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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).

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.

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.

1 The difference between "if" and "iff"

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

1. Mary will eat puddingPudding is either of two general types of food, the second deriving from the first. The older puddings were foods that were presented in a solid mass formed by the amalgamation of various ingredients with a binder that may or may not have been a gelling a today if it's custardCustard is a sweet dessert made from a combination of milk or cream, egg yolks, cornflour, sugar and flavourings such as vanilla. Depending on how much thickener is added, custard may vary in consistency from a thin pouring sauce or Creme Anglaise, to a t.

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

Sentence (1) only states that Mary will eat custard pudding. It does not however preclude the possibility that Mary might also be prepared to eat bread puddingBread pudding is a dessert popular in British cuisine, made using stale (usually left-over) bread, suet, egg, sugar or golden syrup, spices and dried fruit. The bread is soaked (often overnight), squeezed dry and mixed with the other ingredients. The mixt. Maybe she will, maybe she won't. The sentence does not tell us. All it tells us is that we can safely assume that she won't refuse custard pudding today.

Sentence (2) however makes it quite clear that Mary will eat custard pudding and custard pudding only. We need not ask her about any other kind of pudding. We know she will only eat custard pudding.

2 Advanced considerations

A sentence that is composed of two other sentences joined by "iff" is called a biconditionalIn logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent and q is a conclusion (or consequent . The operator is denoted using a doublehea. Iff joins two sentences to form a new sentence. It should not be confused with logical equivalenceIn logic, statements p and q are logically equivalent if they have the same logical content. Syntactically, p and q are equivalent if each can be proved from the other. Semantically, p and q are equivalent if they have the same truth value in every model. 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. QuineWillard Van Orman Quine ( June 25, 1908 December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. Overview Sometimes referred to as the "philosopher's philosopher", Quine is the quintessential model of an's Mathematical Logic, Section 5.

In philosophy and logic, "iff" is used to indicate definitionFor alternative meanings see definition (disambiguation A definition may be a statement of the essential properties of a certain thing, or a statement of equivalence between a term and that term's meaning. The two are not mutually exclusive, nor are theys, since definitions are supposed to be universally quantifiedLogic In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything or every relevant thing. The resulting statement is a universally quantified statement, and we have univer 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).