JOURNAL OF SEMANTICS Volume 23 Number 4
CONTENTS TANIA IONIN AND ORA MATUSHANSKY The Composition of Complex Cardinals
315
BENJAMIN RUSSELL Against Grammatical Computation of Scalar Implicatures
361
ROBERT VAN ROOIJ Free Choice Counterfactual Donkeys
383
Editor’s Note
403
Please visit the journal’s web site at www.jos.oxfordjournals.org
Journal of Semantics 23: 315–360 doi:10.1093/jos/ffl006 Advance Access publication November 16, 2006
The Composition of Complex Cardinals TANIA IONIN USC/UIUC ORA MATUSHANSKY CNRS/Universite´ Paris 8
This paper proposes an analysis of the syntax and semantics of complex cardinal numerals, which involve multiplication (two hundred) and/or addition (twentythree). It is proposed that simplex cardinals have the semantic type of modifiers (ÆÆe, tæ, Æe, tææ). Complex cardinals are composed linguistically, using standard syntax (complementation, coordination) and standard principles of semantic composition. This analysis is supported by syntactic evidence (such as Case assignment) and semantic evidence (such as internal composition of complex cardinals). We present several alternative syntactic analyses of cardinals, and suggest that different languages may use different means to construct complex cardinals even though their lexical semantics remains the same. Further issues in the syntax of numerals (modified numerals and counting) are discussed and shown to be compatible with the proposed analysis of complex cardinals. Extra-linguistic constraints on the composition of complex cardinals are discussed and compared to similar restrictions in other domains.
1 INTRODUCTION The goal of this paper is to propose an account of the cross-linguistic syntax and semantics of complex cardinals. While there has been much work examining the syntax and semantics of simplex cardinals such as three, complex cardinals, which involve multiplication (1) and/or addition (2), have not previously received much attention. (1) a. five hundred thousand b. quatre vingt four twenty ‘eighty’ (French) (2) a. three hundred and five b. twenty-seven c. tri ar ddeg three on ten ‘thirteen’ (Welsh)
Hurford (2003)
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Abstract
316 The Composition of Complex Cardinals d. sto sem’ hundred seven ‘a hundred and seven’ (Russian)
1 We use the term xNP rather than NP or DP to indicate that it is irrelevant which functional layers are projected and which aren’t.
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We will argue that complex cardinals are composed entirely in syntax and interpreted by the regular rules of semantic composition (i.e., construction of complex cardinals is done exclusively by linguistic means). This analysis is independently motivated by syntactic transparency of complex cardinals and their compositional semantics. The paper is organized as follows. In section 2 we argue that simplex cardinals have the semantic type of modifiers (ÆÆe, tæ, Æe, tææ) and show how this accounts for the internal composition of complex cardinals via iterative syntactic complementation. Existential force of cardinal-containing extended NPs (xNPs)1 in argument positions is also treated in this section. Section 3 addresses the semantic atomicity requirement imposed by cardinals on their complements, and shows that such morphosyntactic operations as Case-assignment and number marking in cardinal-containing xNPs provide evidence that complex cardinals are built in the syntax; this section also discusses extralinguistic factors in the composition of complex cardinals. Section 4 presents an analysis of complex cardinals like twenty-two in terms of coordination and discusses some ordering constraints. Section 5 concludes the paper and poses some questions for further research. The Appendices address some further issues in the syntax and semantics of cardinals. Since our goal is to provide an analysis that works for complex cardinals cross-linguistically, we draw upon data from a variety of (typologically different) languages. While some empirical phenomena (e.g. articles, morphological Case assignment, etc.) are visible only in a subset of languages, we will extend the analysis based on these phenomena to other languages, unless there are empirical reasons for not doing so. One caveat is in order: we focus primarily on nonclassifier languages in this paper; however, we show in section 3 that our analysis can be logically extended to classifier languages as well. Due to lack of space, we concentrate here on the semantics of complex cardinals, and discuss their syntax in a relatively superficial manner. More discussion of the thornier issues arising in our analysis can be found in Ionin & Matushansky (in preparation).
