Editing Interior algebra (section)

=== Modal logic<!--'S4 algebra' and 'Lewis algebra' redirect here--> ===
Given a [[theory (logic)|theory]] (set of formal sentences) ''M'' in the modal logic '''S4''', we can form its [[Lindenbaum–Tarski algebra]]:

:'''''L'''''(''M'') = ⟨''M'' / ~, ∧, ∨, ¬, ''F'', ''T'', □⟩

where ~ is the equivalence relation on sentences in ''M'' given by ''p'' ~ ''q'' if and only if ''p'' and ''q'' are [[Logical equivalence|logically equivalent]] in ''M'', and ''M'' / ~ is the set of equivalence classes under this relation. Then '''''L'''''(''M'') is an interior algebra. The interior operator in this case corresponds to the [[modal logic|modal operator]] □ ('''necessarily'''), while the closure operator corresponds to ◊ ('''possibly'''). This construction is a special case of a more general result for [[modal algebra]]s and modal logic.

The open elements of '''''L'''''(''M'') correspond to sentences that are only true if they are '''necessarily''' true, while the closed elements correspond to those that are only false if they are '''necessarily''' false.

Because of their relation to '''S4''', interior algebras are sometimes called '''S4 algebras'''<!--boldface per WP:R#PLA--> or '''Lewis algebras'''<!--boldface per WP:R#PLA-->, after the [[philosophical logic|logician]] [[Clarence Irving Lewis|C. I. Lewis]], who first proposed the modal logics '''S4''' and '''S5'''.