Beginning Conditionals

Conditional statements are the bastard red-headed stepchildren of formal logic. If you've ever taken any formal logic classes, these may make your head explode because technically I do them wrong. However, what I will demonstrate in this and several upcoming posts works on the LSAT, and that's all we're interested in here.

Your basic conditional statement is one of those if/then statements you probably had to suffer through in 9th grade algebra class, only this time we're not doing math. Consider the following:

if A then B

Written shorthand, we can say

A --> B

There are many types of phrases this could symbolize. All A's are B's. When someone has A he must also have B. No A is not a B (ooh, double negative). When you are A you become B. What exactly are A and B? Who knows? Here's a real-world example.

All states are in the northern hemisphere.

S --> NH

With common sense, we can derrive all kinds of information from this. For instance, we could figure out what is disqualified from being a state

If a place is NOT in the northern hemisphere, it is NOT a state.

~NH --> ~S

This is called the contrapositive. Flip and negate. The contrapositive, for purposes of the LSAT, means the exact same thing as the original statement. Knowing either S --> NH or ~NH --> ~S allows is to conclude the other. Or original statement, if A then B, tells us both:

A --> B
~B --> ~A

Notice I did not conclude ~A --> ~B. We know that B is a necessary result of A, but we don't know whether it could exist without A. Perhaps a C would also give us B, and A can take the day off.

An illustration I like to use is a memory of being 4 years old. My mother would say something like:

If you want to go outside, you have to eat your peas.

O --> P

So I'd get all excited, grab my fork, and shovel down my entire pile of peas, imagining all the wonderful things I was going to do once I got outside. Of course, no sooner had the last pea left my fork into my mouth, my mother would exclaim, "OK, time for a bath!"

A BATH?!?!?! But I ate my peas! I'm supposed to be going outside swinging or throwing mud or something!

My mother was a better logician than I was. Her statement said if I wanted to go outside, I had to eat my peas. The contrapositive (the only other thing we know must have been true) was if I DIDN'T eat my peas I COULDN'T go outside.

O --> P
~P --> ~O

Nobody said anything about eating my peas giving my a license to go outside, only that going out side would be denied in the event I failed to eat my peas. My 4-year-old pea-brain assumed that since O --> P then P --> O. This is flawed logic both at my kitchen table and on the LSAT.