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Abstract
Kripke models, interpreted realistically, have difficulty making sense of the thesis that there might have existed things that do not in fact exist, since a Kripke model in which this thesis is true requires a model structure in which there are possible worlds with domains that contain things that do not exist. This paper argues that we can use Kripke models as representational devices that allow us to give a realistic interpretation of a modal language. The method of doing this is sketched, with the help of an analogy with a Galilean relativist theory of spatial properties and relations.
Notes
Apologies to Hilary Putnam for appropriating the title of one of his famous papers (Putnam Citation1980). My topic is not unrelated to his, but I am not going to talk about his paper. I chose the title because it says what my paper is about.
1. Kripke Citation1980.
2. See Lewis Citation1986.
3. I will ignore the binary accessibility relation that is a constituent of a frame in a general Kripke model structure for modal languages, restricting attention to the S5 case where the accessibility relation is universal; necessity (in a given world w) is truth in all possible worlds, and not just in all that stand in some accessibility relation to w.
4. Barwise and Perry Citation1985. 120.
5. The kind of permutation strategy I am sketching is familiar in mathematics, and in measurement theory, but it is important to distinguish the use of this strategy by the Galilean from its use to bring out the conventionality of units of measurement. Newtonians and Galileans will agree, for example, that the choice of a coordinate system for physical space and the unit of measurement for representing distance relations are conventional. Any permutation of the numbers used to represent the distance between two bodies that preserves the ratios between distances will be an equivalent representation. What is permuted, in this case, is the relation between a number and the relational property that holds between two bodies iff they are a certain fixed length apart. The relation, x being 2 meters from y is the same relation as x being 4 half-meters from y, but the Newtonian and the Galilean agree that this is a real relation. Consider a possible world like ours, except that all the distance relations between bodies were doubled. This is very different from considering a world in which we meant by ‘one meter’ a distance relation that was what we actually mean by ‘one half meter’ (cf. Grünbaum Citation1964). The Newtonian claims that there are location properties, even though it is conventional what quadruples of real numbers we use to represent them. The Galilean, in contrast, denies that there are such properties. This disagreement is reflected in the permutation functions that are permitted by the Galilean’s theory, which cannot be interpreted as permuting just the numerical representation of spatio-temporal locations. The disagreement is about whether there is a real difference between rest and uniform motion at constant velocity.
6. It is an irony that Leibniz is both the one who propounded this grounding thesis, and also a prominent defender of a relational theory of space. His views are reconcilable, at least on the surface, since on his metaphysical theory, substances do not stand in spatial relations. His relationism about space is more radical (at least on one way of reading him) than that of the Galilean: space itself is merely ideal. The view seems to be that spatial relations between phenomena are ultimately reducible to the intrinsic properties of substances.
7. I think one should think of physical space itself as a structure of properties and relations of the things in space. On this way of thinking, for the absolutist, spatial locations, and regions defined by sets of specific locations, are intrinsic properties of the things at the location, or in the region. Spatial relations are determined by these intrinsic properties: once you have identified the specific locations of the relata, you have determined how they are spatially related to each other. But for the Galilean, the Leibnizian principle fails.
8. Stalnaker Citation2000.
9. Gallin Citation1975.
10. See Williamson Citation2013, 221ff.
11. As I understand him, Williamson interprets this domain as the domain of absolutely everything, including all of the members of the domains that are the other types. But as I suggested in section 3, we need not take a realistic specification of the subject matter of some language – even a language for doing metaphysics – to be comprehensive. So we could understand the type e as just a class of basic individuals, and take the other types to be classes of entities that are disjoint from the things of type e.
12. The motivation for the label, and for the idea that this type is really a derived type, is the idea that just as n-ary relations can be thought of as properties of n-tuples, so propositions can be thought of as 0-ary relations, properties of 0-tuples.
13. So our model structure is what Williamson calls an ‘inhabited model structure’. There is a ‘fixed point’ constraint on the permutation functions, ensuring that in any of world w’s permutation functions will take w to itself, so the actual world of the model structure , with its domain, will be, by definition, invariant.
14. I say that if the primitive predicates sentence letters and names, all denote real entities of the appropriate kind, then so will the complex closed predicates and sentences. As Bruno Jacinto has shown (in as yet unpublished work), this is not true for open expressions, which have as their semantic values propositional functions, propositional function-functions, functions from individuals to properties, etc. Virtual propositional functions seem to play an ineliminable role in the compositional process by which complex predicates and quantificational constructions are interpreted, even though all of the complex predicates and closed quantificational constructions get an invariant interpretation. I should perhaps not have found this as surprising as I did, since the whole detour through virtual models (that in a sense, represent reality as more fine-grained than it is), together with permutation functions (that serve to bring us back to the right grain in our account of what is real) would be unnecessary if we could do the compositional semantics directly in terms of the real entities that our language is talking about.
15. I spell out some of the details in two appendices to my book, Stalnaker Citation2012, but the appendices are overly compressed, and contain some errors, which have been pointed out and corrected in work by Peter Fritz (Fritz, Citationforthcoming).
16. Jones Citation2016.
17. I take some responsibility for the latter conflation, since as Williamson notes in a footnote (Williamson Citation2013, 189, note 48), I confusingly use the label ‘model’ for what is in fact a model structure. He assumes I must really have meant to talk about a model (the structure plus a valuation function), but what I meant was just a model structure.
18. Williamson Citation2013, 191.
19. Thanks to Peter Fritz, Bruno Jacinto, Nicholas Jones and Tim Williamson for very helpful discussion and correspondence about these issues.