Abstract
In this paper, I argue that for the purposes of ordinary reasoning, sentences about properties of concrete objects can be replaced with sentences concerning how things in our universe would be related to inscriptions were there a pluriverse. Speaking loosely, pluriverses are composites of universes that collectively realize every way a universe could possibly be. As such, pluriverses exhaust all possible meanings that inscriptions could take. Moreover, because universes necessarily do not influence one another, our universe would not be any different intrinsically if there were a pluriverse. These two facts enable anti-realists about abstract objects to replace, e.g. talk of anatomical features with talk of the inscriptions concerning anatomical structure that would exist were there a pluriverse. The availability of such replacements enables anti-realists to carry out essential ordinary reasoning without referring to properties, thereby making room for a consistent anti-realist worldview. The inscriptions of the would-be pluriverse are so numerous and varied that sentences about them can play the roles in ordinary reasoning served by simple sentences about properties of concrete objects.
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Notes
For example, Benacerraf (1973) raises the epistemological objection to platonism, and Field (1989) advances it (see especially pp. 25–30). Furthermore, Lewis (1986, Ch. 3, pp. 176–190) gives an argument that “magical ersatz” theories of possible worlds are unintelligible, and Nolan (Forthcoming) adapts it into a problem for ‘instantiates’ (understood broadly as predicate applying to both properties and relations). Although Nolan proposes a solution to the problem he raises, I do not think his proposed solution to the problem is successful: he proposes grasping abstract predicates by means of sentences utilizing higher-order quantification, and I doubt that one can understand higher-order quantification without already understanding ‘instantiates’ or related predicates concerning relations and their relata.
This is assuming that the things we say that seem to be about abstract objects have their apparent entailments. A nominalist might hold that platonist sentences do not have the entailments their surface grammar suggests, much as ‘the sun rises every morning’ does not entail that the sun moves around the earth. I will set this possibility aside as a concession to platonists.
Van Inwagen (2004) raises this challenge to nominalism about properties, but it has its precedent in the much-discussed Quine-Putnam indispensability argument. My response to one is much the same as my response to the other. Field offers a good discussion of the latter argument (pp. 14–20). Note that while I will focus on properties in this paper, I am only addressing the problem of ontological commitment to properties that is carried by quantifying over them and singular terms for them: I am not concerned here with, for example, arguments for their existence based on their utility in explaining resemblance or in accounting for successful predication. Regarding resemblance and predication, for present purposes I am happy to take the standard austere nominalist line and regard these as primitive (see Loux, Ch.3, pp. 52–55 on predication).
An intuitive understanding of simple sentences about properties of concrete objects suffices for most of what follows, but I can be more precise. Simple sentences about properties of concrete objects are sentences that [i] are about properties of concrete objects, [ii] are not about any other abstract objects, [iii] are such that the results of applying steps 1–5 of the paraphrase strategy (§4) to them are nominalistically acceptable and satisfy conditions (a)–(c) and (e) of Pluriverse Preservation (§4), and [iv] do not contain any expressions linking expressions for properties together (e.g. ‘is identical to’ in ‘some property is identical to some property’, ‘redness is identical to some property’, etc.).
A prominent example is Dorr’s (2008) strategy. It can be roughly characterized as follows: sentences like ‘A and B instantiate a common property’ are replaced with sentences like ‘if there were properties, A and B would instantiate a common property’. Dorr concedes that “nominalism is metaphysically necessary if true” (p. 37). If Dorr is right, his paraphrases are counterpossibles if nominalism is true. His strategy then controversially requires that counterpossibles are not trivially true (see Williamson 2017 for a defense of the trivial view). It also requires that ‘instantiates’ is genuinely meaningful, which is questionable (see fn. 2). Like Dorr’s, my paraphrase strategy makes use of counterfactuals, though not ones with impossible antecedents.
For present purposes, I assume that ‘conspecific’ is a primitive predicate. However, some might take the sentences to tacitly quantify over species. On some theories of species, species are nominalistically acceptable entities, so this poses no extra difficulty for the nominalist (see Ghiselin 1974; Hull 1978 for early representatives of the view that species are composed of the organisms belonging to it; see also Brogaard 2004). Moreover, the paraphrase strategy developed here can be used to avoid committing to species if they are regarded as properties instead (e.g. homo sapiens = the property of being human).
