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 the Rule of Not-Elimination 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.