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Full Idea
In some intuitionist semantics modus ponens is not sanctioned. At any given time there is likely to be a conditional such that it and its antecedent have been proved, but nobody has bothered to prove the consequent.
Gist of Idea
Intuitionism only sanctions modus ponens if all three components are proved
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 6.7)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.207
A Reaction
[He cites Heyting] This is a bit baffling. In what sense can 'it' (i.e. the conditional implication) have been 'proved' if the consequent doesn't immediately follow? Proving both propositions seems to make the conditional redundant.
| 8078 | Modus ponens is one of five inference rules identified by the Stoics [Chrysippus, by Devlin] |
| 20309 | If our ideas are adequate, what follows from them is also adequate [Spinoza] |
| 5395 | Demonstration always relies on the rule that anything implied by a truth is true [Russell] |
| 3094 | You don't have to accept the conclusion of a valid argument [Harman] |
| 13617 | MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock] |
| 13614 | MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock] |
| 10257 | Intuitionism only sanctions modus ponens if all three components are proved [Shapiro] |
| 14184 | In modus ponens the 'if-then' premise contributes nothing if the conclusion follows anyway [Read] |
| 15341 | Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms [Horsten] |