How does the law of excluded middle work?

The law of excluded middle is a fundamental principle in classical logic stating that for any proposition p, either p is true or its negation (not p) is true. Symbolically, this is expressed as $p\vee \neg p$, meaning there is no third or middle truth-value between them.1

Historically, Aristotle identified this as an axiom, though he noted exceptions for "future contingents"—statements about uncertain future events that are not yet determinately true or false. In the 20th century, the Dutch mathematician L.E.J. Brouwer challenged the universal application of this law, particularly regarding infinite sets. His school of intuitionism argues that if a statement cannot be proven or computed, it cannot be assumed to be true or false. Consequently, some modern logical systems, such as many-valued logics, allow for truth-values beyond the binary of true and false, where the law of excluded middle does not hold.12