Monotonicity of entailment

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: :\frac{\Gamma \vdash C}{\Gamma, A \vdash C } This can be read as saying that if, on the basis of a set of assumptions \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: • Relevance logic, where every premise is necessary for the conclusion. • Linear logic, which lacks monotonicity and idempotency of entailment. == See also ==