Conditionals

- Sufficient versus Necessary Conditions
- Converse
- “If and only if”
- Contrapositive
*reductio ad absurdum*- Review

If you have a statement of the form If P then Q,

the whole thing is called a **conditional**.
P is called the **antecedent** of the conditional, and Q is called the **consequent** of the conditional.

In this class, you can take all of the following to be variant ways of saying the same thing:

- If P then Q
- P implies Q
- P -> Q
- P is sufficient (or: a
**sufficient condition**) for Q - If you’ve got P you must have Q
- A
**necessary condition**for having P is that you have Q - Q is necessary for having P
- It’s only the case that P if it’s also the case that Q
- P only if also Q

Here are some examples:

If Fred is a dog, then Fred is a fish.

This conditional is presumably false.If Fred is a dog, then Fred is a mammal.

(Or:Fred is a dog only if Fred is a mammal.

) This conditional is presumably true — regardless of whether Fred is in fact a dog.

If P then Q

may sound like P

has to come first; whereas P only if also Q

may sound like it’s Q

that has to come first. But as philosophers use these words, they’re not making any commitment about which comes first.

Note the terms **sufficient condition** and **necessary condition**.

To say that one fact is a *sufficient* condition for a second fact means
that, *so long as* the first fact obtains, *that’s enough* to guarantee that
the second fact obtains, too. For example, if you have ten children,
that is sufficient for you to be a parent.

To say that one fact is a *necessary* condition for a second fact means
that, *in order for* the second fact to be true, *it’s required that* the
first fact also be true. For example, in order for you to be a father,
it’s necessary that you be male. You can’t be a father unless you’re
male. So being male is a necessary condition for being a father.

When P entails Q, then P is a sufficient condition for Q (if P is true, that’s enough for Q to be true, too); and Q is a necessary condition for P (in order for P to be true, it’s required that Q is also true).

Exercise |
---|

Consider the following pairs and say whether one provides sufficient and/or necessary conditions for the other: |

1. a valid argument, a sound argument |

2. knowing that it will rain, believing that it will rain |

The conditional If Q then P

(which could also be written
Q → P

or P if Q

) is called the **converse** of If P then Q

(which as we said, could be written P only if Q

).
For example:

- The converse of
If Fred is a dog, then Fred is a mammal

isIf Fred is a mammal, then Fred is a dog.

It should be intuitive that the converse says something different than the original conditional. Presumably the original conditional *has* to be true, since every dog is a mammal. But If Fred is a mammal, then Fred is a dog

might or might not be correct. It’d be correct if, for example, we’re doing detective work on what kind of animal Fred is, and maybe he might be a mammal or a fish, but we’ve settled that if he’s a mammal, then the only species he could be is a dog. But more likely, this conditional would be false. In normal circumstances, the premise that Fred is a mammal leaves it open that he might be any number of species: a dog, a cat, a monkey, and so on…

So in general, a conditional and its converse say different things. They might both be true, or they might both be false, or it might be that one is true and the other is false. There might be special cases where if one of them is true, the other one must be true too. But we shouldn’t in general expect claims of the form If P then Q

and If Q then P

to say equivalent things.

Sometimes it’s true *both* that If P then Q

*and* that If Q then P.

When both conditionals are true, philosophers express this by saying P

which they often abbreviate as **if and only if** Q,P

This claim that both conditionals hold is also called a **iff** Q.**biconditional**.
One might also write P just in case Q,

P is necessary and sufficient for Q,

or P <-> Q.

For example, being a male parent is both necessary and sufficient for being a father. If you’re a father, it’s required that you be a male parent. And if you’re a male parent, that suffices for you to be father. So we can say that someone is a father if and only if he’s a male parent.

We were discussing the difference between a conditional and its converse. Now, on the other hand, consider these two conditionals:

If Fred is a dog, then Fred is a mammal.

If Fred

*isn’t*a mammal, then Fred*isn’t*a dog.

It should be intuitive that if either of these is true, the other will be true as well. The second conditional is called the **contrapositive** of the first. That’s a bit of technical logical jargon, but hopefully the basic idea here is clear enough without studying logic. Suppose that Fred is a dog. Then the first claim says that he has to be a mammal, and the second claim implies this too, a bit indirectly, because if he weren’t a mammal, then he couldn’t be a dog. On the other hand, suppose that Fred isn’t a dog. Then the first claim is silent about whether Fred is or isn’t a mammal. And the second claim tells us only that if he isn’t a mammal, then he isn’t a dog, which we were already supposing. So in the case where Fred isn’t a dog, neither conditional adds any new constraints.
It makes sense then that these should be saying the same thing. They just package it in a different form.

Summarizing:

- the claim
If P then Q

*won’t in general*be equivalent to its converse (If Q then P

) — though there can be special cases where they both must have the same truth-value - but
*it will always*be equivalent to its contrapositive (If not-Q then not-P

)

Given the relation between a conditional and its contrapositive,
the following is a valid form of argument: If P, then Q. But not-Q. So not-P.

Some students initially have difficulty understanding why this is a
valid form of argument. Think of it this way: We know that if P, then Q.
Now let’s hypothetically suppose for a moment, and just for the sake of argument,
that P is true. Then Q would have
to be true, too, right? Since if P, then Q. But we already have that Q is not
true! — this is one of our premises. So *our hypothesis that P is true
must be wrong*: it leads us to something that we know is false. It must be the case, then, that not-P.

This kind of reasoning is known as * reductio ad absurdum*: you
assume some hypothesis for the sake of argument, and then you show that
the hypothesis leads to something “absurd” — perhaps to a contradiction, or at least
to some other conclusion you already know to be false. Hence the hypothesis can’t be true. It
has to be rejected.

It can be disorienting when you come across a philosopher employing a
*reductio,* if you misunderstand him as actually subscribing to the
“absurdity” he derives. You have to recognize that the philosopher who
offers a *reductio* does *not* endorse the “absurdity” himself. He’s
arguing that it’s something that follows from *his
opponent’s* view.

Here’s an example of a *reductio*. (I got this wonderful example from my
former colleague Tim Maudlin.)

A computer scientist announces that he’s constructed a computer program that can play the perfect game of chess: he claims that this program is guaranteed to win every game it plays, whether it plays black or white, with never a loss or a draw, and against any opponent whatsoever. The computer scientist claims to have a mathematical proof that his program will always win, but the proof runs to 500 pages of dense mathematical symbols, and no one has yet been able to verify it. Still, the program has just played 20 games against Magnus Carlsen and it won every game, 10 as white and 10 as black. Should you believe the computer scientist’s claim that the program is so designed that it will always win against every opponent?

No. Here’s why:

Suppose for the sake of argumentthat a perfect chess program that always winswerepossible. Then we could program two computers with that program and have them play each other. By hypothesis, the program is supposed to win every game it plays, no matter who the opponent is, and no matter whether it plays white or black. So when the program plays itself, both sides would have to win. But that’s impossible! In no chess game can both white and black be winners. (The game is claimed always to win, not to draw.) Sothe supposition that a perfect chess program is possible leads to an absurd/impossible/obviously incorrect result.So that supposition must be false. A perfect chess program with the abilities the computer scientist claims must not be possible.

This is a *reductio.* We assumed some hypothesis for the sake of
argument and showed that it leads to an absurd result. Hence the
hypothesis must be false.

This page tried to explain, and enable you to understand, the following concepts:

- antecedent and consequent of a conditional
- sufficient condition
- necessary condition
- “if and only if” (iff)
- difference between the converse and contrapositive of a conditional
*reductio*