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:

$\vdash p \lor \neg p$


Explanation

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.


Intuitionist Perspective

The Law of the Excluded Middle 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 Law of the Excluded Middle.

This is because from the perspective of intuitionism:

an object cannot be known to exist unless it can be constructed in a finite number of steps
a statement cannot be known to be true if its proof needs an argument requiring an infinite number of steps.

Thus:

while it is sufficient to prove a statement is not true by demonstrating that it is not true
it is insufficient to prove a statement is true by demonstrating it is not false.


Also known as

The law of (the) excluded middle is otherwise known as:


Also see


Sources

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