Axiom:Rule of Not-Elimination
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Context
The rule of not-elimination is one of the axioms of natural deduction.
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.
The rule
If we can conclude both $p$ and $\neg p$, we may infer a contradiction:
- $p, \neg p \vdash \bot$
It can be written:
- $\displaystyle {p \quad \neg p \over \bot} \neg_e$
- 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.
Truth Table Demonstration
$\begin{array}{|cccc||c|} \hline p & \land & \neg & p & \bot \\ \hline F & F & T & F & F \\ T & F & F & T & F \\ \hline \end{array}$
$\blacksquare$
As can be seen by inspection, the truth value of the main connective, that is $\land$, is $F$ for each model of $p$.
