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{{Transformation rules}} | |||
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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. | |||
An example in [[English language|English]]: | |||
:It's raining and it's pouring. | |||
:Therefore it's raining. | |||
The rule can be expressed in [[formal language]] as: | |||
:<math>\frac{P \land Q}{\therefore P}</math> | |||
or as | |||
:<math>\frac{P \land Q}{\therefore Q}</math> | |||
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. | |||
== Formal notation == | |||
The ''simplification'' rule may be written in [[sequent]] notation: | |||
: <math>(P \land Q) \vdash P</math> | |||
or as | |||
: <math>(P \land Q) \vdash Q</math> | |||
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]]; | |||
and expressed as a truth-functional [[tautology (logic)|tautology]] or [[theorem]] of propositional logic: | |||
:<math>(P \land Q) \to P</math> | |||
and | |||
:<math>(P \land Q) \to Q</math> | |||
where <math>P</math> and <math>Q</math> are propositions expressed in some logical system. | |||
== References == | |||
{{reflist}} | |||
[[Category:Rules of inference]] | |||
[[Category:Theorems in propositional logic]] | |||
[[sv:Matematiskt uttryck#Förenkling]] | |||
Revision as of 16:56, 26 February 2013
Template:Transformation rules
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In propositional logic, simplification[1][2][3] (equivalent to conjunction elimination) is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction A and B is true, then A is true, and B is true. The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself.
An example in English:
- It's raining and it's pouring.
- Therefore it's raining.
The rule can be expressed in formal language as:
or as
where the rule is that whenever instances of "" appear on lines of a proof, either "" or "" can be placed on a subsequent line by itself.
Formal notation
The simplification rule may be written in sequent notation:
or as
where is a metalogical symbol meaning that is a syntactic consequence of and is also a syntactic consequence of in logical system;
and expressed as a truth-functional tautology or theorem of propositional logic:
and
where and are propositions expressed in some logical system.
References
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