Number Sentences and Specificational Sentences

Transcription

1 Philosophical Studies manuscript No. (will be inserted by the editor) Number Sentences and Specificational Sentences Reply to Moltmann Robert Schwartzkopff the date of receipt and acceptance should be inserted later 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. Keywords Easy Ontology Frege Numbers Number Words Specificational Sentences 1 Introduction Call sentences such as the following Number-sentences: 1 1 The number of planets is eight. In 57 of Grundlagen, Gottlob Frege proposed that Number-sentences be analysed as mathematical identity statements in the following sense: in a Numbersentence (i) is expresses identity between objects, (ii) both the the-number-of Work in progress. Comments welcome. Do not cite without my permission. Robert Schwartzkopff Research Project Nominalizations: Philosophical and Linguistical Perspectives, Department of Philosophy, University of Hamburg, Von-Melle-Park 6, Hamburg, Germany Tel.: +49 (40) Fax: +49 (40) robert.schwartzkopff@gmail.com 1 That is, Number-sentences are sentences of the form The number of F s is n in which F s and n are instantiated by a plural noun phrase and a number word, respectively.

2 2 Robert Schwartzkopff term (e.g. the number of planets ) and the number word or verbal numeral (e.g. eight ) function as (singular) referring expressions that (iii) refer to special mathematical objects viz. numbers, conceived of as abstract objects if they refer at all. 2 In her paper Reference to Numbers in Natural Language, Friederike Moltmann argues that, pace Frege, Number-sentences should not be analysed in the way he suggested. On the one hand, Moltmann contends that they should be analysed as so-called specificational sentences. And on the other, Moltmann argues that a specificational analysis of Number-sentences treats neither their the-number-of terms nor their numerals as arithmetical referring expressions. In particular, Moltmann argues that the numeral eight in (1) has the same non-referring semantic function as it has in sentences like the following, which I will refer to as Count-sentences: 3 2 There are eight planets. (As for the exact semantic function of numerals in Count-sentences, philosophers tend to regard them as quantificational determiners like, say, some in There are some planets ; see e.g. Hofweber (2005); Balcerak Jackson (2013); Felka (2014). However, from a linguistic point of view, it is more plausible that they function along the lines of predicational adjectives like volcanic in There are volcanic planets ; see e.g. Krifka (1999); Landman (2003); Geurts (2006). 4 For today s purposes, it does not matter which treatment is the correct one.) The claim that the numerals contained in Number- and Count-sentences such as (1) and (2) uniformly function as non-referring expressions has recently been defended by Thomas Hofweber. 5 In particular, Hofweber urges that Number-sentences should be afforded an anti-fregean analysis according to which their truth does not require the existence of numbers. Unlike Moltmann, though, Hofweber does not supplement his arguments with a concrete proposal about how Number-sentences are to be analysed that would vindicate this view. 6 Thus, if cogent, Moltmann s specificational analysis of Number-sentences would constitute a significant advancement of our understanding of such sentences. 2 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) 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).

3 Number Sentences and Specificational Sentences 3 In addition to being of linguistic interest, the question of how to analyse Number-sentences is also of philosophical significance. This is because Frege s analysis plays a crucial role in certain so-called easy arguments for the existence of numbers. 7 Roughly, such arguments rest on two key premisses. First, that Number-sentences like (1) are entailed by certain corresponding and uncontroversially true Count-sentences such as (2). Second, that the truth of Number-sentences requires the existence of numbers. Typically, proponents and detractors of such easy arguments agree on the second premiss but disagree on the first. 8 Their shared acceptance of the second premiss is usually fuelled by their shared acceptance of Frege s analysis. For on Frege s analysis, the truth of Number-sentences requires that their ingredient numerical singular terms discharge their semantic function i.e. that of effecting reference to numbers. And this in turn means that, on Frege s analysis, their truth requires that the numbers to which these terms aim to refer do, in fact, exist. However, as we will see below, Moltmann s analysis has the anti-fregean consequence that the truth of Number-sentences does not require the existence of numbers. Thus, the question of how to analyse Number-sentences is of philosophical significance because answering it will serve to either vindicate or undermine the second premiss of easy arguments for the existence of numbers. The aim of this paper is two-fold. First, I argue that Moltmann fails to establish that Number-sentences are specificational sentences. Second, I show that, even if Number-sentences are specificational sentences, Moltmann s analysis is untenable. The plan is as follows. In 2, I explain the notion of a specificational sentence that informs Moltmann s analysis of Number-sentences. In 3, I assess Moltmann s argument for the claim that Number-sentences are specificational sentences. In response, I show that Moltmann s argument does not support this conclusion. In 4, I scrutinize Moltmann s proposed specificational analysis of Number-sentences and argue that it is untenable. In 5, I conclude and look ahead. 2 Specificational Sentences According to Moltmann, Number-sentences are specificational sentences. Specificational sentences are a subclass of what linguists call copular clauses. 9 Copular clauses are comprised of at least the following three sentence types: 10 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 Roughly, copular clauses are sentences of the form x X P be Y P y, i.e. sentences in which a phrase of some suitable syntactic type X followed by an inflection of the verb be followed by a phrase of some suitable syntactic type Y. 10 The examples below are Mikkelsen s (2011, 1806). There is a fourth type of copular clause, viz. so-called identificational sentences like That (woman) is Sylvia. Identificational sentences usually involve a demonstrative in the pre-copula phrase and... are typically

4 4 Robert Schwartzkopff Equational 3a 3b The morning star is the evening star. Tully is Cicero. Predicational 4a 4b The hat is big. What I bought for harvey is big. Specificational 5a 5b The director of Anatomy of a Murder is Otto Preminger. Who I met was Otto Preminger. Copular clauses of these three types are marked by, inter alia, the following somewhat intuitive differences concerning their standard uses. The standard use of predicational sentences consist in predicating the property denoted by the post-copula phrase (e.g. being big) of the denotation of the pre-copula phrase (e.g. the hat). The standard use of equational sentences consists in communicating that the denotation of the pre-copula phrase (e.g. Tully) is identical to that of the post-copula phrase (e.g. Cicero). And the standard use of specificational sentences consists in using the denotation of the post-copula phrase (e.g. Otto Preminger) to specify who or what is the denotation of the pre-copula phrase (e.g. the director of Anatomy of a Murder). 11 Most linguists agree that predicational and equational sentences comprise sentence types in their own right. 12 However, there is an ongoing debate about whether specificational sentences can be reduced to either of them. 13 The most influential sui generis analysis of specificational sentences is the so-called Question-in-Disguise account (QID). 14 Roughly, QID s guiding idea is that, semantically, a specificational sentence equates the proposition denoted by some answer-sentence A, contributed by its post-copula phrase, with used for teaching the names of people or of things. ; Higgins (1973, 237). Going forward, I ignore identificational sentences. This four-way taxonomy traces back to Higgins (1973, ). Higgins equational sentences include the philosophers identity statements. For an introduction to copular clauses see, e.g., Mikkelsen (2011). 11 Note, though, that at least some copular sentences might have other uses besides these standard ones. For instance, given that both the director of Anatomy of a Murder and Otto Preminger can be used as referring expressions and is can be used to express (numerical) identity, (5a), which is typically used as a specificational sentence, can also be used as an equational one; see Mikkelsen (2005, ch. 5). 12 However, see Partee (1986), who holds a position on which both equational and specificational sentences reduce to predicational ones. 13 Heycock and Kroch (1999) propose that specificational sentences are a subtype of equational sentences. den Dikken (2006) contends that equational sentences are a subtype of specificational sentences. And Mikkelsen (2005, ) suggests that specificational sentences are a subtype of predicational sentences. 14 For another kind of sui generis account, see e.g. Romero (2005).

5 Number Sentences and Specificational Sentences 5 the proposition denoted by some question-sentence Q, contributed by its precopula phrase. 15 QID-theorists typically employ Groenendijk and Stokhof s (1997) framework for the semantics of question- and answer-sentences. According to this framework, an answer sentence A denotes the same proposition, conceived of as a set of possible worlds, with respect to all possible worlds w, viz. the set of worlds in which A is true. In the case of a sentence like I met Otto Preminger this means that, for any world w: 16 6 I met Otto Preminger w tw 1 : I met Otto Preminger in w 1 u. As for the denotation of question-sentences, the framework has it that, with respect to some world w, a question-sentence Q denotes that proposition that in w is the unique exhaustive true answer to Q. For instance, with respect to some world w, a direct question-sentence like Who did I meet? denotes the set of worlds w 1 in which I met exactly those people (or things) that I have met in w: 7 Who did I meet? w tw 1 I met x in w 1 Ø I met x in wqu. (Note that, unlike answer sentences, the denotation of question sentences is not world-invariant. That is, with respect to different possible worlds, question sentences can denote different propositions.) In a specificational sentence like (5b), the question of how the pre-copula phrase can contribute the proposition denoted by some question-sentence has a straightforward answer. It is in virtue of being an indirect questionsentence. 17 i.e. in virtue of being the indirect counterpart of the direct question-sentence Who did I meet? that (5b) s pre-copula phrase Who I met denotes (relative to some w) and hence contributes a proposition. In order to have a specificational sentence s post-copula phrase contribute a proposition, QID assumes that, syntactically, it is the Surface Structure remainder of a complete answer-sentence which has undergone partial deletion. 18 For instance, QID would give the following syntactic analysis of (5b): 19 5b* [Who I met] was [I met Otto Preminger]. According to QID, a specificational sentence x X P be Y P y is true with respect to a world w just in case, with respect to w, the pre-copula X P and the 15 For details on QID see, e.g. den Dikken et al (2000); Schlenker (2003). The above makes it sound as if QID reduces specificational sentences to equational ones. However, linguists agree that QID is not reductionist in this sense. 16 Read α w as the w-denotation of α. 17 The question of how pre-copula phrases that do not look like questions, such as the ones in (5a), contribute propositions will be addressed presently. 18 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). 19 QID s most recommending feature is its ability to explain so-called connectivity effects displayed by specificational sentence. That is, QID provides an explanation for the fact that in What John likes is himself, the reflexive pronoun himself appears to be connected to John ; see Schlenker (2003); Mikkelsen (2011).

