# Clavius's Law

## Contents

## Theorem

If, from the negation of a proposition $p,$ we can derive $p$, we may conclude $p$:

### Formulation 1

- $\neg p \implies p \vdash p$

### Formulation 2

- $\vdash \left({\neg p \implies p}\right) \implies p$

## Also known as

**Clavius's Law** is also known as **consequentia mirabilis**.

## Also see

- Reductio ad Absurdum, of which this is a particular case.

## 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.

## Source of Name

This entry was named for Christopher Clavius.

The name **consequentia mirabilis** is Latin for "**marvellous** (or **admirable**) **consequence**".