The received view regarding the debate between substantivalists and relational-ists is that the former are committed to a larger set of possibilities than the latter are.45 It is the purpose of this chapter to show this to be false. There are already good arguments to the effect that substantivalists can be deflationists;46 but I have seen no analogous argument in the literature to the effect that relationalists are not necessarily committed to deflationism.47 The conclusions of this chapter will be: (1) substantivalists can be deflationists, and (2) relationalists can be inflationists. More generally: differences in possibility counting are not an adequate way of characterizing these interpretations. This will be of increasing importance in the chapters that follow, and I will attempt to deepen the result by focusing on the nature of the geometric spaces used to represent the possibility structures of theories and their interpretations.
I begin by getting clear on the central terms of the debate—'substantivalism' and 'relationalism'—setting them up using the tools of the previous chapter. I then introduce a further distinction that is often taken to be implicated in the debate between substantivalists and relationalists; I call the opposing views 'inflationism' and 'deflationism' (later to be connected to the modal notions of 'haecceitism' and 'anti-haecceitism' respectively). Next, I outline the so-called 'Leibniz-shift' argument, and show how it is supposed to vindicate relationalism and undo substanti-valism on the basis of possibility counting. Finally, I present the general argument to show that substantivalists can help themselves to deflationism, and follow this up with an analogous argument demonstrating that relationalists are not necessarily bound to deflationism either. The diagnosis is identical for both cases: an unwarranted assumption about modality slipped in to each thesis about spacetime ontology. Specifically: the connection between inflation and substantivalism depends on an assumption of haecceitism, and that between deflation and rela-
45 This claim can be found in the Leibniz-Clarke correspondence [Alexander, 1956], Earman and Norton's hole argument , various papers by Belot [1996; 2000; 2001; 2003b], and in Belot and Earman [1999; 2001]. This difference in the number of possibilities, one large the other small, corresponds to what I call inflation and deflation respectively (see §2.2 below).
46 See, for example, Butterfield , Maudlin , Maidens , Hoefer , and Pooley [in press]. Belot is the most avid of dissenting voices against these arguments (see especially Belot [1996; 2000], Belot and Earman , and §2.5 below).
47 Saunders [2003b; 2003a] comes close and in fact claims that his "eliminative relationalism" is a "non-reductive" and "deflationary" variety of relationalism. However, as I intend the term 'deflationism' (meaning the cutting of certain indiscernible possibilities out of one's ontology), Saunders' relationalism is as reductive and deflationist as the rest.
tionalism on anti-haecceitism. This dependency can easily be questioned, and is concomitant with neither thesis.
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