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Number sentences and specificational sentences

Reply to Moltmann

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Abstract

Frege proposed that sentences like ‘The number of planets is eight’ be analysed as identity statements in which the number words refer to numbers. Recently, Friederike Moltmann argued that, pace Frege, such sentences be analysed as so-called specificational sentences in which the number words have the same non-referring semantic function as the number word ‘eight’ in ‘There are eight planets’. The aim of this paper is two-fold. First, I argue that Moltmann fails to show that such sentences should be analysed as specificational sentences. Second, I show that even if they are to be analysed in this way, Moltmann’s proposed specificational analysis is unsatisfactory.

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Notes

  1. That is, Number-sentences are sentences of the form ‘The number of Fs is n’ in which ‘Fs’ and ‘n’ are instantiated by a plural noun phrase and a number word, respectively.

  2. See Frege (1884, §57). Frege’s own example of a Number-sentence was ‘The number of Jupiter’s moons is four’. Henceforth, I will use ‘number word’ and ‘verbal numeral’ interchangeably and leave the qualifier ‘verbal’ mostly tacit.

  3. Note, though, that the scope of Moltmann’s paper is considerably broader as is constitutes an all-out attack of the view that, in natural language, reference to numbers conceived of as abstract objects is effected by the-number-of terms and numerals. According to Moltmann, this view is ‘fundamentally mistaken’ because reference to numbers is only effected by expressions of the form ‘the number n’ that Moltmann calls ‘explicit number referring terms’. Moltmann’s arguments for her overall view merit attention in their own right. However, this paper is only concerned with those of her considerations that pertain to Number-sentences.

  4. Knowles (2015, 2758–62) summarises the relevant arguments for the adjectival account of numerals.

  5. See Hofweber (2005, 2007). For a critique of Hofweber’s arguments see, e.g., Brogaard (2007), Balcerak Jackson (2013).

  6. The point that this constitutes a weak spot of Hofweber’s arguments has recently been pushed by Balcerak Jackson (2014).

  7. For a recent book-long treatise on easy arguments for ontological conclusions, see Thomasson (2015).

  8. For a rejection of easy arguments for numbers along these lines, see e.g. Field (1984), Yablo (2000).

  9. This taxonomy, which also includes the fourth type of so-called identificational sentences (e.g. ‘That (woman) is Sylvia’), traces back to Higgins (1973, 203–93). For an introduction to copular clauses, see e.g. Mikkelsen (2011). The examples below are Mikkelsen’s (2011, 1806).

  10. See e.g. Heycock and Kroch (1999), den Dikken (2006), and Mikkelsen (2005, 118–30) for reductive analyses along this line.

  11. For another kind of sui generis account, see Romero (2005).

  12. For details on QID see, e.g. den Dikken et al. (2000), Schlenker (2003). Although the above makes it sound as if QID reduces specificational sentences to equational ones, linguists do not regard QID as reductionist in this sense.

  13. Read ‘\(|\alpha |_w\)’ as ‘the w-denotation of \(\alpha \)’.

  14. In passing, let me note that there is some debate about the adequacy of QID’s application to concrete cases; see e.g. Schlenker (2003, 183–90). For instance, (8) will give the wrong truth-conditions for (5b) if the latter’s w-truth is taken to require that, in w, I met only Otto Preminger. (Ascertaining this, I leave as an exercise). However, for the case at hand, this issue can be safely ignored.

  15. This is evidenced by the fact that, sometimes, it is possible for the post-copula position to be occupied by a complete sentence: [What I did then] was [I called the grocer]; see Ross (1972, 89).

  16. This kind of syntactic analysis also enables QID’s most recommending feature. By analysing a specificational sentence like ‘What John likes is himself’ as ‘[What John likes] is [John likes himself]’, QID is able to explain the sentence’s so-called ‘connectivity effect’, i.e. the phenomenon that the reflexive pronoun ’himself’ appears to be bound by ‘John’; see e.g. Schlenker (2003), Mikkelsen (2011).

  17. I say ‘main argument’ because Moltmann (2013, 520f) offers the following as a ‘first indication’ that (1) should not be analysed along Frege’s lines. On Frege’s analysis, ‘eight’ in (1) refers to the number eight. Thus, one would expect that the following is true (as well as acceptable):

    • i   The number of planets is the number eight.

