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Encyclopedia > If and only if

logical symbols
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. Image File history File links Merge-arrow. ... In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... Image File history File links Question_book-3. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Philosophy (disambiguation). ... In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ...

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. This article discusses only the formal meanings of necessary and sufficient causal meanings see causation. ...

The statement "(P iff Q)" is equivalent to the statement "not (P xor Q)" or "P == Q" in computer science. Exclusive disjunction (usual symbol xor) is a logical operator that results in true if one of the operands (not both) is true. ...

In logic formulas, logical symbols are used instead of these phrases; see the discussion of notation. In mathematical logic, a formula is a formal syntactic object that expresses a proposition. ...

The truth table of p iff q (also written as p ↔ q) is as follows: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

 p q p ↔ q T T T T F F F T F F F T

## 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). Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... The metalogic of a system of logic is the formal proof supporting its soundness. ...

Another term for this logical connective is exclusive nor. In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... XNOR Logic Gate Symbol Exclusive nor (usual symbol XNOR occasionally XAND <exclusive and>) is a logical operator in Boolean algebra. ...

### 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 Q, then P", 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. Logic, from Classical Greek Î»ÏŒÎ³Î¿Ï‚ (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Contraposition is the concept of how two qualities or statements relate to each other. ... Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... In logic a truth function is a function generated from sentences of the language. ...

### 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, but in his autobiography he states that he borrowed it from puzzlers. John Leroy Kelley (December 6, 1916 â€“ November 26, 1999) was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis. ... Year 1955 (MCMLV) was a common year starting on Saturday (link displays the 1955 Gregorian calendar). ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Paul Halmos Paul Richard Halmos (March 3, 1916 â€” October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ... Cover of the first English edition of 1793 of Benjamin Franklins autobiography. ... A puzzle undone, which forms a cube Puzzle cube; a type of puzzle For other uses, see Puzzle (disambiguation). ...

## The difference between if, only if, and iff

### Examples

1. Madison will eat pudding if the pudding is a custard. (equivalently: If the pudding is a custard, then Madison will eat it)
2. Madison will eat pudding only if the pudding is a custard. (equivalently: If Madison is eating pudding, then it must be a custard)
3. 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.)

### 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.

### 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. In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... In logic, statements p and q are logically equivalent if they have the same logical content. ...

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. W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...

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. (Some authors, nevertheless, explicitly indicate that the "if" of a definition means "iff"!) For other uses, see Definition (disambiguation). ... In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing. ... This picture illustrates how the hours on a clock form a group under modular addition. ...

### 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.

Logical conjunction (usual symbol and) is a logical operator that results in true if both of the operands are true. ...

### 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, also known as exclusive or and symbolized by XOR or EOR, is a logical operation on two operands that results in a logical value of true if and only if one of the operands, but not both, 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)."

## 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.) For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... This article discusses only the formal meanings of necessary and sufficient causal meanings see causation. ... The field of mathematics has a vast vocabulary of specialist and technical terms. ...

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." The domain of discourse, sometimes called the universe of discourse, is an analytic tool used in deductive logic, especially predicate logic. ...

XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ... In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... Results from FactBites:

 If and only if - definition of If and only if in Encyclopedia (714 words) In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for "if and only if". However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the former is used as a symbol in logic formulas, while the latter -- in reasoning about those formulas (e.g., in metalogic). 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.
More results at FactBites »

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