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In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for "if and only if". It is often, not always, written italicized: iff. Although "P iff Q" is most standard, common alternative phrases include "P is necessary and sufficient for Q" and "P precisely if Q".
If and only if
Notation
The corresponding logical symbols are "↔" and "⇔". 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 former is used as a symbol in logic formulas, while the latter -- in reasoning about those formulas (e.g., in metalogic).
Proofs
When proving the statement "P iff Q", it is equivalent to prove both of the statements "if P, then Q" and "if Q, then P". Alternatively, one can prove both "If P, then Q" and "If not P, then not Q", the latter being a contrapositive of (and thus equivalent to) "If Q, then P". Proving this pair of statements sometimes (but of course not always) leads to a more natural proof.
Start of the abbreviation
The abbreviation appeared in print for the first time 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:
- Mary will eat pudding if it is custard. (equivalently: If it is custard, then Mary will eat pudding)
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- Mary will eat pudding if and only if it is custard.
Sentence (1) states only that Mary will eat custard pudding. It does not however preclude the possibility that Mary might also be prepared 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 not refuse custard pudding.
Sentence (2) however makes it quite clear that Mary will eat custard pudding and custard pudding only. She will not eat any other type of pudding.
Advanced considerations
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:
- Mary 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.
In philosophy and logic, "iff" is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals. In mathematics, however, the word "if" is often used in definitions, rather than "iff". Here are some examples of true statements that use "iff" - true biconditionals (the first is an example of a definition):
- 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").
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, as well, in mathematics publications and talks in general. 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.)
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