6 6 Robert Schwartzkopff post-copula Y P contribute the same proposition. Note, though, that the application of QID to concrete cases is not always straightforward; see Schlenker (2003, ). For instance, a straightforward application to (5b) yields the inadequate result that (5b) is true in, say, the actual world α iff Who I met α = I met Otto Preminger α. Intuitively, (5b) should be true in α iff in α I met Otto Preminger and no one else. Not so, on the straightforward truthconditions specified above. The problem is that whilst Who I met α is an exhaustive proposition (in the sense of entailing the falsity of all rival propositions ), I met Otto Preminger α is not because its truth is compatible with my having met other people besides Otto Preminger. I only mention this problem so as not to misrepresent QID. However, in what follows it will not play a significant role. To sustain QID in cases like (5a) in which the pre-copula phrase does not look like a question, QID-theorists assume that it is a so-called concealed question, i.e. that somehow determiner phrases like the director of Anatomy of a Murder manage to contribute a proposition that would be denoted by an (appropriately related) question-sentence. The pertinent notion of concealment can be understood either syntactically or semantically; see e.g. den Dikken et al (2000, 80f). Understood syntactically, concealment is a matter of ellipsis. For instance, in (5b), the director of Anatomy of a Murder would, relative to some world w, contribute a proposition in virtue of being the Surface Structure remainder of a partially elided indirect question Who the director of Anatomy of a Murder was whose w-denotation it is. 20 Understood semantically, concealment is a matter of systematic ambiguity. Thus understood, concealed question uses of determiner phrases like the director of Anatomy of a Murder are not partially elided uses of the indirect question-sentence Who the director of Anatomy of a Murder was in which the embedded phrase has its usual reference. Rather, it is the determiner phrase itself that in this context co-denotes with the question-sentence Who was the director of Anatomy of a Murder?. In this section, I have introduced the notion of a specificational sentence. I now turn to the question of whether Number-sentences are, in fact, specificational sentences. 3 Are Number-Sentences Specificational Sentences? Why think that Number-sentences are specificational sentences? In this section, I will sketch Moltmann s argument to the conclusion that they are and show that it fails to substantiate this claim. 20 Although the syntactic treatment is most often proposed for specificational sentences like The director I met was Otto Preminger whose pre-copula phrases are headed relative clauses see e.g. Mikkelsen (2011, 1819) it readily extends to cases like (5a). Thus, Schlenker (2003, 191) holds that in They revealed the winner of the contest the determiner phrase the winner of the contest is a concealed indirect question in that the containing sentence is equivalent to They revealed who the winner of the contest was.

7 Number Sentences and Specificational Sentences The Argument from Pronominalisation Moltmann s main argument for the claim that Number-sentences are specificational sentences relies on the application of the so-called pronominalisation test for whether a sentence is a specificational one. 21 The pronominalisation test has been proposed by, for instance, Mikkelsen (2005, 64-86). Mikkelsen observes that there is a contrast between the ways in which the pre-copula phrases of equational, predicational, and specificational copular clauses pronominalise, i.e. a contrast pertaining to what pronouns can be felicitously used to replace them. Mikkelsen s observation is as follows. On the one hand, pre-copula phrases of equational and predicational sentences that are gender-marked for female (male) felicitously pronominalise only with female she (male he ) but not with neuter it (or that ). On the other, pre-copula phrases of specificational sentences felicitously pronominalise only with it ( that ) but not with she ( he ), even when marked for female (male). This contrast is brought out in, for instance, contexts in which the pertinent sentences are used to answer preceding questions. 22 Equational 8 Is Tully Seneca? No, Tully is Cicero. No, {he/?it} is Cicero. Predicational 9 Is the tallest girl in the class German? No, the tallest girl in the class is Swedish. No, {she/?it} is Swedish. Specificational 10 Is the tallest girl in the class Polly? No, the tallest girl in the class is Molly. No, {it/?she} is Molly. 21 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 should 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. 22 Compared to their?-less counterparts,?-marked sentences are less acceptable and somewhat degraded. Moltmann s (2013, 521) example (62) illustrates the same contrast.

