Oliver Kutz

The problem of finding adequate semantics for languages of first-order modal logic, both from a mathematical and philosophical point of view, turned out to be rather difficult. The 1990ies have seen a number of attempts to attack this problem from a new angle, by introducing semantics that extend the usual framework of Kripkean possible worlds semantics. In this paper, I briefly introduce the most important of these semantics and state the main theoretical results that are known so far, concentrating on the (frame-) completeness problem and the role of substitution principles. It is argued that while the mathematical generality of the proposed semantics is a great step forward, a satisfying philosophical interpretation of ``Kripke-type'' semantics has still to be accomplished.