If and only if


If and only if

↔ ⇔ ≡

Logical symbols
representing iff.

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 ("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. Of course, there is nothing to stop us 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 "if and only if" 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. Many authors regard "iff" as unsuitable in formal writing; others use it freely.[citation needed]

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

Contents

Definition

The truth table of p ↔ q is as follows:[1]

Iff
p q
pq
T T T
T F F
F T F
F F T

Note that it is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.

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). In Łukasiewicz's notation, it is the prefix symbol 'E'.

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". 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 iff

Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[2] 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."[3]

Distinction from "if" and "only if"

  1. "If the pudding is a custard, then Madison will eat it." or "Madison will eat the pudding if it is a custard." (equivalent to "Only if Madison will eat the pudding, is it 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.
  2. "Only if the pudding is a custard, will Madison eat it." or "Madison will eat the pudding only if it is a custard." (equivalent to "If Madison will eat the pudding, then 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.
  3. "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 to eat the pudding.

Sufficiency is the inverse of necessity. That is to say, given PQ (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 PQ, 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 PQ, 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 PQ, where P is "the pudding in question is a custard" and Q is "Madison will eat 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 which A and B describe. By contrast "A is logically equivalent to B" mentions both sentences: it describes a logical relation between those two sentences, and not a factual relation between whatever matters they describe. See use–mention distinction for more on the difference between using a sentence and mentioning it.

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.

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.
  • A subset containing n elements of an n-dimensional vector space is linearly independent iff it spans the vector space.
  • The triangular number n(n+1)/2 is an even perfect number iff n = 2p-1 is a Mersenne prime, with p being a prime number. As of the year 2011 only 47 such even perfect numbers and Mersenne primes have been discovered.

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."

In other languages

Following a similar structure, in French iff is written "ssi" ('si et seulement si'). In Spanish it is also written "ssi" ('si y sólo si'). In Portuguese it is written "sse" ('se e somente se').

Notes

  1. ^ p <=> q. Wolfram|Alpha
  2. ^ General Topology, reissue ISBN 9780387901251
  3. ^ Nicholas J. Higham (1998). Handbook of writing for the mathematical sciences (2nd ed.). SIAM. p. 24. ISBN 9780898714203. http://books.google.com/books?id=9gQd2fJA7Y4C&pg=PA24. 

See also

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