Tania Ionin and Ora Matushansky 317
2 SEMANTICS OF CARDINALS This section is dedicated to the semantics of complex cardinals involving multiplication. We ask how complex cardinals such as three hundred, four hundred thousand, etc., are composed semantically (on cardinals involving addition, such as forty-two, see section 4). The background assumption we start with is that the semantics of cardinals is the same cross-linguistically, at least in languages that have complex cardinals.2 We follow the natural hypothesis that complex cardinals are derived from simplex ones: that four hundred should be semantically related to four as well as hundred.3
Furthermore, we strive to capture the basic intuition that the four in (3a) is semantically the same as the four in (3b). The meaning of a complex cardinal should be derived in such a way that each cardinal inside it is also semantically compatible with a lexical xNP: the same four should be able to combine with books as easily as with hundred books.
2.1 Semantic type of cardinals: cardinals are modifiers The above intuition is captured straightforwardly if simplex cardinals have the semantic type of modifiers (ÆÆe, tæ, Æe, tææ). Although the proposal that cardinals are modifiers has been much discussed in the literature (see Link 1987; Verkuyl 1993; Carpenter 1995; Landman 2003, among others), no distinction has previously been made between simplex and complex cardinals. We propose that simplex cardinals are of type ÆÆe, tæ, Æe, tææ, and derive the meaning of xNPs containing complex cardinals compositionally. In order to derive the meaning of complex cardinals, we need full recursivity, which we derive from the semantic type ÆÆe, tæ, Æe, tææ, as illustrated in the structure in (4), where the lexical xNP is the sister 2
The semantics that we propose for simplex cardinals is necessary only for languages that have complex cardinals. The main motivation for the semantic type ÆÆe, tæ, Æe, tææ (see below) is the compositional semantics of complex cardinals; if a language has only simplex cardinals, they can be type Æe, tæ and combine with the lexical xNP via Predicate Modification—see section 2.2.2. It is quite likely that cardinals historically developed from type Æe, tæ (see also Hurford 2001); in some languages simplex cardinals originate as predicates synchronically as well, and are converted to the modifier type (see Ionin and Matushansky (in preparation)). Since the issue is orthogonal to our concerns, we will not address it here. 3 One option that we do not discuss here, in view of total lack of morpho-syntactic evidence for such a hypothesis, is that not all simplex cardinals inside a complex one have the same semantic type (i.e. that four is ÆÆe, tæ, Æe, tææ in (3a) and Æe, tæ in (3b)).
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(3) a. four hundred books b. four books
318 The Composition of Complex Cardinals of the innermost cardinal. (5) is a sample lexical entry for simplex cardinals.
ð4Þ
tæ
. kx 2 De . dS 2 DÆe,
tæ
[P(S)(x) ^ jSj ¼ 2 ^
S is a partition P of an entity x if it is a cover of x and its cells do not overlap (cf. Higginbotham 1981: 110; Gillon 1984; Verkuyl & van der Does 1991; Schwarzschild 1994): (6) P(S)(x) ¼ 1 iff partition S is a cover of x, and "z, y 2 S [z ¼ y _ :da [a Qx]. The challenge is to account for this equivalence, without giving up our standard dynamic account of indefinites. Suppose that we want to interpret a sentence of the form dx/ > w in possibility Æw, gæ. According to the standard Lewis/Stalnaker analysis of counterfactuals, we should then select among those possibilities that verify dx/ the ones that are closest to Æw, gæ and check whether they also make w true. Because / might contain free variables that should be interpreted by means of g, the natural context of interpretation of dx/ is the set W(g) ¼ fÆv, hæ : v 2 W & h ¼ gg.14 After the interpretation of dxPx, for instance, we end up with a set of world-assignment pairs like Æv, hæ where variable x is in the domain of assignment h, and h(x) is an element of the set denoted by P in world v. Let us denote this set of 13
I assume here that assignments are partial functions. Just like Lewis (1973), we might want to limit the worlds considered by means of an accessibility relation. I will ignore this possibility in this paper. Also, I will assume for simplicity that all worlds have the same domain. 14
