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logic 3-25 - Pf BY CONTRADICTION(See notes Def A...

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Midterm Coverage: Chapters 6 – 8 (Fitch), 15.1 – 15.6, 16.1 – 16.4, 17.1 – 17.2 Topics: Proofs in propositional logic (2) Set theory: definitions (for example, define subset), followed by proofs using those definitions, induction (similar to homework) Soundness & Completeness: Completeness theorem is an existence proof. What it shows is the following. We have the proof system F T . Then we have the question, as we have just constructed F T : If Γ proves S, then can we prove S from Γ in F T ? Completeness theorem says yes. Def : A set of sentences Γ is satisfiable if there exists a valuation v such that (v)*(Γ) = True. Compactness Theorem : Suppose Γ is such that all finite sets Δ which are subsets of Γ are satisfiable. Then Γ is satisfiable.
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Unformatted text preview: Pf : BY CONTRADICTION. (See notes.) Def : A computational problem is a specification of two sets A and U, such that A is a subset of U. We say a computational problem is decidable if there exists a finitely describable computer program (M) or an algorithm such that for all x U, M(x) = {yes if ϵ x A, no if x is not in A) ϵ Example (See notes.) Theorem : Propositional logic is decidable. Def : Let S Sent ϵ PL . |S| = number of symbols in S. Question: How many steps does the truth table test for tautology take as a function of |S|. Answer: Take 2 ^ (1/3) |S| Conjecture: For every n N (natural numbers) there exist sentences S Sent ϵ ϵ PL such that the shortest proof of S (Say in F T ) is 2 ^ R |S|, R > 0. Claim: (See notes.)...
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