘((p ∧ q) ∧ r)’, it can have a combination of eight possible truth-values under each propositional variable, and these truth-values can be distributed among three propositional variables in the following order. If ‘p’, ‘q’, and ‘r’ are three propositional variables of a symbolic proposition, then p would have TTTTFFFF values, q would have TTFFTTFF values, and r would have TFTFTFTF values to determine truth functions of the main logical connective of the symbolic proposition. A truth table illustration is given below. p

q

r

T

T

T

T

T

F

T

F

T

T

F

F

F

T

T

F

T

F

F

F

T

F

F

F

In this way, if a symbolic proposition has more than three propositional variables, then we can apply the ‘2n ’ formula to determine the possible truth-values of it. In the case of four propositional variables of a symbolic proposition, we can have 24 = 16 possible truth-values. In the case of five propositional variables of a symbolic proposition, we can have 25 = 32 possible truth-values, so on and so forth. In the case of one propositional variable and a logical connective, to determine its truth function, we draw a truth table consisting of two rows and two columns. In the case of two propositional variables and a logical connective (a compound proposition), to determine its truth functions, we draw a truth table consisting of five rows and three columns. In this way, we may suggest that depending on the total number of propositional variables of a symbolic proposition, we can find out the possible combination of truth-values it would have, and accordingly, we can draw those many rows in a truth table to determine truth functions of the main logical connective of the symbolic proposition. But how many columns are required to determine the truth function of the main logical connective of a symbolic proposition is based on how many truth functions of logical connectives we need to find out in the proposition itself. The following sections determine the truth-values of symbolic propositions by adapting the truth table method. We would draw the truth tables to

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determine truth functions of conjunction, disjunction, implication, and equivalence logical connectives followed by dagger and stroke logical connectives.

10.6 The Conjunctive Function and the Disjunctive Function Two simple propositions joined with ‘and’ are known as compound and conjunctive proposition. For example, ‘Ram is a logic student and Sita is a singer’. This compound proposition is symbolised as ‘p ∧ q’. Since this symbolic proposition has two propositional variables and a logical connective ‘∧’, it has a combination of four possible truth-values to determine the truth function of ‘p ∧ q’. The truth function of ‘p ∧ q’ is mentioned below through the truth table method. p

q

p∧q

T

T

T

T

F

F

F

T

F

F

F

F

The truth table shows that conjunctive function ‘p ∧ q’ is true when both ‘p’ and ‘q’ are true. If either of the propositional variables is false, then the truth function of conjunction becomes false. In the case of a disjunctive function, a compound proposition is formulated with an ‘either-or’ expression. For example, ‘Either Ram is a logic student or Sita is a singer’. In disjunctive proposition, two simple propositions are connected with a logical connective ‘∨’. The disjunctive compound proposition is symbolised as ‘p ∨ q’. Since this symbolic proposition has two propositional variables and a logical connective ‘∨’, it has a combination of four possible truth-values to determine the truth function of ‘p ∨ q’. The truth function of ‘p ∨ q’ is placed below through the truth table method. p

q

p∨q

T

T

T

T

F

T

F

T

T

F

F

F

The truth table suggests that a disjunctive function is true when at least one of its propositional variables is true. But when both the propositional variables are false, the truth function of the disjunctive proposition is false.

10.7 Implicative Function and Equivalence Function

167

10.7 Implicative Function and Equivalence Function The implication is a logical connective that connects with antecedent and consequent parts of a compound proposition and formulated through ‘if-then’ expression. For example, ‘If he will come to logic class then he will get the course materials’. This compound proposition is symbolised as ‘p → q’. Since this symbolic proposition has two propositional variables and a logical connective ‘→’, it has a combination of four possible truth-values to determine the truth function of ‘p → q’. The truth function of ‘p → q’ is mentioned below through the truth table method. p

q

p→q

T

T

T

T

F

F

F

T

T

F

F

T

The truth table conveys that an implicative function is false when the antecedent is true and the consequent is false. In the rest of the other cases, the truth-value of the implicative function is true. In the case of an equivalence logical connective, two atomic propositions are combined with an ‘if and only if’ expression. It is a compound proposition that connects with the logical connective ‘≡’. For example, ‘Her fingers will burn if and only if she plays with fire’. This compound proposition is symbolised as ‘p ≡ q’. Since this symbolic proposition has two propositional variables and a logical connective ‘≡’, it has a combination of four possible truth-values to determine the truth function of ‘p ≡ q’. The truth function of ‘p ≡ q’ is illustrated in the truth table method. p

q

p≡q

T

T

T

T

F

F

F

T

F

F

F

T

The truth table states that if both the propositions are true then the truth function of the equivalence is true. Further, if both the propositions are false, then the truth function of the equivalence is also true. But if one of the propositions is true and another is false, then the truth function of equivalence is false.

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10.8 Dagger Function and Stroke Function In the case of the dagger function, a compound proposition is formulated with ‘neither nor’ expression. For example, ‘Neither he is a science student nor a commerce student’. In this proposition, two simple propositions are connected with a logical connective ‘↓’. This proposition is symbolised as ‘p ↓ q’. Since this symbolic proposition has two propositional variables and a logical connective ‘↓’, it has a combination of four possible truth-values to determine truth function of ‘p ↓ q’. The truth function of ‘p ↓ q’ is stated below in a truth table method. p

q

p↓q

T

T

F

T

F

F

F

T

F

F

F

T

The truth table expresses that when both the propositions are false the truth function of the dagger is true, and if one of the propositions is true, then the truth function of the dagger is false. Stroke function is popularly known as Sheffer stroke function. It is a binary logical connective that connects with two simple propositions and forms a compound proposition. This logical connective states that not both the propositions of a compound position are true. It suggests that at least one of the propositions out of two propositions of the compound proposition must be false. The expression ‘not both’ in a compound proposition identifies that two simple propositions are joined with the stroke function. For example, ‘It is not both a chair and a table’. This compound proposition is symbolised as ‘p | q’. Since this symbolic proposition has two propositional variables and a logical connective ‘|’, it has a combination of four possible truth-values to determine the truth function of ‘p | q’. The truth function of ‘p | q’ is elucidated below through the truth table method. p

q

p|q

T

T

F

T

F

T

F

T

T

F

F

T

The truth table conveys that when both the propositions are true the truth function of Sheffer stroke is false, but when one of the propositions is false, the truth function of Sheffer stroke is true. Let us sum up the truth functions of the binary logical connectives through the truth table method.

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169

p

q

p∧q

p∨q

p→q

p≡q

p↓q

p|q

T

T

T

T

T

T

F

F

T

F

F

T

F

F

F

T

F

T

F

T

T

F

F

T

F

F

F

F

T

T

T

T

This truth table is presented alternatively in the following manner. Conjunction: ∧ = TT/T Disjunction: ∨ = FF/F Implication: → = TF/F Equivalence: ≡ = TT/T and FF/T Dagger: ↓ = FF/T Stroke: | = TT/F

10.9 Truth-Value: Tautology, Contradiction, and Contingent While determining the truth-value of a symbolic proposition by adopting the truth tabular method, under the main logical connective column, we may find that all the values are true or false or a combination of true and false. If the main logical connective column contains only ‘true’ values, then we say the truth-value of the proposition is a tautology. If it contains only ‘false’ values, then we say that the truth-value of the proposition is logically false or self-contradictory. But if it contains a combination of true and false values, then we say the truth-value of the symbolic proposition is contingent. Let us find out the truth-value of the following symbolic propositions through the truth tabular method. Example-1: {(~q → ~p) ≡ (p → q)} The main logical connective of this symbolic proposition is ‘≡’. To find out the truth-value of the symbolic proposition, we need to count how many propositional variables this proposition has. Since the proposition has two propositional variables, we will have a combination of four possible truth-values. To find out the truth-values under the main logical connective, we need to find out the truth function of ‘(~q → ~p)’ and ‘(p → q)’. Further, to find out the truth function of ‘(~q → ~p)’, we need to find out the truth-value of ‘~q’ and ‘~p’. The truth-value of ‘~q’ and ‘~p’ depends on the truth-value of ‘q’ and ‘p’. Again, we need to find out the truth function of ‘(p → q)’. So, we need to have seven columns and five rows to find out the truth-value of the symbolic proposition through the truth tabular method. The truth table is illustrated below.

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p

~p

q

~q

(p → q)

(~q → ~p)

{(~q → ~p) ≡ (p → q)}

T

F

T

F

T

T

T

T

F

F

T

F

F

T

F

T

T

F

T

T

T

F

T

F

T

T

T

T

Under the main logical connective of the symbolic proposition, we found TTTT truth-values. Hence, the proposition is logically true and thereby tautology. Example-2: {(p → q) ∨ (~(p ≡ ~q))} This symbolic proposition has two propositional variables, ‘p’ and ‘q’. Hence, we will have a combination of four possible truth-values. The main logical connective of the symbolic proposition is ‘∨’. To find out the truth-value of the symbolic proposition, we need to find out the truth function of ‘(p → q)’ and ‘(~(p ≡ ~q))’. To find out the truth function of ‘(~(p ≡ ~q))’, we need to find out the truth-value of ~q and truth function of ‘(p ≡ ~q)’. By keeping in mind, the logical connective of a compound proposition, we need to draw the truth table. The truth table is presented below. p

q

~q

(p → q)

(p ≡ ~q)

~(p ≡ ~q)

{(p → q) ∨ (~(p ≡ ~q))}

T

T

F

T

F

T

T

T

F

T

F

T

F

F

F

T

F

T

T

F

T

F

F

T

T

F

T

T

Under the main logical connective of the symbolic proposition, we found TFTT truth-values. It is a combination of true and false values. Hence, the symbolic proposition is regarded as contingent. Example-3: {(p ∧ (~q → p)) ∧ ~((p ≡ ~q) → (q ∨ ~p))} This symbolic proposition has two propositional variables ‘p’ and ‘q’. So, we can have a combination of four possible truth-values. The main logical connective of the proposition is ‘∧’. To determine the truth-value of the main logical connective of the symbolic proposition, we need to find out the truth function of ‘(p ∧ (~q → p))’ and ‘~((p ≡ ~q) → (q ∨ ~p))’. Further, to find out the truth function of ‘(p ∧ (~q → p))’, we need to find out the truth-value of ~q and the truth function of ‘(~q → p)’. Again, to find out the truth function of ‘~((p ≡ ~q) → (q ∨ ~p))’, we need to find out the truth-value of ~p and the truth function of ‘(p ≡ ~q)’ and ‘(q ∨ ~p)’. Now, we need to put all the simple propositions, compound propositions, and complex propositions in the truth table columns to determine the truth-value of the whole symbolic proposition. The below truth table (see the page number 172) illustrates the truth-value of the symbolic proposition.

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171

Under the main logical connective of the symbolic proposition, we found FFFF truth-values. Thus, the truth-value of the symbolic proposition is regarded as contradictory. We have discussed the truth table method in detail and analysed the procedure to determine the truth function of a symbolic proposition. We also explained the conditions under which a symbolic proposition would be treated as a tautology, contradiction, and contingent. But it is found that if a symbolic proposition has more than two propositional variables and a few logical connectives, then it would not be a pleasant exercise to determine its truth function through the truth tabular method, as there would be many columns and many rows. Hence, logicians adopted an indirect method of truth table decision procedure to determine the truth function of complex symbolical propositions. The detailed analysis of an indirect method of truth table decision procedure is delineated in the next chapter.

~p

F

F

T

T

p

T

T

F

F

F

T

F

T

q

T

F

T

F

~q

F

T

T

F

(p ≡ ~q)

T

T

T

F

(q ∨ ~p)

F

T

T

T

(~q → p)

F

F

T

T

(p ∧ (~q → p))

T

T

T

T

(p ≡ ~q) → (q ∨ ~p)

F

F

F

F

~((p ≡ ~q) → (q ∨ ~p))

F

F

F

F

{(p ∧ (~q → p)) ∧ ~((p ≡ ~q) → (q ∨ ~p))}

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Chapter 11

Symbolic Logic-02

In the previous chapter, we discussed the truth function of unary logical symbol, binary logical connectives, truth-value of a propositional variable, translation of logical propositions to symbolic propositions, and vice versa, determination of truthvalue of a symbolic proposition through truth table method, and so on. In continuation of the previous chapter, in this chapter, we will discuss an indirect method of truth table decision and Evert Willem Beth’s truth tree method to determine the validity of an argument. Further, we will elaborate on the methods to formulate arguments in conjunctive and disjunctive forms. In the end, we will illustrate the derivation of the conclusion from the premises of a symbolic argument.

11.1 Indirect Method of Truth Table Decision In the truth table method, to find out the truth-value of the symbolic proposition, we need to draw columns and rows depending on the number of propositional variables and logical connectives it has. If a proposition has more than three propositional variables and a few logical connectives, then the truth table exercise results in inconvenience, nuisance, and awkwardness. Further, adapting the truth table method procedure to determine the validity or invalidity of an argument would require a big table having many rows and columns. To avoid the awkwardness and clumsy exercise, logicians developed a shorter method known as reductio ad absurdum or indirect method of truth table decision to determine the validity and invalidity of an argument. In a truth tabular method, truth-values found under the main logical connective of a proposition determine whether the proposition is regarded as a tautology, or contradiction, or contingent. But in the case of the indirect method of truth table decision, we start by assuming that the premises or propositions of the argument are true (T) and the conclusion is false (F). More precisely, we assume that the argument form is invalid and we write ‘F’ under the main logical connective of the argument. © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_11

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In the truth tabular method, one finds the truth-value of the main logical connective of the argument at the end of the exercise, but in the case of the indirect method of truth table decision, the exercise has to start with the main logical connective of the argument. After assigning ‘F’ under the main logical connective of the argument, we apply the basic rules of truth functions of logical connectives and find out truth-values of all the propositional variables of the argument. Thereafter, we need to examine whether the assumption ‘F’ occurs in the main logical connective of the argument has led to a contradiction of the basic rules of truth table procedure. If we do not find any contradiction at the end of the exercise, then the argument is invalid. It is so because the hypothesis will turn out to be correct. But if the contradiction is found at the end of the exercise, then the hypothesis assumed in the beginning ‘F’ is contradicted. Hence, the hypothesis ‘F’ will be treated as incorrect. The assumption ‘F’ should have been ‘T’ instead. Therefore, the argument is treated as a valid argument. The indirect method of truth table decision is explained with the help of the following examples. Example-1: (((p → q) ∧ p) → q) This argument consists of two premises ‘(p → q)’ and ‘p’ and one conclusion ‘q’. The conclusion ‘q’ is derived from two premises. There are two propositional variables ‘p’ and ‘q’, and two logical connectives ‘→’ and ‘∧’ are associated with this argument. The main logical connective of the argument is ‘→’. This argument states that if ‘(p → q)’ and ‘p’ then we can draw ‘q’ conclusion. Now, we will find out whether the argument is valid or not step by step by adopting the indirect method of truth table decision. Step-1:

We begin by assuming that the argument form is invalid. Hence, we assigned ‘F’ under the main logical connective of the argument. Step-2:

The main logical connective column can be false only when the antecedent ‘(p → q) ∧ p)’ is true and consequent ‘q’ is false. So, we have got the truth-value of ‘q’, that is ‘F’.

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175

Step-3:

About the antecedent of the argument, the main logical connective ‘∧’ is true when ‘(p → q)’ is T and ‘p’ is T. Here, we have obtained the truth-value of ‘p’. Step-4:

We got the truth-value of ‘p’ as T and ‘q’ as F. The truth-values of these propositional variables are assigned in ‘(p → q)’. After assigning the truth-values to ‘(p → q)’, we find that there is inconsistency in the expression ‘(p → q)’. That is when ‘p’ is T and ‘q’ is F, the truth function of ‘(p → q)’ is T. The (p → q) truth-value should have been F. Hence, we find a contradiction in the result based on our assumption. The assumption should have T instead of F. Since the assumption under the main logical connective is T, the whole argument is treated as a valid argument. Example-2: {(p → q) → ~(p ∧ (q ∧ r))} This argument consists of one premise ‘(p → q)’ and one conclusion ‘~(p ∧ (q ∧ r))’. The conclusion ‘~(p ∧ (q ∧ r))’ is derived from the premise. The argument has three propositional variables; ‘p’, ‘q’, and ‘r’, two logical connectives; ‘→’ and ‘∧’, and a logical constant ‘~’. The main logical connective of the argument is ‘→’. This argument states that if ‘(p → q)’ then ‘~(p ∧ (q ∧ r))’, as the conclusion. Now, we will find out the validity of the argument by adopting the indirect method of truth table decision. Step-1:

We begin by assuming that the argument form is invalid. Hence, we assigned ‘F’ under the main logical connective of the argument. Step-2:

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The main logical connective column can be false when the antecedent ‘(p → q)’ is true and consequent ‘~(p ∧ (q ∧ r))’ is false. Step-3:

About the consequent of the argument, if the truth-value of ‘~(p ∧ (q ∧ r))’ is F, then the truth-value of ‘(p ∧ (q ∧ r))’ is T. Step-4:

The truth-value of ‘(p ∧ (q ∧ r))’ is T when ‘p’ is T and ‘(q ∧ r)’ is T. Here, we got the truth-value of ‘p’ as T. Step-5:

The truth-value of ‘(q ∧ r)’ is T if and only if ‘q’ is T and ‘r’ is T. In this step, we obtained the truth-value of ‘q’ as T and ‘r’ as F. Step-6:

We got the truth-value of ‘p’ as T and ‘q’ as T. The truth-value of these propositional variables is assigned to ‘(p → q)’. After assigning the truth-values in ‘(p → q)’, we find that there is no inconsistency in the expression ‘(p → q)’. That is, when ‘p’ is T and ‘q’ is T, the truth function of ‘(p → q)’ is T. Hence, we do not find any contradiction in the result based on our assumption. The assumption F was correct. Since the assumption under the main logical connective was F and it is correct, the whole argument is treated as an invalid argument. Example-3: [{((p → q) ∧ (s → r)) ∧ (q → s)} → (p ∨ r)] This argument consists of three premises ‘(p → q)’, ‘(s → r)’, ‘(q → s)’ and one conclusion ‘(p ∨ r)’. The conclusion ‘(p ∨ r)’ is derived from three premises. This

11.1 Indirect Method of Truth Table Decision

177

argument has four propositional variables ‘p’, ‘q’, ‘r’, and ‘s’ and three logical connectives ‘→’, ‘∧’, and ‘∨’. The main logical connective of the argument is ‘→’. This argument states that if ‘(p → q)’, ‘(s → r)’, and ‘(q → s)’ then ‘(p ∨ r)’ is the conclusion. Now, we will find out the validity of the argument by adopting the indirect method of truth table decision. Step-1:

We begin by assuming that the argument form is invalid. Hence, we assigned ‘F’ under the main logical connective of the argument. Step-2:

The main logical connective column can be false when the antecedent ‘{((p → q) ∧ (s → r)) ∧ (q → s)}’ is true and consequent ‘(p ∨ r)’ is false. Step-3:

Concerning the consequent ‘(p ∨ r)’ of the argument, the truth-value of ‘(p ∨ r)’ is F when ‘p’ is F and ‘r’ is F. In this step, we got the truth-values of ‘p’ and ‘r’ as F. Step-4:

The main logical connective of the antecedent ‘{((p → q) ∧ (s → r)) ∧ (q → s)}’ is ‘∧’. The antecedent consists of two parts ‘((p → q) ∧ (s → r))’ and ‘(q → s)’. We already obtained that the truth-value under the main logical connective of the antecedent is T. The truth-value of the antecedent is T when both of its constituent parts’ truth-values are T.

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Step-5:

The truth-value of the compound proposition ‘((p → q) ∧ (s → r))’ is T when ‘(p → q)’ is T and ‘(s → r)’ is T. Step-6:

We have already obtained the truth-value of ‘p’ and ‘r’, which we put in the ‘(p → q)’ and ‘(s → r)’ propositions. Step-7:

Concerning to ‘(s → r)’ proposition, when ‘s’ is F and ‘r’ is F, the truth-value of ‘(s → r)’ proposition is T. In this step, we got the truth-value of ‘s’ as F. We have mentioned that the truth-value of ‘s’ is F in ‘(q → s)’ proposition. Step-8:

Upon the consideration of the compound proposition ‘(q → s)’, we can assign that the truth-value of ‘q’ is F. It is so because the truth-value of ‘(q → s)’ is T when ‘q’ is F and ‘s’ is F. In this step, we got the truth-value of ‘q’ as F. The truth-value of ‘q’ is mentioned in the proposition ‘(p → q)’. After assigning the truth-value in ‘(p → q)’, ‘(s → r)’, and ‘(q → s)’, we find that there is no inconsistency in these premises. Hence, we do not find any contradiction in the result based on our assumption. The assumption F was proved to be correct. Since the assumption under the main logical connective was F and it is correct, the whole argument is treated as an invalid argument. Besides the indirect method of truth table decision, there is another method known as ‘analytic tableaux’ or ‘trees’ that exists to determine the validity and invalidity of an argument. The ‘trees’ method is popularly known as ‘Beth trees’, as Professor Evert

11.1 Indirect Method of Truth Table Decision

179

Willem Beth (1908–1964) introduced it to logic and logical discourse. Professor Beth is known for his contribution to the development of the ‘method of semantic tableaus (1955)’ known as ‘Beth tree’ and ‘truth tree’.

11.2 Beth Tree (Truth Tree) Beth tree is a method that assists in determining the validity of an argument. To examine an argument, we need to arrange its premises and conclusion in a list. We need to negate the conclusion while mentioning it in the list. The reason is we need to verify whether the argument is satisfied or not with a negative conclusion. Upon our examination, if we find the premises are not true together then the argument is valid. The reason is that at least one of its premises contradicts the negative conclusion, which was taken as an assumption while listing the conclusion before starting of verification. If all the premises are true, then the argument is invalid because the negative conclusion stands correct. To test an argument through the truth tree method, we need to abide by the following rules and procedures. Truth tree rules The Beth tree rules are employed to develop the branches or close them. For that, we need to branch out a proposition or stack the branching in the following manner. A proposition form

Reading the proposition

Stacking

Branching

~~a

Double negation of A

Not applicable

(a ∧ b)

a and b

Not applicable

~(a ∧ b)

Negation of a and b

Not applicable

(a ∨ b)

Either a or b

Not applicable

~(a ∨ b)

Negation of either a or b

Not applicable

(continued)

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(continued) A proposition form

Reading the proposition

Stacking

(a → b)

a implies b

Not applicable

~(a → b)

Negation of a implies b

(a ≡ b)

a equivalence b

Not applicable

~(a ≡ b)

Negation of a equivalence b

Not applicable

Branching

Not applicable

Truth tree procedures (i) (ii) (iii) (iv) (v)

(vi) (vii) (viii) (ix)

Truth tree construction begins with the main logical connective of a symbolic proposition. Truth tree construction proceeds downward, while branch evaluation proceeds upward. Propositions that entail stacking should be decomposed before propositions that entail branching. Whenever a proposition is decomposed, a checkmark is placed to the right of it. Checked propositions are ignored at the time of branch evaluation. A truth tree construction is said to be finished when the given proposition is completely decomposed into a simple proposition or negated simple proposition. A branch is closed if and only if it contains a contradiction. That branch is blocked by placing an ‘x’ at the bottom end. Any branch that is not closed is said to be open. Such branches are notified by placing a ‘0’ at the bottom end. A truth tree is closed if and only if its branches are closed. When all the branches of a truth tree are closed due to self-contradiction, the argument is treated as valid; otherwise, the argument is considered invalid.

With these truth tree procedures and rules, let us find out the validity of the following arguments that are presented through symbolic propositions.

