Methods of Proof for Boolean Logic

1. Language, Proof and Logic Methods of Proof for Boolean Logic Chapter 5

2. 5.0 Beyond truth tables Why truth tables are not sufficient: • Exponential sizes • Inapplicability beyond Boolean connectives Need: proofs, whether formal or informal. For informal proofs, it is relevant who your listener is. This section talks about some informal proof methods.

3. 5.1 Valid inference steps in informal proofs • In giving an informal proof from some premises, if Q is already • known to be a logical consequence of some already proven sentences, • then you may assert Q in your proof. • 2. Each step should be significant and easily understood (this is where • your audience’s level becomes relevant). Valid patterns of inference that generally go unmentioned: • From PQ, infer P (conjunction elimination) • From P and Q, infer PQ (conjunction introduction) • From P, infer PQ (disjunction introduction)

4. 5.2 Indirect proof (proof by contradiction) Contradiction --- any claim that cannot possibly be true. Proof of Qby contradiction: assume Q and derive a contradiction. Proving that “2 is irrational”: Suppose 2 isrational. So, 2= a/b for some integers a,b. We may assume at least one of a,b is odd, for otherwise divide both a and b by their greatest common divisor. From 2=a/b we find 2=a2/b2. Hence a2=2b2. So, a is even. So, a2 is divisible by 4. So, b2 is even. So, b is even. Contradiction.

5. 5.3 Proof by cases (disjunction elimination) • To prove Q from a disjunction, prove it from each disjunct separately. • “There are irrational numbers b,c such that bc is rational”. • 22is either rational or irrational. • If rational, then take b=c= 2, already known to be irrational. • If irrational, take b=22 and c= 2.

6. 5.4 Arguments with inconsistent premises Premises from which a contradiction follows are said to beinconsistent. You can prove anything from such premises! An argument with inconsistent premises is always valid yet never sound!