Rule of Material Implication with Amphiboly

Source Work

2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems:

Chapter $1$: Propositional logic

$1.2.1$ 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.

Sources

• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems: $1.2.1$ Rules for natural deduction: Exercises $1.5 \ 2 \ \text{(i)}$