8 8 Robert Schwartzkopff The pronominalisation test appeals to this difference in the pronominalisation behaviour of the pre-copula phrases of the different types of copular clauses to determine whether or not a given such sentence is a specificational sentence. Since English the-number-of terms like the number of planets are univocally neuter and thus lack the prerequisite gender-marking, the test cannot be directly applied to Number-sentences. However, these terms German translations such as, for instance, Die Zahl der Planeten are univocally marked for female. Moltmann thus applies the test to the German translation of (1), i.e. to Die Zahl der Planeten is acht. Moltmann has it that, since only the first of the following is felicitous, this shows that the only pronoun that can replace [ Die Zahl der Planeten ] is es (neutral) or (more colloquial) das in a German Number-sentence: Die Zahl der Planeten ist acht. Früher dachte man, es wären neun. 24 The number of planets is eight. Before it was thought it was pl nine. 12 Die Zahl der Planeten ist acht. Früher dachte man, sie wäre neun. The number of planets is eight. Before it was thought she wassg nine According to Moltmann, (11) is perfectly fine whereas (12) is unacceptable. Thus, she concludes, by the pronominalisation test, that (1) and Numbersentences in general are specificational sentences. 3.2 Dislodging the Evidence What are we to make of Moltmann s argument? I do not wish to quarrel with Moltmann s data too much. (11) is perfectly fine. And although I do not think that (12) is unacceptable, it is somewhat infelicitous and, in particular, considerably worse than (11). Nevertheless, these data do not support Moltmann s conclusion. To conclude, by the pronominalisation test, that the higher acceptability of (11) supports the conclusion that (1) is a specificational sentences requires that (11) constitutes a proper application of this test. However, (11) is not a proper application. Thus, (12) s lower acceptability when compared to (11) does not support Moltmann s conclusion that (1) and Number-sentences in general are specificational sentences. The problem is this. A proper application of the pronominalisation test would run as follows. First, take a copular clause S(die Zahl der Planeten), i.e. one whose pre-copula phrase is the female die Zahl der Planeten. Second, replace die Zahl der Planeten in S(die Zahl der Planeten) with the neuter es and the female sie to obtain sentences S(es) and S(sie). Third, check 23 See Moltmann (2013, 523). (11) and (12) correspond to her (63a) and (63b). Since English was is univocally singular whereas German wären is univocally plural, Moltmann s was pl in (11) should strictly speaking be were pl. 24 Moltmann s claim that its das -variant is as or even more colloquial than (11) 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.

9 Number Sentences and Specificational Sentences 9 whether S(sie) is infelicitous when compared to S(es). If so, the test would support the claim that S(die Zahl der Planeten) is a specificational sentence. Thus, a proper application of the pronominalisation test requires that es and sie replace die Zahl der Planeten in the same sentence. This is not the case. The crucial es wären neun in (11) and sie wäre neun in (12) are not the result of replacing die Zahl der Planeten with es and sie in the same (well-formed) sentence. If they were, reverse-substituting die Zahl der Planeten for es and sie in (11) and (12) would yield the same (well-formed) sentence. But it does not. In fact, since the result of replacing es in (11) is ungrammatical whereas the result of doing the same with sie in (12) is not, only the latter substitution yields a well-formed sentence (I use * to indicate proper ungrammaticality): *die Zahl der Planeten wären neun.... *The number of planets was pl nine die Zahl der Planeten wäre neun.... the number of planets wassg nine. This is because in the case of (11) the result of the relevant substituition i.e. (13) is ungrammatical since the grammatical number of the singular die Zahl der Planeten fails to disagree with that of the plural wären. Of course, there is a copular clause (and only one) viz. (14) that (i) contains die Zahl der Planeten in pre-copula position, which (ii) yields sie wäre neun when die Zahl der Planeten is replaced by sie, and that (iii) also yields a well-formed sentence when die Zahl der Planeten is replaced by es. Carrying out the latter substitution yields: 15 Die Zahl der Planeten ist acht. Früher dachte man, es wäre neun. The number of planets is eight. Before it was thought it wassg nine Thus, a proper application of the pronominalisation test would compare the acceptability of (12) with that of (15) rather, as Moltmann did, with that of (11). Consequently, the fact that (11) is more acceptable than (12) does not support the claim that (1) is a specificational sentence. In fact, given the following results of comparing the acceptability of (12) and (15), the opposite seems to be the case. At best, (12) and (15) are equally (un)acceptable. At worst, (12) is more acceptable than (15). These judgments are supported by the vast majority of German native speakers I have consulted and who predominantly prefer (12) over (15). The consequences for Moltmann s argument are dire. At best, the pronominalisation test remains inconclusive and thus fails to show that (1) is a specificational sentence. At worst, the test shows that it is not a specificational sentence. Thus, Moltmann s argument for the specificational nature of (1) and Number-sentences in general fails to establish its conclusion. Despite the failure of Moltmann s argument, let me end this section on a conciliatory note. The fact that the plural (11) is more acceptable than either of the singular (12) and (15) i.e. the fact that Moltmann s misapplication of the

