Williamson on necessitism

Abstract

I critically discuss some of the main arguments of Modal Logic as Metaphysics, present a different way of thinking about the issues raised by those arguments, and briefly discuss some broader issues about the role of higher-order logic in metaphysics.

Keywords:

• Metaphysics
• modality
• higher-order logic
• necessitism
• Williamson

Acknowledgements

Thanks to Cian Dorr and Peter Fritz for comments on an earlier version of this paper and to audiences at the 2014 Logical and Modal Space workshop at NYU, the 2015 Williamson on Modality workshop in Montreal, and the 2015 Williamson, Logic and Philosophy conference at Peking University.

Notes

1 All page references are to Williamson (2013).

2 See Sider (2011) and Dorr and Hawthorne (2013) for discussion. It is also not entirely clear how to formulate the relevant supervenience theses in a higher-order setting. If we assume that necessarily equivalent propositions are identical (as Williamson does), then there is only one necessary proposition, and so necessity can be defined in purely logical higher-order terms as being a tautology (e.g. ). And normally, when we formalize the claim that a property (strongly) supervenes on a set of properties and relations, we really mean that it supervenes on the set of properties and relations obtained by closing the original set under certain logical operations. (Sometimes such theses are instead formulated in terms of isomorphisms between possible worlds, but as Williamson points out only necessitists can take such formulations at face value.) In the present higher-order setting, this would then have the effect of making necessity part of every supervenience base, in which case the modal will trivially supervene on the non-modal.

3 See Fine (1977b) and Fritz and Goodman (forthcoming-b) for an exploration of such views in the setting of higher-order modal logic.

4 Consideration of modal truths specified in counterfactual terms, rather than merely in terms of what is necessary or possible, also suggests a way of formulating the ground-theoretic objection in a non-hyperintensional setting, since such truths are not invariably non-contingent.

5 This principle is a consequence of (the modal analogue of) the tensed mereology developed in Hovda (2013); for related ideas, see Hawthorne (2006). Fine (1999) also seems to be committed to such a principle, although since he also takes the relevant notion of parthood to be anti-symmetric, he thereby flirts with paradox; see Goodman (xxx-b) for discussion, where I explore a slightly weakened (but for present purposes strong enough) principle restricted to properties X that, necessarily, have extensions not too big to form a set.

6 An object-language argument for this conclusion would be quite involved, but one can give a relatively straightforward model-theoretic argument relative to the class of structures discussed in Fritz and Goodman (forthcoming-b).

7 See Goodman (xxx-a) and Fritz and Goodman (Forthcoming) for discussion.

8 Of course, those who, unlike Williamson, accept hyperintensional theories of properties might claim that, although it could turn out to be true ‘by accident’ that necessarily everything necessarily has a haecceity in Williamson’s sense, purely modal-metaphysical considerations establish that there could not be the property of being me without there being such a thing as me.

9 The second language also contains polyadic generalizations of plural quantifiers, which we might think of as plurally quantifying over ordered n-tuples of individuals; I will ignore this subtlety in what follows. Williamson also interprets the plural variables in such a way that they can have empty extension – i.e. there are some things (the ‘empty plurality’) of which nothing is one.In addition to Boolean connectives and necessity and possibility operators, both languages also include two operators and which can be seen as generalizing the more familiar rigidifying ‘actually’ operator: has the effect of making the subformula it embeds to be evaluated at the modal scope of any binding it, where an occurrence of binds an occurrence of just in case the latter is in the scope of the former and no occurrence of has intermediate scope.

10 For purposes of illustration, we might take chunkiness to be being grounded in the concrete and chunkiness* to be instantiating some fundamental property or standing in some fundamental relation, on the assumption that these are distinct properties.

11 Williamson mentions making special provision for sets, but if what he has in mind is something like the proposal in Fine (1977a), it would solve the problem for the ‘is a member of’ predicate but not for the ‘is the favorite set of’ predicate. Williamson also justifies the focus on chunky-style necessitsm by saying:

[I]t is in the spirit of the preferred form of necessitism explained in Chapter 1 to characterize non-chunky things mainly in modal or temporal terms, through the properties they would or could have if they were chunky, or did have when they were chunky, or will have when they are chunky. Such necessitists may be happy to confine their primitive non-logical predicates to those meeting the constraint [of being chunkiness-entailing]. [...] Thus we envisage a necessitist who asserts [the relevant chunkiness-entailment principle] for every primitive non-logical predicate in the language. (326,327).

In reply, the ‘favorite set’ example shows that, although it may be natural for necessitists to characterize non-chunky objects ‘mainly in modal or temporal terms’, they are not likely to characterize them only in such terms, which is what matters in the present context.

12 The interpretation of chunkiness as contingent non-concreteness has the advantage of making the chunky-style necessitist’s claim that everything is possibly chunky uncontroversial, since for any condition F it is uncontroversial (assuming S5) that necessarily everything is possibly not contingently not F. The issue is then whether, so understood, the realm of the chunky does in fact constitute ‘neutral ground’ in Kit and Bob’s metaphysical dispute.

13 One might think that, when formulated in supervenience-theoretic terms, the floating-free argument for contingentism can be seen as a version of CSC with chunkiness interpreted as instantiating some fundamental property or standing in some fundamental relation. But this is not so, since the floating-free objection so construed would then not apply to the conjunction of necessitism with the claim that identity is a fundamental relation, a view to which the objection clearly does apply when construed in terms of supervenience.

14 In particular, they discuss ways of adding a ‘qualitative’ operator to modify their schema Comp so as to formalize the Fine-inspired view they call the ‘qualitative generation view.’

15 Of particular relevance are the main result of Fritz (Forthcoming) and the connection Fritz (2013) establishes between such results and the existence of neutral equivalents given CSN, although the latter paper only considers first-order languages.

16 Briefly, here is why. Let abbreviate the formula . One feature of the relevant class of models is that, at each world of every model, the proposition true at only that world is in the -domain of that world. And relative to an assignment of this ‘world proposition’ to p, is true at the world of evaluation just in case has universal extension at that world – not merely in the sense of being satisfied by everything in the t-domain of that world, or even the t-domain of some world, but rather that it be satisfied by any type-t intension definable in the model, even ‘impossible’ ones in the t-domain of no world. It is this feature that lets behave like a ‘classical’ quantifier, in the sense that the -analogue of the unrestricted comprehension schema comes out valid. This in turn allows us to use -quantification of variables of type to simulate necessitists’ plural quantification in a reasonably mechanical way.

17 To say that the quantifiers are not interpreted as ‘unrestricted’ is not to say that they receive a restricted interpretation; see Dorr (2005, 2008).

18 For the relevant notion of a generalized quantifier, (see Westerståhl 2011).

19 See Fritz and Goodman (forthcoming-b, [Section 3.4]) for an explanation of how to make this notion precise.

20 My own idiosyncratic views about granularity make but not behave in an entailment-like way, but that is a long story.

21 The argument is inspired by Dorr (2014); I develop it in Goodman (2016) [Chapter 5].