2010_lecture11 - CS245 - Winter 2010, Lecture 11 Shai...

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CS245 - Winter 2010, Lecture 11 Shai Ben-David Formal proofs are part of the syntax aspect of propositional logic. Formal provability is defined by a strict rules concerning the characters in a proposition without addressing their meaning (the truth values). There is yet another important syntactic notion - consistency. Definition A set of propositions, Σ is consistent if, there exists no proposition α for which both Σ α and σ ( ¬ α ). A set is inconsistent if it is not consistent. Intuitively speaking, a set of propositions is consistent if it domes no con- tradiction can be derived from it. Examples: The following sets are inconsistent. 1. Σ 1 = { p, ( ¬ p ) } . 2. Σ 2 = { a, ( a b ) , ( ¬ b ) } . I leave it to you to prove that these sets are inconsistent by showing that Σ 1 p , Σ 1 ( ¬ p ) } , Σ 2 b and Σ - 2 ( ¬ b ). Can we come up with examples of consistent sets? This seems to be a bit more difficult, since, showing that a set Σ is consistent amounts to showing that, for every proposition α if Σ α then no sequence of propositions consists a formal proof of ( ¬ α ) from Σ. Recalling the monotonicity of propositional logic (Namely, that Σ
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2010_lecture11 - CS245 - Winter 2010, Lecture 11 Shai...

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