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Heim (1982) showed that we could maintain a uniform analysis of indefinites and pronouns, and still get the truth conditions of donkey sentences right, while Groenendijk and Stokhof (1991) and others have demonstrated that such an analysis is actually no threat to compositionality, if we are willing to change our static possible-world conception of the meaning of a sentence. According to the alternative dynamic view, we interpret sentences with respect to a context that is represented by a set of world-assignment pairs, and the meaning of the sentence itself can be thought of as the update of this context, where possibilities are eliminated when the sentence is false, and the assignment of the possibilities is enriched if a new variable, or discourse referent, is introduced by way of an indefinite.13 According to this analysis, the formula dx½Px/Qx is predicted to be equivalent with "x½Px/Qx, which means that we can account for (standard) donkey sentences in a systematic and compositional way. Although a considerable amount of attention has been devoted to donkey sentences in the past, only a particular branch of donkey sentences were actually inspected: indicative ones. To account for these indicative donkey sentences it was no problem to assume that conditional sentences should be analyzed (basically) in terms of material implication. But donkey sentences not only show up in indicative mood; we have counterfactual donkey sentences as well:
392 Free Choice Counterfactual Donkeys
3 SOLUTION
3.1 Counterfactual donkey sentences Fortunately, there is a natural way to define an ordering ‘ Qx is true in Æw, gæ we have to select among the possibilities in /dxPx/g those that are closest to Æw, gæ, and see whether they also verify Qx. But this means that we need an ordering relation, < Æw;gæ , between worldassignment pairs with respect to possibility Æw, gæ : Æu, kæ < Æw;gæ Æv, hæ. Let us assume that we can analyse counterfactuals in terms of selection functions, as before, and that / > w is true in possibility i just in case w is true in all selected /-possibilities closest to i, i.e. in all j 2 fi ð=/=g Þ: It ð=dxPx=g Þ to be is clear what we want the result to be: we want fÆw;gæ [d 2 D fÆw;gæ ðfÆv,g½x =d æ : d 2 Iv ðPÞgÞ: On such an analysis, we would be able to account for (3) without necessarily giving up on a nonmonotonic analysis of counterfactuals.15 On the other hand, it is clear 16 that we don’t want to define fÆw ,gæ ð=dxPx=g Þ to be the desired set. For in that case, our analysis wouldn’t be compositional anymore. What we want, instead, is to first determine the (dynamic) interpretation of dxPx, i.e. /dxPx/g, and then define the selection function f, or the ordering relation < Æw,gæ ; such that the set of selected world-assignment x pairs is identical with [d 2 D fÆw ,gæ ðfÆv; g½ =d æ : d 2 Iv ðPÞgÞ:
Robert van Rooij 393
assignment function. But suppose that / is of the form dxPx. In that case, all the assignments in /dxPx/g differ from g in that they also assign an object to x. Let Æv, hæ and Æu, kæ be two possibilities in /dxPx/g. According to definition 1, to check whether the one is more similar to Æw, gæ than the other only makes sense in case h assigns the same individual to x as k, h(x) ¼ k(x).18 But this means that we check for each individual d separately what are the closest possibilities to Æw, gæ that make Px true. We define the selection function as follows:
def
ð=A=g Þ ¼ fÆv;hæ 2 =A=g : :dÆu;kæ 2 =A=g : Æu;kæ < Æw;gæ Æv;hæg; fÆw;gæ
where =A=g ¼ ½AðfÆv; hæ : v 2 W & h ¼ ggÞ; and j < i k iff j < i k but not k < i j: It is easy to see that it now follows that fÆw;gæ ð=dxPx=g Þ comes out x to be equivalent with [d 2 D fÆw;gæ ðfÆv; g½ =d æ : d 2 Iv ðPÞgÞ: As we have seen above, this is exactly what we want, but now we don’t define the selection function this way (which would give rise to a noncompositional analysis), but we still end up with the same happy result that dx[Px] > Qx is equivalent with "x[Px > Qx].19,20,21 I conclude that we can account for counterfactual donkey-sentences in a natural and compositional way.
