If and only if
From Wikipedia, the free encyclopedia
↔ ⇔ ≡
Logical symbols
representing iff.
In
logic and related fields such as
mathematics and
philosophy,
if and only if (shortened
iff) is a
biconditional logical connective between statements.
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.
[edit] Definition
The
truth table of
p ↔ q is as follows:
[1]
Iff
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Note that it is equivalent to that produced by the
XNOR gate, and opposite to that produced by the
XOR gate.
[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
↔ ⇔ ≡
Logical symbols
representing iff.
In
logic and related fields such as
mathematics and
philosophy,
if and only if (shortened
iff) is a
biconditional logical connective between statements.
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.
[edit] Definition
The
truth table of
p ↔ q is as follows:
[1]
Iff
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Note that it is equivalent to that produced by the
XNOR gate, and opposite to that produced by the
XOR gate.
[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
↔ ⇔ ≡
Logical symbols
representing iff.
In
logic and related fields such as
mathematics and
philosophy,
if and only if (shortened
iff) is a
biconditional logical connective between statements.
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.
[edit] Definition
The
truth table of
p ↔ q is as follows:
[1]
Iff
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Note that it is equivalent to that produced by the
XNOR gate, and opposite to that produced by the
XOR gate.
[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
↔ ⇔ ≡
Logical symbols
representing iff.
In
logic and related fields such as
mathematics and
philosophy,
if and only if (shortened
iff) is a
biconditional logical connective between statements.
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.
[edit] Definition
The
truth table of
p ↔ q is as follows:
[1]
Iff
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Note that it is equivalent to that produced by the
XNOR gate, and opposite to that produced by the
XOR gate.
[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
↔ ⇔ ≡
Logical symbols
representing iff.
In
logic and related fields such as
mathematics and
philosophy,
if and only if (shortened
iff) is a
biconditional logical connective between statements.
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.
[edit] Definition
The
truth table of
p ↔ q is as follows:
[1]
Iff
| p | q | p ↔ q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Note that it is equivalent to that produced by the
XNOR gate, and opposite to that produced by the
XOR gate.
[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.")