Formal Proofs and Formal Proofs and Boolean Logic Boolean Logic

Chapter 6

Language, Proof and

Logic

Conjunction rules6.1

Elim:

P1…Pi…Pn

…

Pi

Intro:

P1

Pn

…P1…Pn

1. ABC

2. B Elim: 13. C Elim: 14. CB Intro: 3,2

Disjunction rules6.2a

Intro:

Pi

…

P1…Pi…Pn

Elim:

P1…Pn

…

…S

P1

…S

Pn

…S

Example6.2b

1. (AB) (CD)

2. AB 3. B Elim: 2 4. BD Intro: 3

5. CD 6. D Elim: 5 7. BD Intro: 6

8. BD Elim: 1, 2-4, 5-7

You try it, page 152

Contradiction and negation rules6.3

Elim:

…

P

Intro:

P … P…

Elim:

P …

P

Intro:

P …

PYou try it, p.163

The proper use of subproofs6.4

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 citeanything from within that subproof.

Strategy and tactics6.5

When looking for a proof, the following would help:

1. Understand what the sentences are saying.

2. Decide whether you think the conclusion follows from the premises.

3. If you think it does not follow, look for a counterexample.

4. 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.

Proofs without premises6.6

The conclusion of such a proof is always logically valid!

1. PP

2. P Elim: 1 3. P Elim: 1 4. Intro: 2,35. (PP) Intro: 1-4