Boxes and Diamonds An Open Introduction to Modal Logic F19 Boxes and Diamonds The Open Logic Project Instigator Richard Zach, University of Calgary Editorial Board Aldo Antonelli,y University of California, Davis Andrew Arana, Université de Lorraine Jeremy Avigad, Carnegie Mellon University Tim Button, University College London Walter Dean, University of Warwick Gillian Russell, Dianoia Institute of Philosophy Nicole Wyatt, University of Calgary Audrey Yap, University of Victoria Contributors Samara Burns, Columbia University Dana Hägg, University of Calgary Zesen Qian, Carnegie Mellon University Boxes and Diamonds An Open Introduction to Modal Logic Remixed by Richard Zach Fall 2019 The Open Logic Project would like to acknowledge the gener- ous support of the Taylor Institute of Teaching and Learning of the University of Calgary, and the Alberta Open Educational Re- sources (ABOER) Initiative, which is made possible through an investment from the Alberta government. Cover illustrations by Matthew Leadbeater, used under a Cre- ative Commons Attribution-NonCommercial 4.0 International Li- cense. Typeset in Baskervald X and Nimbus Sans by LATEX. This version of Boxes and Diamonds is revision ed40131 (2021-07- 11), with content generated from Open Logic Text revision a36bf42 (2021-09-21). Free download at: https://bd.openlogicproject.org/ Boxes and Diamonds by Richard Zach is licensed under a Creative Commons At- tribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Cre- ative Commons Attribution 4.0 Interna- tional License. Contents Preface xi Introduction xii I Normal Modal Logics1 1 Syntax and Semantics2 1.1 Introduction..................... 2 1.2 The Language of Basic Modal Logic....... 4 1.3 Simultaneous Substitution............. 5 1.4 Relational Models.................. 7 1.5 Truth at a World .................. 8 1.6 Truth in a Model .................. 10 1.7 Validity........................ 10 1.8 Tautological Instances ............... 11 1.9 Schemas and Validity................ 14 1.10 Entailment...................... 16 Problems .......................... 18 2 Frame Definability 21 2.1 Introduction..................... 21 2.2 Properties of Accessibility Relations . 22 2.3 Frames........................ 25 2.4 Frame Definability ................. 26 v CONTENTS vi 2.5 First-order Definability............... 29 2.6 Equivalence Relations and S5 . 30 2.7 Second-order Definability ............. 33 Problems .......................... 36 3 Axiomatic Derivations 38 3.1 Introduction..................... 38 3.2 Normal Modal Logics ............... 40 3.3 Derivations and Modal Systems.......... 42 3.4 Proofs in K ..................... 44 3.5 Derived Rules.................... 46 3.6 More Proofs in K . 49 3.7 Dual Formulas.................... 50 3.8 Proofs in Modal Systems.............. 51 3.9 Soundness...................... 53 3.10 Showing Systems are Distinct ........... 54 3.11 Derivability from a Set of Formulas . 56 3.12 Properties of Derivability.............. 56 3.13 Consistency..................... 57 Problems .......................... 58 4 Completeness and Canonical Models 60 4.1 Introduction..................... 60 4.2 Complete O -Consistent Sets............ 62 4.3 Lindenbaum’s Lemma ............... 63 4.4 Modalities and Complete Consistent Sets . 65 4.5 Canonical Models.................. 68 4.6 The Truth Lemma ................. 68 4.7 Determination and Completeness for K . 69 4.8 Frame Completeness................ 70 Problems .......................... 74 5 Filtrations and Decidability 75 5.1 Introduction..................... 75 5.2 Preliminaries .................... 78 5.3 Filtrations...................... 79 CONTENTS vii 5.4 Examples of Filtrations............... 82 5.5 Filtrations are Finite ................ 84 5.6 K and S5 have the Finite Model Property . 85 5.7 S5 is Decidable................... 86 5.8 Filtrations and Properties of Accessibility . 87 5.9 Filtrations of Euclidean Models.......... 89 Problems .......................... 91 6 Modal Tableaux 93 6.1 Introduction..................... 93 6.2 Rules for K ..................... 94 6.3 Tableaux for K ................... 97 6.4 Soundness for K . 98 6.5 Rules for Other Accessibility Relations . 102 6.6 Soundness for Additional Rules . 103 6.7 Simple Tableaux for S5 . 106 6.8 Completeness for K . 107 6.9 Countermodels from Tableaux . 110 Problems ..........................113 II Intuitionistic Logic 115 7 Introduction 116 7.1 Constructive Reasoning . 116 7.2 Syntax of Intuitionistic Logic . 118 7.3 The Brouwer-Heyting-Kolmogorov Interpretation 119 7.4 Natural Deduction . 123 7.5 Axiomatic Derivations . 126 Problems ..........................128 8 Semantics 129 8.1 Introduction.....................129 8.2 Relational models..................130 8.3 Semantic Notions..................132 8.4 Topological Semantics . 