# Determinables, location, and indeterminacy

## Abstract

Discussions about determinables and determinates, on the one hand, and discussions about (formal) theories of location, on the other, have thus far proceeded without any visible interaction, in substantive mutual neglect. This paper aims to remedy this situation of neglect. It explicitly relates (theories of) determinables and (theories of) location. First, I argue that some well known principles of location turn out to be instances of principles relating determinables and determinates. Building on this I then argue that theories of location present formidable counterexamples to those principles about determinables and determinates. One such counterexample in particular is used as an argument against disjunctivism. Finally, I relate the entire discussion to yet another crucial debate in metaphysics, that of metaphysical indeterminacy.

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## Notes

1. 1.

For an introduction see Wilson (2017b) and references therein.

2. 2.

There is no common agreed core list of such features, yet some are more central than others.

3. 3.

See Wilson (2017b, §2.1).

4. 4.

Some other formulations employ modal operators. I will not consider such a complication in the paper.

5. 5.

This formulation might be too strong. Consider the minimally specific determinable $$D = \textit{mass}$$. And consider the two determinates $$D'$$ = having a mass between 2 and 6 pounds, and $$D'' = \textit{having a mass between 3 and 7 pounds}$$. Now, suppose x has a maximally specific determinate $$d = \textit{having a specific mass value = 4 pounds}$$. Arguably, $$D'$$ and $$D''$$ are at the same level of specification of D. And x has both of them at the same time. This would constitute a counterexample to Unique Determination, as it is formulated in the text. One solution would be to restrict Unique Determination to the maximally specific level, i.e., to the level of maximally specific determinates. In what follows I focus explicitly on the maximally specific level.

6. 6.

The formalization adopted in what follows was chosen in order to make the relation between principles about determinables and principles about location as transparent as possible. I should notice that said formalization is loosely inspired by the classic reference in the literature on determinables, that is, Johnson (1921, pp. 173–185). Johnson writes: “We propose to use a capital letter to stand for a determinable, and the corresponding small letter with various dashes to stand for its determinates” (Johnson 1921, p. 179). Semi-formal classic treatments of determinables are found in Prior (1949a, b) and Yablo (1992). Classic formalizations that are different from the one proposed here are in Denby (2001) and Funkhouser (2006). These treatments differ from mine insofar as their formalizations entail Requisite Determination and Unique Determination. Yet a further different formal treatment is in Fine (2011). Fine’s formalization crucially depends on his notion of state. It would be interesting to inquire how Fine’s Directed Completeness relates to failures of Requisite Determination and more in general to disjunctivism, which will be the focus of Sect. 4.4.

7. 7.

As a matter of fact, I will focus on finite cases for the sake of simplicity, but $$D^*$$ can be an infinite, even uncountable set. Details about the cardinality of $$D^*$$ will play a role in the argument in Sect. 4.4.

8. 8.

Slightly abusing formal notation.

9. 9.

For an introduction see Varzi (2016). Following Varzi I take parthood as a primitive, and write $$x \sqsubseteq y$$ for “x is part of y”. Other mereological notions I will use include proper parthood$$x\sqsubset y = x \sqsubseteq y \wedge x \ne y$$—, and overlap$$x \circ y = (\exists z) (z \sqsubseteq x \wedge z \sqsubseteq y)$$.

10. 10.

See e.g. Casati and Varzi (1999), Hudson (2001), Sattig (2006), Hawthorne (2008) and Donnelly (2010).

11. 11.

See Parsons (2007) and Eagle (2010).

12. 12.

The names, especially Functionality, might strike as not particularly explanatory. The thought behind it is that, as Parsons (2007) puts it, Functionality makes location a function. As a matter of fact, if assumed together with Exactness, it makes location a total function over the domain of spatial entities, that is, those entities that are weakly located in space. A perhaps more immediately transparent name for Functionality would be Uniqueness of Exact Location. This would also signal its intimate relation with Unique Determination, which will be explored in what follows. Given that Functionality has become the somewhat standard term in the literature on formal theories of location though, I will stick to it.

13. 13.