For a similar use of ‘paraphrase’, see van Inwagen (2004). As I use ‘paraphrase’, I do not intend for paraphrases to have the same meaning as the original sentences. A paraphrase of a sentence is just a replacement for it that is available when one is pressed to be metaphysically serious. The idea can be made more precise. Let an ideal platonist argument be a valid argument that (i) may but need not have nominalistic premises, (ii) has platonistic premises that are unimpugnable in ordinary life and no other platonistic premises, and (iii) displays crucial ordinary reasoning. A paraphrase strategy is successful for a given language if it replaces every unimpugnable platonistic sentence of that language with a true nominalistic sentence—its paraphrase—and, for every ideal platonist argument of that language, replacing its platonistic sentences how the paraphrase strategy recommends results in an argument such that its conclusion is entailed by its premises and, if needed, some other true nominalistic principles. (Here, a “language” could be a fragment of a larger language.) Paraphrases thus preserve the reasoning important for ordinary affairs while dispensing with everything that is problematic for nominalists.
Understand all references to concrete linguistic entities below to be references to ones that are singular in meaning. They should exhibit no context-sensitivity or other variability in meaning.
Sellars (1963) offers a nominalist system that gives analyses of the sentences about properties in Argument 1 that very loosely resemble the replacements of them I will offer on behalf of the obvious nominalist response. Like the obvious nominalist response, in my estimation Sellars’ system cannot adequately handle cases where there are important properties without any corresponding inscriptions.
Geach (1962/1980, p. 151).
Platonists might reasonably complain that I have not nominalized the notion of Tarskian English. The enumerated anatomical indicators are not intrinsically meaningful: their being meaningful involves some relation to English speakers who understand first-order languages. I will leave nominalizing English, first-order languages, and the relevant semantic and syntactic relations as a further project.
That said, a one-place open sentence-token can have more than one free variable-token provided that every free variable-token in it is a semantic and syntactic duplicate of its first one, as with *x loves x*.
The tricky part is with what to replace ‘the property of having a prosoma’, a singular term. I choose to replace it with a phrase indicating universal and existential quantification over open sentence-tokens that solely indicate of things that they have prosomas.
For another nominalist system that appeals to possibilities for open sentence-tokens, see Charles Chihara (1990). Chihara uses “constructibility quantifiers” and facts about what open sentence-tokens can be constructed to provide a nominalist foundation for mathematics. I object to Chihara’s constructibility quantifiers, but I suspect that his system could be modified along the lines of my proposal by making it exploit facts about what open sentence-tokens would exist were there a pluriverse (see §4) rather than exploit facts about what open sentence-tokens can be constructed. Demonstrating this is a task for another time.
Universes match Lewis’ (1986, Ch. 1) view of possible worlds. Unlike Lewis, I don’t use universes to analyze possibility.
Or rather, all expressions corresponding to qualitative properties of concrete objects are tokened, plus some expressions corresponding to non-qualitative properties as indicated in fn. 18. Kaplan’s-Paradox-like considerations reveal that if every property is expressible, it’s impossible for all possible expressions to be jointly tokened. Necessarily, if some token exists, there are more non-qualitative properties concerning pluralities of tokens (e.g. being among the tokens x, y, and z; not being among the tokens x, y, and z) than there are tokens. Happily, only those non-qualitative properties that can be expressed by tokens created via the method described in fn. 18 are of importance, and the possibility of those tokens existing alongside tokens that express every qualitative property does not result in contradiction.
Or rather, for every important such property. I assume that every qualitative property of concrete objects can be expressed by some token. Moreover, it is possible for non-qualitative properties of concrete objects belonging to one universe (e.g. the property of being identical to Cleopatra) to be expressed by tokens in other universes: all that is needed is for there to be some universe U in which there is a description of the first universe down to the microscopic level, followed by the introduction of descriptions for the individuals the non-qualitative properties concern. Names can then be affixed to the individuals through the descriptions in U, and open sentences involving those names can then be tokened in U. For example, an immortal being or eternally running machine with a vocalization apparatus might declare “there is a universe such that… [fill in a description uniquely satisfied by our universe]. Let ‘A’ be a name for the thing in that universe such that [fill in a description uniquely satisfied by Cleopatra in our universe]” and then inscribe the sentence ‘x is identical to A’. The resulting open sentence-token then expresses the property of being identical to Cleopatra. (See Sinhababu (2008, pp. 255–256) for a similar point concerning de re thought.) Note that this suggestion does not entail that every non-qualitative property of concrete objects is expressible by tokens: no gerrymandered, uncountable plurality of concrete things is such that any token expressing the property of belonging to it could be generated through even an idealized extension of this process.
Abstract possible worlds aren’t, anyway. There aren’t enough universes to serve as concrete possible worlds, but if there were, the obvious nominalist response would be adequate.
See fn. 18 to see how the pluriverse captures de re content about objects within universes. For every object composed out of parts that exist in multiple universes, I assume that if possibly, there is a pluriverse, it exists, and it is not referred to by any expression-token, then necessarily, it is not referred to by an expression-token. As for merely possible individuals, it is plausible that they cannot be referred to until they come into being, and even the existence of a pluriverse does not guarantee that every possible individual is brought into existence.
