lec23b-PredL-Satisfiability-Validity.pdf

# lec23b-PredL-Satisfiability-Validity.pdf - BITS Pilani...

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BITS Pilani Pilani Campus MODULE: PREDICATE LOGIC Semantics - Satisfiability and Validity - Validity is undecidable Sundar B. CS&IS, BITS Pilani 0 29-09-2016 CS/IS F214 Logic in Computer Science

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Predicate Logic: Satisfiability Recall: |= l M denotes that evaluates to TRUE given the set of premises under model M and look-up table l We say a predicate logic formula is satisfiable if it evaluates to TRUE (without any premises) under some model M and some look-up table l : i.e. |= l M for some M and some l 29-09-2016 1 Sundar B. CS&IS, BITS Pilani
Predicate Logic: Validity Recall: |= denotes that evaluates to TRUE given the set of premises under all models (and all look-up tables). We say a predicate logic formula is valid if it evaluates to TRUE (without any premises) under all models : i.e. |= 29-09-2016 2 Sundar B. CS&IS, BITS Pilani

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Predicate Logic: Validity: Example 1 X (p(X) --> q(X)) |= ( X p(X)) --> ( X q(X)) Justification: Let M be any model such that X (p(X) --> q(X)) |= M ( X p(X)) --> ( X q(X)) Suppose that for not every element of (the universe in) M, p(X) is true : then we are done. (Why?) Otherwise suppose X p(X) . Therefore, for every element a
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