Editing Monotonicity of entailment

{{Short description|Property of many systems of logic}}

'''Monotonicity of entailment''' is a property of many [[logical system]]s such that if a sentence follows deductively from a given set of sentences then it also follows deductively from any superset of those sentences. A corollary is that if a given argument is deductively [[Validity_(logic) | valid]], it cannot become invalid by the addition of extra premises.{{sfn|Hedman|2004|p=14}}{{sfn|Chiswell|Hodges|2007|p=61}}

Logical systems with this property are called monotonic logics in order to differentiate them from [[non-monotonic logic]]s. [[Classical logic]] and [[intuitionistic logic]] are examples of monotonic logics. 

==Weakening rule==

Monotonicity may be stated formally as a rule called '''weakening''', or sometimes '''thinning'''. A system is monotonic if and only if the rule is [[admissible_rule| admissible]]. 
The weakening rule may be expressed as a natural deduction sequent: 
:<math>\frac{\Gamma \vdash C}{\Gamma, A \vdash C } </math>
This can be read as saying that if, on the basis of a set of assumptions <math>\Gamma</math>, one can prove C, then by adding an assumption A, one can still prove C. 

== Example ==

The following argument is valid: "All men are mortal. Socrates is a man. Therefore Socrates is mortal." This can be weakened by adding a premise: "All men are mortal. Socrates is a man. Cows produce milk. Therefore Socrates is mortal." By the property of monotonicity, the argument remains valid with the additional premise, even though the premise is irrelevant to the conclusion. 

==Non-monotonic logics==
{{main|Non-monotonic logic}}
In most logics, weakening is either an inference rule or a [[metatheorem]] if the logic doesn't have an explicit rule. Notable exceptions are:

* [[Relevance logic]], where every premise is necessary for the conclusion.
* [[Linear logic]], which lacks monotonicity and [[idempotency of entailment]].

== See also ==

* [[Idempotency of entailment|Contraction]]
* [[Exchange rule]]
* [[Substructural logic]]
* [[No-cloning theorem]]

== Notes ==
{{Reflist }}

== References ==

* {{cite book |last1=Hedman |first1=Shawn |title= A First Course in Logic |date=2004 |publisher=Oxford University Press}}

* {{cite book |last1=Chiswell |first1=Ian |last2=Hodges |first2=Wilfrid |title= Mathematical Logic |date=2007 |publisher=Oxford University Press}}

{{Classical logic}}

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


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