If Pigs Could Fly

[Dan Christensen]

Pigs cannot really fly. As such I can truthfully say, "If pigs could fly, then I am the King of France." And it wouldn't matter if I was actually the King of France or not. This is not a joke, but a legitimate method of proof in mathematics, the so-called Principle of Explosion: From a falsehood (pigs can fly in this case), all things follow. Note that, since pigs cannot fly, the above if-then statement can never be used to infer anything about me. It could not be used to infer that I am the King of France, nor can it be used to infer that I am not the King of France. Here's how it works:

Let P be the proposition that pigs can fly. Let Q be the proposition that I am the King of France.

Theorem:  ~P => (P => Q)

Proof:

1.  Suppose P is false i.e. ~P  (meaning pigs cannot fly)

2.  Suppose for the sake of argument, that P was true, i.e. P  (meaning pigs could fly)

3.  Suppose further that, Q is false i.e. ~Q  (meaning I am not the King of France) 

4.  We have a contradiction: P and ~P  (joining statements 1 and 2)

5.  Using Proof by Contradiction, the premise on line 3 must be false, i.e. ~~Q

6.  Removing the double negation on line 5, we obtain simply Q

7.  Using Conditional Proof on lines 2 and 6, we obtain P => Q.

8.  Using Conditional Proof once more on lines 1 and 7, we obtain ~P => (P => Q) as required.

Comments? Protests?

Dan