It is worth remarking that the ideology of the paraphrases is only modestly more complex than the ideology involved in property discourse. Where property discourse involves ‘instantiates’, the paraphrases have ‘satisfies’; where property discourse has ‘anatomical property’, the paraphrases have ‘anatomical indicator’; etc. The ideological additions that lack a platonic parallel are the counterfactual operator and ‘are as varied as it is possible for any universes to be’, which is used in defining ‘pluriverse’. But while platonists could analyze the counterfactual operator in terms of a nearness relation between possible worlds and define ‘pluriverse’ in terms of ways universes can possibly be, the addition of two ideological primitives is a small price to pay for avoiding the extremely abundant ontology of platonism.
Note that this is intended to include trivial cases, e.g. where P and Q are both ‘Tom is excitable’, or are both ‘Sarah is resilient’, or… etc.
The structure of argument displayed here resembles Lukas Skiba’s (Forthcoming) adaptation of Richard Woodward’s (2010) general proof of inferential safety for fictionalists. Woodward and Skiba interpret the property fiction as Dorr’s paraphrase strategy, and their proofs therefore inherit all the same problematic assumptions (see fn. 6)—though Woodward is more sanguine than I or Dorr on the possibility of properties. However, because the structure of argument here so closely resembles Skiba’s, it is likely that a very similar general proof of inferential safety could be provided for the paraphrase strategy I am proposing.
One might think that universes are necessarily causally isolated de dicto—i.e. one might think that necessarily, no part of one universe causally interacts with a part of another universe—while denying it de re—i.e. while denying that necessarily, every universe u is such that necessarily, no part of u causally interacts with a part of a universe that is not u. For example, if we call our universe Alpha, the de re reading is false if it is metaphysically possible for Alpha to be a part of a larger universe. If such a possibility were to obtain, Alpha would not be a universe at all, but rather a proper part of one. Parts of Alpha could then causally interact with parts of the universe of which Alpha is a part that are not themselves parts of Alpha. I grant that the de re reading might be false, but nonetheless maintain that were there a pluriverse, Alpha would retain its universe status and remain causally isolated. This is because (speaking platonistically) of all the possible worlds that contain a pluriverse, the ones nearest to the actual world are the ones in which the actual things are only spatiotemporally related to each other and thus still compose a universe. Changing what spatiotemporal relations the actual things stand in, even with respect to concrete objects not among them, reflects a greater departure from what is actually the case than holding those relations fixed. The different possible worlds in which enough and only enough concrete objects are added to the actual things for there to be a pluriverse are qualitatively indistinguishable: the only aspect in which these worlds can differ in their closeness to the actual world is in how similar the actual things are in them compared to the actual world, and composing a universe is a significant dimension of similarity. If the point remains in doubt, the antecedents of the paraphrases could be replaced with ‘there is a pluriverse and the actual things are only spatiotemporally related to each other’, which guarantees that Alpha remains a universe in the relevant nearby possibilities.
This is not surprising, for as mentioned in fn. 24, it is likely that a general proof of inferential safety can be provided for the paraphrase strategy that resembles the one Skiba provides for fictionalists.
Note that while I do not endorse any particular conception of the intrinsic/extrinsic distinction, I take it as a constraint on any plausible construal of that distinction that if something has or lacks an intrinsic property when the domain of discourse is taken to be unrestricted, it must have or lack that property on every restricted domain as well. Thus, however the intrinsic/extrinsic distinction is understood, a sentence about properties—intrinsic or extrinsic—is appropriate for acceptance in ordinary life iff its paraphrase is true under the restricted interpretation described above. This point holds even if there is disagreement about which properties are intrinsic and which properties are extrinsic. Much the same can be said about the internal/external distinction, sentences about properties containing internal/external expressions, and their paraphrases.
See the Appendix of Lewis’ Parts of Classes (1990), co-written with John P. Burgess and A.P. Hazen; Lewis’ (1993) “Mathematics is Megethology”; and Chapter 7 of Nolan’s Topics in the Philosophy of Possible Worlds (2002). Nolan’s variation would be particularly helpful. More speculatively, Chihara’s (1990) system is perhaps modifiable to fit within a pluriverse-counterfactual framework (see fn. 15).
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Acknowledgements
I am extremely grateful to Peter van Inwagen, Cian Dorr, Daniel Nolan, Jeff Speaks, Sara Bernstein, Geoffrey Hall, Benjamin Middleton, and my anonymous reviewers for very useful comments on this manuscript and related discussion.
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Himelright, J. Paraphrasing away properties with pluriverse counterfactuals. Synthese 198, 10883–10902 (2021). https://doi.org/10.1007/s11229-020-02757-2
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DOI: https://doi.org/10.1007/s11229-020-02757-2