2010_lecture12 - CS245 - Winter 2010, Lecture 12 Shai...

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Unformatted text preview: CS245 - Winter 2010, Lecture 12 Shai Ben-David We start with yet another definition of a syntactic notion. Definition A set of propositions is maximally consistent if, is consistent and, for every proposition , either or ( ). Why is such a set called maximally consistent? Note that for every propo- sition , if 6 then { } is inconsistent (why?). As it turns out, the converse of this claim is also true. Namely, Claim 1 A set of propositions is maximally consistent if and only if for every proposition , if 6 then { } is inconsistent. The proof of this claim relies on the following Lemma. Lemma 1 For every set of propositions and every proposition , if 6 then { ( ) } is consistent. Proof of the lemma: Assume, by way of contradiction, that { ( ) } is not consistent. Then it proves every proposition, in particular { ( ) } . By the deduction theorem we therefor get (( ) ). However, ((( ) ) ), for every proposition (this is part of the collection of formal proofs that we need. Ill make a list of these separately). Now, applying MP we getthat we need....
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2010_lecture12 - CS245 - Winter 2010, Lecture 12 Shai...

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