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Symbolic Logic Symbolic Logic J. Dmitri Gallow Draft of January 5, 2016 Contents 1 Basic Concepts of Logic 1 2 Sentence Logic 5 2.1 Syntax for SL ................................ 6 2.1.1 Vocabulary ............................. 6 2.1.2 Grammar ............................. 6 2.1.3 Main Operators and Subformulae ................ 7 2.2 Semantics for SL .............................. 10 2.2.1 The Meaning of the Statement Letters .............. 10 2.2.2 The Meaning of ‘∼’ ........................ 10 2.2.3 The Meaning of ‘&’ ........................ 11 2.2.4 The Meaning of ‘_’ ........................ 11 2.2.5 The Meaning of ‘⊃’ ........................ 12 2.2.6 The Meaning of ‘≡’ ........................ 12 2.2.7 Determining the Truth-value of a wff of SL ........... 12 2.3 Translation from SL to English ...................... 15 2.4 Translation from English to SL ...................... 18 2.4.1 Negation .............................. 18 2.4.2 Conjunction ............................ 19 2.4.3 Disjunction ............................ 20 2.4.4 The Material Conditional and Biconditional ........... 21 2.5 What a Truth-Table Represents ...................... 21 2.6 SL Tautologies, Contradictions, and Contingencies ............ 22 2.7 SL Equivalence ............................... 23 2.8 Disjunctive Normal Form and Expressive Completeness ......... 24 2.8.1 Disjunctive Normal Form ..................... 24 2.8.2 Expressive Completeness ..................... 28 2.9 SL-Validity and SL-Invalidity ....................... 30 2.10 SL-Validity and SL-Tautologies ...................... 35 2.11 SL-Consistency & SL-Inconsistency .................... 36 2.12 SL-Consistency and the Other Logical Properties of SL ......... 38 i Contents ii 3 Sentence Logic Trees 42 3.1 Truth Trees: The Basics .......................... 42 3.2 Rules for Truth Trees ............................ 45 3.3 Summary of Rules ............................. 53 3.4 Strategies for Applying Rules ....................... 54 3.5 ST-Consistency ............................... 55 3.6 Reading Truth-Value Assignments off of Open Branches ........ 57 3.7 ST-Validity ................................. 61 3.8 ST-Tautologies, ST-Contradictions, and ST-Contingencies ....... 64 3.8.1 ST-Tautologies ........................... 64 3.8.2 ST-Contradictions ......................... 65 3.8.3 ST-Contingencies ......................... 67 3.9 ST-Equivalence ............................... 68 4 Predicate Logic 70 4.1 The Language PL .............................. 70 4.1.1 The Syntax of PL ......................... 71 4.1.2 Semantics for PL ......................... 79 4.1.3 More on PL-Interpretations of Predicates ............ 81 4.1.4 Truth on an Interpretation .................... 83 4.1.5 Notation .............................. 86 4.2 PL-Validity ................................. 87 4.3 Proving PL-validity and the method of Semantic Proof ......... 90 4.4 Translations from PL into English ..................... 92 4.4.1 Translating Atomic wffs of PL .................. 92 4.4.2 Translating Simple Quantified wffs of PL ............ 93 4.4.3 Translating More Complicated Quantified wffs of PL ...... 94 4.5 Translations from English into PL ..................... 97 4.6 Overlapping Quantifiers and Relational Predicates ............ 99 4.6.1 Changing the Order of the Quantifiers .............. 100 4.6.2 Changing the Order of the Bound Variables ........... 104 4.6.3 Translating ‘(8x)(9y)Rxy’ and ‘(8x)(9y)Ryx’ .......... 106 4.6.4 Changing the Order of the Quantifiers and the Order of the Bound Variables .......................... 107 5 Predicate Logic Trees 109 5.1 Notation and Terminology ......................... 109 5.2 Predicate Logic Trees ............................ 110 5.2.1 Rule for Universally Quantified Wffs ............... 111 5.2.2 Rule for Existentially Quantified Wffs .............. 115 5.2.3 Rules for Negations of Quantified Wffs .............. 117 5.3 Strategies for Applying Rules ....................... 119 5.4 Sample Predicate Logic Trees ....................... 120 5.5 Logical Properties of PL .......................... 121 5.6 PT-Consistency .............................. 122 5.7 Reading Partial PL Interpretations off of Open Branches ........ 124 Contents iii 5.8 PT Validity ................................. 125 5.9 PT Tautologies, PT Contradictions, and PT Contingencies ....... 128 5.10 PT-Equivalence .............................. 131 5.11 Infinite Trees ................................ 133 6 Predicate Logic with Identity and Functions 140 6.1 The Language PLI ............................. 140 6.1.1 Preliminary Orientation ...................... 140 6.1.2 Syntax for PLI ........................... 141 6.1.3 Semantics for PLI ......................... 