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  • Online publication date: March 2017

The semantics of modal predicate logic II. Modal individuals revisited


Abstract. We continue the investigations begun in [10]. We shall define a semantics that is built on a new kind of frames, called coherence frames. In these frames, objects are transcendental (world-independent), as in the standard constant-domain semantics. This will remove the asymmetry between constants and variables of the counterpart semantics of [10]. We demonstrate the completeness of (general) coherence frames with respect to first- and certain weak second-order logics and we shall compare this notion of a frame to counterpart frames as introduced in [10] and the metaframe semantics of [13].

Introduction. In [10] we have developed a semantics that is complete with respect to first- and weak second-order modal predicate logics. This semantics was in addition quite elementary, which was already a great step forward from the previous semantics by Ghilardi [6] and by Skvortsov and Shehtman [13]. Still, from a philosophical point of view this semantics left much to be desired. The introduction of counterpart relations—although in line with at least some philosophical ideas, notably by Lewis — is not always very satisfactory since it makes the notion of an object a derived one. The things we see become strictly world bound: there is no sense in which we can talk of, say, the town hall of Berlin, rather than the town hall of Berlin in a particular world, at a particular point of time. The traditional semantics for modal predicate logic held the complete opposite view. There, objects are transcendental entities. They are not world bound, since they do not belong to the worlds. The difference between these views becomes clear when we look at the way in which the formula is evaluated. In the standard semantics, we simply go to some accessible world and see whether holds. In counterpart semantics, we not only have to choose another world but also some counterparts for the things thatwe have chosen as values for the variables in this world. In the traditional semantics the question of counterparts does not arise because of the transcendental status of objects. We may view this as a limiting case of counterpart semantics, in which the counterpart relation always is the identity.

[1] Sebastian, Bauer, Metaframes, Typen und Modelle der modalen Prädikatenlogik, Master's thesis, FachbereichMathematik, Humboldt–Universität zu Berlin, 2000.
[2] Max J., Cresswell, How to complete some modal predicate logics, Advances inModal Logic. Volume 2 (Michael, Zakharyaschev et al., editors), CSLI Publications, Stanford, 2001.
[3] Melvin, Fitting, Types, Tableaus and Gödel's God, Trends in Logic, vol. 12, Kluwer Academic Publishers, Dordrecht, 2002.
[4] Melvin, Fitting, First-order intensional logic, Annals of Pure and Applied Logic, vol. 127 (2004), no. 1-3, pp. 171–193. Provinces of logic determined. Essays in the memory of Alfred Tarski. Parts IV, V and VI. Edited by Z., Adamowicz, S., Artemov, D., Niwinski, E., Orlowska, A., Romanowska and J., Wolenski.
[5] Melvin, Fitting and Richard L., Mendelsohn, First–Order Modal Logic, Kluwer Academic Publishers, Dordrecht, 1998.
[6] Silvio, Ghilardi, Quantified extensions of canonical propositional intermediate logics, Studia Logica, vol. 51 (1992), pp. 195–214.
[7] George, E.Hughes and Max J., Cresswell, ANew Introduction toModal Logic,Routledge, London, 1996.
[8] Steven C., Kleene, Introduction to Metamathematics, Wolters-Noordhoff Publishing, Groningen, 1971.
[9] Marcus, Kracht, Tools and Techniques inModal Logic, Studies in Logic and the Foundations ofMathematics, vol. 142, Elsevier Science Publishers, Amsterdam, 1999.
[10] Marcus, Kracht and Oliver, Kutz, The Semantics of Modal Predicate Logic I. Counterpart–Frames, Advances in Modal Logic. Volume 3 (Frank, Wolter, Heinrich, Wansing, Maarten de, Rijke, and Michael, Zakharyaschev, editors), World Scientific Publishing Company, 2002.
[11] Gerhard, Schurz, The Is-Ought Problem: An Investigation in Philosophical Logic, Trends in Logic, vol. 1, Kluwer Academic Publishers, Dordrecht, 1997.
[12] Dana, Scott, Advice on Modal Logic, Philosophical Problems in Logic. Some Recent Developments (Karel, Lambert, editor), Reidel, Dordrecht, 1970, pp. 143–174.
[13] Dimiter P., Skvortsov and Valentin B., Shehtman, Maximal Kripke–Type Semantics for Modal and Superintuitionistic Predicate Logics, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 69–101.
[14] Richmond H., Thomason, Some completeness results for modal predicate calculi, Philosophical Problems in Logic. Some Recent Developments (Karel, Lambert, editor), Reidel, Dordrecht, 1970, pp. 56–76.
[15] Jacques van, Leeuwen, Individuals and Sortal Concepts. An Essay in LogicalMetaphysics, Ph.D. thesis, University of Amsterdam, 1991.