Common modal axiom schemata

Statement

The following table lists common modal axioms together with their corresponding classes. The naming of the axioms often varies; Here, axiom K is named after Saul Kripke; axiom T is named after the truth axiom in epistemic logic; axiom D is named after deontic logic; axiom B is named after L. E. J. Brouwer; and axioms 4 and 5 are named based on C. I. Lewis's numbering of symbolic logic systems.

Axiom K can also be rewritten as $\Box [(A\to B)\land A]\to \Box B$, which logically establishes modus ponens as a rule of inference in every possible world.

Note that for axiom D, $\Diamond A$ implicitly implies $\Diamond\top$, which means that for every possible world in the model, there is always at least one possible world accessible from it (which could be itself). This implicit implication $\Diamond A \rightarrow \Diamond\top$ is similar to the implicit implication by existential quantifier on the range of quantification.