The Meaning of "If and Only If"

Chapter Five
Philosophy 404
Summer 1999

In technical discussions, you will often find people saying something like, "A proper name stands for an item in the world if and only if it refers to it." What exactly does "if and only if" (sometimes shortened to "iff") mean? Consider "P if and only if Q". This can be decomposed into "P if Q" and "P only if Q". Consider each of these in turn:

1. "P if Q". This is simply a different way of saying "If Q then P", which is translated into propositional logic as Q > P.

2. "P only if Q". This is trickier. It says that P can be true only if Q is true, which is to say that when Q is false, P must also be false. Notice that it does not tell us anything about the truth value of P if Q is true.

1. If we know that P is true, then we know Q must also be true; however, if we know Q is true, we do not necessarily know anything about the truth value of P.

2. Consider the example: "The light bulb will go on only if the light switch works." If the light bulb goes on, then the switch must have worked, since failure of the switch would have meant darkness; however, if the light switch works, you don't necessarily know that the light bulb will go on, since there could be something wrong in the wiring or the bulb itself that keeps it from illuminating.

3. This is translated into FOL as P > Q.

So P if and only if Q resolves into P > Q and Q > P, which is to say that

P iff Q is logically equivalent to (P > Q) & (Q > P) .

Produce the truth tables for the two conditional statements and use those to convince yourself that this logical equivalence holds.

Another way to explain the meaning of this connective is in terms of necessary and sufficient conditions. The sentence P > Q tells us that the truth of P is sufficient for the truth of Q; that is, if we know that P is true, then we know all we need to assert that Q is true also. So P is a sufficient condition for Q. The sentence also tells us that Q must be true in order for P to be true; that is, if Q is false, P must also be false in order for the sentence as a whole to remain true. But this is to say that P can only be true when Q is, or in other words, that Q is a necessary condition for P. So in a conditional statement, the antecedent is a sufficient condition for the consequent and the consequent is a necessary condition for the antecedent.

In an iff statement (also known as a "biconditional", since it is a conditional in two directions) such as P iff Q , we know that P is both a necessary and sufficient condition for Q, and likewise Q for P. That is, if P is true, then that is sufficient for the truth of Q (i.e., P is a sufficient condition for Q); on the other hand, we know that Q cannot be true unless P is, which is to say that P is also a necessary condition for Q.