# Principle of Non-Contradiction

## Contents

## Theorem

The **Principle of Non-Contradiction** is a valid deduction sequent in propositional logic.

### Proof Rule

- If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.

### Sequent Form

- $p, \neg p \vdash \bot$

## Explanation

The **Principle of Non-Contradiction** can be expressed in natural language as follows:

This means: if we have managed to deduce that a statement is both true and false, then the sequence of deductions show that the pool of assumptions upon which the sequent rests contains assumptions which are mutually contradictory.

Thus it provides a means of eliminating a logical not from a sequent.

## Also known as

The **Principle of Non-Contradiction** is otherwise known as:

**Principium contradictionis**, Latin for**principle of contradiction****Rule of not-elimination****Law of contradiction****Law of non-contradiction**

## Also see

## Sources

- 1978: Alan G. Hamilton:
*Logic for Mathematicians*... (previous) ... (next): $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(b)}$