Countability in Natural Language

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Countability in Natural Language Edited by

Hana Filip Heinrich Heine Universität Düsseldorf

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Countability in Natural Language

This book focuses on current theoretical and empirical research into countability in the nominal domain and, to a lesser extent, the verbal domain. Most studies in this book presuppose compositional semantics combined with the theory of mereology, and they draw on a wealth of data, some hitherto unnoticed, from a number of typologically distinct languages. Some of the contributions propose enrichments of classical extensional mereology with topological and temporal notions, as well as with type theory and probabilistic models. Others present analyses that rely on cutting-edge empirical (experimental, corpus-based) research into meaning in language. The book is suitable as a point of departure for original research or as material for seminars in semantics, philosophy of language, psycholinguistics, and other fields of cognitive science. It is of interest not only to semanticists, but also to anybody who wishes to gain insight into contemporary research into countability. hana filip is Professor of Semantics at Heinrich Heine University Düsseldorf (Germany). Her research focuses on aspect, genericity, and nominal semantics.

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Contents

List of Contributors Preface Introduction hana filip

page vi ix 1

1

Proportional Many/Much and Most carmen dobrovie-sorin and ion giurgea

14

2

Quantity Systems and the Count/Mass Distinction jenny doetjes

52

3

Counting Aggregates, Groups and Kinds: Countability from the Perspective of a Morphologically Complex Language scott grimm and mojmı´ r dočekal

85

4

Individuating Matter over Time manfred krifka

121

5

Reduplication as Summation charles lam

145

6

Iceberg Semantics for Count Nouns and Mass Nouns: How Mass Counts fred landman

161

7

Indexical Inference: Counting and Measuring in Context alice g. b. ter meulen

199

8

Counting and Measuring and Approximation susan rothstein

217

9

The Count/Mass Distinction for Granular Nouns peter r. sutton and hana filip

252

Index

292 v

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Contributors

carmen dobrovie-sorin is a researcher at the Centre national de la recherche scientifique (CNRS), Paris. She works on syntax, the syntax– semantics interface and morphosyntax. Her areas of interest include Romanian and Romance languages, genitives, clitics, genericity, bare names, quantification with mass terms, definite articles and possessive pronouns. mojmı´r docˇ ekal is an Associate Professor at the Department of Linguistics and Baltic Languages at Masaryk University in Brno. His work in semantics, syntax and the philosophy of language is on topics such as polarity items, swarm expressions, degree constructions, group nouns and different types of numerals in Czech. jenny doetjes is Professor of Semantics and Language Variation at Leiden University. Her main research interests are in the domain of semantics and the syntax–semantics interface, encompassing the mass/count distinction (including work on classifier languages) and the cross-categorial distribution of degree modifiers/quantity expressions, and the interface between syntax, semantics and prosody. hana filip is Professor of Semantics at the Heinrich Heine University Düsseldorf. Her research focuses on the semantics of natural languages, and its connections with pragmatics, morphology and syntax, as well as related issues in philosophy of language and cognitive science. Her specific areas of research are aspect, genericity and nominal semantics, and she has also worked in the areas of sentence processing and natural language processing. ion giurgea is a researcher at the Institute of Linguistics (ILIR) in Bucharest. He works on syntax, semantics, diachronic linguistics, etymology and historical linguistics with a principle focus on Romanian. His research topics include polar questions, topic marking and topicalization, and argument structure. scott grimm is Assistant Professor in the Department of Linguistics at the University of Rochester, NY. His research in semantics and typology has vi

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List of Contributors

vii

focused on mereotopological analyses of countability and grammatical number, and on the individuation of abstract entities. He has also worked on case marking and has done field work on Dagaare, especially in relation to inverse number marking. manfred krifka is Professor of General Linguistics at Humboldt University, Berlin, and director of the Leibniz-Centre for General Linguistics (ZAS), also in Berlin. His areas of research include nominal reference, aspectual classes, focus and information structure, questions and speech acts, and precise and approximate number words. charles lam is Assistant Professor in the Department of English at the Hang Seng University of Hong Kong (formerly Hang Seng Management College). His research on syntax, semantics and the syntax–semantics interface focuses on how sentence structure and meaning work together. This includes work on countability in Cantonese, reduplication, and the interaction between classifier and nominal systems. fred landman is Professor of Semantics in the Linguistics Department at Tel Aviv University. He has published on a wide variety of topics in semantic theory, including topics such as aspect, countability, groups, plurals and plurality, pseudo-partitives and the progressive. alice g. b. ter meulen is Professor of Linguistics at the University of Geneva. She has worked on a number of topics and areas in syntax and semantics including representations of time, (in)definiteness, tense and aspect, (temporal) reasoning and dynamic semantics. susan rothstein was Professor of Theoretical Linguistics at the Department of English at Bar-Ilan University, Ramat Gan. Her research concerned many topics within the scope of the syntax–semantics interface, including aspect, theories of counting and measuring, noun phrase structure, event semantics, cross-linguistic semantics and Biblical Hebrew. peter r. sutton is a postdoctoral researcher in semantics at the Heinrich Heine University Düsseldorf. His main research areas are semantics, pragmatics and the philosophy of language. He has published on probabilistic semantics, including representations of vague predicates, and on the mass/ count distinction, including work on countability in the abstract domain.

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Preface

The papers in this volume originated as presentations at a workshop held at the Heinrich Heine University Düsseldorf in 2013. The workshop was funded by a grant from the Deutsche Forschungsgemeinschaft (DFG, German Research Association) awarded to Hana Filip for the project entitled A Frame-Based Analysis of Countability. This project was a part of the DFG funded Collaborative Research Centre 991: The Structure of Representations in Language, Cognition, and Science at the Heinrich Heine University. This large-scale interdisciplinary endeavor united researchers from the departments of linguistics, several language departments (English, German and Romance), philosophy and psychology, as well as computer and informational sciences. It was funded from 2011 to 2020. The goal of the countability workshop was to address theoretical or empirical issues in the study of the count/non-count distinction in the nominal domain, and also the reflexes of the parallel distinction(s) in the verbal domain. The topics of particular interest included (but were not limited to) the cognitive basis of countability, its grammatical manifestations and sources for the crosslinguistic variation in its encoding, relations between count and non-count meanings, sort-shifting, type-shifting, coercion, vagueness and gradations of individuation, so-called ‘fake’ mass nouns (also known as ‘object’ mass nouns), parallels and differences in the counting and measuring for noun and verb meanings, corpus-based and psychological evidence for the distinction between count and non-count meanings, and their computational modeling. This countability workshop focused on new developments in the theories of countability that began to appear around 2010 prompted by the need to reevaluate and improve on a semi-lattice approach originating in Link (1983, 1987), which, in formal semantics, provided the dominant perspective on countability in the nominal and verbal domain. The participants covered a broad swath of interdisciplinary research with the goal of identifying convergences in the foundational insights and in what might lead to novel promising venues for future research. Moreover, the idea was to bring together leading researchers and also young scholars at the beginning of their academic careers. ix

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x

Preface

Although ultimately only about a half of the presentations are published in this volume, the spirit of exploration and discovery remains as originally intended. Special acknowledgment and thanks must be given to Christian Horn (a researcher affiliated with the Collaborative Research Centre 991 at the Heinrich Heine University), who co-organized the workshop with me. The preparation of the final manuscript would not have been possible without the substantial help of Peter Sutton, and also of Kurt Erbach, Eleni Gregoromichelaki and Jon Ander Mendia. The workshop and also the preparation of this volume were supported by the funding provided by the grant from the Deutsche Forschungsgemeinschaft. We all, contributors to this volume as well as participants in the workshop, owe a great, memorial debt of gratitude to Susan Rothstein, whose untimely death in the summer of 2019 jolted the worldwide linguistic community and the many friends all over the world with whom she collaborated. Susan was also my friend and collaborator, who enriched me in many ways that go well beyond academic, intellectual matters. Susan was a Professor of Theoretical Linguistics and a Fellow in the Gonda Multidisciplinary Brain Research Center at Bar-Ilan University, Israel. During the workshop, her incisive and constructive comments generated an atmosphere of intellectual rigor and excitement, while her formidable body of work on the count/non-count distinction in the previous twelve years or so significantly shaped the ideas presented in many papers in this volume.

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Introduction Hana Filip

1

Countability

Early theories of the mass/count distinction in formal semantics and philosophy attempted to analyze the meanings of mass and count nouns in terms of properties like cumulativity, divisivity (due to Quine 1960), atomicity, and homogeneity, among others (see Pelletier 1979 and references therein). Many also agree that such properties are best represented in algebraic or mereological terms. This idea was introduced into formal semantics by Link (1983). His main innovation is to propose that the domain of (concrete) entities has the algebraic structure of a complete join semi-lattice. This allows him to model the differences between mass and count nouns, on the one hand, as well as similarities between mass and plural nouns, on the other hand. There is a sortal semantic distinction between mass and count nouns, which is based on the atomic and non-atomic ontological distinction (see, e.g., Link 1983, 1998), and modeled by means of an atomic and a non-atomic join semi-lattice, respectively. Count nouns are interpreted in the atomic lattice, mass nouns in the non-atomic one. Mass nouns pattern with plurals in having the property of cumulative reference (the term coined by Quine (1960) for the semantics of mass terms, realized as nouns or adjectives). In algebraic (mereological) theories inspired by Link (1983), it is standard to define cumulativity, as well as other semantic properties that characterize the mass/count distinction, in terms of a second-order predicate over algebraically (or mereologically) structured sets, as we see below: (1)

CUMðPÞ $ 8x8y½ðPðxÞ ^ PðyÞÞ ! Pðx  yÞ (Champollion and Krifka 2016, p. 52)

A predicate P is cumulative iff, whenever P applies to any x and y, it also applies to their sum (the ‘upward’ closure property.) For instance, whenever wine applies to two quantities x and y in the winelattice, it also applies to their mereological sum x  y. Similarly, adding entities that fall under apples in the apple(s)-lattice to other entities that also fall under apples will yield a joined element in the lattice that represents their 1

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Hana Filip

sum, which is also describable by apples. In contrast, singular count nouns like (an) apple fail to have the cumulative reference. If (an/one) apple is true of two different entities x and y, their sum xy is describable by two apples, and not an/one apple; the meaning of two apples corresponds to the set of sums of two apples in an apple(s)-lattice. The first early modifications of Link’s (1983, 1984, 1987) algebraic (mereological) theory can be found in Bach (1986), Krifka (1989), Landman (1989a, 1989b) and Pelletier and Schubert (1989/2003), among others. The main impetus for the workshop at which the papers in this volume were presented came from innovations in more recent studies, including Chierchia (2010, also 1998a, 1998b), Rothstein (2010), Landman (2011) and Grimm (2012), to name just a few. One fundamental issue that has come under close scrutiny in these studies, both early and more recent ones, is the relation between the linguistic mass/count categories and ontological or conceptual categories. Link (1983, 1984, 1987) is inspired by Quine’s (1960) basic idea that count nouns divide their reference and mass nouns do not, which in his lattice theory is recast in terms of an atomic semi-lattice for count nouns and a non-atomic one for mass nouns. This amounts to a one-to-one correspondence between the count/mass syntax (and semantics) and the atomic/non-atomic ontological distinction. Subsequently, many pointed to mismatches between the linguistic count/ mass distinction and the conceptual/semantic distinction, and/or also the distinction between undifferentiated stuff and individuated objects, which make Link’s one-to-one correspondence in the form–meaning mapping untenable. One of the most commonly adduced arguments concerns the observation that the structure of the matter in the actual world does not necessarily determine whether a noun (describing some entity/entities in the world) will be grammatically count or non-count, or conceptualized as individuated or not. For instance, rice, beans, lentils, shoes, window curtains, leaves, and the like, each consist of natural units that are perceptually and cognitively salient. However, such entities are not uniformly describable by count nouns in natural languages, but may also be referred to by mass nouns. Moreover, we observe an intriguing variation in the mass/count lexicalization patterns in a particular language and cross-linguistically. In English, for instance, rice is mass, but lentil(s) is count, even if both have entities in their denotation, discrete grains which are perceptually salient. Second, we have lexical mass/count doublets (which are near synonyms) like foliage/leaves, footware/shoes, change/coins, drapery/curtains and mail/letters (Quine 1960, p. 91; McCawley 1975). Although both members of the pair may refer to the same types of entities in the actual world, they also differ in their uses/senses, as we see in rake leaves into a pile versus *rake foliage into a pile (see Grimm 2012). Third, we find pairs of lexical items that are derivationally related and based on the same root,

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Introduction

3

such as carpeting/carpets or drapery/drapes, where one has a mass use/sense and the other a count use/sense, and yet they may refer to the same entities in the real world. Fourth, a single lexical item, such as hair(s), rope(s) and stone(s), can be realized as either mass or count in a particular language, and possibly associated with different uses/senses: e.g., A man with hair on his head has more hair than a man with hairs on his head (Lederer 1998). Equally puzzling is the cross-linguistic variation in the mass/count encoding. Mass expressions in one language may have count near-synonyms in another: e.g., furniture (mass, English) versus le meuble/les meubles (count, French); hrách (mass, Czech) versus pea/peas (count, English). One conclusion that we can draw from such ‘word–world’ mismatches is that the linguistic and conceptual (pre-linguistic) distinctions are independent of each other. This disconnect is clearly manifested by the class of nouns that have been labelled ‘fake’ mass nouns, ‘object’ mass nouns or ‘superordinate aggregates’ (among other terms). Some examples are furniture, kitchenware and jewelry. For instance, the denotation of furniture has clearly individuable entities at the ‘bottom’ of the atomic semi-lattice from which it takes its denotation, that is, individual chairs, stools, tables and the like. This should prima facie satisfy the conditions for a grammatically count treatment. Why then should many languages opt for its lexicalization as a mass noun? Due to their puzzling properties, object mass nouns provide a sine qua non testing ground for theories of the mass/count distinction (see also Chierchia 2010, p. 111) and, for some, a key data point in the formulation of some prominent theories (Chierchia 1998a, 1998b; Landman 2011 are good examples; see further below). Object mass nouns speak against Quine’s (1960) view (adopted in Link 1983) that only count nouns denote discrete individuals. If a count noun like chair(s) divides its reference, then we should also accept that a mass noun like furniture divides its reference, given that, notionally speaking, the denotation of furniture is no less atomic than that of count nouns like table and chair or phrases like a piece of furniture (Chierchia 2010, p. 140). Moreover, furniturelike mass nouns differ from prototypical mass nouns like water in so far as they exhibit different grammatical (semantic and distributional) properties (e.g., big furniture, big apples versus ?big water; Stubborn Distributivity, see Schwarzschild 2011). Indeed, Barner and Snedeker (2005) show that object mass nouns (like furniture) can be used to refer to individuated things, because they pattern with count nouns in so far as they allow for number-based comparison judgments. In contrast, prototypical mass nouns (like mud) only support volume-based comparison judgments. They conclude that the mass/count distinction is asymmetric in so far as count syntax specifies quantification over individuals, while mass syntax is unspecified in this respect. Mass syntax does not force a construal of objects as unindividuated, because there are mass

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nouns, namely object mass nouns (furniture), that have discrete individuals in their denotation. Such insights raised some fundamental questions about the nature of atomicity (as originally conceived of by Link 1983) and its role in the characterization of the semantics of the mass/count distinction. Given the existence of object mass nouns, atomicity (in Link’s original sense) is not sufficient for the grammatical status of a noun as count, and mass nouns as a whole class cannot be uniformly interpreted in the non-atomic lattice, because, as we have just seen, the denotation of object mass nouns like furniture cannot be treated on a par with prototypical mass nouns like water. Not only is Link’s notion of non-atomicity insufficient to model the properties of all mass nouns, but it remains somewhat unclear (Chierchia 2010). According to Link (1983), mass terms take their denotation from the domain that is ‘non-atomic or not known to be atomic’. Partee (1999, p. 95) takes ‘nonatomic’ to mean ‘may but need not have atoms’. Certainly, ‘non-atomic’ cannot mean ‘atomless’ (see the axiom of atomlessness, ‘Everything is infinitely divisible’). Take, for instance, water; what it is true of cannot be infinitely subdivided into proper parts all of which can count as water. Subdivisions of material objects must eventually come to an end (Chierchia 2010). If the domain from which mass nouns take their denotation is ‘not known to be atomic’, or if we are agnostic as to whether it has atoms or not, such ignorance should not be a sufficient reason for the non-countability of mass nouns. “Domains we know very little about but that are clearly countable abound (elementary particles, solutions to problems, . . .)” (Chierchia 2010, p. 144). Another problem with Link’s (1983) original proposal is that the two-sorted disjoint ontology of entities, atomic and non-atomic, leads to certain counterintuitive predictions. Take again, for instance, the mass/count lexical doublets (near-synonyms) like footware/shoes, foliage/leaves, change/coins and drapery/curtains. The mass members of such doublets do not denote the material (mass) counterpart of the count member denotations (related by means of the materialization function h), contrary to Link’s prediction. It would also be counterintuitive to think that footware and shoes, for example, differ in their ontological nature, or that “your furniture changes its ontological nature by referring to it in English [with the mass noun furniture, HF] versus French [with the count noun le meuble/les meubles]” (Chierchia 2010, p. 144). Moreover, the two-sorted atomic/non-atomic domains for entities raise an intractable contradiction, as Bach’s (1986) ‘snowman puzzle’ shows: (2)

The snow making up this snowman is quite new but the H20 making it up is very old (and the H and O even older!) Interpretation: The x such that x constitutes the snowman and x is snow is quite new but the y such that y constitutes the snowman and y is water is very old (not new).

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Introduction

5

Given that the materialization function h is a function (Link 1983 and elsewhere), and given that it maps snowman into the stuff its denotation is made of, namely x (snow) and y (water), it follows that x (snow) and y (water) must be identical. However, x and y have contradictory properties: x (snow) is new and y (water) is old. Two things with contradictory properties cannot be identical (see Leibniz’s principle of the identity of indiscernibles). Therefore, Bach (1986, p. 14) suggests that we “remove altogether the entities in D [the nonatomic domain containing the portions of matter or stuff] from the domain of individuals.” This should suffice to illustrate some of the reasons why Link’s ‘double domain’ (Chierchia 2010) approach was abandoned. Specifically, Bach’s (1986) suggestion later became reflected in ‘one domain’ atomistic mereologies. Some good examples are Landman (1989, 2011), Chierchia (1998a, 1998b, 2010) and Rothstein (2010). Atomistic mereologies are endowed with an additional axiom of atomicity (‘All things are made up of atoms’) requiring that everything in the domain be composed of atoms (see further below). An alternative solution which assumes a single domain mereology is one that is undetermined with respect to atomicity. Early on it was proposed by Krifka (1986, 1989): “we do not have to commit ourselves to an atomic or a non-atomic conception of the world” (Krifka 1989, p. 81). Krifka (1986, 1989 and elsewhere) defines a typal distinction between mass and count nouns. While mass nouns denote cumulative predicates, count nouns denote quantized predicates which are derived by means of an extensive measure function (expressed by a measure term) applied to what is the intension of mass nouns. (3)

QUANTIZED ðPÞ $ 8x, y½PðxÞ ^ y < x ! PðyÞ (Krifka 1986, 1989)

A predicate P is quantized if and only if whenever it holds of something, it does not hold of any of its proper parts. Basic lexical count nouns like cat are quantized due to the extensive measure function NU (for ‘natural unit’) in their lexical semantic structure. It is intended to do the individuating job of determining singular, clearly discrete objects in their denotation. Quantized nominal predicates are expressed not only by basic lexical count nouns (an apple), but also by numerical noun phrases (NPs) (three apples) and measure NPs (three pounds of apples, three liters of water). Just as no proper part of an entity that falls under an apple is an apple, so no proper part of an entity that falls under three liters of water is also describable by three liters of water. For Chierchia (1998a, 1998b) and Landman (2011), the proponents of a single atomic domain, the basic question to be answered is ‘Why cannot we directly count the denotations of mass nouns like meat and salt?’ Why cannot

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mass nouns like meat and salt be directly combined with numericals in counting constructions like ?three meats and ?three salts, unless they are first coerced into a suitable count interpretation? Chierchia’s (2010) answer is couched in a vagueness-based model that is enriched with a form of supervaluationism. The domain in which count and mass nouns are interpreted is atomic (also motivated by the properties of object mass nouns, see Chierchia 1998a, 1998b), but what it means to be atomic is a vague and context-dependent notion (Chierchia 2010, p. 120). Count nouns are generated from stable atoms, where ‘stable’ means that they have entities in their denotation that are atoms in every context, at every total precisification (i.e., making the things precise in all the relevant ways). Mass nouns have vague or ‘unstable’ minimal individuals in their denotation. For instance, mass nouns like rice are vague in the relevant sense, because it is not the case that, across all contexts, a few grains, single grains, half grains and rice dust, for example, always count as rice. Such various quantities of rice are in the vagueness band of rice, and so there is no entity in the denotation of rice that is a rice atom at every total precisification of rice. What counts as one (atom) of rice under one precisification may not count as one (atom) of rice under another. Consequently, there are no stable, same atoms across all precisifications. Generally, if a noun lacks stable atoms, there is no entity that is an atom in the denotation of the predicate at all contexts, and so is uncountable. Counting is counting of stable atoms only. Intuitively, for Chierchia, the reason why the denotation of mass nouns prohibits counting is that it is vague in such a way that “[w]e don’t know what to count, not even in principle (Chierchia 2010, p. 118). For Landman (2011) instead, the key driving intuition is that there are ‘too many’ things to count in the mass noun denotations. That is, the set of entities in their denotation that we would want to count as ‘one’ overlap, which leads to overdetermination of how many entities there are that we may wish to count. Just like Chierchia (1998a, 1998b, 2010), Landman (2011) assumes that the domain in which mass and count nouns are interpreted is atomic. Landman (1989, 2011) relies on sets used to model sum individuals. Ordinary set theory implies a commitment to everything ultimately being composed of atoms, where mereological atoms are represented by singleton sets (see also Scha 1981; Schwarzschild 1996). In Landman (2011), the domain in which mass and count nouns are interpreted is a complete atomic Boolean algebra, which consists of the set of generators of regular set X: gen(X). This is the set of semantic building blocks, or the things that we would count as one P in a single context. Three types of noun denotations are distinguished. Count nouns have denotations built from non-overlapping generators. All mass nouns have denotations built from overlapping generators, and Landman’s key innovation is to distinguish two subcategories of mass nouns: namely, mess mass nouns (e.g., mud, water, salt, meat), on the one hand, and neat mass nouns, also known as object or fake

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Introduction

7

mass nouns (e.g., furniture, kitchenware, jewelry, silverware), on the other hand. Mess mass nouns have denotations built from overlapping minimal generators (‘horizontal’ overlap). For instance, minimal elements in the denotation of water, H2O molecules, always overlap, and the overlap is located in their non-essential structure, that is, in the space between the individual H2O molecules of water. While the application of the overlap idea to mess mass nouns is problematic, and Landman abandons it in his subsequent work, it does a compelling job of motivating the puzzle of object mass nouns, Landman’s neat mass nouns. In a nutshell, Landman’s idea is that the relevant overlap is not located in the minimal generators (‘horizontal’ overlap), but rather along the ‘vertical’ dimension of the Boolean domain. It is this ‘vertical’ overlap among entities that leads to overdetermination of how many entities there are. Given that this overlap cannot be made irrelevant or eliminated, counting with neat mass nouns fails. For instance, when it comes to kitchenware, what we may want to count as ‘one’ in its denotation is not only an individual minimal generator like a teacup, a saucer or a teapot, but also an element that is their sum. A teacup together with a saucer may count as one item of kitchenware. Another partition of the domain may have a teacup, a saucer and a teapot together constituting one item, namely, a tea set. But this means that we have a multiplicity of different partitions, Landman’s variants, of the same domain into generators, i.e., into what we may view as counting as one. Collected together into one set of generators, such variants will contain mutually overlapping elements, because sums and their parts may simultaneously count as one. Counting fails, because if we try to count what we might view as one, we will always count overlapping elements. Both Chierchia’s (2010) and Landman’s (2011) analyses of the mass/count distinction rely on context. For Chierchia (2010), atomicity is a vague and context-dependent notion, while Landman’s (2011) analysis relies on the notion of a (non-)overlapping set in a context. In Rothstein (2010), ‘context’ is understood as a set of entities that count as atoms (i.e., count as one), and this notion of context takes center stage in her theory. Based on the observation that grammatical counting often depends on context, Rothstein (2010) argues that a count noun denotes a set of atoms, and what is one atom is always relative to a particular context. “[T]his contextual dependency must be grammatically encoded in the meaning of the count noun” (Rothstein 2017, p. 108). There is, then, a typal distinction between count and mass nouns. Mass nouns are of type (predicates of individuals). Count nouns are indexed to counting contexts and are of type (predicates of indexed individuals); they are ‘semantically atomic’. The key empirical data are count nouns like twig, sequence or fence, which pose a challenge to any theory of the mass/ count distinction. The problem is that such nouns are clearly grammatically count, i.e., they can be directly counted (e.g., two twigs), but what is a single

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unit for counting is not uniquely determined by their lexical meaning, must be fixed relative to a particular context, and can be arbitrary. The problematic nature of such nouns was noticed by Krifka (1989, p. 87, due to Partee, p.c.), because they are counterexamples to his claim that count nouns denote quantized predicates, as defined in (3). For example, (a) twig fails to denote a quantized predicate, because entities that it is true of have proper parts that also count as (a) twig. However, once we fix what is one fence relative to a particular context, i.e., one ‘semantic atom’ in Rothstein’s (2010) sense, for instance, none of its proper parts can count as one twig at the same time in the same context. Taking Rothstein’s paradigm example, a rectangular enclosure around a field can count as one fence, two fences or four fences (Rothstein 2010). That is, the denotation of fence may have a different unit structure in different contexts, and what we take to be ‘one’ for the purposes of grammatical counting operations depends on a particular context. Rothstein (2010) generalizes the proposal of building the ‘counting context’ dependency into the lexical semantics of fence-like count nouns to all count nouns, including those like cat that have ‘stable’ atoms (in Chierchia’s sense, see above) across all contexts.1 Further important extensions of the classical extensional mereology come from Grimm (2012), who adds topological notions. Predicates of discrete individuals like cat or grain of rice have the mereotopological property of a maximally strongly self-connected (MSSC) individual, i.e., an entity for which every part internally overlaps with the whole. This mereotopological notion allows him to articulate, within lexical entries, certain properties which must be taken as basic but cannot be analyzed in standard 1

Rothstein’s (2010) proposal that the ‘counting context’ dependency of nouns such as twig, sequence or fence should be directly built into their lexical semantics is also motivated by the discussion concerning their behavior in aspectual composition. Although they fail to be quantized, according to the definition given in (3), when analyzed on their own, they behave like quantized predicates with respect to in/for adverbial phrases in sentences exhibiting aspectual composition: e.g., Write a sequence of letters in five seconds/?for five seconds. In order to address this problem, Krifka (1998, p. 221) proposes that quantificational indefinite DPs (subject to existential closure) like a sequence or a twig take a wide-scope reading with respect to other scope-taking elements, such as time-span adverbials (e.g., in an hour). However, this wide-scope solution predicts odd or anomalous readings for sentences with quantificational indefinites. For instance, Every guest ate some muffins would mean that every guest ate the same muffins (Zucchi and White 1996, p. 241ff.). Instead, Zucchi and White (1996, 2001) propose to treat such problematic indefinites in terms of the notion of a maximal participant, i.e., the largest individual (among the individuals in the relevant denotation) in an event in which it takes part at the reference time tR. However, because the maximization is directly built into the meaning of indefinite DPs, we get wrong interpretations for non-quantized (atelic) sentences, as Rothstein (2004, p. 154). For instance, the requirement that the indefinite DP fewer than five countries denote a maximal participant predicates that The emperor has ruled fewer than five countries for the last ten years (Rothstein 2004, p. 153) is true just in case the same maximal set of fewer than five countries be ruled by the emperor at all times in the last ten years.

