Elementary Logic

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Consistency Recall that a set Γ of formulae is consistent if there exists some formula B such that the sequent Γ ⊢ B is not provable. Otherwise, Γ is inconsistent. Lemma For System ND, a set Γ of formulae is inconsistent if and only if there is some formula A such that both Γ ⊢ A and Γ ⊢ ¬A are provable. Theorem For System ND, a set Γ of formulae is satisfiable if and only if Γ is consistent. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 60 / 64
Theory Assume a fixed first-order language. A set S of sentences is closed under provability if S = {A | A is a sentence and S ⊢ A is provable}. A set of sentences is called a theory if it is closed under provability. A theory is typically represented by a smaller set of sentences, called its axioms. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 61 / 64
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Elementary Logic (Based on [Gallier
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Propositions A proposition is a sta
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Symbolic Propositions Propositions
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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 and 56: Quantifier Rules of Natural Deducti
Page 57 and 58: Equality Rules of Natural Deduction
Page 59: Compactness Theorem For any (possib
Page 63 and 64: Theorems A theorem is just a statem
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Elementary Logic

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