11.2 Beth Tree (Truth Tree)

(a) (b) (c) (d)

181

(((p → q) ∧ ~q) → ~p) (((p → q) ∧ p) → q) {(p → (p → q)) → ((p → q) → (q → p))} {((p → q) ∧ (~r → ~q)) → (p → r)}

Example-1: (((p → q) ∧ ~q) → ~p) This argument has two premises ‘(p → q)’ and ‘~q’ and a conclusion ‘~p’. Let us list the premises and conclusion as per the truth tree procedures. Step-1

In step-1, after listing the premises, we negate the conclusion ‘~p’, which becomes ‘~~p’. Step-2

In step-2, we put the checkmark right to the ‘~q’ and ‘p’ as both propositions are decomposed. The proposition ‘~~p’ is stacked to ‘p’. Step-3

In step-3, we decompose ‘(p → q)’ proposition and place the checkmark right of it. We found the ‘~p’ and ‘q’ branches, and there is no further proposition left to decompose. Upon branch evaluation of this truth tree through upward movement, we

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found ‘p’ and ‘~p’, ‘q’, and ‘~q’. Hence, the truth tree contains self-contradiction. As a result, we block the branch ‘~p’ and ‘q’ by placing ‘x’ at their bottom. Since all the branches of this truth tree are closed and they involve in selfcontradiction, the argument is not satisfied. It means, negation to the conclusion, which placed while listing the argument, is not correct. Thus, the given premises (propositions) and the conclusion together are treated as a valid argument. Example-2: (((p → q) ∧ p) → q) This argument has two premises ‘(p → q)’ and ‘p’ and a conclusion ‘q’. Let us list the premises and conclusion as per the truth tree procedures. Step-1

In step-1, after listing the premises, we negate the conclusion ‘q’ that becomes ‘~q’. Step-2

In step-2, we put the checkmark right to the ‘p’ and ‘~q’, as both the propositions are decomposed. Step-3

In step-3, we decompose the ‘(p → q)’ proposition and put the checkmark right to it. We found the ‘~p’ and ‘q’ branches, and there is no further proposition left out to decompose further. Upon branch evaluation of this truth tree through upward movement, we found in the left branch ‘p’ and ‘~p’, in the right branch ‘q’ and ‘~q’.

11.2 Beth Tree (Truth Tree)

183

Hence, the truth tree contains self-contradiction. As a result, we block the branch ‘~p’ and ‘q’ by placing ‘x’ at their bottom. Since all the branches of this truth tree are closed and they involve in selfcontradiction, the argument is not satisfied. It means, negation to the conclusion, which placed while listing the argument, is not correct. Thus, the given premises (propositions) and its conclusion together are regarded as a valid argument. Example-3: {(p → (p → q)) → ((p → q) → (q → p))} Let us list the premise and conclusion as per the truth tree procedures. Step-1

In step-1, after listing the premise, we negate the conclusion, that is ‘~((p → q) → (q → p))’. Step-2

In step-2, we apply the ‘~(a → b)’ truth tree rule and stack the propositions. We also placed a checkmark right to the proposition for its decomposition. Step-3

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In step-3, we apply the ‘~(a → b)’ truth tree rule and stack the propositions. We also placed a checkmark right to the compound proposition for its decomposition. Step-4

In step-4, we apply the ‘(a → b)’ truth tree rule and branch out the propositions. We also placed a checkmark right to the compound proposition for its decomposition. Step-5

In step-5, we apply the ‘(a → b)’ truth tree rule and branch out the propositions. We also placed a checkmark right to the compound proposition for its decomposition. But decomposition of the compound proposition is not yet completed. Upon evaluation

11.2 Beth Tree (Truth Tree)

185

of the left branch ‘~p’, we found that this proposition does not involve in selfcontradiction; hence, we place ‘0’ at its bottom end. It is a self-consistent proposition. Again, upon consideration of the right branch ‘~p’ of the tree, it is found that it is not resulting in self-contradiction. Hence, the branch is marked with ‘0’ at its bottom end. Step-6

In step-6, we apply the ‘(a → b)’ truth tree rule and branch out the propositions. We also placed a checkmark right to the compound proposition for its decomposition. In this step, all the propositions are decomposed into either a simple proposition or a negated proposition. Upon evaluation of the extreme left branch ‘~p’ and ‘q’, we found that these propositions do not involve in self-contradiction. They are selfconsistent propositions. Similarly, upon evaluation of the ‘~p’ and ‘q’ of the right branch of the truth tree, we found that these propositions also do not involve selfcontradiction. They are self-consistent propositions. Since all the branches of this truth tree are not closed and they do not involve in self-contradiction, the argument is satisfied. It means, negation to the conclusion, which placed while listing the argument, stands correct. Thus, the given premises (propositions) and its conclusion together are regarded as an invalid argument. Example-4: {((p → q) ∧ (~r → ~q)) → (p → r)} Let us list the premise and conclusion as per the truth tree procedures.

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Step-1

In step-1, after listing the premises, we negate the conclusion, that is ‘~(p → r)’. Step-2

In step-2, we apply the ‘~(a → b)’ truth tree rule and stack the propositions. We also placed a checkmark right to the compound proposition for its decomposition. Step-3

In step-3, we apply the ‘(a → b)’ truth tree rule and branch out the propositions. We also placed a checkmark right to the compound proposition for its decomposition. We further apply ‘~~a’ truth tree rule to decompose the left side branch, that is ‘~~r’. We placed a checkmark right to the double negative proposition for its decomposition.

11.2 Beth Tree (Truth Tree)

187

Step-4

In step-4, we apply the ‘(a → b)’ truth tree rule and branch out the propositions. We also placed a checkmark right to the compound proposition for its decomposition. Upon evaluation of the extreme left branch ‘r’ and two right branches ‘~p’ and ‘q’, we found that these propositions involve in self-contradiction. Hence, we put ‘x’ at its bottom. Since all the branches of this truth tree are closed and they involve in self-contradiction, the argument is not satisfied. It means negation to the conclusion, which placed while listing the argument, stands incorrect. Thus, the given premises (propositions) and its conclusion together are regarded as a valid argument.

11.3 Propositional Derivation Formulae Propositional derivation formulae are known as ‘rules of replacement’. It states that a proposition can be derived from another proposition of having a similar truth-value. In other words, a proposition can replace another proposition, as they are equivalent to each other and having a similar truth-value. Formula-1: ∼∼ a ≡ a This formula states that negation of the negation of a proposition is the proposition itself. So, double negation becomes an affirmation. Formula-2: (a ∧ b) ≡ (b ∧ a) This formula is known as commutative law. It enunciates that a and b is equal to b and a. Formula-3: (a ∨ b) ≡ (b ∨ a)

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This formula is known as also commutative law. It enunciates that a or b is equal to b or a. Formula-4: (a ∧ (b ∧ c)) ≡ ((a ∧ b) ∧ c) This formula is known as associative law. It expresses that a, and b and c is equal to a and b, and c. Formula-5: (a ∨ (b ∨ c)) ≡ ((a ∨ b) ∨ c) This formula is also known as associative law. It expresses that a, or b or c is equal to a or b, or c. Formula-6: (a ∧ (b ∨ c)) ≡ ((a ∧ b) ∨ (a ∧ c)) This formula is called distributive law. It is read as a, and b or c is equal to a and b or a and c. Formula-7: (a ∨ (b ∧ c)) ≡ ((a ∨ b) ∧ (a ∨ c)) This formula is called distributive law. It is read as a or, b and c is equal to a or b and a or c. Formula-8: (a ∧ b) ≡ ∼ (∼ a ∨ ∼ b) This formula is termed as De Morgan’s law. It is read as a and b is equal to negation of the ‘negation of a’ or ‘negation of b’. The truth-value of ‘(a ∧ b)’ and the truthvalue of ‘~(~a ∨ ~b)’ are identical to each other. In this case, a disjunctive compound proposition is derived from a conjunctive proposition. To explain, a proposition in which conjunction occurs may be expressed through disjunction if we negate both the components of conjunction and the conjunction itself. Formula-9: (a ∨ b) ≡ ∼ (∼ a ∧ ∼ b) This formula is also termed as De Morgan’s law. It is read as a or b is equal to negation of the ‘negation of a’ and ‘negation of b’. The truth-value of ‘(a ∨ b)’ and the truth-value of ‘~(~a ∧ ~b)’ are identical to each other. In this case, a conjunctive compound proposition is derived from a disjunctive proposition. To explain, a proposition in which a disjunction occurs may be expressed through conjunction if we negate both the components of disjunction and the disjunction itself. Formula-10: (a → b) ≡ (∼ b → ∼ a) This formula is known as the law of contraposition. It is read as a implies b is equal to negation of b implies the ‘negation of a’. The truth-value of ‘(a → b)’ is identical

11.3 Propositional Derivation Formulae

189

with the truth-value of ‘(~b → ~a)’. It expresses that an implication compound proposition can be reduced to an implication compound proposition provided that we need to interchange its antecedent and consequent by negating them both. Formula-11: (a → b) ≡ (∼ a ∨ b) This formula is a definition of material implication. It is read as a implies b is equal to negation of a or b. The truth-value of ‘(a → b)’ is identical with the truth-value of ‘(~a ∨ b)’. It states that an implication compound proposition can be expressed in the disjunctive proposition form provided that the antecedent of the implicative proposition gets negated in the disjunctive compound proposition. Formula-12: (a → b) ≡ ∼ (a ∧ ∼ b) This formula is also a definition of material implication. It is read as a implies b is equal to negation of a and negation of b. The truth-value of ‘(a → b)’ is identical with the truth-value of ‘~(a ∧ ~b)’. This formula enunciates that an implication compound proposition can be expressed in conjunctive proposition form provided that the consequent of the implicative proposition gets negated in the conjunctive proposition and the conjunctive proposition gets negated itself. Formula-13: {((a → b) ∧ (b → c)) → (a → c)} This formula is known as transitivity of implication. It is read as if a implies b and b implies c then a implies c. It states that if a proposition ‘a’ implies another proposition ‘b’ and the proposition ‘b’ implies ‘c’, then the proposition ‘a’ implies ‘c’ necessarily. Formula-14: (i) ((a → b) ∧ a) → b (ii) ((a → b) ∧ ∼ b) → ∼ a These formulas are known as the law of detachment. It conveys that if the antecedent of an implicative proposition is affirmed together with the implicative proposition then consequent of the implicative proposition follows. Further, it states that if the consequent of an implicative proposition is negated together with the implicative proposition then the negation of the antecedent of the implicative proposition follows. Formula-15: (a ≡ b) ≡ ((a → b) ∧ (b → a)) This formula is called material equivalence. It is read as an equivalence b is equal to a implies b and b implies a. The truth-value of ‘(a ≡ b)’ is identical with the truth-value of ‘((a → b) ∧ (b → a))’. It expresses that an equivalence compound

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proposition can be reduced to a combination of implication and conjunction logical proposition form. Formula-16: (a ≡ b) ≡ ((a ∧ b) ∨ (∼ a ∧ ∼ b)) This formula is also called as material equivalence. It is read, an equivalence b is equal to a and b or negation of a and negation of b. The truth-value of ‘(a ≡ b)’ is identical to the truth-value of ‘((a ∧ b) ∨ (~a ∧ ~b))’. It expresses that an equivalence compound proposition can be reduced to a combination of conjunction and disjunction logical proposition form. Formula-17: ((a ∧ b) → c) ≡ (a → (b → c)) This formula is also called as law of exportation. It is read if a and b then c is equivalent to a implies, b implies c. The truth-value of ‘((a ∧ b) → c)’ is identical with the truth-value of ‘(a → (b → c))’. It conveys that a combination of implication and conjunction proposition can be reduced to an implication proposition form. Formula-18: (a ∨ a) ≡ a This formula is called as law of tautology. It is read that a or a is equivalent to a. Formula-19: (a ∧ a) ≡ a This formula is also called as law of tautology. It is read that a and a is equivalent to a. Formula-20: (i) (a ∧ b) → a (ii) (a ∧ b) → b We have mentioned two formulas under formula-20 because of their similar nature. The first one is read as a and b implies a, while the second one is read as a and b implies b. This formula enunciates that conjunction implies either of the propositions conjoined. Formula-21: (p ∨ ∼ p) This formula is known as the law of excluded middle. It is read p or negation p. It states that every proposition must be either true or false. If one option is denied out of two alternatives, then the other option is to be accepted. The above formulae are mentioned in the table below for immediate reference and use.

11.3 Propositional Derivation Formulae

191

Sr. No.

Name of formula

Formula

Formula-1

Double negation

∼∼ a ≡ a

Formula-2

Commutative law

(a ∧ b) ≡ (b ∧ a)

Formula-3

Commutative law

(a ∨ b) ≡ (b ∨ a)

Formula-4

Associative law

(a ∧ (b ∧ c)) ≡ ((a ∧ b) ∧ c)

Formula-5

Associative law

(a ∨ (b ∨ c)) ≡ ((a ∨ b) ∨ c)

Formula-6

Distributive law

(a ∧ (b ∨ c)) ≡ ((a ∧ b) ∨ (a ∧ c))

Formula-7

Distributive law

(a ∨ (b ∧ c)) ≡ ((a ∨ b) ∧ (a ∨ c))

Formula-8

De Morgan’s law

(a ∧ b) ≡ ∼ (∼ a ∨ ∼ b)

Formula-9

De Morgan’s law

(a ∨ b) ≡ ∼ (∼ a ∧ ∼ b)

Formula-10

Contrapositive law

(a → b) ≡ (∼ b → ∼ a)

Formula-11

Material implication

(a → b) ≡ (∼ a ∨ b)

Formula-12

Material implication

(a → b) ≡ ∼ (a ∧ ∼ b)

Formula-13

Transitive implication

{((a → b) ∧ (b → c)) → (a → c)}

Formula-14

Detachment law

(i) ((a → b) ∧ a) → b (ii) ((a → b) ∧ ∼ b) → ∼ b

Formula-15

Material equivalence

(a ≡ b) ≡ ((a → b) ∧ (b → a))

Formula-16

Material equivalence

(a ≡ b) ≡ ((a ∧ b) ∨ (∼ a ∧ ∼ b))

Formula-17

Law of exportation

((a ∧ b) → c) ≡ (a → (b → c))

Formula-18

Law of tautology

(a ∨ a) ≡ a

Formula-19

Law of tautology

(a ∧ a) ≡ a

Formula-20

Law of conjunction

(i) (a ∧ b) → a (ii) (a ∧ b) → b

Formula-21

Law of excluded middle

(p ∨ ∼ p)

We will use these formulae to reduce symbolic propositions to either conjunctive normal form (CNF) or disjunctive normal form (DNF). The objective of reducing a given symbolic proposition into either DNF or CNF is to find out the truth-value of the proposition, whether it is a tautology or not. The DNF and CNF are regarded as ‘decision procedure methods’ to determine the truth-value of the symbolic compound and complex propositions.

11.4 Disjunctive Normal Form (DNF) Disjunctive normal form (DNF) is one where a symbolic proposition can be transformed into an expression consists of a disjunction of conjunctions. In this case, conjunction consists of propositional variables and their negations, and each conjunctive proposition is connected with another conjunctive proposition through a disjunctive logical connective. The disjunction built up from these conjunctions portrays a

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standard ordering and bracketing. For example, . . . (∼ a ∧ b) ∨ (b ∧ p) ∨ (∼ p ∧ a) ∨ ∼ p . . . By applying the above formulae, we can transform a symbolic proposition into the DNF form. While converting the symbolic proposition into the DNF form, we need to do the following things. First, strike out all double negations by applying formula1. Second, apply material implication formulas (formulas 11 and 12) and convert the proposition into its equivalent form. Third, apply De Morgan’s laws (formulas 8 and 9) to get rid of brackets. We need to proceed further by applying other formulas as and when required while converting the symbolic proposition into DNF. In the DNF, we must find a proposition or its negation that connects with ‘∨’. If there is no atomic propositional variable found, then there must be compound conjunctive propositions that consist of atomic propositional variables or their negation connected with the logical connective ‘∧’, and each conjunctive proposition is connected with another conjunctive proposition with a disjunction ‘∨’. Further, there shall not be a negative compound conjunctive proposition in the DNF form. A DNF is regarded as tautology when one of its elements is true. It means that the truth-value of one of the conjuncts must be true. Again, a conjunctive compound proposition is true when the truth-values of its propositional variables are true. Now, let us convert the following symbolic propositions into DNF by applying the above formulas and find out whether these propositions are tautology or not. (i) (ii) (iii)

{((p → q) ∧ ~q) → ~p} {((p → q) ∧ p) → q} ~(p → q) → (~r ∨ p)

Example-1: {((p → q) ∧ ~q) → ~p} {((p → q) ∧ ~q) → ~p} = ~((p → q) ∧ ~q) ∨ ~p = ~((~p ∨ q) ∧ ~q) ∨ ~p = (~(~p ∨ q) ∨ ~~q) ∨ ~p = (~(~p ∨ q) ∨ q) ∨ ~p = ((p ∧ ~q) ∨ q) ∨ ~p = ((p ∨ q) ∧ (~q ∨ q)) ∨ ~p = ((q ∨ p) ∧ (q ∨ ~q)) ∨ ~p = (q ∨ (p ∧ ~q)) ∨ ~p = (p ∧ ~ q) ∨ ~p ∨ q

This proposition is given to us Applied formula-11 Applied formula-11 Applied formula-9 Applied formula-1 Applied formula-8 Applied formula-7 Applied formula-3 Applied formula-7 Applied formula-3 (This is DNF)

This expression is now in disjunctive normal form. It is so because, within the bracket, we find ‘p’ and ‘~q’ are connected with the conjunctive logical operator. The bracketing proposition and other propositional variables are connected with a disjunctive logical operator. We find ‘p’ and ‘~p’, ‘q’ and ‘~q’ in the DNF; those are connected with disjunction, and the truth-value of ‘p ∨ ~p’ and ‘q ∨ ~q’ is true

11.4 Disjunctive Normal Form (DNF)

193

when at least one of the propositional variable is true. Thus, the whole expression is a tautology. Example-2: {((p → q) ∧ p) → q} {((p → q) ∧ p) → q} = ~((p → q) ∧ p) ∨ q = ~((~p ∨ q) ∧ p) ∨ q = (~(~p ∨ q) ∨ ~p) ∨ q = (~~(p ∧ ~q) ∨ ~p) ∨ q = ((p ∧ ~q) ∨ ~p) ∨ q = (p ∧ ~q) ∨ ~p ∨ q

This proposition is given to us Applied formula-11 Applied formula-11 Applied formula-8 Applied formula-9 Applied formula-1 Applied formula-15 (This is DNF)

This expression is now in disjunctive normal form. It is so because, within the bracket, we find ‘p’ and ‘~q’ are connected with the conjunctive logical operator. The bracketing proposition and other propositional variables are connected with a disjunctive logical operator. We find ‘p’ and ‘~p’, ‘q’ and ‘~q’ in the DNF; those are connected with disjunction, and the truth-value of ‘p ∨ ~p’ and ‘q ∨ ~q’ is true when at least one of the propositional variables is true. Thus, the whole expression is a tautology. Example-3: ~(p → q) → (~r ∨ p) ~ (p → q) → (~r ∨ p) = ~~(p → q) ∨ (~r ∨ p) = (p → q) ∨ (~r ∨ p) = (~p ∨ q) ∨ (~r ∨ p) = (~p ∨ q ∨ ~ r ∨ p)

This proposition is given to us Applied formula-11 Applied formula-1 Applied formula-11 Applied formula-5 (This is DNF)

This expression is now in disjunctive normal form. It is so because we find propositional variables and their negation are connected with the disjunctive logical operator. We find ‘p’ and ‘~p’ in the DNF, which are connected with disjunction. The truthvalue of ‘p ∨ ~p’ is true when at least one of the propositional variables is true. Thus, the whole expression is a tautology.

11.5 Conjunctive Normal Form (CNF) Conjunctive normal form (CNF) is one where a symbolic proposition can be converted into an expression consists of a conjunction of disjunctions. In this case, disjunctive propositions consist of propositional variables and their negations, and each disjunctive proposition is connected with another disjunctive proposition through a conjunctive logical operator. The conjunction built up from the disjunctions portrays a standard ordering and bracketing. For example, . . . (a ∨ ∼ b) ∧ (∼ c ∨ ∼ b) ∧ (∼ c ∨ a) ∧ ∼ a . . .

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By applying the propositional derivation formulae, we can convert a symbolic proposition into the CNF form. While converting the symbolic proposition into CNF form, we need to do the following things. First, strike out all double negations by applying formula-1. Second, apply material implication formulas (formulas 11 and 12) and convert the proposition into its equivalent form. Third, apply De Morgan’s laws (formulas 8 and 9) to get rid of brackets. We need to proceed further by applying other formulas as and when required while converting the symbolic proposition into CNF. In the CNF, if we find a proposition that connects with the same proposition with ‘∧’, the CNF is treated as a tautology. If there is no atomic propositional variable found in the CNF, then there must be compound disjunctive propositions that consist of atomic propositional variables or their negation connected with the logical connective ‘∨’, and each disjunctive proposition is connected with another disjunctive proposition with a conjunction ‘∧’. Further, there shall not be a negative compound conjunctive proposition in the CNF. A CNF is regarded as tautology when the truth-value of both the disjunctive propositions is true. A disjunctive compound proposition is true when the truth-value of one of its propositional variables is true. Now, let us convert the following symbolic propositions into CNF by applying the propositional derivation formulae and find out whether these propositions are tautology or not. (i) (ii) (iii) (iv)

{((p → q) ∧ p) → q} (p → (q ∧ r)) ∧ (~p → (~q ∧ ~r)) (q ∨ (p ∧ r)) ∧ ~((p ∨ r) ∧ q) ((p → q) ≡ (~p ∨ q))

Example-1: {((p → q) ∧ p) → q} {((p → q) ∧ p) → q} = ~((p → q) ∧ p) ∨ q = ~((~p ∨ q) ∧ p) ∨ q = (~(~p ∨ q) ∨ ~p) ∨ q = ((p ∧ ~q) ∨ ~p) ∨ q = (~p ∨ (p ∧ ~q)) ∨ q = ((~p ∨ p) ∧ (~p ∨ ~q)) ∨ q = (~p ∨ p ∨ q) ∧ (~p ∨ ~q ∨ q)

This proposition is given to us Applied formula-11 Applied formula-11 Applied formula-9 Applied formula-8 Applied formula-3 Applied formula-7 Applied formula-7 (This is CNF)

This expression is now in conjunctive normal form. It is so because we find propositional variables and their negation are connected with a disjunctive logical operator within the bracket and disjunctive propositions are connected with a conjunctive logical operator. In the CNF, we find ‘p’ and ‘~p’ in the left side disjunctive proposition and ‘q’ and ‘~q’ in the right-side disjunctive proposition. Since ‘p’ and ‘~p’, ‘q’ and ‘~q’ are connected with disjunction, the truth-value of the left side disjunctive proposition and right-side disjunctive proposition would be true even if one of its propositional variables is true. Further, it does not matter what would be the value of ‘q’ with regard to the left side disjunctive proposition and ‘~p’ with regard to right-side disjunctive proposition. The reason is that ‘q’ and ‘~p’ are connected with

11.5 Conjunctive Normal Form (CNF)

195

their propositional variables with disjunction. Since ‘p ∨ ~p’ is true in the left side disjunctive proposition and ‘q ∨ ~q’ is true in the right-side disjunctive proposition, the truth-value of the whole disjunctive proposition would be true. Since both the disjunctive propositions are joined with conjunction and their truth-values are true, the truth-value of the whole expression is true. It is so because the truth-value of a conjunctive proposition is true when both of its propositional variables are true. Thus, the symbolic expression is a tautology. Example-2: (p → (q ∧ r)) ∧ (~p → (~q ∧ ~r)) (p → (q ∧ r)) ∧ (~p → (~q ∧ ~r)) = (~p ∨ (q ∧ r)) ∧ (~~ p ∨ (~q ∧ ~r)) = (~p ∨ (q ∧ r)) ∧ (p ∨ (~q ∧ ~r)) = (~p ∨ q) ∧ (~p ∨ r) ∧ (p ∨ ~q) ∧ (p ∨ ~r)

This proposition is given to us Applied formula-11 Applied formula-1 Applied formula-7 (This is CNF)

This expression is now in conjunctive normal form. It is so because we find propositional variables and their negation are connected with a disjunctive logical operator within the bracket, and disjunctive propositions are connected with a conjunctive logical operator. In the CNF, we do neither find ‘p’ and ‘~p’; ‘q’ and ‘~q’; nor ‘r’ and ‘~r’ joined with a disjunctive logical operator. Hence, the truth-value of the disjunctive propositions is not true. This, in turn, the truth-value of the conjunctive proposition is false. Thus, the whole proposition is not a tautology. Example-3: (q ∨ (p ∧ r)) ∧ ~((p ∨ r) ∧ q) (q ∨ (p ∧ r)) ∧ ~((p ∨ r) ∧ q) = ((q ∨ p) ∧ (q ∨ r)) ∧ ~((p ∨ r) ∧ q) = ((q ∨ p) ∧ (q ∨ r)) ∧ (~(p ∨ r) ∨ ~q) = ((q ∨ p) ∧ (q ∨ r)) ∧ ((~p ∧ ~r) ∨ ~q) = ((q ∨ p) ∧ (q ∨ r)) ∧ (~q ∨ (~p ∧ ~r)) = (q ∨ p) ∧ (q ∨ r) ∧ (~q ∨ ~p) ∧ (~q ∨ ~r)

This proposition is given to us Applied formula-7 Applied formula-8 Applied formula-9 Applied formula-3 Applied formula-3 (This is CNF)

This expression is now in conjunctive normal form. It is so because we find propositional variables and their negation are connected with a disjunctive logical operator within the bracket, and disjunctive propositions are connected with a conjunctive logical operator. In the CNF, we do neither find ‘p’ and ‘~p’; ‘q’ and ‘~q’; nor ‘r’ and ‘~r’ joined with a disjunctive logical operator. Hence, the truth-value of the disjunctive propositions is not true. It thereby asserts that the truth-value of the conjunctive proposition is false, as the truth-value of the disjunctive propositions that are connected with a conjunctive logical operator in the CNF is not true. Thus, the whole proposition is not a tautology.

