⇔
≡
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. Logic is the study of the principles of valid demonstration and Inference. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence 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.
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.
The statement "(P iff Q)" is equivalent to the statement "not (P xor Q)" or "P == Q" in computer science.
In logic formulas, logical symbols are used instead of these phrases; see the discussion of notation. In Mathematical logic, a formula is a type of Abstract object a token of which is a Symbol or string of symbols which may be
Contents |
Definition
The truth table of p iff q (also written as p ↔ q) is as follows:
| 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". A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional 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. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" g. , in 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
Another term for this logical connective is exclusive nor. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence Logical equality is a Logical operator that corresponds to equality in Boolean algebra and to the Logical biconditional in Propositional
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 P, then Q", i. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques For contraposition in the field of traditional logic see Contraposition (traditional logic. 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. Truth functional' redirects here for the truth functional conditional see Material conditional.
Origin of the abbreviation
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology. John Leroy Kelley ( December 6 1916, Kansas – November 26 1999, Oakland California) was an American mathematician at Year 1955 ( MCMLV) was a Common year starting on Saturday (link displays the 1955 Gregorian calendar) Its invention is often credited to the mathematician Paul Halmos. A mathematician is a person whose primary area of study and research is the field of Mathematics. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician
The difference between if, only if, and iff
Examples
- Madison will eat pudding if the pudding is a custard. (equivalently: If the pudding is a custard, then Madison will eat it)
- Madison will eat pudding only if the pudding is a custard. (equivalently: If Madison is eating pudding, then it must be a custard)
- 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.
Advanced considerations
Philosophical interpretation
A sentence that is composed of two other sentences joined by "iff" is called a biconditional. In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements "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. In Logic, statements p and q are logically equivalent if they have the same logical content 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.
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. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van"
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. A definition is a statement of the meaning of a Word or Phrase. In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every 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. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element (Some authors, nevertheless, explicitly indicate that the "if" of a definition means "iff"!)
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"). In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of
- For any real numbers x and y, x=y+1 iff y=x−1.
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). "
More general usage
Iff is used outside the field of logic, wherever logic is applied, especially in mathematical discussions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. The Language of mathematics has a vast Vocabulary of specialist and technical terms (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. The domain of discourse, sometimes called the universe of discourse, logical discourse, or simply discourse, is an analytic tool used in Deductive "
See also
Logical equality is a Logical operator that corresponds to equality in Boolean algebra and to the Logical biconditional in Propositional In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statementsDictionary
if and only if
-conjunction
- (logic) Is equivalent to; implies and is implied by; is true and false in the same cases as.
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