Elementary Logic

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Substitutions Let t be a term and A a formula. The result of substituting t for a free variable x in A is denoted by A[t/x]. Consider A = ∀x(P(x) → Q(x, f (y))). When t = g(y), A[t/y] = ∀x(P(x) → Q(x, f (g(y)))). For any t, A[t/x] = ∀x(P(x) → Q(x, f (y))) = A, since there is no free occurrence of x in A. A substitution is admissible if no free variable of t would become bound (be captured by a quantifier) after the substitution. For example, when t = g(x, y), A[t/y] is not admissible, as the free variable x of t would become bound. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 46 / 64
Substitutions (cont.) Suppose we change the bound variable x in A to z and obtain another formula A ′ = ∀z(P(z) → Q(z, f (y))). Intuitively, A ′ and A should be equivalent (under any reasonable semantics). (Technically, the two formulae A and A ′ are said to be α-equivalent.) We can avoid the capture in A[g(x, y)/y] by renaming the bound variable x to z and the result of the substitution then becomes A ′ [g(x, y)/y] = ∀z(P(z) → Q(z, f (g(x, y)))). So, in principle, we can make every substitution admissible while preserving the semantics. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 47 / 64
Page 1 and 2: Elementary Logic (Based on [Gallier
Page 3 and 4: Propositions A proposition is a sta
Page 5 and 6: Symbolic Propositions Propositions
Page 7 and 8: Propositional Logic: Syntax Vocabul
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: Bound Variables Formally Defined Fo
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

Elementary Logic

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