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| In [[propositional calculus|propositional logic]], the '''commutativity of conjunction''' is a [[validity|valid]] [[argument form]] and truth-functional [[tautology (logic)|tautology]]. It is considered to be a law of [[classical logic]]. It is the principle that the conjuncts of a [[logical conjunction]] may switch places with each other, while preserving the [[truth-value]] of the resulting proposition.<ref>{{cite book|title=Introduction to Mathematical Logic|author=Elliott Mendelson|year=1997|publisher=CRC Press|isbn=0-412-80830-7}}</ref>
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| == Formal notation ==
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| ''Commutativity of conjunction'' can be expressed in [[sequent]] notation as:
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| : <math>(P \and Q) \vdash (Q \and P)</math>
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| and
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| : <math>(Q \and P) \vdash (P \and Q)</math>
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| where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>(Q \and P)</math> is a [[logical consequence|syntactic consequence]] of <math>(P \and Q)</math>, in the one case, and <math>(P \and Q)</math> is a syntactic consequence of <math>(Q \and P)</math> in the other, in some [[formal system|logical system]];
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| or in [[rule of inference|rule form]]:
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| :<math>\frac{P \and Q}{\therefore Q \and P}</math>
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| and
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| :<math>\frac{Q \and P}{\therefore P \and Q}</math>
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| where the rule is that wherever an instance of "<math>(P \and Q)</math>" appears on a line of a proof, it can be replaced with "<math>(Q \and P)</math>" and wherever an instance of "<math>(Q \and P)</math>" appears on a line of a proof, it can be replaced with "<math>(P \and Q)</math>";
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| or as the statement of a truth-functional tautology or [[theorem]] of propositional logic:
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| :<math>(P \and Q) \to (Q \and P)</math> | |
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| and
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| :<math>(Q \and P) \to (P \and Q)</math>
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| where <math>P</math> and <math>Q</math> are [[proposition]]s expressed in some formal system.
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| == Generalized principle ==
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| For any propositions H<sub>1</sub>, H<sub>2</sub>, ... H<sub>''n''</sub>, and permutation σ(n) of the numbers 1 through n, it is the case that:
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|
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| :H<sub>1</sub> <math>\land</math> H<sub>2</sub> <math>\land</math> ... <math>\land</math> H<sub>n</sub>
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| is equivalent to
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| :H<sub>σ(1)</sub> <math>\land</math> H<sub>σ(2)</sub> <math>\land</math> H<sub>σ(n)</sub>.
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| For example, if H<sub>1</sub> is
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| :''It is raining''
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| H<sub>2</sub> is
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| :''[[Socrates]] is mortal''
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| and H<sub>3</sub> is
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| :''2+2=4''
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| then
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| ''It is raining and Socrates is mortal and 2+2=4''
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| is equivalent to
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| ''Socrates is mortal and 2+2=4 and it is raining''
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| and the other orderings of the predicates.
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| ==References==
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| {{reflist}}
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| [[Category:Classical logic]]
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| [[Category:Rules of inference]]
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| [[Category:Theorems in propositional logic]]
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