Elementary Logic

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Semantics of First-Order Logic (cont.) We say M, s |= A when A M [s] is T (true) and M, s ̸|= A otherwise. Alternatively, |= may be defined as follows (propositional part is as in propositional logic): M, s |= ∀xA ⇐⇒ M, s[x := m] |= A for all m ∈ M. M, s |= ∃xA ⇐⇒ M, s[x := m] |= A for some m ∈ M. where s[x := m] denotes an updated assignment s ′ from s such that s ′ (y) = s(y) for y ≠ x and s ′ (x) = m. For example, (Z, {+, 0}), s |= ∀x(x + 0 = x) holds, since (Z, {+, 0}), s[x := m] |= x + 0 = x for all m ∈ Z. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 52 / 64
Satisfiability and Validity A formula A is satisfiable in M if there is an assignment s such that M, s |= A. A formula A is valid in M, denoted M |= A, if M, s |= A for every assignment s. For instance, ∀x(x + 0 = x) is valid in (Z, {+, 0}). M is called a model of A if A is valid in M. A formula A is valid if it is valid in every structure, denoted |= A. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 53 / 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 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: Semantics of First-Order Logic Sinc
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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