Principle of Non-Contradiction

Theorem
The rule of not-elimination is a valid deduction sequent in propositional logic:

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

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

Explanation
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.

It is one of the cornerstones of Aristotelian logic, along with the law of the excluded middle.

Variants
The following forms can be used as variants of this theorem:

Also known as
Otherwise known as:
 * Principium contradictionis, Latin for principle of contradiction
 * Rule of Not-Elimination

Also see

 * Law of Excluded Middle

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



or:

where:
 * is the number of the line on the tableau proof where the Principle of Non-Contradiction is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the first of the two lines of the tableau proof upon which this line directly depends
 * is the second of the two lines of the tableau proof upon which this line directly depends
 * is the (optional) comment that is to be displayed in the Notes column.