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Frege on Identity and Identity-Statements: A Reply to Thau and Caplan

Published online by Cambridge University Press:  01 January 2020

Richard G. Heck Jr.
Harvard University, Cambridge, MA 02138, USA


In ‘What's Puzzling Gottlob Frege?’ Michael Thau and Ben Caplan argue that, contrary to the common wisdom, Frege never abandoned his early view that, as he puts it in Begriffsschrift, a Statement of identity ‘expresses the circumstance that two names have the same content’ and thus asserts the existence of a relation between names rather than a relation between (ordinary) objects. The arguments at the beginning of ‘On Sense and Reference’ do, they agree, raise a problem for that view, but, they insist, Frege does not, as the ‘standard’ Interpretation has it, take these arguments to refute it. Rather, they claim, Frege is out to defend (a version of) his earlier view against these objections: indeed, the defense he there offers is pretty much the same defense offered in Begriffsschrift against what are pretty much the same objections.

Research Article
Copyright © The Authors 2003

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1 Michael, Thau and Ben, CaplanWhat's Puzzling Gottlob Frege?Canadian Journal of Philosophy 31 (2001) 159–200Google Scholar. Future references will be marked ‘T&C’ and included in the text.

2 Frege, Gottlob Begriffsschrift: A Formula Language Modeled Upon That of Arithmetic, for Pure Thought, trans. Heijenoort, J. van in Heijenoort, J. van ed., From Frege to Gödel: A Sourcebook in Mathematical Logic (Cambridge, MA: Harvard University Press 1967) 5–82Google Scholar. The quotation is taken from §8, in which Frege discusses identity. Further references are marked Bg and are included in the text.

3 Frege, GottlobOn Sense and Meaning,’ trans. Black, M. in Collected Papers on Mathematics, Logic, and Philosophy (Oxford: Basil Blackwell 1984) 157–77Google Scholar. Further references are marked: SB p. n, op. m, where m is the original page number.

4 Frege, Gottlob Function and Concept, trans. P. Geach, in Collected Papers 137-56, at 138, op. 3Google Scholar. Further references are marked: FB p. n, op. m. Three further passages come from Grundgesetze der Arithmetik (Hildesheim: Georg Olms Verlagsbuchhandlung 1962), vol. II, §§64,103, and 138. (Further references are in the form: Gg II §64, where the Roman numeral indicates the volume number, and the Arabic numeral, the section number.) Another is from a letter to Peano: see Frege, Gottlob Philosophical and Mathematical Correspondence, trans. Kaal, H. Gabriel, G. et al., eds. (Chicago: University of Chicago Press 1980), 128Google Scholar; letter XIV/11; xxxiv/11, in the German edition. The last passage is from ‘Formal Theories of Arithmetic’, in Collected Papers, 112-21, at 118, op. 101. Note that this paper was first published in 1885 and so predates the development of the distinction between sense and reference.

5 Russell, Bertrand The Principles of Mathematics, 2nd ed. (New York: W.W. Norton 1937), 63Google Scholar. The passage is discussed in Fiengo, Robert and May, RobertIdentity Statements,’ in Preyer, G. and Peters, G. eds., Logical Form and Language (Oxford: Clarendon Press 2001) 169–203.Google Scholar

6 For some relevant discussion, see my ‘Grundgesetze der Arithmetik I §10,’ Philosophia Mathematica 7 (1999) 258-92.

7 Correspondence, 204.

Another possible piece of evidence is found in the Introduction to Grundgesetze, where Frege is detalling the ‘internal changes’ he has made to his formal language since Begriffsschrift. He writes:

The primitive signs used in Begriffsschrift occur here also, with one exception. Instead of the three parallel lines I have adopted the ordinary Symbol of equality, since I have persuaded myself that it has in arithmetic precisely the meaning that I wish to symbolize. That is, I use the word ‘equal’ to mean the same as ‘coinciding with’ or ‘identical with'; and the sign of equality is actually used in arithmetic in this way. (Gg I ix)

One might take this passage to imply that Frege's conception of what identity was, in Begriffsschrift, has since been jettisoned and, with it, his old sign for identity of Content. But the question is complicated by the fact that the change in question may have more to do with Frege's understanding of the notion of arithmetical equality than with the notion of identity. Jourdain explicitly lists this change of mind as the second of Frege's ‘advances,’ separately from the fourth, quoted above. See Correspondence, 204.

8 Robert May has suggested to me that Frege himself speaks of cognitive value only in binary terms: thoughts either have it or lack it. Most of his uses of this terminology probably do fit the binary reading, but I am not sure all do. See, in particular, the remark in the letter to Peano quoted below (see note 10 and the accompanying text). In any event, we need some term to be used as I am using ‘cognitive value’ here, as admitting of degrees. Since this use of the term has gained some currency, I shall continue so to use it.

