Normal modal logics can be given a nice semantics by means of Kripke models, also known as possible worlds semantics.

Probably the most important reason for the popularity of
possible-worlds semantics is that common modal axioms correspond
exactly to certain algebraic properties of Kripke models in the
following sense: an axiom is valid in a model if and only if the
alternativeness relation of satisfies some algebraic
condition. (In fact, the correspondence holds on a higher abstraction
level, the level of *frames*, consult [vB84] for details.)
In particular:

**(T)**holds iff is reflexive**(D)**holds iff is serial**(4)**holds iff is transitive**(5)**holds iff is Euclidean**(G)**holds iff is directed

The common normal modal logics can be characterized by appropriate classes of Kripke models. In the following theorem, a Kripke model is said to be reflexive iff its accessibility relation is reflexive, and so on.

The common normal propositional modal logics are conservative extensions of classical logic: if a formula does not contain any occurrence of the modal operator then it is provable in a system mentioned in the previous theorem if and only if it is provable in the propositional calculus.