Abstract
One well-known objection to supersubstantivalism is that it is inconsistent with the contingency of location. This paper presents a new objection to supersubstantivalism: it is inconsistent with the vagueness of location. Though contingency and vagueness are formally similar, there are important philosophical differences between the two. As a result, the objection from vague location will be structurally different than the objection from contingent location. The paper explores these differences and then defends the argument that supersubstantivalism is inconsistent with the plausible thesis that it is vague where I am located.
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Notes
This quick argument will be developed with more care in due course.
Or with the Rayo gloss: being located just is being identical. Officially, Dorr would formulate the thesis as: x is located at \(y \ \equiv _{x,y} \ x\) is identical to y, where the subscripts indicate that x and y are both bound by ‘\(\equiv \)’. For simplicity of notation, I will not include the subscripts throughout the paper, but will leave them implicit. I am not just taking these identities to hold case by case; rather, I take this to say that the relations are identical. Moreover, I’ve formulated the theory in what Dorr calls the sentential style. But we can also formulate (Simple-Identity) in the predicative style as follows: \(\lambda (x, y) (x\) is located at \(y) \equiv \lambda (x, y) (x\) is identical to y). Or more briefly, location is identity.
One natural question for those supersubstantivalists who also think that all regions of spacetime are material objects is whether this is necessarily the case. If so, one natural way of stating the view would be as a metaphysical analysis: \(Mx \ \equiv \ Ry\), which says that for x to be a material object is for x to be a spacetime region (and equivalently, given that ‘\(\equiv \)’ is symmetric, to be a region is to be a material object). See Dorr (2016), p. 43.
Which regions count as material objects? Nolan discusses a number of possible answers to this question. See p. 95.
Here is the argument. Consider (G-Harmony). If I am located at R, then I am identical to R, by (SS). And by Leibniz’s Law, R and I have the same properties (including geometrical properties). So (G-Harmony) is true. Parallel reasoning shows that (P-Harmony) is true.
See Leonard (2021) for more details.
My way of setting up issues concerning vagueness follows the setup in unpublished work by Jeff Russell.
Obviously, this is not the place to defend this point in any detail. However, see Williamson (1994), pp. 187–198, for an argument for why the temptation to reject classical logic for vagueness related reasons is mistaken. Moreover, see Bacon (2018), Chapter 1, for some additional problems with weakening classical logic for vagueness related reasons. Also note that supervaluationists standardly accept the directly relevant principles of classical propositional logic, like the law of excluded middle, as do the ontic views in Barnes and Williams (2009); Wilson (2017).
Though it doesn’t matter for our purposes which we take as primitive, I will assume that the following linking-biconditionals definitely hold:
$$\begin{aligned}&\nabla \varphi \ \leftrightarrow \ (\lnot \Delta \varphi \wedge \lnot \Delta \lnot \varphi )\\&\Delta \varphi \ \leftrightarrow \ (\varphi \wedge \lnot \nabla \varphi ) \end{aligned}$$Why think that (K\(_\Delta \)) is true? Here is a little argument on its behalf. Suppose that it is false; specifically, the following instance: If it’s definitely the case that if it rains, then I get wet, then if it definitely rains, then I definitely get wet—or, \(\Delta (P \rightarrow Q) \rightarrow (\Delta P \rightarrow \Delta Q)\). Suppose that is false. In other words, it’s definitely the case that if it rains, then I get wet (\(\Delta (P \rightarrow Q)\)), and yet it’s not the case that if it definitely rains, then I definitely get wet (\(\sim (\Delta P \rightarrow \Delta Q)\)). That’s equivalent to saying that it definitely rains (\(\Delta P\)) but it’s not definite that I get wet (\(\sim \Delta Q)\). But if it’s not definite that I get wet, then I may not get wet. Here’s why. Say that it may \(\varphi \) just in case it’s not definite that not \(\varphi \) (or, in symbols, \(\diamond \varphi =_{df} \ \sim \Delta \sim \varphi \)). To say that it’s not definite that I get wet is equivalent to saying that I may not get wet (that is, ‘\(\sim \Delta Q\)’ is equivalent to ‘\(\diamond \sim Q\)’). But that can’t be right. We started by assuming that it was definite that if it rains, I get wet. But if that’s true, and it definitely rains, it must be true that I definitely get wet, as well.
One can either take this as an assumption (which is what I officially will do), or one can add the Rule of Necessitation as an additional assumption, and then derive \(\Box (a=a)\) from \(a=a\) and Necessitation.
Again, I’ll take this as an assumption. But one can also derive it from Necessitation and from ‘\(a=a\)’. Necessitation says that if \(\varphi \) is a theorem, then so is \(\Delta \varphi \). Since ‘\(a=a\)’ is a theorem, we can infer that definitely \(a=a\).