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mereological accounts. Among the innovations that Grimm (2012) introduces is the notion of cluster, which differentiates a mereological sum counting as a clustered entity from a mereological sum viewed as one individual entity. Such mereotopological innovations are motivated by the insight that the morphosyntactic organization of grammatical number systems reflects the semantic classification of noun types according to the degree of individuation of their referents, ordered along a scale of individuation: substances < granular aggregates < collectives < individual entities. Noun types which are less individuated are at the lower end of the scale and are cross-linguistically less likely to signal grammatical number. 2

Papers in the Volume

Foundational issues are addressed in the papers by Landman, Rothstein, Krifka and Doetjes. Sutton and Filip focus on ‘granular’ nouns like rice or lentil(s) which have so far remained largely understudied. Other papers address the theories of the mass/count distinction from the point of view of puzzles posed by specific data. Grimm and Dočekal examine the little-known data from Czech, which has morphological means to derive nouns that denote aggregates, groups and kinds. Lam explores reduplication for Cantonese nouns, classifiers, verbs and adjectives and argues for a unified account in terms of summation. The relation of the mass/count distinction to the proportional and cardinality readings of determiners, such as much and many, is addressed by Dobrovie-Sorin and Giurgea, while ter Meulen focuses on the inferences of adverbs such as still and already in sentences with counting constructions. In “Iceberg Semantics for Count Nouns and Mass Nouns: How Mass Counts,” Fred Landman defines an extension to his Iceberg semantics to determiner phrases (DPs) and NPs. It relies on adding to his notion of an i-set (a pair of a body and base set) the notion of an i-object in which the body is a single entity or a single sum entity. This allows him to derive a uniform notion of count/mass DPs and NPs in terms of overlapping bases. Landman uses these tools to model a number of ways in which ‘mass counts’ in terms of portions and hunks, which motivates how and why counting and count comparison is possible in the mass domain. Moreover, in Iceberg semantics with singular shift (a form of semantic singularization), the analysis of ‘Gillon’s problem’ becomes fully tractable. The problem concerns the divergent behavior of count and object mass DPs with respect to reciprocal operators: e.g., The curtains and the carpets resemble each other allows several readings (involving distributive and sum readings for sums of count DPs) that are unavailable for sums of object mass DPs, as in The drapery and the carpeting resemble each other. In both

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cases, singular shift accounts for the presence of the reading comparing the body of curtains with the body of carpets. In her contribution entitled “Counting and Measuring and Approximation,” Susan Rothstein explores the differences between the counting and measuring semantic operations. The main puzzle Rothstein addresses is why we should be able to readily access cardinality comparisons for object mass nouns (e.g., furniture) in comparative more than constructions, despite the fact that object mass nouns are genuine mass nouns and so are infelicitous in counting constructions. Rothstein proposes an account for quantity evaluations (including measure comparisons for mass nouns) in terms of comparison of cardinality for count nouns and measure on a scale for mass nouns. Cardinality comparisons for object mass nouns are assessed in terms of measure comparisons relative to a cardinality scale. In “Individuating Matter over Time,” Manfred Krifka develops a haptomereological semantics, which is a mereology enriched with a basic touch relation and a spatio-temporal component. This, as he argues, allows us to account for puzzles that relate to individuating matter relative to time. For example, Krifka’s haptomereological framework can distinguish solids from liquids and so account for why parts of solids (even massy solids) can be individuated over time in a way that parts of liquids or gasses cannot. Jenny Doetjes’ “Quantity Systems and the Count/Mass Distinction” is an exploration of the mass/count distinction in terms of how the quantity systems are encoded in natural languages. Cross-linguistic variation in the realization of the mass/count distinction comes from the differences in the actual inventory of quantity expressions (including quantifiers and numerical determiners) in different languages. The main claim that Doetjes defends is that, even for languages that have been said not to have a clear grammatical, lexicalized mass/count distinction (such as Mandarin or Yudja), a mass/count distinction can be found within the quantity systems of such languages. “The Count/Mass Distinction for ‘Granular’ Nouns” by Peter Sutton and Hana Filip addresses the puzzle of so-called ‘granular’ nouns, which may be either mass like rice, barley and gravel (‘granular aggregates’ in Grimm 2012), or count like lentils, beans and pebbles. Their denotation consists of collections of similar discrete, non-overlapping single items, that is, single grains, granules, filaments and the like. Furthermore, whenever such nouns are lexicalized as count, it is precisely the single grains, granules or flakes that are denoted by the singular form. However, parts of these grains, granules or flakes are also in the mass denotations of these nouns. For example, a pile of broken-up rice grains or lentils still falls under rice or lentils, respectively, and rice flour also counts as rice. Granular nouns are puzzling, because they display a large amount of variation in their mass/count encoding across different languages, and also within a single language. Sutton and Filip pose

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the following questions: Why should there be mass granular nouns, such as rice, given that such nouns denote entities that are saliently separable and discrete (i.e., the individual grains)? Why can entities with similar granularity properties also be expressed by count nouns, such as lentils? They present a compositional account that captures the fact that granular nouns – be they mass or count – denote discrete grains, and also why this information may fail to be passed up to and so made available to grammatical counting operations. They then use this model to address what they label the ‘accessibility puzzle’: namely, why is it that mass granular nouns (e.g., rice) cannot be coerced into count readings that pick out the single saliently separable and discrete entities in their denotation? Or why is *three rices unacceptable when the intended reading concerns the counting of individual grains? In their joint paper, “Counting Aggregates, Groups and Kinds: Countability from the Perspective of a Morphologically Complex Language,” Scott Grimm and Mojmír Dočekal focus on the little-known data from Czech to explore the grammatical counting operations for nouns denoting aggregates, groups and kinds. Czech provides nominal derivational morphology for deriving aggregate nouns (which are mass) from certain count nouns, as well as morphological means for deriving group numerals, aggregate numerals and taxonomic numerals. Building on Grimm’s previous formal results, they provide a mereotopological semantic model to cover their data, but also the novel empirical data from Czech provide the background for their exploration of three fundamental issues in the theories of the mass/count distinction: (i) What must theories of countability assume as primitive? (ii) Given that Czech provides morphology for explicitly counting aggregates, what can be determined about the nature of aggregates? (iii) To what extent is countability lexically specified as opposed to dependent on pragmatic context? In their paper “Proportional Many/Much and Most,” Carmen DobrovieSorin and Ion Giurgea analyze proportional and cardinality readings of determiners such as much and many (e.g., Many students are intelligent versus John read many books), and also the proportional reading for most. They argue that proportional many is roughly comparable in its semantics to that of scalar quantity predicates, which are analyzed in terms of degrees. In contrast, proportional most, as they propose, is to be analyzed in terms of Generalised Quantifier Theory. “Reduplication as Summation” encapsulates the key thesis of Charles Lam’s paper. He argues that reduplication for Cantonese nouns, classifiers, verbs and adjectives can be given a unified account in terms of summation. For nouns and classifiers, reduplication conveys universal quantification, while for bare adjectives, it conveys a diminutive effect. As far as verbs are concerned, their reduplication has the effect of either iteration or duration. Lam’s analysis makes use of the cumulative/quantized distinction to capture these effects.

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Summation amounts to pluralization for quantized predicates, which accounts for pluralization in nouns and iteration for verbal predicates, while for cumulative predicates it accounts for the effect of either temporal extension (verbal predicates) or a low-intensity inference (adjectives). Alice ter Meulen’s paper “Indexical Inference: Counting and Measuring in Context” addresses adverbs such as still and already and the inferences they give rise to in sentences that contain counting constructions. For example, There are already three students here implies, among other things, that students must be arriving, and Three students are not yet here implies that the three students are arriving. The focus of the paper is indexical inferences insofar as such sentences give rise to inferences expressed in a tense/aspect that have no antecedent in the premise. REFERENCES Bach, Emmon (1986). The algebra of events. Linguistics and Philosophy 9.1: 5–16. Barner, David, and Jesse Snedeker (2005). Quantity judgments and individuation: Evidence that mass nouns count. Cognition, 97.1: 41–66. Champollion, Lucas, and Manfred Krifka (2016). Mereology. In Maria Aloni (ed.), The Cambridge Handbook of Formal Semantics, pp. 513–541. Cambridge: Cambridge University Press. Chierchia, Gennaro (1998a). Plurality of nouns and the notion of “semantic parameter.” In Susan Rothstein (ed.), Events and Grammar: Studies in Linguistics and Philosophy Vol. 7, pp. 53–103. Dordrecht: Kluwer. (1998b). Reference to kinds across languages. Natural Language Semantics 6.4: 339–405. (2010). Mass nouns, vagueness and semantic variation. Synthese 174.1: 99–149. Grimm, Scott (2012). Number and Individuation. PhD Dissertation, Stanford University. Krifka, Manfred (1986). Nominalreferenz und Zeitkonstitution. Zur Semantik von Massentermen, Individualtermen, Aspektklassen. Ph.D. dissertation, University of Munich, Germany. Published in 1989. Munich: Wilhelm Fink Verlag. (1989). Nominal reference, temporal constitution and quantification in event semantics. In Renate Bartsch, Johan van Benthem, and Peter van Emde Boas (eds.), Semantics and Contextual Expression, pp. 75–115. Dordrecht: Foris Publications. (1998). The origins of telicity. In Susan Rothstein (ed.), Events and Grammar: Studies in Linguistics and Philosophy Vol. 7, pp. 197–236. Dordrecht: Kluwer. Landman, Fred (1989a). Groups, i. Linguistics and Philosophy 12.5: 559–605. (1989b). Groups, ii. Linguistics and Philosophy 12.6: 723–744. (2011). Count nouns – mass nouns – neat nouns – mess nouns. In Michael ‘ Glanzberg, Barbara H. Partee, and Jurgis Škilters (eds.), Formal Semantics and , Pragmatics: Discourse, Context and Models. The Baltic International Yearbook of Cognition, Logic and Communication 6, 2010, pp. 1–67, http://thebalticyearbook .org/journals/baltic/issue/current.

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Lederer, Richard (1998). Crazy English. New York, NY: Pocket Books. Link, Godehard (1983). The logical analysis of plurals and mass terms: A latticetheoretic approach. In Reiner Bäuerle, Christophe Schwarze, and Arnim von Stechow (eds.), Meaning, Use, and the Interpretation of Language, pp. 302–323. Berlin: de Gruyter. (1984). Hydras. On the logic of relative clause constructions with multiple heads. In Fred Landman and Frank Veltman (eds.), Varieties of Formal Semantics, GRASS 3, pp. 245–247. Dordrecht: Foris Publications. (1987). Algebraic semantics for event structures. In Jeroen A. G. Groenendijk, Martin J. B. Stokhof, and Frank J. M. M. Veltman (eds.), Proceedings of the Sixth Amsterdam Colloquium, pp. 153–173. Amsterdam: ITLI. (1998). Algebraic Semantics in Language and Philosophy. CSLI Lecture Notes 74. New York, NY: Cambridge University Press. McCawley, James D. (1975). Lexicography and the count–mass distinction. Proceedings of the First Annual Meeting of the Berkeley Linguistics Society 1975, 314–321. Partee, Barbara H. (1999). Nominal and temporal semantic structure: Aspect and quantification. In Eva Hajičová, Tomáš Hoskovec, Oldřich Leška, Petr Sgall, and Zdena Skoumalová (eds.), Prague Linguistic Circle Papers, Vol. 3, pp. 91–106. Amsterdam: John Benjamins. Pelletier, Francis J. (1979). Mass Terms: Some Philosophical Problems. Dordrecht: D. Reidel. Pelletier, Francis J., and Lenhart Schubert (1989/2003). Mass expressions. In Franz Guenthner and Dov M. Gabbay (eds.), Handbook of Philosophical Logic. Volume x, 2nd ed., pp. 249–336. Dordrecht: Kluwer. Updated version of the 1989 version in the 1st ed. of Handbook of Philosophical Logic. Quine, Willard v. O. (1960). Word and Object. Cambridge, MA: MIT Press. Rothstein, Susan (2004). Structuring Events: A Study in the Semantics of Aspect. Malden, MA: Blackwell. (2010). Counting and the mass/count distinction. Journal of Semantics 27.3: 343–397. (2017). Semantics for Counting and Measuring: Key Topics in Semantics and Pragmatics. Cambridge: Cambridge University Press. Scha, Remko (1981). Distributive, collective and cumulative quantification. In Jeroen A. G. Groenendijk, Theo M. V. Janssen, and Martin B. J. Stokhof (eds.), Formal Methods in the Study of Language, Part 2, pp. 483–512. Amsterdam: Mathematisch Centrum. Schwarzschild, Roger (1996). Pluralities. Dordrecht: Kluwer Academic. (2011). Stubborn distributivity, multiparticipant nouns and the count/mass distinction. In Suzi Lima, Kevin Mullin, and Brian Smith (eds.), NELS 39: Proceedings of the North East Linguist Society, pp. 661–678. Amherst, MA: GLSA. Zucchi, Sandro, and Michael White (1996). Twigs, sequences and the temporal constitution of predicates. In Teresa Galloway and Justin Spence (eds.), Proceedings of SALT 6, pp. 329–346. Ithaca, NY: Cornell University. (2001). Twigs, sequences and the temporal constitution of predicates. Linguistics and Philosophy 24.2: 223–270.

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Proportional Many/Much and Most Carmen Dobrovie-Sorin and Ion Giurgea

This chapter is concerned with proportional many and proportional most.* We will argue that despite their common lexical basis (many), these two elements are syntactically distinct (a quantity modifier in SpecMeasP vs. a Quantificational Determiner, respectively, which occupy distinct syntactic positions inside the DP). Correlatively, these elements have different denotations: proportional most denotes a function from sets into sets of sets (as in Generalized Quantifier theory, henceforth GQT, contra Hackl 2009), whereas many introduces degree operators over quantity modification, and this type of modification can take both a cardinality and a proportional reading. The proportional reading of many typically arises in those contexts in which the constituent [DP[DetØ] many NP] has a partitive interpretation, i.e., it introduces a new referent that is a part of a larger plurality given in the common ground (the ‘whole’); in such contexts, the cardinality of the entity denoted by [DP[DetØ] many NP] is evaluated relative to the cardinality of the ‘whole’. Under this proposal, proportional many is not a quantificational determiner (contra Partee 1989). The semantics of proportional many is rather comparable to the semantics of evaluative adjectives. This is crucially different from the semantics of proportional most, which cannot be derived from the semantics of superlative adjectives. Although it is less frequently discussed, much itself has both a cardinal and a proportional reading which are completely parallel to the two readings of many. This contrasts with the proportional determiner most, which can take mass and plural DPs, as well as plural NPs, but, as we will show, not mass NPs as complements. 1.1

Introduction

Many allows two distinct interpretations, a ‘purely cardinal’ and a proportional one, which disregard and take into account respectively the cardinality of the NP set: *

The present paper was supported by the French National Research Agency, ANR-10-LABX0083, (Labex EFL) and by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, project number PN-II-RU-TE-2014-4-0372.

14

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Proportional Many/Much and Most (1)

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a. Many students are intelligent. b. John read many books. c. Many houses collapsed during the hurricane.

(1a) is (or at least tends to be) interpreted proportionally: the ratio of intelligent students to students (intelligent-students : students) is high (compared to some contextually relevant ratio). (1b), on the other hand, has a cardinal reading: the number of books read by John is high (compared to some contextually relevant number); the cardinality of the overall set of books is irrelevant for the interpretation. Both construals are possible in (1c). The truth conditions corresponding to the two readings can be described as in (2): (2)

a. j{x: student(x) & intelligent(x)}j : j{x: student(x)}j  r b. j{x: book (x) & was-read-by-John (x)}j  n

(r a fraction) (n a number)

The question is how this difference is compositionally obtained. One or two LF representations? Which LF representations? Is the difference in LF representations compatible with a uniform, non-ambiguous analysis of many as a cardinality predicate, meaning ‘high cardinality’, where ‘high’ needs to be contextually evaluated (de Hoop and Solà 1996)? Or do we need to assume that many has two distinct lexical entries, a cardinal and a proportional (Partee 1989; Cohen 2001; Romero 2015)? In this chapter, we will try to suggest answers to these questions by comparing proportional many with proportional most. The hypothesis that proportional many is a lexical element distinct from cardinal many predicts (or at least leads us to expect) that proportional many and proportional most occupy the same syntactic position and have the same semantic type. We will argue against this view by bringing up two types of contrasts between proportional many and proportional most (Section 1.2). We will then evaluate the two competing analyses of proportional most, the GQT analysis and Hackl’s (2009) adjectival analysis, and we will conclude in favor of the former (Section 1.3). Section 1.4 will be devoted to the analysis of proportional many. Throughout this chapter we will introduce evidence from various languages. Use of capital letters for MOST, MANY, etc. refers to the counterparts of most, many, etc. across languages. 1.2

Contrasts between Proportional MANY and Proportional MOST

The examples in (3) show that the proportional reading of French le plus (‘the more’ meaning ‘the most’) is ruled out, whereas the proportional reading of beaucoup (‘much/many’) is allowed: (3)

a. Beaucoup d’enfants respectent leurs parents. much of children respect their parents ‘Many children respect their parents.’

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Carmen Dobrovie-Sorin and Ion Giurgea b. *Le plus d’enfants respectent leurs parents. the more of children respect their parents. Intended meaning: ‘Most children respect their parents.’

The relative superlative reading of le plus is unproblematic: (4)

C’est Jean qui a lu le plus de livres. it is Jean who has read the more of books ‘It’s Jean who read the most books.’

Romanian differs from French insofar as it allows the proportional reading of cei mai mult, i ‘the more many’ meaning ‘the most’: (5)

a. Mult, i copii sunt inteligent, i. many children are intelligent b. Cei mai mult, i copii sunt inteligent, i. the more many children are intelligent ‘Most children are intelligent.’

Also allowed is the relative superlative reading of cei mai mult, i ‘the most’, as well as the proportional reading of mult, i ‘many’. These generalizations are not illustrated here because they hold cross-linguistically: if a language has an element meaning ‘many’, the proportional reading will be allowed; and if a language allows the proportional reading of (the) most, then it also allows the relative superlative reading. For our present purposes, it is the French data that are relevant: they show that in certain languages (in fact in many languages, e.g., most Slavic languages, Turkish, Mandarin Chinese, Japanese, Kannada, and Punjabi; see Živanovi
c 2007), the superlative of MANY can take the relative reading, but not the proportional reading; nevertheless, in these same languages, the proportional reading of the basic (positive) form of MANY is possible. Going back to Romanian, it is noteworthy to observe the contrast between mult ‘much’ and its superlative form, cel mai mult, which, respectively, can and cannot take a proportional reading: (6)

a. În ziua de azi multă apă e poluată. in day-the of today much water is polluted ‘Nowadays much water is polluted.’ b. ?* În ziua de azi cea mai multă apă e poluată. in day-the of today the more much water is pollute. Intended meaning: ‘Nowadays most water is polluted.’

Note that the relative superlative reading of cel mai mult/cea mai multă is unproblematic: (7)

Cine a băut cel mai mult vin? who has drunk the more many wine ‘Who drank the most wine?’

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Proportional Many/Much and Most

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The intended meanings of (3) and (6) can be rendered by using an expression of the form ‘the largest part of’ and with the expression for the ‘whole’ represented by a definite DP: (8)

a. La plupart des enfants respectent leurs parents. the more‑part of‑the children respect their parents ‘Most children respect their parents.’ b. În ziua de azi cea mai mare parte a apei e poluată. in day‑the of today the more big part gen water‑the.gen is polluted ‘Nowadays most water is polluted.’

In sum, the proportional reading of MANY/MUCH is possible in all those languages that have these lexical items, whereas the proportional readings of the corresponding superlative forms are allowed only in some languages, and, moreover, in some of those languages they can only be expressed with count NPs. Section 1.3 is devoted to explaining the constraints on MANYsuperl =MUCHsuperl , and Section 1.4 will be concerned with choosing, among the available analyses of proportional MANY/MUCH, one that can explain why this element is not subject to the constraints observed for MANYsuperl =MUCHsuperl . 1.3

Proportional MOST

Most is the superlative form of many, which led Hackl (2009) to propose that the quantificational analysis of most is inadequate. Under Hackl’s analysis, proportional most occupies the syntactic position of an adjective and its semantics is that of absolute superlative adjectives. In what follows we will bring up some problems with Hackl’s proposal and argue in favor of the quantificational analysis of proportional most: this element sits in a Det(erminer) position and denotes a relation between two sets.1 This analysis will account for the constraints on the distribution of most observed in Section 1.2. Since proportional many is not subject to those constraints, we will conclude that proportional many is not a quantificational Determiner (it does not sit in Det and it does not denote a relation between sets) but, rather, a cardinality predicate. 1.3.1

Hackl (2009): Proportional Most as a Superlative Adjective

The empirical generalizations in (9) seem to support the adjectival analysis of most in English: (9)

1

a. Most has the form of the superlative of many/much. b. Most has both proportional and relative readings, which are parallel to the absolute and relative readings of superlative qualitative adjectives (the best, the nicest, the highest).

For the time being we leave aside partitive most, e.g., most of the students.

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Based on these observations, Hackl (2009) proposed that the proportional reading of most has both the syntax and the semantics of an absolute superlative adjective. According to Hackl’s syntactic analysis, proportional most sits in an Adj position, just like highest in (11) below, the Det position being filled with an empty category interpreted as 9; in the case of qualitative superlatives, the D position is occupied by the definite article the: (10)

a. John climbed [[D the] [[AdjP highest] [NP mountain]]. b. John climbed [[D Ø] [[AdjP most] [NP mountains]].

(11)

a. b.

the d-high mountain 9 d-many mountains

Given this syntactic representation, the compositional semantics of proportional most would mimic the compositional semantics of absolute qualitative adjectives (see Section 1.3.5 below). Hackl’s proposal is confronted with various problems. One problem that immediately comes to mind is the fact that the definite article is excluded with proportional most but required with absolute superlative adjectives. Hackl argues that the lack of the definite article follows from the denotation of MANYsuperl, but the denotation he proposes is a modified version of the denotation of superlative adjectives: the ‘non-identity’ constraint on the elements of the comparison class is replaced by ‘non-overlap’, which has the result that any plurality in a plural denotation which has more members than the sum of all the others will satisfy the absolute superlative of MANY; since there is no unique plurality satisfying MOST, the absence of the is expected. In Section 1.3.3, we will argue that Hackl’s analysis of the semantics of superlative quantitatives is problematic. Two other empirical problems for Hackl’s proposal are related to the data brought up in Section 1.2. Indeed, Hackl cannot explain why le plus in French cannot take the proportional reading, despite its being perfectly unproblematically compatible with the relative superlative reading:

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Proportional Many/Much and Most (12)

19

a. C’est Jean qui a lu le plus de livres. (relative reading) ‘It’s John who read the most books.’ b. *Le plus d’enfants sont intelligents. (proportional reading) Intended meaning: ‘Most children are intelligent.’

One might think that the problem with the French example is that the superlative in French requires the definite article, which, according to Hackl, would be incompatible with the proportional reading (cf. *The most children are intelligent, vs. John read the most books, with a relative superlative reading). But, first, notice that the same contrast appears in languages that do not have definite articles (Živanovi
c 2007): (13)

a. Nej-víc lidí pije pivo. (Czech) (Živanovi
c 2007) superl-more people.gen drink beer b. Naj-więcej ludzi piło piwo. (Polish) superl-more people.gen drank beer c. Naj-više ljudi pije pivo. (Serbo-Croatian) superl-more people drink beer d. Naj-več ljudi pije pivo. (Slovenian) superl-more people.gen drink beer = ‘More people are drinking beer than anything else’ (relative superl.) 6¼ ‘Most people drink beer’ (proportional)

Secondly, there are languages where MOST cannot occur without an article, as in French, and yet the proportional reading is allowed – this is the case of German and Bulgarian:2 (14)

2

a. Die meisten Leute trinken Bier. (German) the most people drink beer ‘Most/The most people drink beer.’ (proportional or relative) b. Maria pročete po-veče-to statii (Bulgarian) Maria read -er-many(suppl)-the articles ‘Maria read most articles.’ (proportional)

In Romanian, the situation is at first glance similar – we glossed the element cel which appears in superlatives before the comparative marker as ‘the’, and this item does indeed function as a strong definite article form elsewhere (e.g., cei doi băiet, i ‘the two boys’, cel de acolo ‘the (one) over there’). However, it can be shown that although cel can mark definiteness when DP-initial, it has become an obligatory part of superlative forms, appearing in contexts where a determiner analysis is excluded (see Croitor and Giurgea 2013; Giurgea 2013 for details): (i)

există întotdeauna [un cel mai mic divizor comun a două elemente] exists always a cel more small factor common of two elements ‘There always is a smallest common factor of two elements.’ (ro.wikipedia.org/wiki/Algoritmul_lui_Euclid) (ii) [cei doi [cei mai puternici] oameni din stat] the two cel.mpl more powerful persons in state ‘the two more powerful persons in the state’

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The second problem with Hackl’s analysis is the fact that in Romanian, proportional cel mai mult ‘the more most’, meaning ‘(the) most’, cannot take a mass NP as a complement (see example (6)). The same constraint can be observed in English non-generic sentences, e.g., *Most water in this tub is dirty. 1.3.2

The Relative Reading of le plus de NP

We agree with Hackl that the relative reading of the superlative quantitative most should be analyzed exactly on a par with the relative reading of superlative qualitatives of the type illustrated in (16): (15)

(16)

De tous mes élèves, Jean a lu le plus de livres. of all my students, Jean has read the more of books ‘Out of all my students John read the most [of] books.’ De tous mes élèves, Jean a lu le plus long livre. of all my students, Jean has read the more long book ‘Out of all my students John read the longest book.’

