Monotonicity of entailment
Monotonicity of entailment is a property of many logical systems 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 valid, it cannot become invalid by the addition of extra premises.Template:SfnTemplate:Sfn
Logical systems with this property are called monotonic logics in order to differentiate them from non-monotonic logics. Classical logic and intuitionistic logic are examples of monotonic logics.
Weakening ruleEdit
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. 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.
ExampleEdit
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 logicsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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.