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An argument against nominalism

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Abstract

Nominalism in formal ontology is still the thesis that the only acceptable domain of quantification is the first-order domain of particulars. Nominalists may assert that second-order well-formed formulas can be fully and completely interpreted within the first-order domain, thereby avoiding any ontological commitment to second-order entities, by means of an appropriate semantics called “substitutional”. In this paper I argue that the success of this strategy depends on the ability of Nominalists to maintain that identity, and equivalence relations more in general, is first-order and invariant. Firstly, I explain why Nominalists are formally bound to this first-order concept of identity. Secondly, I show that the resources needed to justify identity, a certain conception of identity invariance, are out of the Nominalist’s reach.

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Notes

  1. It is impossible to give a full fledged account of this semantics here, so I refer to Dunn and Belnap (1968), Barcan-Marcus (1978) for the characterization and discussion of the substitution semantics in logic and philosophy.

  2. S. Shapiro provides a detailed introduction to varieties of others semantics for second-order languages in Shapiro (1991).

  3. x is a first-order variable and it is taken to denote first-order objects of whatever sort, i.e., individuals, abstract objects, concrete particulars, abstract particulars or tropes.

  4. A different attempt to avoid higher-order entities may be that to appeal to plurals. See (Ferrari 2018) for a discussion about Boolos’ approach to plural reference and about a proof of its model-theoretic equivalence to standard second-order semantics.

  5. This is not totally fair. As it will be clearer later, Nominalism claims that universals do exist but just as linguistic entities.

  6. A distinction among CN and TN is that tropes, despite they are the fundamental entities, do not behaves as concrete individuals. My argument against the Nominalist strategy is purported to attack both ontological models.

  7. As a result, second-order logic is infinitary in the following sense: this logic is semantically incomplete, meaning that there are logical truths of second-order language that we do not know how to prove and, so, it is not axiomatizable. On the contrary, first-order logic is semantically complete, meaning that the finitary resources of the syntax are sufficient to prove all the first-order tautologies and vice versa and is axiomatizable. (“finitary” does not mean ‘effective’. First-order logic is not a decidable calculus.) Albeit Godel-incomplete, first-order theories still suppose first-order logic and, so, present the formal advantage of being logically finitary. This is desirable for nominalists. In fact, it is the so-called “canonical model” that makes first-order logic complete, and this model is a mere product of syntax, meaning that first-order logic does not need to appeal to extra-linguistic models to prove completeness.

  8. Of course, “[t]his means that nominalism is committed to an extensional logic, and in particular to a non-modal form of the thesis of anti-essentialism; specifically, the thesis that no nominalistic universal is necessarily true of some of the things of which it can be truly predicated without being necessarily true of all” (Cocchiarella 1989, p. 256).

  9. It is worth noticing that SA is formally provable by induction on formulas (Cocchiarella 2007, p. 32).

  10. For a technical discussion about a minimally nontrivial versions of PII, see (Wörner 2021).

  11. Despite the proper semantic formulation of EP is pretty different:

    • \(\forall _{x^M_1...x^M_n} (P^M(x^M_1...x^M) \Longleftrightarrow Q^M(x^M_1...x^M)) \Longrightarrow P^M=Q^M)\),

    where \(x^M, y^M\) and \(P^M, Q^M\) are the interpretation in the model M of the respective first- and second-order variables, while quantifiers and logical constants are meta-quantifiers and meta-constants. After all, as Keranen argues, PII is a “direct counterpart”’ of EP and viceversa. EP “tells us just what makes a set, it codifies a key ingredient in our understanding of what sets are in the first place. Thus, it seems that when Zermelo-Fraenkel set theory is considered as a philosophical (rather than a mathematical) theory” as it happens in CN, EP “is indispensable to it: without such an axiom, the theory would not really tell us what sets are” (Keranen 2001, p. 328).

