View source for Philosophy:Idempotency of entailment

'''Idempotency of entailment''' is a property of logical systems that states that one may derive the same consequences from many instances of a hypothesis as from just one. This property can be captured by a [[Structural rule|structural rule]] called '''[[Structural rule#Common structural rules|contraction]]''', and in such systems one may say that [[Logical consequence|entailment]] is idempotent if and only if contraction is an [[Admissible rule|admissible rule]].

Rule of contraction: from

:''A'',''C'',''C'' → ''B''

is derived
 
:''A'',''C'' → ''B''.

Or in [[Sequent calculus|sequent calculus]] notation,
:<math>\frac{\Gamma,C,C\vdash B}{\Gamma,C\vdash B}</math>

In [[Linear logic|linear]] and [[Philosophy:Affine logic|affine logic]], entailment is not idempotent.
== See also==
* [[No-deleting theorem]]

{{Classical logic}}

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[[Category:Logical consequence]]
[[Category:Theorems in propositional logic]]



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