General revision throughout the page. Improved inline citations. Wikilinked "first-order logic". Removed "one source" template accordingly. Removed contractions in the article (save within quotes). Put period after full sentences in bullet points. Rephrased sentences to prevent flow disruption. Minor punctuation fixes. Broken down lengthy sentences. Tags: Reverted Visual edit |
Undid revision 976629688 by Miaumee (talk) Per User talk:Miaumee, this is apparently the preferred response to poor editing Tag: Undo |
||
Line 1: | Line 1: | ||
{{one source|date=June 2013}} |
|||
[[File:Venn1001.svg|220px|thumb|[[Venn diagram]] of <math>P \leftrightarrow Q</math><br />(true part in red)]] |
[[File:Venn1001.svg|220px|thumb|[[Venn diagram]] of <math>P \leftrightarrow Q</math><br />(true part in red)]] |
||
In [[logic]] and [[mathematics]], the '''logical biconditional''', sometimes known as the '''material biconditional''', is the [[logical connective]] used to conjoin two statements <math>P</math> and <math>Q</math> to form the statement "<math>P</math> [[if and only if]] <math>Q</math>", where <math>P</math> is known as the ''[[antecedent (logic)|antecedent]]'', and <math>Q</math> the ''[[consequent]]''.<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/#iff|title=The Definitive Glossary of Higher Mathematical Jargon — If and Only If|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-11-25}}</ref><ref name=":1">{{Cite web|url=http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|title=Conditionals and Biconditionals|last=Peil|first=Timothy|date=|website=web.mnstate.edu|url-status=live|archive-url=|archive-date=|access-date=2019-11-25}}</ref><ref>{{Cite book|title=Handbook of Logic|last=Brennan|first=Joseph G.|publisher=Harper & Row|year=1961|isbn=|edition=2nd|location=|pages=81}}</ref> This is often abbreviated as "<math>P</math> iff <math>Q</math>".<ref name=":2">{{Cite web|url=http://mathworld.wolfram.com/Iff.html|title=Iff|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-25}}</ref> The operator is denoted using a doubleheaded arrow (↔ |
In [[logic]] and [[mathematics]], the '''logical biconditional''', sometimes known as the '''material biconditional''', is the [[logical connective]] used to conjoin two statements <math>P</math> and <math>Q</math> to form the statement "<math>P</math> [[if and only if]] <math>Q</math>", where <math>P</math> is known as the ''[[antecedent (logic)|antecedent]]'', and <math>Q</math> the ''[[consequent]]''.<ref name=":0">{{Cite web|url=https://mathvault.ca/math-glossary/#iff|title=The Definitive Glossary of Higher Mathematical Jargon — If and Only If|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-11-25}}</ref><ref name=":1">{{Cite web|url=http://web.mnstate.edu/peil/geometry/Logic/4logic.htm|title=Conditionals and Biconditionals|last=Peil|first=Timothy|date=|website=web.mnstate.edu|url-status=live|archive-url=|archive-date=|access-date=2019-11-25}}</ref><ref>{{Cite book|title=Handbook of Logic|last=Brennan|first=Joseph G.|publisher=Harper & Row|year=1961|isbn=|edition=2nd|location=|pages=81}}</ref> This is often abbreviated as "<math>P</math> iff <math>Q</math>".<ref name=":2">{{Cite web|url=http://mathworld.wolfram.com/Iff.html|title=Iff|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-25}}</ref> The operator is denoted using a doubleheaded arrow (↔<ref>{{Cite web|url=https://www.mathgoodies.com/lessons/vol9/biconditional|title=Biconditional Statements {{!}} Math Goodies|website=www.mathgoodies.com|access-date=2019-11-25}}</ref> or ⇔<ref>{{Cite web|url=https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Book%3A_A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/2%3A_Logic/2.4%3A_Biconditional_Statements|title=2.4: Biconditional Statements|date=2018-04-25|website=Mathematics LibreTexts|language=en|access-date=2019-11-25}}</ref>), a prefixed E "E''pq''" (in [[Polish notation#Polish notation for logic|Łukasiewicz notation]] or [[Józef Maria Bocheński|Bocheński notation]]), an equality sign (=), an equivalence sign (≡)<ref name=":2" />, or ''EQV''. It is logically equivalent to both<math>(P \rightarrow Q) \land (Q \rightarrow P)</math> and <math>(P \land Q) \lor (\neg P \land \neg Q) </math>, and the [[XNOR gate|XNOR]] (exclusive nor) [[Logical connective|boolean operator]], which means "both or neither". |
||
Semantically, the only case where a logical biconditional is different from a [[material conditional]] is the case where the hypothesis is false but the conclusion is true. In which case, the result is true for the conditional, but false for the biconditional.<ref name=":1" /> |
