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:

• 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.

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This page was last edited on 23 January 2024, at 13:18