If and Only If - Distinction From "if" and "only If"

Distinction From "if" and "only If"

1. "If the fruit is an apple, then Madison will eat it." or "Madison will eat the fruit if it is an apple." (equivalent to "Only if Madison will eat the fruit, is it an apple;" or "Madison will eat the fruit ← fruit is an apple")

   This states only that Madison will eat fruits that are apples. It does not, however, preclude the possibility that Madison might also have occasion to eat bananas. Maybe she will, maybe she will not—the sentence does not tell us. All we know for certain is that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficient condition for Madison to eat the fruit.

2. "Only if the fruit is an apple, will Madison eat it." or "Madison will eat the fruit only if it is an apple." (equivalent to "If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → fruit is an apple")

   This states that the only fruit Madison will eat is an apple. It does not, however, preclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. In this case, that a given fruit is an apple is a necessary condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat any and all apples she is given.

3. "If and only if the fruit is an apple will Madison eat it" or "Madison will eat the fruit if and only if it is an apple" or "Madison will eat the fruit ↔ fruit is an apple."

   This, however, makes it quite clear that Madison will eat all and only those fruits that are apple. She will not leave any such fruit uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit.

Sufficiency is the inverse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. "not"). This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways:

P is sufficient for Q

Q is necessary for P

¬Q is sufficient for ¬P

¬P is necessary for ¬Q

As an example, take (1), above, which states P→Q, where P is "the fruit in question is an apple" and Q is "Madison will eat the fruit in question". The following are four equivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madison will eat it.

Only if Madison will eat the fruit in question, is it an apple.

If Madison will not eat the fruit in question, then it is not an apple.

Only if the fruit in question is not an apple, will Madison not eat it.

So we see that (2), above, can be restated in the form of if...then as "If Madison will eat the fruit in question, then it is an apple"; taking this in conjunction with (1), we find that (3) can be stated as "If the fruit in question is an apple, then Madison will eat it; AND if Madison will eat the fruit, then it is an apple".