Graph-theoretic Models of Dispositional Structures

Abstract

The focus of this article is the view about fundamental natural properties known as dispositional monism. This is a holistic view about nature, according to which all properties are essentially interrelated. The general question to be addressed concerns what kinds of features relational structures of properties should be thought to have. I use Bird's graph-theoretic framework for representing dispositional structures as a starting point, before arguing that it is inadequate in certain important respects. I then propose a more parsimonious graph-theoretic system, which, among other things, overhauls Bird's distinction between stimulus and manifestation relations.

Acknowledgements

An early version of this paper was presented in Melbourne at the metaphysics of science conference organized by Brian Ellis and Howard Sankey, and in Nottingham for the Royal Institute of Philosophy Seminar organized by Uri Leibowitz. I would like to thank the audiences at those events for helpful comments and to Stephen Mumford for feedback on an earlier draft. Finally, thanks go to the two anonymous referees of this journal and to the editor James McAllister for his assistance.

Notes

This is an example of concrete dispositional interaction which I will continue to use during this article. It is useful for two reasons. First, this example involves only two entities and so is a relatively simple case to deal with. Second, it is a scientifically respectable case and so conclusions drawn from it will not be hostage to the peculiarities of macroscopic examples.

Property graphs could also serve a useful purpose for those who, while accepting that all properties have an irreducibly dispositional nature, deny that properties are purely dispositional. For example, this article should be of interest to those, such as Heil (2003) and Martin (2008), who hold a two-sided view of properties. On their view, the graphs discussed in this paper could be useful as a tool for representing the features of the dispositional ‘sides’ of properties.

More precisely, according to Bird, none of the subgraphs consisting of those vertices and arcs lying on the directed walk (sequence of arcs) leading away from each node must be structurally identical (Bird 2007b, 531).

Interestingly I see evidence in Heil's work to suggest that he, as a two-sided theorist, would also embrace something like the asymmetry requirement that is involved in the identity constraint. The asymmetry requirement would have to be accepted by a two-sided theorist if it were claimed that no two distinct kinds of categorical qualities could have the same dispositional features (thus precluding graphs with symmetries). Heil seems to be pushing in this direction when he suggests that one cannot change the structural features of a thing (typically associated by the two-sided theorists with the ‘categorical side’) without also changing the dispositional features of that thing. In other words, no two distinct kinds of structural feature bring the same kind of dispositional side. Heil discusses the case of fragility, for example, and says that ‘light bulbs, ice cubes, and kneecaps’, which are all structurally different, ‘shatter in very different ways’. Heil then writes: ‘Try changing a fragile object qualitatively, without altering it dispositionally. The object might remain fragile, but become fragile in a different way’ (Heil 2003, 116).

It should be noted that Armstrong's strategy of explaining causal patterns in terms of his second-order laws has not been without its critics (see e.g. Mellor 1991, 168). Given the structural similarity of Bird's account, it may be that similar objections will have to be answered. Unfortunately, I must postpone discussion of this general issue to another day.

In one place Bird does briefly acknowledge the possibility that dispositions may sometimes come in reciprocal groups (Bird 2007b, 533), but the notion of reciprocity he invokes is different to that discussed here. In the three-node graph which Bird suggests represents reciprocal properties, the properties are not directed towards the same kind of manifestation.

One might think that, equally, the dispositional monist has no independent reasons for denying the possibility of symmetrical property structures. I think, however, that there is an important difference between the dispositional monist's denial of symmetrical worlds, and the denial of ‘odd’ worlds. As I suggest in section 2, the denial of symmetrical worlds is an essential feature of the dispositional view, whereas the denial of ‘odd’ worlds is entailed by a combination of non-essential constraints. This means that the rejection of ‘odd’ worlds is avoidable in principle.

To avoid confusion, note that, just because the top node in Figure 6(b) has as its manifestation one of its reciprocal stimulus dispositions, this does not imply there are causal loops involving backwards causation at the first-order level. To emphasize again, the manifestation relations represented in the graphs are second-order relations of ontological dependence between property types. If an instance of the disposition represented by the top node manifests, the property instance manifested will be different to the property instance responsible for bringing about that disposition (though those property instances would be of the same type, according to the graph in question).