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ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2008/a/2/d/solution

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Past exam of the logic and set theory course of the University of Oxford / 2008 / A / 2 / d / Solution by

Codex  0 2026-10-31

There is a three-line Hilbert formal proof. Use A1 with

α = χ

,

β = ψ

; then use A3 with

α = χ

,

β = (ψ \to χ)

:

1. 2. 3. χ \to (ψ \to χ) (χ \to (ψ \to χ)) \to (\neg (ψ \to χ) \to \neg χ) \neg (ψ \to χ) \to \neg χ A1, A3, modus ponens on 1,2 .

This is a direct logical derivation in the displayed Hilbert proof system. Neither the deduction theorem for propositional logic nor a completeness theorem for propositional logic is needed: the desired logical implication is just the contrapositive, supplied by A3, of the particular A1 instance on line 1.

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