Elementary Logic

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Propositional Logic: Syntax (cont.) Propositional Formulae: Any A ∈ P is a formula and so is ⊥ (these are the “atomic” formula). If A and B are formulae, then so are ¬A, (A ∧ B), (A ∨ B), (A → B), and (A ↔ B). A is called a subformula of ¬A, and A and B subformulae of (A ∧ B), (A ∨ B), (A → B), and (A ↔ B). Precedence (for avoiding excessive parentheses): A ∧ B → C means ((A ∧ B) → C). A → B ∨ C means (A → (B ∨ C)). A → B → C means (A → (B → C)). More about this later ... Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 8 / 64
About Boolean Expressions Boolean expressions are essentially propositional formulae, though they may allow more things as atomic formulae. Boolean expressions: (x ∨ y ∨ z) ∧ (x ∨ y) ∧ x (x + y + z) · (x + y) · x (a ∨ b ∨ c) ∧ (a ∨ b) ∧ a etc. Propositional formula: (P ∨ Q ∨ ¬R) ∧ (¬P ∨ ¬Q) ∧ P Yih-Kuen Tsay (SVVRL @ IM.NTU) Elementary Logic FLOLAC 2011 9 / 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: Propositional Logic: Syntax Vocabul
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 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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