If and only if
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⇔
≡
representing iff.
If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a logical connective between statements which means that the truth of either one of the statements requires the truth of the other. Thus, either both statements are true, or both are false. To put it another way, the first statement will always be true when the second statement is, and will only be true under those conditions.
In writing, common alternative phrases to "if and only if" include iff, Q is necessary and sufficient for P, P is equivalent to Q, 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.
The statement "(P iff Q)" is equivalent to the statement "not (P xor Q)" or "P == Q" in computer science.
In logic formulas, logical symbols are used instead of these phrases; see the discussion of notation.
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[edit] Definition
The truth table of p iff q (also written as p ↔ q) is as follows:
| p | q |
p ↔ q
|
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
[edit] Usage
[edit] 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.
[edit] 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.
[edit] 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.
[edit] The difference between if, only if, and iff
[edit] Examples
- Madison will eat pudding if the pudding is a custard. (equivalently: If the pudding is a custard, then Madison will eat it)
- Madison will eat pudding only if the pudding is a custard. (equivalently: If Madison is eating pudding, then it must be a custard)
- Madison will eat pudding if and only if (iff) the pudding is a custard. (equivalently: If the pudding is a custard, then Madison will eat it. AND If Madison is eating pudding, then it must be a custard.)
[edit] Analysis
Sentence (1) 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 custard pudding.
Sentence (2) 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 sentence (1), which requires Madison to eat any available custard.
Sentence (3), however, makes it quite clear that Madison will eat custard pudding and custard pudding only. She will eat all such puddings, and she will not eat any other type of pudding.
A further difference is that "if" is used in definitions (except in formal logic); see more below.
[edit] Advanced considerations
[edit] 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:
- Madison will eat pudding if and only if it is custard.
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.
[edit] 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, nevertheless, explicitly indicate that the "if" of a definition means "iff"!)
[edit] 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.
[edit] 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)."
[edit] 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."