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Example-4: ((p → q) ≡ (~p ∨ q)) ((p → q) ≡ (~p ∨ q)) = ((~p ∨ q) ≡ (~p ∨ q)) = ((~p ∨ q) ∧ (~p ∨ q)) ∨ (~(~p ∨ q) ∧ ~(~p ∨ q)) = (~p ∨ q) ∨ ~(~p ∨ q) = (~p ∨ q) ∨ (p ∧ ~q) = (~p ∨ q ∨ p) ∧ (~p ∨ q ∨ ~q)

This proposition is given to us Applied formula-11 Applied formula-16 Applied formula-19 Applied formula-8 Applied formula-7 (This is CNF)

This expression is in the conjunctive normal form. It is so because we find propositional variables and their negation are connected with a disjunctive logical operator within the bracket, and disjunctive propositions are connected with a conjunctive logical operator. In the CNF, we find ‘p’ and ‘~p’ in the left side disjunctive proposition and ‘q’ and ‘~q’ in the right-side disjunctive proposition among other propositional variables. Since ‘p’ and ‘~p’, ‘q’ and ‘~q’ are connected with the disjunctive logical operator, the truth-value of the left side disjunctive proposition and the right-side disjunctive proposition would be true even if the truth-value of one of the propositional variables is true. Further, it does not matter what would be the value of ‘q’ concerning the left side disjunctive proposition and ‘~p’ concerning the right-side disjunctive proposition. The reason is that ‘q’ and ‘~p’ are connected with their respective propositional variables with disjunction. Since ‘p ∨ ~p’ is true in the left side disjunctive proposition and ‘q ∨ ~q’ is true in the right-side disjunctive proposition, the truth-value of the disjunctive propositions would be true. Since both the disjunctive propositions are joined with conjunction and their truth-values are true, the truth-value of the whole expression is true. It is so because the truth-value of a conjunctive proposition is true when both of its propositional variables are true. Thus, the whole symbolic expression is a tautology.

11.6 Proving the Validity of Arguments To test the validity of an argument, we need to find out whether the conclusion of the argument can be derived from its premises or not. To do so, there is a decision procedure we need to adapt known as ‘proving the validity of argument’. Proof of validity is a decision procedure to determine the validity or invalidity of an argument. Proving the validity of an argument is like solving a puzzle for which there will not be a fixed modus operandi or one strategy. We have to treat each argument differently, as each argument requires a different strategy to find out whether the given conclusion follows from its premises or not. To prove the validity of an argument, we need to apply the above-mentioned propositional derivation formulae. These formulae are the self-evident rules. We use these formulae to derive the conclusion from the premises. Let us test the following symbolical arguments and find out their validity. Argument-1 Premise-1: a → b

11.6 Proving the Validity of Arguments

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Premise-2: b → c Premise-3: a Conclusion: c Now, arrange the premises and the conclusion in a list and find out whether the conclusion can be derived from these premises by applying propositional derivation formulae. 1: 2: 3: 4. 5.

a→b b→c a Conclusion: c a→c c

(from 1 and 2 by applying the formula-13) (from 4 and 3 by applying the formula-14)

This argument is proved as a valid argument. Argument-2 Premise-1: a → ~b Premise-2: ~b → c Premise-3: c → d Premise-4: (a ∧ d) → e Conclusion: a → e Now, arrange the premises and conclusion in a list and find out whether the given conclusion can be derived from these premises by applying propositional derivation formulae. 1. 2. 3. 4. 5. 6. 7. 8.

a → ~b ~b → c c→d (a ∧ d) → e Conclusion: a → e a→c a→d a → (a ∧ d) a→e

(from 1 and 2 by applying the formula-13) (from 5 and 3 by applying the formula-13) (from 6 by applying the formula-20) (from 7 and 4 by applying the formula-13)

This argument is proved to be a valid argument. Argument-3 Premise-1: a → b Premise-2: b → c Premise-3: b → d Premise-4: ~d Premise-5: a ∨ b Conclusion: c

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Now, arrange the premises and conclusion in a list and find out whether the given conclusion can be derived from these premises by applying propositional derivation formulae. a→b b→c b→d ~d a∨b Conclusion: c ~b (from 3 and 4 by applying the formula-14) ~a (from 1 and 6 by applying the formula-14) b (from 5 and 7 by applying formula-21) c (from 2 and 8 by applying formula-14)

1. 2. 3. 4. 5. 6. 7. 8. 9.

This argument is proved to be a valid argument. Argument-4 Premise-1: a → b Premise-2: b → (c ∨ d) Premise-3: d → e Premise-4: e → f Premise-5: c → g Premise-6: a Premise-7: ~g Conclusion: f Now, arrange the premises and conclusion in a list and find out whether the given conclusion can be inferred from these premises by applying propositional derivation formulae. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

a→b b → (c ∨ d) d→e e→f c→g a ~g Conclusion: f a → (c ∨ d) c∨d ~c d d→f f

(from 1 and 2 by applying the formula-13) (from 8 and 6 by applying the formula-14) (from 5 and 7 by applying the formula-14) (from 9 and 10 by applying the formula-21) (from 3 and 4 by applying the formula-13) (from 12 and 11 by applying the formula-14)

11.6 Proving the Validity of Arguments

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This argument is proved as a valid argument. Let us now consider the following passages as arguments. Each argument consists of premises and a conclusion. We will symbolise the premises and conclusion of the argument by using the English alphabets and logical operators. Thereafter, we will find out whether the given conclusion is derived from the premises or not. In brief, we will find out the validity of the arguments by applying the propositional derivation formulae to them. Argument-5 If an administrator makes an important decision, and if she wants to implement it, then she must be rational in her decision. The administrator does not make an important decision. It is false that the administrator is rational in her decision. Therefore, the administrator will not implement her decision. Let a—An administrator makes an important decision. i—She wants to implement it. r—She must be rational in her decision. 1. 2. 3. 4. 5. 6.

(a ∧ i) → r ~a ~r Conclusion: ~i ~(a ∧ i) ~a ∨ ~i ~i

(from 1 and 3 by applying formula-14) (from 4 by applying formula-8) (from 5 and 2 by applying formula-21)

This argument is proved to be a valid argument. Argument-6 If the majority is not willing to accept the minority decision, there can be no democracy. If the minority does not respect the majority’s rights, then the majority will not be willing to accept the minority decision. The minority does not respect the rights of the majority. Therefore, there cannot be a democracy. Let m—The majority is willing to accept the minority decision. d—There can be a democracy. r—The minority respects the rights of the majority. 1. 2. 3. 4. 5.

~m → ~d ~r → ~m ~r Conclusion: ~d ~m ~d

(from 2 and 3 by applying formula-14) (from 1 and 4 by applying formula-14)

This argument is proved as a valid argument.

Part IV

Predicate Logic

Chapter 12

Predicate Logic

In the previous section, we discussed symbolic logic in detail. Symbolic logic is a part of sentential logic where a sentence is considered as the basic unit of an argument. While discussing symbolic logic, we discussed propositional variables, logical connectives, and the truth tabular method to determine the truth-value of a proposition. We also elucidated the Beth tree method, conjunctive normal form, and disjunctive normal form to determine the truth-value of complex propositions and judge the validity of arguments. In addition to these topics, we explained the methods to prove the validity of arguments by applying propositional derivation rules. In continuation to the previous section, in this section, we will deliberate over predicate logic. We will explain universal quantifier and existential quantifier. We will discuss the method to translate logical sentences to predicate logic form by considering the semantics of the sentences. We will also illustrate the opposition of propositions by considering categorical propositions of predicate logic. Further, we will test the validity of arguments by adopting propositional derivation rules.

12.1 What is Predicate Logic? In predicate logic, a proposition and its subject matter are taken into consideration while translating a proposition into symbolic form. Here, we need to consider the quality and quantity of a proposition while translating it into symbolic form. The aim of predicate logic is to explain the inner logical structure of a proposition while translating it to symbolic form. Gottlob Frege (1848–1925), a German philosopher who is believed to be the founder of predicate logic and analytic philosophy, writes that sentential logic does not express the concept of a proposition accurately when it is translated into symbolic logic. Thus, a symbolic logic sentence must consider the quality and quantity of a proposition to convey the subject matter of the proposition correctly. This would assist in determining the validity of an argument correctly.

© Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_12

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Frege explains the notion of ‘quantification’ in his work Concept Script (Begriffsschrift) (1879). Following Frege, Professor Alfred North Whitehead (1861–1947) and Bertrand Russell (1872–1970) elaborate the work on ‘quantification’ of predicate logic in their manuscript Principia Mathematica (Volume-1; 1910, Volume-2; 1912, Volume-3; 1913). These treatises highlight the flaws of sentential logic. In sentential logic, a proposition is considered the basic unit of an argument. The subject matter of the proposition is not taken into consideration while translating it to symbolic form. For example, Sita is a dancer—‘p’. Sita is not a dancer—‘~p’. All ladies are beautiful—‘p’. All ladies are not beautiful—‘~p’. In these examples, we find that ‘~p’ represents two propositions and ‘p’ represents two propositions. Even though the two propositions are symbolised as ‘p’, they both have affirmative quality but differ in quantity. A similar situation exists with ‘~p’ propositions. In this context, a question arises: is it correct to symbolise a proposition by considering its quality alone and ignoring its quantity? In sentential logic, each proposition is symbolised as either ‘p’ or ‘m’, or ‘s’ or a small letter of the English alphabet as a propositional variable. In this case, we consider the structure of the proposition when we symbolise the proposition. We do not consider the meaning of the proposition for its symbolisation. For example, we can symbolise the proposition ‘All swans are white’ as ‘p’ and ‘Grass is green’ as ‘p’. Here, ‘All swans are white’ and ‘Grass is green’ are affirmative propositions and symbolised as ‘p’, but do not convey a similar meaning. Further, the proposition ‘Some students are tall’ can be symbolised as ‘p’ and ‘All students are bachelors’ too can be symbolised as ‘p’. These two propositions are also judged as affirmative propositions but differ from each other concerning their meanings. It is so because, when the proposition ‘Some students are tall’ is conveying about a few students, the proposition ‘All students are bachelors’ is stating about all the students. Further, when the former proposition is talking about the tallness of students, the latter proposition is stating about students’ bachelorhood. Even though these two propositions significantly differ from each other, yet, in sentential logic, we symbolise these two propositions as ‘p’. In sentential logic, we consider the quality (affirmative or negative) of a proposition only while symbolising the proposition. This leads to a critical situation where, if we consider an affirmative proposition as ‘p’ in an argument irrespective of its quantity (universal or particular), then the argument may appear to be valid but it would not be valid indeed. For example, All students are cricket players. Some swimmers are students. Therefore, some swimmers are cricket players. This argument consists of three propositions: a major premise, a minor premise, and a conclusion. All these propositions are affirmative propositions. In the sentential

12.1 What is Predicate Logic?

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logic, let us symbolise the major premise as ‘p’, the minor premise as ‘q’, and the conclusion as ‘r’. So, we can write p q Therefore, r. We can write the argument in the symbolic proposition form as ((p ∧ q) → r). Now, the problems are neither ‘p’ nor ‘q’ symbol reveals the inner structure of the proposition (i.e. quantity of the propositions). Also, the symbols ‘p’ and ‘q’ do not state anything about the predicate of the proposition. Rather, they represent the whole proposition as such. Due to lack of clarity in the symbolic propositions of the major premise and minor premise, the derivation of the conclusion from the major and minor premise together will lead to confusion and bewilderment. Since we consider only the structure of the proposition, not the meaning of the proposition, in the sentential logic, we may end up in deriving invalid conclusion from the true premises. To fix the lacunas of sentential logic, predicate logic is introduced in the logic subject. Predicate logic is also known as predicate calculus. Predicate logic considers the quality and quantity of a proposition while translating the proposition into symbolic form. The objective of predicate logic is to make the internal structure of the proposition correct and vivid. In short, in predicate logic, a proposition’s quality (affirmative or negative) and quantity (universal or particular) are made clear while translating it into symbolic form. In this sense, predicate logic offers exact and precise information about the internal structure of the proposition. In predicate logic, importance is given to the predicate. It means we need to find out whether the predicate affirms or denies the subject in the proposition and further, whether the predicate affirms the subject partly or wholly. There may be cases where the predicate of a proposition states something about two different subjects of a proposition. For example, Mita and Mira are lecturers of Odisha government colleges. There are also situations in which a proposition has two predicates, and they refer to a subject and state something either affirmatively or negatively about it. For example, Miku is an intelligent and sincere student. Due to these peculiar characteristics of propositions, in predicate logic, both subject and predicate are viewed as properties of the propositions. In predicate logic, a proposition is thus explained with its properties. For example, ‘All swans are white’. This proposition is explained in predicate logic as if there is a thing that has the property of being a swan; then, it has the property of being white. Consider another example: ‘All logic students are intelligent’. This proposition in predicate logic is analysed as if a person (thing) is a logic student, then he/she has the property of being intelligent. Note that in predicate logic, we are concerned with a thing that has property(ies). Predicate logic gives more information about the proposition compared to sentential logic. However, predicate logic borrows many things from sentential logic. For example, in sentential logic, we use logical constant and logical connectives while translating a compound proposition into the symbolic proposition. In predicate logic also, we use logical constant and logical connectives while translating compound propositions into symbolic propositions. In sentential logic, to prove the validity of

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an argument, we use propositional derivation rules, etc. We use the propositional derivational rules in the predicate logic also to prove the validity of an argument. Thus, it would not be inappropriate to say that predicate logic is an extension and advancement of sentential logic. In predicate logic, the following symbols are used to translate propositions to symbolic propositions. (i) (ii) (iii) (iv) (v)

Individual variables: x, y, z, … Individual constants for proper names: a, b, c, … Universal quantifier: ∀x Existential quantifier: ∃x Predicates: L, M, N, …

The symbol ‘∀x’ is conveyed as ‘for all values of x’, and ‘∃x’ is read as ‘there is at least one thing x such that’. The universal quantifier ‘∀x’ relates to A and E categorical propositions, whereas existential quantifier ‘∃x’ deals with I and O categorical propositions. Now, let us discuss four categorical propositions A, E, I, and O of predicate logic and analyse those with appropriate examples. A: All logic students are rational beings. E: No logic students are astronauts. I: Some logic students are swimmers. O: Some logic students are not cricket players.

12.2 Universal Quantifier The universal quantifier is symbolised as ‘∀x’. It is read as ‘for all x’. Universal quantifier ‘∀x’ is used in the case of A (universal affirmative) and E (universal negative) propositions. In the case of A and E propositions, all the members of the subject are taken into consideration while symbolising the propositions. With regard to A proposition, all the members of the subject have a quality or thing that the predicate states about, whereas in the case of E proposition, all the members of the subject deny a quality or a thing that the predicate states about. For example, ‘All logic students are rational beings’. This is an A proposition. Now, let us symbolise it in the predicate logic. A: All logic students are rational beings. = (Whatever x may be) (If x is logic student, then he/she is a rational being) = (∀x) (If x is L then x is R) = (∀x) (Lx → Rx) In this case, we have considered L for ‘logic student’ and R for ‘rational being’, and ‘if then’ proposition is symbolised with logical connective ‘implication’. An example of E proposition is ‘No logic students are astronauts’. Now, we will symbolise the proposition in the predicate logic.

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E: No logic students are astronauts. = (Whatever x may be) (If x is a logic student, then x is not an astronaut) = (∀x) (If x is L then x is not an A) = (∀x) (Lx → ~Ax) In this case, we have taken L for ‘logic student’ and A for ‘astronaut’. The word not is symbolised as ‘~’.

12.3 Existential Quantifier The existential quantifier is symbolised as ‘∃x’. It is read as ‘for some x’. Existential quantifier ‘∃x’ is used for I (particular affirmative) and O (particular negative) propositions. In the case of I and O propositions, at least one member of the subject is taken into consideration while symbolising the propositions. In the case of existential quantifier, a propositional function affirms that at least one true substitution instance exists as true. It means that we are substituting an individual constant for its individual variable. With regard to I proposition, at least one member of the subject has a quality or thing that the predicate states about, whereas in the case of O proposition, at least one member of the subject denies a quality or a thing that the predicate states about. For example, ‘Some logic students are swimmers’. This is an I proposition. Now, let us symbolise it in the predicate logic. I: Some logic students are swimmers. = (There is something x) (This x is a student and this x is a swimmer) = (∃x) (This x is a S and this x is M) = (∃x) (Sx ∧ Mx) In this case, we have considered S for ‘student’ and M for ‘swimmer’, and the proposition is symbolised with the conjunction logical connective. Now, we will consider an O proposition and symbolise the proposition in the predicate logic. O: Some logic students are not cricket players. = (There is something x) (This x is a student and this x is not a cricket player) = (∃x) (This x is a S and this x is not a C) = (∃x) (Sx ∧ ~Cx) In this symbolic proposition, we have taken S for ‘student’ and C for ‘cricket player’. The word not is symbolised as ‘~’.

12.4 Atomic Proposition An atomic proposition is one that is not combined with any other proposition, and it cannot be divided into further simple propositions. Atomic propositions are known as

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elementary propositions and simple propositions. Examples of atomic propositions are: Miku is a good boy, the author of Nicomachean Ethics is a tall man, etc. These propositions are not combined with any other proposition. They do not contain any logical connective, such as ‘and’, ‘either or’, and ‘if then’ to determine their truth function. The subject part of an atomic proposition is either a proper name (Miku) or a definite description (the author of Nicomachean Ethics) of the subject. In the predicate logic, it is understood that the subject of a proposition must be a thing or a person that has a quality that the predicate is stating about. Thus, the symbolisation of an atomic proposition in predicate logic is carried out in a unique way where both quality and quantity of the proposition are taken into consideration. We use small alphabets like a, b, c, etc., for a person or a thing, and capital letters like A, B, C, etc., for quality of the subject. The quality symbol is used to the left of the person or a thing. For example: Example-1 Miku is a good boy. = m is a G = Gm In this symbolic proposition, ‘m’ stands for Miku and ‘G’ stands for ‘good boy’. Example-2 Susama is a singer. = s is a N = Ns In this symbolic proposition, ‘s’ stands for Susama and ‘N’ stands for ‘singer’. Now, let us consider a compound proposition where a subject has two qualities, and symbolise it in the predicate logic. Example-3 Mira is a singer and a dancer. = m is a S and m is a D = Sm ∧ Dm In this symbolic proposition, ‘m’ stands for Mira, ‘S’ stands for ‘singer’, and ‘D’ stands for ‘dancer’. Example-4 Mita is a badminton player and a writer. = m is a B and m is a W = Bm ∧ Wm In this symbolic proposition, ‘m’ stands for Mita, ‘B’ stands for ‘badminton player’, and ‘W’ stands for ‘writer’.

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12.5 Opposition of Propositions In Chap. 4, we discussed the opposition of propositions as part of the sentential logic. We borrow the rules and conditions of the opposition of propositions and use those in the predicate logic. We borrow A, E, I, and O propositions of sentential logic to explain them in the predicate logic. Four forms of the opposition of propositions are found in the predicate logic that are related to A, E, I, and O atomic propositions. These are: (i) (ii) (iii) (iv)

Sub-alternation Contrary Sub-contrary Contradictory

In Fig. 12.1, A and E propositions are related to each other on contrary relation, and A and I, and E and O propositions are related to each other on sub-alternation relation. The propositions I and O are having the sub-contrary relation. Further, A and O, and E and I propositions are related to each other on contradictory relation. The categorical propositions A, E, I, and O are dealt with quantity and quality of the propositions. The propositions A and E are symbolised in universal quantifiers, whereas I and O propositions are symbolised in existential quantifiers. Now, let us symbolise A, E, I, and O propositions in the predicate logic with subject X and predicate Y. Symbol of categorical propositions

Sentential logic

Predicate logic

A

All X is Y

(∀x) (Xx → Yx)

E

No X is Y

(∀x) (Xx → ~Yx)

I

Some X is Y

(∃x) (Xx ∧ Yx)

O

Some X is not Y

(∃x) (Xx ∧ ~Yx)

Fig. 12.1 Opposition of Propositions

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12.6 Translation of Logical Propositions to Predicate Logic Unlike sentential logic, in predicate logic, both quality and quantity of the proposition are taken into consideration while symbolising it. The aim of predicate logic is to reveal the inner structure of the proposition through its symbolic form. As of now, we have discussed the methods to symbolise atomic propositions and compound propositions in the predicate logic. But in the linguistic discourse, we may not always find simple or compound propositions. Rather, there are occasions, wherein propositions are not found in a standard logical form. In order to symbolise these propositions in the predicate logic, we need to reduce these propositions into the logical standard form and need to identify quality and quantity of these propositions. We will elucidate some examples of this sort of proposition below for your reference, understanding, and discernment. For example, Example-1 None of the civil engineers in the city could save the flyover. = No civil engineer in the city is a person who could save the flyover. = (Whatever x may be) (If x is a civil engineer in the city, then x is a person who could not save the flyover) = (∀x) (Cx → ~Fx) Example-2 Not all students like logic. = Some students are not liking logic. = (There is something x) (This x is a student and x is not liking logic) = (∃x) (Sx ∧ ~Lx) Example-3 Only professors are doctorate holders. = All those who are doctorate holders are professors. = (Whatever x may be) (If x is a doctorate holder, then x is a professor) = (∀x) (Dx → Px) In the linguistic discourse, it is also found that propositions are having a subject and a predicate, and the subject and predicate have more than one quality. To symbolise this sort of proposition in predicate logic, we need to find out the quality and quantity of the proposition. For example, Example-1 Girls who are swimmers are strong. = (Whatever x may be) (If x is a girl and x is a swimmer, then x is strong) = (∀x) ((Gx ∧ Mx) → Sx) In this symbolic proposition, G stands for ‘girl’, M stands for ‘swimmer’, and S stands for ‘strong’. Example-2 Boys who play badminton are strong and active.

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= (Whatever x may be) (If x is a boy and x plays badminton, then x is strong and x is active) = (∀x) ((Bx ∧ Nx) → (Sx ∧ Ax)) In this symbolic proposition, B stands for ‘boy’, N stands for ‘badminton’, S stands for ‘strong’, and A stands for ‘active’. Example-3 It is false that some students are illiterate. = ~(There is something x) (This x is a student and this x is illiterate) = ~(∃x) (Sx ∧ Ix) In this symbolic proposition, S stands for ‘student’ and I stands for ‘illiterate’. There are also propositions consisting of more than one thing (or a person) and a predicate. In order to symbolise this sort of proposition in the predicate logic, we need to find out the names of the things or persons and predicates used in the proposition. For example, Example-1 Everyone likes something. = (Whatever x may be) (If x is a person, (there is something y, y is a thing), then x likes y) = (∀x) (Px → (∃y) Lxy) In this symbolic proposition, ‘everyone’ is reduced to ‘every person’. Hence, it is represented through a universal quantifier. We take variable ‘x’ to represent ‘every person’. The word ‘something’ is represented in existential quantifier. In this case, we take the variable ‘y’ to represent ‘something’. The word ‘like’ is a predicate that connects ‘everyone’ with ‘something’. As a result, the proposition states that if there is a person and there is a thing, then the person likes that thing. Example-2 Someone loves everyone. = (There is something x) (This x is a person and (whatever y may be, if y is a person then x loves y)) = (∃x) (Px ∧ (∀y) (Py → Lxy)) In this symbolic proposition, ‘someone’ is reduced to ‘a person’. Hence, it is symbolised in existential quantifier. The word ‘everyone’ is reduced to ‘every person’ and represented in a universal quantifier. For universal quantifier and existential quantifier, we have taken two separate variables, x and y, respectively. The word ‘love’ is the predicate that connects ‘a person’ with ‘every person’. As a result, this proposition states that there is a person and if there are other persons, then a person loves every other person. Example-3 Everyone loves someone. = (Whatever x may be) (If x is a person then (there is something y, this y is a person and x loves this person))

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= (∀x) (Px → (∃y) (Py ∧ Lxy)) In this symbolic proposition, ‘someone’ is reduced to ‘a person’. Hence, it is represented through existential quantifier. The word ‘everyone’ is reduced to ‘every person’ and represented through a universal quantifier. For universal quantifier and existential quantifier, we have taken two separate variables, x and y, respectively. The word ‘love’ is a predicate that connects ‘every person’ with ‘a person’. This proposition thus states that if there is a person, then there is another person and each and every person loves that person.

