# Necessary and Sufficient conditions

Class notes for 18 January

## An example

Suppose the Dodgers are playing the Angels in the World Series. (The World Series is a set of seven baseball games. The team that wins four games wins the Series).

Suppose also that the Dodgers have won three games and the Angels have won one game.

Finally, suppose that:

• A = winning the fifth game and
• B = winning the World Series.

## Sufficient conditions

Winning the fifth game is a sufficient condition of winning the World Series for the Dodgers. That means that if the Dodgers win the fifth game, they will win the World Series.

Winning the fifth game is not a sufficient condition of winning the World Series for the Angels. If the Angels win the fifth game, they will have won only two games. They have to win four in order to win the Series.

If A is a sufficient condition of B,

1. Whenever A is true, B is too.
2. Whenever B is false, A is too.

Thus it is impossible for the Dodgers to win the fifth game and not win the World Series: it would be their fourth win and the team that wins four games wins the Series.

It is also impossible for the Dodgers not to have won the Series and to have won the fifth game.

If you tell me that the Dodgers lost the World Series after being ahead 3-1, I know that they did not win the fifth game. If they had won the fifth game, they would have won the Series. But you just told me that they did not win the Series. Therefore, they must not have won the fifth game.

## Necessary conditions

Winning the fifth game is a necessary condition of the Angels' winning the World Series. They cannot win the World Series without winning the fifth game. If they lose the fifth game, the Dodgers will have won four games and, thus, the Series.

Winning the fifth game is not a necessary condition of the Dodgers winning the World Series. The Dodgers can lose the fifth game and still win the Series by winning either the sixth or seventh games.

If A is a necessary condition of B,

1. Whenever A is false, B is too.
2. Whenever B is true, A is too.

Thus it is impossible for the Angels to lose the fifth game and win the World Series.

If you tell me that the Angels won the World Series after being behind 3-1, I know that they did win the fifth game. If they had not won the fifth game, the Dodgers would have won their fourth game and thus the Series. But you told me that the Angels won the Series. Therefore, they must have won the fifth game.

## Refuting proposed conditions

Often, we debate about whether proposed necessary and sufficient conditions are genuine. Here’s how to do it. Look for cases that would be ruled out if the proposed necessary or sufficient condition were true. If they’re possible, or, even better, actual, then the proposal is incorrect.

So, I said that if A is a sufficient condition of B, then whenever A is true, B is too. That means that you should look for cases in which A is true and B is, or could be, false.

I also said that if A is a sufficient condition of B, then whenever B is false, A is too. So another way of challenging a proposed sufficient condition is to find cases in which B is false and A is, or could be, true.

If A is a necessary condition of B, then whenever A is false, B is too. In order to challenge a proposed necessary condition, look for cases in which A is false and B is, or could be, true.

I also said that if A is a necessary condition of B, then whenever B is true, A is too. So another way of challenging a proposed necessary condition is to find cases in which B is true and A is false.

## Back and forth

You can reverse things in the following way: the second part is the other condition of the first part. Thus,

1. A is a sufficient condition of B =
B is a necessary condition of A
2. A is a necessary condition of B =
B is a sufficient condition of A

Explaining how this works will expose one very important thing about necessary and sufficient conditions. They aren’t causes.

### 1. Sufficient to necessary conditions

Suppose that winning the fifth game (A) is a sufficient condition of winning the World Series (B) for the Dodgers. That means that winning the World Series (B) is a necessary condition of winning the fifth game (A) for the Dodgers.

Let that sink in a bit. Switch the A's and B's around from our earlier statement.

If B is a necessary condition of A, then:

1. Whenever B is false, A is too.
2. Whenever A is true, B is too.

If the Dodgers fail to win the World Series (B is false), then they did not win the fifth game either (A is false). If they win the fifth game (A is true), then they win the Series (B is true).

Given that the Dodgers have a 3-1 lead, these two statements have to be true.

### 2. Necessary to sufficient conditions

Suppose that winning the fifth game (A) is a necessary condition of winning the World Series (B) for the Angels. That means that winning the World Series (B) is a sufficient condition of winning the fifth game (A).

If B is a sufficient condition of A, then:

1. Whenever B is true, A is too.
2. Whenever A is false, B is too.

If the Angels win the World Series (B is true), then they must have won the fifth game (A). If they did not win the fifth game (A is false), then they did not win the Series either (B is false).

### So these aren’t causes then, right?

Right. Winning the World Series is a sufficient condition of winning the fifth game for the Angels. But winning the Series isn’t what causes them to win the fifth game. If anything, it's the other way around: they win the Series because they win the fifth game.

And if you think that winning the World Series is somehow a necessary pre-condition for the Dodgers' winning the fifth game, as if it had to come first in the causal order, you really don't understand how this works. You only win the Series as a whole by winning the games.

In general, it’s tempting to think about necessary and sufficient conditions in terms of causes and effects. That temptation should be resisted.

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