    However, according to Moltmann, (i) is ‘much less acceptable than (1) and in fact not true’; (i) = Moltmann’s (59b). Her (59c) and (59d) are in a similar spirit. I concur that, compared to (1), (i) is less acceptable. Whether this must be taken to mean that (i) is not true is a different question. Granted, the oddity of (i) poses a challenge to Frege’s analysis, a challenge that may or may not be met. Taking up this mantle is not today’s project. Moreover, since Moltmann only regards this as a first indication that Frege’s analysis is mistaken, it leaves open the question of whether it must be regarded as a specificational sentence. This Moltmann seeks to establish by way of the argument to be discussed in the main text.

  18. Compared to their ?-less counterparts, ?-marked sentences are less acceptable and somewhat degraded. Moltmann’s (2013, 521) example (62) illustrates the same contrast.

  19. See (Moltmann 2013, 523). (12) and (13) correspond to her (63a) and (63b). Since English ‘was’ is univocally singular whereas German ‘wären’ is univocally plural, Moltmann’s ‘was\(_{\hbox {pl}}\)’ in (12) should strictly speaking be ‘were\(_{\hbox {pl}}\)’.

  20. Moltmann’s claim that its ‘das’-variant is as or even more colloquial than (12) itself is somewhat contentious. To me as well as to all native German speakers to whom I have put this question, it sounds considerably worse.

  21. See Higgins (1973, 199). Romero (2005) agrees. Note, however, that unlike Moltmann’s, Romero’s specificational analysis of Number-sentences treats the numerals they contain as numerical singular terms.

  22. In addition to the analysis in (1*), Moltmann presents an alternative analysis of (1), according to which |‘the number of planets’\(|_w\) = |‘How many are the planets?’\(|_w\) and ‘eight’ is spelled out (not as the Count-sentence (2) but) as the numerical predication ‘The planets are eight’. My main arguments against Moltmann’s official analysis also apply to this alternative.

  23. See Moltmann (2013, 501ff). The basic idea of the-number-of terms as trope-referring expressions is the following. If there are eight planets, then according to trope-theory, not only are there the planets and the number-property of being eight they jointly exemplify, there is also that property’s particular instance—a number-trope—which inheres in the planets. On Moltmann’s view, it is the primary semantic function of ‘the number of planets’ to refer to this trope.

  24. ‘\(\forall {{\varvec{n}}}\)’ is an appropriately restricted higher-level quantifier binding variables in the position of numerals in Count-sentences.

  25. [A] = Moltmann (2013, 502f), [B] = Moltmann (2013, 522). (21), (22), (23) and (24) below correspond to Moltmann's (2a), (2b), (i), and (3), respectively.

  26. The claim that they are chimes well with Higgins’s (1973, 202) claim that the following two sentences are equivalent:

    • i I counted the number of planets.

    • ii I established how many planets there are by counting them.

  27. Thanks to an anonymous referee for Linguistics and Philosophy, who in relation to a different paper of mine pointed this out to me. For more on intensional verbs, see e.g. Moltmann (2008).

  28. Conceiving of ‘how many’ as a variable-binding device is in the spirit of Groenendijk and Stokhof (1997).

  29. Similarly, for Moltmann’s alternative Number-analysis on which ‘the number of planets’ is understood as co-denoting with ‘How many are the planets?’; see p. 10 fn. 21. Thus, the former would have to be spelled out as ‘[[the number of n][the planets are n]]’. However, ‘The planets are eight’ translates into German as ‘Die Planeten sind acht’, a sentence in which the plural description stands in the nominative case. Thus, it is equally unlikely that ‘die Anzahl der Planeten’, in which the plural description stands in the genitive case, can be spelled out in the suggested way, i.e. as [[Die Anzahl\(_{{{\varvec{n}}}}\)] [die Planeten ]].

  30. For a different non-QID specificational analysis of Number-sentences, which allegedly is anti-Fregean, see Knowles (2015).

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Acknowledgments

I am grateful for helpful discussions with and comments from Arvid Båve, Michael Clark, Christian Folde, Stephan Krämer, Stefan Roski, Benjamin Schnieder, and Nathan Wildman, to audiences in Hamburg, Konstanz, Turin, and Stockholm, where I have presented versions of this paper, and to an anonymous referee of this journal. For the financial support during the early and middle stages of this paper’s development, I wish to express my gratitude to the Arts and Humanities Research Council and the Royal Institute of Philosophy.

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

    Robert Schwartzkopff

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Schwartzkopff, R. Number sentences and specificational sentences. Philos Stud 173, 2173–2192 (2016). https://doi.org/10.1007/s11098-015-0603-1

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