Reasoning With Rabbits

I took very little formal logic in school, but it was enough to know that there's more to the subject than I'd guessed. I've since forgotten a lot of it, but for some reason, one bit has stuck with me: it's called the contrapositive.

When you take a logical statement and turn it inside-out, you have the contrapositive. There is, however, a specific way of doing this, in order to preserve the logical consistency.

When working with logic, sometimes it is helpful to use "nonsensical" sentences in order to be sure you're really thinking the thing through and not taking any shortcuts. That is what I am going to do to demonstrate how to derive the contrapositive.

We begin with this logical statement, describing a make-believe world of red rabbits:

If it is red, then it is a bunny.

That is our starting point, and we must treat it as fact in order to logically manipulate it. To find the contrapositive of the statement, change both terms to a negative, and reverse the order. You should wind up with a statement which looks something like this:

If it is not a bunny, then it is not red.

But, but - what about the fire hydrants and stop signs and - and - ketchup? Sorry. Not a bunny, not red.

What's enjoyable about this particular logical maneuver is that if you get confused you can easily step back and think your way through it. All red things belong to the class of bunnies: check. All non-bunnies belong to the class of non-red things: check. But not all non-red things necessarily belong to the class of non-bunnies. This fact is not deducible from our original statement, which said only that the bunnies had monopolized the color red. We do not know if they similarly hijacked other colors. You could possibly have blue bunnies, for instance.

Tidy, isn't it?

Image courtesy of ericclaridge.com