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Full Idea
Modus Ponens is equivalent to the converse of the Deduction Theorem, namely 'If Γ |- φ→ψ then Γ,φ|-ψ'.
Gist of Idea
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ
Source
David Bostock (Intermediate Logic [1997], 5.3)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.207
A Reaction
See 13615 for details of the Deduction Theorem. See 13614 for Modus Ponens.
Related Ideas
Idea 13615 'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
Idea 13614 MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
| 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] |