Abstract
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.
I should like to thank Melvin Fitting, Hartmut Fitz, Marcus Kracht and Frank Wolter for helpful discussions, as well as two anonymous referees for their suggestions. The work of the author was supported by DFG grant Wo 583/3–1.
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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.
For further simple examples of incomplete logics and proof sketches, compare, e.g., [Hug1Cre96].
For a treatment of this in a classical setting, cf. [Fit2Men98].
Thus, the modal operators behave quite similar to what is known as an unselective binder in linguistics.
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.
A logic is said to be canonical, if the frame underlying its canonical model is a frame for the logic.
For an analysis of this compare [Kra0Kut∞b].
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.
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© 2003 Springer Science+Business Media Dordrecht
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Kutz, O. (2003). New Semantics for Modal Predicate Logics. In: Löwe, B., Malzkom, W., Räsch, T. (eds) Foundations of the Formal Sciences II. Trends in Logic, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0395-6_11
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DOI: https://doi.org/10.1007/978-94-017-0395-6_11
Publisher Name: Springer, Dordrecht
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