If and only if

If and only if

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In logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 and related fields such as mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

, if and only if (shortened iff) is a biconditional logical connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

 between statements.

In that it is biconditional, the connective can be likened to the standard material conditional
Material conditional
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a 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
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...

 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.

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

Definition


The truth table
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...

 of p ↔ q is as follows:
Iff
p q
T T T
T F F
F T F
F F T


Note that it is equivalent to that produced by the XNOR gate
XNOR gate
The XNOR gate is a digital logic gate whose function is the inverse of the exclusive OR gate. The two-input version implements logical equality, behaving according to the truth table to the right. A HIGH output results if both of the inputs to the gate are the same...

, and opposite to that produced by the XOR gate
XOR gate
The XOR gate is a digital logic gate that implements an exclusive or; that is, a true output results if one, and only one, of the inputs to the gate is true . If both inputs are false or both are true , a false output results. Its behavior is summarized in the truth table shown on the right...

.

Notation


The corresponding logical symbols are "↔", "⇔" and "≡", and sometimes "iff". These are usually treated as equivalent. However, some texts of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

 (particularly those on first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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
Metalogic
Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves...

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

Another term for this logical connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...

 is exclusive nor.

Proofs


In most logical systems, one proves
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...

 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
John L. Kelley
John Leroy Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis....

's 1955 book General Topology.
Its invention is often credited to Paul Halmos
Paul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...

, who wrote "I invented 'iff,' for 'if and only if'—but I could never believe I was really its first inventor."

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".

Philosophical interpretation


A sentence that is composed of two other sentences joined by "iff" is called a biconditional
Logical biconditional
In logic and mathematics, the logical biconditional is the logical connective of two statements asserting "p if and only if q", where q is a hypothesis and p is a conclusion...

. "Iff" joins two sentences to form a new sentence. It should not be confused with logical equivalence
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...

 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 definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

s, since definitions are supposed to be universally quantified
Universal quantification
In predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....

 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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 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
    Subset
    In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

     containing n elements of an n-dimensional vector space
    Vector space
    A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

     is linearly independent
    Linear independence
    In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...

     iff it spans
    Linear span
    In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

     the vector space.
  • The triangular number
    Triangular number
    A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

     n(n+1)/2 is an even perfect number
    Perfect number
    In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

     iff n = 2p-1 is a Mersenne prime
    Mersenne prime
    In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

    , with p being a prime number
    Prime number
    A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

    . 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
Exclusive disjunction
The logical operation exclusive disjunction, also called exclusive or , is a type of logical disjunction on two operands that results in a value of true if exactly one of the operands has a value of true...

).

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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 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
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...

 for the other. This is an example of mathematical jargon
Mathematical jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal...

. (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
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...

, z is in X if and only if z is in Y."

In other languages


Following a similar structure, in French
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

 iff is written "ssi" ('si et seulement si').
In Spanish
Spanish language
Spanish , also known as Castilian , is a Romance language in the Ibero-Romance group that evolved from several languages and dialects in central-northern Iberia around the 9th century and gradually spread with the expansion of the Kingdom of Castile into central and southern Iberia during the...

 it is also written "ssi" ('si y sólo si'). In Portuguese
Portuguese language
Portuguese is a Romance language that arose in the medieval Kingdom of Galicia, nowadays Galicia and Northern Portugal. The southern part of the Kingdom of Galicia became independent as the County of Portugal in 1095...

 it is written "sse" ('se e somente se'). In German it is written “gdw.” ('genau dann, wenn').

See also


  • Logical biconditional
    Logical biconditional
    In logic and mathematics, the logical biconditional is the logical connective of two statements asserting "p if and only if q", where q is a hypothesis and p is a conclusion...

  • Logical equality
    Logical equality
    Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus...

  • Necessary and sufficient condition