133 CONTENTS viii Problems ..........................135 9 Soundness and Completeness 136 9.1 Soundness of Axiomatic Derivations . 136 9.2 Soundness of Natural Deduction . 137 9.3 Lindenbaum’s Lemma . 139 9.4 The Canonical Model . 141 9.5 The Truth Lemma . 142 9.6 The Completeness Theorem . 144 9.7 Decidability.....................144 Problems ..........................145 III Counterfactuals 146 10 Introduction 147 10.1 The Material Conditional . 147 10.2 Paradoxes of the Material Conditional . 149 10.3 The Strict Conditional . 150 10.4 Counterfactuals...................152 Problems ..........................153 11 Minimal Change Semantics 155 11.1 Introduction.....................155 11.2 Sphere Models....................157 11.3 Truth and Falsity of Counterfactuals . 159 11.4 Antecedent Strengthenng . 160 11.5 Transitivity .....................162 11.6 Contraposition ...................164 Problems ..........................164 IV Appendices 166 A Sets 167 A.1 Extensionality....................167 A.2 Subsets and Power Sets . 169 CONTENTS ix A.3 Some Important Sets . 170 A.4 Unions and Intersections . 172 A.5 Pairs, Tuples, Cartesian Products . 175 A.6 Russell’s Paradox..................177 Problems ..........................179 B Relations 180 B.1 Relations as Sets ..................180 B.2 Special Properties of Relations . 182 B.3 Equivalence Relations . 184 B.4 Orders........................185 B.5 Graphs........................188 B.6 Operations on Relations . 190 Problems ..........................191 C Syntax and Semantics 192 C.1 Introduction.....................192 C.2 Propositional Formulas . 194 C.3 Preliminaries ....................196 C.4 Valuations and Satisfaction . 198 C.5 Semantic Notions..................200 Problems ..........................201 D Axiomatic Derivations 202 D.1 Introduction.....................202 D.2 Axiomatic Derivations . 204 D.3 Rules and Derivations . 206 D.4 Axiom and Rules for the Propositional Connectives208 D.5 Examples of Derivations . 209 D.6 Proof-Theoretic Notions . 211 D.7 The Deduction Theorem . 213 D.8 Derivability and Consistency . 215 D.9 Derivability and the Propositional Connectives . 216 D.10 Soundness......................218 Problems ..........................219 CONTENTS x E Tableaux 220 E.1 Tableaux.......................220 E.2 Rules and Tableaux . 222 E.3 Propositional Rules . 223 E.4 Tableaux.......................224 E.5 Examples of Tableaux . 225 E.6 Proof-Theoretic Notions . 230 E.7 Derivability and Consistency . 233 E.8 Derivability and the Propositional Connectives . 235 E.9 Soundness......................238 Problems ..........................242 F The Completeness Theorem 243 F.1 Introduction.....................243 F.2 Outline of the Proof . 245 F.3 Complete Consistent Sets of Sentences . 246 F.4 Lindenbaum’s Lemma . 248 F.5 Construction of a Model . 249 F.6 The Completeness Theorem . 250 Problems ..........................251 About the Open Logic Project 252 Preface This is an introductory textbook on modal logic. I use it as the main text when I teach Philosophy 579.2 (Modal Logic) at the University of Calgary. It is based on material from the Open Logic Project. The main text assumes familiarity with some elementary set theory and the basics of (propositional) logic. This material is part of a prerequisite for my course, Logic II. The textbook for that course, Sets, Logic, Computation, is also based on the OLP, and so is available for free. The required material is included as appendices in this book, however. I assign these appendices for background reading whenever I teach the material. Part I is originally based in part on Aldo Antonelli’s lec- ture notes on “Classical Correspondence Theory for Basic Modal Logic,” which he contributed to the OLP before his untimely death in 2015. I heavily revised and expanded these notes, e.g., the material on frame definability and tableaux is new. xi Introduction Modal logics are extensions of classical logic by the operators □ (“box”) and ^ (“diamond”), which attach to formulas. Intu- itively, □ may be read as “necessarily” and ^ as “possibly,” so □p is “p is necessarily true” and ^p is “p is possibly true.” As necessity and possibility are fundamental metaphysical notions, modal logic is obviously of great philosophical interest. It allows the formalization of metaphysical principles such as “□p ! p” (if p is necessary, it is true) or “^p ! □^p” (if p is possible, it is necessarily possible). The operators □ and ^ are intensional. This means that whether □A or ^A holds does not just depend on whether A holds or doesn’t. An operator which is not intensional is extensional. Negation