Elementary Logic

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A Proof in First-Order ND Below is a partial proof of the validity of ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)) → ∀x(p(x) → ¬r(x)) in ND, where γ denotes ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)). . γ, p(y), r(y) ⊢ r(y) → q(y) γ, p(y), r(y) ⊢ q(y) (Ax) γ, p(y), r(y) ⊢ r(y) (→E) . (∧I ) ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)), p(y), r(y) ⊢ q(y) ∧ ¬q(y) (¬I ) ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)), p(y) ⊢ ¬r(y) (→I ) ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)) ⊢ p(y) → ¬r(y) (∀I ) ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)) ⊢ ∀x(p(x) → ¬r(x)) (→I ) ⊢ ∀x(¬p(x) ∨ ¬q(x)) ∧ ∀x(r(x) → q(x)) → ∀x(p(x) → ¬r(x)) Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 56 / 64
Equality Rules of Natural Deduction Let t, t 1 , t 2 be arbitrary terms; again, assume all substitutions are admissible. Γ ⊢ t = t (= I ) Γ ⊢ t 1 = t 2 Γ ⊢ A[t 1 /x] Γ ⊢ A[t 2 /x] (= E) Note: The = sign is part of the object language, not a meta symbol. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 57 / 64
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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 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: Quantifier Rules of Natural Deducti
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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