Possible-world Semantics
Modal logic is most commonly interpreted in terms of possible world semantics or Kripke structures. This semantics carries over naturally to dynamic logic by interpreting worlds as states of a computer in the application to program verification, or states of our environment in applications to linguistics, AI, etc. One role for possible world semantics is to formalize the intuitive notions of truth and validity, which in turn permit the notions of soundness and completeness to be defined for axiom systems. An inference rule is sound when validity of its premises implies validity of its conclusion. An axiom system is sound when all its axioms are valid and its inference rules are sound. An axiom system is complete when every valid formula is derivable as a theorem of that system. These concepts apply to all systems of logic including dynamic logic.
Read more about this topic: Dynamic Logic (modal Logic)