145 6.1.4 More on PLI-Interpretations of Function Symbols ....... 147 6.1.5 Truth on a PLI Interpretation ................... 150 6.2 Logical Notions of PLI ........................... 152 6.3 Trees for PLI ................................ 152 6.3.1 The Rule (=) ........................... 153 6.3.2 The Rule (, ×) .......................... 155 6.3.3 Completing Trees ......................... 156 6.4 PT I Consistency .............................. 160 6.5 PT I Validity ................................ 161 6.6 PT I Tautologies, Contingencies, and Contradictions ........... 163 6.6.1 Properties of Identity ....................... 164 6.7 PT I Equivalence .............................. 166 6.8 Infinite PLI Trees ............................. 167 6.9 Number Claims ............................... 168 6.10 The Only .................................. 176 6.11 Definite Descriptions ............................ 177 7 Metatheory for Sentence Logic 179 7.1 Use & Mention ............................... 179 7.2 Object Language & Metalanguage .................... 180 7.3 Metavariables ................................ 181 7.3.1 Use and Mention with Metavariables ............... 182 7.4 Syntax and Semantics ........................... 183 7.5 Syntactic Validity and Semantic Validity ................. 186 7.6 Soundness and Completeness of the Tree Method ............ 187 7.7 Mathematical Induction .......................... 190 7.7.1 Terminology ............................ 191 7.7.2 Examples of Mathematical Induction ............... 193 7.8 Varieties of Mathematical Induction ................... 195 7.9 Choosing the Right Inductive Property .................. 197 7.10 More Examples of Mathematical Induction ................ 199 7.10.1 Number of Parentheses ...................... 201 7.10.2 Substitution of SL-equivalents .................. 204 7.10.3 Duals ................................ 208 Contents iv 8 Soundness of the Tree Method for Sentence Logic 211 8.1 Preliminaries ................................ 211 8.2 The Proof in Broad Outline ........................ 213 8.3 Proof that the Tree Method for SL is Sound ................ 215 9 Completeness of the Tree Method for Sentence Logic 225 9.1 The Proof in Broad Outline ........................ 226 9.2 Proof that the Tree Method for SL is Complete .............. 227 Chapter 1 Basic Concepts of Logic 1. Logic is the study of arguments. (a) An argument is (for our purposes), just a collection of statements (or declarative sentences), one of which is designated as the conclusion, the remainder of which are designated as premises. (b) A statement (declarative sentence) is a sentence of which it makes sense to say that it is true or false. i. It’s true/false that it is Monday. X −! ‘It is Monday’ is a state- ment. ii. It’s true/false that close the door! × −! ‘Close the door!’ is not a statement. 2. There are many good-making features that an argument can have. (a) The premises can give good (though not conclusive) reason to accept the conclusion. (b) The premises can be true. (c) The premises can be probable. 3. In this course, we’ll just be focusing on one good-making feature that an argument can have: the property of deductive validity. Deductive Validity An argument is deductively valid if and only if (‘iff’) there is no possible scenario in which the premises are all true while the conclusion is false. 1 2 An argument is deductively invalid iff it is not deductively valid. Deductive Invalidity An argument is deductively invalid iff there is a possible scenario in which the premises are all true while the conclusion is false. 4. We may also characterize deductive validity in terms of the notion of a counterex- ample. Counterexample A counterexample to the deductive validity of an argu- ment is a specification of a possibility in which the premises of the argument are all true yet the conclusion is simultaneously false. Then, Deductive Validity (2) An argument is deductively valid iff it has no coun- terexample. Deductive Invalidity (2) An argument is deductively invalid iff it has a counterexample. 5. What we want is a theory that tells us, of any given argument, whether it is de- ductively valid (or just ‘valid’) or deductively invalid. 6. Other important logical notions: (a) Logical properties of statements: Statements can be logical tautologies, logi- cal contradictions, or logical contingencies. p q Logical Tautology A statement A is a logical tautology iff there is no p q p q possibility in which A is false (i.e., iff A is true in every possibility). p q Logical Contradiction A statement A is a logical contradic- p q p q tion iff there is no possibility in which A is true (i.e., iff A is false in every possibility). p q Logical Contingency A statement A is a logical contingency iff p q there is some possibility
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