I did not know this expression.

It stands for “If and only if”.

大西泰斗、ポール・マクベイ『一億人の英文法』（東進ブックス、2011年）625ページ。

I did not know this expression.

It stands for “If and only if”.

大西泰斗、ポール・マクベイ『一億人の英文法』（東進ブックス、2011年）625ページ。

必要条件と十分条件

If and only if

“In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.”–Wikipedia

its used a lot in philosophy

Distinction from “if” and “only if”

(1) “Madison will eat the fruit if it is an apple.”

(equivalent to “Only if Madison will eat the fruit, it is an apple;” or “Madison will eat the fruit ← fruit is an apple”)

This states simply that Madison will eat fruits that are apples. It does not, however, exclude the possibility that Madison might also eat bananas or other types of fruit. All that is known 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) “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, exclude 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 all the apples she is given.

(3) “Madison will eat the fruit if and only if it is an apple”

(equivalent to “Madison will eat the fruit ↔ fruit is an apple”)

This statement makes it clear that Madison will eat all and only those fruits that are apples. She will not leave any apple 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 converse 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”.

I am a native speaker, and I have never seen this before. However, I know that people who study logic use lots of specialized symbols and abbreviations.

私は英語のネーティブ・スピカーですが、この表現をみたことがありません。しかし、論理学を専門とする人はこのような専門的な記号と省略をよく使用するらしい。

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Me either.