Relative readings are characterized by the fact that the entities which are compared (books in (15)–(16)) are related via a (potentially complex) predicate, provided by the clause (be read by in (15)–(16)), to a set of individuals that includes the individual denoted by a constituent in the clause (Jean in (15)–(16)). In its relative reading, the superlative qualitative adjective in (16) says that the length of the book read by John is the largest in a comparison class made up of the books read by people other than John. In a parallel fashion, the superlative quantitative adjective (the) most in example (15) says that the cardinality of the plurality of books read by John is the largest in a comparison class made up of pluralities of books read by people other than John. This much seems to be unanimously agreed upon in the current literature and, for the purposes of this chapter, we do not need to take a position with respect to the debate as to whether the ‑est morpheme raises at LF (Heim 1999; 2000; Hackl 2009; Romero 2015, among others) or not (Farkas and Kiss 2000; Sharvit and Stateva 2002; Coppock and Beaver 2014). 1.3.3

No Absolute Reading for Superlative Quantitatives

In absolute superlatives, the comparison class (the set of entities which are compared) is established using only the DP-internal material: it is supplied by the NP (with possible implicit contextual restrictions, e.g., the set of books denoted by livre in (17) may be contextually restricted to a salient set, e.g., the set of books in my library): (17)

Jean a lu le plus long livre (dans la bibliothèque/au monde) Jean has read the more long book in the library in-the world ‘Jean read the longest book (in the library/in the world).’

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Proportional Many/Much and Most

21

Longest book denotes the singleton set that contains a book x such that x is longer than any other book (in the world/in the library). Following Heim’s (1995) analysis of superlatives, Hackl derives this reading by raising -est to a DP-internal position, above the [A+N(P)] constituent: (18)

〚[-est C]i [di-long book]〛= λx. 8y 2C [y6¼x ! max {d: x is a d-long book} > max {d: y is a d-long book}]

The singleton set is shifted to the unique entity that it contains at the DP level (by applying the definite article; in languages such as Romanian and French, where the article arguably is inside the superlative, cf. Giurgea 2013; see note 2), it may be assumed that DP-initial superlatives license a null definite D). Hackl (2009) proposes to extend this analysis to superlative quantitatives: (19) (20)

John climbed most mountains. 〚[-est C]i [di-many mountains]〛= λx. 8y2C [y6¼ x ! max{d:mountains(x) = 1 & |x|  d} > max{d: |y|  d}]

Note, however, that if we compare all the pluralities in the NP set, the largest one will be the one that comprises all the others (the supremum), which means that absolute most is equivalent to all (since the sets of pluralities denoted by modified NPs have the algebraic structure of join semi-lattices, the plurality that has the highest cardinality in the set is the supremum of the join semilattice). In order to derive the proportional reading, Hackl proposes that ‘6¼’ in the formula in (20) should be read as ‘non-overlap’ (“for the purpose of counting, . . . two pluralities are non-identical only if there is no overlap between y and x” [Hackl 2009, p. 81]). This means that the comparison class consists of x, the external argument of the [DegP + NP] constituent, and all the pluralities in the NP set which do not overlap with x; x must be bigger than the biggest of these pluralities, which is, in set-theoretical terms, 〚NP〛\ {x} (the set of elements in the NP that are not parts of x). This is equivalent to the “more-than-half” interpretation. Besides the empirical problems noted in Section 1.2, the replacement of the ‘non-identity’ condition by the stronger ‘non-overlap’ condition is also empirically problematic: the claim that “two pluralities are non-identical (for the purpose of counting) only if there is no overlap between them” (Hackl 2009, p. 81) is disproven by examples involving comparison between groups. Take a context where pluralities based on nationality are compared, say Nigerians, Congolese, French, and Russians, and somebody has a double nationality, e.g., Nigerian and French. In such a situation, according to Hackl, the group of Frenchmen is excluded from the comparison class of the group of Nigerians. Therefore, the sentence in (21) should be true if Nigerians outnumber the Russians and the Congolese but the Frenchmen outnumber the Nigerians, which is clearly not the case:

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(21)

Cei mai mult, i candidat, i sunt nigerienii. (Ro.) the more many applicants are Nigerians-the ‘The most numerous/the largest group of applicants are the Nigerians.’

The same problem appears if we apply Hackl’s rule to relative superlatives (for those analyses of relative readings that rely on comparison classes containing individuals (e.g., Heim 1999; Farkas and Kiss 2000; Sharvit and Stateva 2002; Hackl 2009; Coppock and Beaver 2014). The problem does not arise for those theories that rely on comparison classes containing degrees (e.g., Krasikova 2011; Romero 2015): No matter whether we compare the correlate, i.e., the Russians in (22) below, with other entities (as in Heim 1999; Farkas and Kiss 2000; Hackl 2009; Coppock and Beaver 2014), or the referents of the DPs that contain the relative superlative, i.e., the money in (22) (as in Sharvit and Stateva 2002; Teodorescu 2009), we can see that overlap relations are allowed. In (22), we show this for the comparison class which involves the correlate, in (23), for a comparison class which comprises the DPs that contain the relative superlative: (22)

The RUSSIANS spent the most money. Relevant context: the compared groups are the Russians, the Frenchmen, and the Nigerian, and one individual has double nationality, Russian and French.

(23)

MARY read the most books. Relevant context: Mary, John, and Alice are compared, and one of the books read by Mary was also read by John and Alice.

We are thus led to conclude that the proportional reading of most is not an absolute superlative reading. As had been observed in the literature before Hackl (see Szabolcsi 1986 and especially Gawron 1995), most simply lacks the absolute reading. Based on examples of the type in (21), we conclude that comparison of cardinality is only possible between groups that are independently individuated – in (21), the context provides a set of groups that are compared, whose elements are known (the French candidates, the Nigerian candidates, etc.); in the relative reading, the groups are individuated via the relation they have with other entities (or, in the Degraising analysis, it is the correlate itself that is compared with other entities in terms of the degree property). In the cases we want to rule out, all members satisfying a plural property are compared – i.e., in (21), not just the groups formed by applicants of a certain nationality, but any sum of applicants. In other words, we aim at a constraint that excludes comparison classes that are join semi-lattices. A possible formulation of this constraint is as follows: (24)

C cannot function as a comparison class if, for every element x in C, there is a y 6¼ x in C such that x and y stand in a part–whole relation (i.e., C should not be partially ordered by the part–whole relation)

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Proportional Many/Much and Most

23

Summing up, (i) the non-overlap constraint on comparison classes of pluralities that Hackl postulates for the semantics of the proportional (viewed as ‘absolute superlative’) most is a departure from the definition of the comparison classes of superlative adjectives, based on ‘non-identity’; no such departure is needed for relative superlative the most, the semantics of which is exactly parallel to the semantics of relative superlative adjectives; (ii) the replacement of ‘non-identity’ with ‘non-overlap’ when comparing pluralities is empirically inadequate; (iii) hence, we are left with an ‘impossible’ absolute reading of the superlative of many: by comparing all the elements in a plural NP in terms of their cardinality, we would obtain the reading in which most means all; (iv) this reading of most is ruled out by a constraint that rules out join semi-lattices as comparison classes. Relative readings are possible because, in their case, even assuming the insitu analysis of relative superlatives in which the comparison class consists of elements characterized by the NP property (pluralities of books in (15)), the comparison class is not a join semi-lattice; it is an unordered set that consists of plural entities defined by an independent criterion – e.g., books read by various people. 1.3.4

The Proportional Reading of MOST

Given the observations of the previous section, Hackl’s analysis must be abandoned: proportional most in English cannot be analyzed as an absolute superlative adjective. This leads us back to the generalized quantifier analysis: (25)

Proportional most is a quantificational Determiner (type )

We will furthermore argue that, in order to be able to function as a quantificational Determiner, most must be able to sit in the (highest) D position (the one that is normally occupied by the definite article and quantifiers of the everytype). This analysis is supported by the distribution of the definite article: (26)

a. John climbed *(the) most mountains. (relative reading) ‘the group of mountains climbed by John is more numerous than the groups of mountains climbed by other people (out of a contextually given set)’ b. John climbed (*the) most mountains in Romania. (proportional reading) ‘the number of mountains in Romania climbed by John is bigger than the number of mountains in Romania that John did not climb’

In (26) the definite article sits in the D position and most in an adjectival position, a syntactic configuration that underlies the relative superlative reading, which is allowed not only for qualitative but also for quantitative modifiers: no maximality problem arises, because the groups of mountains that are compared are individuated relative to their respective climbers (in other

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words, the superlative operator applies to an unordered set of individuated groups rather than to a join semi-lattice). In (260 ), on the other hand, the absence of the arguably indicates that most occupies the D position: (260 )

a. [DP[D the] [XP[Adj most] [NP mountains]]] b. [DP[D most] [NP mountains]]

(relative most) (proportional most)

(260 ) is crucially different from absolute superlatives insofar as most is not an NP modifier and, correlatively, the DP headed by most denotes a generalized quantifier that takes the main predicate as an argument (hence the QP label used in (2600 )): (2600 )

b. [DP/QP [D/Q most] [NP mountains]] (λx. John climbed (x))

Given this LF representation, most itself is a quantificational determiner that denotes the relation between the NP set and the set obtained by abstracting over the position of most NP (Mostowski 1957; Rescher 1964, 2004). Thus, (26b) is true iff (27) below holds, i.e., iff the cardinality of the mountains (in Romania) climbed by John is bigger than the cardinality of the mountains (in Romania) that John did not climb: (27)

|{x: mountains(x) ^ climb(john,x)}| > |{{x: mountains(x)} {x: climb(john,x)}}|

This analysis was attacked by Hackl on the grounds that it is not compositional, i.e., that the superlative morpho-syntax is not taken into account. Note, however, that the formula in (27) does take into account the semantics of superlatives, which involves two pieces of meaning: the comparative operator and another component that roughly means ‘than all others’. The comparative component is exactly as in run-of-the-mill superlatives, but the ‘than all others’ bit is different: whereas absolute qualitative superlatives single out one element (a singular or plural entity) in the comparison class determined by the (contextually restricted) NP, proportional most compares the cardinality of one set (the NP set that satisfies the main predicate) to the cardinality of its complement set (the NP set that does not satisfy the main predicate). Since the two sets together exhaust the NP set, to say that one set is larger than the other is equivalent to saying that that set is the largest of all NP sets. Crucially, this semantics depends on most being a semantic determiner, i.e., as denoting a relation between sets, rather than a modifier. If most sits in a modifier position (as in the relative reading), it cannot have the semantics of quantificational Ds (it cannot compare two complement sets), but instead it has the semantics of superlative modifiers, i.e., it applies to a set of pluralities and yields the singleton set that contains the plurality with the highest cardinality. We may conclude that the superlative morpho-syntax of most does not force us to assume that proportional most is a modifier: this morpho-syntax is compatible with – in fact necessary for – the set-relational analysis of most.

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Proportional Many/Much and Most

25

Romanian poses a prima facie problem for the D-analysis of proportional most, as, in this language, the corresponding element – see (5) – looks like a complex phrase – cei mai mult, i ‘the.mpl more many.mpl’, cele mai multe ‘the.fpl more many.fpl’ – which arguably indicates that it does not sit in a head position such as D . However, upon closer examination, Romanian turns out to support our analysis: what is crucial for a generalized quantifier analysis is that most is the highest element in the DP, closing off the nominal projection and allowing the denotation to access the main predicate. This requirement may be satisfied not only by a D but also by a phrase sitting in SpecDP. The distributional evidence shows, indeed, that proportional cei mai mult, i sits in SpecDP. Note that superlatives in general may occur both pre- and postnominally in Romanian, and when DP-initial, they mark the DP as definite; with quality adjectives, the relative and absolute readings are possible in both positions: (28)

a. cel mai frumos cadou the most beautiful present b. cadoul cel mai frumos present-the the most beautiful

(✓ relative, ✓ absolute) (✓ relative, ✓ absolute)

Cei mai mult, i ‘(the) most’ also has this variable placement, but the postnominal position only allows the relative superlative reading. The proportional reading requires DP-initial placement, which is predicted by our hypothesis, according to which MANYsuperl can get a proportional reading only if it sits in D or Spec,DP positions: (29)

a. Cele mai multe cărt, i sunt de la Ion (✓ relative, ✓ proportional) the more many books are from Ion (i) ‘the books from Ion are more numerous than the books from other people’ (relative) (ii) ‘the majority of the books are from Ion’ (proportional) b. Cărt, ile cele mai multe sunt de la Ion (✓ relative, * proportional) books-the the more many are from Ion only: ‘the books from Ion are more numerous than the books from other people’

We conclude that cei mai mult, i is ambiguous in Romanian, with two interpretations, either a quantitative superlative or a proportional quantifier. The latter reading can only occur in SpecDP, due to semantic compatibility reasons (quantifiers close off the DP and have access to the predicate of the clause). 1.3.5

Explaining Why le plus Cannot Take the Proportional Reading

Let us now go back to the ungrammaticality of the absolute construal of le plus de NP in French (see (12)):

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(12)

b. *Le plus d’enfants sont intelligents. ‘The most [of] children are intelligent.’

This example is built in such a way that le plus ‘the most’ must take an absolute reading (the material in the clause does not offer a set of entities which may be paired, via the predicate, to groups of children). So, if Hackl’s analysis of most were right, we would expect (12) to be grammatical and interpreted as involving a proportional reading of le plus. What we see is that (12) is ungrammatical. As we have seen in the previous section, one part of the explanation is that, contra Hackl (2009), superlative quantitatives cannot take absolute readings. Note, however, that le plus occurs at the beginning of the DP, like Romanian cei mai mult, i in (29), and we may wonder why it cannot function as a quantificational determiner (i.e., denote a function from sets into sets of sets) with the proportional meaning characteristic of English most and Romanian cei mai mult, i. We propose that the impossibility of French le plus taking the denotation of a quantificational Det (i.e., a function from sets into sets of sets) is due to the use of de ‘of’, which forces (le) plus to sit in a quantity modifier position (SpecMeasP).3 Note, indeed, that whereas Romanian cei mai mult, i agrees with the noun, in French neither le nor plus agrees, and of-insertion is required. This makes le plus parallel to measure phrases based on nouns:4 (30)

a. [trois grammes] de beurre, [deux verres] de vin, [trois mètres] de tissu three grams of butter, two glasses of wine, three meters of tissue b. [trois douzaines] d’enfants three dozens of children

Note that of-insertion and absence of agreement characterizes not only le plus, but also the positive degree of the quantitative, beaucoup: (31)

3

4

a. [beaucoup] de beurre, much of butter b. [le plus] d’enfants, the more of children

[beaucoup] d’enfants much of children [le plus] de beurre the more of butter

These quantitative constructions, in which de is followed by NP, currently called ‘pseudopartitives’, should be distinguished from genuine partitives, i.e., constituents of the form XP of/de DPs. It may be assumed that de in these examples occupies the head of a projection devoted to quantity, QuantP, and the measure phrase sits in its specifier. Alternatively, the noun of the measure phrase is a light/semi-functional N head that takes an of + NP complement. Under either of these accounts, we can explain the fact that le plus is exclusively a quantitative: as a Spec of QuantP, (le) plus must be a MeasP. If de-constructions rely on a semi-functional N head, plus may be analyzed as the spell-out of a complex MORE + BIG + QUANTITY (see Kayne 2005 for this type of decomposition of quantitatives, based on the silent nouns NUMBER and AMOUNT).

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Proportional Many/Much and Most

27

Thus, the positive beaucoup must be assumed to always be a quantity modifier, rather than a D, even on its proportional reading. The fact that the MeasureP projection, which is devoted to quantity (hosting quantity modifiers in its Spec), must be divorced from the D-level is supported by the possibility of (at least a part of ) quantitatives to be preceded by Ds:5 (32)

a. ces [trois grammes] de beurre these three grams of butter b. les deux enfants the two children

As for the article le in le plus, we may analyze it as a special superlative marker, part of the DegP, rather than as D, as it can also appear in adverbial superlatives: (33)

C’est lui qui a le plus souffert. It’s he who has the more suffered ‘It’s him who suffered most.’

Alternatively, we may assume that le occupies the D position, in which case we would need to say that the comparative form plus acquires a superlative interpretation in the environment of a definite D. In a nutshell, our explanation of the data in (12a–b) relies on the following two generalizations: (34)

a. Superlative quantitative modifiers can take relative, but not absolute readings. b. Le plus is necessarily a quantity modifier, which sits in a syntactic position that is lower than Det (namely, SpecMeasP); correlatively, le plus has the semantics of a superlative quantitative modifier of the NP (type ).

The generalization in (34) is a semantic universal (see Section 1.3.2) whereas (34) is a syntactic peculiarity that distinguishes French le plus from English most or Romanian cei mai mult, i, which are able to access the Det syntactic position and correlatively function as quantificational Determiners with a proportional semantics (see Section 1.3.4 above). Thus, the impossibility of the proportional construal of le plus can be explained by making crucial use of its superlative morpho-syntax, combined with its modifier syntax. Importantly, beaucoup resembles le plus regarding its modifier syntax, but, unlike le plus, it can take a proportional reading. The analysis of the 5

Beaucoup lacks this possibility, in contrast to Engl. many, e.g., the many children that arrived yesterday. This interesting issue will be left aside here because it is not crucial for our main purposes, which concern the analysis of DP-initial MANY. Note that Romanian mult, i/multe ‘manymasc/fem’ can be preceded by a demonstrative (aceste multe erori‚ ‘these many errors’) or can host the suffixal definite article (multele sale greşeli ‘many-the his mistakes’).

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proportional reading of beaucoup given in Section 4 will be based on its modifier syntax. The general conclusion will be that, cross-linguistically, proportional MANY and proportional MOST have clearly distinct syntactic and semantic representations. 1.3.6

Entity-Restrictor Proportional Quantifiers

The French examples below are built with proportional quantifiers that take DPs in their restriction. Such quantifiers express the intended proportional meaning of le plus: (35)

a. La plupart des enfants sont intelligents (Fr.) the more-part of-the children are intelligent b. La plus grande partie de l’eau est polluée. the more big part of the water is polluted ‘Most water is polluted.’

Many other languages, e.g., Spanish, Italian, Albanian, and most Slavic languages, are like French in having DP-restrictor (or Nmax-restrictor in languages without articles, as Slavic)6 but not NP-restrictor MOST, and the reverse probably never holds: there is no language we know of that has an NP-restrictor but not a DP-restrictor proportional MOST. The larger cross-linguistic distribution of DP/Nmax-restrictor quantifiers can be explained as follows: the superlative of MANY/MUCH can always occupy a quantity modifier position, but only in some languages can it sit in the Det position, which is necessary for the proportional reading to be possible. Slavic languages are particularly telling in that the impossibility of the proportional reading of MOST + NP constituents can be related to the absence of the D-level of representation. By way of contrast, constituents of the type ‘the largest part of’ are cross-linguistically widespread because they occupy a position above a full DP (or above Nmax in Slavic) and have acquired a special grammaticalized proportional meaning, possibly on the model of ratio expressions of the type 20 percent of DP/Nmax, which are arguably widespread. The use of such expressions is probably available in all languages, regardless of their DP syntax, as soon as the corresponding mathematical concept exists. 6

As shown in Živanovi
c 2007, those Slavic languages that do not have articles do not use MOST as a proportional quantifier, but resort to a nominal form (of the type ‘majority’) followed by a genitive NP to express this meaning. The fact that the genitive-marked NP is a Nmax can be shown, besides the parallel with languages with articles (where majority-nouns are followed by DPs), by the possibility to use demonstratives: Většina těchto lidí (Czech) majority.nom these.gen people.gen ‘most of these people’

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29

In sum, proportional ‘majority’ quantifiers come in two types of syntactic configurations, SUP + NP and SUP + DP (note the superlative form in both configurations). These two configurations are associated with different semantic types: proportional quantifiers that combine with NPs, which we will call ‘set-restrictor’ quantifiers, denote relations between sets – type – whereas proportional quantifiers which take DPs, which we will call ‘entity-restrictor’ quantifiers, have the type 9x [read by John (x) ^ books(x) ^ many(x)] For problems with this analysis, see Dobrovie-Sorin and Beyssade 2013; Dobrovie-Sorin and Giurgea 2015. For our present purposes, the choice of a particular analysis of weak indefinite DPs is not directly relevant.

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1.4.2

Strong Indefinite DPs Are E-Type Expressions

Besides the ambiguity problem noted above, the quantificational analysis of Dlinked cardinal indefinites is confronted with empirical problems regarding scope effects (Fodor and Sag 1982; Farkas 1997a, 1997b): indefinites headed by cardinals can have specific readings in which they have scope outside a finite clause (cf. Fodor and Sag 1982). As noticed by Ruys (1992), in such cases only their existential scope is outside the clause, while their distributive scope remains clause-bound (they cannot distribute over something outside their finite clause, as it would be expected if the wide scope had been the result of Quantifier Raising; for the distinction between existential and distributive scope, see Szabolcsi 2010): (48)

If three relatives of mine die, I will inherit a house. Possible wide-scope reading: There is a group of three relatives of mine such as if they all die, I will inherit a house. Impossible wide-scope reading: There are three relatives of mine such as for each x among them, if x dies I will inherit a house.

Extra-wide-scope indefinites are not necessarily referential (as in Fodor and Sag’s initial proposal); they can also have intermediate scope, as noticed by Farkas (1981): (49)

Each student has to come up with three arguments which show that some condition proposed by Chomsky is wrong. Possible reading: each student > some condition > three arguments ‘For each student x, there is a specific condition proposed by Chomsky such that x must come up with three arguments showing that this condition is wrong.’

Reinhart (1997) proposed an analysis of extra-wide-scope indefinites as entity-denoting expressions obtained by applying a choice function to the NP property. She proposes that the choice function is a variable that gets existential closure at various levels (thus explaining the intermediate readings). The existential scope of the indefinite is given by the level where the choice function is existentially bound: (50)

a. 9f (all w such that die(f(three relatives) in w are such that I inherit a house in w)) b. 8x [student(x) ! 9f (x must come up with three arguments showing that f (condition proposed by Chomsky) is wrong)]

Kratzer (1998) eliminates existential quantification over choice functions, treating choice functions as free variables (being contextually given) and proposing parameterized choice functions for intermediate readings as in

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Proportional Many/Much and Most

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(49). Parameterized choice functions take an additional argument of e-type, represented by the variable with which the indefinite covaries:13 (500 )

b. 8x [student(x) ! x must come up with three arguments showing that f(x) (condition proposed by Chomsky) is wrong)]

Choice functions were initially introduced only for specific indefinites. Winter (1997) extended the choice-functional analysis to all readings of indefinites. The same extension is found in Steedman (2006, 2012), who uses Skolem functions and Skolem terms. In this wider use, choice functions do not imply specificity.14 Under the choice-functional analysis, it is not the cardinal itself that acts as a determiner (contrary to the classical GQT theory, where the cardinals have the semantic type of quantifiers, characterizing the intersection of the NP set and the set denoted by the sister of the DP); rather, the choice function applies to a plural property (a property/set of sums), and the cardinal restricts this plural property: (51)

Two children are intelligent.

(52)

This indicates that examples built with strong cardinal indefinites rely on predicate saturation: two children is an individual-denoting term (obtained by applying a choice function to the NP set) that saturates a unary predicate of type : (53)

λx intelligent (x) (fchoice (two children)) = intelligent (fchoice (two children))

Given this analysis, the D-linking that characterizes some of the strong readings of cardinal indefinites is due to pragmatic principles, e.g., referential anchoring (von Heusinger 2007) or Topic-marking (Büring 1996) rather than to the quantificational status of cardinal numerals. In this view, D-linked indefinites resemble definites in that their existence is not asserted but given in the background knowledge or can be accommodated based on this knowledge. D-linking (covert partitivity), i.e., anchoring to a context-given

13 14

Chierchia (2001) argues that both mechanisms are needed for intermediate scope readings. Bare NPs are not covered by this account.

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set, is one type of anchoring a new discourse referent in the shared knowledge that constitutes the common ground of the conversation. In the case of complex descriptions, such as a girl I met yesterday, the common ground is first augmented with the information “the Speaker met a girl yesterday,” and the indefinite is then anchored with respect to this information (see ErteschikShir 1997, 2007). The non-quantificational analysis of D-linked cardinal indefinites has the conceptual advantage of dispensing with a systematic ambiguity of cardinals: in both weak (non-D-linked) and strong (D-linked) indefinite DPs, the cardinal itself is a quantity expression that characterizes the cardinality of a sum (with the general meaning λx.|x|=n) and, when it heads an indefinite DP, a sum-entity with the cardinality n is introduced in the semantic analysis via the same mechanisms as those needed for other indefinites (built with a or some), e.g., existential closure, existential quantifier in D, choice functions. The semantic difference between weak (generalized existential quantifiers) and strong cardinal indefinite DPs (e-type referential terms) is due to the null D : following Dobrovie-Sorin and Giurgea (2015), we assume that the null D of indefinite DPs denotes either a generalized existential Q or a choice function. The choice between these two possible denotations is determined by DP-external syntax (i.e., syntactic position, information structure, semantic properties of the main predicate, e.g., i-level vs s-level, entity-predicate vs. existential predicate, etc.). Checking mechanisms can ensure the correlation between DP-external syntax and the semantics of D . Under this view, the null determiner of strong cardinal indefinites can be analyzed as introducing a choice function: (54)

½½D Øi  ¼ λP fiðPÞ where f is a choice function

1.4.3

From Strong Cardinal Indefinites to Proportional MANY

Extending this analysis to many-DPs, we get the following two representations for the strong and weak readings: (55)

    a. DP D Østrong ½many NP b. ½DP ½D Øweak ½many NP

Besides general considerations (i.e., that it is preferable to have a single lexical entry for many), a further argument against treating proportional many as a quantificational determiner, provided by Solt (2009, ch. 4), is based on the observation that the proportional reading of many can occur after an overt determiner:

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Proportional Many/Much and Most (56)

37

The few children who like spinach do not need vitamin supplements. (Solt 2009, p. 208, ch.4 ex. 75) Possible reading: ‘the proportion of children who like spinach to the total number of children is small’

In (56), many has a proportional reading despite the fact that it clearly does not occupy the D position, and as such cannot be analyzed as a quantificational determiner. Solt (2009, ch. 3) argues nevertheless that strong DPs built with many cannot be analyzed on a par with strong indefinites built with cardinals (as we have proposed in (55)), because they have different scopal properties. The first observation is that many-indefinites lack the extra-wide scope found with cardinals and indefinite determiners (a, some etc.) in examples such as (48):15 (57)

If many relatives of mine die, I’ll inherit a million dollars. (Solt 2009, ch. 3 ex. 54) Impossible reading: ‘there is a specific sum of relatives of mine x such that x has a large number (or a large proportion among my relatives) and if x dies, I’ll inherit a million dollars’

Solt’s second observation is that many-DPs resist inverse scope, which is acceptable with unmodified cardinals and the indefinite article: (58)

Every student read five/a book(s). a. every > five/a ✓ b. five/a > every ✓

(59)

Every student read many books. (Solt 2009, ch. 3 ex. 59) a. every > many ✓ b. many > every *

Based on these observations, Solt rejects the choice function analysis, treating all many-DPs essentially as weak DPs, more precisely, as propertydenoting expressions whose unsaturated position is identified with the unsaturated position of the predicate by Variable Identification, followed by existential closure. We agree that it is very difficult to obtain inverse scope from the object position, but it is not always impossible. In Romanian, where direct objects 15

The context in (57) (subject of die) does not force a weak reading, so the impossibility of the extra-wide scope in (57) is not due to the fact that the indefinite is necessarily weak. The same impossibility is found in contexts that impose a strong reading: If many students are intelligent, I will be very pleased. Impossible reading: ‘there is a specific sum of students, proportionally large w.r.t. the total number of students (in the class), such that if the members of this sum are intelligent, I will be very pleased’

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may be marked as specific via differential object marking, we may obtain inverse scope of many-DPs in object position by using this marking (in such examples, many bears the main stress, as is normal in covert partitive environments, where the NP is part of the given material): (60)

Fiecare profesor i-a examinat pe MULTI ¸ student, i. every professor cl.3pl.acc-has examined dom many students ‘Every professor examined many (of the) students.’ many>every : = ‘There are many students who were examined by every professor.’

