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Does everything resemble everything else to the same degree?

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Abstract

According to Satosi Watanabe’s “theorem of the ugly duckling”, the number of predicates satisfied by any two different particulars is a constant, which does not depend on the choice of the two particulars. If the number of predicates satisfied by two particulars is their number of properties in common, and the degree of resemblance between two particulars is a function of their number of properties in common, then it follows that the degree of resemblance between any two different particulars is also constant, which is absurd. Avoiding this absurd conclusion requires questioning assumptions about infinity in the proof or interpretation of the theorem, adopting a sparse conception of properties, or denying degree of resemblance is a function of number of properties in common. After arguing against both the first two options, this paper argues for a version of the third which analyses degree of resemblance as a function of properties in common, but weighted by their degree of naturalness or importance.

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Notes

  1. See also Watanabe (1965) and Watanabe (1985, p. 82).

  2. The name “abundant” comes from Lewis (1983, p. 346); see also Lewis (1986b, p. 59). Some writers prefer “deflationary”, following Hale, who writes “According to the abundant or, as I prefer to call it, deflationary conception of properties, every meaningful predicate stands for a property or relation, and it is sufficient for the actual existence of a property or relation that there could be a predicate with appropriate satisfaction conditions” (Hale , 2013, p. 132); see also Cook (2019) and Hale (2015). I prefer “abundant” not only because of the contrast with “sparse”, but also because I do not wish to imply that, according to the abundant conception, properties are any less real or fundamental.

  3. For predicate nominalism in this sense see especially Armstrong (1978a, pp. 11–24). For class nominalism, see, for example, Armstrong (1978a, pp. 28–34), Lewis (1983) and Lewis (1986b, pp. 50–69).

  4. See, for example, Niiniluoto (1987, p. 37), Armstrong (1989, p. 40), Rodriguez-Pereyra (2002, 66-7) and Ott (2016, 140).

  5. For analyses of degree of dissimilarity as a function of number of properties in common and not in common see, for example, Niiniluoto (1987, pp. 22–35), Oliver (1996, p. 52), Rodriguez-Pereyra (2002, pp. 65–69), Paseau (2012, p. 365), Blumson (2014b, pp. 179–193), Paseau (2015, p. 110), Blumson (2018), and Blumson (2019b). Yi (2018) criticises some of these analyses. For scepticism of whether resemblance is measurable by numerical degree at all, see Lewis (1973, p. 50), Williamson (1988, pp. 457–460), Blumson (2019a), and Paseau (2020); for defence see, for example, Bigelow (1976, 1977), Tversky (1977), Suppes et al. (1989, pp. 159–225), Weisberg (2012), Kroedel and Huber (2013, pp. 459–462), and Enflo (2020). Section 4 is also a partial defence of this presupposition. Morreau (2010) argues against the cogency of overall comparative similarity on different grounds.

  6. For similar passages see Guigon (2014, p. 390), Cowling (2017, p. 4) and the references in footnote 7.

  7. The argument is also raised outside metaphysics by, amongst others, Towster (1975), Watanabe (1985), Oddie (1986, pp. 164–165), Niiniluoto (1987, pp. 35–37), Medin et al. (1993, p. 255), Mundy (1995, pp. 35–36), Feldman (1997, p. 150), Byrne (2003, p. 641), Priest (2008, p. 97), Sterrett (2009, p. 803), Decock and Douven (2011, p. 68), Isaac (2013, p. 685), Blumson (2014b, pp. 182-196), Ott (2016, pp. 140-141), and Harnad (2017, pp. 36–37). Most of these authors stress the infinite case, and many draw radical conclusions.

  8. For a related point see, for example, Oddie (2005, pp. 151–152).

  9. For a related point see, for example, Oddie (2005, pp. 151–152).

  10. Note that an “atom” in this context is not a syntactically simple predicate: ‘lands’, for example, is not an atom, because it is strictly entailed by ‘lands one’, whereas ‘lands on an even number and lands on a prime number’ is an atom, because it is strictly entailed by ‘does not land’ but not by any other predicate. Atoms are akin to Sider’s “profiles” (1993, p. 50); see also Dorr and Hawthorne (2013, p. 23).

  11. In contrast to predicates in the infinitary languages discussed by Cook (2019).

  12. For discussion of whether there are infinite sentences in natural language see, for example, Langendoen and Postal (1986), Collins (2010) and Blumson (2014a).

  13. See especially Cook (2019, pp. 2572–2581). For linguistic supertasks see also Blumson (2015, pp. 129-130), and for supertasks in general see Benacerraf (1962) and Thomson (1954).

  14. Bjerring and Schwarz (2017, p. 26) make a similar point about analysing propositions as sets of worlds.

  15. Gardenfors (2000, pp. 59–100) and Oddie (2005, pp. 152–158), for example, present a sparse conception of properties which is like Armstrong’s in accepting that all conjunctions of sparse properties are properties, but differs from Armstrong’s in only denying that some negative and disjunctive properties are. But Blumson (2019b) proves this conception does not overcome the problems with Armstrong’s.

  16. Although no conceptions in the literature correspond exactly to this conception, some are just as unremittingly sparse. According to Rodriguez-Pereyra (2002, pp. 48–52), for example, the sparse properties are “lowest determinate properties”, and there are no negative, conjunctive or disjunctive sparse properties. I intend to consider Rodriguez-Pereyra’s theory in more detail elsewhere.

  17. The name “ultra-additivity” comes from Skyrms (1983, 227).

  18. This problem, suggested by anonymous referee, generalises objections to Rodriguez-Pereyra (2002) given by Yi (2018). Blumson (2018, pp. 34–36) also raises similar problems of infinity for the analysis of degree of similarity as proportion of properties in common, some of which generalise to analysis of degree of similarity as weighted proportion of properties in common. I intend to take this up again elsewhere.

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Acknowledgements

Thanks to John Baez, Zach Barnett, Bob Beddor, Jens-Christian Bjerring, Antony Eagle, Jay Garfield, Jeremiah Joven Joaquin, David Kovacs, Dan Marshall, Neil Mehta, Qu Hsueh Ming, Daniel Nolan, Josh Parsons, Alexander Paseau, Michael Pelczar, Abelard Podgorski, Alex Sandgren, Kranti Saran, Nicholas Silins, Nick Smith, Neil Sinhababu and Weng Hong Tang, as well as audiences at the National Chung Cheng University, the University of Delhi, the University of Sydney and the Australasian Association of Philosophy conference at the University of Queensland in 2013 for discussion of this paper.

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    Ben Blumson

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Blumson, B. Does everything resemble everything else to the same degree?. AJPH 1, 22 (2022). https://doi.org/10.1007/s44204-022-00023-5

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