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09ma6cHW5

# 09ma6cHW5 - S ‘ A for any wﬀ A(30 2 Prove Proposition...

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Ma/CS 6c Assignment #5 Due Thursday, May 6 at 1 p.m. IN ALL PROBLEMS BELOW YOU CANNOT USE THEOREM 1.11.5 IN THE NOTES (since the exercises below form parts of the proof of that theorem). ALSO “FORMULA” OR “WFF” MEANS “WELL-FORMED FORMULA IN PROPOSITIONAL LOGIC BUILT USING PROPOSITIONAL VARIABLES, PARENTHESES, AND ONLY ¬ , .” (40%) 1. * (i) Show that for any formula A , ( ¬¬ A A ) . (ii) Prove Proposition 1.11.8 in the notes: Let S be any set of wffs and A any wff. If S ∪ { A } is formally inconsistent, then S ‘ ¬ A . (iii) Show that if S is formally inconsistent, then
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Unformatted text preview: S ‘ A for any wﬀ A . (30%) 2. Prove Proposition 1.11.9 in the notes: Let S be any set of wﬀs and A,B any wﬀs. Then: S ∪ { A } ‘ ¬ B iﬀ S ∪ { B } ‘ ¬ A. (40%) 3. (i) Show that ¬ ( A ⇒ B ) ‘ A and ¬ ( A ⇒ B ) ‘ ¬ B (ii) Prove Lemma 1.11.12 in the notes: If ¯ S is a formally consistent and complete set of formulas and ν is the valuation deﬁned by ν ( p i ) = ( 1 , if p i ∈ ¯ S , if p i 6∈ ¯ S, then for any formula A , ν ( A ) = 1 iﬀ A ∈ ¯ S. 1...
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