Interestingly enough, if we take @ as primitive and define $$@_{\circ }$$ as: $$x@_\circ r \equiv _{df} (\exists s) (x@s \wedge r \circ s)$$, then Exactness follows. One might press the point that this is problematic, given the discussion in Sect. 4. As I pointed out, Parsons (2007) and Eagle (2010) take $$@_{\circ }$$ as a primitive and go on to define @ in its terms. Their definitions are however substantially different. It turns out that, given Parsons’ definition Functionality follows, whereas this is not true for Eagle’s definition. Thus, Eagle’s theory admits the possibility of multilocation. I believe that the theory of location Eagle presents faces independent problems, but clearly such a discussion goes beyond the scope of this paper.

14. 14.

The picture I just sketched is substantivalist insofar as it does not attempt to reduce spacetime regions to something else.

15. 15.

See Sect. 4.3.

16. 16.

Another way to express it is to use generalized identity ($$\equiv _x$$). The claim would then be “to have position just is to be weakly located at a region”, formalized as: $$P(x) \equiv _x (\exists r) (x@_\circ r$$). I am not taking a stand here, nor I need to. For generalized identity see Dorr (2016) and Correia and Skiles (2019).

17. 17.

I used the superscript because, strictly speaking $$p_i$$-s are maximally determinate properties, whereas $$p'_i$$-s are spatial regions.

18. 18.

Naturally enough, we could use generealized identity here as well.

19. 19.

There is a worry that I want to briefly discuss. It is a general unease in picking two principles out of two broadly axiomatic systems and discuss them independently from such broader systems. As I mentioned in Sect. 2, Wilson (2017b) lists fourteen different principles relating determinables and determinates. What if they make some sort of “a package deal”, so that cherry-picking some of them is not a viable option? I find this suggestion interesting. If it is on the right track it seems I am not allowed to discuss counter-examples to some principles without embedding them into the larger context. I am not sure how to make this worry more precise so as to turn it into a fully-fledged objection. But I think a few things can be said in response to the general strategy here. In order to do so, I will compare this situation to similar ones we encounter when dealing with other broadly axiomatic systems. The first example I have in mind is mereology. It is usually considered part and parcel of the notion of part that it obeys some form of supplementation. Different supplementation principles have been proposed: quasi-supplementation, weak company, weak supplementation, strong supplementation and the like. As a matter of fact, the philosophical discussion centers around how to pick the “right” axiom, so to speak—i.e., that axiom that captures the relevant supplementation intuition without thereby committing one to (allegedly) unwarranted consequences, such as extensionality. It is crucial to this discussion that we can discuss logically independent principles independently from one another. The anti-extensionalist that objects to strong supplementation by way of counter-examples does not need to bring into the discussion the partial ordering axioms for parthood. As a matter of fact, some of the arguments crucially depends on holding the partial ordering axioms fixed. That is to say, when it comes to mereology, we do not look at logically independent principles as a package deal. The second example is identity. There is a raging controversy about the Identity of Indiscernable. In the discussion about alleged counter-examples to the principle, other principles about identity such as its being an equivalence relation, or even the Indiscernibility of Identicals are held fixed. In this case, too, we discuss logically independent principles independently from one another. We don’t look at them as a package deal. I am confident these are not the only examples: dependence and grounding come to mind as well. I admit that this falls short of an argument. Perhaps the case of determinables and determinates is relevantly different from the cases I discussed. But I think that at this stage of the argument, this is enough to shift the burden of the proof. The examples show that we usually do not treat broad axiomatic systems as a package deal. We can, and in fact do, discuss individual principles independently from one another, especially when considering possible counter-examples. And this is exactly the strategy I followed. It is up to the objector to make the case that the principles relating determinables and determinates are an exception to this widespread practice. Thanks to a referee for this journal for pushing this point.

20. 20.

See e.g. Parsons (2007, §3).

21. 21.

For an introduction see, e.g. Gruszczynski and Pietruszczak (2009). For consequences of Whitehedean space for theories of location see Leonard (2018).

22. 22.

Parsons takes an omnipresent object to be an object that pervades every region—where pervasion ($$@_>$$) is defined in terms of weak location as: $$x@_> r \equiv _{df} (\forall s) (s \circ r \rightarrow x@_\circ s)$$. Yet, according to his own formal system, if something pervades a region it has an exact location.

23. 23.

Some might think that there is in fact only one ominpresent object, namely God. Sometimes this view is called panentheism. McTaggart considers panentheism in his Some Dogmas of Religion. For a discussion see Geach (1979, p. 163). See also Simons (2014). Thanks to Kevin Mulligan here.

24. 24.

That is, not overlapping.

