consistency_soundness_march5

consistency_soundness_march5 - CS245 Winter 2010 Lecture of...

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CS245 - Winter 2010, Lecture of March. 5th - Consistency in FOL Shai Ben-David In this lecture we continue the discussion of syntactic notions in first or- der logic (FOL). Basically, everything we discuss here is very similar to the analogous discussion that we carried out for propositional logic. Definition A set Γ of formulas is said to be consistent if, there exists no formula α such that both Γ α and Γ ‘ ¬ α . In other words, Γ is consistent if it does not imply any contradiction. As is the case for propositional logic, we can provide an equivalently definition of this notion that does not assign any special role to the connective ” ¬ ”. Claim 1 A set of formulas is consistent if and only if there exists a formula that it cannot prove. Namely, Γ is consistent if and only if, for some β , Γ 6‘ β . Proof First, assume that Γ is consistent, pick any formula α . Since Γ cannot prove both α and ¬ α one of these two formulas is not provable form Γ. For the other direction, let us show that if a set of formulas Γ is not con-

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