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 corresponding logical symbols are ↔ and ⇔. The abbreviation appeared in print for the first time in 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 the puzzles community. A statement that is composed of two other statements joined by 'iff' is called a biconditional. In philosophy and logic, for example, 'iff' is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals.

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 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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-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''.
+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 corresponding logical symbols are &harr; and &hArr;.
 The abbreviation appeared in print for the first time in 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 the puzzles community.
+A statement that is composed of two other statements joined by 'iff' is called a [[biconditional]].
+In philosophy and logic, for example, 'iff' is used to indicate [[definition]]s, since definitions are supposed to be [[universal quantification|universally quantified]] biconditionals.
+Here are some examples of true statements that use 'iff'--true biconditionals.  The first is an example of a definition:
-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.
+*A 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').