(61)

Ion nu i-a invitat pe MULTI ¸ colegi. Ion not cl.3pl.acc-has invited dom many colleagues many>not : = ‘There are many (specific) colleagues that Ion did not invite.’

Even for English, inverse scope of many is given as acceptable by Frawley (2013, p. 472) for the PP in (62): (62)

A bird was seen by many children.

Note that extra-wide scope remains excluded even if we use differential object marking: (63)

* Dacă îi examinez pe mult, i student, i, voi fi mult, umit. if cl.3mp.acc examine.1sg dom many students will.1sg be pleased Intended reading: ‘there is a specific sum of students, proportionally large wrt the total number of students, such that if I examine the members of this sum are intelligent, I will be pleased’

The contrast between many and cardinals can be related to the fact that many (on a par with much, few, little) does not introduce an exact quantity, but is a scalar quantitative adjective, whose interpretation relies on degree quantification: POS (for the unmarked forms many, few, much, little), COMP for the comparatives more and fewer, and SUP for the superlative. In the positive, many (in combination with the positive degree head POS) characterizes a sum as larger than a standard value (on a cardinality or proportion scale; see the following sub-section). Assuming that POS is a degree-quantifier which as such undergoes QR, the lack of extra-wide scope illustrated in (57) and (63) can be explained by assuming the following principle: (64)

16

An indefinite cannot take a higher scope than a degree-quantifier over its measure.16

This is clear with the comparative of many. Thus, (i) – which contains a strong more-DP, being the subject of an i-level predicate – cannot have the reading in (ib), according to which there is a sum of intelligent students of this year which is larger than some sum of intelligent students of

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39

In (57) and (63), the degree-quantifier POS cannot raise out of the if-clause, because quantifiers are clause-bound. Therefore, by (64), the many-indefinite cannot take extra-large scope. Inside the clause, we have seen that inverse scope is very difficult, but not totally excluded: if many-DPs are explicitly marked as specific, as in (60), they may take scope from the object position, over the subject. For such cases, we may assume that first the entire object is raised above the subject, and then the POS head raises outside the DP. To conclude, strong DPs introduced by many may be analyzed as relying on the null D found in other strong indefinites; the restriction on their possible scope (which sets them apart from plural indefinites with cardinals or some) follows from independent principles and therefore does not need to be encoded in the form of a special D. We shall not decide here among the various possible analyses of the null D of strong indefinites. For convenience we have adopted a choice-functional analysis, but we do not exclude other possible analyses, e.g., a strong existential Q:17 (65)

½DP ½D Ø9 ½many students ½D Ø9  ¼ λP λQ:9x:PðxÞ ^ QðxÞ

What is important for our purposes is that one of those possibilities may apply to strong many DPs, and therefore we need not assume that many in strong DPs is itself a determiner. An important conclusion of this set-up is that Milsark’s notion of ‘strong’ vs. ‘weak’ only applies to indefinite DPs (and to the element, possibly null, that occurs in D ), not to quantity expressions such as cardinals or many/much, all of which are modifiers (cardinality/quantity predicates) rather than elements that would be ambiguous between cardinality/quantity predicates and determiners. 1.4.4

The Proportional Reading of Strong MANY

The analysis proposed so far, in which many-DPs are treated on a par with cardinal DPs, still does not explain why many in strong DPs has a proportional last year (the time specification – last year, this year – is represented as modifying the predicate student, because it is used to select persons according to the time when they are students): This year, more students are intelligent than last year. a. max{d: 9x (*student(x)(this year) ^ |x| = d ^ *intelligent(x))} > max{d´:9x (*student(x)(last year) ^ |x| = d´ ^ *intelligent(x))} b. # 9f 9g [max{d: *intelligent(f(λx.*student(x))(this year) ^|x|=d))} > max{d´: *intelligent(g(λx.*student(x)(last year) ^|x|=d´))}] 17

A determiner of this type was proposed by Krifka (1999) for indefinites built with cardinals, regardless of the weak vs. strong distinction. We restrict this possible analysis of the null D of indefinites to their strong readings.

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interpretation. The distinction between the proportional and the (purely) cardinal reading of scalar quantity adjectives is most transparent in the case of few, as shown by Partee (1989). If the total number of entities which satisfy the NP property is small, cardinal few can be used even if all members of the NP class satisfy the VP property. Thus, on its cardinal reading, (66) below can be true, even if I know all the linguists with a second degree in chemistry, in case the total number of such linguists is small. In the partitive interpretation of this example, overtly marked in (66), the sentence is false in the aforementioned context; it is required that out of the total number of linguists with a second degree in chemistry, the proportion of the linguists I know is small: (66)

a. I know few linguists with a second degree in chemistry. b. I know few of the linguists with a second degree in chemistry.

The contrast in (66) suggests that the proportional reading is correlated with partitivity: the proportional interpretation of the positive degree of many is imposed by (covert or overt) partitivity. The correlation between the strong/ weak distinction and proportionality comes from two facts: (i) strong indefinites are often interpreted as covert partitives (i.e., as introducing an entity included in a contextually salient group); (ii) because (covert or overt) partitive indefinites are existentially presupposed by virtue of being parts of already established discourse referents, their existence is not asserted in the sentence, therefore they do not function as weak indefinites. The fact that partitivity rather than the strong status is crucial for the proportional interpretation is proven by the behavior of many-DPs in object position, illustrated in (67) below. As shown by (67), the object position of respect does not allow existential bare nouns, which shows that it is a strong DP position. We must therefore conclude that the many-DP in (67) is a strong DP. Nevertheless, it can have a cardinal reading, in addition to the proportional reading: (67)

a. John respects many professors. (✓ cardinal, ✓ proportional) b. John respects professors. (✓ generic, ?? existential)

The correlation between the strong status and the proportional interpretation holds for the subject position in examples such as (68), because in such cases the covert partitive interpretation is compulsory: such sentences have as a topic a group or class in which the referent of the indefinite is included (as such sentences cannot be ‘thetic’, they need an aboutness topic introduced by one of the arguments, typically the subject, cf. Erteschik-Shir 1997, 2007). (68)

a. Many students of history are intelligent. b. Many girls were blond.

Another interesting observation, due to Solt (2009, 2017), is that, even in subject position, the proportional interpretation is obligatory only in the

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Proportional Many/Much and Most

41

positive degree of scalar quantity adjectives. The comparative more in the example (69) allows both a proportional and a cardinal reading. Yet know requires a strong DP in its subject position (existential bare nouns are impossible in that position, see (70) below). (69)

More residents of Ithaca than New York City know their next-door neighbors. (Solt 2009, p. 208, ch. 4 ex. 74a) (i) The proportion of residents of Ithaca that know their neighbors (the ratio of the residents of Ithaca who know their neighbors to the total number of residents of Ithaca) is larger than the proportion of New Yorkers that know their neighbors. (ii) The total number of residents of Ithaca who know their neighbors is larger than the total number of New Yorkers who know their neighbors.

(70)

Linguists know many languages. (✓ generic, * existential)

The two readings of (69) have distinct truth conditions, as can very easily be seen due to the large difference between the numbers of residents of Ithaca and New York: it is likely that, under the reading in (i), (69) is true, given that residents of small towns usually know their neighbors, and, under the reading in (ii), it is false, because of the large overall number of residents of New York. In order to analyse these facts, we need to refine our analysis of many, which has so far been treated as a quantity modifier. It is widely assumed that quantity modifiers, although they are not themselves Ds (because they can co-occur with determiners; see the three girls, those many problems), belong nevertheless to the functional domain of the noun phrase. This is because they must precede quality adjectives, license N ellipsis (e.g., I took three/many vs. I took new *(ones)), and allow a null strong D (see (54)– (55)). We follow Schwarzschild (2006) and Solt (2009) in analyzing quantity modifiers as specifiers of a functional head (called Mon0 by Schwarzschild and Meas0 by Solt) that is specialized in introducing a measure function monotonic on the part-whole structure of the entity to which it applies (as shown by Schwarzschild 2006, this is the common property of the various dimensions – e.g., cardinality, volume, mass, etc. – that may be linguistically manifested in the functional projection devoted to ‘quantity’). Solt (2009) proposes the denotation in (71) below for Meas0, which needs a special rule of semantic composition (Variable Identification) in order to combine with the NP. (71)

  Meas0 ¼ λx λd:μDIM ðxÞ  d

(Solt 2009, p. 105, ch. 3 ex. 47)

We propose a slightly modified version, which dispenses with the special rule of Variable Identification:

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42 (72)

Carmen Dobrovie-Sorin and Ion Giurgea   Meas0 ¼ λN λd λx:NðxÞ ^ μðxÞ  d, where μ is a contextually identified measure function which must be monotonic on the part–whole structure of x

In examples with numerals, the dimension is cardinality, and the composition unfolds as follows: = λd λx. students (x) ^ #(x)  d

(73)

〚Meas [students]〛

(74)

〚[three [Meas [students]]〛= λx. students (x) ^ max {d: #(x)  d} = 3 = λx. students (x) ^ #(x) = 3

Scalar quantitatives (many/much and few/little) introduce degree quantification on the variable which is the degree argument of Meas0. As is clear in the case of comparatives as in (69), these degree operators may take scope in the clause (for now, we only illustrate the cardinal reading; we will address proportional readings later on):18 (75)

More boys than girls are intelligent. LF: [-ER [than d1 [[d1 Meas girls] are intelligent)]] [d2 [[d2 Meas boys] are intelligent]] Interpretation: max(λd. (9f. intelligent(f (λx. boys(x) ^ #(x) = d)) > max (λd. (9f. intelligent(f (λx. girls(x) ^ #(x) = d))

Solt proposes that quantity adjectives denote relations between degrees and degree intervals, as shown in (76).19 Their first argument, d in (76), serves as an argument for the degree operators with which these adjectives obligatorily combine. The second argument, I, is a predicate over degrees obtained by degree abstraction when the adjective raises in the clause, leaving a degree variable in the base position. Inv is a function that maps an interval to the join-complementary interval on the same scale:20 18

Solt (2009) argues that not only degree operators but also the adjectives many/much and few/ little themselves raise outside the DP. The main argument for this claim comes from few/little, which arguably introduce a negation that may take scope over the DP in which they occur (cf. Heim 2006) – this scopal possibility is illustrated by the reading in (c) of the example below: They need few reasons to fire you. (a) ‘There are some (specific) reasons X such that the number of X is not large and they need X to fire you.’ (b) ‘To fire you, they need there not to be a large number of reasons.’ (c) ‘It is not the case that they need a large number of reasons to fire you.’

19

20

(76) is a simplified version of Solt’s formulae, which distinguish between many and few, which require the measuring dimension to be cardinality, and much and little, for which other dimensions such as Volume, Weight, etc., are used. For few/little, we may also use Heim’s (2006) entry for little: ½½little ¼ λd λP: ¬PðdÞ: Solt (2009: 98) notes a subtle difference between the two formulations (a (semi)closed interval I shares a point with Inv(I)), but does not explain why she prefers the formulation using Inv.

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Proportional Many/Much and Most (76)

a. ½½many=much ¼ λd λI:d 2 I b. ½½few=little ¼ λd λI:d 2 InvðIÞ

43 (adapted from Solt 2009, ch. 3 ex. 35)

In view of the proportional reading of comparatives illustrated in (69), we propose that the measuring function itself, introduced by Meas, may have a proportional version,21 in which the measure of the entity across a certain dimension is evaluated with respect to the measure of a ‘whole’ in which that entity is included. The degrees compared in (69) are proportions, which presupposes the existence of proportional Measure functions, which make use of proportional scales. Such scales represent transformations of the various dimensions introduced by Meas, obtained by the following rule: (77)

PROP ðμDIM Þ ðyÞ ðxÞ ¼ μDIM ðxÞ=μDIM ðyÞ, defined iff there is an established ysuch that x is a part of y. referent  Meas0 ¼ λN λd λx:NðxÞ ^ PROP ðμDIM Þ ðyÞ ðxÞ ¼ d, defined iff there is an established referent y such that x is a part of y.

The parameter y, in this formulation (the ‘whole’) is contributed by the context. Compositionally, we may assume that it is provided by the maximal sum of the NP property (the sister of Meas), with relevant contextual restrictions (which may be introduced via situation variables) – see (78), which is similar to Solt’s (2017) implementation: (78)

½½Meas%  ¼ λN λd λx:NðxÞ ^ μDIM ðxÞ=μDIM ððNÞÞ

However, this does not explain why proportional interpretations are not found with weak indefinites, as in (79): (79)

a. There were many books on the table. (✓ cardinal, * proportional) b. There were more books on the table than on the floor. (✓ cardinal, * proportional)

In our view, covert or overt partitivity is a necessary condition for proportional readings, and weak DPs cannot be covert partitives. Hence the impossibility of the proportional reading in (79). Notice now that in a strong context, more allows not only a proportional, but also a cardinal interpretation (see the readings of (69) earlier), which means

21

Solt (2009) does not make this proposal, and her analysis of the example is untenable: she proposed that (i) the compared groups (residents of Ithaca who know their neighbors and residents of New York who know their neighbors) are measured on bounded scales whose maxima are the number of all residents of Ithaca and of all residents of New York, respectively, and (ii) the endpoints of these two bounded scales are aligned. But there is no way in which the 20,000 residents of Ithaca and the 8 million of New York can be ‘aligned’ on a scale of cardinality. The necessity of introducing proportional scales was independently recognized by Solt (2017), an article that came to our knowledge after the submission of the first version of our chapter (June 2016).

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that a cardinal Meas function is allowed for the interpretation of MORE in strong contexts. This contrasts with the intuition concerning positive many, which is always proportional: (80)

Many residents of New York know their next-door neighbors.

Given that for the proportional reading of the comparative of many we need to assume a proportional Meas function, it is clear that the proportional interpretation of the positive can be derived using a proportional Meas. The question that arises is whether the proportional reading, which is the only possible one in (80) earlier, can also be derived via the cardinal Measure, which is also needed for more in strong contexts (as shown by the cardinal reading paraphrased in (69)). We suggest that it can, via a certain way of establishing the standard involved in the interpretation of the positive degree. The positive degree states that the degree is above a neutral range of the scale, which is that region of the scale for which neither the positive nor the positive form of the antonym holds (e.g., between short and tall men, there is a region characterizable as ‘neither short nor tall’).22 The neutral range is established contextually, taking into account various factors – for quality adjectives, the NP property is usually relevant, e.g., tall for children depends on their age, big for a dog is different from big for a mouse. For many, in weak contexts, we take into account expectations concerning the presence of that type of objects in the situation, e.g., how many flowers are in average gardens, for (81), or how many books people like John may write, in (81): (81)

a. There were many flowers in the garden. b. John wrote many books.

It is reasonable to assume that in strong contexts with overt or covert partitivity, the neutral range of the cardinality scale is established relative to the cardinality of the whole: for an example such as (82), we establish the neutral range based on the total number of Scandinavians (which is the topic of the sentence): (82)

Many Scandinavians have blue eyes.

Since establishing the neutral range based on the measure of the whole should probably involve an average ratio (possibly around 1 : 3, as suggested by Solt 22

Cf. the semantic entry for POS proposed by von Stechow (2005) and adopted by Solt (2009), among others: ½½POS ¼ λD ð8d 2 LC Þ DðdÞ, where LC is a contextually established neutral range

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2009), in practice this procedure will have the same effect as using a proportional scale.23 In sum, the proportional reading of the positive form many is due to the fact that the neutral range necessary for the interpretation of POS is established relative to the measure of the whole, regardless as to whether we use proportional or cardinal Meas functions. For the comparative form more, the two types of Meas functions yield truth-conditionally distinct readings. The standard involved in the interpretation of POS can also be overtly expressed, using phrases like compared to. Note that, in such cases, the compared degrees can be proportions, which supports our conclusion that the scale introduced by Meas can be proportional: (83)

a. Few Americans speak Finnish, as compared to the number of Europeans who do. b. Few Americans speak Finnish, as compared to French. c. Compared to what we predicted, few Americans speak Finnish. d. Compared to the situation fifty years ago, today few Americans speak Finnish. (Solt 2009, p. 228, ch. 4 ex. 113)

An apparent residual cardinality meaning in proportional many is that, besides the condition of being a sufficiently high proportion, it also requires the cardinality not to be too small (cf. Solt 2009, p. 172). Thus, if the overall number is small, many cannot be used even if the proportion of the discoursegiven set that satisfies the predicate is very big, whereas most can be used in such a situation (cf. Solt 2009, p. 172):24 (84)

[context: there are only three students in my seminar this year, two come from Brazil] #Many/Most students in my seminar come from Brazil.

On our analysis, this phenomenon can be accounted for by a constraint on the use of proportions, which requires that the relevant sets should not be too small. 23

24

Solt (2017) proposes another way in which the proportional reading of positive many in strong contexts can be achieved based on the cardinality scale: (overt or covert) partitivity would trigger the use of a domain-restricted measure function – a measure function whose domain extends from the bottom of the scale up to the measure of the whole; this would ensure that the standard interval used for the interpretation of the positive belongs to the bounded segment of the scale (which has the measure of the whole as its upper bound). It is not clear to us whether domain restriction is sufficient to yield the proportionality effect: if the cardinality of the whole is very large, e.g., 327 millions, that is, the population of the United States, using a neutral range around 1,000, let’s say 500–1,500, is perfectly compatible with domain restriction, predicting that Many Americans have Huntington’s disease is true if 1,900 Americans suffer from this conditions, contrary to intuitions. A similar argument has been put forth by Büring (1996) for the so-called ‘reverse’ proportional readings.

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1.5

Conclusions

The negative result of our investigation is that proportional many does not sit in a Det position, and, correlatively, it does not denote a function from sets into sets of sets. Proportional many sits in the Spec of MeasP, exactly the same position occupied by cardinal many, and its semantics is basically the same: a scalar quantity predicate. The difference between the proportional and cardinal readings of many DPs is due to the partitivity induced by certain syntactic contexts (which force a choice-functional analysis combined with topicality or specificity): When many DPs are (covert or overt) partitives, the neutral range of POS is obligatorily established relative to the measure of the whole to which the referent of the many DP belongs. This analysis was implemented by making use of the general tools of degree semantics, which has led us to consider not only the positive, but also the comparative forms of MANY. For those we have argued that proportional Measure functions are needed, in addition to cardinality Measure functions. For the positive MANY, the two types of Measure functions yield truth-conditionally identical readings, which, in intuitive terms, are ‘proportional’, i.e., say that the measure of the referent of the many DP constitutes a large proportion out of the measure of the maximal sum in the NP. Arguably, the difference between proportional and cardinal Meas functions is read off two distinct syntactic configurations, depending on whether the head of MeasP is an abstract noun NUMBER (Kayne 2005) or PROPORTION.25 In intuitive terms, this amounts to saying that MANY is ambiguous between meaning ‘a large number’ or ‘a large proportion’, but this ambiguity is not due to the existence of two distinct lexical items, MANYcard and MANYprop. Rather it is due to the two abstract meanings that the null abstract head of MeasP can bear, either NUMBER or PROPORTION. REFERENCES Beyssade, C., and C. Dobrovie-Sorin (2005). A syntax-based analysis of predication. In E. Georgala and J. Howell (eds.), Proceedings of SALT 2004, pp. 44–61. Ithaca, NY: Cornell University Press. Bhatt, Rajesh (2002). The raising analysis of relative clauses: Evidence from adjectival modification. Natural Language Semantics 10: 43–90. Bianchi, Valentina (1999). Consequences of Antisymmetry: Headed Relative Clauses. Berlin: Mouton de Gruyter.

25

This does not imply that the lexical noun number necessarily refers to cardinality. ‘A large/ small number’ is equivalent to many and can also have a proportional interpretation: Compared to Americans, a large number of Romanians are practicing Christians.

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Bobaljik, Jonathan (2012). Universals in Comparative Morphology. Cambridge: MIT Press. Braşoveanu, A., and D. Farkas (2011). How indefinites choose their scope. Linguistics and Philosophy 34: 1–55. Büring, Daniel (1996). A weak theory of strong readings. In Teresa Galloway and Justin Spence (eds.), Proceedings of SALT vi, pp. 17–34. Ithaca, NY: Cornell University, http://journals.linguisticsociety.org/proceedings/index.php /SALT/ article/view/2779/2519. Carlson, Gregory N. (1977). A unified analysis of the English bare plural. Linguistics and Philosophy 1: 413–457. Chierchia, Gennaro (1998). Reference to kinds across languages. Natural Language Semantics 6.4: 339–405. (2001). A puzzle about indefinites. In Carlo Cecchetto, Gennaro Chierchia, and Maria Teresa Guasti (eds.), Semantic Interfaces, pp. 51–90. Stanford, CA: CSLI. Cohen, Ariel (2001). Relative readings of many, often and generics. Natural Language Semantics 69: 41–67. Cohen, Ariel, and Nomi Erteschik-Shir (2002). Topic, focus and the interpretation of bare plurals. Natural Language Semantics 10: 125–165. Coppock, E., and D. Beaver (2011). Sole sisters. In N. Ashton, A. Chereches, and D. Lutz (eds.), Proceedings of the 21st Semantics and Linguistic Theory Conference. New Brunswick, NJ: Rutgers University Press. (2014). A superlative argument for a minimal theory of definiteness. In Proceedings of SALT 24, pp. 177–196. Ithaca, NY: Cornell University. Croitor, Blanca, and Ion Giurgea (2013). Relative superlatives and Deg-raising. Acta Linguistica Hungarica 63.4: 1–32. Diesing, Molly (1992). Indefinites. Cambridge, MA: MIT Press. Dobrovie-Sorin, C. (1990). Clitic doubling, wh-movement and quantification in Romanian. Linguistic Inquiry 21.3: 351–397. (1993). The Syntax of Romanian: Comparative Studies in Romance. Berlin: Mouton de Gruyter. (1995). On the denotation and scope of indefinites. Venice Working Papers in Linguistics 5: 67–114. (1997a). Types of predicates and the representation of existential readings. In A. Lawson (ed.), SALT vii, pp. 117–134. Ithaca, NY: Cornell University Press. (1997b). Classes de prédicats, distribution des indéfinis et la distinction thétiquecatégorique. Le gré des langues 12: 58–97. (2009). Existential bare plurals: From properties back to entities. Lingua 119.2: 296–313. (2013). Most: The view from mass. In Maria Aloni, Michael Franke, and Floris Roelofsen (eds.), Proceedings of the 19th Amsterdam Colloquium (AC), pp. 99–107. Berlin: Springer. (2014). Collective quantification and the homogeneity constraint. In Proceedings of SALT 2014, pp. 453–472. Dobrovie-Sorin, C., and C. Beyssade (2003). Définir les indéfinis. Paris: CNRS Editions. (2011). Redefining Indefinites. Berlin: Springer. Dobrovie-Sorin, C., and Ion Giurgea (2015). Weak reference and property denotation. Two types of pseudo-incorporated bare nominals. In Olga Borik and Berit Gehkre

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(eds.), The Syntax and Semantics of Pseudo-Incorporation, pp. 88–125. Leiden and Boston, MA: Brill. (forthcoming). Quantity Superlatives and Proportional Quantification. A Crosslinguistic Analysis of “Most.” Oxford: Oxford University Press. Enç, M. (1991). The semantics of specificity. Linguistic Inquiry 22.1: 1–25. Erteschik-Shir, M. (1997). The Dynamics of Focus Structure. Cambridge: Cambridge University Press. (2007). Information Structure: The Syntax–Discourse Interface. Oxford: Oxford University Press. Farkas, D. (1981). Quantifier scope and syntactic islands. In R. Hendrik et al. (eds.), Papers from the Seventh Regional Meeting, Chicago Linguistic Society (CLS), pp. 59–66. Chicago, IL: University of Chicago Press. (1997a). Dependent indefinites. In F. Corblin, D. Godard, and J.-M. Marandin (eds.), Empirical Issues in Formal Syntax and Semantics. Selected Papers from the Colloque de Syntaxe et de Sémantique de Paris (CCSP 1995), pp. 243–268. Bern: Lang. (1997b). Evaluation indices and scope. In A. Szabolcsi (ed.), Ways of Scope Taking, pp. 183–215. Dordrecht: Springer. Farkas, D., and E. Katalin Kiss (2000). On the comparative and absolute readings of superlatives. Natural Language and Linguistic Theory 18: 417–455. Farkas, D., and H. de Swart (2003). The Semantics of Incorporation: From Argument Structure to Discourse Transparency. Stanford, CA: CSLI. Fodor, Janet, and Ivan Sag (1982). Referential and quantificational indefinites. Linguistics and Philosophy 5: 355–398. Frawley, W. (2013). Linguistic Semantics. London and New York, NY: Routledge. Gawron, Jean Mark (1995). Comparatives, superlatives, and resolution. Linguistics and Philosophy 18: 333–380. van Geenhoven, V. (1996). Semantic Incorporation and Indefinite Descriptions: Semantic and Syntactic Aspects of Noun Incorporation in West Greenlandic. PhD dissertation, Tübingen. Published in 1998 by CSLI. Giurgea, Ion (2013). Originea articolului posesiv-genitival al şi evolut, ia sistemului demonstrativelor în română. Bucharest, Editura Muzeului Nat, ional al Literaturii Române. Glasbey, Sheila (1998). Bare plurals, situations and discourse context. In L. Moss, J. Ginzburg, and M. de Rijke (eds.), Logic, Language and Computation 2, pp. 85–105. Stanford, CA: CSLI. Hackl, Martin (2000). Comparative Quantifiers. PhD dissertation, MIT. (2009). On the grammar and processing of proportional quantifiers: Most versus more than half. Natural Language Semantics 17: 63–98. Hartmann, Jutta (2008). Expletives in Existentials. English there and German da. PhD dissertation, University of Utrecht. Heim, Irene (1985). Notes on comparatives and related matters. Ms., University of Texas at Austin. (1999). Notes on superlatives. Ms., MIT. (2000). Degree Operators and Scope. Proceedings of Semantics and Linguistic Theory (SALT) x. Ithaca, NY: CLC Publications. (2006). Little. Proceedings of Semantics and Linguistic Theory (SALT) xvi. Ithaca, NY: CLC Publications.

