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Full Idea
The Deduction Theorem says ψ is derivable in classical predicate logic from ψ iff the sentence φ→ψ is a theorem of classical logic. Hence inferring φ to ψ is truth-preserving iff the axiom scheme φ→ψ is provable.
Gist of Idea
Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
Source
Leon Horsten (The Tarskian Turn [2011], 02.2)
Book Ref
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.18
A Reaction
Horsten offers this to show that the Tarski bi-conditionals can themselves be justified, and not just the rule of inference involved. Apparently you can only derive something if you first announce that you have the ability to derive it. Odd.
| 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] |