25. 25.

For some insightful remarks on location in quantum mechanics, see Pashby (2016). Disclaimer: I do not agree with everything Pashby writes about quantum location. In any event, even Pashby agrees that a physical system, say a particle, that is not confined to any bounded region—i.e., a region with finite Lebesgue measure—has a weak but not an exact location. This is enough for Exactness to fail.

26. 26.

Kleinschmidt (2016) presents yet another violation of Exactness. Her example—the almond in the void—is a little more cumbersome. Take @ as a primitive and define weak location as follows: $$x@_{\circ } r \equiv (\exists y) (y \sqsubseteq x) \wedge (\exists r_1) (x@r_1 \rightarrow r \circ r_1)$$. Now imagine an extended simple region r that contains an almond a and its parts. a is smaller than r. In this case, Kleinschmidt argues, a does not have any exact location, yet it is weakly located at r. This constitutes a violation of Exactness.

27. 27.

See e.g. Hudson (2001), Barker and Dowe (2003), Beebee and Rush (2003), McDaniel (2003), Calosi and Costa (2015), Eagle (2016) to mention a few.

28. 28.

One might also argue that multilocation theorists should reject the claim that having a precise exact location is a maximally determinate property. Instead, they could claim that there are multiple ways to have a particular exact location. An object can have it uniquely, an object can have it as one of many exact locations, and so on. This would arguably undermine the argument from failures of Functionality to failures of Unique Determination. I am not sure I completely understand the suggestion, and how to spell it out precisely. As I pointed out already, orthodoxy has it that position (location) is relevantly similar to, say, mass. Should we say that there are multiple ways of having mass, rather than having particular mass values as maximally determinate properties? At this juncture, it is simply fair to shift the burden of the proof. It is multilocation theorists that owe us a clear, fleshed-out account of this controversial suggestion.

29. 29.

Whatever that might mean.

30. 30.

See e.g. Heil (2003).

31. 31.

See e.g. Wilson (2013), and French (2014).

32. 32.

For a discussion see Wilson (2017a, pp. 32–34).

33. 33.

The case of infinite disjunction is more complicated for infinite disjunction are not equivalent to existential statements. However, one counterexample is enough.

34. 34.

Compare this with the remark in footnote 11.

35. 35.

It would be more hazardous to generalize from position to every determinable D.

36. 36.

See Wilson (2013). For other accounts of metaphysical indeterminacy see Akiba (2004), Barnes (2010), and Barnes and Williams (2011).

37. 37.

One might think that cases of gappy MI violate Unique Determination as well. The reason is roughly the following. Cases of gappy MI are cases in which an object x has a determinable D but does not have any determinate $$d_i$$ of D. A fortiori, x does not have a unique determinate $$d_i$$ of D. Therefore it violates Unique Determination as well. This is a compelling line of thought that brings to light some subtleties about how to formulate Unique Determination. It turns out that, given the formalization in (2), cases of gappy MI do not violate it. This is because in gappy cases the antecedent of (2) will be false, thus rendering (2) vacuously true. To accommodate that cases of gappy MI do violate Unique Determination, one might give a different formalization of the latter. A proposal is the following, Unique Determination*: $$(\forall D) (D(x) \rightarrow (\exists d_i \in D^*) (d_i (x)) \wedge (\forall d_j \in D^*) (d_j (x) \rightarrow d_i = d_j)$$. In other words, Unique Determination* requires the object in question to have at least one determinate. Cases of gappy MI violate Unique Determination*, but not Unique Determination. At the bottom, this is because Unique Determination* entails Requisite Determination, whereas Unique Determination does not. This does not play any significant role in what follows insofar as our interest in cases of gappy MI will be restricted to violations of Requisite Determination. Thanks to a referee for this journal for pushing this point.

38. 38.

For this particular example see Calosi and Wilson (2018).

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## Acknowledgements

I want to thank Fabrice Correia, Damiano Costa, Alessandro Giordani, Matt Leonard, Kevin Mulligan and Jessica Wilson for several discussions on previous drafts of this paper. I also want to thank two referees for this journal for their insightful suggestions. I also want to acknowledge the generous support of the Swiss National Science Foundation (SNF), Project Number PCEFP1-181088.

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Calosi, C. Determinables, location, and indeterminacy. Synthese (2019). https://doi.org/10.1007/s11229-019-02336-0