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Uva-DARE (Digital Academic Repository) UvA-DARE (Digital Academic Repository) Logic in Action van Benthem, J.; van Ditmarsch, H.; van Eijck, J.; Jaspars, J. Publication date 2016 Document Version Final published version Link to publication Citation for published version (APA): van Benthem, J., van Ditmarsch, H., van Eijck, J., & Jaspars, J. (2016). Logic in Action. (New ed.) Logic in Action. http://logicinaction.org/ General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:11 Oct 2021 Logic in Action –New Edition, November 23, 2016– Johan van Benthem, Hans van Ditmarsch, Jan van Eijck, Jan Jaspars 0-2 Contents 1 General Introduction 1-1 1.1 Inference, Observation, Communication . 1-1 1.2 The Origins of Logic . 1-3 1.3 Uses of Inference . 1-5 1.4 Logic and Other Disciplines . 1-9 1.5 Overview of the Course . 1-11 Classical Systems 2-1 2 Propositional Logic 2-1 2.1 Reasoning in Daily Life . 2-1 2.2 Inference Patterns, Validity, and Invalidity . 2-3 2.3 Classification, Consequence, and Update . 2-5 2.4 The Language of Propositional Logic . 2-8 2.5 Semantic Situations, Truth Tables, Binary Arithmetic . 2-13 2.6 Valid Consequence and Consistency . 2-18 2.7 Proof . 2-22 2.8 Information Update . 2-24 2.9 Expressiveness . 2-26 2.10 Outlook — Logic, Mathematics, Computation . 2-28 2.11 Outlook — Logic and Practice . 2-32 2.12 Outlook — Logic and Cognition . 2-34 0-3 0-4 CONTENTS 3 Syllogistic Reasoning 3-1 3.1 Reasoning About Predicates and Classes . 3-1 3.2 The Language of Syllogistics . 3-4 3.3 Sets and Operations on Sets . 3-5 3.4 Syllogistic Situations . 3-10 3.5 Validity Checking for Syllogistic Forms . 3-12 3.6 Outlook — Satisfiability and Complexity . 3-18 3.7 Outlook — The Syllogistic and Actual Reasoning . 3-21 4 The World According to Predicate Logic 4-1 4.1 Learning the Language by Doing . 4-2 4.2 Practising Translations . 4-8 4.3 Reasoning Patterns with Quantifiers . 4-13 4.4 Formulas, Situations and Pictures . 4-17 4.5 Syntax of Predicate Logic . 4-25 4.6 Semantics of Predicate Logic . 4-30 4.7 Valid Laws and Valid Consequence . 4-35 4.8 Proof . 4-38 4.9 Identity, Function Symbols, Algebraic Reasoning . 4-41 4.10 Outlook — Mathematical Background . 4-46 4.11 Outlook — Computational Connection . 4-49 4.12 Outlook — Predicate Logic and Philosophy . 4-51 Knowledge, Action, Interaction 4-57 5 Logic, Information and Knowledge 5-1 5.1 Logic and Information Flow . 5-1 5.2 Information versus Uncertainty . 5-3 5.3 Modeling Information Change . 5-10 5.4 The Language of Epistemic Logic . 5-12 5.5 Models and Semantics for Epistemic Logic . 5-15 CONTENTS 0-5 5.6 Valid Consequence . 5-21 5.7 Proof . 5-25 5.8 Information Update . 5-30 5.9 The Logic of Public Announcement . 5-37 5.10 Outlook — Information, Knowledge, and Belief . 5-42 5.11 Outlook – Social Knowledge . 5-44 5.12 Outlook – Secrecy and Security . 5-47 6 Logic and Action 6-1 6.1 Actions in General . 6-1 6.2 Sequence, Choice, Repetition, Test . 6-6 6.3 Viewing Actions as Relations . 6-10 6.4 Operations on Relations . 6-13 6.5 Combining Propositional Logic and Actions: PDL . 6-17 6.6 Transition Systems . 6-20 6.7 Semantics of PDL . 6-23 6.8 Axiomatisation . 6-26 6.9 Expressive power: defining programming constructs . 6-30 6.10 Outlook — Programs and Computation . 