12.7 Proving Validity of Arguments In sentential logic, we explained the techniques and procedures to test the validity of symbolic arguments by applying propositional derivation formulae. The propositional derivation formulae are regarded as self-evident rules. We use these formulae in predicate logic to test the validity of symbolic arguments. For our convenience and immediate reference, we mention the propositional derivation formulae to test the validity of the arguments. S. No.

Name of formula

Formula

Formula-1

Double negation

~~a ≡ a

Formula-2

Commutative law

(a ∧ b) ≡ (b ∧ a)

Formula-3

Commutative law

(a ∨ b) ≡ (b ∨ a)

Formula-4

Associative law

(a ∧ (b ∧ c)) ≡ ((a ∧ b) ∧ c)

Formula-5

Associative law

(a ∨ (b ∨ c)) ≡ ((a ∨ b) ∨ c)

Formula-6

Distributive law

(a ∧ (b ∨ c)) ≡ ((a ∧ b) ∨ (a ∧ c))

Formula-7

Distributive law

(a ∨ (b ∧ c)) ≡ ((a ∨ b) ∧ (a ∨ c))

Formula-8

De Morgan’s law

(a ∧ b) ≡ ~(~a ∨ ~b)

Formula-9

De Morgan’s law

(a ∨ b) ≡ ~(~a ∧ ~b)

Formula-10

Contrapositive law

(a → b) ≡ (~b → ~a)

Formula-11

Material implication

(a → b) ≡ (~a ∨ b)

Formula-12

Material implication

(a → b) ≡ ~(a ∧ ~b)

Formula-13

Transitive implication

{((a → b) ∧ (b → c)) → (a → c)}

Formula-14

Detachment law

((a → b) ∧ a) → b ((a → b) ∧ ~b) → ~a

Formula-15

Material equivalence

(a ≡ b) ≡ ((a → b) ∧ (b → a))

Formula-16

Material equivalence

(a ≡ b) ≡ ((a ∧ b) ∨ (~a ∧ ~b))

Formula-17

Law of exportation

((a ∧ b) → c) ≡ (a → (b → c))

Formula-18

Law of tautology

(a ∨ a) ≡ a

Formula-19

Law of tautology

(a ∧ a) ≡ a (continued)

12.7 Proving Validity of Arguments

213

(continued) S. No.

Name of formula

Formula

Formula-20

Law of conjunction

(i) (a ∧ b) → a (ii) (a ∧ b) → b

Formula-21

Law of excluded middle

(p ∨~p)

Along with these formulae, we need another four rules to prove the validity of arguments in predicate logic. These rules are: Universal instantiation (UI) Universal generalisation (UG) Existential instantiation (EI) Existential generalisation (EG) The expression ‘universal instantiation’ is abbreviated as UI. The rule UI expresses that in a proposition, whatever is predicated (either affirmatively or negatively) universally to a class, it may be predicated (either affirmatively or negatively) to all its member of the class. For example, if ‘All crows are black’ is true, then ‘Crow x is black’, ‘Crow y is black’, ‘Crow z is black’, etc., are also true. Further, if ‘No swans are black’ is true, then ‘Swan x is not black’, ‘Swan y is not black’, and ‘Swan z is not black’ are also true. The expression ‘universal generalisation’ is abbreviated as UG. The UG rule states that if a given proposition is true of any thing or a person, then by applying the rule of inference, from this proposition, we can infer the universal valid proposition. For example, if ‘The sum of the three interior angles of a triangle is equal to two right angles’ is true, then ‘The sum of the three interior angles of all the triangles is equal to two right angles’ is also true. The expression ‘existential instantiation’ is abbreviated as EI. The rule EI expresses that in a proposition whatever is predicated (either affirmatively or negatively), particularly to a class, may be predicated (either affirmatively or negatively) to at least one member of the class. For example, if ‘Some students are swimmers’ is true, then at least one student say ‘Miku is a swimmer’ is also true. Further, if ‘Some students are not cricket players’ is true, then at least one student say ‘Smita is not a cricket player’ is also true. The expression ‘existential generalisation’ is abbreviated as EG. The EG rule states that if a given proposition is true of any thing or a person, then by applying the rule of inference, from this proposition, we can infer the existential quantification of the proposition. For example, if ‘The sum of the three interior angles of a triangle is equal to two right angles’ is true, then ‘The sum of the three interior angles of all the triangles is equal to two right angles’ is also true. Now, let us prove the validity of the following arguments in the predicate logic. Example-1 All teachers are learners. All researchers are teachers. Therefore, all researchers are learners.

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The symbolic translation of this argument is as follows. (∀x) (Tx → Lx) (∀x) (Rx → Tx) Therefore, (∀x) (Rx → Lx) In this argument, T stands for ‘teacher’, L stands for ‘learner’, and R stands for ‘researcher’. While translating the proposition into symbolic form, we have taken into account the quality and quantity of all the propositions of the argument. We can prove the validity of this argument in two ways. First, apply the rule of UI to all the premises of the argument, as these are universal affirmative propositions (A propositions). We can replace an individual constant, say ‘a’ with the variable ‘x’. Hence, we need not mention the quantifier in the argument. Once we formulate the argument with individual constant, we can apply the propositional derivation formulae to test the validity of the argument. By doing so, we are reducing the predicate logic argument to a sentential logic argument. 1. 2. 3.

(Ta → La) (Ra → Ta) Conclusion (Ra → La) (Ra → La) (from 2 and 1 by applying the formula-13)

This argument is proved as a valid argument. Second, we need to reduce the quantifier of the premises of the argument one by one by using UI and EI rules, as demanded. At the end of this process, we find predicate logic argument reduced to sentential logic form. Then, we apply the propositional derivation formulae to test the validity of the argument. Thereafter, we translate the sentential logic proposition (conclusion of the argument) into predicate logic proposition form by using UG and EG rules as demanded. The above argument is proved in the following way. 1. 2. 3. 4. 5. 6.

(∀x) (Tx → Lx) (∀x) (Rx → Tx) Conclusion (∀x) (Rx → Lx) (Ta → La) (from 1 by using UI) (Ra → Ta) (from 2 by using UI) (Ra → La) (from 2 and 1 by applying formula-13) (∀x) (Rx → Lx) (from 5 by using UG) This argument is proved as a valid argument. Example-2

1. 2.

3.

(∀x) (Ax → Bx) (∃x) (Ax ∧ Cx) Therefore, (∃x) Bx By applying the first method, we are proving the validity of the argument. (Aa → Ba)

12.7 Proving Validity of Arguments

4. 5. 6. 7.

215

(Aa ∧ Ca) Conclusion Ba Aa (from 2 by applying formula-20) Ca (from 2 by applying formula-20) Ba (from 1 and 3 by applying formula-14) This argument is proved as a valid argument. Now, we shall apply the second method to prove the validity of the argument.

1. 2. 3. 4. 5. 6. 7. 8.

(∀x) (Ax → Bx) (∃x) (Ax ∧ Cx) Conclusion, (∃x) Bx (Aa → Ba) (from 1 by using UI) (Aa ∧ Ca) (from 2 by using EI) Aa (from 4 by applying formula-20) Ca (from 4 by applying formula-20) Ba (from 3 and 5 by applying formula-14) (∃x) (Bx) (from 7 by using EG) This argument is proved as a valid argument.

Part V

Basic Sets and Laws of Algebra

Chapter 13

Basic Sets

In this chapter, we will discuss the significance of set theory in the logic subject. We will illustrate notation, forms of a set, and different kinds of the set such as equality of sets, null set, subset, proper subset, universal set, power set, finite and infinite sets amid others. Further, we will discuss the line diagram, Venn diagram, and other related topics of the set theory.

13.1 Introduction to Set Theory In the universe, innumerable living creatures and non-living objects exist, and each one has certain features. Since innumerable entities (living and non-living objects) exist in the universe, plenty of structures are found in the universe. A structure of an object can be explained through language. Language assists in describing the objects, highlighting features of the objects and explaining the structure of the object. In this regard, Wittgenstein (1922), an analytic philosopher in Tractatus LogicoPhilosophicus, writes that the limits of my language are the limits of my world. There are three types of language found in logical discourse: first-order language, ordinary language, and meta-language. First-order language is one through which we describe the structure of the objects of the world. Ordinary language is regarded as the language of logic that explains an object in detail with its features. Meta-language deals with characterising the ordinary language. A language has two components: syntax and semantics. Syntax deals with the production of a word and a sentence. It is related to the structure of a word and a sentence. Thus, the syntax of a sentence is judged as either correct or incorrect. But the semantics of a word and a sentence is judged as either true or false because it concerns with the meaning of the word and sentence. By using the correct syntax and true semantics of a language, we communicate the entities of the world to others. Georg Cantor (1845–1918), a German mathematiciancum-philosopher, has introduced the set theory to the logic subject. He says the © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_13

219

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concept of ‘set’ is the most fundamental notion of mathematics. The reason is a set theory deals with the entities of the world, whether it is a person of a village, or a cricket player of a country, or a colour of an object, or a shape of a moving creature, or a size of an object, or a commodity of farmland, etc. According to Georg Cantor, a set is a collection of distinct and distinguishable entities. According to Singh and Goswami (1998), ‘a set is a collection of some definite objects and the collection itself forms an entity’ (p. 2). In Lipschutz’s (1964) view, ‘a set is a well-defined list, collection or class of objects’ (p. 1). Some say a set is a collection of distinct objects of our intuition and thought. Thus, instead of the term ‘set’, logicians alternatively use its equivalent expressions, such as ‘ensemble’, ‘collection’, or ‘class’ of objects. Examples of a few sets are a set of red roses in a garden, the collection of rice bags in a warehouse, the collection of fish in an aquarium, the class of 2019 batch logic students of Indian Institute of Technology Madras, the colours of a rainbow, etc. It is to be noted here that a set may have concrete or abstract entities. Concrete objects are pine trees of a forest, and abstract objects are dots of a line, etc. Further, it is to be mentioned that objects are considered as elements of a set and an object may belong to many sets.

13.2 Notation As a rule, we denote a set by a capital letter, say, A, B, C, D, P, K, etc., and elements or members of a set by lower case letters, such as a, b, c, f, g, l, and m. A set is expressed with the help of a pair of curly brackets {…}. For example, A is a set of freedom fighters of India, namely Gandhi, Nehru, Patel, Sardar Vallabhbhai Patel. This set is written as A = {Gandhi, Nehru, Patel, Sardar Vallabhbhai Patel}. Another example is B is a set of a, f, g, h, and u elements. We write this set as B = {a, f, g, h, u}. Since ‘a’ is an element of the set B, we write ‘a ∈ B’, and it is read as ‘a is an element of B’ or ‘a belongs to B’. The symbol ‘∈’ stands for the expressions ‘an element of’, ‘belongs to’, and ‘is in’. If an element or an object is not a member of a set, let us say ‘n’ is not an element of the set B, then we write ‘n ∈ / B’. It is read as ‘n does not belong to B’ or ‘n is not in B’. The symbol ‘∈’ / stands for the expressions ‘not a member of’, ‘not an element of’, and ‘not belong to’.

13.3 Forms of a Set There are two forms to express a set. One is the ‘tabular form’, and the other one is the ‘set-builder form’. The tabular form of a set is expressed denotationally or extensionally. In this form, we define a set by specifying its members. For example, A is a set consisting of three members p, m, and n. We write this set as A = {p, m, n}. In contrast to tabular form, in set-builder form, a set is expressed connotationally or intensionally. In this form, we define a set by considering the property or condition

13.3 Forms of a Set

221

of every member of the set, for example, A = {x/x is an international cricket player of India} and B = {x/x is a woman}. In these two examples, ‘x’ is treated as an arbitrary element. It means, concerning the first example, in the place of ‘x’ we may take any international cricket player of India and concerning the second example, we may take any woman as an element of the set. The line ‘/’ is read as ‘such that’. So, these two sets are read as ‘A is a set of international cricket players of India x such that x is an international cricket player of India’ and ‘B is a set of women x such that x is a woman’. Some sets are mentioned below in the tabular form and the set-builder form for your reference and discernment. Tabular form P = {Rama, Bhima, Rita, Reena} Q = {1, 2, 3, 4} R = {2, 4, 6, 8, 10} L = {1, 3, 5, 7, 9} M = {mango, banana, jackfruit, pineapple, apple} N = {logic students, economics students, science students} By considering these sets, we can write 2 ∈ Q, 6 ∈ R, 3 ∈ L, 3 ∈ Q, apple ∈ M, etc. We can also write 12 ∈ / R, 2 ∈ / L, engineering students ∈ / N, 6 ∈ / L, etc. Set-builder form A = {x/x is student of logic class} B = {x/x is a ripe fruit} C = {x/x is a doctor} G = {x/x is a river in India} V = {x/x is a heroine of Bollywood cinema industry} L = {x/x is a red colour flower}. By considering these sets, we can write: Miku ∈ A, Dr. Asthana ∈ C, Ganga River ∈ G, Madhuri ∈ V, etc. We can also write Amazon River ∈ / G, jasmine ∈ / L, Samantha ∈ / V, green cucumber ∈ / B, etc. Different kinds of sets are found in the set theory and are discussed below.

13.4 Equality of Sets and the Null Set Two sets, let us say P and Q, are treated as equal or identical if and only if for all values of x if x ∈ P then x ∈ Q and for all values of y if y ∈ Q then y ∈ P. In simple terms, P and Q sets are equal when both the sets contain the same members. It suggests that P must contain all the members of Q and Q must contain all the members of P. We write the equality of sets of P and Q as ‘P = Q’ and read as ‘P is equal to Q’. Examples of equality of sets are:

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Example-1 P = {a, b, c, d} Q = {b, a, c, d} Example-2 P = {4, 2, 1, 5} Q = {1, 2, 5, 4, 5} Example-3 P = {2, 1} Q = {1, 1, 2, 1} R = {x/x2 − 3x = −2} With regard to example-1, it is stated that even though elements of a set are rearranged, the set does not change. Hence, P is treated as equal to Q and vice versa. In example-2, it is suggested that a set does not change even though its elements are repeated. Thus, P and Q sets are equal to each other. In example-3, if we put x’s value as 1 and 2, we will get ‘−2’. So, the elements of R set are 1 and 2. It is stated that, so long as the elements of two sets are the same members, irrespective of tabular form or set-builder form, the two sets are to be treated equal to each other. In this example, the set P is treated equal to Q and R sets, the set Q is treated equal to P and R sets, and R set is treated equal to P and Q sets. We can write the equality of sets of P, Q, R as P = Q = R. When members of a set L (i.e. L = {1, 2, 3}) are not the same as the members of the set P (i.e. P = {6, 7, 8}), we say that L is not equal to P and we write L = P. The null set is known as an ‘empty set’ or ‘void set’. It does not contain any member in it. In other words, a null set refers to a set that has no members or elements. It is denoted by the Greek letter ‘φ’ (phi). Examples of a null set are: Example-1 X = {x/x is a sky flower} Example-2 M = {x/x is eighth day in a week} Example-3 B = {x/x2 = 4, x is an odd number > 1} In example-1, we know that no sky flower exists. Hence, X is an empty set, that is, X = { }. Note, when we convey X is a null set, we must not write X = {φ}, or X = 0, or X = {0}. Since the null set does not possess any member, we write φ = { }. With reference to example-2, there is no such week that has eight days. Hence, the set M is treated as a null set, that is, M = φ. In example-3, there will not be an odd number greater than 1 that satisfies the property of x. Hence, B is treated as a null set, that is, B = φ.

13.5 Subset, Proper Subset, and Superset

223

13.5 Subset, Proper Subset, and Superset If every member of a set P is a member of set Q, then P is a subset of Q. We write ‘P ⊆ Q’. The symbol ‘⊆’ represents the concept of a subset. More precisely, P is a subset of Q; if x ∈ P then x ∈ Q. ‘P ⊆ Q’ is read as ‘P is contained in Q’. If P is a subset of Q, then we can also write ‘Q ⊇ P’. It is read as ‘Q contains P’ or ‘Q is a superset of P’. Some examples of the subset are mentioned below: Example-1 P = {a, b, c, d} Q = {a, b, c, d, e} Example-2 L = {1, 2, 3, 5} M = {1, 2, 3, 5} Example-3 A = {x/x is an odd number} B = {1, 3, 5, 7, …} In example-1, all the members of the set P belong to the set Q. Hence, we can say, ‘P ⊆ Q’. In example-2, as per the definition of subset, all the members of L belong to the set M. Hence, ‘L ⊆ M’. Although L is a subset of M, yet these two sets are equal to each other as they have the same members, that is, ‘L = M’. From this example, we can assert that every set is a subset of itself. About example-3, it is evident that A contains B. So, B is a subset of A, that is, ‘B ⊆ A’. In set theory, if P is not a subset of Q, then at least one member of P is not a member of Q. We write it as P  Q. P  Q is read as ‘P is not contained in Q’ or ‘Q does not contain P’. If P  Q, then we state that ‘Q is not a superset of P’, which we write as Q  P. From the above analyses, we assert that a set R is equal to a set S, if and only if R is a subset of S and S is a subset of R. That is, ‘R ⊆ S’ and ‘S ⊆ R’. Further, an empty set φ is considered as a subset of every set. A set M is a proper subset of a set L when it satisfies two conditions. First, M is a subset of L, that is, M ⊆ L. Second, set M is not equal to L, that is, M = L. It means that there must be at least one member of the set L which is not a member of M and all the members of the set M must be members of the set L. When M is the proper subset of L, we write M ⊂ L. A few examples of the proper subset are mentioned below. Example-1 A = {a, b, 4, 9} B = {a, b, 4, 5, 9} Example-2 P = {x/x is an even number} Q = {2, 4, 6, 8}

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In example-1, the number 5 of set B is not contained in set A, but all the members of set A are contained in set B. Thus, A is a proper subset of B, that is, A ⊂ B. In example-2, all members of the set Q belong to the set P. Further, P and Q sets are not equal to each other. Thus, Q is a proper subset of P. We write it as Q ⊂ P. If a set P is not a proper subset of Q, then we write it as ‘P ⊂ Q’.

13.6 Universal Set, Power Set, and Comparability of Sets All applications of the theory of sets are defined under a set known as the ‘universal set’. The universal set is also called the ‘universe of discourse’ and denoted by the symbol ‘U’. Some examples of the universal set are mentioned below. Example-1 U = {x/x is an object of the universe} Example-2 U = {x/x is a living being on this earth} Example-3 U = {x/x is a river of the planet} With regard to example-1, all the objects of the universe are members of the set U. In example-2, all living beings on this earth are members of the set U. In example-3, all rivers across the globe are members of the set U. The family of all subsets of a set is called the power set of that set. To explain, let us consider a set A. The power set of A is P(A) = {x/x ⊆ A}. If set A has n members, then P(A) will have 2n members. Let us consider set A to have three members, a, b, and c. We write set A = {a, b, c}. The power set of A, that is, P(A), is 23 , that is, eight members. The members are P(A) = {{a, b, c}, {a, b}, {b, c}, {c, a}, {a}, {b}, {c}, φ}. In the power set of A, every member of the set A has two possibilities, either being or not being a member of subsets of A. An empty set is a member of P(A) because it is the subset of every set. Some examples of the power set are given below. Example-1 L = {1, 2} P(L) = {{1, 2}, {1}, {2}, φ} Example-2 S = {a, b, c, d} P(S) = {{a, b, c, d}, {a, b, c}, {b, c, d}, {c, d, a}, {d, a, b}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a}, {b}, {c}, {d}, φ} In example-1, the set L has two members; hence, the power set of L, that is P(L), has 22 members. Therefore, P(L) has totally four members, as mentioned in the

13.6 Universal Set, Power Set, and Comparability of Sets

225

example. Concerning example-2, set S has four members. So, the power set of S, that is, P(S), must have 24 members, that is, sixteen members, as mentioned in the example. Comparability of sets takes place between two sets when a set is the proper subset of the other set. To explain, set A and set B are comparable to each other when all the elements of B are the members of A but an element of A is not in B, that is, B ⊂ A. Two sets P and Q are not comparable when an element of P is not a member of Q and an element of Q is not a member of P. In this case, P is not a proper subset of Q (P ⊂ Q) and Q is not a proper subset of P (Q ⊂ P). A few examples of comparable and not comparable sets are placed below. Comparable sets: Example-1 A = {1, 2, 4} B = {1, 2, 4, 5} Example-2 L = {chair, table, wood, desk} M = {chair, table, desk} Not comparable sets: Example-1 P = {1, 2, 4} Q = {3, 2, 4} Example-2 R = {chair, table, wood, desk} S = {table, fan, desk} With regard to the examples of comparable sets, we find that A is a proper subset of B (A ⊂ B) and M is a proper subset of L (M ⊂ L). It means at least one member of B and L sets does not belong to the sets A and M, respectively. However, all the members of A and M are the members of B and L sets, respectively. Concerning the examples of not comparable sets, we find that an element of P is not in Q and an element of Q is not in P. Further, at least one element of R does not belong to S, and an element of S does not belong to R. Thus, neither P and Q sets nor R and S sets are comparable to each other.

13.7 A Set of Sets, and Finite and Infinite Sets When a set consists of a few sets as its members, then the set is regarded as a set of sets. ‘A set of sets’ is popularly known as ‘a class of sets’ and ‘a family of sets’. An example of a set of sets is B = {{a, b}, {1, 2}, {f, g, t}, {t}}. In this example, B is a set of four sets. These are {a, b}, {1, 2} {f, g, t}, and {t}. Since we learned that a set is denoted by a capital letter A, B, C, etc., instead of writing elements of the sets in a set of sets, we can write capital letters as elements of the set of sets, for example,

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A = {{a, b}, D, {f, g, t}}. In this example, set A has three sets. These are {a, b}, D, and {f, g, t}. Please note that D is not mentioned within a pair of curly brackets. The reason is it is a set that has some members in it. However, in the case of {a, b} and {f, g, t}, they stand for members of two different sets. Only in one condition, we cannot treat a set as a family of sets, that is, when a set has members that are a combination of individual member and sets, for example, B = {{a, b, v}, n, D, {t, l}}. The set B consists of three sets: {a, b, v}, D, and {t, l} and one element ‘n’. As a result, we cannot treat B as a set of sets. In other words, when elements of B set are sets and individual members, set B will not be treated as ‘a family of sets’. Concerning a finite set, it is stated that a set is regarded as a finite set when all the members of the set are countable. More precisely, when a person wishes to count the members of a set and he/she comes to an end of his/her counting, the set will be treated as a finite set. But if the counting does not come to an end, then the set is treated as an infinite set. Examples of a finite set K = {x/x is a logic student of IIT Madras, Chennai, India} L = {x/x is a mango tree in a Salem mango orchard} P = {x/x is a fish of a pond} N = {x/x is a river on this earth} T = {x/x is a pine tree of Ooty town in Tamil Nadu} Examples of an infinite set S = {1, 3, 5, 7, …} P = {2, 4, 6, 8, …} M = {x/x is sand particles of Marina beach} H = {x/x is a star in the galaxy}

13.8 Disjoint Sets To understand the concept of disjoint sets, we need a minimum of two sets for our consideration. Two sets are disjoint from each other when no common element is found in both sets. To explain, if L and M are two sets and at least one member of L is not found in M and at least one member of M is not found in L, then we call L and M as disjoint sets. Examples of disjoint sets are: Example-1 A = {2, 3, 4, 5} B = {8, 7, 9, 1} Example-2 S = {x/x is an odd number} Y = {x/x is an even number}

13.8 Disjoint Sets

227

With regard to example-1, we do not find a common element between A and B sets. Thus, they are disjoint sets. In the example-2, S set refers to odd numbers, such as 1, 3, 5, and 7, and Y set refers to even numbers, such as 2, 4, 6, and 8. Thus, not even a single element is found that is common to both sets. As a result, we treat S and Y as disjoint sets.

