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Quantity Tropes and Internal Relations

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Abstract

In this article, we present a new conception of internal relations between quantity tropes falling under determinates and determinables. We begin by providing a novel characterization of the necessary relations between these tropes as basic internal relations. The core ideas here are that the existence of the relata is sufficient for their being internally related, and that their being related does not require the existence of any specific entities distinct from the relata. We argue that quantity tropes are, as determinate particular natures, internally related by certain relations of proportion and order. By being determined by the nature of tropes, the relations of proportion and order remain invariant in conventional choice of unit for any quantity and give rise to natural divisions among tropes. As a consequence, tropes fall under distinct determinables and determinates. Our conception provides an accurate account of quantitative distances between tropes but avoids commitment to determinable universals. In this important respect, it compares favorably with the standard conception taking exact similarity and quantitative distances as primitive internal relations. Moreover, we argue for the superiority of our approach in comparison with two additional recent accounts of the similarity of quantity tropes.

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Notes

  1. Some trope theorists [such as Maurin (2002) and Simons (2003)], introduce relational tropes as a category additional to property tropes.

  2. Ehring (2011, p. 179) employs Armstrong’s (1989a, pp. 43–44) characterization of internal relations. Since Armstrong’s characterization can be considered a disjunction of [IR1] and [IR2] below, we will not discuss it here.

  3. Cf. Campbell (1990, pp. 111–113), Vallicella (2002, p. 5), and Wieland and Betti (2008, p. 517) for similar examples. Campbell and Wieland and Betti do not restrict the corresponding properties to the monadic properties of the relata, but we do not discuss the other options in this article.

  4. Essential properties as further entities are ruled out, e.g. by Armstrong (1989b, 1997) and Campbell (1990). The advocates of the first conception seem to follow this lead. An advocate of the first conception can admit that entities have their formal features (such as particularity and individuality) necessarily, but they are not considered as further entities.

  5. Campbell’s (1990, p. 112) “founded external relations” provide a variant of the same conception of internal relations. Cf. also Maurin (2002, pp. 90–91) and Wieland and Betti (2008).

  6. In both [IR1] and [IR2], the relata of internal relations are called “objects”. The relata of internal relations in [IR2] could be entities of any kind. By contrast, in [IR1], the reference to objects (in the sense of bearers of contingent properties) is indispensable. The account of such entities depends on the details of the specific trope bundle theory.

  7. Although giving internal relations a Moorean characterization [“internal relations are essential to the identity of their (both) relata”] (Moore 1960), Campbell (1990, p. 112) considers this equivalent to the modal characterization. Cf. Maurin (2002, pp. 87–89).

  8. Arguably, any ontological category system must introduce formal ontological relations to avoid infinite regresses (cf. Smith and Mulligan 1983; Simons 2003; Lowe 2006, Sect. 3).

  9. Cf. Keinänen and Hakkarainen (2010), Keinänen (2011, Sect. 4).

  10. This assumption is made in order to make it explicit that in clause [PIR], we restrict to relations R holding of entities a1, …, an. An alternative definition of proto internal relations would be a clause like [IR2]: entities a1, …, an are related by proto internal relation R iff, necessarily, R holds of a1, …, an if and only if a1, …, an exist.

  11. In Sect. 3, we replace exact similarity with the relation of order as a primitive basic internal relation between tropes. Internal relations in the sense of [IR1] are derived internal relations, which can be shown by means of a similar example. For reasons of space, we will leave the discussion of these derived internal relations for another occasion.

  12. Cf. Bigelow and Pargetter (1990, pp. 55–62) for a similar suggestion to use the relations of proportion to spell out the relations between determinate quantities. However, Bigelow and Pargetter introduce proportions as second-degree relation universals, i.e. relations between relation universals. By contrast, proportions are considered here as basic internal relations between quantity tropes.

  13. These basic units include both the charges of leptons and their anti-particles as well as the charges of quarks and anti-quarks.

