Quine, New Foundations, and the Philosophy of Set Theory by Sean Morris (review)

This book has two main goals: first, to show that Quine's New Foundations (NF) set theory is better motivated than often assumed; and second, to defend Quine's philosophy of set theory. It is divided into three parts. The first concerns the history of set theory and argues against readings that see the iterative conception of set being the dominant notion of set from the very beginning. The second part concerns Quine's philosophy of set theory. Part 3 is a contemporary assessment of the philosophical status of NF. Here one of the central targets is Boolos's defense of the iterative conception of set and his dismissal of NF as completely unnatural.

As this is a very short review, I will avoid specific points and focus on what I see as a tension between Morris's two goals. I will also focus on how this tension relates to a feature of Morris's interpretation of Quine. Morris considers Quine's account of explication and takes Quine's position on set theory to be that the various set theories are explications of what we mean by 'sets.' Quine's most detailed discussion of explication is in §53 of Word and Object. The ordered pair, Quine claims here, is a paradigm of philosophical analysis because we can identify the central feature of an ordered pair that any explication must preserve: <x, y> = <w, z> only if x = w and y = z. Whether we explicate ordered pairs as Wiener recommends or as Kuratowski recommends is of no importance. The explications may be inconsistent with one another, but they diverge only over what Quine calls "don'tcares" (Word and Object [Cambridge, MA: MIT, 1960], 259). Both are acceptable because they both preserve the central feature of the intuitive notion.

After mentioning the absence of objects for sets to be explicated in terms of, Morris writes, "Still, I think explication is the right way to describe Quine's approach to set theory" (124). There is, however, another reason, beyond sets' being ontologically fundamental, for why the various set theories fall short of explicating the notion of set. For Quine, as we saw, an explication begins by identifying the central feature of the concept that ought to be preserved (for more on this, see my "On the Quinean-Analyticity of Mathematical Propositions," Philosophical Studies 159 [2012]: 299–319). We cannot identify the central defining feature of our ordinary conception of set, in Quine's words, "because the natural scheme is the unrestricted one that the antinomies discredit" (The Ways of Paradox and Other Essays [Cambridge, MA: Harvard University Press, 1976], 16). It is simply not the case that the various set theories agree on a common core that underlies our intuitive understanding of sets, and disagree only over "don't-cares." In many places, Quine claims that the intuitive notion of set is the inconsistent one, and that the various set theories are not guided by intuition: "Intuition here is bankrupt" (Set Theory and Its Logic, rev. ed. [Cambridge, MA: Harvard University Press, 1969], x).

The question of whether any of the various set theories are an explication of the/a notion of set becomes important when the discussion turns to Boolos and the iterative conception of set. The problem with Boolos's view, as Morris sees things, is not the argument that ZF (Zermelo-Fraenkel set theory) is well motivated, but that ZF alone is a well-motivated set theory. Boolos writes, "ZF alone (together with its extensions and subsystems) is … an independently motivated theory of sets: there is so to speak, a 'thought behind it' about the nature of sets which might have been put forth even if, impossibly, naïve set theory had been consistent" (Logic, Logic, and Logic [Cambridge, MA: Harvard University Press, 1998], 17). Morris seeks to defend Quine by arguing that other set theories, including NF, are also well motivated. Quine, as Morris shows, displayed a pluralistic...