New Semantics for Modal Predicate Logics

  • Oliver Kutz
Part of the Trends in Logic book series (TREN, volume 17)


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.


Predicate Symbol Modal Algebra World Semantic Kripke Frame Counterpart Theory 
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  1. 2.
    By “standard semantics” I mean here standard Kripke frames enriched by an assignment of domains to worlds, meeting some extra conditions. Actually, if the CBF schema is omitted, one has to deal with non-denoting terms and to move to a quantificational base in free logic, e.g., by introducing an existence predicate. Google Scholar
  2. 3.
    For further simple examples of incomplete logics and proof sketches, compare, e.g., [Hug1Cre96].Google Scholar
  3. 4.
    For a treatment of this in a classical setting, cf. [Fit2Men98].Google Scholar
  4. 7.
    Thus, the modal operators behave quite similar to what is known as an unselective binder in linguistics.Google Scholar
  5. 8.
    But compare [Fit2∞a] for a variation of counterpart semantics that can be understood as a special case of the counterpart frames to be introduced below.Google Scholar
  6. 9.
    A logic is said to be canonical, if the frame underlying its canonical model is a frame for the logic.Google Scholar
  7. 11.
    For an analysis of this compare [Kra0Kut∞b].Google Scholar
  8. 12.
    Because we work with free logic, either assume that the language contains an existence predicate, or that the first-order domains consist of an inner and an outer domain—the former being the domain for the actualist quantifiers.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Oliver Kutz
    • 1
  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany

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