Law of Excluded Middle

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Proof Rule

The law of (the) excluded middle is a valid argument in certain types of logic dealing with disjunction $\lor$ and negation $\neg$.

This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.

Proof Rule

$\phi \lor \neg \phi$ for all statements $\phi$.

Sequent Form

The Law of Excluded Middle can be symbolised by the sequent:

$\vdash p \lor \neg p$


The law of (the) excluded middle can be expressed in natural language as:

Every statement is either true or false.

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

The LEM is rejected by the intuitionistic school, which rejects the existence of an object unless it can be constructed within an axiomatic framework which does not include the LEM.

Also known as

The law of (the) excluded middle 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).
  • the principle of bivalence.

Also see