3.1.1 Identifying and weak counterfactual donkey sentences Although I believe that a counterfactual donkey sentence is in general equivalent to a formula with wide scope universal quantification, there is a particular 18
Though the relation ‘X w is true in Æw, gæ iff ;X ;X "Æv, hæ 2 fÆw;gæ ð=/=g Þ : dÆu; kæ 2 fÆw;gæ ð=/=g Þ : Æu; kæ;X Æv, hæ and Æu, kæ verifies w. Now we can account for the truth of the weak counterfactual donkey (5) where in the closest counterfactual world(s) I have more than one dime in my pocket. In the rest of this paper I won’t come back to identifying or weak readings of counterfactual donkey sentences and will always assume unselective binding.
Now we know how to account for counterfactual donkey sentences, it becomes straightforward how to account for counterfactuals with disjunctive antecedents. The reason is, of course, that disjunctive sentences can simply be represented by existential sentences (cf. Alonso-Ovalle 2004). Let ‘P’ denote the property that Spain fought on the x-side. In that case we can represent (1a) by dx[Px ^ (x ¼ allied _ x ¼ nazi)] > Spain bankrupt.24 (1a) If Spain had fought on either the Allied side or the Nazi side, it would have made Spain bankrupt. Given our analysis of counterfactual donkey sentences above, it is quite clear that we now predict that from (1a) we can indeed infer that (1b) and (1c) follow. (1b) If Spain had fought on the Allied side, it would have made Spain bankrupt. (1c) If Spain had fought on the Nazi side, it would have made Spain bankrupt.
24 In the main text I illustrate the proposal to account for the problem of simplication of disjunctive antecedents in terms of donkey anaphora in counterfactuals by means of disjunctions of type e. But this is for illustrative purposes only: the analysis works for disjunctions of any type. And this is needed as well: as mentioned by one reviewer, from an example like ‘If John had bought a car or borrowed a motorcycle, he’d be on time’ we intuitively infer ‘If John had bought a car, he’d be on time and if John had borrowed a motorcycle, he’d be on time’. In fact, Alonso-Ovalle (2004) proposed his analysis for disjunctions of type Æe, tæ.
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3.2 Counterfactuals with disjunctive antecedents
Robert van Rooij 397
25 McKay and van Inwagen (1977), however, present the following example, which shows that we don’t want SDA to be valid in general: from (i) we don’t want to conclude (ii):
(i)
If Spain had fought on either the Allied side or the Nazi side, it would have fought on the Nazi side. (ii) If Spain had fought on the Allied side, it would have fought on the Nazi side. I conclude that counterfactuals with disjunctive antecedents do not falsify the Lewis/Stalnaker account: we cannot conclude If /, then w from all instantiations of (subjunctive) conditionals of the form If / or v, then w. On the analysis suggested in section 3.1 this means that either not all counterfactuals with disjunctive antecedents should be represented by means of existential quantifiers (see Alonso-Ovalle 2004 for this type of move), or that the variable introduced by the quantifier representing the disjunction is irrelevant for the ordering relation (as proposed in section 3.1.1). On neither analysis, SDA is guaranteed to be valid. One reviewer noticed that for counterfactuals of the form ‘:(/ ^ v) > w’ we typically make an SDA-type of inference: :/ > w and :v > w, although that is not (immediately) predicted by the proposed analysis: (iii) If Jack had not seen both Mary and John, he would be unhappy. To account for such examples—as also suggested by the reviewer—I could, and would, either represent them as I would represent (iv), i.e. as (v): (iv) If Jack had not seen Mary or had not seen John, he would be unhappy. (v) dx[:Px ^ (x ¼ m _ x ¼ j)] > w. or would represent them more in line with their surface form, in which case the simplification inference is not guaranteed to go through, but depends on the ordering relation. 26
Alonso-Ovalle (p.c.) has the intuition that ‘Might’-counterfactuals of the form (/ _ v) > )w should intuitively entail both / > )w and v > )w. Neither his own analysis from 2004, nor my analysis can account for this. Fortunately, there is an easy way out of this problem. In the main text we have assumed that fÆw;gæ ð=/=g Þ is a set of world-assignment pairs. But we might redefine the selection function such that it rather denotes a set of sets of world-assignment pairs: + fÆw;gæ ð=/=g Þ ¼ ffÆv; hæ 2 =/=g : :dÆu; kæ 2 =/=g : Æu; kæ w is true in Æw, gæ iff w is entailed by each set in fÆw;gæ a compositional analysis, but one where ‘Might’-counterfactuals have Alsono-Ovalle’s desired truth conditions (at least, if disjunctions are analysed in terms of dynamic quantification).
