If and only if

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In mathematics, philosophy, logic and technical fields that depend on them, "if and only if" is a connective between statements which means that the truth of either one of the statements requires the truth of the other. Thus, either both statements are true, or both are false.

In writing, common alternative phrases to "if and only if" include iff, "Q is necessary and sufficient for P", "P is equivalent to Q", "P precisely if Q", and "P precisely when Q". (Many authors regard "iff" as unsuitable in formal writing; others use it freely.)

In logical formulae, logical symbols are used instead of these phrases; see the discussion of notation.

Usage

Notation

The corresponding logical symbols are "↔" and "⇔" and "≡", and sometimes "iff". These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those formulas (e.g., in metalogic).

Another term for the logical connective is exclusive nor.

Proofs

In most logical systems, one proves a statement of the form "P iff Q", through the more roundabout route of separately proving both of the statements "if P, then Q" and "if Q, then P". (or its contrapositive, "If not P, then not Q"). Proving this pair of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts--that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have both been shown true, or both false.

Origin of the abbreviation

Usage of the abbreviation iff first appeared in print in John Kelley's 1955 book General Topology. Its invention is often credited to the mathematician Paul Halmos, but in his autobiography he states that he borrowed it from puzzlers.

The difference between "if" and "iff"

Put simply, the difference between if and iff can be explained with the following two sentences:

  1. Madison will eat pudding if the pudding is a custard. (equivalently: If the pudding is a custard, then Madison will eat it)
  2. Madison will eat pudding if and only if (iff) the pudding is a custard.

Sentence (1) states only that Madison will eat custard pudding. It does not however preclude the possibility that Madison might also have occasion to eat bread pudding. Maybe she will, maybe she will not. The sentence does not tell us. All we know for certain is that she will eat custard pudding.

Sentence (2) however makes it quite clear that Madison will eat custard pudding and custard pudding only. She will not eat any other type of pudding.

A further difference is that "if" is used in definitions (except in formal logic); see more below.

Advanced considerations

Philosophical interpretation

A sentence that is composed of two other sentences joined by "iff" is called a biconditional. Iff joins two sentences to form a new sentence. It should not be confused with logical equivalence which is a description of a relation between two sentences. The biconditional "A iff B" uses the sentences A and B, describing a relation between the states of affairs A and B describe. By contrast "A is logically equivalent to B" mentions the two sentences: it describes a relation between those two sentences, and not between whatever matters they describe.

The distinction is a very confusing one, and has led many a philosopher astray. Certainly it is the case that when A is logically equivalent to B, "A iff B" is true. But the converse does not hold. Let's reconsider the sentence:

Madison will eat pudding today if and only if it's custard.

There is clearly no logical equivalence between the two halves of this particular biconditional. For more on the distinction, see W. V. Quine's Mathematical Logic, Section 5.

Definitions

In philosophy and logic, "iff" is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals. In mathematics and elsewhere, however, the word "if" is normally used in definitions, rather than "iff". This is due to the observation that "if" in the English language has a definitional meaning, separate from its meaning as a propositional conjunction. This separate meaning can be explained by noting that a definition (for instance: A group is "abelian" if it satisfies the commutative law; or: A grape is a "raisin" if it is well dried) is not an equivalence to be proved, but a rule for interpreting the term defined. (Some authors, nevertheless, explicitly indicate that the "if" of a definition means "iff"!)

Examples

Here are some examples of true statements that use "iff" - true biconditionals (the first is an example of a definition, so it should normally have been written with "if"):

  • A person is a bachelor iff that person is an unmarried but marriageable man.
  • "Snow is white" (in English) is true iff "Schnee ist weiß" (in German) is true.
  • For any p, q, and r: (p & q) & r iff p & (q & r). (Since this is written using variables and "&", the statement would usually be written using "↔", or one of the other symbols used to write biconditionals, in place of "iff").

Analogs

Other words are also sometimes emphasized in the same way by repeating the last letter; for example orr for "Or and only Or" (the exclusive disjunction).

More general usage

Iff is used outside the field of logic, wherever logic is applied, especially in mathematical discussions. It has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon. (However, as noted above, if, rather than iff, is generally used in statements of definition.)de:Logische Äquivalenz es:Si y sólo si fr:SSI he:אם ורק אם is:Eff it:Sse lt:Tada ir tik tada (teiginys) nl:Dan en slechts dan als ja:同値 pl:Gddy pt:Se e somente se sr:Акко sv:Om och endast om zh:当且仅当