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 (a statement of material equivalence),^{[1]} the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("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). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning. There is nothing to stop one from stipulating that we may read this connective as "only if and if", although this may lead to confusion.
In writing, phrases commonly used, with debatable propriety, as alternatives to P "if and only if" Q include 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.^{[2]} Some authors regard "iff" as unsuitable in formal writing;^{[3]} others use it freely.^{[4]}
In logical formulae, logical symbols are used instead of these phrases; see the discussion of notation.
The truth table of P Q is as follows:^{[5]}^{[6]}
P | Q | |||
---|---|---|---|---|
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | F |
F | F | T | T | T |
It is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.
The corresponding logical symbols are "", "", and "?", and sometimes "iff". 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 first, , is used as a symbol in logic formulas, while is used in reasoning about those logic formulas (e.g., in metalogic). In ?ukasiewicz's notation, it is the prefix symbol 'E'.
Another term for this logical connective is exclusive nor.
In most logical systems, one proves a statement of the form "P iff Q" by proving "if P, then Q" and "if Q, then P". Proving this pair of statements sometimes leads to a more natural proof since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts--that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have both been shown true, or both false.
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.^{[7]} Its invention is often credited to Paul Halmos, who wrote "I invented 'iff,' for 'if and only if'--but I could never believe I was really its first inventor."^{[8]}
It is somewhat unclear how "iff" was meant to be pronounced. In current practice, the single 'word' "iff" is almost always read as the four words "if and only if." 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'". The authors of one discrete mathematics textbook suggest:^{[9]} "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if'", implying that "iff" could be pronounced as /?f:/.
Technically, definitions are always "if and only if" statements and many texts like Kelley's General Topology follow the strict demands of logic and use "if and only if" or iff in definitions of new terms (for instance, from General Topology, p. 25: "A set is countable iff it is finite or countably infinite" [boldface in original]). However, this usage of "if and only if" is not universal and often, mathematical definitions follow the special convention that "if" is interpreted to mean "if and only if" (for example, one might say, "A topological space is compact if every open cover has a finite subcover").^{[10]}
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Sufficiency is the converse of necessity. That is to say, given P->Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P->Q, it is true that ¬Q->¬P (where ¬ is the negation operator, i.e. "not"). This means that the relationship between P and Q, established by P->Q, can be expressed in the following, all equivalent, ways:
As an example, take (1), above, which states P->Q, where P is "the fruit in question is an apple" and Q is "Madison will eat the fruit in question". The following are four equivalent ways of expressing this very relationship:
So we see that (2), above, can be restated in the form of if...then as "If Madison will eat the fruit in question, then it is an apple"; taking this in conjunction with (1), we find that (3) can be stated as "If the fruit in question is an apple, then Madison will eat it; and if Madison will eat the fruit, then it is an apple".
Euler diagrams show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and "P->Q" all mean that P is a subset, either proper or improper, of Q. "P if Q", "if Q then P", and Q->P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.
Iff is used outside the field of logic. 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. This is an example of mathematical jargon. (However, as noted above, if, rather than iff, is more often used in statements of definition.)
The elements of X are all and only the elements of Y is used to mean: "for any z in the domain of discourse, z is in X if and only if z is in Y."
While it can be a real time-saver, we don't recommend it in formal writing.
It is common in mathematical writing.
Theorems which have the form "P if and only Q" are much prized in mathematics. They give what are called "necessary and sufficient" conditions, and give completely equivalent and hopefully interesting new ways to say exactly the same thing..