A nonhybrid logic for crossworld predication


E.V. Borisov


Some sentences of natural languages, such as John might have been taller than Mary is, ascribe relations to objects each of which is associated with a possible world. In this example, Mary is associated with the actual world whereas John is associated with a possible world that can be distinct from the actual one. This phenomenon — the phenomenon of crossworld predication — cannot be reflected by means of standard modal logic. Some nonstandard logics accommodating crossworld predication have been proposed; the most expressive among them is Kocurek’s hybrid logic. All of them make use of nonstandard operators such as the operators of hybrid logic. In this paper, I propose a nonhybrid first-order modal logic for crossworld predication, CWPL, that demonstrates significant expressive power (though is less expressive than Kocurek’s hybrid logic) and has that advantage over other proposals that it makes no use of any nonstandard operators. Semantically, CWPL is based on crossworld interpretation of predicates that assigns extensions to each n-ary predicate with respect to n-tuples of possible worlds rather than single possible worlds. To be able to employ crossworld interpretation of predicates when evaluating formulae, I suggest relativizing truth values to partial functions from variables to possible worlds. In the paper, I describe CWPL’s syntax and semantics, provide a translation of CWPL into a two-sorted first-order logic, and compare it with Kocurek’s hybrid logic in terms of expressive power.




Non-classical logics



Borisov, 2020 – Borisov, E.V. “Logika dlya kross-mirovoy predikatsii” [A logic for crossworld predication], Nauka kak obshchestvennoye blago [Science as a public good], Vol. 4, ed. by I.T. Kasavin and L.V. Shipovalova. Moscow: Russian Society for History and Philosophy of Science, 2020, pp. 205–209. (In Russian)
Butterfield, Stirling, 1987 – Butterfield, J., Stirling, C. “Predicate modifiers in tense logic”, Logique et Analyse, 1987, Vol. 30, pp. 31–50.
Fitting, Mendelsohn, 1998 – Fitting, M.C., Mendelsohn, R. First-Order Modal Logic. Dordrecht: Springer Science+Business Media, 1998. 287 pp.
Fitting, 2017 – Fitting, M. “On height and happiness”, Rohit Parikh on Logic, Language and Society, ed. by C. Baskent et al. Cham: Springer, 2017, pp. 235–258.
Kemp, 2000 – Kemp, G. “The Interpretation of Crossworld Predication”, Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, 2000, Vol. 98, pp. 305–320.
Kocurek, 2016 – Kocurek, A.W. “The problem of cross-world predication”, Journal of Philosophical Logic, 2016, Vol. 45, pp. 697–742.
Lewis, 1968 – Lewis, D.K. “Counterpart Theory and Quantified Modal Logic”, The Journal of Philosophy, 1968, Vol. 65, pp. 113–126.
Priest, 1998 – Priest, G. An Introduction to Non-Classical Logic. From If to Is. Cambridge: CUP, 2008. 613 pp.
Russell, 1905 – Russell, B. “On Denoting”, Mind. New Series, 1905, Vol. 14, pp. 479493.
van Benthem, 1983 – van Benthem, J. Modal Logic and Classical Logic. Naples: Bibliopolis, 1983. 234 pp.
von Stechow, 1984 – von Stechow, A. “Comparing semantic theories of comparison”, Journal of Semantics, 1984, Vol. 3, pp. 1–77.
Wehmeier, 2012 – Wehmeier, K. F. “Subjunctivity and cross-world predication”, Philosophical Studies, 2012, Vol. 159, pp. 107–122.
Wehmeier, R¨uckert, 2019 – Wehmeier, K., R¨uckert, H. “Still in the Mood: The Versatility of Subjunctive Markers in Modal Logic”, Topoi, 2019, Vol. 38, pp. 361–377.