Law of Excluded Middle (Propositional Logic)

Definition

Law of Excluded Middle (Propositional Logic)

The law of excluded middle (LEM) is the claim that every propositional formula is either true or false: for every formula $P$,

$$\models P\vee \neg P.$$

Equivalently, for every interpretation $v$,

$$v(P\vee \neg P)=v(\top ).$$

There is no third truth value: $P$ and $\neg P$ exhaust the two possibilities, so exactly one of the disjuncts is true under every interpretation.

What LEM Does Not Say

Warning

The negation laws of the verum and falsum,

$$\neg \top \equiv \perp ,\neg \perp \equiv \top ,$$

already follow from the truth table of negation alone and do not require LEM. They also hold in intuitionistic logic, where LEM fails. LEM is strictly stronger: it asserts $P\vee \neg P$ for arbitrary $P$, not merely for the constants.

Relation to Classical Logic

LEM is one of the axioms that separate classical logic from intuitionistic logic. Over the intuitionistic rules, LEM is equivalent to double negation elimination $\neg \neg P\to P$: accepting either one makes the logic classical.