Law of Excluded Middle

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

All statements have a truth value of either true or false.

Sequent Form

 * $\vdash p \lor \neg p$

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.

Tableau Form
In a tableau proof, the law of the excluded middle can be invoked in the following manner:


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

Also see

 * Principle of Non-Contradiction


 * Double Negation Elimination implies Law of Excluded Middle, where the Law of Excluded Middle is derived from the Law of Double Negation Elimination. Thus the latter can be treated as axiomatic instead.