In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("if") combined with its reverse ("only if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other, i.e., either both statements are tr...
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In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional ("if") combined with its reverse ("only if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other, i.e., either both statements are true, or both are false. The connective is thus an "if" that works both ways.
In writing, common alternative phrases to "if and only if", Q is necessary and sufficient for P, P is equivalent (or materially equivalent) to Q (compare material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Many authors regard "iff" as unsuitable in formal writing; others use it freely.
In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.
The truth table of...
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