Philosophy: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:
- [math]\displaystyle{ \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]\displaystyle{ \Gamma }[/math], 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
- Contraction
- Exchange rule
- Substructural logic
- No-cloning theorem
Notes
References
Hedman, Shawn (2004). A First Course in Logic. Oxford University Press.
Chiswell, Ian; Hodges, Wilfrid (2007). Mathematical Logic. Oxford University Press.
![]() | Original source: https://en.wikipedia.org/wiki/Monotonicity of entailment.
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