Evans (1978) gave a much-discussed proof of a related conclusion: \(\nabla (a=b) \rightarrow a \not = b\). In other words, if it is vague whether a is identical to b, then a is distinct from b. However, his argument showed less than he thought. Evans seems to think that he provides a reductio on the assumption that \(\nabla (a=b)\), thereby showing that it can never be vague whether a is identical to b. However, he never actually derives a contradiction (without S5, that is, which is totally implausible for \(\Delta \), a point which is made in Heck (1998) and McGee (1997)), and thus fails to show that there can never be cases of borderline identity. He did, however, correctly show something interesting: borderline identity implies distinctness.
See, for instance, McGee (1997), p. 152, who writes “Just to make sense of the attachment of the word ‘determinately’ to an open sentence containing free variables is a bit of a stretch, since we primarily think of determinacy as an attribute of sentences.”
See, for instance, Quine (1953).
Though some further problems for metalinguistic approaches to vagueness are worth mentioning. First, there are well-known issues concerning higher-order vagueness discussed in Williamson (1994). Second, many take Montague’s Theorem to show a problem with metalinguistic approaches to modality. But as Bacon (2018) notes, the same problem arises for metalinguistic approaches to vagueness which take vague expressions to be implicitly quotational, so long as \(\Delta \) obeys the modal principles K and T. Also see Bacon (2018, Chap. 2) for additional problems with supervaluationist approaches to vagueness.
See Dorr (2016), pp. 48, 49, for a couple of different versions of this principle. And see pp. 50, 51 for a discussion of how these principles relate to the issue of opaque contexts, generated for instance by attitude ascription reports, such as “Lois believes Superman can fly.”
See, however, Garson (2014) for some hesitation regarding (B). Also see Bacon (forthcoming).
Of course, if one does accept (B\(_\Delta \)), then this argument would go through. But for those of us who find higher-order vagueness plausible, we will have to look elsewhere.
A fiddly issue arises: might the identity theory be only vaguely true? If I am located at R, and it is vague whether the identity theory is true, then it would be vague whether I am located at R. So one might think that the identity theory is consistent with it being vague whether I am located at R after all. But this would also show that the identity theory is false. By paralleling the argument for the definiteness of identity, we can use higher-order Leibniz’s Law to show that the identity theory implies the definiteness of the identity theory (and thus the vagueness of the identity theory implies the falseness of the identity theory).
One supervaluationist rejects Existential Generalization when vague names are present; see McGee (1997). For what it’s worth, rejecting (EG) seems to be a very high a price to pay.
Unfortunately, this paper is not the place to explore these interesting issues. However, here is one relevant observation. I’ve been talking about my exact location, but other authors discuss other locative relations. It’s interesting to note that even if we assume that I have a definite exact location, very little follows about the definiteness of other locative relations (defined mereologically) I bear to regions. For instance, following Parsons (2007), say that x pervades y just in case there is some region R such that x is located at R and y is a part of R. Even if I am definitely located at R, it doesn’t follow that I definitely pervade every part of R (in particular, I do not definitely pervade the particle shaped region y on the outskirts of my body where is it vague whether y is part of R).
Nolan (2014) takes something like this very seriously. He then argues that since being a part of something just is being a material object that is a subregion of another material object, since it can be vague whether some region is a material object, it could be vague whether that region is a part of something.
Here is one more (albeit exotic) example. Suppose that p is a subatomic particle having no proper parts, and suppose there is an object A such that it is vague whether p is a part of A. Also suppose that it is vague whether the number seven is a part of A, and that A has no other definite proper parts or vague proper parts. If supersubstantivalism is true, then it seems to be vague whether A is a region.
Interestingly, we can show that if the simple theory is true, then even if it could be vague where some object is located, we could never know it or never assert it. Bacon (2018) argues that the following result is derivable from the modal logic T: vague identity implies vague identity at every order. As Bacon notes, if vagueness precludes knowledge and forbids assertion, if it is vague whether \(x=y\), then we can never know it and can never assert it. But given (Simple-Identity) and higher-order Leibniz’s Law, we can substitute and show that vague location implies vague location at every order. So if there is an object such that it is vague where it is located, the simple theory implies that one can never know it and can never assert it.
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Acknowledgments
Thanks to Andrew Bacon, Maegan Fairchild, Jeremy Goodman, Dana Goswick, Hannah Kim, Shieva Kleinschmidt, L.A. Paul, Alex Skiles, Trevor Teitel, and Rohan Sud for helpful discussion. I’d also like to thank two anonymous referees for this journal for their helpful feedback. Special thanks to John Hawthorne, Jeff Russell and Gabriel Uzquiano for helpful comments and criticism on countless drafts.
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Leonard, M. Supersubstantivalism and vague location. Philos Stud 179, 3473–3488 (2022). https://doi.org/10.1007/s11098-022-01828-z
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DOI: https://doi.org/10.1007/s11098-022-01828-z