Abstract
According to the sortal conception of the universe of individuals every individual falls under a highest sortal, or category. It is argued here that on this conception the identity relation is defined between individuals a and b if and only if a and b fall under a common category. Identity must therefore be regarded as a relation of the form \(x=_{Z}y\), with three arguments x, y, and Z, where Z ranges over categories, and where the range of x and y depends on the value of Z. An identity relation of this kind can be made good sense of in Martin-Löf’s type theory. But identity so construed requires a reformulation of Hume’s Principle that makes this principle unfit for explaining the sortal concept of cardinal number. The Neo-Logicist can therefore not appeal to the sortal conception in tackling the Julius Caesar problem, as proposed by Hale and Wright (The reason’s proper study. Oxford University Press, Oxford, pp. 335–396, 2001b).
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Notes
Lowe (1989b, p. 2) thinks already Locke had ‘grasped the key point that that in which identity “consists” for things of one sort (say, parcels of matter) may not be the same as that in which it “consists” for things of another sort (say, living organisms).’ It may be argued that what Frege gave were only certain suggestive remarks, spelled out in more detail by his readers Geach (1962, pp. 38–40) and Dummett (1973, e.g. pp. 75–80).
Anscombe and Geach (1961, p. 86) remark that the relevant distinction was made by Aquinas at Summa Theologica Ia. q. 39 art. 3, here in the Blackfriars translation:
nouns signify something existing as substance (per modum substantiae); adjectives, something existing as an accident (per modum accidentis), i.e. as inhering in a subject. Substance, having esse on its own, also has unity or multiplicity on its own; accordingly we put a noun in the singular or plural on the basis of the form signified by the word itself. Accidents, having esse in a subject, so also derive from it their unity or multiplicity; that is why we put adjectives in the singular or plural to agree with their subjects.
From Greek krinein, one meaning of which is ‘to judge’.
A similar notion of category has more recently been adopted by Linnebo (2005, p. 218).
Cf. e.g. the distinction drawn by Horsten (2010, p. 415) between a semantical, an epistemic, and a metaphysical take on criteria of identity.
The relation between basic domains and sorts is discussed by Klev (2014, pp. 162–167).
The reader will understand this even if no corner quotation marks are used.
For this line of argument, see Russell (1908, pp. 231–235).
This argument is, as far as I can see, not discussed by Heck (1997), who investigates Hume’s Principle in the setting of standard many-sorted logic.
The idea is suggested by Frege in § 64 of the Grundlagen. See Wright (1997, pp. 277–279) for a useful summary of the Neo-Logicist view on the matter.
I take that to be one conclusion of the techical considerations in Heck (1997, pp. 291–292, 302–304). Heck emphasizes that it is not required that the range of these variables equals the whole universe of individuals: it is both sufficient and necessary that it includes the natural numbers. Dummett’s objection goes through in any case.
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In writing this paper I have benefited greatly from comments on, and criticisms of, earlier drafts by Stefan Roski and three referees at Erkenntnis.
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Klev, A. Identity and Sortals (and Caesar). Erkenn 82, 1–16 (2017). https://doi.org/10.1007/s10670-016-9801-2
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DOI: https://doi.org/10.1007/s10670-016-9801-2