  14. According to some trope theorists (e.g. Simons 1994, 1998), only simple objects are trope bundles. By contrast, Keith Campbell (1990) postulates complex tropes, which are properties of complex objects.

  15. In other words, all charge tropes are connected by a greater than or equal to relation; the relation is transitive and non-symmetric (neither asymmetric nor symmetric).

  16. That we call this quantity “e” and “charge” is, of course, based on our conventions, which presuppose that tropes are in the required relations of proportion and order, cf. below.

  17. Thus, we adopt the standard conception of tropes as particular natures defended by Campbell (1990), Maurin (2002, 2005), and Simons (2003). Moreover, as any basic entities of a category system, such as property universals in a system postulating universals, tropes stand in different formal ontological relations, which are also basic internal relations, cf. Sect. 2.

  18. This kind of derived internal relation would fulfil [PIR], [DIR], and [IR2], but not [BIR]. Ellis (2001) and Lowe (2006, 2012) adopt this type view by claiming that all tropes (or modes) instantiate the respective property universals (e.g. that of −e charge). Lowe (2012, p. 412) is also explicit in maintaining that, necessarily, if modes and certain property universals exist, modes instantiate these property universals.

  19. We can give quantitative distances only relative to some conventionally chosen unit. But as we saw above, the unit can be given without reference to the determinable.

  20. In saying this, trope theorists may remain uncommitted to entities other than just tropes connected by these basic internal relations.

  21. It might be attractive to consider these determination relations as metaphysically necessary, for instance, that charge tropes in certain relative locations necessarily generate certain kinds of attractive or repulsive forces between objects. One variant of such a view would be a dispositionalist conception of tropes (cf. Whittle 2008). However, the details of such a view need to be worked out.

  22. Cf. Balashov (1999) and Morganti (2009) for useful discussions of the basic monadic physical quantities.

  23. Cf. Maurin (2002, Ch. 6, 2011) and Wieland and Betti (2008) for an interesting recent account of relational tropes as relata-specific relations.

  24. Cf. Maudlin (2007, p. 86ff.) for a discussion of colour charge.

  25. In the present case, one may, e.g. consider quantitative distance a derived internal relation between tropes v and w; it holds because v and w instantiate the respective determinate kind universals and the kind universals are internally related. Cf. Ellis (2001, Ch.2) for a more detailed view of tropes as instances of determinate and determinable kind universals.

  26. To be more precise, according to Campbell (1990), every trope t exists independently of any other trope that is not a proper part of t. The only exceptions of this rule are positions of space–time, which he claims to be separate "quasi-tropes" or "place tropes": every trope t must be accompanied by some (although not any specific) place trope.

  27. As a generalization of his conception of conjunctive compresences, Campbell allows for the distinct maximal aggregates of co-located tropes falling under a determinable in different locations to form conjunctive compresences, whose value amounts to the sum of the values of all of their constituent tropes.

  28. However, the advocate of conjunctive compresences may consider the restrictions to the composition of tropes in her theory as brutal. Therefore, she can avoid all reference to determinables in her account of complex tropes.

  29. A similar approach to inexact resemblances between quantitative properties is advocated by Eddon (2013), who follows Mundy (1987). However, she considers quantities as universals and order and summation as second-order relation universals.

  30. In constructing her original approach, Eddon (2013) assumes that monadic properties are Platonic universals, which need not be instantiated in order to exist.

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Acknowledgements

We wish to thank Gonzalo Rodriguez-Pereyra, Tuomas Tahko, and two referees of Erkenntnis for helpful comments on earlier versions of this paper. Moreover, we thank Finnish Cultural Foundation for funding this research.

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  1. Faculty of Social Sciences (Philosophy), University of Tampere, 33014, Tampere, Finland

    Markku Keinänen

  2. University of Tampere, Tampere, Finland

    Antti Keskinen & Jani Hakkarainen

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  1. Markku Keinänen
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Keinänen, M., Keskinen, A. & Hakkarainen, J. Quantity Tropes and Internal Relations. Erkenn 84, 519–534 (2019). https://doi.org/10.1007/s10670-017-9969-0

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