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| {{Unreferenced|date=May 2008}}
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| '''[[Non-monotonic logic|Monotonicity]] of [[entailment]]''' is a property of many [[logical system]]s that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In [[sequent calculus|sequent calculi]] this property can be captured by an inference rule called '''weakening''', or sometimes '''thinning''', and in such systems one may say that [[logical consequence|entailment]] is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called ''[[monotonic logics]]'' in order to differentiate them from [[non-monotonic logic]]s.
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| ==Weakening rule== | |
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| To illustrate, starting from the [[natural deduction]] [[sequent]]:
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| <center>Γ <math>\vdash</math> C</center>
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| weakening allows one to conclude:
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| <center>Γ, A <math>\vdash</math> C</center>
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| ==Non-monotonic logics== | |
| {{main|Non-monotonic logic}}
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| In most logics, weakening is either an inference rule or a [[metatheorem]] if the logic doesn't have an explicit rule. Notable exceptions are:
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| * [[Strict logic]] or [[relevant logic]], where every hypothesis must be necessary for the conclusion.
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| * [[Linear logic]] which disallows arbitrary [[Contraction (operator theory)|contraction]] in addition to arbitrary weakening.
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| * [[Bunched logic|Bunched implications]] where weakening is restricted to additive composition.
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| * Various types of default reasoning.
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| * [[Abductive reasoning]], the process of deriving the most likely explanations of the known facts.
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| * Reasoning about knowledge, where statements specifying that something is not known need to be retracted when that thing is learned.
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| == See also ==
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| * [[Idempotency of entailment|Contraction]]
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| * [[Exchange rule]]
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| * [[Substructural logic]]
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| {{logic-stub}}
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| [[Category:Logical consequence]]
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| [[Category:Theorems in propositional logic]]
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