Formal Proofs and Boolean Logic

1. Language, Proof and Logic Formal Proofs and Boolean Logic Chapter 6

2. 6.1 Conjunction rules Intro: P1  Pn … P1…Pn Elim: P1…Pi…Pn … Pi • ABC • B Elim: 1 • C Elim: 1 • CB Intro: 3,2

3. 6.2a Disjunction rules Elim: P1…Pn …  … S Intro: Pi … P1…Pi…Pn P1 … S Pn … S

4. 6.2b Example 1. (AB)  (CD) 2. AB 3. B  Elim: 2 4. BD  Intro: 3 5. CD 6. D  Elim: 5 7. BD  Intro: 6 8. BD  Elim: 1, 2-4, 5-7 You try it, page 152

5. 6.3 Contradiction and negation rules Intro: P … P …  Elim:  … P Intro: P …  P Elim: P … P You try it, p.163

6. 6.4 The proper use of subproofs A subproof may use any of its own assumptions and derived sentences, as well as those of its parent (or grandparent, etc.) proof. However, once a subproof ends, its statements are discharged. That is, nothing outside that subproof (say, in its parent or sibling proof) can cite anything from within that subproof.

7. 6.5 • When looking for a proof, the following would help: • Understand what the sentences are saying. • Decide whether you think the conclusion follows from the premises. • If you think it does not follow, look for a counterexample. • If you think it does follow, try to give an informal proof first, and • then turn it into a formal one. • 5. Working backwards is always a good idea. • 6. When working backwards, though, always check that your • intermediate goals are consequences of the available information. • You try it, page 170. Strategy and tactics

8. 6.6 Proofs without premises The conclusion of such a proof is always logically valid! 1. PP 2. P Elim: 1 3. P Elim: 1 4.  Intro: 2,3 5. (PP) Intro: 1-4