Subjunctivity and cross-world predication

Kai F. Wehmeier 0 0 K. F. Wehmeier (&) Department of Logic and Philosophy of Science, School of Social Sciences, University of California , Irvine , 3151 Social Science Plaza , Irvine, CA 92697-5100, USA The main goal of this paper is to present and compare two approaches to formalizing cross-world comparisons like ''John might have been taller than he is'' in quantified modal logics. One is the standard method employing degrees and graded positives, according to which the example just given is to be paraphrased as something like ''The height that John has is such that he might have had a height greater than it,'' which is amenable to familiar formalization strategies with respect to quantified modal logic. The other approach, based on subjunctive modal logic, mimics the mixed indicative-subjunctive patterns typical of cross-world comparisons in many natural languages by means of explicit mood markers. This latter approach is new and should, for various reasons, appeal to linguists and philosophers. Along the way, I argue that attempts to capture cross-world comparison by means of sentential operators are either inadequate or subject to substantive logical and philosophical objections. The notion of cross-world comparison is perhaps best introduced by contrasting it with its intra-world cousin. Consider the sentences in groups 1 and 2 below. - (2a) John might have been taller than Mary. (2b) Johannes hatte groer sein konnen als Maria. (2c) Jean aurait pu etre plus grand que Marie. The sentences in group 1 make cross-world comparisons: They say that there is a possible world w such that John, as he is in w, is taller than Mary, as she is in the actual world w . The sentences in group 2, though they are sometimes used to convey what their counterparts in group 1 express, are most naturally read as making intra-world assertions to the effect that there is a world w such that in w, John is taller than Mary. Standard quantified modal logic (QML) is built on the assumption that all predication is intra-world: Its predicate symbols R are assigned extensions Rw relative to each possible world w, and whether the relation expressed by R holds of some tuple of arguments a1; . . .; an at w is determined by whether the argument tuple belongs to Rw. By contrast, the formalization of cross-world comparison seems to require, on the face of it, that predicate symbols be assigned extensions relative not just to one world, but across worlds. The plan of this paper is as follows. I will first define, in Sect. 2, a notion of Kripke structure that incorporates cross-world extensions, and consider attempts to implement cross-world comparison in QML by means of sentential operators. Such attempts will turn out to be either inadequate to the task or logically and philosophically unattractive. In Sect. 3, I develop an extension of the subjunctive modal logic (SML) introduced in Wehmeier (2004) that allows for a simple and natural way of formalizing cross-world comparisons. Section 4 is dedicated to the received strategy, going back to Russell (1905), of analyzing cross-world comparison in terms of abstract objects (degrees) added to QML. This approach is based on a somewhat different notion of Kripke structure, which I define precisely. In Sect. 5, I compare the methods discussed in the preceding two sections and argue that the SML-approach developed in Sect. 3 has a leg up over the QMLbased analysis: First, it can make do without the latters ontological commitments to abstract objects, and second, when combined with a degree-analysis, it is able to represent as substantive certain inferences that the QML-analysis must relegate to the level of informal paraphrase. A note about terminology: By comparison I understand, for the purposes of this paper, the attribution of a comparative relation (strict or non-strict), or of an equivalence relation, to a pair of individuals. For the sake of exposition, I will discuss only the case of one strict cross-world comparative relation. The case of non-strict comparatives is entirely analogous, and that of equivalence relations is simpler. I will briefly indicate in the notes the modifications that would need to be made in order to cover those other cases. 2 Cross-world extensions and sentential operators By a signature R we will mean a pair R C; P, consisting of a set C of individual constants and a set P of predicate symbols. Each P 2 P has a unique arity ]P 1, and the binary predicate symbol = is always a member of P. A (standard) Kripke structure K for such a signature R is a quintuple where K is Ks set of worlds; w 2 K is the actual world of K; for each w 2 K, Dw is a set (the domain of individuals existing in w) such that D : SfDwjw 2 Kg 6 ;; for each constant c 2 C, cK 2 Dw ; and for each n-ary predicate symbol P 2 P, Pw (the extension of P in w) is a subset of Dwn. It is required that =w is true identity on Dw.1 I will assume familiarity with the notion of a formula of the language LR of QML for the signature R, as well as the notion of the truth of such a formula / at a world w of a Kripke structure K under some variable assignment r, K w/r . Two points are perhaps worth recalling: First, by definition, / is true in K (under r) if it is true in K at the actual world w of K (under r). Second, in order to remedy certain expressive defects (of which more in Sect. 3), one often sees included in the language of QML a so-called actuality operator A.2 This is a unary sentential operator with the property that a formula A/ is true at w in K under the assignment r just in case / is true at the actual world w of K under r. We will assume that the language of QML contains the actuality operator, unless otherwise indicated. The most straightforward way of interpreting cross-world predications semantically is to construe them as asserting relations between individuals in different possible worlds (typically, if not invariably, the actual world and a counterfactual one). For instance, the sentences in group (1) above make a comparison between John in some possible world w and Mary in the actual world w . Suppose, then, that R is a binary predicate intended to stand for a cross-world comparative (say the predicate of being taller than, for definiteness). Standard Kripke structures K assign R an extension with respect to each possible world w, so that we may compare the individuals in any Dw with respect to R. For cross-world predications, we also want to compare individuals from any Dv with individuals in any Dw with respect to R, and hence we will assign an extension Rv;w Dv Dw to R for any pair hv, wi of possible worlds. To say that John, in w, is taller than Mary is in w then just means that the pair hjK; mKi belongs to the extension Rw;w . Formally, we proceed as follows. Let R C; P be a standard signature, and let R be a binary predicate symbol. A cross-world Kripke structure (CW-structure for short) K for R with the cross-world comparative R i (...truncated)