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In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
and related fields such as
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
and
philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such questio ...

philosophy
, "if and only if" (shortened as "iff") is a
biconditional In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
logical connective In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
between statements, where either both statements are true or both are false. The connective is
biconditional In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
(a statement of material equivalence), and can be likened to the standard
material conditional The material conditional (also known as material implication) is an binary operator, operation commonly used in mathematical logic, logic. When the conditional symbol \rightarrow is semantics of logic, interpreted as material implication, a fo ...
("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 Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
for P'', ''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 \leftrightarrow and \Leftrightarrow, are used instead of these phrases; see below.


Definition

The
truth table A truth table is a mathematical table Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy ...

truth table
of ''P'' \Leftrightarrow ''Q'' is as follows: It is equivalent to that produced by the XNOR gate, and opposite to that produced by the
XOR gate XOR gate (sometimes EOR, or EXOR and pronounced as Exclusive OR) is a digital logic gate A logic gate is an idealized model of computation or physical electronic device implementing a Boolean function, a logical operation performed on one ...

XOR gate
.


Usage


Notation

The corresponding logical symbols are "↔", "\Leftrightarrow", 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 formal sys ...
(particularly those on
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantificat ...
, rather than
propositional logic Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, ...
) 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 is the study of the metatheory A metatheory or meta-theory is a theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational th ...
). In 's
Polish notation Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast t ...
, it is the prefix symbol 'E'. Another term for this
logical connective In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
is
exclusive nor Logical equality is a logical operator that corresponds to equality in Boolean algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struct ...
. In
TeX TeX (, see below), stylized within the system as TeX, is a typesetting system which was designed and mostly written by Donald Knuth and released in 1978. TeX is a popular means of typesetting complex mathematical formulae; it has been noted ...
, "if and only if" is shown as a long double arrow: \iff via command \iff.


Proofs

In most
logical system A formal system is 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 formal system is essentiall ...
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 In logic, disjunction is a logical connective typically notated \lor whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic, it is given a truth functional semantics of logic, sema ...
"(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-function In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
al, "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 Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a HungarianHungarian may refer to: * Hungary, a country in Central Europe * Kingdom of Hungary, state of Hungary, existing between 1000 and 1946 * Hungarians, ethnic g ...
, 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 Phonaesthetics (also spelled phonesthetics in North America North America is a continent entirely within the Northern Hemisphere and almost all within the Western Hemisphere. It can also be described as the northern subcontinent of the A ...

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 always "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" is relatively uncommon, as the majority of textbooks, research papers and articles (including English Wikipedia articles) follow the special 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 diagram An Euler diagram (, ) is a diagrammatic means of representing Set (mathematics), sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramm ...

Euler diagram
s 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
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 Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
for the other. This is an example of
mathematical jargon The language of mathematics has a vast vocabulary A vocabulary, also known as a wordstock or word-stock, is a set of familiar words within a person's language. A vocabulary, usually developed with age, serves as a useful and fundamental tool ...
(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 In the formal sciences Formal science is a branch of science studying formal language disciplines concerned with formal system A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used f ...
, ''z'' is in ''X'' if and only if ''z'' is in ''Y''."


See also

*
Equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
*
Logical biconditional In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as the ...
*
Logical equality Logical equality is a logical operator that corresponds to equality in Boolean algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struct ...
*
Logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same truth value in every model (logic), model. The logical equivalence of p and q is sometimes ...
*
Polysyllogism A polysyllogism (also called multi-premise syllogism, sorites, climax, or gradatio) is a string of any number of proposition In logic and linguistics, a proposition is the meaning of a declarative sentence (linguistics), sentence. In philosophy, ...


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