6-31 6.11 Outlook — Equivalence of Programs and Bisimulation . 6-35 7 Logic, Games and Interaction 7-1 7.1 Logic meets Games . 7-1 7.2 Evaluation of Assertions as a Logical Game . 7-4 7.3 Zermelo’s Theorem and Winning Strategies . 7-8 7.4 Sabotage Games: From Simple Actions to Games . 7-11 7.5 Model Comparison as a Logic Game . 7-13 7.6 Different Formulas in Model Comparison Games . 7-16 7.7 Bisimulation Games . 7-19 7.8 Preference, Equilibrium, and Backward Induction . 7-21 7.9 Game logics . 7-31 0-6 CONTENTS 7.10 Games with imperfect information . 7-33 7.11 Logic and Game Theory . 7-36 7.12 Outlook — Iterated Game Playing . 7-44 7.13 Outlook — Knowledge Games . 7-46 7.14 Outlook — Games and Foundations . 7-47 7.15 Outlook — Games, Logic and Cognition . 7-48 Methods 8-1 8 Validity Testing 8-1 8.1 Tableaus for propositional logic . 8-2 8.1.1 Reduction rules . 8-5 8.2 Tableaus for predicate logic . 8-9 8.2.1 Rules for quantifiers . 8-13 8.2.2 Alternative rules for finding finite counter-models . 8-17 8.2.3 Invalid inferences without finite counter-examples . 8-19 8.2.4 Tableaus versus natural reasoning . 8-20 8.3 Tableaus for epistemic logic . 8-22 9 Proofs 9-1 9.1 Natural deduction for propositional logic . 9-2 9.1.1 Proof by refutation . 9-5 9.1.2 Introduction and elimination rules . 9-7 9.1.3 Rules for conjunction and disjunction . 9-9 9.2 Natural deduction for predicate logic . 9-13 9.2.1 Rules for identity . 9-18 9.3 Natural deduction for natural numbers . 9-18 9.3.1 The rule of induction . 9-20 9.4 Outlook . 9-23 9.4.1 Completeness and incompleteness . 9-23 9.4.2 Natural deduction, tableaus and sequents . 9-23 CONTENTS 0-7 9.4.3 Intuitionistic logic . 9-23 9.4.4 Automated deduction . 9-23 10 Computation 10-1 10.1 A Bit of History . 10-1 10.2 Processing Propositional Formulas . 10-3 10.3 Resolution . 10-7 10.4 Automating Predicate Logic . 10-12 10.5 Conjunctive Normal Form for Predicate Logic . 10-15 10.6 Substitutions . 10-17 10.7 Unification . 10-19 10.8 Resolution with Unification . 10-24 10.9 Prolog . 10-26 Appendices A-1 A Sets, Relations and Functions A-1 A.1 Sets and Set Notation . A-1 A.2 Relations . A-3 A.3 Back and Forth Between Sets and Pictures . A-5 A.4 Relational Properties . A-6 A.5 Functions . A-9 A.6 Recursion and Induction . A-11 B Solutions to the Exercises B-1 0-8 CONTENTS Chapter 1 General Introduction 1.1 Inference, Observation, Communication Much of our interaction with each other in daily life has to do with information processing and reasoning about knowledge and ignorance of the people around us. If I ask a simple question, like “Can you tell me where to find the Opera House?”, then I convey the infor- mation that I do not know the answer, and also, that I think that you may know. Indeed, in order to pick out the right person for asking such informative questions, we need to reason about knowledge of others. It is our ability to reason in the presence of other reasoning agents that has made us historically so successful in debate, organization, and in planning collective activities. And it is reasoning in this broad sense that this course is about. We will study informational processes of inference and information update – and while we can start dealing with these for single agents, our theories must also work interac- tively.
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