THE CONTRAPOSITIVE

The contrapositive is a term of logic. The contrapositive of the if-then sentence "if he is a king, then he is a man" is "if he is not a man, then he is not a king". The two are, logically speaking, the same (logically equivalent, in the jargon).

That makes the contrapositive a useful little tool, particularly when it comes to the question of proof. I give an example. In mathematics, it is important to know that if you can find a solution to a problem, then that solution will be unique. That allows you to muck about with the problem until you find a solution. It doesn't matter whether it is a simple or a complicated solution, or how you find it, if you can find one that works, then that's good enough.

Proving that a solution is unique from first principles is difficult, if not impossible. So we use the contrapositive. We assume that the solution is not unique; and (like the man/king sentence at the top) work backwards to show that that would then breach the assumptions in the problem. Because the contrapositive is logically equivalent to the original statement, we have now proved the latter.

We can also use the tool in everyday life, to test the truth of politicians and others. "If he really believed x, then he would do y" is a statement. If that statement holds true, and y doesn't happen, then the contrapositive demonstrates conclusively that he doesn't really believe x.

Try it some time. You will have logic on your side.

Walter Blotscher