# 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[edit]

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:

This can be read as saying that if, on the basis of a set of assumptions , one can prove C, then by adding an assumption A, one can still prove C.

## Example[edit]

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[edit]

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[edit]

## Notes[edit]

## References[edit]

Hedman, Shawn (2004). *A First Course in Logic*. Oxford University Press.

Chiswell, Ian; Hodges, Wilfrid (2007). *Mathematical Logic*. Oxford University Press.