13.9 Line Diagram A line diagram is used to demonstrate the relation between two sets. It is one of the most effective and useful tools to exhibit the relationship between two sets. If P is a proper subset of Q (i.e. P ⊂ Q), then we place Q on a higher level and P on the lower level and connect P and Q by a line. The line diagram of ‘P ⊂ Q’ is presented below.

In the line diagram, we need to find out which set(s) is a proper subset of which set(s) and accordingly we need to arrange the sets in a line diagram. A few examples of line diagram are mentioned below. Example-1 Let L = {m}, M = {n}, P = {m, n, q} In this example, ‘m’ is a member of L set and P set. But the set P has additional members besides ‘m’. Thus, we can say L is a proper subset of P, that is, L ⊂ P. Similarly, the set M has a member ‘n’ and the set P too has the ‘n’ member besides other members. Thus, we can write M ⊂ P. Further, M = L, M = P, and L = P. Since we derive L ⊂ P and M ⊂ P from the above-mentioned sets, the line diagram of these sets is drawn in the following way.

Example-2 Let P = {1, 2} Q = {1, 2, 3} R = {1, 2, 4, 3} S = {5, 3, 4, 1, 2} In this example, by considering all the sets and their members, we find that P = Q, Q = R, R = S, S = Q, R = P, and P = S. Further, the members of P are contained

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in Q, the members of Q belong to R, and the members of R belong to S. Thus, we can write P ⊂ Q, Q ⊂ R, and R ⊂ S. The line diagram of the sets mentioned above is presented below.

Example-3 Let P = {1, 2, 3} Q = {1} R = {1, 2, 4} In this example, by considering all the sets and their members, we find that P = Q, Q = R, and R = P. The member of Q is contained in P and R, but members of P do not belong to R and members of R do not belong to P. Thus, we have Q ⊂ P and Q ⊂ R. The line diagram of the above-mentioned sets is presented below.

Example-4 Let P, Q, R, and S be sets such that P ⊃ Q, P ⊃ R, S ⊂ Q, and S ⊂ R. Since P is the superset of Q (P ⊃ Q) and P is the superset of R (P ⊃ R), we can write it in a proper subset form, that is, Q ⊂ P and R ⊂ P, respectively. The line diagram of P, Q, R, and S sets is drawn below.

13.9 Line Diagram

229

Example-5 Let A, B, C, D, and E be sets where A ⊂ B, A ⊂ E, D ⊂ E, and D ⊂ C. The line diagram of these sets is drawn below.

13.10 Venn Diagram A Venn diagram is a simple and important method to elucidate the relationship between two sets and among many sets. A set in a Venn diagram is represented by either a circle, or a square, or a triangle, etc. Mostly, it is found that logicians use circles to represent sets and convey the relationship between two sets and among many sets in a Venn diagram. To draw a Venn diagram of two sets, we need to find out which set is a proper subset of which set. This would assist us to draw a bigger circle that has more members in relation to the other set. A small circle is to be drawn within the bigger circle, as members of the small circle (i.e. a set) belong to the bigger circle (i.e. a set). For example, A is a set consisting of three members a, b, and c. We write it as A = {a, b, c}. B is a set consisting of members a, b, c, d, and e. We write B = {a, b, c, d, e}. The members of set A belong to set B, but some members of B are not found in A. Thus, we write A ⊂ B and A = B. The relationship between the sets A and B is represented in the Venn diagram below.

B A

A few more examples of drawing Venn diagram of sets are mentioned below. Example-1 Let A = {2, 3} B = {1, 2} C = {1, 2, 4, 3} In this example, we find three sets A, B, and C, where A = B, B = C, and A = C. There is a common element ‘2’ found in A and B sets. Further, B ⊂ C and A ⊂ C. The relationship among these sets is represented in the following Venn diagram.

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4

A

B C

Example-2 Let A, B, C, and D be sets, where C ⊂ B, D ⊂ B, B ⊂ A, C ⊂ A, and D ⊂ A. Further, C and D sets are mentioned as disjoint sets. In this regard, the Venn diagram below is drawn to represent the relationship among all the sets.

A B C

D

Example-3 Let A, B, C, and D be sets where C ⊂ B, D ⊂ B, B ⊂ A, C ⊂ A, D ⊂ A, and D ⊂ C. The Venn diagram below represents all the sets and the relationship between them.

A

B

C

D

Chapter 14

Basic Set Operations

In the previous chapter, we introduced and explained the basic concepts of set theory and the relation of sets. We elucidated different types of sets and the relationship between two sets through the line diagram and Venn diagram. In continuation to the previous chapter, in this chapter, we will illustrate the operation of sets such as the union of sets, the intersection of sets, the relative complement of sets, the absolute complement of a set, and some proofs of the set theory. Further, we will mention the laws of the algebra of sets and carry out a few exercises by applying these laws. It is important to note that a minimum of two sets is required in carrying out an operation of sets. The basic set operations are similar to the concept of addition, subtraction, and multiplication of the arithmetic subject. In arithmetic, addition takes place between two numbers, but in set theory, the union of sets takes place between two sets. The relative complement of sets is similar to the subtraction of the arithmetic subject and so on. Let us begin the discussion of set operations with the union of sets and the intersection of sets.

14.1 Union and Intersection The concept of the union of sets is an operation on two sets. If P and Q are two sets, then the union between P and Q is written as ‘P ∪ Q’, which is read as ‘P union Q’ or ‘P cup Q’. The elements of P and Q put together are the elements of ‘P ∪ Q’. In other words, the union of P and Q sets comprises all the elements belonging to both P and Q. The union of P and Q sets is defined as P ∪ Q = {x/x ∈ P or x ∈ Q}. With regard to P ∪ Q, it is always found that P ⊆ (P ∪ Q) and Q ⊆ (P ∪ Q). From these findings, we can assert that (P ∪ Q) ⊇ P and (P ∪ Q) ⊇ Q. An example of the union of sets is mentioned below. Let P = {a, b, c} Q = {s, t, r} © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_14

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P ∪ Q = {a, b, c, s, t, r} In this example, P ∪ Q is the smallest possible set of which both P and Q are subsets. P ∪ Q is treated as the least upper bound (LUB) for P and Q. From the union of P and Q sets, two laws are derived. These are: P ∪ P = P; Q ∪ Q = Q (Idempotent law) P∪Q=Q∪P (Commutative law) The Venn diagram of ‘P ∪ Q’ is drawn below. In this diagram, the square is treated as a universal set denoted by U. In the square, two circles are drawn representing P and Q sets.

P Q is shaded If we add R set to the union of P and Q set, we obtain the following law known as the associative law. That is, (P ∪ Q) ∪ R = P ∪ (Q ∪ R) (Associative law) The concept of the intersection of sets is an operation on two sets. If P and Q are two sets, then the intersection between P and Q is written as ‘P ∩ Q’, which is read as ‘P intersection Q’. The common elements of P and Q sets are the elements of ‘P ∩ Q’. In other words, the intersection of P and Q sets comprises all the common members belonging to P and Q sets. The intersection of P and Q sets is defined as P ∩ Q = {x/x ∈ P and x ∈ Q}. Concerning P ∩ Q, P ∩ Q is a subset of both P and Q sets. That is, (P ∩ Q) ⊆ P and (P ∩ Q) ⊆ Q. From these findings, we can assert that P ⊇ (P ∩ Q) and Q ⊇ (P ∩ Q). Further, it may be stated that if P ⊆ Q, then P ∪ Q = Q and P ∩ Q = P. An example of the intersection of two sets is described below. Let P = {a, b, c, d} Q = {b, d, e, g} P ∩ Q = {b, d}

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In this example, P ∩ Q is called the greatest lower bound (GLB) for P and Q. From the intersection of P and Q sets, the following laws are derived. P ∩ P = P; Q ∩ Q = Q (Idempotent law) P∩Q=Q∩P (Commutative law) The Venn diagram of ‘P ∩ Q’ is drawn below. In this diagram, the square is treated as a universal set denoted by U. Within the square, two circles are drawn; one circle is denoted by P that represents the P set, and the other circle represents the set Q, hence named Q.

P Q is shaded If we have three sets P, Q, and R, and there is an intersection of sets among these three sets, then we obtain the following law. (P ∩ Q) ∩ R = P ∩ (Q ∩ R) (Associative law) If both P and Q sets do not have a common element(s), then the intersection of P and Q sets is regarded as the null set. The reason is that a null set does not have any member. Hence, we write P ∩ Q = ∅. Since a null set (∅) is considered as a subset of every set, we can write ∅ ⊆ P and ∅ ⊆ Q.

14.2 Relative Complement The relative complement of the two sets is known as the difference between the two sets. The difference of sets P and Q is denoted by ‘P − Q’, which is read as ‘P minus Q’ or ‘P difference Q’. ‘P − Q’ is also denoted by ‘P/Q’ or ‘P ~ Q’. ‘P − Q’ has elements belonging to P and not belonging to Q. It may be precisely defined as P – Q = {x/x ∈ P and x ∈ / Q}. Concerning ‘P − Q’, it is found that ‘P − Q’ is the subset

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of P (i.e. (P – Q) ⊆ P), and thereby, P is the superset of ‘P − Q’ (that is, P ⊇ (P − Q)). An example of ‘P − Q’ is presented below. Let P = {1, a, r, t, 5, 4} Q = {1, r, 5, 4} P − Q = {a, t} The Venn diagram of ‘P − Q’ is depicted below. In this diagram, the square is treated as a universal set denoted by U. Inside the square, there are two circles, one is named P as it represents the set P, and the other one represents the set Q, hence named Q.

Q is shaded

14.3 Absolute Complement and Double Complement The complement of a set, let us say S, is called the absolute complement of the set S. It is denoted by S . It states that members of the set S do not belong to S. It explains that S is a set comprising all those members that are obtained from the difference of the universal set U and the set S. The absolute complement of the set S is concisely / S} or it may be defined as S = {x/x ∈ / S}. In the defined as S = {x/x ∈ U and x ∈ absolute complement of the set S, both S and S are subsets of the universal set U. An example of the absolute complement of a set is given below. Let S = {Monday, Wednesday, Friday} U = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} S = (U − S) S = {Tuesday, Thursday, Saturday, Sunday} From the explanations and analyses of the absolute complement of a set, the following laws are derived.

14.3 Absolute Complement and Double Complement

(i) (ii) (iii)

235

The union of a set S and its complement set S is the universal set U. That is, S ∪ S = U. The intersection of a set S and its complement set S is the null set, as these two sets are disjoint. That is, S ∩ S = ∅. The absolute complement of a universal set U is a null set (i.e. ∅). We write U = ∅. Further, the absolute complement of a null set (i.e. ∅) is the universal set U. That is, ∅ = U.

The Venn diagram of the absolute complement of the set S is depicted below. In the diagram, the square is considered as a universal set denoted by U, and within the square, there is a circle named S, which represents the set S.

S' is shaded The double complement of a set, let us say S, is explained as a complement of the complement of the set S. It is denoted by S . It states that the members of the set S belong to S. In other words, the complement of the complement of the set S is the set S itself. That is, (S ) or S = S. Summing up the above-mentioned operations of sets in the below Venn diagram. In this Venn diagram, the square represents a universal set U. There are two circles within the universal set that represent P and Q sets. The universal set U has four elements: I, II, III, and IV. Set P consists of II and III elements, and set Q consists of III and IV elements.

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P ∪ Q = Q ∪ P = II, III, IV P ∩ Q = Q ∩ P = III P − Q = II Q − P = IV P = I, IV Q = I, II P = II, III Q = III, IV

(Union of sets) (Intersection of sets) (Relative complement) (Relative complement) (Absolute complement) (Absolute complement) (Double complement) (Double complement)

14.4 Some Proofs Prove that ∅ is a unique set.

(i)

It is true that there is one and only one empty set. Let us assume that there are two empty sets: φ and If it was so, then φ ⊆ (empty set (φ) is a subset of every set) and ⊆ φ (empty set ( ) is a subset of every set) Hence, = φ. (Since they are a subset of each other) Two empty sets are treated as one and the same. Hence, we write any one of them instead of writing two. Therefore, φ is a unique set. (ii)

Prove (B − A) is a subset of A .

Let x belong to (B − A). If it is so, then x ∈ B, and x ∈ / A. Since x ∈ / A, x ∈ A . Therefore, if x ∈ (B − A), then x ∈ A . Hence, (B − A) ⊆ A . (iii)

Prove that (A ∪ B) = (B ∪ A) Let x ∈ (A ∪ B) An assumption x ∈ A or x ∈ B by definition x ∈ B or x ∈ A by Commutative law x ∈ (B ∪ A) by definition (A ∪ B) ⊆ (B ∪ A) (A ∪ B) = (B ∪ A) (Proved)

(iv)

Prove (B − A ) = (B ∩ A) / A } (B − A ) = {x/x ∈ B, x ∈ = {x/x ∈ B, x ∈ A} = (B ∩ A) (Proved)

14.5 Laws of the Algebra of Sets

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14.5 Laws of the Algebra of Sets While discussing the operation of sets, we derived some laws mentioned in the table below. These laws are considered the laws of the algebra of sets. We apply these laws to evaluate expressions and perform calculations of set theory operation. These laws are regarded as the fundamental properties of set operations and set relations. In order to evaluate a set-theoretic operation of an expression that has left-hand side (LHS) and right-hand side (RHS) sets operation, we need to find out whether LHS sets operation is equal to RHS sets operation. If LHS is found equal to RHS of an expression at the end of the calculation of sets operation, then we write Quod Erat Demonstrandum (hereafter, QED). It means that RHS is proved equal to LHS. To prove that LHS is equal to RHS, we consider LHS sets operation and their relations. We apply laws of the algebra of sets to LHS until we get the RHS. To do so, no fixed procedure or method exists. But to obtain RHS of an expression of the sets operation, we have to apply laws of the algebra of sets to LHS as required. It is like the way a driver negotiates his/her vehicle on a congested road. It is like a cricket game where depending on the type of ball bowled to a batsman; the batsman hits the ball, etc. Name of the law

Details of the law

Idempotent

X ∪ X = X, X ∩ X = X

Commutative

X ∪ Y = Y ∪ X, X ∩ Y = Y ∩ X

Absorption

X ∪ (X ∩ Y) = X, X ∩ (X ∪ Y) = X

Distribution

X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z) X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z)

De Morgan

(X ∩ Y) = X ∪ Y (X ∪ Y) = X ∩ Y

Association

X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z

Identity

X ∪ ∅ = X, X ∪ U = U X ∩ ∅ = ∅, X ∩ U = X

Complement

∅ = U, U = ∅ X ∪ X = U, X ∩ X =∅

Double complement

X = X

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14.6 Exercises (i)

Prove that A ∪ (B ∩ A) = A Serial number 1 2 3

(ii)

Statement A ∩ (A ∪ U) ∩ (B ∪ ∅) (A ∩ U) ∩ (B ∪ ∅) A∩B

Justification Left-hand side (LHS) Identity Identity (QED)

Prove that (A ∩ B ∩ C) ∪ (A ∩ B ∩ C) ∪ (B ∪ C ) = U

Serial number 1 2 3 4 5 6 7 8 9 (iv)

Justification Left-hand side (LHS) Commutative Absorption (QED)

Prove that (A ∩ (A ∪ U)) ∩ (B ∪ ∅) = A ∩ B Serial number 1 2 3

(iii)

Statement A ∪ (B ∩ A) A ∪ (A ∩ B) A

Statement (A ∩ B ∩ C) ∪ (A ∩ B ∩ C) ∪ (B ∪ C ) (A ∩ (B ∩ C)) ∪ (A ∩ (B ∩ C)) ∪ (B ∪ C ) ((B ∩ C) ∩ A) ∪ ((B ∩ C) ∩ A ) ∪ (B ∪ C ) (B ∩ C) ∩ (A ∪ A ) ∪ (B ∪ C ) (B ∩ C) ∪ (B ∪ C ) ∩ (A ∪ A ) (B ∩ C) ∪ (B ∩ C) ∩ (A ∪ A ) (B ∩ C) ∪ (B ∩ C) ∩ U U∩U U

Justification Left-hand side (LHS) Association Commutative Distribution Commutative De Morgan Complement Complement Idempotent (QED)

Prove that ((A ∪ B) ∩ (B ∪ U)) ∩ (A ∪ ∅) = A Serial number 1 2 3 4 5 6

Statement ((A ∪ B) ∩ (B ∪ U)) ∩ (A ∪ ∅) ((A ∪ B) ∩ U) ∩ (A ∪ ∅) (A ∪ B) ∩ (A ∪ ∅) (A ∪ B) ∩ A A ∩ (A ∪ B) A

Justification Left-hand side (LHS) Identity Identity Identity Commutative Absorption (QED)

14.6 Exercises

(xxii)

239

Prove that ((A ∪ B) ∩ (B ∪ A)) ∪ (A ∪ B) = A ∪ B Serial number 1 2 3 4

(vi)

Statement ((A ∪ B) ∩ (B ∪ A)) ∪ (A ∪ B) ((A ∪ B) ∩ (A ∪ B)) ∪ (A ∪ B) (A ∪ B) ∩ (A ∪ B) (A ∪ B)

Prove that (A ∩ B ) ∪ B = A ∪ B Serial number 1 2 3 4 5

(vii)

Justification Left-hand side (LHS) Commutative Idempotent Idempotent (QED)

Statement (A ∩ B ) ∪ B (A ∪ B ) ∪ B (A ∪ B) ∪ B A ∪ (B ∪ B) A ∪ B

Justification Left-hand side (LHS) De Morgan Double complement Association Idempotent (QED)

Prove that ((A ∪ B) ∩ (A ∪ U)) ∪ ((A ∪ B) ∩ (B ∩ ∅)) = A ∪ B

Serial number 1 2 3 4

Statement ((A ∪ B) ∩ (A ∪ U)) ∪ ((A ∪ B) ∩ (B ∩ ∅)) ((A ∪ B) ∩ U) ∪ ((A ∪ B) ∩ ∅) (A ∪ B) ∪ ∅ (A ∪ B)

Justification Left-hand side (LHS) Identity Identity Identity (QED)

Part VI

Induction

Chapter 15

Induction

In this chapter, we will discuss the relationship between deduction and induction as well as the differences between them. We will analyse the types of induction and lacunas of induction. Further, we will elucidate scientific induction, induction by analogy, induction by simple enumeration, and induction by complete enumeration with suitable examples.

15.1 What is Induction? Induction is a method that assists in drawing a conclusion from a few observed empirical evidence. The conclusion of induction is mostly a universal proposition in which the predicate term either affirms or denies all the members (entities) of the subject term. Francis Bacon (1561–1626), an empiricist philosopher, his work on Novum Organon (1620) highlights the lacunas of ‘deduction’ and mentioned ‘induction’ in a precise and concise manner. John Stuart Mill (1806–1873), at a later period, explained the notion of ‘induction’ in detail in his work A System of Logic, Vol. 1 (1843). In Bacon’s view, unlike deduction, in induction, we derive axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms at last (Novum Organum, 1620, Aphorism XIX). According to J. S. Mill (1843), ‘Induction is the name given to the operation of the mind, by which we infer that what we know to be true in particular case or cases, will be true in any other case or cases of the similar kind’ (p. 1103). To put it in simple terms, inductive inference suggests that what is true at certain times will be true at all times or at least in the next time in a similar circumstance, and what is true of certain individuals (entities) of a class is true of the whole class or at least of the next individual. In deductive logic, the premises are given to us to which we apply syllogistic rules to derive a conclusion from the premises. If the given premises support the conclusion, then the argument is treated as valid, and if the given premises did not support the © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_15

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conclusion, then the argument is treated as invalid. In the deductive argument, the structure (form) of the propositions plays a vital role to validate the argument, but not the content of the propositions. Thus, in deductive logic, a valid argument must consist of formally correct propositions, not materially true propositions. It implies that a deductive argument is treated as valid even if its constituent premises and conclusion are materially false propositions. For example, All quadruped animals are having horns. (Major premise) All horses are quadruped animals. (Minor premise) Therefore, all horses are having horns. (Conclusion) This argument is a valid mood of the first figure of syllogism known as BARBARA. It satisfies all the syllogistic rules, and the valid conclusion is derived from the given two premises. However, the conclusion does not correspond to facts of the world, so it is treated as a materially false proposition. Unlike a deduction, in induction, the content of the premises corresponds to the true state of affairs of the world. Thus, they are treated as materially true propositions. In the inductive argument, the constituent propositions are regarded as synthetic propositions because they are verified through empirical observations and found to be true propositions. These true propositions together assist in inferring the conclusion of the argument, but they do not establish the truth-value (either true or false) of the conclusion (i.e. a universal proposition) conclusively. The reasons are explained in the example given below. Swan X is white perceived in place1. Swan Y is white perceived in place2. Swan Z is white perceived in place3. -------------------------{Inductive leap} -------------------------Therefore, perhaps all swans are white. It is to be noted here that in an inductive argument, premises and conclusion are of the same kind. The above-mentioned inductive argument states that Swan X is white, and it is verified through perception in place1, swan Y is white and it is verified through perception in place2, swan Z is white and it is verified through perception in place3. From these three empirical pieces of evidence, we conclude a universal proposition that is ‘perhaps all swans are white’, even though we have not perceived all the swans of the past, present, and future. In the inductive argument, there is a movement from particular premises to a universal conclusion, wherein a universal conclusion includes contents of the verified premises that are mentioned in the argument and contents of the unverified premises that are not mentioned in the argument. Thus, it is stated that a universal conclusion is inferred from only a few observed and verified instances (i.e. premises) only and not from all the instances of past, present, and future. Induction is a method adopted widely in social sciences, natural sciences, and applied sciences. It plays a pivotal role to derive a conclusion from certain observed

15.1 What is Induction?

245

and verified premises of a research study; however, the induction method is not used in pure mathematics and formal logic. In these subjects, mostly rules and laws are considered as the primary means to validate the conclusion of the arguments. It is to be mentioned here that it is wrong to believe that social scientists, basic scientists, and applied scientists adopt only inductive inference to draw a conclusion from the data set (verified true premises) in their research studies. Rather, they also adopt the deductive method besides the inductive method in their research studies. They adopt theories and laws to draw a conclusion from the premises. There are a few occasions where scientists use deductive reasoning to establish the laws, such as the law of planetary motion. The planetary motion law states that all the planets move around the sun in a fixed elongated orbit. For instance, the earth is a planet that moves around the sun in a designated elliptical orbit. Inductive and deductive inferences are used as methods to search and discover the truth of worldly affairs. These methods are treated as fundamental methods to study topics of every subject, and they should be viewed as complementary or supplementary to each other instead of viewing them as contradictory or antithetical methods. The reason is the conclusion of the inductive method may serve as the premise of the deductive method, and a premise of a deductive argument may be required to be verified through an application of inductive inference. In inductive arguments, ‘inductive leap’ is found invariably. It suggests that there could be a few more pieces of evidence (premises) of a similar kind that could be added to the argument to infer a true universal conclusion from the premises. The possibility of adding a few more shreds of evidence (premises) of a similar kind to the existing premises raises the probability of the conclusion of the argument. This probability brings uncertainty to the content of the conclusion. The reason is the conclusion depends on the strength of the content of the premises. Thus, the relationship between conclusion and premises is not of necessity but of suggestion that is associated with the probability of an inductive argument. Hence, it would be appropriate to state that inductive arguments are treated as weaker or stronger, convincing or unconvincing, but certainly neither valid nor invalid. The difference between inductive and deductive arguments is based on the following grounds. The inductive inference is probable, but the deductive inference is demonstrative. The conclusion of inductive inference is associated with a degree of probability, whereas the conclusion of deductive inference is treated as either true or false, as it is derived from its premises upon application of rules of deduction (syllogistic rules). In the deductive argument, premises are given in the argument, but in the case of inductive argument, premises are to be collected as verified pieces of evidence of worldly affairs. In a deductive argument to find out the truth-value of the conclusion, we need to check the structure of the premises, but in the case of an inductive argument, to find out the truth-value of the conclusion, we need to find out the truth-value of the premises through empirical observation or verification. In an inductive argument, premises must be formally and materially true, but in the case of deductive argument, premises need not be materially true but ought to be formally true.

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However, the difference between induction and deduction does not sustain on the following ground. That is, in the inductive argument, there is a movement that takes place from particular premises to a universal conclusion, and in the deductive argument, there is a movement that takes place from universal premises to a particular conclusion. The reason is, in the inductive argument, we can infer a particular conclusion from particular premises and, in the deductive argument, we can derive a universal conclusion from universal premises. The below examples illustrate these claims. An example of a deductive argument All students are wise beings. All cricketers are students. Therefore, all cricketers are wise beings. An example of an inductive argument X is a logic student and a member of the Humanities and Social Sciences Department. Y is a logic student and a member of the Humanities and Social Sciences Department. Z is a logic student. –---------------------–-----–---––-------Therefore, perhaps Z is a member of the Humanities and Social Sciences Department. There is no such case of deductive logic where premises are true but the conclusion is false, but in the case of inductive logic, even though premises are true, the conclusion may not be judged as true. It will not be an error in accepting true premises and false conclusion of inductive arguments, as each inductive argument is associated with an ‘inductive leap’. For example, X is a logic student and she scored good marks. Y is a logic student and she scored good marks. L is a logic student and she scored good marks. Z is a logic student and she scored good marks. ----------------------------------------------------------Therefore, perhaps all logic students score a good mark. This inductive inference is weak and not convincing. The reason is there may be a case where M is a logic student and she does not score a good mark. So, the conclusion of the argument may be treated as false even though the premises are verified as true. Another example, X is addicted to smoking and diagnosed with cancer. Y is addicted to smoking and diagnosed with cancer. M is addicted to smoking and diagnosed with cancer.

15.1 What is Induction?

247

Z is addicted to smoking and diagnosed with cancer. ----------------------------–------–-––---–-----Therefore, perhaps any person who is addicted to smoking is diagnosed with cancer. This inductive inference sounds strong and convincing, but we cannot treat the argument as valid or invalid. The reason is there might be a possibility that all the individuals addicted to smoking are diagnosed with cancer, as smoking is injurious to health. In this example, there is a high probability that any person who smokes is getting diagnosed with cancer. So, the conclusion of the argument has a high probability of being treated as true even though only a few premises are verified and mentioned as true propositions. Since the conclusion of induction is not treated as valid and invalid, but to be judged as convincing or unconvincing, strong or weak, etc., and further the conclusion is drawn from a few observed pieces of evidence (i.e. premises) only, logicians find induction is of different types.

15.2 Types of Induction One of the essential features of induction is the ‘inductive leap’. Due to this salient feature, there is a probability to derive a true conclusion from a few observed and verified true premises. In the case of induction, on some occasions, the conclusion is not convincing and very weak, and on some other occasions, the conclusion is stronger and convincing. Due to the probability of obtaining a true conclusion from a few observed and verified premises of inductive arguments, logicians divided inductive arguments into two heads: scientific induction and unscientific induction. Further unscientific induction is divided into two kinds: enumerative and analogical inductions. Enumerative induction is further divided into ‘induction by simple enumeration’ and ‘induction by complete enumeration’. The picture below depicts the types of induction that is found in the logic and logical discourse.

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Types of Induction

15.3 Scientific and Unscientific Induction Scientific induction deals with empirical observations and experiments. The empirical observations or data are used as premises, and from these premises, a universal conclusion is inferred. The most important part of scientific induction is that it establishes a causal link between two phenomena or empirical facts mentioned in the premises. The causal link is treated as an invariable relation between a cause and its effect. The cause and effect relation is explained with an example: let us say, ‘milk’ is the cause and ‘curd’ is the effect. If there is milk, then there is curd. Curd is made out of milk. If milk does not exist, then curd does not exist. Symbolically speaking, the causal link between two phenomena A and B states that when A is, B is; when A is arising, B arises; when A is ceasing, B ceases; and when A does not exist, B’s existence is ruled out. These arguments express that there is a specific cause for every effect. Thus, the cause is an antecedent that precedes the effect, and the effect is the consequence that succeeds the cause. Although the cause is an antecedent for effect, every antecedent may not be the cause or part of the cause of an effect. Even though A precedes B in time sequence, A may not be a cause, and B may not be the effect. For example, the logic class starts at 09:00 a.m. on Monday of every week. So, it would be wrong to say 09:00 a.m. the cause of Monday morning logic class. It is also wrong to state that Monday morning logic class is an effect of the cause 09:00 a.m. It is so because there may be cases that logic classes are taking place in

15.3 Scientific and Unscientific Induction

249

different timings on Monday and some other days as well. So, a cause is defined as an invariable and unchangeable antecedent of an effect. If we hold the definition of ‘cause’, we may end up referring to something that is invariable and unchangeable antecedent and becomes the cause of an event. Take an example: there are seven days in a week, and each day follows from another day. Let us say Tuesday follows after Monday. In this case, Monday is an invariable and unchangeable antecedent of Tuesday. But it would be wrong to say that Monday is the cause of Tuesday. So, this definition is not free from error. Hence, we need to reformulate it. We can define it (cause) as an invariable, unchangeable, unconditional, and immediate antecedent of an effect. In this definition, the term ‘unconditional’ means a cause must be selfsufficient to produce an effect. It does not require any further support to give rise to an effect. The term ‘immediate’ means a cause must have an effect in quick succession. With these explanations and interpretations, logicians explain ‘cause’ is an invariable, unchangeable, unconditional, and immediate antecedent of an effect. And, an effect is an invariable, unchangeable, unconditional, and immediate consequence of a cause. There may be a possibility that a cause may have multiple effects, and multiple causes may together give rise to an effect. Consider an example: a potter can produce a pot, a mug, a glass, and a jug from a lump of clay. Here, a lump of clay is the cause and effects are many, such as pot, mug, glass, and jug. Another example is a person’s death may be caused due to COVID-19, an accident, fire burn on his/her skin, etc. In this case, death is the effect and there are multiple causes behind a person’s death. Scientists and logicians do not agree on the view that a cause may have multiple effects, and multiple causes may together give rise to an effect. They argue that the softness of mud, water, and other conditions are associated with a lump of clay to make a jug different from a glass. Similarly, the death of a person due to COVID-19 is different from the death of a person due to an accident. If all sorts of death had been the same kind, then doctors would not search for the right cause of death of a person through post-mortem. So, each cause is unique to produce its designated effect. In this sense, the theory of causation is more logically consistent and a viable tool for induction. However, a ‘cause’ of a fact shall not be understood as ‘reason’ of that fact. It is so because there is a difference found between ‘cause’ and ‘reason’. A cause produces a designated effect uniformly under similar conditions or circumstances. But a reason explains why a cause gives rise to an effect recursively under similar conditions. For example, mosquitos’ bites are the cause of malaria, but not the reason for malaria. The reasons for malaria are infected anopheles mosquitos carry the plasmodium parasite and bite human beings. The parasite is released into human beings’ bloodstream. The parasite travels to the liver of human beings and gets matured. After three to four days, the mature parasites enter into the bloodstream of the human beings and begin to infect the red blood cells. Thereafter, the parasites inside the red blood cells multiply and burst the infected cells. As a consequence, human beings are diagnosed with malaria disease. For logicians and scientists, ‘cause’ means a condition or a component responsible for producing an immediate effect, but not the reason for an effect. No effect exists without a cause, since an effect is produced under certain conditions. There are two

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types of conditions associated with the theory of causation: necessary condition and sufficient condition. Necessary conditions are those conditions that assist a cause to produce an effect. For example, A is a necessary condition for B. So, B cannot be produced in the absence of A. The absence of a necessary condition can prevent the effect from occurring, but its presence alone cannot guarantee the effect to occur. For example, a subject (a person), an object, and a method of cognition are required to possess knowledge of an object. In the absence of these necessary conditions, knowledge of an object cannot be obtained. Even if these necessary conditions are present, but the person does not have a desire to acquire knowledge of the object, knowledge of the object cannot be gained automatically. On the other hand, sufficient conditions are those conditions that are accompanied by a cause to produce a designated effect all the time. Symbolically speaking, A is a sufficient condition of B if and only if every occurrence of B is accompanied by the occurrence of A. When sufficient conditions are present, a cause will produce a specified effect immediately and invariably. Sufficient conditions include the necessary conditions of a cause in its ambit, but not inversely. Necessary conditions may be considered as the constituents of sufficient condition of a cause. Necessary and sufficient conditions together produce a designated effect from a cause. Thus, whenever a cause produces a designated effect, both necessary and sufficient conditions are present in the cause. Logicians and scientists consider ‘cause’ in terms of sufficient condition to infer the effect (i.e. conclusion of an induction). In scientific research, scientists infer the conclusion by considering a few verified pieces of evidence and establishing the causal link among all this evidence. In social science subjects, research works are often carried out through data collection. In this case, mostly a questionnaire is given to the target audience for their responses. However, the data accumulated for a research study may not be exhaustive, even then researchers conclude with a universal proposition from their research findings. They do so by establishing the link among all the evidence gathered through filled-in questionnaires. Similar to the scientific research and social science research works, scientific induction also establishes the causal link among all the observed facts mentioned in the premises. So, in the scientific induction, the causal link among empirical evidence is found to be strong, and that helps to infer a universal conclusion, which has a high probability of truth. A question may arise: does scientific induction result in a strong and convincing conclusion? Logicians argue that scientific induction may result in a reasonable conclusion, but need not be a rational conclusion. It is so because the causal link among empirical evidence may not be found in future the way it is found in the present time. The future is uncertain, and individuals cannot verify future events or facts. Hence, we can only logically and rationally infer things for the future, but we cannot state anything with truth and certainty about the future. For instance, the fire burns an object (let us say, a log) in the present time, but what guarantees that fire will burn the same object in future. Logicians answer that scientific induction is based on two presuppositions that assist us in inferring facts for the future based on the empirical evidence available to us at present, wherein the empirical evidence of the present time and the inferred fact of the future are of a similar kind. These two

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presuppositions are known as ‘the law of uniformity of nature’ and ‘the principle of causation’. The law of uniformity of nature expresses that under similar conditions, the same cause produces the same effect. To explain, if a certain phenomenon behaves in a certain manner in the past and in the present time, then the same phenomenon will behave the same way in future as well. For example, when someone cried in the past, his/her tears were dropping down from his/her eyes. When someone cries in the present time, his/her tears are dropping down from his/her eyes. Thus, if someone will cry in future, his/her tears will drop down from his/her eyes as well. The principle of causation states that every event has a cause. There is no such thing that happens without a cause in the universe. An effect follows its cause in time succession. If sufficient conditions are involved in a cause in the past and in the present time to produce an effect, these sufficient conditions will produce the same effect in future. Thus, the conclusion of a scientific induction is considered true, strong, convincing, and reasonable. Besides scientific induction, unscientific induction too exists in the logic and logical discourse. Like scientific induction, unscientific induction is also equally important for inductive reasoning. The reason is scientific induction is narrow in its scope, as its premises are the experimental data or the empirically verified data. Experimental data are mostly gathered through research works. Empirically verified data are also very limited in a number concerning the numerous facts and events of worldly affairs. Besides experimental data and empirically verified data, we can consider empirically observed phenomena as data (premises) for induction. The empirically observed phenomena support unscientific induction. Unscientific and scientific inductions together encompass a large number of facts and events of worldly affairs. These two types of induction thus enlarge the boundary of induction. Unscientific induction deals with observed phenomena of worldly affairs. From these observed phenomena, we can infer a universal conclusion by establishing a causal link among the premises (i.e. observed phenomena). Since the premises are the observed phenomena, we cannot establish a strong causal link among them in comparison with the premises of scientific induction. Nevertheless, the scope of unscientific induction is wider than scientific induction. Thus, unscientific induction covers a wide range of induction in comparison with scientific induction. Unscientific induction is of two types: induction by analogy and enumerative induction.

15.4 Induction by Analogy and Enumerative Induction In our everyday lives, there are many occasions; we compare a known object with an unknown object through their similarities and dissimilarities. By doing that, we gain new knowledge about the unknown object. This method is known as ‘analogy’. In simple terms, an analogy is a form of inference where premises (observed phenomena) of a known object and an unknown object are compared with each other to draw a universal conclusion from the premises. It is important to note here that two objects (let us say, A and B) are comparable when all the features of A are

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found in B and a feature of B is not found in A. Two objects are not comparable when a feature of A is not found in B and a feature of B is not found in A. Consider an example of an analogy. X knows about a country cow, but does not know about an animal that lives in a forest known as ‘gavagai’. X desires to know about gavagai and meets a forester who desires to communicate X about ‘gavagai’ animal. The forester states that gavagai is a forest animal that has some similarities with a country cow and dissimilarities as well, such as a long neck and very short tail. At a later date, X went on a forest trip and saw an animal that looks like a country cow, but not a country cow. X compares features of a country cow with the perceived forest animal and attributes the forester’s descriptions about ‘gavagai’ on that animal. X found that the animal features are matching with the forester’s descriptions about ‘gavagai’. As a result, X gains new knowledge about the forest animal known as ‘gavagai’. In the case of analogy, there may be a possibility that X refers to a forest animal as gavagai, but indeed the animal is not a gavagai. Instead, the animal has a different name but has some partial resemblance with a country cow. There are also possibilities that a few animals that live in the forest may have similar features of a country cow, but they have many dissimilarities with a country cow. Further, when X compares a country cow with a forest animal, it all depends on X’s ability, skills, and previous knowledge of a country cow. Hence, the conclusion of an analogical argument is not certain but probable. Induction by analogy is therefore to be treated as neither correct nor incorrect, rather be judged as stronger or weaker, and convincing or unconvincing. Consider another example of an analogy. In the summer season, Y saw varieties of mango in a fruit shop’s shelves. Y found ‘dasheri’ ripe mango is attractive as its colour is golden yellow. Its flesh is tight, the peel is thin, and its size is medium and oval-shaped. Y tasted a dasheri mango and found it very sweet and aromatic. By considering these features of a dasheri mango, Y purchased ten pieces of dasheri mango from the shop, tasted each one of them in her home and found that these are very sweet and aromatic. After a few days, Y went again to the same fruit shop and purchased twelve pieces of dasheri mango by inferring that these twelve mangos would resemble the features of dasheri mango and will therefore taste sweet. But in reality, there may be a possibility that out of twelve mangos, one mango does not taste sweet due to some or other reasons. Thus, inferring a conclusion through analogy may not result in a true conclusion all the time but maybe a probable or appealing conclusion. The analogy is considered as a form of inference where a universal conclusion is inferred from the premises (observed phenomena) by comparing their similar features. Scientists use the analogy on some occasions to discover the causal relationship between two events and facts of the world. For example, physicians test a drug in an animal before prescribing it for human being’s use. They compare animal organ function more or less similar to human being’s organ function. Computer scientists and engineers compare human brain function with a highly sophisticated computer’s central processing unit (CPU). Thus, it may be enunciated that scientists use the analogy method to discover many laws, propose a few theories, and discover

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some new facts of worldly affairs. So, induction by analogy plays an important role in obtaining new information and knowledge about worldly affairs. A poet can use the ‘analogy’ method to describe an unperceived phenomenon; let us say, sky lotus or golden mountain. A poet can also create an imaginary picture in readers’ minds through analogical descriptions of a fact or an event. So, the analogy is a powerful tool that helps us to gain new knowledge about worldly affairs. A scientist can use the analogy to explain some interesting phenomena of scientific discovery. Consider an example, Sir Isaac Newton’s law of universal gravitation. This law is an inferred conclusion from a few premises (observed phenomena) where similarities among the premises are compared to each other. The premises are X is ripe fruit and falls on the soil, a dead dry leaf of a tree falls on the soil, a solid object thrown into the sky will fall on the soil, etc. Unlike poets and scientists, logicians use analogy to formulate arguments and derive conclusions in the arguments from a few observed phenomena (premises). An induction by analogy is presented below for comprehension and discernment. X, Y, and Z are the swimmers of having the characteristics C1 , C2 , C3 , C4 , and C5 . In Y and Z, a new characteristic is found, say C7 . Therefore, perhaps X has the new characteristic C7 . In this argument, the conclusion is treated as neither correct nor incorrect but judged as either strong or weak, and convincing or unconvincing. Induction by analogy is reliable, convincing, and strong if and only if the numbers of similar evidence (observed phenomena) are more. Similarities between the premises play an important role to draw a reliable conclusion in induction by analogy. In induction by analogy, the conclusion of an argument should be based on the true resemblance between premises, not on the number of pseudo-resemblances between the premises. So, the aim should be on finding the true similarities between the observed phenomena (premises), not the number of pseudo-similarities between the observed phenomena. The comparison between a country cow and a gavagai should not be based on how many similarities these two animals have, rather how many appropriate and true similarities these two animals have. The conclusion of induction by analogy is more appealing, stronger, and convincing if there are true similarities found between the premises. ‘Analogy’ should not be confused with ‘example’. An example represents all such things similar to it in a group, but an analogy describes a relationship of similar features between two facts or events or phenomena. Take an example: we need to press the power button of a laptop to switch on the device. It is an example of how to switch on a laptop. A laptop’s power button may not be placed in the right top corner of every laptop. Even then, a person learns how to switch on a laptop by pressing its power button. But to know which laptop keeps the charge for a long time, we need to compare a laptop with another laptop by considering their features, configuration, etc. This is called an analogy. By making this analogy, one can infer the conclusion that which laptop keeps the charge for more time.

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Unlike induction by analogy, there are many cases where a universal conclusion is drawn from a few countable empirical pieces of evidence of a similar kind. This kind of induction is named enumerative induction. In enumerative induction, the conclusion is inferred from a countable number of empirical pieces of evidence or observed phenomena. Enumerative induction is of two types, induction by simple enumeration and induction by complete enumeration. In the case of the former, a universal conclusion is inferred from a few observed phenomena of a particular quality. In the case of the latter, a universal conclusion is inferred from the countable observed phenomena that are very limited in the number of having a particular quality. Thus, induction by complete enumeration is called the ‘perfect induction’. These two types of enumerative induction are discussed below.

15.5 Induction by Simple Enumeration Induction by simple enumeration is a basic form of induction. In this type of induction, the conclusion is derived from a few empirical pieces of evidence (premises) of having a particular quality. Since the conclusion is inferred from a few empirical evidences, it is believed that the universal conclusion also possesses the same quality as the premises possess. So, in induction by simple enumeration, a person’s knowledge of limited empirical evidence is adequate for him or her to infer the universal conclusion. For example, Mr. X is mortal. (Empirical evidence-1) Mr. Y is mortal. (Empirical evidence-2) Mr. Z is mortal. (Empirical evidence-3) --------------------{Inductive leap} --------------------Therefore, perhaps all men are mortal. In this argument, three empirical pieces of evidence (premises) are verified and found true. All the premises have one quality that is ‘mortality’. The quality ‘mortality’ is associated with Mr. X, Mr. Y, and Mr. Z (empirical evidence), and thereby it serves a good reason to establish a universal conclusion of having the same quality for other individuals as well. In this argument, the conclusion ‘perhaps all men are mortal’ is inferred. Note that, in all the premises, no reason is given for Mr. X’s death, Mr. Y’s death, and Mr. Z’s death. So, the relation between ‘Mr. X and mortality’, ‘Mr. Y and mortality’, and ‘Mr. Z and mortality’ is not established. Even then, logicians and scientists believe that induction by simple enumeration is potent to derive a universal conclusion from a few empirical evidences. Consider another example of induction by simple enumeration. X is a bird and it has feathers. Y is a bird and it has feathers.

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Z is a bird and it has feathers. ---------------------------------------{Inductive leap} ---------------------------------------Therefore, perhaps all birds have feathers. Take another example: Fire burns a table that is made of wood. Fire burns a chair that is made of wood. Fire burns a desk that is made of wood. Fire burns a bench that is made of wood. ------------------------{Inductive leap} ------------------------Therefore, perhaps fire burns all the items made of wood. In induction by simple enumeration, a few numbers of empirical evidence of having a particular quality and the absence of any contradictory evidence to the empirical evidence serve the criteria to draw a universal conclusion. In this type of induction, the ‘random sampling’ of a similar phenomenon is considered as premises and that increases the reliability of the universal conclusion. It is quite natural for human beings to accept that if a phenomenon is repeatedly occurring, the same phenomenon will occur in future under the similar circumstance. In short, a few empirical pieces of evidence (premise) of having a particular quality imply the greater reliability of a universal conclusion in induction by simple enumeration. Concerning the above-mentioned example, so long as there is not a single contradictory empirical evidence found against the proposition ‘fire burns objects’, the universal conclusion ‘fire burns objects’ stands true and considers to be a reliable conclusion. Induction by simple enumeration has a few shortcomings as well. These are: there is a threat involved in inferring a universal conclusion from a few empirical pieces of evidence. Since the conclusion is based on some random sampling of empirical evidence, if one premise claims contradictory facts against the random sampling evidence, the conclusion will be treated as false and the whole induction would be treated as inappropriate and unconvincing. For example, if a bird does not have feathers due to diseases, then the general conclusion ‘perhaps all birds have feathers’ would be treated as false. So, inferring a universal conclusion from a few empirical pieces of evidence commits a fallacy known as ‘hasty generalisation’. It is stated that a universal conclusion inferred from countable empirical evidence cannot hold with certainty all the time. So, inferring the truth of the conclusion from a few empirical evidences is not free from risk. It is found that there are cases where a large number of empirical shreds of evidence are not even enough to infer a true universal conclusion. At the same time, it is also true that in some cases, a few empirical pieces of evidence are enough to draw a reliable, convincing, and true universal conclusion. Consider an example: a person observes that ‘water flows down’ on a few occasions and found to be true in all these

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cases. From his/her experience, he/she can infer a universal conclusion that ‘perhaps water flows down’. The universal conclusion ‘perhaps water flows down’ stands true, convincing, and reliable unless there is a negative instance found against the claims made on the premises. In this case, a person need not find out all the possible empirical evidence of water flows down to infer the universal conclusion. Another example is a scientist accumulates some samples (empirical evidence) relating to a fact, that is, when an iron rod is heated with so and so temperature it expands. From these samples, the scientist can infer a universal conclusion that perhaps if any iron rod is heated with so and so temperature it will expand. This conclusion will be considered as a strong, reliable, and true conclusion unless a negative evidence disproves the claim made on the premises.

15.6 Induction by Complete Enumeration In the case of induction by complete enumeration, all the empirical evidences (premises) are accumulated from having a particular quality, and from these premises, a universal conclusion is inferred. The conclusion of induction by complete enumeration is judged as true, reliable, and convincing. For example, A gulab jamun is tasted sweet in the sweet shop-1. A gulab jamun is tasted sweet in the sweet shop-2. A gulab jamun is tasted sweet in the sweet shop-3. A gulab jamun is tasted sweet in the sweet shop-4. A gulab jamun is tasted sweet in the sweet shop-5. Therefore, all gulab jamuns taste sweet in all the sweet shops. Take another example: Logic book-1 of IIT Madras library is stamped and labelled. Logic book-2 of IIT Madras library is stamped and labelled. Logic book-3 of IIT Madras library is stamped and labelled. Logic book-4 of IIT Madras library is stamped and labelled. Logic book-5 of IIT Madras library is stamped and labelled. Logic book-6 of IIT Madras library is stamped and labelled. Therefore, all logic books of IIT Madras library are stamped and labelled. In the above-mentioned examples, it is found that total numbers of empirical shreds of evidence are taken into consideration while inferring the universal conclusion. So, there is no ‘inductive leap’ found in this type of induction. Since all the premises are to be considered to draw a true conclusion, the scope of this induction is very narrow. In other words, this type of induction applies to a few cases of worldly affairs. Since its scope is narrow and limited, it has little use in logic and logical discourse. Nevertheless, this type of induction is free from lacunas, and thereby it is regarded as ‘perfect induction’. In induction by complete enumeration,

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the conclusion expresses a summary of the empirical evidence mentioned in the premises.

