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Notes on the Interpretation of First-Order Logic

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Notes on the Interpretation of First-Order Logic Symbolic Logic Notes on the Interpretation of First-Order Logic Notes for Symbolic Logic Fall 2005 John N. Martin Table of Contents Lecture 1. Set Theory.....................................................................................................1 The Axioms of Naïve Set Theory .................................................................................1 Set Identity and The Principle of Extensionality .......................................................1 Set Membership and the Principle of Abstraction.....................................................2 Inference Rules............................................................................................................4 Abbreviative Definitions ...............................................................................................4 Set Abstracts............................................................................................................4 Defined Relations on Sets........................................................................................5 Defined Sets and Operations on Sets ......................................................................5 Statement of the Axiom System...................................................................................7 Summary of the System...............................................................................................7 Axioms .....................................................................................................................7 Rules of Inference ....................................................................................................7 Abbreviative Definitions............................................................................................7 Reduction of Relations to Sets.....................................................................................8 Properties of Relations and Order .............................................................................12 Functions ...................................................................................................................13 Construction and Inductive Definitions.......................................................................15 Definitions by Necessary and Sufficient Conditions ...............................................15 Inductive Definitions and Sets ................................................................................18 The Natural Numbers.............................................................................................20 Proof by Induction ..................................................................................................24 Construction Sequences ........................................................................................24 Axiomatized Set Theory.............................................................................................29 Lecture 2. Semantics of Sentential Logic .....................................................................32 Sentential Syntax.......................................................................................................32 Modern Symbolic Notation .....................................................................................32 Formation Rules, Generative Grammar, Inductive Sets.........................................35 Grammatical Derivations ........................................................................................37 Truth-Functionality .....................................................................................................40 Truth-Tables for the Connectives ...........................................................................40 Negation.................................................................................................................40 Disjunction..............................................................................................................41 Conjunction ............................................................................................................42 The Conditional ......................................................................................................42 The Biconditional....................................................................................................43 Sentential Semantics .................................................................................................44 Tarski’s Correspondence Theory for Complex Grammars .....................................44 The Strategy for an Inductive Definition ................................................................47 Interpreting Negations............................................................................................48 Interpreting Disjunctions.........................................................................................49 Interpreting Conjunctions .......................................................................................49 Interpreting the Conditional ....................................................................................50 Interpreting the Biconditional..................................................................................50 Page ii Version11/14/2005 The Inductive Definition of Interpretation................................................................51 Truth-Conditions ........................................................................................................52 General Truth-Functions ........................................................................................52 Satisfaction of Tarski’s Adequacy Condition ..........................................................56 Calculating Sentence Values by Truth-Tables .......................................................58 Examples of Truth-functional Computation and Truth-Tables ................................60 The Definition of Logical Concepts.........................................................................64 Summary ...................................................................................................................66 Lecture 3. First-Order Logic..........................................................................................68 Expressive Power ......................................................................................................68 Simple and Complex Sentences in a Single Syntax...............................................68 The Limitations of the Syllogistic and Propositional Logic......................................70 New Notation: Constants, Predicates and the Quantifiers......................................73 Syntax for First-order Logic........................................................................................75 Definition of Well-Formed Formula.........................................................................75 Informal Semantics ....................................................................................................79 Quantifiers and Models ..........................................................................................79 Unrestricted Quantifiers..........................................................................................80 Universal Affirmatives.............................................................................................80 Particular Affirmatives ............................................................................................82 Distribution of the Quantifiers over Connectives ....................................................84 Embedded Quantifiers............................................................................................85 Syllogisms in Modern Notation...............................................................................86 Properties of Relations...........................................................................................86 Lecture 4. Formal Semantics for First-Order Logic.......................................................88 Intuitions about the Truth-Conditions of Each Formula Types ...................................88 Atomic Formulas ....................................................................................................88 Molecular Formulas: The Connectives...................................................................90 Quantified Formulas...............................................................................................90 The Inductive Definition of Interpretation ...................................................................92 Introduction ............................................................................................................92 Formal Definitions ..................................................................................................93 Simultaneous Induction and Impossibility of Truth-Tables .....................................94 Tarski’s Adequacy Condition .....................................................................................97 Calculating Truth-Values Using Truth-Conditions ...................................................98 The Technique .......................................................................................................98 Examples .............................................................................................................100 The Correspondence Theory of Truth for First-Order Logic.....................................106 The Definition of Logical Concepts.......................................................................108
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