Editing Principle of bivalence (section)

== Classical logic ==
The intended semantics of classical logic is bivalent, but this is not true of every [[formal semantics (logic)|semantics]] for classical logic. In [[Boolean-valued semantics]] (for classical [[propositional logic]]), the truth values are the elements of an arbitrary [[Boolean algebra]], "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element. Intermediate elements of the algebra correspond to truth values other than "true" and "false". The principle of bivalence holds only when the Boolean algebra is taken to be the [[two-element Boolean algebra|two-element algebra]], which has no intermediate elements.

Assigning Boolean semantics to classical [[predicate calculus]] requires that the model be a [[complete Boolean algebra]] because the [[universal quantifier]] maps to the [[infimum]] operation, and the [[existential quantifier]] maps to the [[supremum]];<ref name="SørensenUrzyczyn2006">{{cite book|author1=Morten Heine Sørensen|author2=Paweł Urzyczyn|title=Lectures on the Curry-Howard isomorphism|url=https://books.google.com/books?id=_mtnm-9KtbEC&pg=PA206|year=2006|publisher=Elsevier|isbn=978-0-444-52077-7|pages=206–207}}</ref> this is called a [[Boolean-valued model]]. All finite Boolean algebras are complete.