# Principle of Non-Contradiction/Proof Rule

## 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.

As a proof rule it is expressed in the form:

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

It can be written:

$\ds {\phi \quad \neg \phi \over \bot} \neg_e$

### Tableau Form

Let $\phi$ be a well-formed formula in a tableau proof.

The Principle of Non-Contradiction is invoked for $\phi$ and $\neg \phi$ in the following manner:

 Pool: The pooled assumptions of $\phi$ The pooled assumptions of $\neg \phi$ Formula: $\bot$ Description: Principle of Non-Contradiction Depends on: The line containing the instance of $\phi$ The line containing the instance of $\neg \phi$ Abbreviation: $\operatorname {PNC}$ or $\neg \EE$

## Explanation

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

A statement can not be both true and not true at the same time.

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:

## Technical Note

When invoking the Principle of Non-Contradiction in a tableau proof, use the {{NonContradiction}} template:

or: