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), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning.
In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q.
[3] Some authors regard "iff" as unsuitable in formal writing;[4] others consider it a "borderline case" and tolerate its use.
Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly.
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.
However, in the preface of General Topology, Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and euphony demands something less I use Halmos' 'iff'".
[16] However, this usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if".
The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention of interpreting "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover").
Euler diagrams show logical relationships among events, properties, and so forth.
Wherever logic is applied, especially in mathematical discussions, 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.
In their Artificial Intelligence: A Modern Approach, Russell and Norvig note (page 282),[18] in effect, that it is often more natural to express if and only if as if together with a "database (or logic programming) semantics".
They give the example of the English sentence "Richard has two brothers, Geoffrey and John".
The database semantics is analogous to the legal principle expressio unius est exclusio alterius (the express mention of one thing excludes all others).