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5. Theory of Logic / K. Features of Logics / 10. Monotonicity

[if something is proved, nothing new can unprove it]

6 ideas
Valid deduction is monotonic - that is, it remains valid if further premises are added [Psillos]
     Full Idea: Valid deductive arguments have the property of monotonicity; if the conclusion Q follows from the premises P, then it will also follow if further premises P* are added to P.
     From: Stathis Psillos (Causation and Explanation [2002], §9.2.1)
     A reaction: For perversity's sake we could add a new premise which contradicted one of the original ones ('Socrates is a god'). Or one premise could be 'I believe..', and the new one could show that the belief was false. Induction is non-monotonic.
Explanations fail to be monotonic [Rosen]
     Full Idea: The failure of monotonicity is a general feature of explanatory relations.
     From: Gideon Rosen (Metaphysical Dependence [2010], 05)
     A reaction: In other words, explanations can always shift in the light of new evidence. In principle this is right, but some explanations just seem permanent, like plate-tectonics as explanation for earthquakes.
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
     Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7)
     A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science.
In classical logic the relation |= has Monotony built into its definition [Antonelli]
     Full Idea: In classical logic, Monotony follows immediately from the nature of the relation |=, for Γ |= φ holds precisely when φ is true on every interpretation on which all sentences in Γ are true.
     From: G. Aldo Antonelli (Non-Monotonic Logic [2014], 1)
     A reaction: That is, semantic consequence (|=) is defined in terms of a sentence (φ) always being true if some other bunch of sentences (Γ) are true. Hence the addition of further sentences to Γ will make no difference - which is Monotony.
Cautious Monotony ignores proved additions; Rational Monotony fails if the addition's negation is proved [Antonelli]
     Full Idea: Basic Monotony: something stays proved if further premises are added. Cautious Monotony: the addition of something which has been proved makes no difference. Rational Monotony: it stays proved as long as the addition's negation hasn't been proved.
     From: G. Aldo Antonelli (Non-Monotonic Logic [2014], 1)
     A reaction: [A compressed and non-symbolic summary]
Monotonicity means there is a guarantee, rather than mere inductive support [Rumfitt]
     Full Idea: Monotonicity seems to mark the difference between cases in which a guarantee obtains and those where the premises merely provide inductive support for a conclusion.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.3)
     A reaction: Hence it is plausible to claim that 'non-monotonic logic' is a contradiction in terms.