Elementary Logic

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Symbolic Inferences We are given the following assumptions: P (Leslie is a teacher.) not P or not Q (Either Leslie is not a teacher or Leslie is not rich.) R implies Q (If Leslie is a pop singer, then Leslie is rich.) We wish to conclude the following: not R (Leslie is not a pop singer.) Correctness of the inference may be checked by asking: Is (P and (not P or not Q) and (R implies Q)) implies (not R) a tautology (valid formula) Or, is (A and (not A or not B) and (C implies B)) implies (not C) a tautology (valid formula) Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 6 / 64
Propositional Logic: Syntax Vocabulary: A countable set P of proposition symbols (variables): P, Q, R, . . . (also called atomic propositions); Logical connectives (operators): ¬, ∧, ∨, →, and ↔ and sometimes the constant ⊥ (false); Auxiliary symbols: “(”, “)”. How to read the logical connectives. ¬ (negation): not ∧ (conjunction): and ∨ (disjunction): or → (implication): implies (or if . . . , then . . . ) ↔ (equivalence): is equivalent to (or if and only if) ⊥ (false or bottom): false (or bottom) Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 7 / 64
Page 1 and 2: Elementary Logic (Based on [Gallier
Page 3 and 4: Propositions A proposition is a sta
Page 5: Symbolic Propositions Propositions
Page 9 and 10: About Boolean Expressions Boolean e
Page 11 and 12: Truth Assignment and Valuation The
Page 13 and 14: Truth Assignment and Satisfaction W
Page 15 and 16: Object vs. Meta Language The langua
Page 17 and 18: Tautology and Validity A propositio
Page 19 and 20: Relating the Logical Connectives Le
Page 21 and 22: Semantic Entailment Consider two se
Page 23 and 24: Sequents (cont.) A sequent A 1 , A
Page 25 and 26: Proofs A deduction tree is a tree w
Page 27 and 28: Detour: Some Laws for Calculational
Page 29 and 30: Natural Deduction (cont.) Γ, A ⊢
Page 31 and 32: Soundness and Completeness Theorem
Page 33 and 34: Consistency Lemma A set Γ of propo
Page 35 and 36: Inferences We are given the followi
Page 37 and 38: Symbolic Inferences We are given th
Page 39 and 40: First-Order Logic: Syntax (cont.) T
Page 41 and 42: First-Order Logic: Syntax (cont.) W
Page 43 and 44: Free Variables Formally Defined For
Page 45 and 46: Bound Variables Formally Defined Fo
Page 47 and 48: Substitutions (cont.) Suppose we ch
Page 49 and 50: Substitutions Formally Defined (con
Page 51 and 52: Semantics of First-Order Logic Sinc
Page 53 and 54: Satisfiability and Validity A formu
Page 55 and 56: Quantifier Rules of Natural Deducti
Page 57 and 58:
Equality Rules of Natural Deduction
Page 59 and 60:
Compactness Theorem For any (possib
Page 61 and 62:
Theory Assume a fixed first-order l
Page 63 and 64:
Theorems A theorem is just a statem
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Elementary Logic

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