If and only if: Difference between revisions

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In logic and technical fields that depend on it, iff is used for "if and only if". It is often, not always, written italicized: iff. The abbreviation appeared in print for the first time in Kelley's 1955 book "General Topology" and was apparently invented by the mathematician Paul Halmos. The corresponding logical symbols are ↔ and ⇔.

A statement that is composed of two other statements joined by 'iff' is called a biconditional. Here are some examples of true statements that use "iff"--true biconditionals:

That person is a bachelor iff that person is an unmarried man.
'Snow is white' (in English) is true iff 'schnee ist weiss' (in German) is true.
For any p, q, 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').

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 *That person is a bachelor ''iff'' that person is an unmarried man.
 *'Snow is white' (in English) is true ''iff'' '<i>schnee ist weiss</i>' (in German) is true.
-*For any p, q, r: (p & q) & r iff p & (q & r).  (Since this is written using variables and '&', the statement would usually be written using '&harr;', or one of the other symbols used to write biconditionals, in place of 'iff').
+*For any p, q, r: [(p & q) & r] iff [p & (q & r)].  (Since this is written using variables and '&', the statement would usually be written using '&harr;', or one of the other symbols used to write biconditionals, in place of 'iff').