law of excluded middle
Noun
Meaning
law of excluded middle
(logic) A logical principle which states all statements must be either true or false, i.e. in symbols:
P
∨
¬
P
{\displaystyle P\vee \neg P}
.
Synonyms
Examples
In this way, the law of excluded middle is true, but because truth itself, and therefore disjunction, is not exclusive, it says next to nothing if one of the disjuncts is paradoxical, or both true and false.
History The first known classical logician who didn't fully accept the law of excluded middle was Aristotle (who, ironically, is also generally considered to be the first classical logician and the "father of logic" Hurley, Patrick.
An example of an argument that depends on the law of excluded middle follows.
However, there are proofs in classical logic that depend upon the law of excluded middle; since that law is not usable under this scheme, there are propositions that cannot be proven that way.
In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g. the natural numbers).
In the above argument, the assertion "this number is either rational or irrational" invokes the law of excluded middle.