Semantically, the only case where a logical biconditional is different from a [[material conditional]] is the case where the hypothesis is false but the conclusion is true. In which case, the result is true for the conditional, but false for the biconditional.<ref name=":1" /> |
||
In the conceptual interpretation, <math>P = Q</math> means "All <math>P</math>'s are <math>Q</math>'s and all <math>Q</math>'s are <math>P</math>'s". In other words, the sets <math>P</math> and <math>Q</math> coincide: they are identical. However, this does not mean that <math>P</math> and <math>Q</math> need to have the same meaning (e.g., <math>P</math> could be "equiangular trilateral" and <math>Q</math> could be "equilateral triangle"). When phrased as a sentence, the antecedent is the ''subject'' |
In the conceptual interpretation, <math>P = Q</math> means "All <math>P</math>'s are <math>Q</math>'s and all <math>Q</math>'s are <math>P</math>'s". In other words, the sets <math>P</math> and <math>Q</math> coincide: they are identical. However, this does not mean that <math>P</math> and <math>Q</math> need to have the same meaning (e.g., <math>P</math> could be "equiangular trilateral" and <math>Q</math> could be "equilateral triangle"). When phrased as a sentence, the antecedent is the ''subject'' and the consequent is the ''predicate'' of a [[universal affirmative]] proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate). |
||
In the propositional interpretation, <math>P \leftrightarrow Q</math> means that <math>P</math> implies <math>Q</math> and <math>Q</math> implies <math>P</math>; in other words, the propositions are [[logically equivalent]], in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as <math>P</math> could be "the triangle ABC has two equal sides", and <math>Q</math> could be "the triangle ABC has two equal angles". I |
|||
In the propositional interpretation, <math>P \leftrightarrow Q</math> means that <math>P</math> implies <math>Q</math> and <math>Q</math> implies <math>P</math>; in other words, the propositions are [[logically equivalent]], in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as <math>P</math> could be "the triangle ABC has two equal sides" and <math>Q</math> could be "the triangle ABC has two equal angles". In general, the antecedent is the ''premise'', or the ''cause'', and the consequent is the ''consequence''. When an implication is translated by a ''hypothetical'' (or ''conditional'') judgment, the antecedent is called the ''hypothesis'' (or the ''condition'') and the consequent is called the ''thesis''. |
|||
A common way of demonstrating a biconditional of the form <math>P \leftrightarrow Q</math> is to demonstrate that <math>P \rightarrow Q</math> and <math>Q \rightarrow P</math> separately (due to its equivalence to the conjunction of the two converse [[Material conditional|conditional]]s<ref name=":1" />). Yet another way of demonstrating the same biconditional is by demonstrating that <math>P \rightarrow Q</math> and <math>\neg P \rightarrow \neg Q</math>.<ref name=":0" /> |
A common way of demonstrating a biconditional of the form <math>P \leftrightarrow Q</math> is to demonstrate that <math>P \rightarrow Q</math> and <math>Q \rightarrow P</math> separately (due to its equivalence to the conjunction of the two converse [[Material conditional|conditional]]s<ref name=":1" />). Yet another way of demonstrating the same biconditional is by demonstrating that <math>P \rightarrow Q</math> and <math>\neg P \rightarrow \neg Q</math>.<ref name=":0" /> |
||
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a ''theorem'' and the other its ''reciprocal''.{{Citation needed|date=August 2008}} Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the ''hypothesis'' |
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a ''theorem'' and the other its ''reciprocal''.{{Citation needed|date=August 2008}} Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the ''hypothesis'' and whose consequent is the ''thesis'' of the theorem. |
||
It is often said that the hypothesis is the ''[[sufficient condition]]'' of the thesis, and that the thesis is the ''[[necessary condition]]'' of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the [[necessary and sufficient condition]] of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time. |
It is often said that the hypothesis is the ''[[sufficient condition]]'' of the thesis, and that the thesis is the ''[[necessary condition]]'' of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the [[necessary and sufficient condition]] of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time. |
||
Line 22: | Line 22: | ||
===Truth table=== |
===Truth table=== |
||
The following is truth table for <math>P \leftrightarrow Q</math> (also written as <math>P \equiv Q</math>, <math>P = Q</math>, or '''P EQ Q'''): |