9 Of course, in one sense, the problem about identity is a different problem: it is the problem we just finished discussing, about what identity is. But the point here is that what makes that problem pressing, in Frege's mind, is the problem about cognitive value.

10 Correspondence, 127; letter XIV/11; xxxiv/11. The bracketed parts of the quotation fill in references to an earlier part of the discussion. This particular letter is undated, but is presumably from about 1896.

11 Correspondence, 127-8; my emphasis. Similar remarks are contained in letter XV/18 (xxxvi/18), to Russell: see esp. 164-5. In this passage and the preceding one, Frege is tacitly appealing to principles of compositionality he does not State. Since the meaning of the whole is determined by the meanings of the parts, and since ‘24 and ‘4x4’ have the same meaning, ‘24=42’ and ‘4x4=42’ must have the same meaning. Since the sense of a part is part of the sense of the whole (see Gg I §32), and ‘24’ and ‘4x4’ have different senses, ‘24=42’ and ‘4x4=42’ also have different senses.

12 Of course, as said above, someone committed to the view that, for Frege, identity statements express meta-linguistic thoughts could yet insist that (2) and (3) express meta-linguistic thoughts. But that would be sheer desperation.

13 These are taken from the Table of Important Theorems, at the end of Volume I. The translations are my own. I have silently translated Frege's notation into something more familiar and shall continue to do so.

14 These two examples are from Bg §§20 and 21, respectively.

15 Frege, Gottlob The Foundations of Arithmetic, trans. Austin, J.L. 2nd rev. ed. (Evanston, IL: North western University Press 1980)Google Scholar

16 For a discussion of how Frege's views about identity evolve with his conception of his logicism, see May, RobertFrege on Identity Statements,’ in Cecchetto, C. Chierchia, G. and Guasti, M.T. eds., Semantic Interfaces: Reference, Anaphora and Aspect (Stanford: CSLI Publications 2001).Google Scholar

17 These are the passage from Function and Concept, the one from ‘Formal Theories of Arithmetic,’ and the three from Grundgesetze. See note 4 for references. For a detalled, and I think largely accurate, discussion of the relation between Frege's attacks on formalism and the development of his views about identity, see Robert May, ‘Frege on Identity Statements.'

18 There is, however, a similar passage in the letter to Peano (letter XIV/11; xxxiv/11) that T&C claim ‘makes it almost out of the question that Frege is rejecting the name view in the opening paragraph of “Sense and Reference”’ (T&C, 169). But the context makes it clear that Frege is attempting to fore stall any misimpression that, according to him, an identity-statement expresses identity of sense. The fact that Frege uses the verb ‘zu meinen’ is not, by itself, significant.

19 Thanks to Kent Bach and Robert May for emphasizing this point to me.

20 This wonderful way of putting the point is due to Kaplan, David See his ‘Words,’ Proceedings of the Aristotelian Society supp. vol. 64 (1990) 93-119, at 118.Google Scholar

21 They are also worried about why the word ‘essentially’ occurs in the first sentence of what they call section IV. But maybe it's because the sentence reads better that way. Or perhaps Frege is simply wary of making the strong claim that the cognitive value would, in this case, be exactly the same, a claim he just doesn't need. In any event, T&C are wise not ‘to rest too much of [their] case on Frege's use of the word “essentially”’ (T&C, 185).

22 The way in which this Charge is over-stated is unfortunately characteristic of the tone of T&C's paper: many of their arguments evaporate when the rhetoric is replaced by a balanced Statement of what they have actually shown. This particular Charge is especially surprising, given that T&C are forced to the view that, when Frege writes at the end of ‘On Sense and Reference,’ ‘Let us return to our starting point,’ he does not actually return us to our starting point. See T&C, 194-5.

23 Once again, for an extensive discussion of the evolution of Frege's views about identity-statements, see Robert May, ‘Frege on Identity Statements.'

24 Another Option would be to hold, as May does, that Frege is not criticizing the Begriffsschrift view in ‘On Sense and Reference’ at all, but a different view (see ‘Frege on Identity Statements’). I'm not sure he's wrong, but I'm not sure he's right, either, so I offer the response in the text as an alternative or Supplement, as the case may be. My reading of the relevant parts of Begriffsschrifl contradicts T&C's, for which see section IV of their paper.

25 For some ideas, see my ‘Do Demonstratives Have Senses?’ Philosophers’ Imprint (2002),, 6-8.

26 Thanks to Kent Bach, Robert May, Øystein Linnebo, Michael Rescorla, and Jason Stanley for discussion and for comments upon earlier drafts of this material. The comments of an anonymous referee also proved helpful.

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