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:

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

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.

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.

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.

• Contraction
• Exchange rule
• Substructural logic
• No-cloning theorem

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

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