2010_lec13 - CS245 - Winter 2010, Lecture 13 Shai Ben-David...

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Unformatted text preview: CS245 - Winter 2010, Lecture 13 Shai Ben-David We are getting close to completing the picture of relationship between syn- tactic and semantic notions. In the last lecture we proved that the consistency of a set of propositions is equivalent to its satisfiability. We have also seen that provability entails logical implication. Namely, if then | = . This is the Soundness theorem. The last remaining component is the Completeness theorem, which is the reverse of the soundness theorem. Theorem 1 (The Completeness Theorem) For every set of propositions and every proposition , if | = then . Proof Let us prove the equivalent contrapositive statement: If 6 then 6| = . Recall Lemma ?? from the previous lecture. That lemma states that for every set of propositions and every proposition , if 6 then { ( ) } is consistent....
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This note was uploaded on 02/11/2010 for the course ART AFM101 taught by Professor Mr.lushman during the Spring '10 term at University of Toronto- Toronto.

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2010_lec13 - CS245 - Winter 2010, Lecture 13 Shai Ben-David...

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