Law of Excluded Middle

Context
The law of (the) excluded middle is one of the axioms of natural deduction.

The rule
For any statement $$p$$, either $$p$$ is true or $$p$$ is false:
 * $$\vdash p \or \neg p$$

Otherwise known as tertium non datur (Latin for "third is not given", that is, a third possibility is not possible).

It can be written:
 * $${{} \over p \or \neg p} \textrm{LEM} \qquad \text { or } \qquad {\top \over p \or \neg p} \textrm{LEM}$$

where the symbol $$\top$$ (called top) signifies tautology.


 * Abbreviation: $$\textrm{LEM}$$
 * Deduced from: Nothing.
 * Depends on: Nothing.

Explanation
This is one of the Aristotelian principles upon which the whole of classical logic, and the majority of mainstream mathematics rests.

This rule is denied by the intuitionist school.

Truth Table Demonstration
We apply the Method of Truth Tables to the proposition $$\vdash p \or \neg p$$.

As can be seen by inspection, the truth value of the main connective, that is $$\or$$, is $$T$$ for each model of $$p$$.

$$\begin{array}{|cccc|} \hline p & \or & \neg & p \\ \hline F & T & T & F \\ T & T & F & T \\ \hline \end{array}$$

Also see
It is possible to derive the Law of Excluded Middle from the Rule of Double Negation Elimination, and treat the latter as axiomatic instead.