10 10 Robert Schwartzkopff pronominalisation test relies on is puzzling. Thus, it would be worth investigating what exactly is going on in (11) and whether this puzzling difference in acceptability can be appealed to in order to revive Moltmann s argument in a way that does not misapply the pronominalisation test. Doing so, seems particularly prudent because, as Moltmann herself points out, Roger Higgins is adamant that Number-sentences are specificational sentences. 25 And since Higgins is the godfather of specificational sentences, this is a claim is that we should not refuse lightly. In the remainder of this paper I will, thus, grant that Number-sentences are specificational sentences. This does not mean, though, that Moltmann s overall view concerning such sentences would be vindicated. For as we will see presently, Moltmann s specificational analysis of Numbersentences is untenable. 4 Number-Sentences as Specificational Sentences This section investigates the credentials of Moltmann s analysis of Numbersentences as specificational sentences. I begin by sketching Moltmann s analysis and then continue to show why it is unsatisfactory. 4.1 Moltmann s Analysis Moltmann proposes a specificational analysis of Number-sentences in the style of the Question in Disguise (QID) account sketched in 2. Moreover, as indicated in 1, this analysis has it that numerals in a Number-sentence such as 1 The number of planets is eight, do not function as (arithmetical) referring expressions but rather have the same non-referring semantic function as they have in a Count-sentence like: 2 There are eight planets. At least against the background of QID, it is clear that the claim that a specificational analysis of Number-sentences treats numerals in this way requires argument. This is because the QID account of specificational sentences is compatible with their post-copula expressions functioning as referring expressions, when considered as the constituents of the complete sentences whose Surface Structure remainders they are. For instance, the QID analysis of (5b) given in (5b*), spelled out the post-copula Otto Preminger as I met Otto Preminger, a sentence in which Otto Preminger functions as a referring expression. 25 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 arithmetical referring expressions.

11 Number Sentences and Specificational Sentences 11 In general, giving and defending a QID-style specificational analysis of Number-sentences divides into two tasks. Consider the case of (1). On the one hand, one must specify and subsequently defend one s choice of the complete sentence that one takes to occupy (1) s post-copula position and whose Surface Structure remainder is eight. And on the other, one must specify and subsequently defend one s choice of what one considers as the question-denotation of the pre-copula phrase the number of planets and how it acquires this particular denotation. As for the first task, Moltmann analyses eight in (1) as the Surface Structure remainder of the Count-sentence (2). As for the second task, note that the number of planets clearly is not an indirect question. Thus, a QIDstyle specificational analysis of (1) requires that it be understood as a concealed question. Adopting the semantic approach to concealment, Moltmann attempts to discharge the second task by contending that, in the context of (1), it is the number of planets itself that denotes a proposition (relative to some world w). In particular, Moltmann has it that, as a (semantically) concealed question, the w-denotation of the number of planets is the same as that of How many planets are there?. Thus, Moltmann s (2013, 521f) specificational analysis of (1), which evidently is meant to generalise to all Number-sentences, can be captured as follows: 16 [The number of planets] is [there are eight planets], where the number of planets w = How many planets are there? w. 26 What are we to say about this analysis? In particular, how does this analysis fare vis-à-vis the two tasks that are involved in defending a QID-style specificational analysis of Number-sentences? We should grant that spelling out the post-copula numeral eight as the Surface Structure remainder of There are eight planets becomes unobjectionable once the analysis of the pre-copula the number of planets as a semantically concealed question that co-denotes with How many planets are there is vindicated. After all, that there are eight planets is the most straightforward answer to the question of how many planets there are. Thus, the question is whether this proposal concerning the number of planets and the-number-of terms in general can be vindicated. That it can is not trivial. This is because concealed questions are definite determiner phrases that instead of having their usual referents, have a closely related question-denotation; see Grimshaw (1979). Now, on Moltmann s own view the-number-of terms primarily function as referring expressions that aim to refer to what she calls number-tropes. 27 Thus, Moltmann faces the challenge 26 In addition to the analysis in (16), 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. 27 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,

