# 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.[1][2]

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 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. The weakening rule may be expressed as a natural deduction sequent:

${\displaystyle {\frac {\Gamma \vdash C}{\Gamma ,A\vdash C}}}$

This can be read as saying that if, on the basis of a set of assumptions ${\displaystyle \Gamma }$, 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

In most logics, weakening is either an inference rule or a metatheorem if the logic doesn't have an explicit rule. Notable exceptions are: