If and only if
From Wikipedia, the free encyclopedia
"Iff" redirects here. For other uses, see IFF (disambiguation).
 | This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (July 2010) |
↔ ⇔ ≡
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.Logical symbols
representing iff.
representing iff.
In that it is biconditional, the connective 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. 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 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.[citation needed]
In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.
Contents[hide] |
[edit] Definition
The truth table of p ↔ q is as follows:[1]| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
[edit] Usage
[edit] 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). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.
[edit] 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.e. "if not Q, then not 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.[edit] Origin of iff
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[2] 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."[3][edit] Distinction from "if" and "only if"
"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.")If and only if
From Wikipedia, the free encyclopedia
"Iff" redirects here. For other uses, see IFF (disambiguation).
| | This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (July 2010) |
↔ ⇔ ≡
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.Logical symbols
representing iff.
representing iff.
In that it is biconditional, the connective 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. 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 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.[citation needed]
In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.
Contents[hide] |
[edit] Definition
The truth table of p ↔ q is as follows:[1]| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
[edit] Usage
[edit] 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). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.
[edit] 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.e. "if not Q, then not 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.[edit] Origin of iff
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[2] 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."[3][edit] Distinction from "if" and "only if"
"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.")If and only if
From Wikipedia, the free encyclopedia
"Iff" redirects here. For other uses, see IFF (disambiguation).
| | This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (July 2010) |
↔ ⇔ ≡
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.Logical symbols
representing iff.
representing iff.
In that it is biconditional, the connective 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. 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 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.[citation needed]
In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.
Contents[hide] |
[edit] Definition
The truth table of p ↔ q is as follows:[1]| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
[edit] Usage
[edit] 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). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.
[edit] 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.e. "if not Q, then not 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.[edit] Origin of iff
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[2] 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."[3][edit] Distinction from "if" and "only if"
"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.")If and only if
From Wikipedia, the free encyclopedia
"Iff" redirects here. For other uses, see IFF (disambiguation).
| | This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (July 2010) |
↔ ⇔ ≡
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.Logical symbols
representing iff.
representing iff.
In that it is biconditional, the connective 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. 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 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.[citation needed]
In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.
Contents[hide] |
[edit] Definition
The truth table of p ↔ q is as follows:[1]| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
[edit] Usage
[edit] 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). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.
[edit] 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.e. "if not Q, then not 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.[edit] Origin of iff
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[2] 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."[3][edit] Distinction from "if" and "only if"
"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.")If and only if
From Wikipedia, the free encyclopedia
"Iff" redirects here. For other uses, see IFF (disambiguation).
| | This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (July 2010) |
↔ ⇔ ≡
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.Logical symbols
representing iff.
representing iff.
In that it is biconditional, the connective 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. 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 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.[citation needed]
In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.
Contents[hide] |
[edit] Definition
The truth table of p ↔ q is as follows:[1]| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
[edit] Usage
[edit] 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). In Łukasiewicz's notation, it is the prefix symbol 'E'.Another term for this logical connective is exclusive nor.
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