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 Σ is maximally consistent if, Σ is consistent and, for every proposition α , either Σ ‘ α or Σ ‘ ( ¬ α ). Why is such a set called maximally consistent? Note that for every propo- sition α , if Σ 6‘ α then Σ ∪ { α } is inconsistent (why?). As it turns out, the converse of this claim is also true. Namely, Claim 1 A set of propositions Σ is maximally consistent if and only if for every proposition α , if Σ 6‘ α then Σ ∪ { α } is inconsistent. The proof of this claim relies on the following Lemma. Lemma 1 For every set of propositions Σ and every proposition α , if Σ 6‘ α then Σ ∪ { ( ¬ α ) } is consistent. Proof of the lemma: Assume, by way of contradiction, that Σ ∪{ ( ¬ α ) } is not consistent. Then it proves every proposition, in particular Σ ∪ { ( ¬ α ) } ‘ α . By the deduction theorem we therefor get Σ ‘ (( ¬ ) α → α ). However, ‘ ((( ¬ ) α → α ) → α ), for every proposition α (this is part of the collection of formal proofs that we need. I’ll make a list of these separately). Now, applying MP we getthat we need....
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This note was uploaded on 02/11/2010 for the course ART AFM101 taught by Professor Mr.lushman during the Spring '10 term at University of Toronto.

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

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