What "necessary" Means

Yesterday, I discussed the concept of sufficiency via a few examples. Today, I'll discuss the concept of necessity using those same examples. What does it mean for something to be "necessary"?

The first example I gave yesterday was this:

"If I get the loan, then I promise to buy your house."

This means that IF the loan is granted, then it is NECESSARY to buy the house. You can't get the loan without being obligated to buy the house. This doesn't mean that if you don't get the loan you can't still buy the house, or be obligated to buy the house for some other reason. What it DOES mean is that if you're not obligated to buy the house, then you didn't get the loan. Again: buying the house is NECESSARY if you get the loan.

Memorize this:  The necessary condition is always on the right. So the statement above could be diagrammed like this:

Get loan --> Buy house

And the contrapositive (a concept I introduced yesterday) is this:

Buy house --> Get loan

  • So getting the loan is sufficient information to prove that you have to buy the house.
  • Another way of saying this is that buying the house is necessary if you get the loan.
  • Another way of saying this is that if you don't buy the house, that's sufficient information to prove you didn't get the loan.
  • Yet another way of saying this is that not getting the loan is necessary if you don't buy the house.

The four statements above are all logically identical.

We can do the same thing with the other two examples I gave yesterday:

"If you're Warren Buffett, then you're rich."

Since the necessary condition is always on the right, that looks like this:

Buffett --> Rich

and the contrapositive:

Rich --> Buffett

  • So being Buffett is sufficient information to prove that you are rich.
  • Another way of saying this is that being rich is necessary if you are Buffett.
  • Another way of saying this is that if you aren't rich, that's sufficient information to prove you aren't Buffett.
  • Yet another way of saying this is that being someone other than Buffett is necessary if you aren't rich.

The four statements above are all logically identical.

And the final example I discussed yesterday was:

“If you trip and fall headfirst into an industrial sausage grinder, then you are dead.”

Since the necessary condition is always on the right, that looks like this:

Sausage Grinder --> Dead

And the contrapositive:

Dead --> Sausage Grinder

  • So going into the grinder is sufficient information to prove that you are dead.
  • Another way of saying this is that being dead is necessary if you go into the grinder.
  • Another way of saying this is that if you aren't dead, that's sufficient information to prove you didn't go into the grinder.
  • Yet another way of saying this is that having stayed out of the grinder is necessary if you aren't dead.

The four statements above are all logically identical.

More than anything, I promise you that this is all a hell of a lot easier than it might look at first blush. If you have questions, I really hope you'll post them here.

Next time, I'll be back with a look at the very big difference between "if" and "only if."