Law of Excluded Middle

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

The rule
All statements have a truth value of either true or false:
 * $\vdash p \lor \neg p$

Otherwise known as:
 * (Principium) tertium non datur, Latin for third not given, that is, a third possibility is not possible;
 * Principium tertii exclusi, the Principle of the Excluded Third (PET).

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

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 $\vdash p \lor \neg p$.

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

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

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.