lec04-inference

# lec04-inference - EECS 203 Winter 2007 Discrete Mathematics...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 4 Rules of Inference and G¨ odel’s Theorems January 16 Reading: Rosen [1.5] January 16 Rules of Inference and G¨ odel’s Theorems, Page 1

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4.1 Review A model (or interpretation ) M for a first order logic L is a universe U together with an assignment of actual objects in U for each constant, variable, function, and predicate in L . We say that M satisfies φ , denoted by M φ , if φ M = True . A wff φ is said to be valid if it is satisfied by all models. Two wffs φ and ψ are logically equivalent if φ ←→ ψ is valid. The following wffs are valid. A wff obtained from a propositional tautology by replacing each propositional variable by a wff. ( ) -→ φ [ x t ], for all wff φ and variable x , where φ [ x t ] is φ with all free x replaced by a term t . φ -→ ∀ , for all wff φ with no free x . ( x ( φ -→ ψ )) -→ (( ) -→ ( )), for all wffs φ , ψ and variable x . January 16 Rules of Inference and G¨ odel’s Theorems, Page 2
4.1 Review • ¬∀ ≡ ∃ x ¬ φ , • ¬∃ ≡ ∀ x ¬ φ . January 16 Rules of Inference and G¨ odel’s Theorems, Page 3

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4.2 More valid formulas Let φ and ψ be two first-order formulas. Then 1. x ( φ ψ ) ( ) ( ).

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