 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2007/a/2/iii/solution

An instance of L3 is

(α \to β) \to (α \to (\neg β \to γ)) .

Γ ⊢ α \to (\neg β \to γ)

. The L2 instance for these same three propositional formulas is

(α \to (β \to γ)) \to ((α \to (\neg β \to γ)) \to (α \to γ)) .

Use the other assumed logical derivation and then the newly obtained logical derivation in two applications of modus ponens rule. Therefore

Γ ⊢ α \to γ .

This conditional modus ponens construction is the step needed to discharge assumptions from a modus ponens rule line in part (iv).

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