Modal logic deals with sentences that are qualified by modalities, the most typical of which are necessity and its dual possibility. What makes these modal operators essentially different from classical logic operators (NOT, AND, etc.) is that they are not truth functions, i.e. the truth value of a compond formula involving modaities depends on more than just the truth values of its subformulas.

Varieties of modl logics

Modal logics differ in what collections of modalities they allow, how the existing modal operators are interpreted, and how the logic is axiomatized. Below is an incomplete list of various interpretations of modal operators studied in the literature:

Possible readings of []A: "it is necessary that A", "A is provable", "I know that A", "everyone knows that A", "I believe that A", "A is obligatory", "A will always be true", "A has always been true", etc.

Possible readings of A>B: "B is interpretable in A", "A logically implies B", etc.