15.7 David Hume and the Problem of Induction David Hume (1711–1776), a Scottish philosopher, has raised some objections against the inductive method. His criticisms are mentioned in his work An Enquiry Concerning Human Understanding (1748). He argues that in induction, we deal with matters of fact and their existence. We infer unobserved matters of fact and their existence in conclusion from a few observed instances of past and present. So, is it rational and justifiable to assert about all the matters of facts and their existence in the conclusion that are yet to be observed? He states that no matter how much empirical evidence (premises) we consider as true, the inferred unobserved universal conclusion of an inductive argument will become a probable one. The unobserved universal conclusion does not have any rational justification for accepting it as true phenomena. At best, one may consider the universal conclusion as a prediction of facts of the world, but not true facts of the world that are certain and observable. So, any degree of certainty levied on the universal conclusion of an induction the conclusion will not be judged as true. In other words, the limited observed and verified premises of an inductive argument will fail to demonstrate the truth of the conclusion. In inductive arguments, even if the conclusion is false, the premises must not be false, as they are observed and verified empirical facts. It suggests that in induction, from true premises, we can draw a false conclusion, and even then, the argument is not treated as invalid. So, it is rationally unacceptable that true premises and a false conclusion together form an argument. Hume states that ‘if the conclusion of an inductive argument could be deduced from its premises, then the falsehood of the conclusion would contradict the truth of the premises. But the falsehood of its conclusion does not contradict the truth of its premises. Therefore, the conclusion of an inductive argument cannot be deduced from its premises’ (Lange, 2008, pp. 48– 49). In inductive arguments, a risk is involved in inferring unobserved instances in conclusion from a few observed instances (i.e. premises). That is, we are inferring matters of fact from past to future to justify the principle of uniformity of nature, and we are using the principle of uniformity of nature as a basis to justify the induction. According to logicians, an induction must satisfy two features to infer a universal conclusion from the premises. These features are the law of uniformity of nature and the law of causation. The law of uniformity of nature expresses that under similar conditions, the same cause will produce the same effect, and the law of causation states that every event has a cause and nothing happens without a cause. Concerning the law of uniformity of nature, Hume argued that what is the guarantee that past events will occur in the same way in future? In other words, what is the reason to believe that the future will resemble the past? For example, what is the guarantee

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that the sun will rise tomorrow in the east as it has risen in the past? What is the guarantee that fire will burn the solid objects in future the way it is burning the objects in the present time? What is the guarantee that raw mangos will remain raw mangos in future? What is the guarantee that a person’s head’s thick and dense hair strands will remain as it is on his head in future? Hume argues that if we are accepting the principle of uniformity of nature as a basis of inferring a universal conclusion in an inductive argument, then we are committing the fallacy of ‘circular reasoning’. That is, in the principle of uniformity of nature, we are inferring matters of fact from past to future, and we are using uniformity of nature as a basis to justify the induction. We are, therefore, implicitly saying induction justifies induction. Hence, the conclusion of induction is not free from falsehood. About the law of causation, Hume argues that we perceive ‘cause’ on the one hand and ‘effect’ on the other hand. But we do not perceive the causal connection between cause and effect. We only infer that an effect follows from a cause the way we infer unobserved phenomena in conclusion from some of the observed phenomena (premises) in an inductive argument. The relation between cause and effect is established using induction. So, the law of causation commits the fallacy of circular reasoning. That is, the law of causation justifies an inductive argument and induction requires explaining the matters of fact associated with cause and effect and their relationship (i.e. the law of causation). From the above analyses and arguments, it is asserted that a universal conclusion of an inductive argument is not to be treated as true and certain. But the inductive method can be used to form beliefs about unobserved matters of fact of the world and not to establish the truth of unobserved matters of fact of the world.

15.8 Induction and Probability When we say something is ‘probable’, we mean that that thing is not sure, not true, but at the same time, it is not impossible. For example, Mr. Y says, ‘Probably, it may rain tomorrow’. This sentence is associated with the concept of probability; that is, it may rain or may not rain tomorrow. Logically speaking, the term ‘probability’ expresses a matter of degrees between ‘impossibility’ and ‘certainty’. The concept of probability is related to inductive arguments. It is so because, in inductive arguments, the conclusion is inferred from a few premises (i.e. empirical evidence). Since all the empirical evidence is not taken into account while inferring the argument’s conclusion, the truth and certainty of the conclusion are associated with the concept of ‘probability’. Like scientific findings, conclusions of inductive arguments are also associated with a degree of probability concerning their validity, certainty, and truth. The conclusion of an inductive argument is merely a probable and uncertain proposition. The reason is the conclusion of an inductive argument is inferred relying on the theory of causation and the law of uniformity of nature. In this regard, a few questions arise: can we be sure that the causal connection between two facts or

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events that exists at present would remain true and certain in future? How can we be confident and certain about the causal connection between two facts or events? When nothing is permanent on the earth, on what basis can we say that uniformity concerning empirical evidence of today would remain the same in future? Wouldn’t it be possible that a piece of evidence in future not resemble present and past evidence of a similar type? So, a degree of probability is associated with the conclusion of inductive arguments. Since the conclusion of an inductive argument is inferred from a few pieces of evidence, it would not be treated as certain and true. In other words, the conclusion of an inductive argument is probable. The concept of probability is related to ‘experience’ on the one hand and ‘belief’ on the other hand. When related to ‘experience’, we treat probability as an ‘objective’ phenomenon, and when it is related to the notion of ‘belief’, we treat probability as a ‘subjective’ phenomenon. Concerning inductive arguments, probability is treated as an objective phenomenon, as it deals with some of the empirical evidence of worldly affairs. But when we consider the probability of a case as a state of belief, we treat ‘probability’ as a ‘subjective’ phenomenon. In the latter case, truth and certainty of probability are weaker than in the former case (i.e. inductive arguments). Although ‘probability’ means the slightest chance of obtaining the truth of a fact or an event, yet it stands for an opportunity (i.e. a possibility). As a result, the concept of ‘probability’ has significance and relevance in scientific enquiry, inductive logic, and knowledge acquisition from worldly affairs.

Chapter 16

J. S. Mill’s Inductive Methods

In the previous chapter, we examined the difference between deduction and induction. We mentioned types of induction and explained scientific induction, induction by analogy, induction by simple enumeration, and induction by complete enumeration in detail. We also elucidated the lacunas of induction. In continuation of the previous chapter, in this chapter, we will discuss John Stuart Mill’s (1806–1873) arguments on scientific induction and also types of scientific inductive method: for that J. S. Mill’s manuscript, A System of Logic (1872) of its eighth revised edition is considered. This edition is treated as a comprehensive work of J. S. Mill that assists in understanding his contributions to scientific induction. However, it is to be noted here that the manuscript A System of Logic was first published in the year 1843.

16.1 J. S. Mill’s Proposal on Inductive Methods J. S. Mill considers ‘scientific induction’ and explains the scientific inductive methods in his works. He states that in the scientific inductive methods, cause and effect are related to each other in all instances. For example, if Covid-19 is the cause of death of a person, then the effect ‘death’ is invariably related to the cause. The scientific inductive methods are popularly known as the ‘experimental methods’ and ‘inductive canons’. Scientists in their research studies aim to find a causal link between two phenomena, that is, how one thing is caused by another thing and the conditions that are associated with a thing to give rise to the effect? For example, X causes Y. In this case, a scientist will find out a causal link between X and Y, and the conditions that are associated with X to give rise to Y only, not Z, L, M, etc. According to Francis Bacon, an empiricist philosopher, scientists use the ‘inductive method’ to find out the causal link between two facts or many phenomena. By taking a cue from Francis Bacon, J. S. Mill explains that scientific induction is meant to find a causal link among many empirical facts (premises), and based on the causal link, we can infer a universal proposition as a conclusion. In Mill’s view, a causal link © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_16

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is found among premises of scientific induction, and that helps to infer a universal conclusion. He argued that scientific induction plays a pivotal role in finding out certainty as well as the truth of the inductive arguments and his justification for accepting inductive methods is based on the theory of causation. According to him, the theory of causation holds four true suppositions. They are: • • • •

A cause exists therefore an effect exists. A cause does not exist therefore an effect does not exist. A cause can produce a particular designated effect. If there are changes in the cause, there will be a change in the effect.

Mill proposes five types of scientific inductive methods, namely the method of agreement, the method of difference, the joint method of agreement and difference, the method of residues, and the method of concomitant variations. He finds that the method of agreement and the method of difference are the basic methods of scientific induction. The remaining inductive methods are subsidiary to the fundamental methods, although they assist in inferring a universal conclusion from a few empirical pieces of evidence (premises). In Mill’s view, the joint method of agreement and difference is a special modification of the basic inductive methods: the method of agreement and the method of difference. The method of residues is a peculiar modification of the method of difference, whereas the method of concomitant variations is a special form of either the method of agreement or the method of difference depending on the context/situation. Of the two basic inductive methods, the method of agreement suggests a universal conclusion from some observable facts (premises) of having a particular quality. Hence, it is not problematic to establish the causal link between two or more premises (observed facts). But in the method of difference, the universal conclusion is inferred from a few observed facts that have some common and unique qualities. Hence, establishing the causal link between the observed facts (premises) is tricky and challenging. In this sense, the method of difference is more challenging than the method of agreement to infer certainty and truth of the universal conclusion of scientific induction.

16.2 The Method of Agreement The method of agreement is a scientific inductive method where one event (let us say A) gives rise to another event (let us say B) on several occasions. It suggests that whenever A occurs, B occurs immediately. The togetherness of A and B events on multiple occasions induces to relate them causally. That is, if A occurs, B will also take place. However, the strength of the method of agreement is based on a large number of instances (premises) where A and B events occur one after another immediately. The reliability of causal relation between A and B events stands promising when a large number of A and B event occurrences are gathered. Thus, it is considered the most preferable and popular inductive method to infer a true universal conclusion in scientific induction. Logicians, philosophers, scientists, and common people

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mostly use this method to find out the cause of an effect in their empirical discovery. By using this method, they also infer a universal conclusion from a few observed instances. Mill (1882) explains the method of agreement as ‘if two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is the cause (of effect) of the given phenomenon’ (p. 482). An example of the method of agreement X finds that oxygen caused the fire to burn an object in the kitchen. X finds that oxygen caused the fire to burn an object in front of a house. X finds that oxygen caused the fire to burn a wood piece in the backyard of a house. X finds that oxygen caused the fire to burn dry leaves. ------------------------------------------------------------------------------------------------------------Therefore, X infers that perhaps oxygen caused the fire to burn the objects. In this example, X infers the conclusion by applying the method of agreement to the induction. X uses the method of agreement to draw the universal conclusion by finding the causal link between ‘oxygen’ and ‘fire burns objects’. In this induction, X identifies a common factor (i.e. oxygen) that presents in all the observed empirical phenomena (premises), which assists X to conclude that whenever oxygen presents in an object, the fire will burn the object. Take another example of the method of agreement. X is diagnosed with Covid-19 and cancer in the year 2020, and X died. A is diagnosed with Covid-19 and diabetes in the year 2020, and A died. E is diagnosed with Covid-19 and cold in the year 2020, and E died. H is diagnosed with Covid-19 and Ebola in the year 2020, and H died. P is diagnosed with Covid-19 and diabetes in the year 2020, and P died. H is diagnosed with Covid-19 and malaria in the year 2020, and H died. Y is diagnosed with Covid-19 and jaundice in the year 2020, and Y died. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Therefore, probably Covid-19 is the cause of mass death in the year 2020. In this example, the common factor ‘Covid-19’ disease is found in all the premises and there is a causal relationship found between Covid-19 and the death of a person. Due to the existence of a common factor in all the premises, a universal conclusion is drawn, that is, perhaps Covid-19 disease is the cause of mass human beings death in the year 2020. Refer to Table 16.1, and examine all the instances to find out the real cause of death of individuals in the year 2020. By applying the method of agreement to these instances, it is found that ‘Covid-19’ is the cause of the death of individuals in the year 2020 across the globe. Here, Covid-19 is the cause and death is its effect, and both cause and effect are linked to each other in all the instances.

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Table 16.1 Real cause of death of people in eight countries Countries

Disease

Disease

Death

Year

The USA

Covid-19

Cancer

Yes

2020

The UK

Covid-19

Diabetes

Yes

2020

The Netherlands

Covid-19

Cold

Yes

2020

Germany

Covid-19

Ebola

Yes

2020

Italy

Covid-19

Diabetes

Yes

2020

France

Covid-19

Malaria

Yes

2020

India

Covid-19

Jaundice

Yes

2020

China

Covid-19

Ebola

Yes

2020

The method of agreement helps in deriving a universal conclusion from a few observed instances of having a particular quality. The certainty of the conclusion will be stronger if there are more instances to derive the conclusion. However, it is wrong to believe that the method of agreement results in a true, strong, and reliable conclusion on every occasion. The method of agreement has a few shortcomings as well. One may argue that there are cases where collecting empirical evidence of having a particular quality take quite a few years; thus a universal conclusion cannot be inferred so easily. In this case, the causal relation between two events cannot be established so easily, as one has to wait for many years to collect empirical evidence. Also, there is difficulty finding out the real common factor of all the empirical shreds of evidence. As a result, the method of agreement would not suffice to draw a reliable and strong conclusion in all the scientific inductions. Take an example: if a person wants to gather a few evidence about leap years to derive a universal conclusion, then he/she has to wait quite a few years to collect the evidence, as a leap year comes once in 04 years. So, to collect the four pieces of evidence (let us have a minimum of four premises), he/she has to wait for 16 years. Due to the long waiting period to collect the relevant evidence, he/she will not be able to find a true causal link among premises, as many things would be changed during 16 years time period. Hence, the method of agreement is not free from constraints on establishing a causal link among all the evidence to draw a true, strong, reliable, and universal conclusion in scientific induction. A few more questions arise against the method of agreement. These are, how does a person know that a common factor that causally links to the observed instances (premises) stand true? How would a person be sure that the common factor is the relevant factor in determining the causal relationship between two observed phenomena? The following examples explain how the irrelevant common factor is wrongly considered as a cause of the event or fact. As a result, the method of agreement is applied to scientific induction to draw an unreliable and weaker universal conclusion. Example-1 Dog X barked at Y, Y was afraid and ran away. Dog L barked at M, M was afraid and ran away.

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Dog S barked at Q, Q was afraid and ran away. Dog N barked at T, T was afraid and ran away. Dog P barked at R, R was afraid and ran away. ----------------------------------------------------------------------------------------------------------------------------Therefore, perhaps whenever a dog barked at a person, that person was afraid and ran away. The method of agreement is applied to this induction. But it would be wrong to infer that ‘perhaps whenever a dog barked at a person, that person was afraid and ran away’. The reason is there may be a situation where a dog barked at a person and that person was not afraid and therefore did not run away. Example-2 X drank rum by mixing water and got intoxicated. Y drank vodka by mixing water and got intoxicated. Z drank brandy by missing water and got intoxicated. L drank gin by mixing water and got intoxicated. ----------------------------------------------------------------------------------------------------------------------------Therefore, water is the cause of X, Y, Z, L intoxication. This example is borrowed from Copi and Cohen (1995). They convey that the method of agreement on some occasions will not find the real cause (true cause) of an effect. In this example, water is wrongly considered as the common factor for all the instances. And, it is wrong to infer that ‘water is the cause of X, Y, Z, L intoxication’. The reason is that even though the method of agreement applies to this induction, it does not establish the right causal relationship among all the instances. As a result, the irrelevant common factor is wrongly considered as the cause of the instances. Example-3 X did not attend the Monday logic class because his bicycle tube was punctured while coming to the class. Y did not attend the Monday logic class because her bicycle tube was punctured while coming to the class. Z did not attend the Monday logic class because his bicycle tube was punctured while coming to the class. M did not attend the Monday logic class because her bicycle tube was punctured while coming to the class. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Therefore, all the students who could not attend the Monday logic class were due to the puncture of their bicycle tubes. In this example, we apply the method of agreement to infer the universal conclusion from a few observed instances. But, it would be wrong to treat the conclusion

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as true. The reason is there may be a possibility that P is a student who could not attend the Monday logic class due to her illness. Thus, the common factor that is believed to be the cause of all the premises is not a genuine and true cause. In this sense, the method of agreement is not free from establishing an unreliable, weaker, and unconvincing universal conclusion in scientific induction. In the method of agreement, there is a possibility that we may find two probable causes of an effect. In this case, we would wonder which cause is to be treated as the real and genuine cause and which one is an accidental cause. Consider an example, let us say, after returning from a dinner party, four persons fall sick. A doctor wants to know the real cause of their sickness. The doctor finds these four persons ate two food items (i.e. paneer curry and brinjal curry) that are not eaten by other guests at the party. In this case, even though the doctor applies the method of agreement to find out the real cause for their sickness, but confusion arises as to which curry is the real cause and which curry is the accidental cause for their sickness? Take another example, Covid-19 and diabetes are the cause of the death of many people in India. But what is the real cause and what is the accidental cause for human beings death in India will not be able to find out through the method of agreement. Further, a few questions arise. How do we know that which common factor is reliable and genuine of a scientific induction? How do we find the true causal connection among all the evidence mentioned in the premises of scientific induction? To answer these questions, J. S. Mill suggests another inductive method known as ‘the method of difference’. This inductive method assists in finding out the true causal connection among all the instances of scientific induction. In brief, the method of difference assists in finding out a common isolated factor among all the instances that is regarded as the real cause of the instances of scientific induction.

16.3 The Method of Difference The method of difference is also known as the method of disagreement. The method of difference is a fundamental scientific inductive method used by scientists, philosophers, and common people for their empirical discoveries. Unlike the method of agreement, this method does not require a large number of instances to infer a conclusion. The method of difference requires only two true verified instances to infer the conclusion, provided the two instances must resemble each other in every respect except one. For example, at a birthday party, A ate p, q, r, s foods, and B ate p, q, r, s, t foods. After eating foods, B falls sick and visits a doctor for treatment. The doctor, by speaking to B, learns about the food items eaten by A and B at the birthday party. The doctor eliminated the food items eaten by A from food items eaten by B, as A did not fall sick after eating the foods at the birthday party. The doctor concludes that it is the t food that makes B sick. Mill (1882) explained the method of difference as, ‘if an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance save one in common, that one occurring only in the former; the circumstance in which alone

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the two instances differ, is the effect, or cause, or an indispensable part of the cause, of the phenomenon’ (p. 483). J. S. Mill enunciates that the method of difference can find the causal relation between two instances and assists in obtaining a strong and reliable conclusion in scientific induction. Some logicians also claim that the method of difference is the strongest method among other inductive methods to derive a reliable and convincing conclusion in scientific induction. However, the method of difference has the following limitations. The method of difference requires two instances of one kind, which resemble each other in every respect except one. It is indeed difficult to accumulate two kinds of evidence/instances from our mundane life experiences and experimental studies. Hence, meeting the requirements to apply the method of difference is not feasible all the time pragmatically. Thus, it is stated that the method of difference applies to a few situations of our mundane life and some scientific research works.

16.4 The Joint Method of Agreement and Difference The joint method of agreement and difference is a combination of the method of agreement and the method of difference. This method applies to a scientific induction where the method of agreement and the method of difference are applied together. It is believed that this method is a stronger and superior method than the independent method ‘the method of agreement’ and ‘the method of difference’. Since two methods together apply to an induction, it establishes a strong causal connection between facts or events mentioned in the premises. Further, it increases the reliability and certainty of the conclusion drawn from the premises. Mill (1882) explains the joint method of agreement and difference in the following lines, ‘If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of the circumstance; the circumstance in which alone the two sets of instances differ is the effect, or cause, or a necessary part of the cause, of the phenomenon’ (p. 489). Let us consider an example to elucidate the joint method of agreement and difference. X ate a, b, c, d foods, and fell sick. Y ate b, d, c foods, and not sick. Z ate c, d, b foods, and not sick. Therefore, food ‘a’ causes X sick and foods ‘b’, ‘c’, ‘d’ do not cause X sick. Upon applying the method of the agreement to the above induction, a doctor would find that b, c, and d food items are common to X, Y, and Z. Thus, b, c, d food items are not the causes for X’s sickness. But upon applying the method of difference, a doctor would find that X ate food item a, which was not eaten by Y and Z. Thus, ‘a’ is the cause for X’s sickness. In this induction, the joint method of agreement and difference assists in finding out the real cause for X’s sickness. This method takes

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into consideration the necessary and sufficient conditions of instances to establish the causal link between instances (premises). In this inductive method, even though the conclusion is drawn from the affirmative instances, the negative instance brings certainty and reliability to the conclusion.

16.5 The Method of Residues The method of residues is a special kind of method that applies to scientific inductions to infer a reliable, strong, and convincing conclusion. This method suggests that deduct X (maybe a fact or an event) from Y (maybe a fact or an event) and find out the causal connection between X and Y. Based on the causal connection, derive the conclusion in the induction. Mill (1882) explains this method as ‘subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents (p. 491)’. Hoffman (1962) explains the method of residues through the following symbolical representation (p. 496). AB→DE A∧D Therefore, B ∧ E This example states that if A and B occur, then D and E occur. A and D are causally connected, as D follows from A. From the above two instances, we can conclude that whenever B occurs, E follows from that, as after deduction of ‘A ∧ D’ from ‘A B → D E’, we get only ‘B ∧ E’. Consider an example, Miku went to the market to purchase 10 L of sunflower oil, carrying a barrel to bring the oil. After reaching the store, he asked the shopkeeper to give 10 L of sunflower oil. The shopkeeper weighs the empty barrel first, then pours 10 L of oil into it. To find out 10 L of oil in the barrel, Miku deducts the empty barrel weights from the sunflower oil barrel weights. The difference in weight between the sunflower oil barrel and the empty barrel is the sunflower oil’s actual weight. This is the only method to determine the actual weight of 10 L of sunflower oil in the barrel. Many shopkeepers and individuals use this method to draw a reliable conclusion on various matters in their mundane lives without knowing that it is called the method of residues. So, the method of residues plays a significant role in establishing a causal link between two phenomena and assisting in deriving a convincing conclusion from the observed phenomena. Take another example: most of us might have noticed that to find out the actual weight of rice in a sack, people deduct the weight of the sack from the weight of the rice sack. The weight difference between the rice sack and the empty sack is the actual weight of the rice in the sack. Scientists in their experimental research also use the method of residues to find out a reliable conclusion of their studies. This inductive method helps scientists

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to discover many things in the empirical world, such as extraction of aluminium from bauxite ore, nitrogen gas from the air, iron from iron ore, etc. Although a few philosophers regard the method of residues is a method of deduction where fewer instances are required to draw a conclusion, but it is not true. The reason is, like Mill’s other scientific inductive methods, this method has significance in deriving a reliable and convincing conclusion from its premises. This method finds a causal link between facts or events mentioned in the premises, and the causal link assists in inferring a reliable conclusion in the induction. However, the conclusion of induction may not be considered as true as it lacks foolproof.

16.6 The Method of Concomitant Variations The method of concomitant variations states that when two facts or events vary, they suggest a causal relationship between them. For example, when a product supply increases in the market, its price decreases, and when a product supply is restricted, its price increases in the market. To find out the causal link between the supply of a product and its price variation in the market, we use the method of concomitant variations. This method is used in all fields to draw a reliable and strong conclusion. In the economics subject, researchers use the method of concomitant variations to determine the price of a commodity in a market. Mill (1882) elucidates the method of concomitant variations as, ‘Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation’ (pp. 495– 496). Let us consider a few more examples of the method of concomitant variations. In Asian countries, in the summer season, it is noticed that in the morning time, when the sun rises in the east, the sunrays do not hurt our skin, but in the noontime, sunrays burn our skin, and we get hurt. The causal link between sunrays not burning the skin in the morning and sunrays burning the skin in noontime can be established by adopting the method of concomitant variations. Another example, on full moon day, we find high tides in the sea. From the full moon day to the next fourteen days, we find the height of tides of the sea varies as per the sizes of the moon. The application of the method of concomitant variations to this case assists in finding the causal link between the high and low tides of a sea with sizes of the moon. Thus, the method of concomitant variations helps scientists, researchers, and common people to find out the true causal link between two varied facts. The method of concomitant variations is primarily dealt with inductions of quantitative nature to find out the causal link between two facts or events. This method measures the increase and decrease of the quantity on a scale. The method of concomitant variations is not regarded as a foolproof method to determine the true conclusion of scientific induction. The reason is none of the scales gives an accurate and authentic result. Further, it is argued that there are occasions when two facts or events vary in relation; they may not be causally connected. Instead, they may be the co-effects

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of some other cause. For example, the second stick and the minute stick of a watch vary simultaneously but the movement of the minute stick is not the cause of the movement of the second stick and vice versa. In this case, the method of concomitant variations fails to establish a causal link between two events, and hence it fails to infer a true conclusion from the observed facts (premises). In brief, the method of concomitant variations has two shortcomings. First, it does not apply to qualitative variations cases, and second, it applies to only those facts that are empirically verified. The five kinds of scientific inductive methods are used to derive a reliable, strong, and convincing conclusion of scientific induction. But the conclusion may not be treated as a true conclusion. The reason is these inductive methods may not find a true causal link between observed phenomena (premises) with certainty and foolproof. So, the knowledge obtained through scientific inductive methods is not certain and free from doubt. However, inductive methods are considered important methods and alternatives to deductive methods to derive a strong and convincing conclusion of scientific inductions. Inductive methods thus help in obtaining new knowledge about worldly affairs.

