Law of Excluded Middle

Proof Rule
The law of (the) excluded middle is a valid deduction sequent in propositional logic: All statements have a truth value of either true or false.

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 $\top$ (top) signifies tautology.

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.

Variants
The following forms can be used as variants of this theorem:

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.

Technical Note
When invoking Law of Excluded Middle in a tableau proof, use the ExcludedMiddle template:



or:

where:
 * is the number of the line on the tableau proof where the Law of Excluded Middle is to be invoked
 * is the statement of logic that is to be displayed in the Formula column, without the  delimiters
 * is the (optional) comment that is to be displayed in the Notes column.