If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, 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" include iff, 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.^{[citation needed]}

In logic formulas, logical symbols are used instead of these phrases; see the discussion of notation.

Definition

The truth table of p iff q (also written as p ↔ q) is as follows:

**Iff**
p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

Usage

Notation

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).

Another term for this logical connective is exclusive nor.

Proofs

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" (or the inverse of "if P, then Q", i.e. "if not P, then not Q"). 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.

Origin of the abbreviation

Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology. Its invention is often credited to the mathematician Paul Halmos.

The difference between if, only if, and iff

"If the pudding is a custard, then Madison will eat it." or "Madison will eat the pudding if it is a custard."
- This states only that Madison will eat custard pudding. It does not, however, preclude the possibility that Madison might also have occasion to eat bread pudding. Maybe she will, maybe she will not—the sentence does not tell us. All we know for certain is that she will eat any and all custard pudding that she happens upon. That the pudding is a custard is a sufficient condition for Madison to eat the pudding.
"Only if the pudding is a custard, will Madison eat it." or "Madison will eat the pudding only if it is a custard."
- This states that the only pudding Madison will eat is a custard. It does not, however, preclude the possibility that Madison will refuse a custard if it is made available, in contrast with (1), which requires Madison to eat any available custard. In this case, that a given pudding is a custard is a necessary condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat any and all custard puddings she is given.
"If and only if the pudding is a custard, will Madison eat it." or "Madison will eat the pudding if and only if it is a custard."
- This, however, makes it quite clear that Madison will eat all and only those puddings that are custard. She will not leave any such pudding uneaten, and she will not eat any other type of pudding. That a given pudding is custard is both a necessary and a sufficient condition for Madison eating the pudding.

Sufficiency is the inverse 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:

P is sufficient for Q

Q is necessary for P

¬Q is sufficient for ¬P

¬P is necessary for ¬Q

As an example, take (1), above, which states P→Q, where P is "the pudding in question is a custard" and Q is "Madison eats the pudding in question". The following are four equivalent ways of expressing this very relationship:

If the pudding in question is a custard, then Madison will eat it.

Only if Madison will eat the pudding in question, is it a custard.

If Madison will not eat the pudding in question, then it is not a custard.

Only if the pudding in question is not a custard, will Madison not eat it.

So we see that (2), above, can be restated in the form of if...then as "If Madison will eat the pudding in question, then it is a custard"; taking this in conjunction with (1), we find that (3) can be stated as "If the pudding in question is a custard, then Madison will eat it; AND if Madison will eat the pudding, then it is a custard".

Advanced considerations

Philosophical interpretation

A sentence that is composed of two other sentences joined by "iff" is called a biconditional. "Iff" joins two sentences to form a new sentence. It should not be confused with logical equivalence which is a description of a relation between two sentences. The biconditional "A iff B" uses the sentences A and B, describing a relation between the states of affairs A and B describe. By contrast "A is logically equivalent to B" mentions both sentences: it describes a relation between those two sentences, and not between whatever matters they describe.

The distinction is a very confusing one, and has led many a philosopher astray. Certainly it is the case that when A is logically equivalent to B, "A iff B" is true. But the converse does not hold. Reconsidering the sentence:

If and only if the pudding is a custard, will Madison eat it.

There is clearly no logical equivalence between the two halves of this particular biconditional. For more on the distinction, see W. V. Quine's Mathematical Logic, Section 5.

One way of looking at "A if and only if B" is that it means "A if B" (B implies A) and "A only when B" (not B implies not A). "Not B implies not A" means A implies B, so then we get two way implication.

Definitions

In philosophy and logic, "iff" is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals. In mathematics and elsewhere, however, the word "if" is normally used in definitions, rather than "iff". This is due to the observation that "if" in the English language has a definitional meaning, separate from its meaning as a propositional conjunction. This separate meaning can be explained by noting that a definition (for instance: A group is "abelian" if it satisfies the commutative law; or: A grape is a "raisin" if it is well dried) is not an equivalence to be proved, but a rule for interpreting the term defined. (Some authors,^[1] nevertheless, explicitly indicate that the "if" of a definition means "iff"!)

Examples

Here are some examples of true statements that use "iff" - true biconditionals (the first is an example of a definition, so it should normally have been written with "if"):

A person is a bachelor iff that person is a marriageable man who has never married.
"Snow is white" (in English) is true iff "Schnee ist weiß" (in German) is true.
For any p, q, and 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").
For any real numbers x and y, x=y+1 iff y=x−1.

Analogs

Other words are also sometimes emphasized in the same way by repeating the last letter; for example orr for "Or and only Or" (the exclusive disjunction).

The statement "(A iff B)" is equivalent to the statement "(not A or B) and (not B or A)," and is also equivalent to the statement "(not A and not B) or (A and B)".

It is also equivalent to: not[(A or B) and (not A or not B)],

or more simply:

¬ [ ( ¬A ∨ ¬B ) ∧ ( A ∨ B ) ]

which converts into

[ ( ¬A ∧ ¬B) ∨ (A ∧ B) ]

and

[ ( ¬A ∨ B) ∧ (A ∨ ¬B) ]

which were given in verbal interpretations above.

More general usage

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."

Notes

^ Such as Artin, Michael (1991), Algebra, p. 585.