The following is truth table for <math>P \leftrightarrow Q</math> (also written as <math>P \equiv Q</math>, <math>P = Q</math>, or '''P EQ Q'''): |
||
{| class="wikitable" style="text-align:center; background-color: #ddffdd;" |
{| class="wikitable" style="text-align:center; background-color: #ddffdd;" |
||
Line 47: | Line 47: | ||
:<math>(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow ...) \leftrightarrow x_n</math>, |
:<math>(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow ...) \leftrightarrow x_n</math>, |
||
or |
or may be interpreted as saying that all <math>x_i</math> are ''jointly true or jointly false'': |
||
:<math>(x_1 \land ... \land x_n) \lor (\neg x_1 \land ... \land \neg x_n)</math> |
:<math>(x_1 \land ... \land x_n) \lor (\neg x_1 \land ... \land \neg x_n)</math> |
||
Line 139: | Line 139: | ||
|} |
|} |
||
'''[[Distributivity]]:''' Biconditional |
'''[[Distributivity]]:''' Biconditional doesn't distribute over any binary function (not even itself), but [[Logical disjunction#Properties|logical disjunction distributes]] over biconditional. |
||
'''[[idempotency]]: No'''<br /> |
'''[[idempotency]]: No'''<br /> |
||
Line 213: | Line 213: | ||
==Rules of inference== |
==Rules of inference== |
||
{{Main|Rules of inference}} |
{{Main|Rules of inference}} |
||
Like all connectives in |
Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs. |
||
===Biconditional introduction=== |
===Biconditional introduction=== |
||
Line 219: | Line 219: | ||
Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A [[if and only if]] B. |
Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A [[if and only if]] B. |
||
For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" |
For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically: |
||
B → A |
B → A |
||
Line 232: | Line 232: | ||
Biconditional elimination allows one to infer a [[Material conditional|conditional]] from a biconditional: if A <small>↔</small> B is true, then one may infer either A <small>→</small> B, or B <small>→</small> A. |
Biconditional elimination allows one to infer a [[Material conditional|conditional]] from a biconditional: if A <small>↔</small> B is true, then one may infer either A <small>→</small> B, or B <small>→</small> A. |
||
For example, if it is true that |
For example, if it is true that I'm breathing [[if and only if]] I'm alive, then it's true that ''if'' I'm breathing, then I'm alive; likewise, it's true that ''if'' I'm alive, then I'm breathing. Or more schematically: |
||
<u>A ↔ B </u> |
<u>A ↔ B </u> |
||
Line 242: | Line 242: | ||
==Colloquial usage== |
==Colloquial usage== |
||
One unambiguous way of stating a biconditional in plain English is to adopt the form "''b'' if ''a'' and ''a'' if ''b''"—if the standard form "''a'' if and only if ''b''" is not used. Slightly more formally, one could also say that "''b'' implies ''a'' and ''a'' implies ''b''", or "''a'' is necessary and sufficient for ''b''".<ref name=":0" /> The plain English "if" may sometimes be used as a biconditional (especially in the context of a mathematical definition<ref>In fact, such is the style adopted by [[wikipedia:Manual of Style/Mathematics|Wikipedia's manual of style in mathematics]].</ref>). In which case, one must take into consideration the surrounding context when interpreting these words. |
One unambiguous way of stating a biconditional in plain English is to adopt the form "''b'' if ''a'' and ''a'' if ''b''"—if the standard form "''a'' if and only if ''b''" is not used. Slightly more formally, one could also say that "''b'' implies ''a'' and ''a'' implies ''b''", or "''a'' is necessary and sufficient for ''b''".<ref name=":0" /> The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition<ref>In fact, such is the style adopted by [[wikipedia:Manual of Style/Mathematics|Wikipedia's manual of style in mathematics]].</ref>). In which case, one must take into consideration the surrounding context when interpreting these words. |
||
For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker |
For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining. |
||
==See also== |
==See also== |
Revision as of 15:38, 21 September 2020
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective used to conjoin two statements and to form the statement " if and only if ", where is known as the antecedent, and the consequent.[1][2][3] This is often abbreviated as " iff ".[4] The operator is denoted using a doubleheaded arrow (↔[5] or ⇔[6]), a prefixed E "Epq" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡)[4], or EQV. It is logically equivalent to both and , and the XNOR (exclusive nor) boolean operator, which means "both or neither".
Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis is false but the conclusion is true. In which case, the result is true for the conditional, but false for the biconditional.[2]
In the conceptual interpretation, means "All 's are 's and all 's are 's". In other words, the sets and coincide: they are identical. However, this does not mean that and need to have the same meaning (e.g., could be "equiangular trilateral" and could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).
In the propositional interpretation, means that implies and implies ; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as could be "the triangle ABC has two equal sides" and could be "the triangle ABC has two equal angles". In general, the antecedent is the premise, or the cause, and the consequent is the consequence. When an implication is translated by a hypothetical (or conditional) judgment, the antecedent is called the hypothesis (or the condition) and the consequent is called the thesis.
A common way of demonstrating a biconditional of the form is to demonstrate that and separately (due to its equivalence to the conjunction of the two converse conditionals[2]). Yet another way of demonstrating the same biconditional is by demonstrating that and .[1]
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal.[citation needed] Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the hypothesis and whose consequent is the thesis of the theorem.
It is often said that the hypothesis is the sufficient condition of the thesis, and that the thesis is the necessary condition of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the necessary and sufficient condition of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time.
Definition
Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.[2]
Truth table
The following is truth table for (also written as , , or P EQ Q):
T | T | T |
T | F | F |
F | T | F |
F | F | T |
When more than two statements are involved, combining them with might be ambiguous. For example, the statement
may be interpreted as
- ,
or may be interpreted as saying that all are jointly true or jointly false:
As it turns out, these two statements are only the same—when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:
The left Venn diagram below, and the lines (AB ) in these matrices represent the same operation.
Venn diagrams
Red areas stand for true (as in for and).
|
|
|
Properties
Commutativity: Yes
Associativity: Yes
Distributivity: Biconditional doesn't distribute over any binary function (not even itself), but logical disjunction distributes over biconditional.
idempotency: No
Monotonicity: No
Truth-preserving: Yes
When all inputs are true, the output is true.
Falsehood-preserving: No
When all inputs are false, the output is not false.
Walsh spectrum: (2,0,0,2)
Nonlinearity: 0 (the function is linear)
Rules of inference
Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.
Biconditional introduction
Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A if and only if B.
For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically:
B → A A → B ∴ A ↔ B
B → A A → B ∴ B ↔ A
Biconditional elimination
Biconditional elimination allows one to infer a conditional from a biconditional: if A ↔ B is true, then one may infer either A → B, or B → A.
For example, if it is true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, then I'm alive; likewise, it's true that if I'm alive, then I'm breathing. Or more schematically:
A ↔ B ∴ A → B
A ↔ B ∴ B → A
Colloquial usage
One unambiguous way of stating a biconditional in plain English is to adopt the form "b if a and a if b"—if the standard form "a if and only if b" is not used. Slightly more formally, one could also say that "b implies a and a implies b", or "a is necessary and sufficient for b".[1] The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definition[7]). In which case, one must take into consideration the surrounding context when interpreting these words.
For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining.
See also
- If and only if
- Logical equivalence
- Logical equality
- XNOR gate
- Biconditional elimination
- Biconditional introduction
References
- ^ a b c "The Definitive Glossary of Higher Mathematical Jargon — If and Only If". Math Vault. 2019-08-01. Retrieved 2019-11-25.
{{cite web}}
: CS1 maint: url-status (link) - ^ a b c d Peil, Timothy. "Conditionals and Biconditionals". web.mnstate.edu. Retrieved 2019-11-25.
{{cite web}}
: CS1 maint: url-status (link) - ^ Brennan, Joseph G. (1961). Handbook of Logic (2nd ed.). Harper & Row. p. 81.
- ^ a b Weisstein, Eric W. "Iff". mathworld.wolfram.com. Retrieved 2019-11-25.
- ^ "Biconditional Statements | Math Goodies". www.mathgoodies.com. Retrieved 2019-11-25.
- ^ "2.4: Biconditional Statements". Mathematics LibreTexts. 2018-04-25. Retrieved 2019-11-25.
- ^ In fact, such is the style adopted by Wikipedia's manual of style in mathematics.
External links
- Media related to Logical biconditional at Wikimedia Commons
This article incorporates material from Biconditional on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.