# Contraposition

### A Lesson in Logic

Contraposition is a pretty interesting substitution rule. It’s essentially judo for material conditionals, a complete reversal. With that in mind, let’s get into it, and use it in a couple of proofs.

The basic idea of contraposition is that you can take a material conditional that looks like this:

If it’s raining, then it’s cloudy. (R → C)
And turn it into this:
If it’s not cloudy, then it’s not raining. (~C → ~R)

The two statements have exactly the same truth values, essentially restatements of each other, as you can see from the truth table. Contraposition provides one more way of digging into material conditionals, and a useful way of sifting them out. We can see this logic happening in a lot of investors. The Facebook IPO was today, but it finished trading only slightly above where it began. When the stock starts falling, one line of logic an investor might take is

1. If I don’t want to profit, then I won’t sell (~P → ~S)
2. I’ll sell. (S)
3. Therefore, I want to profit. (P)

Now there’s two ways to prove this, one using double negation and modus tollens, and one using contraposition and modus ponens, which I’m going to show.

4. If I sell, then I want to profit. (S → P) – Contraposition from 1
5. Therefore I want  to profit. (P) – Modus Ponens 2, 4

Here we can see not just the relation of contraposition to rules of inference, but also the relation of value and action. If investors weren’t interested in a profit, and held on to the stock say for sentimental value or as a collector’s item, they wouldn’t have sold and things might have ended differently.

There you have it, contraposition. Also, if you haven’t already, head over to Headshots from the Heart and check out some of the awesome stuff I’m working on for children’s hospitals! And because of Headshots, which happens next weekend, I won’t be doing a logic post next Saturday. On the up side, you can find me undergoing a brutal ordeal playing games for children’s hospitals for 24 hours over at the Headshots website. Watch our webcast, and pledge. Help us help kids.