Chapter 17

Science and Hypothesis

In this chapter, we will discuss the significance of a ‘hypothesis’ in a logical inquiry, a scientific investigation, and research work. We will enumerate some of the definitions of ‘hypothesis’. We will elaborate on the nature and scope of the ‘hypothesis’ and the sources to obtain a hypothesis. Further, we will explain the kinds of hypothesis with suitable examples. In the end, we will illustrate methods to verify a hypothesis in a logical inquiry and a scientific investigation. Let us start with ‘what is a hypothesis?’.

17.1 What is a Hypothesis? A hypothesis is regarded as a provisional supposition. It means a hypothesis is a suggestion or a possible explanation of a logical inquiry. It is a tentative solution and not a real solution to a logical inquiry until tested, verified and proved to be true. Thus, a hypothesis is subject to revision and rejection. In every scientific investigation, researchers (investigators) assume some possible explanations of the research problem at the beginning of the research work. It is so because a scientific investigation or a research study cannot be an unplanned, unorganised, and random activity. The ‘possible explanations’ are called hypotheses. The hypotheses of a research study (a scientific investigation) guide researchers (investigators) to adopt the correct methodology, tools, and techniques for data collection, data analysis and interpretation, and subsequently derivation of conclusion from the data. If the conclusion of the study supports the hypotheses, then the hypotheses are judged as legitimate and valid. And, if the conclusion does not support the hypotheses, then the hypotheses are treated as false and invalid. Thus, it is stated that when a scientific investigation (a research study) is planned to be carried out, a few hypotheses are to be formulated to move in the correct direction of the study and achieve the objectives of the study within a time period. We, human beings, use logic to formulate hypotheses on the social and personal problems of our mundane lives. For example, in a rainy season, if most of the students © Springer Nature Singapore Pte Ltd. 2021 S. S. Sethy, Introduction to Logic and Logical Discourse, https://doi.org/10.1007/978-981-16-2689-0_17

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did not come to a history class, the course teacher formulates the following hypotheses (i.e. possible explanations). (i) (ii) (iii) (iv)

Perhaps, the students did not have an umbrella with themselves. Perhaps, the students got wet while coming to the class and did not feel comfortable attending the class. Perhaps, due to heavy and insistent rain, the students overslept in their hostel rooms and could not come to the class. Perhaps, a few students were misinformed by their classmates that due to heavy and continuous rain, the course teacher might not engage the history class.

These are the possible explanations or suggestions for the above-mentioned problem. These suggestions are not to be treated as valid unless these are verified as true. Thus, it may be stated that a valid hypothesis is an accepted solution to a research problem. A hypothesis is considered as an intelligent guess or a tentative solution to a research problem. It is a presumptive statement made relying on the available evidence (data). A presumptive statement is not to be considered as a wild guess of a researcher. Rather, it is formulated based on the past relevant research studies conducted by a few researchers. According to Van Dalen (1973), a hypothesis is a powerful beacon that lights the way for the researchers. Hypotheses are written in declarative statement form where researchers (investigators) make a conjecture about the outcomes of a research study. From the above discussions, we can assert that a hypothesis plays an important role in a scientific investigation (research study). On the one hand, a hypothesis is required to investigate a research problem, and on the other hand, there are no fixed rules and criteria to formulate a legitimate and valid hypothesis. And, at the same time, the formulation of a hypothesis is not a mechanical task. So, the active role of an investigator (researcher) plays an important role in formulating the hypotheses of a research problem.

17.2 Definition, Nature, and Importance of ‘Hypothesis’ A hypothesis is an indispensable component of a research study. It is an assumption about the relation between two or more variables and a possible explanation of a research problem. Many definitions of ‘hypothesis’ are found in the literature. Some of them are mentioned below. According to Werkmeister (1948), the guesses a researcher makes are hypotheses, which either solve the problem or guide him/her in further investigation (p. 300).1 According to Lundberg (1968), ‘a hypothesis is a tentative generalisation, the validity 1 Werkmeister,

Publication.

W.H. (1948). The basis and structure of knowledge. New York: Haper and Bros

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of which remains to be tested’.2 For Theodorson and Theodorson (1969), ‘a hypothesis is a tentative statement asserting a tentative relationship between certain facts’ (p. 191). Black and Champion (1976) have described a hypothesis ‘as a tentative statement about something the validity of which is usually unknown’.3 According to Goode and Hatt (1971), ‘a hypothesis is a proposition which can be put to test to determine validity’.4 The Merriam-Webster’s dictionary (2020) defines a hypothesis as ‘a tentative assumption made in order to draw out and test its logical and empirical consequences’.5 In a research study or a scientific investigation, hypotheses are formulated mostly by considering ‘induction by analogy’ and ‘induction by simple enumeration’. These two inductions are explained in detail in Chap. 15 of this manuscript. Succinctly speaking, induction by simple enumeration suggests that, when two or more phenomena together occur repeatedly, perhaps there exists a connection between the phenomena; this becomes a ground to formulate a hypothesis. This hypothesis is regarded as the conclusion of the argument. In the case of ‘induction by analogy’, it is stated that when two things resemble each other in certain respects and are dissimilar in certain other respects, this becomes a ground to formulate hypotheses. That is, perhaps these two things may have further resemblances and dissimilarities as well. However, these hypotheses are not considered true unless they are verified as true. A hypothesis is treated to be a valid hypothesis when it satisfies the following conditions (Bailley, 1978; Sarantakos,6 2005): (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

A hypothesis must be empirically testable and verifiable. A hypothesis should be formulated in a precise and clear manner. A hypothesis shall not contradict another hypothesis of a research problem. A hypothesis must establish the relationship between two variables. Each hypothesis must deal with one issue, shall not club all the issues as a whole. A hypothesis shall not be construed based on an investigator’s desires and wishes. A hypothesis must aim to achieve the objectives of the research study. A hypothesis should be testable within a reasonable period.

If a hypothesis is formulated based on these conditions, then it satisfies both the descriptive and rational form of a hypothesis. A hypothesis in its descriptive form describes a fact or an event, and in its rational form, it establishes the relation between two or more variables. 2 Lundberg,

G.A. (1968). Social research: A study in methods of gathering data. New York: Greenwood Press. 3 Black, J. A., and Champion, D.J. (1976). Method and issues in social research. New York: John Wiley & Sons. 4 Goode, W.J., and Hatt, P.K. (1971). Methods in social research. New York: McGraw-Hill Publication. 5 https://www.merriam-webster.com/dictionary/hypothesis. 6 Sarantakos, S. (2005) (3rd Edition). Social research. New York: Palgrave Macmillan.

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Importance of a hypothesis for a research study A hypothesis plays an important role in a scientific investigation or a research study. It provides provisional explanations (suggestions) to a research problem. It guides researchers in finding out the appropriate methodology to carry out the research tasks. A hypothesis directs, monitors, and controls the research efforts. A valid hypothesis adds new information to the existing information of a research area. It not only assists researchers in designing their study but also guides them about data collection, data interpretation, and derivation of conclusion from the results of the study. Further, a hypothesis guides researchers to carry out the research tasks within the scope of the research study. It suggests researchers to relate logically known facts to rational and intelligent conjectures about unknown facts. It guides the researchers to follow the inductive method to verify the validity of the hypothesis. In this sense, a valid hypothesis makes a research report interesting and meaningful. It allures readers to learn new information about the research study. It helps researchers to save their time and energy to collect relevant data that would enable them to establish the relationship among variables. Thus, a research study’s hypotheses constantly guide researchers to eliminate irrelevant literature from relevant literature and collect necessary and relevant data instead of unimportant and irrelevant data. It may be conveyed that hypotheses are formulated to focus only on relevant and pertinent data of a research study or a scientific investigation. Salient features of a valid hypothesis A hypothesis plays a pivotal role in carrying out a research study. It is a provisional supposition made by the researchers based on their intellectual rigour, rational judgement, and logical analysis of the previous research studies related to the research problem. A hypothesis is treated as a predicated declarative statement, but all predicted declarative statements are not to be regarded as hypotheses. The reason is, a hypothesis has the following features, but a predicated declarative statement may not have these features. (i)

(ii)

(iii)

A hypothesis must be consistent with known facts. It should explain facts or events surrounded the researchers. As a result, it can be verified to establish its truth. Hypotheses are to be verified through observation or experiment. If a hypothesis is not verified, it cannot be used in a scientific investigation. But, if a hypothesis is verified and found to be false then it would not be treated as a valid hypothesis. A hypothesis must be directed to solve or resolve the research problem. It must either be a cause or part of a cause of the research problem for which it is formulated. Irrelevant hypotheses have no use in scientific investigations. Since hypotheses are formulated on the basis of the existing literature pertaining to a research problem, they contribute new information to the existing knowledge. A hypothesis shall not be a cluster of assertive propositions. Rather, it should be an assertive proposition that needs to be formulated in a precise, clear,

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(iv)

(v)

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and distinct manner. An obscure and ambiguous assertive proposition is not considered as a hypothesis of a research study. A hypothesis must not contradict the established truths and laws. To explain, a hypothesis is formulated based on the information and evidence available in the literature pertaining to the research problem. If the formulated hypothesis contradicts the established truths and laws, the hypothesis would be suspicious in every respect. If an attempt is made to contradict the previous established truths and laws, the hypothesis is judged as false unless proved true. For example, Copernicus, a Polish astronomer, challenged the established Platonic finding, that is, ‘sun moves around the earth’. Copernicus, in his research works, proved that his hypothesis ‘earth moves around the sun’ is a valid hypothesis and Platonic finding is not a valid truth. So, it is to be noted that a hypothesis must be based on some relevant theories and discovered truth. A hypothesis must be testable. If it were not so, researchers would face difficulty finding out whether the hypothesis contradicts or confirms the relationship between two or more variables. For example, ‘inclusive education brings all-round development of school students’. In this hypothesis, it would be difficult for the researchers to isolate other factors that might contribute to school students’ all-round development. A hypothesis should delimit its scope. If the scope of a hypothesis is not restricted, then it will not be verified in a specific time period. As a consequence, the purpose of the hypothesis will not be fructified. So, it is suggested that a hypothesis must be formulated keeping in mind that it should be testable within the desired period, as a research task has to be carried out and finished in a given period of time. While formulating hypotheses, researchers need to keep in mind the time period for data collection, availability of tools and techniques to carry out and complete the research tasks on time.

17.3 Sources of a Hypothesis Formulation of a valid hypothesis is not a mechanical task. It is the researchers who have earned certain skills (e.g. imaginative, speculative, hands-on experience skills, etc.) and abilities (e.g. analytical mind, logical thinking, rational judgements, etc.) to formulate legitimate and valid hypotheses at the beginning of their research work on research problems. A hypothesis may originate from a variety of sources. The sources of hypotheses have important bearings on the contribution of hypotheses to the research problems. Researchers may formulate hypotheses from the following sources but not limited to these alone. Some of the noticeable sources to formulate hypotheses are mentioned below. (i)

Knowledge of a research domain A researcher or a scientific investigator must be familiar with the existing theories, research findings, and established facts of an educational field. He/she has to study the journal papers and books and attend seminars and conferences on a

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(v)

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subject’s topics to find out the working hypotheses of a research problem. The available and existing literature of a subject concerning the research problem is an important source of hypotheses formulation. The literature of a subject guides researchers to find out the relationship among variables, and the aspects of the relationship that have not been studied. Thus, the background knowledge of an educational field assists researchers to formulate the hypotheses of a research problem. Application of deductive and inductive logic to various phenomena of a research field By using deductive and inductive logic, a researcher can formulate certain working hypotheses of a research problem. In induction, researchers begin with specific observations and combine them to produce a more general statement of relationship, namely ‘hypothesis’. In deduction, the relationship between two or more variables is used to arrive at specific hypotheses. Induction begins with empirical events (data) and proceeds towards the hypothesis, while deduction begins with the general hypothesis and proceeds towards a specific hypothesis (Kabir, 2016, p. 59). Intuition Intuition is an inner feeling and belief of a researcher that tempts to formulate a hypothesis on a research problem. In this case, the researcher conducts the research study to verify whether his/her ideas are true or not. Suppose the hypotheses are found to be true. In that case, the research study adds new information to the existing field of knowledge and integrates that information into the theories of the domain knowledge. If the hypotheses are not found to be true, then they are treated as invalid hypotheses. Analogy An analogy is based on similarities and dissimilarities between two facts or events. A hypothesis can be formulated by finding out some new similarities or dissimilarities between features of two facts. In this sense, an ‘analogy’ is considered as a valid source to obtain hypotheses of a research problem. Conversation with scholars of an academic field Interactions with established researchers of an academic field may give certain clues to the researchers to find a new perspective of a research problem that would assist them in looking into the research problem again, and it may enable them to formulate the working hypotheses of the research problem. Theory A systematic study, review, and analysis of an academic field’s existing theories help researchers formulate working hypotheses of a research problem. On most occasions, these working hypotheses are verified as legitimate and valid hypotheses. It is so because knowledge of an academic field is derived from the existing theories and application of these theories to various research problems.

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17.4 Types of Hypothesis Hypotheses are of different kinds. About nine types of hypothesis are found in the research arena. These are: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

Descriptive hypothesis Directional hypothesis Non-directional hypothesis Associative hypothesis Causal hypothesis Working hypothesis Scientific hypothesis Research hypothesis Null hypothesis.

The descriptive hypothesis predicts the relationship between two or more variables. It is of two types, simple hypothesis and complex hypothesis. A simple hypothesis predicts a relationship between a dependent variable (DV) and an independent variable (IV). For example, those who learn logic lessons would earn good grades in a logic course. In this hypothesis, learning logic lessons is the independent variable and earning a good grade is the dependent variable. A complex hypothesis predicts the relationship between dependent variables and independent variables. For example, attending logic classes (IV) will help a student to learn the lessons (DV), make his/her attendance sufficient to appear in the end semester examination (DV), develop rational thinking (DV) and to make logical arguments on various issues (DV). Directional hypothesis demonstrates a researcher’s intellectual rigour to predict a clear and concise result of a research problem. It indicates a relationship between two variables. For example, students who expect a high score in a literature subject emotionally suffer more compared to students who do not expect a high score in the literature course. This hypothesis is a directional hypothesis, as it predicts the expected difference between two variables, and further it brings out the clear directional relationship between two variables. A non-directional hypothesis is made when sufficient research studies are not available on a research field to predict the relationship between two variables clearly and concisely. In other words, a hypothesis that does not specify the direction of the expected relationship between two variables is judged as a non-directional hypothesis. For example, the academic achievements of Sociology students are related to their participation in college cultural activities. In the case of an associative hypothesis, it is found that a change in one variable will cause a change in another variable. However, the relationship between two variables is not linked to each other as cause–effect relation. For example, more supply of apple in the market results in a decrease in the cost of apple, and less supply of apple in the market results in an increase in the price of apples. The causal hypothesis delineates the causal relation between two variables or more than two variables. For example, logic students who participated in college cultural

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festivals (variable-1) and spent less time on their studies (variable-2) scored low marks in their courses (variable-3). In this hypothesis, a researcher needs to find out the cause–effect relationships among three variables (i.e. variable-1, variable-2, and variable-3) and establish the causal connection between the cause and effect of the hypothesis. Further, the researcher needs to justify that the effect is the consequence (result) of the cause. About ‘working hypothesis’, when researchers plan to work on a research problem and they have preliminary assumptions about the problem, the preliminary assumptions are treated as ‘working hypotheses’. In this stage, researchers do not have adequate information to formulate the correct and legitimate hypotheses. They have not even completed their readings on the available existing literature pertaining to the research problem. With very limited information, researchers formulate the probable hypotheses to design the research plan. These hypotheses are treated as working hypotheses. Thus, working hypotheses are subject to modification, as researchers are in the process of collecting more information about past research studies on the research problem. An example of a working hypothesis, ‘Better salary and perks to the university teachers in India will motivate them to excel in their teaching and research tasks’. Later on, based on the literature review, the researcher modifies this hypothesis as ‘Better salary and academic perks to the university teachers will motivate them to do creative research work and publish papers in reputed journals’. A working hypothesis is accepted as true for the time being until a proper hypothesis is formulated for the research problem. A scientific hypothesis is formulated from the available theoretical and empirical data of a research field. The empirical and theoretical data guide the researchers to formulate a scientific hypothesis for investigation. But in the case of a ‘research hypothesis’, researchers formulate the hypothesis based on their opinions on certain social, political, cultural, economic, and psychological facts without referring to the correct ascriptions of the facts. In this case, investigators believe that their opinions are true, and at the same time, they want to prove their opinions. A research hypothesis guides the investigation that is to be conducted and what tools are to be used to measure the hypothesis’s variables. It may be stated that the whole research work carried out based on a research hypothesis implies the emergence of a new developing theory. An example of a research hypothesis: ‘Indian Institute of Technology (IIT) BTech students, at the end of their studies, wish to be Chief Executive Officers (CEO) of private companies’. This research hypothesis is formulated based on the researcher’s opinion on the observed phenomena. Another kind of hypothesis is known as the ‘null hypothesis’. A null hypothesis is a negative but declarative statement. In this hypothesis, no relation is found between the two variables. For example, there is no difference between the parents in rural and urban areas concerning bringing up their children. It is a null hypothesis because there is no relation found between the two variables. A null hypothesis is denoted as ‘H0 ’.

17.5 Verification of a Hypothesis

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17.5 Verification of a Hypothesis After formulating the hypotheses of a research problem, researchers proceed to test these hypotheses to find out their truth and validity. If a hypothesis is verified and found to be true, then it is treated as a valid hypothesis. And, if a hypothesis is verified and found to be false, then it is treated as an invalid hypothesis. It is to be noted here that, until a hypothesis is verified, it is neither to be treated as valid nor invalid. Since a hypothesis is required to be verified to find out its validity for a research study, researchers aim at finding out whether the conclusion derived from the research study supports the empirical data or not. The objective of verifying a hypothesis is to find out the truth of the hypothesis through empirical data and logical arguments. To verify a hypothesis, we equate the conclusion derived from the research study with the hypothesis. If an agreement is found between the hypothesis and the conclusion of a research study, then the hypothesis is treated as a valid hypothesis. And, if no agreement is found between the hypothesis and the conclusion of a research study, the hypothesis is treated as an invalid hypothesis. There are two ways researchers can verify a hypothesis: (i) (ii)

Direct verification Indirect verification.

Direct verification Researchers verify the hypotheses through direct access to the empirical data. To find out the validity of a hypothesis, researchers access the existing and available empirical data directly. For example, a researcher aims to find out the validity of the hypothesis ‘Indian government has taken all possible measures to solve the polio health-related problems among the poor’. If the empirical data available in government offices suggest that India has taken all possible measures to solve the polio related issues among the poor, then the hypothesis is considered as verified through direct means (i.e. direct verification). Consider another hypothesis, ‘All passed out MA Integrated students of Indian Institute of Technology (IIT) Madras have joined the administrative jobs’. To verify this hypothesis, researchers collect data directly from the passed out MA Integrated students of IIT Madras. If the empirical data suggest that all the MA Integrated passed out students of IIT Madras do not join administrative jobs, then the hypothesis is treated as invalid. In this case, researchers verify the hypothesis through empirical data and find that it is an invalid hypothesis. Indirect verification There are many occasions where researchers cannot verify the hypotheses directly. The reason is, empirical data are not easily available to the researchers. On some occasion, empirical data are also not accessible to the researchers. Researchers, in this situation, infer the data from the actual empirical facts available to them. Researchers search for indirect evidence to verify the hypotheses. In indirect verification, researchers deduce a conclusion from the research study and equate it with

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the hypothesis. If the deduced conclusion agrees with the inferred data collected indirectly from the actual empirical facts, the hypothesis is treated as valid. But if the deduced conclusion does not agree with the inferred data, the hypothesis is treated as invalid. Indirect verification of a hypothesis is based on logical arguments where arguments use the data that are inferred from certain observed facts. In this case, logical arguments are mostly formulated through hypothetical syllogism. For example: Example-1 If there is sun, there is light. There is sun. Therefore, there is light. Example-2 If we cast our vote for an assembly election, an MLA will be elected. We cast our vote for the assembly election. Therefore, an MLA is elected. With reference to example-1, the conclusion (i.e. hypothesis) ‘there is light’ is verified by another fact, that is, ‘there is sun’. In example-2, the conclusion (i.e. hypothesis) ‘an MLA is elected’ is verified by the fact that ‘we cast our vote for the assembly election’. So, in indirect verification, hypotheses are verified on the basis of other facts that are observed as true. The hypothetical syllogism is also used to disprove the rivalry hypothesis of a research study. For example: Example-1 If there is heavy and continuous rainfall, then there will be a flood. There is no flood. Therefore, there is no heavy and continuous rainfall. Example-2 If the monsoon were at the right time, rice production would increase. Rice production did not increase. Therefore, the monsoon was not at the right time. These two examples suggest that when researchers disprove a rival hypothesis, they need to eliminate that hypothesis from the group of hypotheses and thereby increase the truth probability of other hypotheses in the group. Consider another example: Mr. X died in the hospital. Doctors formulated four hypotheses (let us say H1 , H2 , H3 , H4 ) and verified each hypothesis based on the empirical data. Here, the doctors’ aim was to find out which hypothesis was valid and which one was a rival hypothesis. H1 : Mr. X was suffering from diabetes type-2. H2 : Mr. X was suffering from Covid-19.

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H3 : Mr. X was suffering from brain disease. H4 : Mr. X was suffering from blood cancer. Doctors considered H1 and found that Mr. X’s sugar was in control when he died. Hence H1 is a rival hypothesis for his death. They considered H2 and found that Mr. X was not suffering from Covid-19. So, H2 is a rival hypothesis. They further considered H3 and found that Mr. X’s brain was functioning in correct order like a normal person and usual manner. Hence, H3 is also a rival hypothesis for Mr. X’s death. When doctors considered H4 , they found that Mr. X died due to blood cancer. So, in this example, H1 , H2 , H3 are treated as rival hypotheses and H4 is considered as the valid hypothesis, as the conclusion of the research study supports hypothesis H4 . By adapting hypothetical syllogism, we can test the hypotheses in the following way. (a)

(b)

(c)

(d)

If Mr. X was dead, then he was suffering from diabetes type-2. Mr. X was not suffering from diabetes type-2. Therefore, Mr. X was not dead. If Mr. X was dead, then he was suffering from Covid-19. Mr. X was not suffering from Covid-19. Therefore, Mr. X was not dead. If Mr. X was dead, then he was suffering from brain disease. Mr. X was not suffering from brain disease. Therefore, Mr. X was not dead. If Mr. X was dead, then he was suffering from blood cancer. Mr. X was dead. Therefore, Mr. X was suffering from blood cancer.

To prove a hypothesis as a valid hypothesis from a group of hypotheses, researchers need to eliminate the rival hypotheses from the group of hypotheses. After eliminating the rival hypotheses, the most suitable and acceptable hypothesis is considered as a valid hypothesis. There are a few cases where a hypothesis cannot be verified conclusively because empirical data are not available adequately to verify the hypothesis, e.g. the origin of the universe, the origin of human beings on the earth, the origin of human language, etc. Although a few research scholars verified these hypotheses to some extent, yet none of them has proved these hypotheses conclusively. In this context, we may say that no hypothesis is completely verified, but it can achieve a high degree of confirmation. With a high degree of confirmation, researchers or scientific investigators can treat the hypothesis as a legitimate and valid hypothesis.

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