2. The Logic of Conditionals Chapter 8 Language, Proof and Logic

3. Informal methods of proof 8.1.a Valid steps: 1. Modus ponens: From P  Q and P, infer Q. 2. Biconditional elimination: From P and either P  Q or Q  P, infer Q. 3. Contraposition: P  Q   Q   P The method of conditional proof: To prove P  Q, temporarily assume P and derive Q based on that assumption.

4. Informal methods of proof 8.1.b Proving Even(n 2 )  Even(n): Assume n 2 is even. To prove that n is even, assume, for a contradiction, that n is odd. Then, for some k, n=2k+1. Hence n 2 =(2k+1)(2k+1)= =4k 2 +4k+1=2(2k 2 +k)+1. Hence n 2 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 n 2 =(2k+1)(2k+1)= =4k 2 +4k+1=2(2k 2 +k)+1. Hence n 2 is odd

5. Informal methods of proof 8.1.c 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. n 2 is even 3. n 2 is divisible by 4 Cycle: (3)  (2)  (1)  (3)

6. Formal rules of proof for  and  8.2  Elim: P  Q … P … Q  Intro: P … Q P  Q  Elim: P  Q (or Q  P) … P … Q  Intro: P … Q … P P  Q You try it, pages 208, 211

7. Soundness and completeness 8.3.a F T --- the portion of F that contains the rules for , , , , , . P 1,…,P n  - T Q --- provability of Q in F T from premises P 1,…,P n. CLAIM: F T is sound and complete with respect to tautological consequence. Soundness: If P 1,…,P n  - T Q, then Q is a tautological consequence of P 1,…,P n. Completeness: If Q is a tautological consequence of P 1,…,P n, then P 1,…,P n  - T Q. So, once you see that Q is not a tautological consequence of P 1,…,P n, you can be sure that there is no way to F T -prove Q from P 1,…,P n. So, once you see that Q is a tautological consequence of P 1,…,P n, you can be sure that there is an F T -proof of Q from P 1,…,P n, even if you have not actually found such a proof.

8. Soundness and completeness 8.3.b Would the system be sound if it had the following rule for “exclusive OR”  ? How about the same rule with the ordinary  ? P  Q P … S Q … T S  T See page 215 for a proof idea for the soundness of F T.