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Number words and reference to numbers

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Abstract

A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we may deny that the relevant statements commit us to numbers. The present paper argues that the correct linguistic analysis of the relevant number statements supports the anti-realist view that Identity is false. However, as will further be shown, pace the anti-realist, this analysis does not establish that such statements do not commit us to numbers after all.

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Notes

  1. Actually, Jupiter has more than four moon. But since it is an often used example, I will assume that (1) is true.

  2. A famous adherent of Identity is, of course, Frege (1974, p. 69).

  3. At least according to the widely assumed Generalized Quantifier Theory (Barwise and Cooper 1981).

  4. See Hofweber (2005).

  5. The former assumption is often rejected on the grounds that (2), unlike (1), is not literally true but just within a certain fiction (e.g. Balaguer 1998; Field 1984; Yablo 2001). Moltmann (2013) joins Hofweber (2005, 2007) in arguing against Identity. But since Moltmann thinks that there are other statements in which we refer to numbers, she is not an anti-realist in the strict sense.

  6. Brogaard (2007) critically discusses the proposal to resolve the two puzzles with the aid of the question-in-disguise theory. I will discuss her central objection to the question-in-disguise theory in Sect. 5.

  7. For more on focus see e.g. Rochemont and Culicover (1990).

  8. Brogaard (2007).

  9. One may think that (14) is a more appropriate answer to (15) than (13) since it is the more direct answer. I will come back to that in Sect. 5 At the moment, it is just important that the answer is appropriate. This seems to be the case since it picks out a distinguishing property of Wagner.

  10. Hofweber (2007, p. 20).

  11. Hofweber thinks it cannot be the case that in (12) but not in (17) there is a focus on ‘Wagner’. If there were a focus in (12) but not in (17), this would conflict with his assumption that intonation-independent focus is always due to the fact that the focused expression appears in an extraordinary syntactic position. For, in (12) and (17) ‘Wagner’ appears in the same syntactic position (see Hofweber 2007, pp. 19ff.). However, the more plausible diagnosis of the case is that (12) and (17) show that Hofweber’s assumption is false. This is especially important since Hofweber also uses the assumption in his argument for Resolution.

  12. Copular sentences are sentences of the form ‘XP be YP’ where YP ≠ participial VP (see den Dikken 2001, p. 1). The examples are from den Dikken (2001), Higgins (1973), and Mikkelsen (2005, 2011). In some cases I chose a different name or predicate for uniformity reasons.

  13. As you can see, statements which are usually treated as identity statements in philosophy are considered as predications. For a philosophical defense of this view see Fara (2001).

  14. For an overview see Mikkelsen (2011).

  15. Or, in linguistic terminology, as expressions of type e and type 〈e, t〉 respectively. See e.g. Mikkelsen (2005, p. 49).

  16. See for this e.g. Heller (2005), Mikkelsen (2005). While Heller argues that identificational sentences are a special form of identity statements, Mikkelsen thinks they are a special type of specificational sentences. The account offered here precludes only Mikkelsen’s view.

  17. Mikkelsen (2011, p. 1807).

  18. Mikkelsen (2011, p. 1809).

  19. Often only statements with a demonstrative pronoun in subject position are cited as examples of identificational sentences (however, den Dikken 2001, p. 5; Higgins 1973, p. 190). But this is not a necessary condition for being an identificational sentence. What is decisive is that the pre-copular term functions as a kind of pointer (Higgins 1973). However, sentences with a demonstrative pronoun in subject position are certainly the clearest cases of identificational sentences.

  20. Higgins (1973, pp. 247ff.).

  21. Mikkelsen (2005, p. 58).

  22. See e.g. Akmajian (1979), Higgins (1973), Heycock (1994), Heycock and Kroch (1999), Mikkelsen (2005, 2012).

  23. For more on this see Hofweber (2007, p. 18).

  24. Since some copular sentences can be used in different ways, it can depend on the context whether a sentence has a focus on the post-copular term or not. For instance, the intuitions with respect to ‘The man who helps us on Fridays is Otto’ are unclear if the sentence appears out of context. The reason is that this sentence, as mentioned above, can be used predicationally, specificationally, and identificationally.

  25. In linguistics, Higgins (1973, p. 199) and Romero (2005, p. 688) cite (2) as a paradigmatic example of a specificational sentence. Moltmann (2013, pp. 520ff.) also argues that (2) is a specificational sentence. She gives a different argument for this thesis. Her argument is not convincing but, for reasons of space, I cannot discuss it here.

  26. See e.g. den Dikken (2001), Mikkelsen (2011), Schlenker (2003). Brogaard (2007) also deals with this characteristic of specificational sentences.

  27. Specificational sentences show different kinds of connectivity effects. For discussion of other connectivity effects see e.g. Mikkelsen (2011).