  12. Notice that Arenhart, Bueno and Krause recently urged not to confuse numerical identity with the notion of identity in standard logic and mathematics (Arenhart et al. 2019), as the one encoded by RI and SA. According to them, this second notion of identity deflates identity from the “metaphysical content” required to capture the numerical identity coded by principles as PII and EP. For this very reason, PII and EP (and the likes) are sometimes thought as principles that play the specific and distinctive metaphysical role that, with appropriate modifications, Nominalists require.

  13. As I mentioned above, the argument can be generalized from identity to generic equivalence relations. In particular, the argument applies to variants of Nominalism that make equivalence relations occurring in IS in place of identity.

  14. On the contrary, second-order languages are provably categorical, meaning they have isomorphic models.

  15. Indeed, a second-order language with non-full or non-standard semantics are provably equivalent to first-order (denumerable) languages (Shapiro 1991).

  16. Impredicative or standard, or realist CP made use just of conditions (1) and (2).

  17. If the cardinality of the first-order domain is denumerably infinite, the cardinality of its power set is of the order of the continuum.

  18. The other conditions on \(\psi \) are still satisfied for (1) \(=\) doesn’t occur free, (2) x, y are free.

  19. The formal contradiction obtained is crucial for my general argument against Nominalism, but it does not involve IS or the likes even implicitly. Notwithstanding, a further consequence of that contradiction is that, CP! and IS (or the like) can no longer be maintained as two formally independent principles of CN.

  20. Formally, \(\exists _{F_=} \forall _{x,y} (F_=(x,y) \longleftrightarrow \psi (x) \longleftrightarrow \psi (y))\), with \(\psi \) first-order wff.

  21. “Well-definite” in the Quinenan sense of possessing clear identity criteria (see also Keranen 2001).

  22. Thanks to an one of the anonimous referees for calling this reference to my attention.

  23. Notice, that “any occurrence of a name or a predicate symbol in a first-order validity is schematic, so no such symbol will be extracted” (Bonnay and Westerståhl 2012, p. 690).

  24. According to the idea that “When a particular consequence relation is given, certain symbols are to be considered as logical constants because the consequence relation makes them play a special role with respect to validity” (Bonnay and Westerståhl 2012, p. 688).

  25. Thanks to one of the anonymous referees for calling to my attention this reference.

  26. Hellman’s is a reply to Awodey (1996). For a criticism of Hellman’s view see also (Awodey 2004; Pedroso 2009).

  27. Thanks to one of the anonymous referees for pointing this out, and correcting an earlier mistake of mine.

  28. A systematic discussion of the HoTT-based identity concept is out of the scope of the present work, nonetheless the following brief discussion might be helpful and appropriate to fix some point.

  29. Thanks to one of the anonymous referees for calling to my attention this reference.

  30. Notice, this kind of structure may have more than just two elements on each level.

  31. Recall that IIP does not hold in CT.

  32. Such an operation is precisely the truncation procedure mentioned above with respect to extensional MLTT.

  33. In asserting that a category A is univalent, we assert that the equality types \(a = b\) among its objects are equivalent to the sets Iso(a, b) of isomorphisms among its objects. Since the property of ‘being a set’ itself obeys the equivalence principle for types the equality types \(a = b\) are themselves sets.

  34. Shapiro (1991) points out that the first-order kludges for second order logic do not, in the end, succeed in capturing the Reals.

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Acknowledgements

This article was supported by the São Paulo Research Foundation (FAPESP). Grant n°2018/16465-6. I am particularly indebted to my colleague and friend Andrea Raimondi (Thapar University, India) for discussing the content(s) with me and also for having kindly helped me brush up on my English. Thanks also to all the anonymous reviewers for reading and criticizing early versions of this paper.

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  1. Center for Logic, Epistemology and History of Science (CLE), State University of Campinas (UNICAMP), Campinas, Brazil

    Francesco Maria Ferrari

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Ferrari, F.M. An argument against nominalism. Synthese 200, 403 (2022). https://doi.org/10.1007/s11229-022-03520-5

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