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## Semantics for normal modal logic: Kripke models

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.

Next: Montague-Scott semantics Up: Modal logic Previous: Sytax of modal logic   Contents
2001-04-05