# Principle of Non-Contradiction

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## Theorem

The **Principle of Non-Contradiction** is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.

This includes classical propositional logic and predicate logic, and in particular natural deduction.

### 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

- 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(b)}$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**contradiction** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**contradiction**