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2

Quantity Systems and the Count/Mass Distinction Jenny Doetjes

2.1

The Nature of the Count/Mass Distinction

When investigating the linguistic properties of the count/mass distinction, one can do so from two different perspectives.* A first perspective focuses on the meaning of nouns. Whereas some nouns have a count meaning, others do not. Whereas a noun such as sandwich is count, and cannot easily obtain a mass interpretation, a noun like water is normally used with a mass interpretation. The second perspective considers the lexical or syntactic environment of a noun. In English, the use of a plural or an indefinite article will trigger a count interpretation of the noun, while a bare noun without plural marking cannot be a count noun. In this type of environment, a count noun such as sandwich will need to be reinterpreted as a noun with a mass meaning, otherwise the sentence is uninterpretable. Similarly, in a count environment, the mass noun water needs to obtain a count interpretation in order to be licit. Quantity expressions (two, a bit) play an important role in determining whether a context is count or mass.1 (1)

a. John ate sandwiches / a sandwich / two sandwiches. b. #John ate (a bit of ) sandwich.

(2)

a. John drank (a bit of ) water. b. #John drank waters / a water / two waters.

The count/mass distinction is a domain of language that is closely connected to cognitive notions such as individuation and number representation. *

1

I would like to thank Hana Filip and two anonymous reviewers for their thoughtful comments on a previous version of this paper. For help with data, I would like to thank in particular Anikó Lipták, Simanique Moody, Yang Yang, Hang Cheng, Yiya Chen and Lisa Cheng. Yiya Chen also provided me with the tones for the Mandarin examples. Several parts of this research were presented during classes on the count/mass distinction in Leiden and in Curitiba (Brazil), at the Countability Workshop (Düsseldorf, 2013), the Conference on Referentiality (Curitiba, 2013) and the 11th International Symposium of Language, Logic and Cognition (Riga, 2015). I would like to thank the audiences for their input. Thanks also to Johan Rooryck for inspiring discussions on core knowledge systems. All the usual disclaimers apply. The following abbreviations are used: cl: classifier; exp: experiential aspect; prt: particle; perf: perfective; pl: plural; prs: present; sg: singular.

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Properties of the linguistic count/mass distinction may therefore shed light on the relation between language and cognition. A question that is often asked in relation to the count/mass distinction is whether there are languages with or without a lexical count/mass distinction. What does it mean for a language to only have count nouns? Or only mass nouns? And what is the difference between count and mass nouns in the first place? An example of a language that has been claimed to have only mass nouns is the numeral classifier language Mandarin. In numeral classifier languages, the syntax of nouns with typical count meanings (pen, apple, dog etc.) usually resembles the syntax for mass nouns, and numeral classifiers resemble measure words in a language such as English, as illustrated by the contrast between (3) and (4). (3)

a. sān three b. sān three

*(gè) cl gen *(jīn) cl half_kilo

píngguǒ apple mǐ rice

(4)

a. three apples b. three *(kilos of ) rice

[Mandarin]

Based on these types of examples, one may want to conclude that all nouns in Mandarin are mass (see for instance Chierchia 1998a, 1998b, Denny 1986, Li 2013, Ojeda 1993; note that these authors have different perspectives on what it means for a noun to be mass). On the other hand, in Yudja, all nouns combine directly with numerals, without number marking and without use of measure words or numeral classifiers (Lima 2010, 2014), leading Lima to the conclusion that in this language all nouns have count properties: (5)

a. txabïu ali [Yudja] three child ‘three children’ b. txabïu y’a three water ‘three containers/drops/puddles etc. of water’

It is clear that in Mandarin, English and Yudja the types of strategies for combining numerals with nouns are different. However, what should be concluded on the basis of this in terms of the realization of the count/mass distinction in these languages? Do we want to assume that there are mass only (Mandarin), count only (Yudja) and count/mass languages (English)? In this paper, I will approach these questions on the basis of cross-linguistic properties of quantity expressions. As indicated above, quantity expressions play an important role in creating count and mass environments across

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languages, as illustrated in in (1) and (2) above for English. In what follows, I will take the term quantity expression to cover expressions such as numerals, many, several and a lot, but also number markers, measure words and numeral classifiers. All these expressions are related to the expression of quantity, and their distributions influence each other within the language they occur in. I assume that quantity expressions in a particular language form a system in a Saussurian sense. Linguistic variation is due to the differences between the precise lexical inventory of different languages and the language-specific properties of these items. However, quantity systems also show parallels cross-linguistically. In particular, I will argue that quantity expressions can be subdivided into expressions that presuppose countability and ones that do not. This distinction can be seen as a linguistic reflection of the cognitive distinction between number representation on the one hand (Dehaene 1997, Feigenson et al. 2004) and global quantity representation on the other (Lourenco and Longo 2011). The properties of these two types of expressions are tightly connected to the count/ mass distinction and make predictions on the possibilities of ‘count only’ and/ or ‘mass only’ languages; whereas a language in which nouns only have count meanings might exist (as count meanings are compatible with both basic types of quantity expressions), mass only languages are predicted to be impossible. In Section 2.2, I will first introduce the notion of quantity system. After that, I will turn to parallels between quantity systems and I will defend the claim that there exists a fundamental linguistic distinction between count quantity expressions, which only combine with nouns that have count meaning, and global or non-count quantity expressions, which are indifferent with respect to the count/mass status of the meanings of the nouns they combine with (where I take a semantic approach to the count/mass status of nouns). Section 2.3 turns to the interaction between quantity expressions and count and mass meanings of nouns, focusing on the consequences of the existence of the two types of quantity expressions for the types of meanings we expect to occur crosslinguistically. Section 2.4 further investigates the interaction between quantity expressions and noun meaning, based on the idea that the presence of count quantity expressions may trigger an effect of “individuation boosting”. Section 2.5 summarizes the main conclusions. 2.2

Quantity Expressions across Languages

2.2.1

The Notion of “Quantity System”

In what follows, I assume that quantity expressions in a given language form a type of system in a Saussurian sense of the term. Saussure (1916) compares the properties of a language system to those of a chess game in order to explain

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that certain properties of language are internal to the linguistic system, while others are not. Certain properties of a chess game are irrelevant for the system of chess, for instance, from which material the pieces are made, what their exact form is like and where the game originates. However, the number of pieces and the number of checkers, as well as the rules concerning the ways in which the pieces can or cannot move and other rules of the game, are part of the system. Changes in these domains will change “the grammar” of the game, as Saussure puts it (p. 44). When looking at quantity expressions in a given language, one can observe that the conditions they impose interact with each other. Languages may have plural markers or not, and if a language has a plural marker, this marker will have language particular properties, which may affect the way other lexical items in the system function. For instance, in English, number marking is obligatory and interacts with the distribution of other quantity expressions as illustrated in in (1) and (2). Numerals such as two only combine with plural nouns. Nouns such as furniture, which have a countable meaning but which lack a singular–plural opposition, cannot combine with numerals (I will come back to this below). Differences in lexical inventory seem to be at the basis of the large amount of variation in the count/mass distinction, including the three types of strategies illustrated in (3)–(5) for the ways in which nouns combine with numerals (classifier insertion, number marking and direct combination of the numeral and the noun) and many variants where a combination of these strategies is used (see Doetjes 2012 for examples). There is evidence that properties of individual lexical items play an important role in this variation. A nice example illustrating this is the distribution of classifiers in the Mayan language Chol (Bale and Coon 2014). Chol disposes of two sets of numerals, one of which requires the insertion of a classifier, while numerals of the other class directly combine with nouns, showing that the two types of numerals have different lexical properties and interact with classifiers. Despite the differences that can be observed, quantity systems will be argued to have properties that encompass these differences and that I consider to be at the basis of the count/mass distinction. In the next sections, I will argue that quantity expressions (including classifiers, measure expressions and number markers) can be categorized in two basic classes, depending on whether they presuppose countability. 2.2.2

Count, Non-Count and Anti-Count Quantity Expressions

When looking at quantity expressions in English, a clear distinction can be made between three types of expressions: ones that only combine with count nouns, ones that combine with mass nouns and count plurals, and ones that

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only combine with mass nouns. Examples of the three types of expressions are given in (6): (6)

a. John ate three apples/*soup. b. John ate a lot of apples/soup. c. John ate a bit of soup/*apples.

This results in the following classification of quantity expressions (cf. Chierchia 1998a): A. expressions that presuppose the availability of units that can be counted (count quantity expressions); B. expressions that do not presuppose the availability of units that can be counted (global quantity expressions or non-count quantity expressions); C. expressions that presuppose the absence of units that can be counted (‘anti-count’ quantity expressions). Below, I will present cross-linguistic evidence in favor of the idea that the first two types can be found across languages, independently of the strategy a language chooses for combining nouns and numerals.2 Moreover, I will argue that the third type (‘anti-count’) has a different status, and should be seen as a subclass of the non-count quantity expressions. The basic distinction between expressions that presuppose the availability of units that can be counted and an evaluation of quantity in terms of number, and expressions that do not introduce such a presupposition is in my view related to the cognitive distinction between numerical magnitude representation and general magnitude representation (Doetjes 2017b). While the cognitive representation of numerical magnitude involves distinct, individuated units that can be counted (Dehaene 1997; Feigenson et al. 2004), general magnitude representations may depend on different measurement scales, such as weight and volume (Lourenco and Longo 2011). If this is right, the linguistic distinction between the two basic types of quantity expressions is rooted in a cognitive distinction. At the same time, language permits abstractions that make the linguistic distinction different from the cognitive distinction that is at the basis of it. I will come back to this at various points, and in particular in Section 2.3.3. In the next section, the different types of quantity expressions will be illustrated on the basis of data from a number of unrelated languages. Before that, I will briefly go over their defining properties on the basis of English examples. 2

The distinction that I make between count and non-count quantity expressions is different from the difference between counting and measuring in Rothstein (2009, 2011). I will come back to this in Section 2.3.3.

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Count quantity expressions are defined by the property that they presuppose the availability of units that can be counted, which are normally provided by the noun they combine with, which must be compatible with a count interpretation. In a language like English, which has obligatory number marking, count quantity expressions combine with plural count nouns, and they are incompatible with mass nouns, unless a measure term is added: (7)

three/several books/*(kilos of ) sand

Expressions such as three and several presuppose that the noun they combine with introduces units that can be counted and otherwise cannot be interpreted.3 Besides the numerals and vague expressions such as several, various and (how) many, several other quantity expressions presuppose countability of the noun they combine with. Examples are quantity expressions that include the word number (a number of) and distributive universal quantifiers (each, every).4 I also include the plural marker in this category, and I will argue below that the grammatical status of the plural marker has important effects on the ways in which quantity expressions in English interact with nouns and with each other. Traditionally, the defining property of predicates that have a count denotation is atomicity (cf. Krifka 1992): (8)

a. 8x, P½ATOM ðx; PÞ $ PðxÞ & ¬9y½y < x & PðyÞ (x is a P-atom) b. 8P½ATM ðPÞ $ 8x½PðxÞ ! 9y½y  x & ATOM ðy; PÞ (the predicate P has atomic reference)

According to Chierchia (1998a), all nouns have atomic reference. However, he also makes a distinction between atoms that are well-defined and atoms that are not, based on the concept of ‘vague’ atoms. Even though in his view a noun like water has atomic reference, these atoms are vague, and it is not clear what the atoms are. On the other hand, in the case of books, and also in the case of furniture, it is clear what the atoms are, namely the individual books and the individual pieces of furniture. In what follows I assume that a count denotation is an atomic denotation with non-vague atoms (see also Doetjes 1997; Deal 2017).5 3

4

5

The precise semantics of the various count quantity expressions will differ depending on the account that is adopted and is beyond the scope of this paper. Independently of the approach that is taken, their semantics will need to restrict them to an environment that provides units that can be counted. Non-distributive universal quantifiers are usually indifferent with respect to the count/mass distinction. In the rest of this paper, I will not consider universal quantifiers, for reasons of space. A known problem of atomicity is that some nouns that have clearly countable interpretations and that occur freely with numerals are not atomic (Nicolas 2004; Rothstein 2010b; Wiggins 1980; Zucchi and White 2001). Examples are twig, thing, fence and bunch. This problem is accounted for in the literature in various ways, such as context-dependency (Rothstein 2010b) and, within a mereological approach, a connectedness condition (Grimm 2012).

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The incompatibility of furniture and count quantity expressions will be related to the particular properties of plural marking in English. Whereas some nouns are ‘naturally atomic’ in the sense that their primary denotation is count (e.g., bicycle), other nouns, often called ‘notional mass nouns’, typically have a mass meaning. However, this does not mean that they cannot have a count interpretation. Take, for instance, the examples in Yudja in (5), where the noun y’a ‘water’ is used to denote puddles or other ‘individualized’ portions of water.6 This is not possible in English, where three waters would rather refer to three types of water or three glasses of water in a restaurant. This illustrates the language-specific component in the availability of count meanings for notional mass nouns. The noun water has a denotation with vague atoms, while the atoms of the corresponding noun y’a in Yudja are not vague and correspond to contextually defined portions of water.7 What is similar across languages, however, is that count meaning plays a role in the linguistic system and interacts with quantity expressions. I will also assume that measure words introduce countable units. Measure nouns such as kilo and liter also have a count, atomic denotation and behave as count nouns in many languages. If a numeral combines with kilos, the numeral counts the number of kilos, even if the boundaries between the kilos are usually arbitrary. One might argue that in this case the atoms are ‘vague’, but contrary to the vague atoms Chierchia discusses in the context of nouns such as water, the expression kilo makes clear what can count as an atom and what not, namely quantities that have a weight of one kilo. Similarly, in two kilos of sand we may not be able to distinguish two different objects, but still we know that the amount of sand we are talking about can be partitioned in two non-overlapping quantities of one kilo of sand. The assumption that measure nouns can have a count denotation has consequences for the mapping between language and cognition: kilos do not constitute discrete entities, and yet, in linguistic structures, they may be used as entities that we can count (see Section 2.3.3 below). Contrary to count quantity expressions, global or non-count quantity expressions are compatible with nouns that have a count interpretation and with nouns that have a mass interpretation. As such, they do not presuppose countability. Examples of non-count quantity expressions in English are given in (9). In strict number marking languages such as English, these expressions typically combine with mass nouns and plurals: (9) 6

7

more/less/a lot of books/sand

Lima (2014, p. 118) assumes that the function that maps kinds to countable objects can apply to notional mass nouns because “the grammar of the language allows its speakers to treat concrete portions of a kind as atoms.” Note that this difference is not very relevant in the absence of quantity expressions.

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More, less and a lot in (9) are degree expressions. In languages such as English, these expressions usually cannot be used with a singular count noun, unless there is a shift to a mass meaning (an exception being no; no book(s)/sand): (10)

#more/less/a lot of book

As argued by Link (1983), mass nouns and plurals are both characterized by cumulative reference, as defined in (11) (Krifka 1992): (11)

8P ðCUM ðPÞ $ 8x, y½PðxÞ & PðyÞ ! Pðx∨yÞ & 9x, y½PðxÞ & PðyÞ & x 6¼ y

I assume that quantity expressions such as a lot are sensitive to cumulative reference only, which makes them compatible with mass nouns and count plurals. Note that it is not the case that expressions that indicate a degreerelated quantity need to fall in this category, as illustrated by English many, a few and fewer. However, if a quantity expression with a degree-related meaning can also be used outside of the nominal system as a degree expression, it never presupposes that the noun it combines with has a count meaning (Doetjes 1997, p. 172); an expression such as fewer, which presupposes count meaning, cannot function as a degree modifier while less can (cf. *fewer/less complicated, to complain *fewer/less). Measure words constitute another category of expressions that are often insensitive to the mass/count properties of the noun they modify, as illustrated in (12): (12)

two pots/kilos of honey/olives

Even though the numeral two counts pots and kilos in this example, the quantity defined by the pots or kilos can be countable (olives) or non-countable (honey). Measure words do not always behave as non-count quantity expressions. Whereas most are compatible with plurals and mass nouns as illustrated in (12) (and thus constitute non-count quantity expressions), some are restricted to plurals (e.g., dozen) and others to mass nouns (e.g., drop). Measure words do not occur with singular count nouns, suggesting that they necessarily combine with expressions that have cumulative reference, irrespectively of the type of quantity expression they belong to.8 To conclude, in English non-count quantity expressions usually combine with mass nouns and plurals, that is, they combine with expressions that have cumulative reference without imposing further restrictions on these nouns.9 The third type of quantity expressions that I will consider are ‘anti-count’ quantity expressions. These expressions are very similar to the non-count 8 9

Lehrer (1986) observes that measure words (‘quantity related classifiers’ in her terminology) differ in this respect from ‘varietal classifiers’ such as kind and type, as illustrated by a type of pen. Most English quantity expressions, whether count or non-count, can also be used in partitive structures (a lot of these books). A discussion of partitives is beyond the scope of this paper.

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quantity expressions, but contrary to non-count quantity expressions they seem to be incompatible with count nouns altogether. An example is English a bit, as illustrated in (6c) above. If one wants to state that John read a small number of books, one has to use the count quantity expression a few, as in a few books. A similar distinction exists between much and many. A possible way of accounting for anti-count quantity expressions is to assume that these expressions presuppose the absence of countability, and thus the lack of proper, identifiable atoms in the denotation of the nouns they combine with. This would mean that they could only combine with nouns that have divisive reference (Cheng 1973), the downward counterpart of cumulative reference, which implies the absence of atomic parts.10 Under this analysis, they would be the counterparts of count quantity expressions, as both types would introduce a presupposition related to the count/mass status of the nouns they may combine with. However, I would like to hypothesize that the restriction to mass environments has a different explanation, and that anti-count quantity expressions form a subclass of the non-count quantity expressions. The restrictions that are found are due not to selection, but to blocking. In the literature, the distinction between many and much is often seen as an instance of blocking or the elsewhere condition (Di Sciullo and Williams 1987). Under a blocking account, much is not inherently incompatible with count nouns. However, the existence of many, which is incompatible with mass nouns and therefore more specific, blocks much from being used in a count context. In Doetjes (1997), I extended this analysis to the expression a bit, which is replaced by the more restricted expression a few in the context of a count noun.11 It is important to stress that the blocking effect must be due to a pairing of specific expressions in the lexicon. Whereas much gets replaced by many when the noun it combines with is a plural, this does not apply to a lot, which is compatible with both mass nouns and plurals, as illustrated in (9) above.12 Despite the fact that it is hard to predict which elements get blocked and which do not, there are several arguments of treating expressions such as a bit as a special case of non-count expressions rather than as a separate subtype of expressions that presuppose divisivity (cf. also Chierchia 1998a).

10 11 12

See Deal (2017) for a definition of divisivity in line with the assumptions on count meaning in terms of non-vague atoms made above. This could also be stated in terms of a general principle such as Maximize Presupposition (Heim 1991; Sauerland 2003). There is evidence that this is a general property of blocking in the domain of quantity expressions. In French, the distribution of tant ‘so much/many’ is blocked by si ‘so’ in the context of adjectives (si/*tant difficile ‘so difficult’). However, tellement ‘so (much/many)’ is not blocked from this context (tellement difficile ‘so difficult’). See Doetjes (1997, p. 112) for independent evidence for an analysis of these facts in terms of blocking.

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In the first place, the distribution of anti-count expressions is very similar to the one of non-count quantity expressions such as more. Both a bit and more are found not only in the context of nouns, but also in the context of verbs, and they can even be used with adjectives, in which case they indicate a degree rather than a quantity (cf. Doetjes 1997, 2008): a bit/more complicated/to complain a bit/more. In the second place, when the expression a bit is modified, it may lose the property of being restricted to mass expressions. In particular, modification by quite facilitates, for some speakers at least, the use of a plural noun, as in % quite a bit of books. In the third place, research on the acquisition of many and much follows the pattern that is expected under a blocking analysis. Gathercole (1985) shows that children first use much with both count and mass nouns, that is, they use it as if it were a non-count rather than an anti-count expression. Moreover, they start to use it correctly only when they have acquired a correct use of many, which is acquired later than much. This is very similar to the pattern described by Ferdinand (1996) for the acquisition of the verbal paradigm of agreement morphology in French and to the pattern Pinker (1995) describes for the acquisition of irregular verb forms in English, two phenomena for which a blocking type of analysis is commonly assumed. To conclude, there seems to be an asymmetry between count quantity expressions and anti-count quantity expressions. Anti-count expressions do not behave as counterparts of count quantity expressions, but rather seem to be a subcase of non-count quantity expressions. This asymmetry can be accounted for by the hypothesis that the distribution of anti-count quantity expressions is due to a blocking effect, which is triggered by the existence of an equivalent count quantity expression with which it is strongly associated. The special status of anti-count quantity expressions may be a reason why they are cross-linguistically rare compared to the two other types (Chierchia 1998a). In the following sections, I will discuss examples of count, non-count and anti-count quantity expressions in several languages. After this, I will turn back to the notion of quantity systems.13 2.2.3

Count Quantity Expressions across Languages

Numerals are count quantity expressions by definition, as their meaning presupposes the availability of units that can be counted. If a language has 13

Note that I will restrict myself to languages in which quantity expressions form part of a noun phrase. This is clearly not always the case cross-linguistically (see Bach et al. 2013). I expect that the basic semantic distinction between count and non-count quantity expressions also applies to quantity expressions with a different type of syntax/semantics (see Wilhelm 2008 for data that point in this direction).

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numerals, this implies that it has expressions that can only combine with nouns that have a count meaning. As shown in the discussion above, the ways in which numerals interact with nouns vary across languages. What is constant, however, is the type of meaning they require, which needs to be countable. Not all languages have a fully developed system of cardinal numerals (cf. for instance Wiese 2003), but it is quite plausible that all languages have quantity expressions that presuppose countability. Even languages that have very restrictive numeral systems – e.g., Mundurucu (Pica et al. 2004) – seem to have quantity expressions that resemble numerals or expressions such as several in the sense that they presuppose the availability of units that can be counted. Next to numerals, languages often also have expressions that presuppose countability that do not indicate a precise number. English examples are several, various, different, (a) few and many, and the existence of this type of quantity expression turns out not to be restricted to languages with obligatory plural marking in the context of numerals. A first example is the numeral classifier language Mandarin. In Mandarin, vague quantity expressions may have a distribution that is similar to that of numerals, including jǐ ‘how many, a few’. These expressions are usually assumed to occupy the same structural position as numerals (see for instance Hsieh 2008, p. 62; Zhang 2013, p. 85). On a par with numerals, these expressions obligatorily trigger the insertion of a classifier when combined with a noun, and as such they are restricted to count environments; this is illustrated by the examples in (13) (see Zhang 2012, p. 43). I will turn to classifiers and their relation to the meaning of the noun below.14 (13)

a. Nǐ yǒu jǐ *(duǒ) huā? [Mandarin] you have how.many cl flower ‘How many flowers do you have?’ b. Yīgòng yǒu jǐ *(shēng) yoú? total have how.many cl liter oil ‘How many liters of oil are there in total?’

A perhaps more surprising example of a quantity expression that presupposes countability is the expression dàduōshù ‘most’ (lit. ‘bigger number’). This quantity expression directly modifies the noun. It is optionally followed by the marker de, which marks nominal modifiers, and cannot be combined with a classifier. According to Zhang (2013, p. 139), it is incompatible with mass nouns such as mò-shuǐ ‘ink’, a property that can be traced back to the presence of the morpheme shù ‘number’ (for some speakers I consulted, this restriction does not hold; I leave this as an issue for further research): 14

Example (13a) is also given in Zhang (2013, p. 86); example (13b) has been slightly adapted.

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Quantity Systems and the Count/Mass Distinction (14)

a. dàduōshù shū diào-dào dì-shàng Most book fall-to ground-on ‘Most of the books fell on the ground.’ b. *dàduōshù mò-shuǐ liú-dào dì-shàng most ink-water flow-to ground-on ‘Most of the ink flew to the ground.’

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le. prt

[Mandarin]

le. prt

Turning now to a typologically different language, again we can find expressions that are restricted to count environments. Hungarian hány ‘how many’ (Schvarcz and Rothstein 2017) is in this respect similar to Mandarin jǐ ‘some, how many’ and English how many, even though the strategy that is used to combine the quantity expression and the noun differs. In Hungarian, number marking is obligatory in definite noun phrases but absent in the context of numerals and other quantity expressions, which I take to be an indication of the number-neutral nature of these nouns (Farkas and de Swart 2010).15 An example illustrating the distribution of hány ‘how many’ is given in in (15):16 (15)

a. Hány könyv how.many book ‘How many books b. *Hány por how.many dust

áll a polcon? stand.3sg the shelf.on are on the shelf?’ áll a polcon? stand.3sg the shelf.on

[Hungarian]

The impossibility of combining hány with notional mass nouns shows that Hungarian differs from Yudja. The strategy of directly combining nouns and quantity expressions does not imply the freedom of count meanings available in Yudja (cf. Wilhelm 2008). Next to the quantity expressions discussed above, number markers and certain classifiers are themselves also count quantity expressions, as they presuppose a count interpretation of the noun they combine with. Let us first turn to count selecting classifiers. There are two types of classifiers that are only compatible with nouns that have a count interpretation, and thus can be said to presuppose countability: sortal classifiers and group classifiers. An example of a sortal classifier in Mandarin is given in (16): (16)

sān zhī bǐ three cl branch pen ‘three pens’

[Mandarin]

It has been argued in the literature that sortal classifiers introduce count meaning and that in the absence of the classifier the noun is semantically mass (see for instance Denny 1986; Ojeda 1993). However, if this is taken in a literal 15 16

Hungarian disposes of one sortal classifier, which is optionally inserted with numerals and count quantity expressions, see Schvarcz and Rothstein (2017). I would like to thank Anikó Lipták for providing these examples.

