William M. Briggs

Background

You often hear, if you manage to stay awake during the lectures, a mathematician or physicist say, “The following is a necessary but not sufficient condition for my theory to be true.”

We say this so often that we tend to blend the words together: necessarybutnotsufficient, and we forget that it can be a confusing concept.

It means that there is an item or a list of items that must be the case in order for my theory to be true. But just because that item or those items on that list are true, it does not mean that my theory must be true. It could be the case that the item is true but my theory is false.

This is all important in the theory/model building that goes on in the sciences.

For example, it is necessary but not sufficient that my theory be able to explain already observed data that I have collected. If I cannot at least explain that data, then my theory cannot be true. This is necessarybutnotsufficient in the weak sense: all theories must be able to explain their already-observed data.

Again, it is a necessary but not sufficient condition that my theory be able to explain future data. This is necessarybutnotsufficient in the strong sense: if my theory is right, it must be able to explain data that is not yet seen.

Understand: it can still be the case that my theory can predict data that is not yet seen and my theory could be false. This is true in all cases where we cannot deduce (know with certainty) the true of a theory. Most theories (outside math) are not, of course, deduced.

How about an example? Let’s us the following game.

The Game

Go to The Philosopher’s Mag and play this game called “Dealing with Induction”. It asks the question: “how easy is it to draw a wrong conclusion about the future from the evidence of the past?”

The game tests your inductive reasoning skills and asks you to infer the rule that accepts or rejects cards from a standard 52-card deck.

Let me be as clear as possible. The following conditions hold: (1) You believe that a rule that generates the card exists. (2) You will see a sequence of cards from which you will attempt to infer, through induction, the rule. (3) No matter how many cards are shown you will never know the rule with certainty; that is, you will never be able to deduce the rule from a set of premises.

Do not read further until you have played the game fully and discovered its secret.

Did you really play?

Tell the truth. Don’t cheat and read any more until you have played the game.