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Unformatted text preview: Def. is formally complete just in case for all S Sent PL ,(a sentence in propositional logic) either S or /-- S. Lemma 2: Suppose is formally complete and formally consistent. Then: 1. Gamma proves P and Q iff gamma proves P and gamma proves Q. 2. Gamma proves P or Q iff gamma proves P or gamma proves Q. 3. Gamma proves not Q if gamma does not prove Q. 4. Gamma proves P Q iff does prove P or proves Q. 5. proves PQ iff either gamma proves P and gamma proves Q or gamma does not prove P and gamma does not prove Q. Lemma 3: Suppose gamma is formally complete and formally consistent. Then there exists valuation v such that v*() = T. Lemma 4: If gamma is formally complete iff A or gamma does not prove A for all i N. Lemma 5: Every formally consistent set gamma can be extended to some set * and is a subset of * and * is formally complete. FINALLY result is the COMPLETENESS THEOREM...
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- Spring '08