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sense, count meaning should be only available in the context of classifiers. This is not the case, as bare nouns may have count meanings in the absence of a classifier. Moreover, nouns that are compatible with sortal classifiers always can have a count, atomic interpretation in the absence of the classifier (see also (14) and (17)). Nowadays, it is commonly assumed that nouns that are compatible with sortal classifier need to have a count meaning independently of the presence of the classifier (Chao 1968; Cheng and Sybesma 1998, 1999, 2005; Doetjes 1997, 2012, 2017c; Grinevald 2005; Li et al. 2009; Zhang 2013). Besides sortal classifiers, group or plural classifiers also presuppose the presence of units that can be counted in the denotation of the noun they combine with. Consider for instance the examples in (17) (Doetjes 1997, p. 34): (17)

yī dá/qún bái-mǎ one cl dozen/herd white horse ‘a dozen/herd (of ) white horses’

[Mandarin]

Contrary to numerals, these expressions are incompatible with sortal classifiers. As in the case of sortal classifiers, group classifiers are incompatible with nouns that have a mass interpretation. A final class of expressions that presuppose the existence of units that can be counted is the class of number markers. Number markers show up in very different ways across languages. English number marking is obligatory and plays a role in the selection properties of numerals, and English is often seen as the prototypical case of a number marking language.17 As in the case of classifiers, number markers show a large amount of cross-linguistic variation (see, among many others, Cabredo Hofherr forthcoming; Corbett 2000; Dryer 2005; Wiltschko 2008). Both classifiers and number markers seem to be freer in the types of restrictions that they may impose on the expressions they combine with. Besides presupposing count meaning, number markers may also be sensitive to animacy, for instance. In numeral classifier languages, it is usually the case that different nouns combine with different sortal classifiers. To sum up, count quantity expressions are commonly found across languages (in particular numerals, expressions such as several and how many,

17

It is actually not clear how frequent the English type of system is. WALS classifies 133 languages out of a sample of 291 as languages that have always-obligatory number marking on all (count) nouns. However, among these 133 languages many are very different from English. The Austronesian numeral classifier language Mokilese, for instance, falls in this class. In this language number marking on determiners and demonstratives is (mostly) obligatory, but there is no number marking on nouns (Harrison and Albert 1976). In a different vein, the distribution of number markers in languages with obligatory number marking on nouns is not always the same as in English (see for instance Hungarian; Farkas and de Swart 2010).

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sortal classifiers and plural markers). Languages differ in the number and types they make use of and the ways in which they interact with each other, but the general property of being sensitive to the availability of count meaning may well be a basic classificatory property of quantity expressions across languages. In Section 2.2.5 I will turn back to the interaction between quantity expressions, but before that I will first discuss cross-linguistic occurrences of non-count and anti-count quantity expressions. 2.2.4

Non-Count and Anti-Count Quantity Expressions across Languages

The core property of non-count quantity expressions is that they presuppose neither the availability of units that can be counted nor the absence thereof. Again, it seems plausible that expressions that belong to this type are crosslinguistically widespread or even universal. Degree-related quantity expressions often do not presuppose the availability of units to count or the absence thereof. Examples for English are given in (9), repeated in (18): (18)

more/less/a lot of books/sand

As indicated above, expressions of this type in English are claimed to be sensitive to cumulative reference of the expression they combine with. In the context of this type of quantity expression, the presence or absence of a count interpretation depends on the meaning of the noun (see in particular Barner and Snedeker 2005). The conditions on the use of degree expressions depend partly on the distribution and function of numeral classifiers and number marking in a language. In English, where marking of plural is obligatory and where the singular form is usually assumed to have a singular meaning (see for instance Chierchia 1998a, Link 1983), these expressions typically combine with plural count nouns and with mass nouns while being incompatible with count singulars. Deal (2017) claims that the Sahaptian language Nez Perce exhibits a similar pattern for a large class of quantity expressions. At first sight, Nez Perce is very different from English. All nouns in Nez Perce are permitted to be used in count contexts, as in Yudja, and it seems as if all quantity expressions freely combine with all nouns. However, Deal convincingly argues that this view is too simple. Whereas numerals always trigger a count interpretation of the noun, other quantity expressions do not. Moreover, when notional count nouns are used, plural marking on the noun (for human nouns) and/or on the adjective (in other cases) is obligatory in the context of a count interpretation, as illustrated for the quantity expression ’ilexˆ ni ‘a lot’ in (19) (Deal 2017, pp. 149–150):

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(19)

a. ’ilexˆ ni ha-ham/*haama [Nez Perce] a.lot pl-man/*man.sg ‘a lot of men’ b. ’ilexˆ ni ??tiyaaw’ic/ ti-tiyaw’ic wixˆ si’likeecet’es a.lot ??sturdy/ pl-sturdy chair ‘a lot of sturdy chairs’

With notional mass nouns, the presence versus the absence of a plural marker corresponds to the presence versus the absence of a count interpretation of the noun (Deal 2017, p. 152): (20)

a. ’ilexˆ ni cimuuxcimux samq’ayn a.lot black fabric ‘a lot of black fabric’ b. ’ilexˆ ni cicmuxcicmux samq’ayn a.lot pl.black fabric ‘a lot of pieces of black fabric’

[Nez Perce]

The resulting pattern is strikingly similar to the one illustrated in (9) and (10) for English: even though the number markers are realized in a different way, plural marking is obligatory in the context of nouns with a count meaning. Deal, who also assumes that quantity expressions such as ’ilexˆ ni ‘a lot’ are sensitive to cumulative reference, takes this to be an argument in favor of the idea that there is a singular–plural opposition in Nez Perce that is very similar to the one that is commonly assumed for English: whereas plural marked nouns with a count meaning have cumulative reference, the corresponding singular nouns denote sets of atoms, and are quantized. Under the assumption that quantity expressions such as ’ilexˆ ni ‘a lot’ only combine with expressions that have cumulative reference, it follows that nouns with a count interpretation need to be pluralized in this context. As already indicated above, many languages do not require plural marking or classifier insertion in the context of numerals, even though some of these languages have obligatory plural markers in other contexts. Examples are Yudja and Hungarian. For Hungarian, which clearly allows for mass meanings, non-count expressions can be easily found. Whereas the Hungarian quantity expression hány ‘how many’ in (15) above is only compatible with nouns that have a count interpretation, sok ‘a lot’ and mennyi ‘how many/much’ are insensitive to the count/mass status of the noun they combine with; (21a) is from Farkas and de Swart (2010, p. 11) and (21b–d) from Anikó Lipták (p.c.). (21)

a. Sok gyerek gyűlt össze a a-lot child gather.past prt the ‘Many children gathered in the square.’ b. Sok sár gyűlt össze a a-lot mud gather.past prt the ‘A lot of mud accumulated on the square’

téren. square.on

[Hungarian]

téren. square.on

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Quantity Systems and the Count/Mass Distinction c. Mennyi könyv áll a how.many book stand.3sg the ‘How many books are on the shelf?’ d. Mennyi por áll a how.much dust stand.3sg the ‘How much dust is on the shelf?’

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polcon? shelf.on polcon? shelf.on

For a language such as Yudja, it is harder to show that a distinction between count and non-count quantity expressions exists, given that count quantity expressions combine so easily with notional mass nouns. However, Lima (2014) shows that not all quantity expressions behave in the same way in quantity judgment tasks. More in particular, itxïbï ‘many’ requires an evaluation in terms of number, while bitu ‘more’ also permits an evaluation in terms of volume. This suggests that itxïbï is a count quantity expression, while bitu is non-count and as such similar to English more (see also Section 2.3.1). In classifier languages as well, it is possible to find non-count quantity expressions. These are compatible with all nouns, independently of the type of meaning they have and can be assumed to be sensitive to cumulative reference, under the assumption that the noun has a number-neutral interpretation (Cheng and Sybesma 1999, Rullmann and You 2006). In general, noncount quantity expressions can come in two types, depending on the possibility of inserting a classifier. An example of is hěnduō, which is insensitive to the count/mass status of the noun it combines with, as illustrated in the examples in (22) (Zhang 2013, p. 19):18 (22)

a. Wǒ yǐqían mǎi-guò hěnduō máo-pǐ, [Mandarin] 1sg before buy-exp a.lot brush-pen xiànzài shèngxìa liǎoliǎowújǐ. now remain few ‘I bought many brush-pens before, but few of them remain now.’ b. Wǒ yǐqían mǎi-guò hěnduō mò-shuǐ, 1sg before buy-exp a.lot ink-water *xiànzài shèngxìa liǎoliǎowújǐ. now remain few ‘I bought much ink before, *but few of them remain now.’

Depending on the dialect, Mandarin hěnduō may be followed by a classifier (Cheng et al. 2012; Doetjes 2012), but this is never necessary (see also Section 2.3.4 below). Other non-count quantity expressions never permit insertion of a classifier. These expressions function as modifiers and may be followed by de,

18

In the original gloss, hěnduō is translated as many. Given that hěnduō is a non-count quantity expression, I changed this into ‘a.lot’.

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a marker of nominal modification. As in the case of hěnduō, the noun can be either count or mass (Hsieh 2008). Examples are dàliàng ‘a lot’, suǒyǒu ‘all’, quánbù ‘all’, and dàbùfèn ‘most’ (Zhang 2013, p. 84):19 (23)

a. dàliàng (de) (*duō) a.lot (de) (*cl) ‘a lot of flowers’ b. dàliàng (de) (*píng) a.lot (de) *cl bottle) ‘a lot of water’

huā flower

[Mandarin]

shuǐ water

Mensural classifiers (in classifier languages) and measure words (in non-classifier languages) are often indifferent to the count/mass status of the nouns they combine with. This is illustrated in (12) above for English and in (24) for Mandarin: (24)

a. liǎng jīn mǐ/ píngguǒ [Mandarin] two cl half_kilo rice/ apple ‘two pounds of rice/apples’ b. liǎng píng (de) mì/ gǎnlǎn two cl pot (de) honey/ olive ‘two pots of honey/olives’

Measure constructions and measure terms are not found in all languages (see for instance Lima 2014, p. 22, who states that they are absent in Yudja). In classifier languages, measure words usually surface as classifiers, but not necessarily (Doetjes 2017a). As indicated in Section 2.2.2 above, measure words (whether classifiers or not) typically combine with nouns that have cumulative reference. Next to quantity expressions that are indifferent to the count/mass distinction, there are also quantity expressions that are incompatible with nouns that have a count interpretation, as illustrated by English much and a bit. These expressions, which I called anti-count quantity expressions above, typically combine with mass nouns. As indicated in the previous section, I assume that anti-count quantity expressions should be considered to be a special case of non-count degree expressions and that some kind of blocking process plays a role in limiting the distribution of these expressions to mass contexts. As in the case of the other types of expressions discussed above, anti-count expressions can be found in typologically different languages. Pairs like a bit and a few, describing small quantities, seem to be cross-linguistically quite common, even though there also exist languages where a single expression is 19

The semantics of these expressions is often transparent: dà ‘big’ + liàng ‘quantity’, quán ‘complete’ + bù ‘part’ and dà ‘big’ + bù ‘part’ + fèn ‘fragment’ (Yang Yang, p.c.).

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used (e.g., Breton un nebeud ‘a few, a bit’, Hamon 1984).20 In Mandarin, yī diǎnr belongs to the class of expressions that combine typically with nouns that have a mass interpretation (Iljic 1994). In case a count meaning is present, the expression jǐ ‘a few’ is used, which triggers the obligatory insertion of a classifier.21 Some measure words are also restricted to mass nouns. In general, they impose more restrictions on the noun they combine with. Take, for instance, the example drop or liter. A drop/liter of N can only be used if N denotes a liquid, and as liquids are mass nouns in English, the expression is restricted to mass nouns as well.22 In this respect, the nature of the restriction seems different from the restriction we find for a bit or yī diǎnr, of which I argued that they are in principle compatible with count expressions, but blocked in this context by the existence of a more specific expression that presupposes the existence of units to count. 2.2.5

Back to Quantity Systems

In the preceding sections, I introduced a distinction between count and noncount quantity expressions and argued that this distinction exists crosslinguistically. As indicated, cross-linguistic variation comes from the actual inventory of quantity expressions, which I assume to form systems in a Saussurian sense. Number markers and classifiers are an important source of cross-linguistic variation, as they may have different distributions in different languages. The fact that English has a grammatical and obligatory singular–plural opposition affects the way quantity expressions interact with each other and with nouns.

20

21

The fact that anti-count expressions often indicate a small quantity might be related to the fact that there is a separate system of numerical recognition for small quantity (see for instance Feigenson et al. 2004). I leave this as a question for further research. Zhang (2013, p. 88) challenges the claim that yī diǎnr ‘some, a bit’ is an anti-count quantity expression on the basis of the example in (i). According to her judgments, it is possible to use it in the sense of a few with nouns that have a count interpretation (the use of a classifier being still prohibited): Nàlǐ yǒu (yī)-diǎnr (*kē) xiǎo xīguā. [Mandarin] there have one-dian (cl) small watermelon ‘There are a few small watermelons’ (e.g. in the context of talking about the quantity of the storage in a certain place).

22

The reactions of the Mandarin speakers I consulted suggest that judgements vary: some only could get the reading corresponding to ‘a bit of small watermelon’ (in which case, xiao xigua ‘small watermelon’ is necessarily interpreted as a type of watermelon), while others could also get the reading ‘a few small watermelons’. The same could actually be said about many group classifiers: the term flock can only be used in the context of certain types of animals.

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Count quantity expressions in English are also sensitive to the presence of plural marker. In this respect, English plural markers interact with the selection properties of other count quantity expressions. At the same time, in systems with obligatory plural marking, exceptions to number marking may arise (Chierchia 2010; Doetjes 2010), as exemplified by nouns such as furniture.23 Nouns such as furniture are not compatible with numerals and expressions such as many or numerals, despite their count meaning.24 A different example of how language-specific properties of quantity expressions may interact comes from Mandarin. Numeral classifiers are systematically used with numerals and a small set of other quantity expressions. As shown above, expressions such as dàbùfèn ‘most’ occur in a modifier position and are incompatible with classifiers. Given that the measure words in measure constructions are always realized as classifiers in Mandarin, they are incompatible with expressions such as dàbùfèn ‘most’, so that most bottles of water cannot be translated as in (25):25 (25)

dàbùfèn (*píng) shuǐ most (*cl bottle) water ‘most water’

[Mandarin]

This section focused on cross-linguistic properties of quantity expressions. In the next two sections, I will first turn to the consequences of the available types of quantity expressions for the types of noun meanings that need to be available cross-linguistically (Section 2.3). Then I will consider cases where noun meaning interacts with the presence vs. absence of count quantity expressions (Section 2.4). 2.3

Count and Mass in the Lexicon: Properties of Nouns

2.3.1

Mass and/or Count Only Languages?

If it is correct that quantity expressions that presuppose countability are universal, count meaning must be strongly anchored in the nominal system across languages, which is in accordance with the importance of counting and 23

24 25

Pluralia tantum such as oats are also exceptions to the general rule. In the context of these nouns, most count quantity expressions are simply excluded, as they are uninterpretable in the context of mass meaning. However, people hesitate between much and many, the normative rule being that many is required because of the plural marking. The reason for this could be semantic (see, for instance, Bale and Barner 2009; Rothstein 2010b) but also morphological (see Cowper and Hall 2012). With some of the expressions that do not require classifier insertion, some speakers allow for the use of (mensural) classifiers, while other speakers apply stricter rules and do not permit the use of a classifier. If the mensural classifier is prohibited, a literal translation of e.g., most bottles of water is not available. See Rothstein (2010a) for a similar phenomenon in Hebrew.

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individuation in cognition (see for instance Spelke and Kinzler 2007). The cross-linguistic availability of these expressions predicts that all languages should have nouns that have or at least can have a count meaning. It is important to realize that this also applies to classifier languages: classifiers do not create a system in which all nouns have mass meaning, given that sortal classifiers can be shown to presuppose countability themselves. This means that ‘mass only’ languages cannot exist. Non-count quantity expressions do not require either count or mass meaning, and as such do not predict anything with respect to the types of noun meanings that need to be available. If selection on the basis of mass meanings does not exist, as I argued in the previous section, languages could in principle do without mass meaning. An obvious candidate for a language without mass meaning is course Yudja, as illustrated by the examples in (5) above. According to Lima, notional mass nouns in Yudja have an atomic, count meaning, which is similar to the count meaning that notional count nouns have, and she offers conclusive evidence for the possibility of interpreting notional mass nouns as count. However, one may also wonder whether Yudja offers any evidence for the possibility of a mass meaning for notional mass nouns such as y’a ‘water’. According to Deal (2017), who argues that count only languages are not possible (cf. the Nez Perce data in (19) and (20) above), such evidence exists. Her claim is based on the results of a specific quantity judgment task with bitu ‘more’, in which no choice was offered in terms of number (Lima 2014).26 Out of twenty adult Yudja speakers surveyed, 88 percent interpreted (26) in terms of volume and opted for the big pile of flour in the given context (Lima 2014, p. 132): (26)

Context: there is one large pile of flour and one small pile of flour. Ma de bitu asa dju a’u? [Yudja] who more flour have ‘Who has more flour?’

According to Deal, the fact that the volume reading was permitted and even preferred indicates that the noun asa ‘flower’ can have a mass reading. However, this is not a necessary conclusion. As Lima indicates, bitu ‘more’ does not force a count interpretation in the same way as itxïbï ‘many’ does, as shown by the results of her standard quantity judgment tests (in which both number and volume vary). If the results of the test with bitu ‘more’ are compared to the ones with itxïbï ‘many’, an interesting difference shows up. 26

In the standard quantity judgment task, one picture represents the largest volume and the other the largest number, e.g., one large pile of flour and three small ones (cf. Barner and Snedeker 2005).

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Whereas quantity judgments for itxïbï ‘many’ yielded a 100 percent score with all nouns for adults (Lima 2014, p. 187), the results with bitu ‘more’ showed more variety: in about 15–20 percent of cases, the adults would make an estimation based on volume rather than on the number of portions (p. 122). This suggests that itxïbï ‘many’ is a count quantity expression, while bitu ‘more’ is a non-count quantity expression and as such is compatible with a global quantity meaning. The question is then whether the volume reading is possible if the noun has a count interpretation. For Deal, the answer to this question seems to be no. The distinction between an evaluation based on number and one based on global quantity is claimed to be due to the count or mass meaning of the noun, and as such the absence of an evaluation in terms of number implies mass meaning. Under this assumption, the results of Lima’s experiments show that there is only a preference for the count meaning of notional mass nouns in Yudja. However, there is also another way to interpret the data. One could assume that bitu ‘more’, being a non-count expression, does not imply that the evaluation is made in terms of number, even if the noun has a count interpretation. Whereas an evaluation in terms of number is strongly favored in the context of a noun with a count meaning (see also the original experiments of Barner and Snedeker 2005), there is no a priori reason to assume that an evaluation in terms of volume would be excluded (see also Section 2.4). In Lima’s experiments with bitu, evaluation in terms of number is preferred for all nouns, but this is so independently of whether the noun is notional mass or notional count. Evaluation in terms of volume is also attested, contrary to what is found for the quantity judgment experiments with itxïbï ‘many’. This shows that this option is independent of the presence of a mass (that is, divisive) meaning, as it is unlikely that the effect in the examples with notional count nouns is due to a count-to-mass meaning shift. In the example in (26), an evaluation in terms of volume is the only option, as a comparison in terms of number is excluded. This explains the high percentage of answers based on an evaluation in terms of volume. The interpretation of (26) is thus not a sufficient reason to conclude that notional mass nouns in Yudja also can have a mass meaning. Other data that may suggest that nouns in Yudja can have a mass meaning are not conclusive either. First, Lima (2012) reports that the nouns kapïiã ‘fog’ and makasu ‘wind’ are incompatible with modifiers such as itxïbï ‘many’, and assumes that “individualization of portions is not possible.” This does not exclude however, that they are conceived of as nouns with a unique referent (cf. the sun). Second, the quantity expression xinaku ‘a little’ seems to behave as an anti-count quantity expression. Contrary to urahu ‘big’, this expression cannot be used to modify notional count nouns (Lima 2012, example (44)):

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a. xinaku/urahu apeta [Yudja] a little/ big blood ‘(There is) a small/big amount of blood in a single place’ b. #xinaku ali a little child c. urahu ali big child ‘(There is) a big child’

The properties exemplified by urahu ‘big’ are also found in other languages in which notional mass nouns such as blood are easily used as count nouns, such as Nez Perce (Deal 2017). As also indicated by Lima and Dahl, if urahu means ‘big’ in the context of a notional count noun, the interpretation in (27a) is expected under a count interpretation of the noun apeta ‘blood’. The impossibility of (27b) shows that the same reasoning does not apply to xinaku. However, Lima’s translation still suggests that the blood is in a single space, and more evidence seems necessary in order to conclude that xinaku is only compatible with apeta under a mass interpretation of the noun. This means that Yudja still seems to be a candidate of a language in which all nouns have a count meaning. Based on the asymmetric properties of quantity expressions with respect to count meaning, languages are predicted to have either both count and mass meanings for nouns or only count meanings. This excludes the possibility of ‘mass only’ languages, while ‘count only’ languages may well exist. This is in accordance with the observation that Yudja is a serious candidate of a count only language. In the next subsection I will turn to the types of count meanings that are available cross-linguistically, and the large amount of variation in this domain. 2.3.2

Available Count Meanings and Cross-Linguistic Variation

The examples of Yudja and Nez Perce are also important from the point of view of linguistic variation in terms of the types of meanings that nouns may or may not have. The types of count meanings that are allowed in these languages are not available in all languages and cannot be directly translated into English. The grammatical properties of Nez Perce laid out in Deal show, moreover, that the free availability of count meanings for notional mass nouns is possible in a grammatical system that shows a strong resemblance to English (see (19) and (20) above). In general, there is a large amount of variation in the types of count and mass meanings nouns may or may not get, and this is to a large extent arbitrary. The English word for bacon cannot mean ‘slice of bacon’, while the Blackfoot word for bacon can (Franz and Russell 1995, cited in Wiltschko 2012):

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singular plural aiksinoosak aiksinoosakiksi ‘bacon’ ‘bacon’ (slabs or slices of )

[Blackfoot]

Similarly, Cheng et al. (2008) show that English and Dutch differ from Mandarin Chinese and Gungbe (a Kwa language spoken in Benin) in terms of the possibilities they offer for assigning a mass meaning to a typical count noun (e.g., dog). Similarly, the use of a typical mass noun with a count type reading, as in two different golds for two types of gold, is much more limited in Dutch than in English (Doetjes 1997) (*twee gouden ‘two golds’ is impossible under the intended reading) and has been claimed to be absent in Ojibwe (Algonquian, Mathieu 2012). Differences between the lexical choices languages make with respect to noun meaning can also be illustrated by the well-known contrast between the word advice in English and its French translation conseil. Whereas French conseil is used in combination with quantity expressions that presuppose countability (plusieurs conseils ‘several pieces of advice’), English advice can only be used in contexts that do not presuppose countability (more/less advice(*s)). Similarly, success in English is a mass noun, while failure is a count noun. It turns out to be extremely hard to predict the count or mass properties of nouns, especially in the domain of abstract nouns (Gillon 2012, Grimm 2014, Pelletier and Schubert 1989).27 These examples illustrate that available meanings vary cross-linguistically, which implies that there is an important language-specific lexical component involved. Even in flexible meaning approaches (Bale and Barner 2009; Borer 2005; Pelletier 2012), where nouns have one flexible meaning which is compatible with both mass and count grammatical contexts, it will not be possible to do without a lexical component delimiting the possible interpretations of a noun (whatever theoretical status these interpretations have) in a given language. Given the scarce systematic cross-linguistic literature on this topic, more research is necessary. 2.3.3

Count Meaning, “Default Units of Counting” and Abstraction

The acquisition of the count/mass distinction illustrates the way language makes more abstract count meanings available. As pointed out by Brooks et al. (2011), children who acquire the count/mass distinction first strongly rely on the “default units” of counting that are pre-linguistically available (see also 27

The difference between conseil and advice is similar to the difference between French rougir, which is an accomplishment verb, and its Dutch and English translations blozen and to blush, which are activity verbs. The phenomenon these verbs describe can be conceived of as an accomplishment and as an activity.

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Shipley and Shepperson 1990). These units correspond to cognitively salient units, such as physical objects and sounds. During the first stages of acquiring count meaning, counting depends on these default units. Take, for instance, a context where there are three forks and two half forks – that is, one fork that has been broken in two pieces – so that there are five separate physical objects and four forks. In this context, young children who are asked to count the number of forks will tend to count the number of objects instead, while older children and adults will abstract away from the fact that one fork is made up of two half forks. This is an important observation in view of the difference between language and cognition: even though cognition makes us sensitive to counting and default units, language allows us to define new units and to count these instead of the “default units of counting.” In what preceded, I assumed that measure words such as kilo are count nouns and have count meaning. Evidence for this is that they behave like ordinary count nouns: they can be combined with numerals, and if they are, they are marked for number. If this assumption is correct, languages allow for highly abstract count interpretations. This view is different from the one presented by Rothstein (2009, 2011, 2017). Rothstein makes a distinction between measuring and counting based on the two types of readings one can get for expressions such as three bottles/ liters of wine. On the one hand, the measuring expression three bottles/liters can define, or measure, a quantity of wine, resulting in a reading where we talk about a certain quantity of wine, as in they drank three bottles/liters of wine last night. This is the most natural reading for expressions such as kilo or liter. Alternatively, three bottles/liters of wine can denote three actual (liter) bottles filled with wine, as in she put three bottles/liters of wine on the table. In this case, the numeral counts actual bottles of wine.28 Rothstein reserves the term counting to this second use. From the perspective of the current paper, numerals are always count quantity expressions and as such they are interpreted in relation to units that can be counted. In Rothstein’s “measuring” cases, these units are abstract, and in the “counting” contexts, the units are concrete, but in both cases I assume that the numeral has a similar meaning and counts the units. My aim is to make a classification of quantity expressions based on their general distributional properties. In this respect, it is important to realize that not only numerals (which are selected by the measure word in Rothstein’s account), but also other count quantity expressions in English combine with measure words under their abstract reading, as illustrated by the examples in (29). 28

Rothstein argues that this meaning difference involves also a structural difference. Whereas [[three bottles] of wine], in which the word bottle first combines with the numeral, corresponds to measuring, the structure [three [bottles of wine]], corresponds to a reading with three individual bottles.

