Abstract
Nominalists are confronted with a grave difficulty: if abstract objects do not exist, what explains the success of theories that invoke them? In this paper, I make headway on this problem. I develop a formal language in which certain platonistic claims about properties and certain nominalistic claims can be expressed, develop a formal language in which only certain nominalistic claims can be expressed, describe a function mapping sentences of the first language to sentences of the second language, and prove some facts about that function and facts about some sound logics for those languages. In doing so, I prove that, given some plausible metaphysical assumptions, a large class of sentences about properties of concrete objects are “safe” on nominalistic grounds. Whenever some true sentences about concrete objects and some sentences belonging to this class that are true according to platonists collectively entail a conclusion about concrete objects, some nominalistically acceptable sentences are true and entail the same conclusion. Because the proof can itself be formulated without abstract objects, it provides a nominalistic explanation of the success of theories that invoke properties of concrete objects.
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Notes
The business of proving safety results has a long history. John Divers (1999) provides a safety result for possible world talk, while Richard Woodward (2010) and Lukas Skiba (2019) have provided safety results for abstract object talk in general. Unfortunately, Woodward’s proof and Skiba’s proof suffer from a difficulty encountered by the proof of safety I discuss based on Cian Dorr’s (2008) paraphrases of claims about abstract objects: they require that either nominalism is merely contingently true or that counterpossibles are non-trivial. As will be seen, the proof of safety I endorse does not face either limitation. However, it should be noted that both Woodward’s proof and Skiba’s proof have the advantage of applying to talk of composite objects. My proof of safety is of no help to mereological nihilists. (To clarify, Woodward’s proof needs the existence of abstracta to be contingent to apply to talk of abstract objects and needs the existence of composites to be contingent to apply to talk of composite objects. By contrast, Skiba’s proof applies in either case if the relevant phenomenon is contingent or counterpossibles are not trivial.)
More accurately, in any schema instance, let ‘χχ’ be a plural variable for things among the denotation of the instance of ‘X’ in that schema instance, and likewise for the rest. This degree of precision is cumbersome, so I opted for something simpler, albeit strictly speaking incorrect.
Charles Chihara (1990, pp. 162–3) raises the point about Field using set theory in his proof, and Stuart Shapiro (1983) criticizes Field for using a model-theoretic notions in characterizing conservativeness. On the latter issue, see Field (1985) for a reply and Chihara (1990, pp. 153–159) for discussion. Field (1992) provides a nominalistic proof of conservativity, and Field (1998) provides a nominalistic account of conservativeness.
Dorr’s strategy requires that there be some nominalistic way of interpreting counterfactuals, which are standardly analyzed in terms of possible worlds. One way of doing so would be by a strict entailment view of counterfactuals. On this view, a counterfactual of the form φ \(\square\kern-3pt\rightarrow\) ψ really has the logical form of □ [(φ ∧ χ1 ∧ … ∧ χN) → ψ], where χ1 ∧ … ∧ χN are supplied by context. I neither affirm nor deny this view, but I assume that some nominalistic way of interpreting counterfactuals is correct.
For an example of such a logic, see David Lewis (2001/1973, p. 132).
Though widely held, this point is controversial. Several philosophers, such as Field (1993) and Mark Colyvan (2000), have maintained that the truth or falsity of nominalism is contingent (see Kristie Miller [2012] for discussion). Still, even if it is not settled that nominalism is either necessarily true or necessarily false, that there is controversy makes it best to have a solution to the problem of inferential safety that is consistent with nominalism being necessary.
I thank Benjamin Middleton for making this point to me.
Additionally, if counterpossibles are not trivial, it is plausible that deduction within counterfactual conditionals is invalid. However, Skiba (2019) has addressed how a safety result can be obtained using Dorr’s sort of strategy even without deduction within counterfactual conditionals.
Note that Field is explicitly a mathematical fictionalist, and Skiba treats a Dorr-style paraphrase as constituting a fictionalist approach to abstract objects. My approach could be regarded as fictionalist as well, though I don’t think much hangs on whether it counts as a kind of fictionalism or some other kind of nominalist theory.
In my 2020 paper, I alluded to the fact that a safety result is likely available for the paraphrase strategy I offered, though I did not show that this is so (see fn. 24 of that paper). The current paper makes good on that claim for a slightly tweaked version of that paraphrase strategy.
Field and Dorr are not the only philosophers to propose nominalist systems. For example, Chihara has his own, and Geoffrey Hellman (1989) has developed one as well. It would take me too far afield to critique them all. I will note, however, that even if these alternative proposals are successful, they are generally aimed at mathematics. Property discourse has been much neglected by comparison, and part of what I am attempting to do is to fill in some of the gaps left by the focus on mathematics.
The arguments excluded from the scope of this claim are those that cannot be represented in the formal language L to be developed later in this paper.
Of course, I have quantified over sentence-types rather than sentence-tokens in describing the structure of the proof, which might threaten its nominalistic status. I suspect that it would be easy to construct a notion of deductive consequence appropriate to tokens and recast the proof in terms of the possibilities for inscribing sentence-tokens. However, I shall not attempt this here. Goodman and Quine (1947) developed a proof system for tokens that serves as an illustration of how the first part of this task can be done.
At least, this is true in the sense of ‘actual’ captured by the actuality operator.
Core sentences largely overlap with what I have elsewhere described as simple sentences about properties of concrete objects (2020, fn. 5), except that core sentences can also contain an identity predicate for sproperties. What I called simple sentences are some, but not all, of the core sentences.
In my 2020 paper, I offered a relatively simple definition of ‘pluriverse’ (§4). But that definition is not metaphysically neutral. Using that definition, if it turns out that necessarily, no more than one universe exists, then necessarily, every universe is a pluriverse. But this is clearly not satisfactory. While I do not in general believe that definitions must be metaphysically neutral, as a concession to the desire for neutrality I explicate the meaning of ‘pluriverse’ in a neutral way here.