12 12 Robert Schwartzkopff to explain how the the-number-of term the number of planets, which primarily refers to a number-trope, can function as a semantically concealed question that co-denotes with the closely related How many planets are there?. Note, though, that the challenge of explaining how the-number-of terms can acquire the pertinent question-denotation is not germane to Moltmann s view of the-number-of terms as primary trope-referrers. In fact, it arises against the background of every view on which these terms are primarily used as referring expressions. For instance, even on the Fregean view of the-number-of terms as number-referring expressions, the claim that they can also be used as semantically concealed questions invokes the challenge to explain how this is possible. Thus, the plausibility of Moltmann s specificational analysis of Number-sentences seems to turn almost entirely on whether it is possible to discharge this explanatory demand. I will turn to Moltmann s explanation presently. However, before I do so, let us pause to appreciate a distinctive feature of Moltmann s analysis. As indicated in the introduction, Moltmann s specificational analysis of Numbersentences accords their post-copula numerals the same non-referring function as they have in Count-sentences. With Moltmann s analysis now on the table we can see why. It is simply because it spells out the post-copula numerals of Number-sentences as the Surface Structure remainders of partially elided Count-sentences. Moreover, I suggested that, if correct, Moltmann s analysis would be anti-fregean, in the sense of undermining one of the central premisses of certain easy arguments for the existence of numbers, viz. that the truth of Number-sentences requires the existence of numbers. Again, we can now see why this should be so. For one, (2) is a sentence whose truth is commonly regarded as not requiring the existence of numbers. And for another, on Moltmann s analysis, the truth-conditions of (1) are the same as those of (2) Thus, the truth of (1) would not require the existence of numbers either. The reason for the crucial identity of truth-conditions is this. First, with respect to any possible world w, (2) denotes the set of worlds in which it is true: 17 There are eight planets w = {w 1 : there are eight planets in w 1 }. Second, according to Groenendijk and Stokhof s semantics for questionsentences, the w-denotation of How many planets are there? hence, that of the number of planets when used as a concealed question would be the set of worlds w 1 in which there are as many planets as there are in w. Put differently: 18 How many planets are there? w = tw 1 are n planets in w 1 Ø there are n planets in wqu. 28 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. is an appropriately restricted higher-level quantifier binding variables in the position of numerals in Count-sentences.

13 Number Sentences and Specificational Sentences 13 Third, according to Moltmann s QID-style analysis, (1) would be true with respect to any world w iff: 19 How many planets are there? w = There are eight planets w. Now, let w be any world in which (2) is true and in which, thus, there are eight planets. By (18), this means that, with respect to w, How many planets are there? denotes the same world-set as (2), as its members are those worlds w 1 in which there are also eight planets. And by (18), this means that (1) is true in all worlds w in which (2) is true. Conversely, let w be any world in which (2) is false and in which, consequently, there are either more or less than eight planets. Regardless, of how many planets, different from eight, w contains, the w-denotation of How many planets are there? will be distinct from that of (2). Thus, by (19), (1) will be false in all worlds w in which (2) is false. Consequently, (1) is true with respect to all and only the worlds with respect to which (2) is true. Hence, (1) and (2) have the same truth-conditions. Having thus seen how the correctness of Moltmann s Number-Analysis would undermine certain easy arguments for the existence of numbers, let us turn to the question whether it is correct. 4.2 Undermining the Analysis Recall, a defence of Moltmann s analysis requires an explanation of how the number of planets can acquire the question-denotation that the analysis ascribes to it. Moltmann s explanation has two parts: 29 [A]... the number of can be used as a replacement of a numeral. The number of is then on a par with three, several, and many as well as a great number of or a small number of, as below: 20 John kissed a great number of babies. 21 John made a small number of mistakes. The use of the number of as a numeral replacement is also indicated by the possibility of plural agreement rather than singular agreement in English: 22 A great number of women were arrested. The number of appears to be used as a numeral replacement in that sense below in [(23)] (which certainly does not mean that Mary counted a single number trope): 23 Mary counted the number of mistakes she had made. [B] With the number of acting as a numeral replacement, a concealedquestion analysis becomes straightforwardly available: The number of 29 [A] =Moltmann (2013, 502f), [B] =Moltmann (2013, 522).

14 14 Robert Schwartzkopff in the number of planets in [(1)] will then the [sic] same semantic function as how many, thus naturally allowing for a question-denotation of the right type. (Of course the number of and how many have different syntactic functions, the former serving to form a complex definite [determiner] phrase, the latter a wh-clause.) From [B], it is clear that Moltmann s explanation of how the number of planets can function as a concealed question use that co-denotes with How many planets are there? crucially involves the following two assumptions: A.1 the number of can act as a numeral replacement. A.2 If it acts as a numeral replacement in the number of terms, the number of has the same semantic function as how many. The remainder of this paper is dedicated to undermining these assumptions, thereby undermining Moltmann s Number-analysis itself. First off, though, let me note that there may well be something right about the thought that the number of and how many are connected in the way Moltmann alleges. For instance, replacing the number of in (23) with how many yields: 24 Mary counted how many mistakes she had made. If nothing else, it seems as if (23) and (24) are closely related. 30 And this fact may be taken to indicate that, in some way or other, the-number-of terms are capable to function as semantically concealed questions. However, the issue here is complicated. Arguably, count in (23) functions as a intensional verb and as such does not take questions, concealed or otherwise, as complements. Thus, (23) is not an example of a concealed-question-use of the-number-of terms. 31 In what follows, I will therefore put Moltmann s example (23) to the side, and instead concentrate on her other examples. Now, with respect to the first assumption (A.1), Moltmann contends that, when acting as a numeral replacement, the number of is on a par with several, many, and phrases of the form a (Adj) number of (for certain adjectives such as, for instance, small and great ). This claim is rather doubtful. To see why consider the following sentences: 25a 25b 25c There are eight planets. John kissed eight babies. John made eight mistakes. 30 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. 31 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).

