Ground and modality

ABSTRACT

The grounding relation is routinely characterized by means of logical postulates. The aim of this paper is twofold. First, I show that a subset of those postulates is incompatible with a minimal characterization of metaphysical modality. Then I consider a number of ways for reconciling ground with modality. The simplest and most elegant solution consists in adopting serious actualism, which is best captured within a first-order modal language with predicate abstraction governed by negative free logic. I also explore a number of alternative strategies by revising the ground-theoretic postulates, while keeping the modal ones fixed. As I argue, each of those strategies is either unviable, highly contentious, or insufficiently motivated.

KEYWORDS:

• Necessitism
• grounding necessitarianism
• actualism
• modal logic
• existence
• quantification
• natural properties
• totality

Acknowledgments

I would like to thank Axel Barceló, Aliosha Barranco, Daniel Drucker, Kit Fine, Martin Glazier, Jon Litland, Roberto Loss, Agustín Rayo, Carlos Romero, Alex Skiles, Nathan Wildman, and one anonymous referee for this journal for their comments and suggestions, as well as audiences at the 2019 meeting of the American Association of Mexican Philosophers at Florida State University, the workshop Issues on the (Im)Possible VII in Bratislava, and the conference ExLog 2019 at UCLouvain.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Alternatively, grounding has been modeled as a sentential operator (Fine 2012), although none of what follows turns on that distinction.

2 Hughes and Cresswell (1996, 40).

3 Aside from the standard observations, it is worth pointing out that evidence in favor of Normality also comes from the natural sciences. Indeed, the state spaces which are routinely employed in modeling the evolution of dynamical systems in physics, biology, etc are endowed with a structure satisfying the conditions of a normal modal logic (Williamson 2016). A healthy naturalistic attitude toward modal epistemology would suggest that we embrace those principles which are validated by our current best science. Williamson (2016) in fact argues that dynamical systems validate S5, as well as a form of necessitism restricted to propositions, which does not interfere with the present assumption of Contingentism. Although Williamson also makes a case for unrestricted necessitism in dynamical systems, his argument for the general case is more tentative.)

4 The irreflexive of ground has recently been challenged by Woods (2018), and defended by Kovacs (2017).

5 I take Dep to be at the heart of what Raven (2013, 198) calls Irrelevance.

6 See Makin (2017) for a different diagnosis of what goes wrong with Schaffer's argument in terms of rigid vs nonrigid ground.

7 One can then define ‘${\mathrm{\exists }}^{\ast }$’ as follows: ${\mathrm{\exists }}^{\ast }xPx:=\mathrm{\exists }x(E^{\ast }x\wedge Px)$.

8 In fact, ∃-intro${}^{SO}$ is by itself incompatible with the irreflexivity of ground, if ‘$\mathrm{\exists }X(Xa)$’ can be substituted for ‘Pa’. This ‘paradox’ could be avoided by reformulating ∃-intro${}^{SO}$ in such a way as to avoid impredicativity by implementing constraints on the kind of expressions that can be substituted in for ‘Pa’ (cf. Krämer (2013, 88) for a discussion in the analogous case of quantification into sentential position).

9 When this paper had just been accepted for publication, it was brought to my attention that there is a revision in the vicinity of Fine's ∃-intro${}^E$ that allows to block the reductio from Section 1, while retaining the standard analysis of ‘Ex’ as ‘$\mathrm{\exists }y(y=x)$’. The rule says that from Pa and Ea we can infer $|Pa|<|\mathrm{\exists }yPy|$. Since Ea is not a logical truth, this rule prevents the application of the necessitation rule, just like ∃-intro${}^E$. And since $|Ea|$ does not appear as a partial ground, the existence predicate can be unpacked in terms of existential quantification without running afoul of the irreflexivity of ground. It should be noted, however, that this proposal is unable to block a version of Fine's reductio (Section 2) in which Ea is added as an extra premise, since that proof employs Necessitarianism instead of the necessitation rule. This proposal, then, will be seen favorably by those we have independent reasons for rejecting Necessitarianism. Insofar as I am remaining impartial regarding the truth of Necessitarianism (see Section 2), I am not in a position to recommend this proposal.

10 Cowling (2012) and Finocchiaro (2019) argue that qualitative, not quantitative ideological parsimony is an epistemic virtue. If that was indeed the case, my argument against a primitive existence predicate could be resisted by regarding ‘E’ and ‘∃’ as belonging to the same ideological kind. However, reasons for being skeptical about ideological parsimony of the qualitative variety are offered in Torza (2017, 387).

11 Could the use of ‘E’ by $G_2$ be offset by the use of ‘λ’ by $G_1$? Hardly so. As Fitting and Mendelsohn (2012) have shown, there are independent reasons for adopting lambda abstraction in the context of positive free logic as well.

12 Sider's formulation of the principle is in fact more general, as is his notion of naturalness. Since the greater generality of Sider's approach is irrelevant to the present debate, I will stick to the restricted formulation. It is also worth noticing that EP raises a number of concerns which are addressed in Torza (2019). For present purpose those concerns can be set aside.

13 Nat is strictly entailed by a principle which Sider (2013) dubs purity.

14 Notice that if every material object's existence is fully grounded in a location fact, then every composite material object will have (at least) two full grounds: the collection of the existence facts corresponding to its proper parts; and a fact about its location.

15 This presupposes a thesis known as mereological harmony (see Uzquiano 2011), although the argument can be rephrased without assuming harmony.

16 Dasgupta (2014) argues that qualitativism properly construed requires that the grounding relation be plural in both argument places.

17 I am grateful to Jon Litland for suggesting this strategy.