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| {{Transformation rules}}
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| {{other uses}}
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| In [[propositional calculus|propositional logic]], '''simplification'''<ref>Copi and Cohen</ref><ref>Moore and Parker</ref><ref>Hurley</ref> (equivalent to '''conjunction elimination''') is a [[validity|valid]] [[immediate inference]], [[argument form]] and [[rule of inference]] which makes the [[inference]] that, if the [[Logical conjunction|conjunction]] ''A and B'' is true, then ''A'' is true, and ''B'' is true. The rule makes it possible to shorten longer [[formal proof|proofs]] by deriving one of the conjuncts of a conjunction on a line by itself.
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| An example in [[English language|English]]:
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| :It's raining and it's pouring.
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| :Therefore it's raining.
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| The rule can be expressed in [[formal language]] as:
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| :<math>\frac{P \land Q}{\therefore P}</math>
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| or as
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| :<math>\frac{P \land Q}{\therefore Q}</math>
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| where the rule is that whenever instances of "<math>P \land Q</math>" appear on lines of a proof, either "<math>P</math>" or "<math>Q</math>" can be placed on a subsequent line by itself.
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| == Formal notation ==
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| The ''simplification'' rule may be written in [[sequent]] notation:
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| : <math>(P \land Q) \vdash P</math>
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| or as
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| : <math>(P \land Q) \vdash Q</math>
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| where <math>\vdash</math> is a [[metalogic]]al symbol meaning that <math>P</math> is a [[logical consequence|syntactic consequence]] of <math>P \land Q</math> and <math>Q</math> is also a syntactic consequence of <math>P \land Q</math> in [[formal system|logical system]];
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| and expressed as a truth-functional [[tautology (logic)|tautology]] or [[theorem]] of propositional logic:
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| :<math>(P \land Q) \to P</math>
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| and
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| :<math>(P \land Q) \to Q</math> | |
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| where <math>P</math> and <math>Q</math> are propositions expressed in some logical system.
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| == References ==
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| {{reflist}}
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| [[Category:Rules of inference]]
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| [[Category:Theorems in propositional logic]]
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| [[sv:Matematiskt uttryck#Förenkling]]
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