Abstract
Robert Black argues against quidditism on the grounds that the quidditist is either committed to proper-class many possible worlds and proper-class many possible fundamental properties or must adopt an unacceptably arbitrary restriction on the number of possible fundamental properties. In this paper, I examine Black’s cardinality argument against quidditism and argue that quidditists and non-quidditists alike have reason to reject a key premise of that argument. While it may be the case that the quidditist is committed to nomically indiscernible possible worlds that mark distinctions that make no difference to the physicist, it does not appear that the cardinality of the class of possible worlds or the class of possible fundamental properties postulated by the quidditist is any more problematic than that postulated by the non-quidditist.
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Notes
For a nice overview of the debate between dispositionalists and quidditists, see Jennifer Wang (2016).
Hawthorne (2001) argues against quidditism on the grounds that, in addition to properties, the quidditist must also postulate quiddities. Although we have empirical reasons to postulate properties (they are invoked in scientific theories), Hawthorne takes the postulation of quiddities to offend against the principle of parsimony. Locke (2012) makes a distinction between extreme and austere forms of quidditism. Austere forms of quidditism simply identify properties and quiddities and so escape Hawthorne’s objection. (Versions of quidditism according to which fundamental properties are barely, numerically different also escape this objection.) Neither Hawthorne’s objection nor Locke’s response trade on the possibility of mere quiddistic differences between worlds.
Here, it is worth noting that, in Smith (2016), I distinguish between what I call “individuation quidditism” and what I call “recombinatorial quidditism.” Individuation quidditism is simply the view that fundamental properties are individuated by something other than their causal or nomological role. Recombinatorial quidditism is the view that fundamental properties can freely recombine while preserving possibility. I argue that recombinatorial quidditism entails individuation quidditism but that individuation quidditism does not entail recombinatorial quidditism. If that is correct, individuation quidditism is consistent with but does not entail the existence of possible worlds that differ merely quiddistically.
Lewis (2009) dubs this form of skepticism “Ramseyan Humility”.
In Smith (2016), I articulate a version of quidditism that I dub “non-recombinatorial quidditism” in which the relation between a property and its nomological role is metaphysically necessary. Such a view is largely (though not wholly) immune from objections that trade on worlds exhibiting merely quiddistic differences. For various responses to the skeptical objection, see Lewis (2009), Rae Langton (2004), Locke (2009), Schaffer (2005), and Whittle (2006).
Black explicitly uses this argument to motivate the rejection of Lewis’ concrete modal realism (see Lewis 1986) in favor of what he calls abstract modal realism. However, it seems clear that the thrust of the argument, if cogent, would tell against any version of quidditism (whether or not it is combined with modal realism) that allows for merely quiddistic differences between worlds. Black’s cardinality argument also plays a key role in Stephen Mumford’s argument against the external theory of laws (see Mumford 2004, p. 152).
This specific formulation is my own, but each of its premises is either explicit or implicit in Black’s text.
A property is alien to w if and only if it is not instantiated in w.
Cantor’s theorem guarantees that the cardinality K of any set is such that 2K > K.
A proper class is a class whose cardinality is not the cardinality of any set.
Lewis (1986, p. 104) provides an argument in favor of this premise and Black (2000, pp. 97–99) further motivates it. In broad outline, the idea is this: if there are proper-class many possible worlds, then for any set of possible worlds (including the set of all possible worlds), there will be possible worlds that are not members of that set. But, possible worlds, unlike sets, are not generated or otherwise ordered in an iterative hierarchy. So, it is absurd to think that one could not initially collect them all together in a set.
The adoption of non-recombinatorial quidditism (see Smith 2016) would allow one to reject premise 1 of this argument. While non-recombinatorial quidditism is consistent with the idea that distinct fundamental properties would share nomological roles and so consistent with worlds exhibiting merely quiddistic differences, it is not consistent with the free recombination of fundamental properties required by premise 1.
Interestingly, Black’s abstract modal realism may entail that there are proper-class many possible worlds (see Black 2000, p. 99), and Black notes that, although Lewis initially argued that the class of possible worlds should be set-sized, he later seemed open to there being proper-class many possible worlds (ibid, p. 97). See also Daniel Nolan (1996) who argues that taking possible objects to form a proper class allows for an unrestricted principle of recombination that is simpler and more intuitive than that offered by Lewis (1986, p. 89). Although Nolan’s focus is explicitly on the cardinality of the class of possible objects and its implications for the maximal size of any possible world rather than on the cardinality of the class of possible worlds, the claim that there are proper-class many possible objects along with the unrestricted principle of combination together entails that there are proper-class many possible worlds.
This is not a response that would be open to a proponent of Lewis’ concrete modal realism.
Armstrong (1989) was once a notable exception. In that earlier work, Armstrong argued for a combinatorial theory of possibility according to which possible worlds are combinations of actual objects and properties. He also understood fundamental properties to be Aristotelian universals that are distinguished by having distinct intrinsic natures (quiddities). Given those commitments, he explicitly denied that there are any alien fundamental properties.
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Smith, D.C. On the Cardinality Argument Against Quidditism. Acta Anal 38, 275–281 (2023). https://doi.org/10.1007/s12136-022-00524-1
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DOI: https://doi.org/10.1007/s12136-022-00524-1