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

Beyond truth tables 5.0 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.

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)

Proof by cases (disjunction elimination) 5.2 To prove Q from a disjunction, prove it from each disjunct separately. “There are irrational numbers b,c such that b c is rational”.  2  2 is either rational or irrational. 1.If rational, then take b=c=  2, known to be irrational. 2.If irrational, take b=  2  2 and c=  2.

Indirect proof (proof by contradiction) 5.3 Contradiction --- any claim that cannot possibly be true. Proof of  Q by contradiction: assume Q and derive a contradiction. Proving that “  2 is irrational”: Suppose  2 is rational. 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=a 2 /b 2. Hence a 2 =2b 2. So, a is even. So, a 2 is divisible by 4. So, b 2 is even. So, b is even. Contradiction.

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