Abstract
Non-symmetric relations like loves or between can apply to the same relata in non-equivalent ways. For example, loves may apply to Abelard and Eloise either by Abelard’s loving Eloise or by Eloise’s loving Abelard. On the standard account of relations (Directionalism), different applications of a relation to fixed relata are distinguished by the direction in which the relation applies to the relata (e.g., from Abelard to Eloise rather than from Eloise to Abelard). But neither Directionalism nor its most popular rival, Positionalism, offer accounts of differential application that generalize to relations of arbitrary symmetry structure. Here, I develop an alternative account, Relative Positionalism, which distinguishes different applications of a relation to fixed relata in terms of the ways in which the relata are characterized relative to one another. In presenting and defending Relative Positionalism, this paper covers some of the same ground as my [2016], but avoids the latter’s algebraic approach and focuses on interpretative issues—in particular, how to make sense of relative property instantiation—that were not addressed in the earlier paper.
This is a preview of subscription content, log in via an institution to check access.
(betweenMCL) Moe is between Curly and Larry
(betweenLCM) Larry is between Curly and Moe
(betweenCML) Curly is between Moe and Larry
(LineCMLS) Curly, Moe, Larry, and Shemp stand (in that order) in a line
(LineCSLM) Curly, Shemp, Larry, and Moe stand (in that order) in a line
Similar content being viewed by others
Notes
Throughout this paper, I use the following terminology to distinguish different levels of symmetry for an n-ary relation R denoted by the predicate ‘R’:
R is completely symmetric just in case: necessarily, for any x1, …, xn in the domain of R and any permutation P of 1, …, n,
$$\boldsymbol{R}x_{ 1} \ldots x_{\text{n}} \,{\text{iff }}\boldsymbol{R}x_{{{\text{P}}( 1)}} \ldots x_{{{\text{P}}({\text{n}})}} .$$R is non-symmetric just in case: R is not completely symmetric.
R is completely non-symmetric just in case: for any permutation P of 1, …, n, possibly, there are x1, …, xn such that
$$\boldsymbol{R}x_{ 1} \ldots x_{\text{n}} {\text{ and not }}\boldsymbol{R}x_{{{\text{P}}( 1)}} \ldots x_{{{\text{P}}({\text{n}})}} .$$R is partly symmetric just in case: R is neither completely symmetric nor completely non-symmetric.
Here and throughout this paper, I take plural terms like “Abelard and Eloise” to refer to an unordered plurality of individuals (unless the plural term is explicitly modified by the qualifier “in that order”).
A partition of a set S is a set of pairwise disjoint subsets S1, …, Sm of S such that S1 ∪···∪ Sm = S.
I am glossing over some details that I do not have room for in this paper. See (Donnelly 2016). The general idea is that the exact partitioning of permutations depends on which predicate designates R, but I assume that partitions determined by any two predicates designating R are at least isomorphic and thus represent the same structure.
See Russell (1903, Sect. 94):
…it is characteristic of a relation of two terms that it proceeds, so to speak, from one to the other. This is what may be called the sense of the relation, and is, as we shall find, the source of order and series. …We may distinguish the term from which the relation proceeds as the referent, and the term to which it proceeds as the relatum. The sense of a relation is a fundamental notion which is not capable of definition.
In Fine (2000) and elsewhere, Russell's account of relations is called “the standard account”.
Here, I assume that the number of possible symmetry structures for n-ary relations is equal to the number of non-isomorphic subgroups of the symmetric group of order n (where the symmetric group of order n, Sn, is the group of all permutations of n things). See Donnelly (2016) for further discussion of the application of algebra to relational symmetry structures.
This point has been made elsewhere. See, e.g., MacBride (2014, pp. 4–6).
This point is also made at Gaskin and Hill (2012, p. 175).
Even when limited to binary relations—which cannot be partly symmetric—Directionalism’s answer to (DiffApp2) is problematic since it would seem to require that non-symmetric binary relations apply to their relata in an order, while symmetric binary relations apply in no particular order to their relata. But it is not clear why binary relations would apply to their relata in such different ways.
Gaskin and Hill (2012) similarly conclude that there is no general explanation of distinctions among different applications of a relation to fixed relata.
Fine introduces the term “completion” to stand for whatever we take the result of applying a relation to relata to be—a relational state, fact, proposition, or something else. For simplicity, I will just assume that relational completions are states.
See Leo (2016) for a further development of Fine’s account of relations.
See Fine (2000, p. 17, n. 10), Macbride (2007, pp. 41–44), Gaskin and Hill (2012, p. 175). See Donnelly (2016, p. 89, n. 22) for an algebraic characterization of the kinds symmetry structures to which Positionalism is limited.
Relative Positionalism is not an entirely new account of relational claims. In Sect. 2, Part 7 of The Categories, Aristotle introduces the category of relatives, giving as examples property pairs such as inferior/superior, half/double and slave/master and explaining that, e.g., something is said to be “superior” by reference to something else (which, in turn, is said to be “inferior” by reference to the first thing). See Tegtmeier (2004) for a comparison of Aristotle’s account to Russell’s earlier and later accounts of relations. There are similar threads in Hector-Neri Castañeda’s readings of Leibniz’s and Plato’s treatments of relations (Castañeda 1982). Ultimately, however, Castañeda attributes to both Plato and Leibniz versions of Positionalism, not Relative Positionalism.