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a. Every inch of cloth is used, nothing is wasted. b. If you want to make that dress, you will need several yards of fabric.

Neither of these examples assumes that the cloth or fabric is cut into inches/ yards, which shows that they cannot be treated as cases of “counting” in Rothstein’s terms. This shows that the abstract meanings are available irrespectively of the type of count quantity expression (numeral or other). 2.3.4

Properties of Nouns: Summary

The idea that there is a fundamental distinction between quantity expressions that presuppose the existence of units to count and quantity expressions that do not make this presupposition has consequences for the type of noun meanings that we expect to exist in languages. If it is true that languages universally have these two types of quantity expressions, this forces them to have nouns with count meanings and allows them to have nouns with meanings that are not compatible with expressions that presuppose countability. This means that ‘mass only’ languages are not possible, while ‘count only’ languages could exist, and Yudja might well be an example of such a language (Lima 2014). The choices that languages make with respect to possible count meanings are often ontologically motivated, in particular in cases where nouns have a ‘naturally individuated’ referent, such as nouns denoting clearly delimited objects or animate individuals. However, ontology does not provide a means to predict the properties of a noun, which are to a large extent arbitrary. The substance water may be mass from an ontological point of view, but languages such as Yudja and Nez Perce use it with a count meaning that the English word water cannot get. Similarly, languages differ in terms of the mass meanings they allow or do not allow for nouns that denote naturally countable objects or individuals. In addition to this, count meanings can be abstract in the sense that the individual units do not need to be clearly individuated in the actual world, as can be illustrated by measure words and abstract nouns on the one hand, and the acquisition of abstract meaning on the other. In the next section, I will turn to a different and more tentative way in which quantity expressions and word meanings interact, and argue that languages can make use of the system of quantity expressions they dispose of in order to mark the atomic individuals that are present in a count meaning as more or less salient. 2.4

Individuation Boosting and Degrees of Individuation

In this section, we will briefly explore the idea that the presence of an expression that presupposes units that can be counted triggers an effect of what I will call “individuation boosting.” In several cases where subtle

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distinctions in the prominence of individuation are made, these distinctions seem to correlate with the presence vs. absence of count quantity expressions. For instance, nouns such as furniture are felt as less individuated than regular count nouns, and do not take number marking. Similarly, the data from the experiments of Lima (2014) described above show that bitu ‘more’ does not require a count reading while ïxibï ‘many’ does. The hypothesis of individuation boosting attributes these effects to the absence vs. presence of the presupposition of count meaning, which is introduced by count quantity expressions. The idea of the effect of individuation boosting is quite simple. A noun with a count meaning introduces individuals. If, in addition, there is a linguistic expression in the context that requires the noun to have a count meaning, this makes the presence of the individuals more salient.29 Let us first turn to the contrasts between bitu ‘more’ and itxïbï ‘many’. Even though there are good reasons to assume that all nouns have count meaning in Yudja, an interpretation in terms of the number of items is only required in the case of itxïbï. The Yudja data differ in this respect from data with more books, which despite the fact that more is a non-count quantity expression triggers an evaluation in terms of the number of books. However, contrary to Yudja, English is a language with obligatory number marking. The difference between the two languages can be accounted for if we assume that the obligatory evaluation in terms of number is triggered by the presence of a count quantity expression. This hypothesis can be tested on the basis of Brazilian Portuguese. Brazilian Portuguese is a language with optional number marking with a lexical distinction between mass and count nouns (see Ferreira forthcoming, for an overview): (30)

a. duas menina(s) two girls b. #duas areia(s) two sands

[Brazilian Portuguese]

If a count noun is combined with an expression that does not require number marking, such as mais, number marking may be present or absent. In the following examples, taken from Beviláqua and Pires de Oliveira (2014), the first sentence permits both an evaluation in terms of volume and in terms of number while the second triggers obligatorily a reading in terms of number (see also Pires de Oliveira and Rothstein 2011). Beviláqua and Pires de

29

Note that the idea of individuation boosting implies a hybrid relation between language and cognition, as it implies a strong association between the linguistic property of countability and the cognitive concept of individuation, despite the fact that linguistic structures that involve counting permit to abstract away from individuation (e.g., two/many liters of water).

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Oliveira (2014) present experimental data from Beviláqua (2014) that confirm this general pattern: as soon as a number marker is present somewhere in the structure, quantity judgments are almost exclusively made on the basis of number, while in the absence of the plural marker interpretations in terms of volume are permitted (though not preferred). (31)

a. João João b. João João

tem has.prs.3sg tem has.prs.3sg

mais livro more book mais livros more book-pl

que than que than

a Maria. [Brazilian Portuguese] the Maria a Maria. the Maria

The patterns in Brazilian Portuguese and Yudja show that individuation originating from the lexical meaning of a noun is enough for obtaining an evaluation in terms of number (cf. the original experiment of Barner and Snedeker 2005).30 Under the hypothesis of individuation boosting, the presence of an expression that presupposes countability makes the individuated reading even more salient, as a result of which an evaluation in terms of volume becomes virtually impossible. A second example of a phenomenon that could be attributed to individuation boosting is the use of the quantity expression hou2 do1 ‘a lot’, which optionally combines with a classifier (Cheng et al. 2012): (32)

hou2 do1 (bun2) syu1 a lot cl volume book

[Cantonese]

As expected, the books are more clearly individuated when the classifier is present. For instance, if the noun phrase is used to refer to the books in a library, the classifier will normally be absent. In order to describe books on a table, which can clearly be individualized, the classifier will be used. A different domain where the idea of individuation boosting could be exploited is the lexicon. As indicated above, some lexical systems have number marking on some nouns and not on others. From the perspective of individuation boosting, this would be expected to correlate with higher and lower individuation. Whereas nouns that exceptionally lack number marking (e.g., furniture in English) are expected to correspond to a low degree of individuation, nouns that are exceptionally marked for plurality are expected to correspond to nouns with a high degree of individuation such as animate/ human nouns (cf. Smith-Stark 1974). The concept of individuation boosting is related to the recent proposals of Grimm (2018), who argues in favor of distinguishing degrees of individuation 30

An unpublished quantity judgment experiment presented by Scott Grimm and Beth Levin at Sinn und Bedeutung 16, comparing pairs such as jewels and jewelry, showed that the presence of plural morphology correlated with the necessity of an evaluation in terms of number, in line with the Brazilian Portuguese data.

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in natural language. Under the hypothesis of individuation boosting, the type of system that is at the basis of the linguistic expression of degrees of individuation might be severely constrained: in this latter view, degrees of individuation are the result of possibilities that the lexicon of a particular language offers for creating structures with and without count quantity expressions. Obviously, more research is needed to further exploit this idea. 2.5

Conclusions

This chapter studied the count/mass distinction from the perspective of quantity expressions. The properties of quantity expressions can only be understood in the context of the quantity system of which they are part. Language-specific properties of quantity expressions and the ways in which they interact with each other and with the possible noun meanings of the language they occur in are at the basis of the large amount of variation found in the domain of the count/mass distinction. The main claim defended in this chapter is that, behind this linguistic variation, all languages present a distinction between count vs. non-count quantity expressions, which is at the basis of the count/mass distinction across languages. These two types of quantity expressions can be seen as linguistic reflections of numerical magnitude representation (Dehaene 1997; Feigenson et al. 2004) and general magnitude representation (Lourenco and Longo 2011), respectively. Whereas count quantity expressions presuppose countability and therefore only combine with expressions that may have a count meaning, noncount quantity expressions do not introduce such a presupposition, and therefore do not impose restrictions on the count (non-vague atoms) or mass (vague atoms/divisivity) meaning of the nouns they combine with. Quantity expressions that typically combine with mass nouns (e.g., a bit) are claimed to be a special type of non-count quantity expressions. They are analyzed in terms of blocking rather than in terms of sensitivity to the presence of mass meaning. As a consequence, the basic distinction between types of quantity expressions is a two-way distinction, and the count/mass distinction is asymmetric in the sense that quantity expressions make reference to count meaning but not to the absence thereof. Cross-linguistic evidence for this generalization based on different types of quantity systems was presented in Section 2.2. The basic opposition between two types of quantity expressions – ones that need count meanings and ones that are indifferent towards the presence or absence of count meanings – and the hypothesis that these two types are crosslinguistically available, makes predictions on the availability of count and mass meanings across languages. If languages always dispose of expressions that can only be combined with nouns that (can) have a count meaning, count meaning necessarily exists across languages and ‘mass only’ languages are not

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available. ‘Count only’ languages, on the other hand, are not incompatible with the types of quantity expressions that are found. If it is correct that quantity expressions that presuppose mass meaning do not exist, languages in which all nouns always have a count denotation are predicted to be possible. As shown by Lima (2014), Yudja seems to be a candidate for such a language. As the examples of count meaning show, there is a large amount of lexical crosslinguistic variation in this domain, which is currently not well understood. Finally, I suggested that the presence of a grammatical expression that presupposes countability may introduce an effect of individuation boosting. The ways in which this effect shows up in a particular language depend on the specific properties of the quantity expressions and the available noun meanings in that language. I gave several examples of phenomena that could be understood in terms of individuation boosting, and I consider this to be an important domain for further research. REFERENCES Bach, Elke, Eloise Jelinek, Angelika Kratzer, and Barbara Partee (2013). Quantification in Natural Languages, Vol. 54. Dordrecht: Springer. Bale, Alan, and David Barner (2009). The interpretation of functional heads: Using comparatives to explore the mass/count distinction. Journal of Semantics 26.3: 217–252. Bale, Alan, and Jessica Coon (2014). Classifiers are for numerals, not for nouns: Consequences for the mass/count distinction. Linguistic Inquiry 45.4: 695–707. Barner, David, and Jesse Snedeker (2005). Quantity judgments and individuation: Evidence that mass nouns count. Cognition 97: 41–46. Beviláqua, Kayron (2014). Nomes nus e nomes plurais: Um experimento sobre a distinção contável-massivo no PB. MA Thesis, Universidade Federal do Paraná. Beviláqua, Kayron, and Roberta Pires de Oliveira (2014). Brazilian bare phrases and referentiality: Evidences from an experiment. Revista Letras 90: 253–275. Borer, Hagit (2005). Structuring Sense Volume i: In Name Only. Oxford: Oxford University Press. Brooks, Neon, Amanda Pogue, and David Barner (2011). Piecing together numerical language: Children’s use of default units in early counting and quantification. Developmental Science 14: 44–57. Cabredo Hofherr, Patricia (forthcoming). Nominal number morphology. In Patricia Cabredo Hofherr and Jenny Doetjes (eds.), The Oxford Handbook of Grammatical Number. Oxford: Oxford University Press. Chao, Yuen Ren (1968). A Grammar of Spoken Chinese. Berkeley, CA: University of California Press. Cheng, Chung-Ying (1973). Comments on Moravcsik’s paper. In Jaakko Hintikka, Julius Moravcsik, and Patrick Suppes (eds.), Approaches to Natural Language, pp. 286–288. Dordrecht: Reidel. Cheng, Lisa, and Rint Sybesma (1998). Yi-wan tang, yi-ge tang: Classifiers and massifiers. The Tsing Hua Journal of Chinese Studies 28.3: 385–412.

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3

Counting Aggregates, Groups and Kinds: Countability from the Perspective of a Morphologically Complex Language Scott Grimm and Mojmír Dočekal

3.1

Introduction: The Components of Countability and Their Interrelation

Theories of countability face the task of explaining how various nouns’ different participation in grammatical number constructions corresponds to meaning contrasts among those nouns.*,1 What, if anything, in a noun’s meaning impinges on its ability to appear in different morphosyntactic contexts related to counting and/ or measuring? Or in the other direction, how do morphosyntactic contexts impinge on the possible interpretations of a noun? An explanation of countability must show how and why countability distinctions arise, from morphosyntax or lexical meaning or from a combination thereof, and be predictive of grammatical number systems both generally and in particular languages. Different proposals have staked out different positions along the spectrum of possible ways morphosyntax, lexical meaning and countability are related – from arguing that lexical meaning simply does not influence countability patterns (see Borer 2005, among others) to arguing that lexical meaning fully determines countability patterns (Wierzbicka 1988; Wisniewski et al. 2003). The evidence for which analysis is best comes from natural language data, of course, and consequently the choice of which languages to discuss will predetermine the range of tenable conclusions about the nature of countability. This paper adds to this discussion data from Czech and, as a consequence, directs the focus towards languages that have a morphologically rich nominal system. Importantly, more expressive nominal morphology may specifically target entity types which in less rich nominal systems remain morphologically *

1

We would like to thank Markéta Ziková and the audience of OLINCO 2014 for discussion and comments. We would like to thank two anonymous reviewers for questions and comments that aided to clarify the final version of this chapter. Mojmír Dočekal acknowledges the support of a Czech Science Foundation (GAČR) grant to the Department of Linguistics and Baltic Languages at the Masaryk University in Brno (GA17–16111S). Our use of the term “countability” subsumes various phenomena broadly related to grammatical number, including the count/non-count contrast, collectives, duals, pluralia tantum and so forth, following the usage in Payne and Huddleston (2002), Joosten (2003), and Grimm (2012).

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undifferentiated. In the case of Czech, several complex cardinal expressions along with derivational morphology specifically target entity types beyond the object level: namely, aggregates, groups and (taxonomic) kinds, none of which, for instance, in English are morphologically distinguished in such a clear fashion. While the data presented here come mostly from Czech, we note that much of these phenomena appear in Slavic languages more generally: In all three sub-groups of the Slavic language group (West Slavic, East Slavic and South Slavic), there are numerical expressions and derivational morphology processes which restrict the domain of quantification to aggregates, groups and (sub)-kinds. Polish possess all three types of these expressions (only the taxonomic numerals are less productive and more archaic than in Czech, see Wągiel 2015 for details); in Russian there are at least group- and aggregatedenoting numerical expressions (see Khrizman 2016, 2020); finally, Bosnian/ Croatian/ Serbian (BCS) seems to have in its inventory both group numerals and aggregate-denoting expressions. Thus, while Czech is far from being exceptional among Slavic languages in possessing a rich derivational morphology system for nouns and numerals, investigating these constructions in Czech is advantageous, as all three types of expressions can be found in contemporary spoken Czech, whereas only a subset of the three types of expressions can be found in other Slavic languages. The greater complexity of the system of cardinal expressions has implications even at the level of the very diagnostics of countability. One of the clearest diagnostics of countability is the combination of nouns with cardinal expressions, as in two cats (the impossibility of which is named the “signature property” of non-countable nouns by Chierchia 2010). In Czech, certain nouns may combine with some cardinal expressions yet not others, and accordingly be “countable” in different ways. In this paper, we relate Czech’s nominal system to three questions that are central to theories of countability. First, what must theories of countability assume as primitive? There are two sub-questions here. What elements serve as the ontological foundations of countability? And is one countability category, e.g., non-countable, more basic than another? Second, as Czech provides morphology for explicitly counting aggregates, what can be determined about the nature of aggregates? Finally, to what extent is countability lexically specified as opposed to dependent on pragmatic context? Put in other terms, how intrinsic is nominal flexibility to countability? We now provide initial discussion of each question in turn. Primitives of Countability The early theory of Link (1983) took the count/mass distinction to be a lexical contrast grounded in a basic ontological distinction between atomic and non-atomic entities, from which the syntactic patterns of countability followed. From this starting point, researchers have taken more and less parsimonious positions. Chierchia (1998a, 2010), for instance, takes a more parsimonious position wherein all nouns derive from

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a common ontological sort: All nouns have an atomic domain, but mass nouns, due to their inherent vagueness in Chierchia (2010), impede the ability to refer to the atomic parts. Alternately, many researchers account for three (grammatical) types of nouns – substances, individuals and nouns such as furniture that are non-countable yet refer to individuals – while retaining two ontological sorts, namely atomic entities and non-atomic entities, where nouns like furniture are comprised of atomic entities but are non-countable for some reason, e.g., a lexical feature is stipulated, as in Barner and Snedeker (2005) or Bale and Barner (2009). On the other hand, Grimm (2012) accepts an ontological contrast between atomic and non-atomic entities, and further argues that even more ontological contrasts are needed than just the one between atomic and non-atomic (substance) reference, namely reference to aggregates, for example, for nouns such as hair or foliage, while Grimm and Levin (2017) argue for recognizing yet other ontological types in the artifactual domain, distinguishing between individual artifact nouns (hammer, chair) and artifactual aggregates (furniture, laundry). Czech provides more evidence that nominal roots may be ontologically richer than the individual/substance contrast which has garnered the most attention in the literature. This is in part due to the complex cardinals mentioned, which directly target several ontological sorts (aggregates, groups and kinds), but also due to derivational morphology which observably derives aggregates, which form a subclass of non-countable nouns in Czech with particular properties. A related, yet slightly different question facing theories of countability is what to take as basic and what to derive. Countable and non-countable nouns could have equal status, or one class could be derived from the other. For Borer (2005), again an example of a parsimonious theory, all nouns begin as mass or “stuff,” and then are given further structure in syntax through the application of functional heads. To anticipate, the Czech system provides a complicated, and perhaps fatal, data point for theories where one class of nouns is derived from the other. The Nature of Aggregates The discussion of aggregates as a separate ontological sort leads to the more specific question of what, if anything, can be determined about the nature of aggregates, or the various different sorts of things which have been labeled “aggregates.” While the recent literature has often proposed a three-way contrast in nominal semantic types – substance (non-countable), aggregate (non-countable) and individuals (countable), where aggregate nouns are complete atomic join semi-lattices, that is, including atoms and sums (Bale and Barner 2009; Deal 2017) – many questions remain, most critically, why these particular nouns rather than others have aggregate denotations. Other questions include what the nature of any specific lexical semantics in play is and the cross-linguistically commensurability of different classes of aggregates. Czech’s morphology isolates a reasonably coherent

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lexical class of aggregate nouns, prototypically based on spatial proximity of elements, which in turn sheds light on some of the preconditions that associate with aggregate nouns. Nominal Flexibility In English, famous shifts in meaning have been attributed to operations known as ‘grinding’, ‘packaging’ and ‘sorting’ (see Pelletier 1979; Bunt 1985, among others), demonstrated with examples (1), (2) and (3) from Bach (1986). For these cases in English, erstwhile countable nouns appear as non-countable, as in (1), and vice versa, as in (2) and (3), given suitable context. The core question is how the meaning of a word impinges on its ability to appear in different contexts, if at all. (1)

a. There was dog splattered all over the road. b. Much missionary was eaten at the festival.

(2)

ice-creams = ‘portions of ice-cream’

(3)

muds = ‘kinds of mud’

While the existence of contextual shifts is uncontroversial, their importance for theories of countability is a matter of ongoing debate. Some researchers have taken nominal flexibility to be a foundational property of countability (Pelletier and Schubert 2004; Borer 2005; Bale and Barner 2009; Chierchia 2010), which is compatible with the data in English. On the other hand, there are serious questions as to how well the flexibility data in English carry over to other languages. A valuable set of recent studies, including Dalrymple and Mofu (2012) on Indonesian, Lima (2014) on Yudja and Deal (2017) on Nez Perce, have reported on languages with restricted morphological means to grammatically code number distinctions and which simultaneously permit a large degree of nominal flexibility. In these languages, canonical mass nouns, like mud or blood, may be counted when appearing in contexts which license such interpretations. Although the authors have not settled whether in these languages there is a lexically encoded distinction between countable and non-countable nouns (see discussion in Deal 2017), these studies have greatly added to our knowledge about how such languages function. Examining Czech, which uses rich inflectional and derivational morphology to code countability, provides a counterpoint study. As will become clear, Czech presents the opposite behavior than theories that emphasize flexibility would predict: While Czech possesses rich number morphology, it manifests limited nominal flexibility. More generally, we seek to provide evidence bearing on a larger question: As the syntax and morphology varies from language to language, how does this affect the distribution of labor between pragmatically available senses and morpho-syntactically specified senses? This chapter is organized as follows. In Section 3.2, we introduce the core countability data from Czech, including the basic diagnostics as well as

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derivational morphology and complex cardinals. To the best of our knowledge, even this basic data has not received any substantial attention in the countability literature. Further, as we have indicated, the data themselves already provide directions towards answering some of the questions brought up here, which we return to in Section 3.3. Sections 3.4 and 3.5 develop a formal analysis of Czech’s nominal system, providing an extended investigation of how the different semantic distinctions are parceled out to different parts of the morphosyntax. Altogether, this exploration of countability in a morphologically rich language advances the understanding of the interplay between morphosyntax, lexical meaning and countability. 3.2

Countability in Czech: The Core Data

As with the majority of Slavic languages, Czech is an SVO language possessing rich inflectional and derivational morphology. Czech nominals lack overt articles, and those that appear bare can be interpreted in various ways, including definite, indefinite or generic interpretations.2 Czech manifests the familiar distributional contrasts for notionally countable and non-countable nouns. For instance, pes ‘dog’, a prototypical count noun, intuitively denotes individual dogs. Such nouns can be pluralized, as shown in (4b), and also may combine both with cardinal numerals, as shown in (4c) and (4d). Cardinal numerals fall into two syntactic categories: adjectives for cardinal numbers 1, 2, 3 and 4, as in (4c), and nouns for cardinal numerals 5 and greater, as in (4d). There are two reasons for classifying numerals 5 and greater as nouns: (i) they (unlike cardinal numerals for 1, 2, 3 and 4) assign genitive case to their nominal complements, and (ii) if the cardinal numeral for numbers 5 and above is the subject of a sentence, the verb does not agree with the nominal complement (of the numeral). Countable nouns also combine with determiners such as mnozí ‘many’, a vague adjectival determiner agreeing with the noun it modifies in both case and number, as shown in (4e). (4)

a. pes-∅ b. dog-SG ‘dog’ c. dva/tři ps-i d. two/three dog-PL ‘two/three dogs’ e. mnoz-í ps-i DET-NOM.PL dog-NOM.PL ‘many dogs’

ps-i dog-PL ‘dogs’ pět ps-uº five dogs-GEN.PL ‘five dogs’

We take bláto ‘mud’ as an instance of a prototypical non-countable noun, denoting a substance. As shown in (5), these nouns cannot be pluralized, nor do 2

See Dayal (2004), Filip (1999) and Krifka (1992), among others, for discussions of the interpretation of bare nouns in Czech in connection with lexical and grammatical aspect.

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they combine with cardinal numerals or with determiners such as ‘many’. However, they do combine with certain determiners, such as všechno, roughly equivalent to English ‘all’, which requires singular agreement on the noun, shown in (6). (5)

a. blát-o b. *blát-a mud-SG mud-PL ‘mud’ ‘muds’ c. *dvě/tři blát-a d. *mnoh-á blát-a two/three mud-PL DET-NOM.PL mud-NOM.PL ‘two/three mud’ ‘many mud’

(6)

a. všechn-o blát-o b. všechn-a vod-a DET-SG mud-SG DET-SG water-SG ‘all mud’ ‘all water’

Another diagnostic from English, that only non-countable nouns may appear as bare singulars, does not succeed in Czech, as countable nouns may also appear as bare singulars. We also note that Czech bare singulars are able to refer to objects (7a) as well as to kinds (7b) (at least when the bare singulars are arguments of kind-level predicates as in (7b)). (7)

º a. Blesk zpusobil požár a zabil mamut-a. thunderbolt caused wildfire and killed mammoth-ACC.SG ‘A thunderbolt caused a wildfire and killed a mammoth.’ b. Mamut-∅ vymřel i v oblastech, kde bylo málo lidí. mammoth-NOM.SG died-out even in areas where was few people ‘Mammoths died out even in areas where there were only few people.’

For certain prototypical non-countable nouns, such as voda ‘water’, Czech also manifests an additional interpretation, that is, a “packaging” use, as witnessed in (8a). Yet, the ability for non-countable nouns to be used with a packaging interpretation is highly constrained, typically only found in conventionally licensed instances, similar to what has been observed for English. Attempting to establish a parallel packaging interpretation with písky ‘sand’ in (8b) fails: Even when speakers are given a background scenario in which a quarry sells different packages of sand to builders, Czech native speakers reject (8b) and deem it unnatural. (8)

a. mnoh-é vod-y už byl-y vyprod-ané many-NOM.PL waters-NOM.PL already were-PL sold-PST.PART ‘many bottles of water were already sold out’ b. #mnoh-é písk-y už byl-y vyprod-ané many-NOM.PL sands-NOM.PL already were-PL sold-PST.PART Intended meaning: ‘many packages of sand were already sold out’

So far, the nominal classes that can be isolated through distributional tests match up with expectations derived from English and other Western European languages. Yet, the data in Czech are far more involved. The remainder of this section explores the various types and interpretations of nouns, made visible

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through complex morphology on both nouns and cardinal numerals, and then turns to consider the issue of nominal flexibility. Through its derivational morphology, Czech identifies entity types critical to understanding countability, but which have garnered less discussion in the literature than the individual/substance distinction, namely derived non-countable aggregates (list-í ‘foliage’), and complex numerals which count groups (dvoj-ice ‘a group of two’), connected clusters (dvoje hranolky ‘two collections of French fries’) and taxonomic kinds (dvojí metr ‘two kinds of measure’). We discuss each in turn. 3.2.1

Derived Aggregates

In addition to nouns which we consider to be lexically non-countable, Czech possesses derivational morphology, the suffix -í,3 which derives non-countable nouns. Table 3.1 presents the nouns known to be derived by -í.4 While these nouns are restricted to be inanimate, they arise in several semantic domains. Schematically, -í applies to a countable noun root N and results in an interpretation ‘a collection of N’, although some of the examples illustrate that an even more specific (lexical) semantics may be necessary.5 Nouns derived with -í are strongly non-countable. Unlike the root nouns from which they are formed, nouns derived by -í do not pluralize or combine with simple cardinal numerals or vague quantifiers, as shown in (9)–(11),6 nor do they manifest packaging or otherwise countable interpretations, as shown in (12), unlike lexically non-countable nouns such as voda ‘water’. (9)

3 4 5

a. list-í-m leaf-í-INST.SG ‘foliage’

b. *list-í-mi leaf-í-INST.PL ‘foliages’

There are several variant forms of -í, such as -oví, which, as far as we can discern, do not differ semantically. The items presented in the table comprise the majority of Czech nouns derived by -í. A few other lexical items exist but they are archaic or at the periphery of contemporary Czech. For twenty-one of the twenty-two nouns given, -í clearly applies to purely countable nouns, as we verified through applying the standard tests for distinguishing countable and non-countable nouns. These nouns are often naturally found in counting contexts as shown in the following example for krajka ‘lace’, where the word krajkoví ‘lacework’ designates pieces of lacework. (i) přida-l-a ještě další dv-ě krajk-y a vyda-l-a add-PST-3SG still next two-ACC.PL lace-ACC.PL and publish-PST-3SG nový malý sešit se čtyř-mi krajka-mi new small notebook with four-INS.PL lace-INS.PL ‘She added two pieces of lacework and published a new small notebook with four pieces of lacework.’

