Contrapositives (LSAT)

Ideally, contrapositives will come naturally. In other words when you see something such as:

x--> y 

you won't need to also write down the rule ~y --> ~x. Having to write down the contrapositives can quickly muddy up a logic game with a lot of conditional rules. So, this is an important skill to develop. 

In addition to recognizing contrapositives without having to notate them, it's essential not to assume contrapositives where they don't exist. As an example, 

x --> y 

never leads to the contrapositive of ~x --> ~ y. 

If I tell you that if it rains tomorrow, the game will be cancelled, and you learn tomorrow that it does not rain, you do not know whether or not the game is cancelled. But if I tell you that that game is not cancelled you know with certainty that it did not rain. 

Let's try a more difficult contrapositive now:

If the 7th instrument played is not the trumpet, then the guitar will play before the piano. We might symbolize this as follows, but there's certainly not only one way to symbolize conditional language:

t~7 --> g---p. 

Let's say we learn that the trumpet is 4th

_ _ _ t _ _ _ 

Do we know whether the guitar is before the piano? We don't. Maybe the guitar is before the piano, or maybe the piano is before the guitar. We can't know this. 

What if we're told that the piano is first?

p _ _ _ _ _ _ 

If the piano is first then certainly the piano is before the guitar. And the contrapositive of the above statement tells us that if the piano is before the guitar then the trumpet is not 7th. So here' I'd know the following:

p _ _ _ _ _ ~t

That may not seem terribly helpful, but imagine this:

"If the piano is first, which of the following must be true?"

Answer: The trumpet is not 7th

All said, learn those contrapositives, so that they are so natural that you don't need to think about them. That's a step in the right direction to improving at logic games.