Thus, when I say that, e.g., Eloise is beloved from Abelard’s standpoint, this should not be understood to imply that there is, besides Abelard, a distinct entity which is Abelard's standpoint. Rather, Abelard's standpoint is just Abelard himself insofar as he functions as parameter at which things in the world are characterized.
This use of orientation predicates like “in front of” or “in back of” is distinct from another common use of such terms which depends on the intrinsic orientation of one of the relata. On this alternative use of “in front of”, X counts as in front of Y iff X is on Y’s front side. For example, on the intrinsic-orientation use of “in front of”, the Washington Monument is in front of me just in case it is on the front side of my body (i.e., I am facing it). Note that none of the examples used in the body of this paper make any assumption about the intrinsic orientation of relata. In particular, none of the examples involving the stooges include any information about which directions they are facing.
For a rigorous discussion of the notoriously multi-faceted and confusing ordinary uses of spatial terminology, see Herskovits (1986).
It is its significantly wider range of possible distinct relative properties per relation—between 1 and n!—as comparted to Positionalism’s range of 1 to n possible distinct positions per relation that enables Relative Positionalism, unlike Positionalism, to accommodate all possible symmetry structures for finite fixed arity relations.
See Donnelly (2016) for details. Relations with symmetry structures too complex for Positionalism include the ternary stand clockwise in a circle and the quaternary relation holding between x, y, z, and w when the distance between x and y is equal to the distance between z and w.
In Donnelly (2016), I suggest that the relative positionalist might hold that only relative properties, not relations, are fundamental entities. I still find this an appealing option for the relative positionalist, allowing her to minimalize her ontological commitments. But I prefer to focus in this paper on the more important question of whether Relative Positionalism offers a viable general explanation of differential application.
References
Castañeda, H. (1982). Leibniz and Plato’s Phaedo theory of relations and predication. In M. Hooker (Ed.), Leibniz: Critical and interpretative essays (pp. 124–159). Manchester: Manchester University Press.
Dixon, S. (2018). Plural slot theory. In K. Bennett & D. Zimmerman (Eds.), Oxford studies in metaphysics (Vol. 11, pp. 193–223). Oxford: Oxford University Press.
Dixon, S. (2019). Relative positionalism and variable arity relations. Metaphysics, 2, 55–72.
Donnelly, M. (2016). Positionalism revisited. In A. Marmodoro & D. Yates (Eds.), The metaphysics of relations (pp. 80–99). Oxford: Oxford University Press.
Fine, K. (2000). Neutral relations. Philosophical Review, 109, 1–33.
Fine, K. (2005) Tense and reality. In Modality and tense: Philosophical papers. Oxford: Oxford University Press.
Fine, K. (2007). Response to Fraser MacBride. Dialectica, 61, 57–62.
Gaskin, R., & Hill, D. (2012). On neutral relations. Dialectica, 66, 167–186.
Gilmore, C. (2013). Slots in universals. In K. Bennett & D. Zimmerman (Eds.), Oxford studies in metaphysics (Vol. 8, pp. 187–233). Oxford: Oxford University Press.
Gilmore, C. (2014). Parts of propositions. In S. Kleinschmidt (Ed.), Mereology and location (pp. 156–208). Oxford: Oxford University Press.
Herskovits, A. (1986). Language and spatial cognition: An interdisciplinary study of the prepositions in English. Cambridge: Cambridge University Press.
King, J. (2007). The structure and nature of content. Oxford: Oxford University Press.
Kölbel, M. (2003). Faultless disagreement. Proceedings of the Aristotelian Society, 104, 53–73.
Leo, J. (2016). Coordinate-free logic. The Review of Symbolic Logic, 9, 522–555.
Liebesman, D. (2014). Relations and order-sensitivity. Metaphysica, 15, 409–429.
Lipman, M. (2016). Perspectival variance and worldly fragmentation. Australasian Journal of Philosophy, 94, 42–57.
MacBride, F. (2007). Neutral relations revisited. Dialectica, 61, 25–56.
MacBride, F. (2014). How involved do you want to be in a non-symmetric relationship? Australasian Journal of Philosophy, 92, 1–16.
Orilia, F. (2011). Relational order and onto-thematic roles. Metaphysica, 15, 409–429.
Ostertag, G. (2019). Structured propositions and the logical form of predication. Synthese, 196, 1475–1499.
Russell, B. (1903). The principles of mathematics. London: George Allen and Unwin.
Spencer, J. (2016). Relativity and degrees of relationality. Philosophy and Phenomenological Research, 92, 432–459.
Tegtmeier, E. (2004). The ontological problem of order. In H. Hochberg & K. Mulligan (Eds.), Relations and predicates (pp. 149–160). Frankfurt: Ontos Verlag.
van Inwagen, P. (2006). Names for relations. Philosophical Perspectives, 20, 453–478.
Williamson, T. (1985). Converse relations. Philosophical Review, 94, 249–262.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
I am grateful for the helpful comments of Scott Dixon, Francesco Orilia, and Fraser MacBride on an earlier draft of this paper.
Rights and permissions
About this article
Cite this article
Donnelly, M. Explaining the differential application of non-symmetric relations. Synthese 199, 3587–3610 (2021). https://doi.org/10.1007/s11229-020-02948-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-020-02948-x