# Principle of Dilemma

## Theorem

### Formulation 1

$\paren {p \implies q} \land \paren {\neg p \implies q} \dashv \vdash q$

### Formulation 2

$\vdash \paren {p \implies q} \land \paren {\neg p \implies q} \iff q$

## Also known as

This proof structure is also referred to as proof by cases but it is usual to reserve that name for a more general concept.

## Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.