In
logic and related fields such as
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
, "if and only if" (shortened as "iff") is a
biconditional logical connective between statements, where either both statements are true or both are false.
The connective is
biconditional (a statement of material equivalence), and 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), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is false.
In writing, phrases commonly used as alternatives to P "if and only if" Q include: ''Q is
necessary and sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
for P'', ''for P it is necessary and sufficient that Q'', ''P is equivalent (or materially equivalent) to Q'' (compare with
material implication), ''P precisely if Q'', ''P precisely (or exactly) when Q'', ''P exactly in case Q'', and ''P just in case Q''. Some authors regard "iff" as unsuitable in formal writing; others consider it a "borderline case" and tolerate its use.
In
logical formulae, logical symbols, such as
and
,
are used instead of these phrases; see below.
Definition
The
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
of ''P''
''Q'' is as follows:
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
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
(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
Polish notation, it is the prefix symbol 'E'.
Another term for the
logical connective, i.e., the symbol in logic formulas, is
exclusive nor.
In
TeX, "if and only if" is shown as a long double arrow:
via command \iff.
Proofs
In most
logical system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A for ...
s, one
proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these 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 been shown to be both true, or both false.
Origin of iff and pronunciation
Usage of the abbreviation "iff" first appeared in print in
John L. Kelley's 1955 book ''General Topology''.
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."
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: "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 .
Usage in definitions
Technically, definitions are "if and only if" statements; some texts — such as Kelley's ''General Topology'' — follow the strict demands of logic, and use "if and only if" or ''iff'' in definitions of new terms.
However, this logically correct usage of "if and only if" relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention to interpret "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover").
Distinction from "if" and "only if"
* ''"Madison will eat the fruit
if it is an apple."'' (equivalent to ''"
Only if Madison will eat the fruit, can it be an apple"'' or ''"Madison will eat the fruit ''←'' the fruit is an apple"'')
*: This states that Madison will eat fruits that are apples. It does not, however, exclude the possibility that Madison might also eat bananas or other types of fruit. All that is known for certain is that she will eat any and all apples that she happens upon. That the fruit is an apple is a ''sufficient'' condition for Madison to eat the fruit.
* ''"Madison will eat the fruit
only if it is an apple."'' (equivalent to ''"
If Madison will eat the fruit, then it is an apple"'' or ''"Madison will eat the fruit ''→'' the fruit is an apple"'')
*: This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. In this case, that a given fruit is an apple is a ''necessary'' condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat all the apples she is given.
* ''"Madison will eat the fruit
if and only if it is an apple."'' (equivalent to ''"Madison will eat the fruit ''↔'' the fruit is an apple"'')
*: This statement makes it clear that Madison will eat all and only those fruits that are apples. She will not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a ''necessary'' and a ''sufficient'' condition for Madison to eat the fruit.
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:
:''P'' is sufficient for ''Q''
:''Q'' is necessary for ''P''
:''¬Q'' is sufficient for ''¬P''
:''¬P'' is necessary for ''¬Q''
As an example, take the first example 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:
:If the fruit in question is an apple, then Madison will eat it.
:Only if Madison will eat the fruit in question, is it an apple.
:If Madison will not eat the fruit in question, then it is not an apple.
:Only if the fruit in question is not an apple, will Madison not eat it.
Here, the second example 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 the first example, we find that the third example 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".
In terms of Euler diagrams
File:Example of A is a proper subset of B.svg, ''A'' is a proper subset of ''B''. A number is in ''A'' only if it is in ''B''; a number is in ''B'' if it is in ''A''.
File:Example of C is no proper subset of B.svg, ''C'' is a subset but not a proper subset of ''B''. A number is in ''B'' if and only if it is in ''C'', and a number is in ''C'' if and only if it is in ''B''.
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.
More general usage
Iff is used outside the field of logic as well. 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
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
for the other. This is an example of
mathematical jargon (although, as noted above, ''if'' is more often used than ''iff'' in statements of definition).
The elements of ''X'' are ''all and only'' the elements of ''Y'' means: "For any ''z'' in the
domain of discourse, ''z'' is in ''X'' if and only if ''z'' is in ''Y''."
See also
*
Equivalence relation
*
Logical biconditional
*
Logical equality
*
Logical equivalence
*
Polysyllogism
A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of any number of propositions forming together a sequence of syllogisms such that the conclusion of each syllogism, together with the next proposition, ...
References
External links
*
Language Log: "Just in Case"Southern California Philosophy for philosophy graduate students: "Just in Case"
{{Common logical symbols
Logical connectives
Mathematical terminology
Necessity and sufficiency