Rule of Material Implication with Amphiboly

Source Work

 * Chapter $1$: Propositional logic
 * $1.2.1$ Rules for natural deduction: Exercises $1.5 \ 2 \ \text{(i)}$
 * $1.2.1$ Rules for natural deduction: Exercises $1.5 \ 2 \ \text{(i)}$

Mistake
Prove the sequent


 * $\vdash \neg p \lor q \longrightarrow p \longrightarrow q$.

Correction
This propositional formula as it stands is an amphiboly.

It needs parentheses put in:
 * $\vdash \left({\neg p \lor q}\right) \implies \left({p \implies q}\right)$

See Rule of Material Implication/Formulation 2/Reverse Implication.