09ma6cHW5 - S A for any w 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 w A . (30%) 2. Prove Proposition 1.11.9 in the notes: Let S be any set of ws and A,B any ws. Then: S { A } B i 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 is the valuation dened by ( p i ) = ( 1 , if p i S , if p i 6 S, then for any formula A , ( A ) = 1 i A S. 1...
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