  28. See e.g. Haegemann (1994, p. 134). For clarification, it holds that a node x dominates the nodes under it that can be traced back to it.

  29. As mentioned above, copular sentences can be used differently. However, this does not hold for sentences that show connectivity effects. For instance ‘What John is is important’ has a predicational reading (‘what John is’ refers e.g. to John’s job and the sentence says that this job is important) as well as a specificational reading (the sentence says that John himself is important). In contrast to that, the sentence ‘What John is is important to himself’ allows only a specificational reading (see Heller 2005, p. 5).

  30. See Mikkelsen (2011, p. 1807).

  31. QID is defended by, e.g., den Dikken et al. (2000), Schlenker (2003), Yoo (2003). A different syntactic analysis of specificational sentences, the logical form analysis, is defended by Heycock and Kroch (1999), Bošković (1997). Semantic analyses are defended by Heller (2005), Jacobson (1994), Sharvit (1999). For an overview see Mikkelsen (2011).

  32. I borrowed this example from Yablo (1996) who also thinks that (30) is neither an identity statement nor a predication.

  33. For sake of brevity I will often say that a certain sentence is the answer instead of saying that the proposition expressed by that sentence is the answer. A further note is important here: according to Schlenker’s version of QID, the copula expresses the identity between two semantic values, namely between the extension of ‘how Valium soothes’ and the intension of ‘Valium soothes by blocking that neurotransmitter’ (see for the relevant semantics of questions Groenendijk and Stokhof (1994)). Since, according to Schlenker, (30) expresses the identity between semantic values, he calls specificational sentences identity statements. His reasons make clear that he uses ‘identity statement’ in a different sense than the term is used here. See for discussion Hofweber (2007, pp. 21ff.).

  34. In Sect. 5.2.1, I will propose a different analysis.

  35. This example is due to Ross (1972).

  36. Further examples can be found in Aloni (2008), Baker (1968), Grimshaw (1979), Heim (1982), Romero (2005).

  37. See also Brogaard (2007, p. 10).

  38. One may object that ‘the number of moons of Jupiter’ is not a singular term but a quantifier phrase (Russell 1905). But even if this were true, the assertive use of sentences that contain this term would commit us to numbers. For, in that case, the term would existentially quantify over numbers.

  39. This proposal is due to Schwartzkopff (ms.).

  40. The first and the third example is due to Yablo (1996), the second one is due to Moltmann (2013).

  41. For example den Dikken et al. (2000) also argue that in some cases the specificational reading persists if we change the word order.

  42. This is also the case if the post-copular term is a singular term that gets its reference through a precedent singular term like ‘himself’. Otherwise we had to analyze (29) as ‘[Who the person John likes is] is [Himself is the person John likes]’. ‘Himself is the person John likes’ cannot be the relevant answer. Otherwise it would again be mysterious how binding can take place, since ‘himself’ is not c-commanded by ‘John’.

  43. One may think that this term opens up a different possibility for vindicating a realist view of numbers. For, even if (2) is not a statement in which reference to numbers is made, this would still hold for statements like ‘The number four is my favorite number’. However, perhaps a different story can be told about such statements. Moreover, it is easier for an anti-realist to reject the assumption that ‘The number four is my favorite number’ is true, since, in contrast to (2), this statement does not have an equivalent counterpart that is obviously true.

  44. See also Moltmann (2013, p. 521).

  45. Moltmann (2013, pp. 520ff.).

  46. E.g. Baker (1968) endorses the first view about concealed questions, Grimshaw (1979) the second view. For further discussion see den Dikken et al. (2000, pp. 81ff.).

  47. One may object to the thesis that (2*) commits us to numbers that ‘the number of moons of Jupiter’ appears only in a question. Since questions are not true, we cannot claim that the term has to refer in order for the question to be true. However, since the question presupposes that there are numbers and it has a true answer, the presupposition has to be fulfilled. A detailed development of this response would have to appeal to issues in the semantics and pragmatics of questions. It has to be left to future research.

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Acknowledgments

Thanks to an anonymous referee of this journal, Daniel Dohrn, Peter Fritz, Thomas Hofweber, Stephan Krämer, Benjamin Schnieder, Robert Schwartzkopff, and, most of all, Alexander Steinberg for helpful discussions and comments. Thanks also to audiences in Hamburg, Turin, Alghero, Berlin, and Cambridge (especially to Michael Potter who was my commentator) where I presented my paper.

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  1. Nominalizations Research Group, Department of Philosophy, University of Hamburg, Von-Melle-Park 6, 20146, Hamburg, Germany

    Katharina Felka

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Felka, K. Number words and reference to numbers. Philos Stud 168, 261–282 (2014). https://doi.org/10.1007/s11098-013-0129-3

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