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Language, Proof and Logic The Logic of Conditionals Chapter 8 8.1.a Informal methods of proof Valid steps: 1. Modus ponens: From PQ and P, infer Q. 2. Biconditional elimination: From P and either PQ or QP, infer Q. 3. Contraposition: PQ Q P The method of conditional proof: To prove PQ , temporarily assume P and derive Q based on that assumption. 8.1.b Informal methods of proof Proving Even(n2)Even(n): Assume n2 is even. To prove that n is even, assume, for a contradiction, that n is odd. Then, for some k, n=2k+1. Hence n2=(2k+1)(2k+1)= =4k2+4k+1=2(2k2+k)+1. Hence n2 is odd, which is a contradiction. So, n is even. Proving the same using contraposition: Assume n is odd. Then, for some k, n=2k+1. Hence n2=(2k+1)(2k+1)= =4k2+4k+1=2(2k2+k)+1. Hence n2 is odd 8.1.c Informal methods of proof Proving biconditionals: To prove a number of biconditionals, try to arrange them into a cycle of conditionals, and then prove each conditional. 1. n is even 2. n2 is even 3. n2 is divisible by 4 Cycle: (3)(2)(1)(3) 8.2 Formal rules of proof for and Elim: PQ … P … Q Intro: P … Q PQ Elim: PQ (or QP) … P … Q Intro: P … Q Q … P PQ You try it, pages 208, 211 8.3.a Soundness and completeness FT --- the portion of F that contains the rules for ,,,,,. P1,…,Pn -T Q --- provability of Q in FT from premises P1,…,Pn. CLAIM: FT is sound and complete with respect to tautological consequence. Soundness: If P1,…,Pn -T Q, then Q is a tautological consequence of P1,…,Pn. So, once you see that Q is not a tautological consequence of P1,…,Pn, you can be sure that there is no way to FT-prove Q from P1,…,Pn. Completeness: If Q is a tautological consequence of P1,…,Pn, then P1,…,Pn -T Q. So, once you see that Q is a tautological consequence of P1,…,Pn, you can be sure that there is an FT-proof of Q from P1,…,Pn, even if you have not actually found such a proof. 8.3.b Soundness and completeness Would the system be sound if it had the following rule for “exclusive OR” ? How about the same rule with the ordinary ? PQ P … S Q … T ST See page 215 for a proof idea for the soundness of FT.