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
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

Technical Note
When invoking the Rule of Not-Elimination in a tableau proof, use the NotElimination template:



or:

where:
 * is the number of the line on the tableau proof where Principle of Non-Contradiction is to be invoked
 * is the pool of assumptions (comma-separated list)
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the line of the tableau proof upon which this line directly depends, the one with $\bot$ on it
 * is the (optional) comment that is to be displayed in the Notes column.