Here ‘\(\diamondsuit\)’ means ‘it is metaphysically possible that’ and ‘□’ means ‘it is metaphysically necessary that’.
This could be explained at length in terms of tokens to make it nominalistically acceptable.
Regarding possible non-qualitative o-tokens, Neil Sinhababhu (2008) describes how immortal beings in other universes could uniquely describe actualities in ours, and they could likewise uniquely describe actualities in other universes if there are any. The immortal beings could use these descriptions to name actualities and inscribe non-qualitative o-tokens concerning them (e.g. such a being could write down an o-token that is the same in meaning as *x loved Caesar*, or *y is a better painting than the Mona Lisa*, or whatever non-qualitative o-token might be inscribed concerning an actuality.) And since every possibility for an immortal being doing so is an aspect of a way a universe can possibly be, every such possibility would obtain were there a pluriverse. Note that this reasoning assumes that the actualities coexist with the pluriverse. As indicated by the schema for ‘pluriverse’, a pluriverse is a fusion of universes such that every qualitative way a universe could possibly be is a way one of its universe parts is, and that the immortal beings are describing actualities depends on the non-qualitative fact that the actualities are present. No matter what descriptions they may create, immortal beings cannot describe actualities if actualities are not there to be described. However, since all the actualities would exist were there a pluriverse (a point for which I will argue later), this assumption is no real limitation.
Not even in a pluriverse is it possible to refer to all possible individuals. Importantly, this point avoids Kaplan’s-Paradox-like considerations that could be generated from non-qualitative properties pertaining to pluralities including merely possible objects. Thankfully, even platonists do not agree over whether such properties exist.
Here I am discussing possible o-tokens and meanings, both of which are strange bedfellows with nominalism. But this is a choice of convenience rather than necessity. The point could be made by using a predicate for sameness of meaning and explaining how o-tokens cannot be so and so were there a pluriverse, but only laboriously.
If there were a pluriverse, o-fusions would have parts in many different universes. This means that the ‘will’ used in the statement of what an ‘O’ predicate means must pick out, for each i, i’s particular future. Objects in different universes are not only spatially isolated from each other, but temporally isolated as well. While this poses no special difficulty, it worth commenting on so as to make clear the intended meaning.
See John Nolt (2006, pp. 1039–1043) for discussion.
If there is any remaining doubt on this point, consider the following procedure. First, swap the constants of sort ‘p’ with some unindexed dummy constants of sort ‘p’ – ‘ap’, ‘bp’, and so on, adding these into L. Since the indexed ‘p’ constants lack any internal structure, the resulting Γ' and ψ' are notational variants of Γ and ψ, and Γ proves ψ iff Γ' proves ψ'. Then swap every predicate Φa…a with Φ+a…a and every ‘p’ subscript with an ‘o’ subscript. The resulting Γ'' and ψ'' are obviously notational variants of Γ' and ψ', and Γ' proves ψ' iff Γ'' proves ψ''. Now swap every x that is a dummy constant of sort ‘o’ – ‘ao’, ‘bo’, etc. – with the nominalization of the indexed ‘p’ constant replaced by the ‘p’ dummy constant from which x was arrived at via the subscript swapping. The result is nom(Γ) and nom(ψ). Given that the definite descriptions are treated like ordinary classical constants, nom(Γ) and nom(ψ) are notational variants of Γ'' and ψ'', and Γ'' proves ψ'' iff nom(Γ) proves nom(ψ). So Γ proves ψ iff Γ' proves ψ' iff Γ'' proves ψ'' iff nom(Γ) proves nom(ψ), which entails Γ proves ψ iff nom(Γ) proves nom(ψ).
This might appear to depend on the assumption that all change involves causation, which can be challenged. For example, our universe is fine-tuned for carbon-based life. One might think that if there is in fact a pluriverse, then if our universe alone were to exist, it would not be fine-tuned: it would be highly improbable for there to be a fine-tuned universe in the absence of a pluriverse, and that fact in some way lowers the probability of our universe being fine-tuned if our universe alone exists. However, even if eliminating all but one of the universes from a pluriverse would change the properties of the remaining universe, it is difficult to see how adding the universes needed for there to be a pluriverse would make a difference to whatever universe is the starting point. No matter how the original universe is, the pluriverse must contain a universe qualititatively just like it. By the same rationale, even if the starting point contains multiple universes rather than a single one, the addition of the universes needed to make a pluriverse will not change the original universes. Given that fact, it seems unlikely that non-causal factors introduced by the addition of the pluriverse will make any difference to the actualities.
I presented a more detailed explanation of the necessary causal isolation of universes and its impact on pluriverse counterfactuals in my paraphrase paper (2020, §4, especially fn. 26).
Of course, in my justifications of (i)–(iii), I have talked as though properties exist. This is for ease of presentation. A more careful, nominalistic justification could be given by making clear that there is a parallel between facts about properties according to platonists and facts about o-fusions in pluriverses, being very careful to always affix “according to platonists” where necessary.
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Acknowledgements
I am grateful to Cian Dorr, Geoffrey Hall, and Benjamin Middleton for their helpful comments on earlier versions of this manuscript and to David Mark Kovacs and the Israel Science Foundation for their support. I also thank Peter van Inwagen, Daniel Nolan, and Jeff Speaks for related discussions that improved this paper.
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Himelright, J. Safety first: making property talk safe for nominalists. Synthese 200, 262 (2022). https://doi.org/10.1007/s11229-022-03714-x
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DOI: https://doi.org/10.1007/s11229-022-03714-x