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Our analysis explains why counterfactuals with disjunctive antecedents allow for simplification of the antecedent.25,26 Let us now turn to example (2) and see whether our new analysis can account for the appropriateness of negative polarity item any in the antecedent of a counterfactual. Obviously, we cannot account for the licensing of any on the standard DE-analysis: although we slightly changed the analysis of counterfactuals, it is still not predicted on our analysis that the antecedent of a counterfactual forms a downward entailing context. Fortunately, the DE-analysis is not the only analysis of NPI-licensing around. For both empirical and conceptual reasons, Kadmon and Landman (1993) and Krifka (1995) have argued in favour of a more pragmatic analysis of licensing NPIs. Kadmon and Landman (1993), for instance, have argued that the semantic meaning of any is just the same as that of an indefinite like some—i.e. that of the existential quantifier—but with a wider domain of quantification. To account for licensing, they claim that the NPI any can be used
398 Free Choice Counterfactual Donkeys
3.3 Permission sentences As it turns out, a very similar change of the ordering relation relevant for the (performative) analysis of permission sentences as what we used above for counterfactuals solves our remaining problems in section 2.1 as well. Let us represent ‘You may take any apple’ by a formula of the form ‘May( j,dxPx)’. As above, I will assume that existential quantifiers should be analysed dynamically. This means that /dxPx/g is not the set of possible worlds where there is an object that has property P, but rather the set of world-assignment pairs, Æv, hæ, where the object h(x) has property P in v. Given that the truth-set of an existential formula involves not only worlds but also assignments, we are forced to adjust our performative analysis of permission sentences as sketched in section 2.1, because that analysis was based on an ordering relation ‘ q. For instance, if D# ¼ fd1g and D ¼ fd1, d2g, then dxD[Px] > q has the same truth conditions as the conjunction ðPðd1 Þ > qÞ ^ ðPðd2 Þ > qÞ, while dxD#[Px] > q just means Pðd1 Þ > q, and is thus weaker (I assume here that d is the name of d). But this means that according to the widening analysis, in combination with our analysis of counterfactual donkey sentences, any is predicted to be licensed in antecedents of counterfactuals, just as desired.
Robert van Rooij 399 def
¼ fv 2 Wjdh : Æv;hæ 2 =/=g :dÆu;kæ 2 =/=g : Æu;kæ < Æv;hæg: P=/= g
27 Just as for counterfactuals with disjunctive antecedents, this dual reading might be a virtue rather than a vice: the free choice inference involving disjunctive permissions might be cancelled.
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This minor adjustment has the important consequence that if the domain of quantification involves only three apples, a1, a2, and a3, the permission ‘You may take any apple’, represented abstractly by ‘May(j, dxPx)’, is predicted to give rise to the inference that the hearer may take apple a1, apple a2 and may take apple a3 whatever the reprehensibility relation is. Thus, the permission is predicted to give rise to the free choice inference and this inference cannot be cancelled anymore. The reason is that although the reprehensibility relation ‘
