Negating Required & Sufficient Conditions (LSAT)

If it rains prior to 1pm today, then the game is cancelled. But it's now after 1pm and it has not rained. Thus, the game is not cancelled. ❌

If it rains prior to 1pm today, then the game is cancelled. The game is not cancelled. Thus, it did not rain prior to 1pm today. ✅ 

The first example does not follow logically, but the second example does. Why? A perfectly acceptable answer is that the second example contains the contrapositive, and the first example does not. Although that's fine, it's a bit technical. 

A more thorough understanding requires knowing the difference between necessary and sufficient conditions. In the first example, raining prior to 1pm today is sufficient for guaranteeing that the game is cancelled. But just because something is sufficient for guaranteeing another thing, doesn't mean that it's necessary for the occurrence of the other thing. In other words, there could very well be reasons other than rain for why the game is cancelled; knowing that it did not rain does not allow drawing any conclusion about the cancellation of the game.

In the second example, the game being cancelled is a required or necessary condition once we know that it rains prior to 1pm today. It has to happen. And so because it's required once we know that it rains, knowing that the game was not cancelled allows for us to know with certainty that it did not rain prior to 1pm today. 

Negating a sufficient condition does not allow for drawing the conclusion drawn here. Negating a necessary condition does. 

This is why the following holds. 

Imagine an "if --> then" statement: x --> y

~x --> ~y ❌

but

-y --> ~x ✅