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Unformatted text preview: Def. Γ is formally complete just in case for all S ε Sent PL ,(a sentence in propositional logic) either Γ├ S or Γ├/-- S. Lemma 2: Suppose Γ 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 Γ does prove P or Γ proves Q. 5. Γ proves P↔Q 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*(Γ) = T. Lemma 4: If gamma is formally complete iff Γ├ A or gamma does not prove A for all i ε N. Lemma 5: Every formally consistent set gamma can be extended to some set Γ* and Γ is a subset of Γ* and Γ* is formally complete. FINALLY – result is the COMPLETENESS THEOREM...
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- Spring '08
- Logic, Metalogic, Proof theory, GAMMA