Principle of Non-Contradiction

Theorem
The rule of not-elimination is one of the axioms of natural deduction.

If we can conclude both $p$ and $\neg p$, we may infer a contradiction:

It can be written:
 * $\displaystyle {p \quad \neg p \over \bot} \neg_e$

It is equivalent to the Principle of Non-Contradiction, which is a formulation of the concept that a statement can not be both true and not true at the same time.

Tableau Form
In a tableau proof, the rule of not-elimination can be invoked in the following manner:


 * Abbreviation: $\neg \mathcal E$
 * Deduced from: The pooled assumptions of $p$ and $\neg p$.
 * Depends on: The lines containing $p$ and $\neg p$.

Explanation
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 see

 * Principle of Non-Contradiction


 * Rule of Not-Elimination Equivalent to Principle of Non-Contradiction