15 Number Sentences and Specificational Sentences 15 25d Eight women were arrested. (Of course, (25a) is the same as (2). (25b-d) have been obtained by replacing a small/great number of in Moltmann s (20) (22) with the numeral eight.) With respect to (25a-d), we can observe the following contrast. Replacing eight with several/many/a (Adj) number of, yields perfectly acceptable sentences: 26a 26b 26c 26d There are {several/many/a (Adj) number of } planets. John kissed {several/many/a (Adj) number of } babies. John made {several/many/a (Adj) number of } mistakes. {Several/Many/A (Adj) number of } women were arrested. Thus, there is a clear sense in which several/many/a (Adj) number of are on a par with the numeral eight in (25a-d) and, thus, a clear sense in which they can act as numeral replacements. In contrast, replacing the numeral eight in (25a-d) with the supposed numeral replacement the number of yields sentences that, although perhaps not strictly speaking ungrammatical, are decidedly odd: 27a 27b 27c 27d?There are the number of planets.?john kissed the number of babies.?john made the number of mistakes. *The number of women were arrested. The oddity of (27a) may be explained by the fact that the complement position of be in existential sentences such as (25a) display so-called indefiniteness-effects which prevents it from being occupied, without ill effect, by certain phrases including those headed by the ; see e.g. Partee (1999). However, this explanation does not carry over to (27b-d). In particular, (27d) is downright ungrammatical due to an agreement violation with respect to grammatical number between the singular the number of planets and the plural were arrested. Thus, it is clear that, unlike several/many/a (Adj) number of, the number of is not a par with the numeral eight in (25a-d) and, thus, clear that it cannot act as a numeral replacement in quite the same way. Now, I do not wish to claim that this contrast constitutes decisive evidence against (A.1). In particular, it is worth noting that there are sentences which, on the face of it, seem rather close to the above but which are perfectly acceptable: 27b* John kissed eight babies and Mary kissed the same number of babies. 27c* John made eight mistakes and Mary made the same number of mistakes.

16 16 Robert Schwartzkopff Thus, it may well be that the above contrast can, somehow, be explained along lines that would vindicate assumption (A.1). However, the existence of this contrast does show that (A.1) stands in need of further clarification and support. And since (A.1) is crucial to Moltmann s defence of the proposed specificational analysis of Number-sentences, this means that we should at least be agnostic whether her defence is successful. However, let us grant that the number of can, as per (A.1), act as a numeral replacement. Even then, though, Moltmann s analysis would falter if it turned out that (A.2), her second crucial assumption, is unwarranted. And as we will see now, this is, in fact, the case. Recall, (A.2) was the assumption that, if the number of acts as a numeral replacement in the-number-of terms, it has the same semantic function as how many. It is clear that (A.2) requires argument even if (A.1) is granted. After all, not all numeral replacements share the semantic function of how many. For if they did, the above (26a-d), which were obtained from (25ad) by replacing the numeral eight with several/many/a (Adj) number of, would have to have the denotations of corresponding how many -sentences. Clearly, though, they do not. In order to assess the credentials of assumption (A.2), we therefore need a clearer grasp on what, exactly, the semantic function of how many is. To answer this question, it is beneficial to first take a look at the syntactic function of how many. As Moltmann herself observes, replacing numerals with how many serves the syntactic function of forming wh-clauses, i.e. sentences headed by explicit interrogatives like what, who, or how many. For instance, replacing the numeral eight with how many in (25a-d) yields the wh-clauses: 28a There are how many planets? («How many planets are there?) 28b John kissed how many babies? («How many babies did John kiss?) 28c John made how many mistakes? («How many mistakes did John make?) 28d How many women were arrested? Now, within the framework of Groenendijk and Stokhof (1997), we already know what these wh-clauses denote relative to some world w. (28a) denotes the set of possible worlds w 1 in which there are as many planets as there are in w. (28b) denotes the set of possible worlds w 1 in which John kissed as many babies as he kissed in w. (28c) denotes the set of possible worlds w 1 in which John made as many mistakes as he made in w. And (28d) denotes the set of possible worlds w 1 such that there are (exactly) as many women who were arrested in w 1 as there are women who were arrested in w. However, although we thus know what these wh-clauses denote, we do not as of yet know how they manage to do so. That is, we do not know how replacing numerals n in non-wh-clauses like (25a-d) with how many yields wh-clauses with the pertinent denotations. Answering this question is to answer the question of the semantic function of how many.

17 Number Sentences and Specificational Sentences 17 One attractive way to think of the semantic function of how many is the following. How many is a higher-level variable-binding device, which we can symbolise as?n: and that is restricted to the kind of semantic values that numerals have in Count-sentences. 32 If so, (28a-d) can be given the following semi-formalisations, just like Romeo kissed someone can be semi-formalised as Dx Romeo kissed x : 28a*?n: there are n planets. 28b*?n: John kissed n babies. 28c*?n: John made n mistakes. 28c*?n: n women were arrested. Conceiving of how many along these lines has two consequences one syntactic, the other one semantic. On the syntactic side, how many would operate on certain open sentences, i.e. sentences in which an restricted numeral variable n occurs unbound. On the semantic side, the semantic function of how many can now be specified as follows. Let Spnq be a sentential context in which the numeral n functions the same way as numerals in Countsentences. And let Spnq be the result of replacing the numeral n with a numeral variable n. Then the semantic function of how many is to turn Spnq into a sentence Sphow manyq =?n : Spnq such that: 29?n : Spnq w = tw 1 in w 1 Ø Spnq in wqu. Equipped with a clearer grasp of the semantic function of how many, we can now assess the credentials of Moltmann s second assumption (A.2): if the number of acts as a numeral replacement in the-number-of terms, it has the same semantic function as how many. For (A.2) would be vindicated if it was possible to construe the number of in the-number-of terms in Numbersentences along the lines of the variable-binding device?n:. At first glance, the claim that it can seems questionable. For recall, given our understanding of how many as the variable-binding device?n:, how many operates on complete sentences that contain a free numeral variable. Because of this, replacing how many in a wh-clause with a numeral once more results in a sentence. For instance, replacing how many in the wh-clauses (28ad) results in the sentences (25a-d). Not so it seems, in the case of the number of planets. For, on the face of it, replacing the number of in the number of planets with the numeral eight yields (not a sentence but) the noun phrase eight planets. Thus, it may seem as if the number of in the-number-of terms cannot function in the same way as how many simply because, within the context of the-number-of terms, it would lack the required input viz. certain open sentences that an expression with this semantic function operates on. If so, this would constitute strong evidence against the viability of Moltmann s explanation of how the-number-of terms can acquire the question-denotation Moltmann s analysis of Number-sentences ascribes to them. 32 Conceiving of how many as a variable-binding device is in the spirit of Groenendijk and Stokhof (1997).

18 18 Robert Schwartzkopff Perhaps, though, appearances are deceptive. For is it not possible that there is more to the-number-of terms than meets the eye, at least when they are used as concealed questions? That is, is it not possible to analyse for instance the number of planets in a way that spells out planets in the number of planets as the Surface Structure remainder of the open sentence There are n planets and the number of as a variable binding device in the mould of?n:? Roughly, such an analysis would have it that, when used as a concealed question, the syntactic structure of the number of planets is: 30 [[the number ofn] [There are n planets]] If viable, this analysis would go a long way to substantiate Moltmann s assumption (A.2) according to which the number of in the-number-of terms can have the same semantic function as how many. However, although the analysis in (30) would indeed vindicate (A.2), there is strong cross-linguistic evidence that undermines it. Moltmann (2013, 509) herself observes that [i]n some languages, the number of is followed by a definite plural..., for example in German ( die Anzahl der Planeten/*von Planeten the number of the planets/of planets ). Moreover, the German translation of (1), appealed to in Moltmann s argument for the specificational nature of (1), is: 31 Die Anzahl der Planeten ist acht. Thus, assuming that Moltmann s Number-Analysis carries over to German, (31) is to be analysed as: 32 [Die Anzahl der Planeten] ist [Es gibt acht Planeten], where Es gibt acht Planeten is the German translation of the English Countsentence There are eight planets. Note, though, that in (31) die Anzahl ( the number (of) ) is followed by the definite plural der Planeten, which is not a constituent of Es gibt acht Planeten. Thus, the German equivalents of the-number-of terms cannot be understood along the lines suggested above. That is, die Anzahl der Planeten in (32) cannot be spelled out as: 33 [[die Anzahln] [es gibt n Planeten]]. On the reasonable assumption that the impossibility of analysing concealed question uses of the German equivalents of the-number-of terms in the above manner is evidence that English the-number-of terms cannot be analysed in this way either, this is evidence against the viability of the analysis proposed in (30) 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. 11 fn. 26. Thus, the former would have to be spelled out as The numbern of [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 Anzahln] [die Planeten sind n].