Elementary Logic

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Satisfaction Alternatively (in a more generally applicable format), the satisfaction relation |= may be defined as follows: v ̸|= ⊥ v |= P ⇐⇒ v(P) = T , for all P ∈ P v |= ¬A ⇐⇒ v ̸|= A (it is not the case that v |= A) v |= A ∧ B ⇐⇒ v |= A and v |= B v |= A ∨ B ⇐⇒ v |= A or v |= B v |= A → B ⇐⇒ v ̸|= A or v |= B v |= A ↔ B ⇐⇒ (v |= A and v |= B) or (v ̸|= A and v ̸|= B) Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 14 / 64
Object vs. Meta Language The language that we study is referred to as the object language. The language that we use to study the object language is referred to as the meta language. For example, not, and, and or that we used to define the satisfaction relation |= are part of the meta language. Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 15 / 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: Truth Assignment and Satisfaction W
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

Elementary Logic

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