6

The only exceptional noun is dříví, derived from dřevo ‘wood’, which is ambiguous between countable and non-countable interpretations. We assume that -í applies to the countable form. Since case syncretism often obscures the morphological patterns in Czech, we often present nouns in the instrumental case, which is morphologically more transparent.

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Table 3.1 Non-countable nouns derived by -í Noun

Gloss

Derivational source

Trees stromoví boroví habroví olšoví vrboví

‘clump of ‘clump of ‘clump of ‘clump of ‘clump of

Plants listí rákosí trní jahodí lískoví maliní ostružiní

‘foliage’ ‘rushes’ ‘thorns, brambles’ ‘clump of strawberry plants’ ‘clump of hazel bushes’ ‘clump of raspberry plants’ ‘clump of blackberry plants’

list ‘leaf’ rákos ‘reed’ trn ‘thorn’ jahoda ‘strawberry’ líska ‘hazel tree’ malina ‘raspberry’ ostružina ‘blackberry’

Complex objects cihloví krajkoví lat’koví nádobí sít’oví

‘brickwork’ ‘lacework’ ‘fence (made from laths)’ ‘dishes’ ‘netting/nets’

cihla ‘brick’ krajka ‘lace’ lat’ka ‘lath/slat’ nádoba ‘container’ sít’ ‘net’

Nautical terms lanoví plachtoví ráhnoví

‘rigging/ropes’ ‘sails’ ‘spars’

lano ‘rope’ plachta ‘sail’ ráhno ‘spar’

Other dříví kamení

‘firewood’ ‘rocks’

dřevo ‘wood’ kámen ‘rock’

trees’ pine trees’ hornbeam trees’ alder trees’ willow trees’

strom ‘tree’ borovice ‘pine tree’ habr ‘hornbeam tree’ olše ‘alder tree’ vrba ‘willow tree’

(10)

a. dva list-y b. *dvě list-í CARD.Masc leaf-Masc.PL CARD.Neut leaf-í.Neut ‘two leaves’ ‘two foliages’

(11)

*mnohé list-í už spadl-y many leaf-í already fell-PL ‘many foliage already fell’

(12)

??břízy a smrky shodil-y mnohé list-í birches and spruces shed-PL many leaf-í ‘birches and spruces shed many foliage’

As the examples in Table 3.1 indicate, the derived nouns implicate multiple elements designated by the root noun, that is, a clump of pine trees implicates multiple pine trees. Further, nouns derived by -í pattern in their semantic properties with plural nouns: They are cumulative (list-í + list-í = list-í) and non-divisive, since there are atomic parts for which divisiveness

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does not hold – parts of list-í are list-í but the atomic parts (list ‘leaf’) are not list-í.7 Despite these similarities, the meanings of nouns derived by -í contrast strongly with ordinary plural meanings. As the glosses given in Table 3.1 express, the resultant meaning of nouns derived by -í is not simply a set of, for example, pine trees, but a set in which the members are coherently related. This is most often in terms of spatial proximity, as in the case of clumps of trees or plants, but also may be in terms of functional interdependence, as in the case of lanoví, where it is not the individual ropes at issue but their coherent organization as part of a ship’s rigging. Similarly, cihloví, derived from cihla ‘brick’, does not signify a random collection of bricks, rather they must be related in some manner, for instance, used together in an architectural motif.8 Additionally, -í is not nearly as productive as one would expect if it were only to signal plurality. As the examples in (13) demonstrate, the ability to derive new nouns with -í is limited. Such limited productivity is, however, consistent with the lexical restrictions typically found with collective morphology (see Acquaviva 2008 for discussion of various systems). (13)

a. *stol-í b. *kabát-í table-í jacket-í ‘collection of tables’ ‘collection of jackets’ c. *planet-í planet-í ‘collection of planets’

Given these observations, the non-countability, at least at the object level, of derived aggregates is expected, since these nouns do not simply designate entities, but coherently related groupings of entities. As will be seen in detail in Section 3.2.2.2, Czech provides morphological mechanisms for counting precisely these more complex composite entities. 3.2.2

Complex Numerals

Czech, like other Slavic languages, possesses simple cardinal numerals, such as those shown in (14), as well as morphologically derived complex numerals. 7

For reference, we present definitions of predicates which are cumulative, divisive and atomic (relative to a property) in (i). a. CumulativeðPÞ $ ½PðxÞ ^ PðyÞ ! Pðx t yÞ b. DivisiveðPÞ $ 8x½PðxÞ ! 8y½y < x ! PðyÞ c. Atomicðx; PÞ $ PðxÞ ^ ¬9y½y < x ^ PðyÞ

8

We confirmed these intuitions by examining uses of -í nouns along with their related images on the Internet. For instance, in one illustrative example, the noun cihloví designates various clusters of bricks used in an architectural pattern, differentiated by color (www.novinky.cz/bydleni/tipya-trendy/240130-karnevalove-vile-rekonstrukce-vratila-puvodni-rozverny-styl.html).

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(14)

a. tři ps-i b. tři hrnk-y three dog-NOM.PL three cup-NOM.PL ‘3 dogs’ ‘3 cups’

This section lays out the basic data from complex numerals in Czech through which speakers make reference to groups, aggregates and taxonomic kinds. 3.2.2.1 Derived Group Numerals We first examine the complex numeral derived with the suffix -ice, which we term group numerals. These numerals produce an interpretation of “group of n Xs,” such as “group of three sailors” given in (15b). The distribution of the -ice suffix is restricted, both in the numerals it combines with, limited to basic numerals from 2 to 8, and in the nouns it combines with, restricted to animate nouns. (15)

a. tři námořníc-i b. troj-ice námořník-uº three sailor-NOM.PL three-ICE sailor-GEN.PL ‘3 sailors’ ‘a group of 3 sailors’

We first make some remarks on the basic syntactic configuration of these numeral phrases.9 The derivational suffix -ice applies to a non-cardinal numeral stem and takes a nominal complement in the genitive case (námořníkuº ‘sailors’ in the example in (15b)). The derived complex numeral, rather than the noun, behaves as the head of the construction. First, -ice numerals assign genitive case to their nominal complement, and there is no further agreement between the derived numeral and the nominal complement. Additionally, if such a numeral is the subject of a sentence, the verb agrees with the numeral, not with the nominal complement. In these respects, the -ice complex numerals parallel syntactically the cardinal numerals for numbers 5 and above, since both cardinal and complex numerals are syntactically not modifiers, but rather behave like heads of the extended NP projection. But the two classes differ semantically: The cardinal numerals 5 permit collective and distributive interpretations, whereas the -ice complex numerals only permit a collective interpretation.10 9

10

We present here only properties which will be critical to our analysis of these constructions, while acknowledging that the literature on Slavic numerals and their syntactic properties is intricate. See Veselovská (2001) and Ionin and Matushansky (2006), among others, for discussion. Some properties of the Czech group numeral are shared by a similar type of complex cardinal number in Russian, the collective numeral, discussed in Khrizman (2016, 2020). First, both types of numerals favor a “collective” interpretation, and second, Russian collective numerals are also restricted to combining with animate nouns (more specifically, the nouns must denote humans or young animals). Yet, there are many important differences between the Russian collective and Czech group numerals. Most critically, Russian collective numerals allow distributive readings (when they act as antecedents of dependent indefinites) and cumulative readings, while such readings are simply unacceptable for Czech group numerals. The Czech examples in (ia) and (ib) witness this fact. In (ia), the obligatory distributive marker po heading the dependent indefinite phrase clashes with the collectivity of the -ice numeral construction,

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The complex numeral formed with -ice may occur in both singular and plural, as shown in (16). (16)

a. s troj-ic-í námořník-uº with three-ICE-INST.SG sailor-GEN.PL ‘with a/the group of three sailors’ b. s troj-ice-mi námořník-uº with three-ICE-INST.PL sailors-GEN.PL ‘with groups of three sailors’

Complex numeral phrases formed through the application of -ice can be further quantified over with cardinal numerals, as shown in (17). Further, they are compatible with the vague adjectival determiners mnozí ‘many’, as shown in (18). At the same time, unlike simple cardinal numerals, the use of -ice derived numerals is incompatible with the singular universal quantifier všechno ‘all’, as shown in (19). (17)

dvě troj-ice námořník-uº two three-ICE sailor-GEN.PL ‘two groups of three sailors’

(18)

mnohé troj-ice námořník-uº many three-ICE sailor-GEN.PL ‘many groups of three sailors’

resulting in the sentence being unacceptable. In (ib), the sentence has only a collectivedistributive interpretation: the group of three sailors has been lost twice (in different harbors). The cumulative interpretation, for example where Sailor A has been lost in a Harbor 1 and Sailors B and C have been lost in a Harbor 2, is unacceptable. Consequently, Czech group numerals seem to be a clear case of group interpretation, unlike Russian collective numerals. In terms of the classification of collective predicates introduced by Winter (2001), Czech group numerals act like collective set predicates as indicated by the differing acceptability of Czech group numerals with quantifiers all and every, shown in (ii), similarly to other collective set predicates like meet, gather, be brothers or be similar. This is formalized in Section 3.5.2 accordingly. More data concerning Czech group numerals can be found in Dočekal (2012). vyhrál-a po litru rum-u. a. *Troj-ice námořník-uº three-ICE sailor-GEN.PL won-3SG.F DIS liter rum-GEN.SG ‘A group of three sailors won a liter of rum.’ se ztratil-a ve dvou přístavech. b. Troj-ice námořníkuº three-ICE sailors REFL lost-3SG.F in two harbors ‘A group of three sailors has been lost in two harbors.’ º (ii) a. ?Všichni podezřelí byl-a troj-ice námořník-u. all suspects AUX-3.SG three-ICE sailor-GEN.PL ‘All suspects were a group of three sailors.’ b. *Každý podezřelý byl troj-ice námořník-uº every suspect was three-ICE sailor-GEN.PL ‘*Every suspect was a group of three sailors.’ (i)

On the other hand, Polish collective numerals like dwójka ‘group of two’ appear to behave exactly like Czech -ice numerals, as reported in Wągiel (2015).

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(19)

*všechna troj-ice námořník-uº all three-ICE sailor-GEN.PL ‘all groups of three sailors’

Accordingly, counting groups through complex numerals provides only a partial match for the diagnostics of countability present for simple cardinal numerals, or countable nouns for that matter. The other complex cardinals, to which we now turn, also show only partial matching, but in different fashions than the group numerals. 3.2.2.2 Complex Numerals for Aggregates A second complex numeral, which we call an aggregate numeral, is formed through combining the suffix -oje with cardinal numeral roots and designates a number of collections of an entity. For instance, applied to the cardinal numeral root dv- ‘two’ yields dv-oje, which roughly translates to ‘two collections’, as shown in (20). This complex numeral adds (morphologically) the derivational suffix -oje to a cardinal stem. Unlike the derived group numerals -oje derived numerals are much more productive and appear to be derivable for any cardinal numeral, though there is allomorphy: the morpheme -oje is used with cardinal numerals 2 and 3, but for the cardinal numerals 4 and up the morpheme -ery is used. (The distribution of the morphemes -oje/-ery is standardly described in the academic and descriptive grammars of Czech as allomorphy: see Komárek, Kořenský ad. 1986, Karlík, Nekula ad. 1995). Second, the noun is the head of the numeral phrase. The numeral must agree with its head noun in case and number, as shown in (20-a), in contrast to the group numerals. (This is often difficult to see since the agreement may be morphologically syncretic – for example, the phonologically zero NOM.PL morpheme on the numeral in (20a) is syncretic with ACC.PL.) When the numeral is part of a subject of a clause, the verb agrees with its head noun, as in (20a). The allomorphs -ery and -oje are identical in terms of morphosyntactic behavior (see again Komárek, Kořenský ad. 1986, Karlík, Nekula ad. 1995 for more details), as shown in (20e). We will mostly use examples with the morpheme -oje (numerals 2–3) since smaller cardinality of aggregates typically yields more comprehensible and intuitively acceptable examples, yet our analysis and argumentation extends to complex numerals with -ery as well. (20)

a. Dv-oje-∅ kart-y ležel-y na stole. two-OJE-NOM.PL card-NOM.PL lie-3PL on table ‘Two sets of cards were lying on the table.’ b. dv-oje klíč-e c. dv-oje bot-y two-OJE key-PL two-OJE shoe-PL ‘two rings/sets of keys’ ‘two pairs of shoes’ d. dv-oje schod-y two-OJE stair-PL ‘two staircases’

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On the other hand, this complex numeral is quite restricted in the sorts of nouns it may combine with, primarily nouns derived by -í, pluralia tantum, and entities that typically come together in multiples. Examples of each type are shown in (21)–(23), respectively. There is clearly a lexical semantic theme concerning which nouns may combine with -oje numerals: The nouns must designate entities which are sets of individuals that typically come together in groups or are “connected” to one another in some manner. (21)

nouns derived by -í: nádobí ‘dishes’, dříví ‘firewood’, kamení ‘rocks’

(22)

pluralia tantum: dveře ‘doors’, housle ‘violins’, brýle ‘glasses’

(23)

entities that typically come together in multiples: sirky ‘matches’, hranolky ‘french fries’, schody ‘stairs’

In the case of nouns derived by -í, native speakers always prefer the use of the complex numeral over the simple cardinal numeral, despite a slight oddity for some combinations. A naturally occurring example is given in (24a) where the sequence of counting with cardinal numerals is changed from simple cardinals to an -oje numeral for a noun derived by -í. The use of a simple cardinal numeral would be unacceptable for native speakers. This is further illustrated in (24b), a constructed example representing in a typical fast-food ordering scenario where the -oje numeral is the only felicitous and natural option. In this context, dvě hranolky (feminine), or dva hranolky (masculine), is odd, and infelicitous, because it is naturally interpreted as referring to two individual French fries, not to two portions of French fries. (24)

a. Dvě kuchyně, dva nákupy, dvě lednice, dv-oje nádobí two kitchens two purchases two refrigerators two-oje dishes ‘two kitchens, two purchases, two refrigerators, two sets of dishes’ Czech National Corpus (2018) b. [context: fast-food order] dvě kávy, dv-oje/#dvě hranolky two coffees two-oje/#two French.fries ‘two coffees, two portions of French fries / #two individual French fries’

Applying -oje to nouns other than those denoting entities which are sets of individuals that typically come in groups or are connected to one another in some manner typically result in infelicities, as given (25) (but see note 21, further below). (25)

a. ??dv-oje stol-y b. ??dv-oje kabát-y two-OJE table-PL two-OJE jacket-PL ‘two sets of tables’ ‘two sets of jackets’ c. ??šest-ery aut-a six-ERY car-PL ‘six sets of cars’

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Combining -oje numerals with všechny ‘all’, as in (26), or with simple cardinals, as in (27), results in unacceptability: (26)

(27)

a. ??všechny dv-oje DET two-OJE ‘all two sets of violins’ b. ??všechny dv-oje DET two-OJE ‘all two sets of keys’ c. ??všechny sedm-ery DET seven-ERY ‘all seven staircases’

housl-e violin-PL klíč-e key-PL schod-y stair-PL

a. *tři dv-oje klíč-e three two-OJE key-PL ‘three two sets of keys’ b. *tři dv-oje dveř-e three two-OJE door-PL ‘three two sets of doors’ c. *tři sedm-ery schod-y three seven-ERY stair-PL ‘three seven staircases’

Thus, the sort of object delivered through combination with an aggregate numeral is not one that can be further counted, at least not in a standard manner. 3.2.2.3 Taxonomic Numerals The final type of complex numeral, derived with the suffix -ojí, yields taxonomic plurals, namely, a “different kinds” reading, which we call the taxonomic numeral. This complex numeral is morphologically parallel to -oje: The derivational suffix combines with a cardinal numeral root and then with a noun. The noun is the head of the numeral phrase, and the numeral must agree with its head noun in case and gender. Like -oje, -ojí appears to be derivable for any cardinal numeral. Applying -ojí to the cardinal numeral root dv- ‘two’ yields dv-ojí, roughly translated as ‘two kinds’. As in the case of -oje, there is an allomorphy: for numerals 2–3, the suffix -ojí is used, for numerals 4 and up the morpheme -ero is the derivational suffix of the taxonomic numerals (and again we follow the standard allomorphy analysis as described in more detail in the academic/ descriptive grammars of Czech: see Komárek, Kořenský ad. 1986, Karlík, Nekula ad. 1995). Examples are given in (28). Note that sýry ‘cheese’ in example (28b) does not combine with ordinary cardinals, and by that diagnostic is uncountable. Notice that the morpheme -ero in (28d) applied to the numeral becomes the syntactic head and assigns genitive to its complement (analogous to the case of derived group numerals).

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a. dv-ojí život two-OJI life ‘two kinds of life’ (as in a Jekyll and Hyde scenario) b. dv-ojí sýry c. dv-ojí tvář two-OJI cheese two-OJI face ‘two kinds of cheese’ ‘two faces’ (as in a Janus face) d. čtv-ero sýr-uº four-ERO cheese-GEN.PL ‘four kinds of cheese’

Unlike the previous two complex numerals, the nouns with which -ojí combines are not as restricted. Still, there are some noun types for which usage is more usual than for others, namely liquid and substance nouns, pluralia tantum and abstract nouns. Examples of each type are shown in (29)–(31), respectively. (29)

Liquid and substance nouns: krev ‘blood’, víno ‘wine’, sýr ‘cheese’

(30)

pluralia tantum: housle ‘violin’, šaty ‘dress’

(31)

abstract nouns: život ‘life’, metr ‘measure’, občanství ‘citizenship’

Similarly to what was found with complex aggregate numerals derived by -oje, further combination with všechny ‘all’ or simple cardinals is disallowed, as shown in (32) and (33), respectively. (32)

a. *všechny dv-ojí sýry DET two-OJI cheese ‘all two kinds of cheese’ b. *všechen dv-ojí život DET two-OJI life ‘All two kinds of life’ c. *všechno čtv-ero sýr-uº DET four-OJI cheese-GEN.PL ‘All four kinds of cheese’

(33)

a. *tři dv-ojí sýry three two-OJI life ‘three two kinds of cheese’ b. *deset dv-ojí-ch životuº ten two-OJI-GEN.PL lives ‘ten two kinds of life’ c. *devět čtv-ero sýr-uº nine four-ERO cheese-GEN.PL ‘nine four kinds of cheese’

Having set out the data for complex numerals in Czech, we now turn to examining the ability of nouns in Czech to shift their interpretation as a function of the context.

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3.2.3

Restricted Nominal Flexibility

In comparison with a language such as English, the grammatical elements impacting countability and nominal interpretation in Czech are far more elaborate. At the same time, and, as we will argue, relatedly, the interpretative possibilities of nouns appear to be more impoverished than in English. At issue is the reduced number of interpretations a noun licenses. As discussed at the beginning of this paper, packaging is permitted in Czech, if there is sufficient conventional use associated with the entity at issue. Yet, two other operations frequently claimed to be “universal,” namely the Universal Grinder (Pelletier 1979) and the Universal Sorter (Bunt 1985), are even more restricted. “Grinding” appears to be broadly rejected by Czech native speakers. The examples in (34) and (35) provide standard grinding contexts in Czech, the first providing a “splattered animal” context and the second providing an “animal-as-food-product” context. Both are anomolous in Czech.11 (34)

#Po celé silnici byla kráva. on whole road was cow Intended meaning: ‘There was cow all over the road.’

(35)

#V salátu bylo prase. in salad was pig Intended meaning: ‘There was (a) pig in the salad.’

Turning to taxonomic interpretations, Czech’s grammatical means to reference taxonomic sub-kinds are more elaborate, and this corresponds to greater intricacy in both distribution and interpretation. First, in Czech, interpreting plural nouns as referring to “different kinds” of the relevant noun is often not possible to the extent it is in English. While the sentence in (36a) is unremarkable in English, its Czech counterpart distinctly odd. (36)

a. I used two oils in this salad. b. # V salátu jsem použil dva oleje. to salad AUX.1SG used two oils Intended meaning: ‘I used two oils in this salad.’

While taxonomic sub-kind readings may occur for typically non-countable nouns both as bare plurals and with simple cardinals, their distribution is restricted, both in terms of their syntactic context and in terms of their interpretation. Acceptable examples of taxonomic sub-kind uses with bare plural noun phrases and with a simple cardinal phrase are given in (37) and (38),12 respectively. 11

12

We tested these sentences with thirty-two native speakers of Czech recruited from Masaryk University. The first sentence was accepted as a mass interpretation by only two of the thirtytwo, and only eight out of thirty-two consented to a mass interpretation for the second sentence. We thank an anonymous reviewer for providing example (38a) and several other examples of simple cardinal phrases with taxonomic readings.

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(38)

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a. Prodává-me vína lahvová i stáčená sell-1PL wine.PL in-bottles and wine-on-tap ‘We sell wines in bottles and on tap.’ b. Prodává-me oleje pro osobní, nákladní a užitková vozidla sell-1PL oil.PL for personal cargo and utility car.PL ‘We sell oils for personal cargo and utility cars.’ V Brně mají na čepu další tři piva in Brno have.3PL on tap next three beer.PL ‘They have three beers on tap in Brno.’

Countable nouns appear also to receive taxonomic sub-kind interpretations, although usually manifest post-nominal modification which ensures a taxonomic interpretation, as in (39). (39)

º Chováme psy (ruzných ras) breed.we.PL dog.PL (different.GEN.PL types.GEN.PL) ‘We breed (different types of ) dogs.’

A common trait of all the examples of taxonomic uses of non-countable nouns was they were found in generic, existential or otherwise non-episodic contexts. This contrasts with clearly episodic uses of nouns in their taxonomic interpretation in English, as in (36a). This further contrasts with the use of the taxonomic numeral, which is licensed both in episodic and non-episodic contexts. This is shown in (40), where simple cardinal phrases used in generic and episodic descriptions produce a contrast in acceptability, shown in (40a) and (40b), while the use of the taxonomic numeral is acceptable in both contexts, as shown in (40c) and (40d). (40)

a. Naše benzínka prodává tři paliva. our gas-station sells.IMPERF-HAB three fuel.PL ‘Our gas station sells three fuels.’ b. ??Naše benzínka včera prodala tři paliva our gas-station yesterday sold.PERF three fuel.PL ‘Yesterday our gas station sold three fuels.’ c. Naše benzínka prodává trojí palivo. our gas-station sells.IMPERF-HAB three-kind fuel.SG ‘Our gas station sells three fuels.’ d. Naše benzínka včera prodala trojí palivo. our gas-station yesterday sold.PERF three-kind fuel.SG ‘Yesterday our gas station sold three fuels.’

A second difference between the use of simple cardinals or bare plurals to indicate taxonomic sub-kinds and the use of taxonomic numerals is in the type of sub-kind that is identified. With the use of simple cardinals or bare plurals, specific, named, and well-established “sub-specimens” appear to be the most salient, and possibly the only, type of taxonomic entities referenced. For

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instance, in the example (40a), the use of the simple cardinal with a taxonomic interpretation would be interpreted as contrasting, for example, specific types of gasoline, such as with different octane levels (e.g., 85, 95 or 98). The use of the taxonomic numeral, however, typically brings about upper-level taxonomic contrasts, such as between diesel, gasoline and natural gas, although the more specific types are also a possible interpretation. We now take stock of the implications of Czech’s nominal system for theories of countability. 3.3

Interim Discussion

All theories of countability have at their core a claim of what countability is about. By measuring the claims of different theories against the Czech data, we can work towards narrowing down the space of theoretical possibilities. We return to the three issues introduced in Section 3.1: (i) whether nouns are ontologically uniform or multi-sorted, (ii) the nature of aggregates, and (iii) the trade-off between a language’s potential for nominal flexibility and its capacity for expression through morphology. To gain insight into the first issue, we consider what it means within different theories to be non-countable and then compare those conjectures with Czech data from the non-countable nouns derived by -í. In brief, we will show that the Czech data poses challenges for theories in which there is one overarching explanation of how (non-)countability arises. In particular, these explanations falter when extended to account for the second form of noncountable nouns in Czech derived by -í. First, we consider the theory of Borer (2005), for which all nouns begin as noncountable, and then through combination with a functional head, e.g., Div0, may become countable. Two facets of the Czech system in conjunction undermine this claim: the lack of grinding and the nouns in Czech derived by -í. Instead, the opposite trajectory is observed. Non-countable aggregate nouns such as listí ‘foliage’ begin as nominal roots for which there is only evidence that they are fully countable nouns, that is, list ‘leaf’ only has a countable interpretation and no effects from ‘grinding’ may be observed. Through combination with morphology, these nouns become then non-countable with respect to combination with simple cardinal numerals, which is exactly opposite of the prediction in Borer (2005). This is not to say that the system in Borer (2005) could not be altered to account for the immediate data under consideration, in the way that others have expanded her system to account for different countability phenomena (De Belder 2013; Mathieu 2012). Rather, the phenomenon of -í nouns in Czech stands in contradistinction to the conceptual predictions of Borer (2005). Similar unmet expectations are found when considering how the data aligns with the theory of Chierchia (2010), for which non-countable nouns differ from countable ones in that non-countable nouns designate entities for which the

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atomic parts are vague. Applying this line of reasoning to -í nouns would also be counterintuitive, since the parts, designated by the derivational source, e.g., list‘leaf/leaves’, are non-vaguely atomic and fully countable. One would have to argue either that the semantic contribution of -í brings about vagueness into the derived forms, or that there is a secondary cause for non-countability.13 In either case, this subclass of non-countable nouns would go against the grain of the explanation of non-countability in Chierchia (2010). More generally, the subclass of non-countable nouns derived by -í provides a strong argument that non-countability may arise from more than one source. Accordingly, theories for which there is a single path of explanation that gives rise to (non-)countable interpretations are simply not expressive enough. In other words, the domain of nominal meaning is not of a single ontological sort, but many-sorted, containing at least substances and what we have termed aggregate nouns, those derived by -í, the nature of which we turn to now. Granting the existence of aggregates nouns as separate from substance nouns, many questions remain about their nature. Again, measuring the theoretical possibilities by the Czech data proves illuminating. The grammatical number system of Czech forces the recognition of (i) aggregate nouns as a distinct class, both through devoted derivational morphology (-í) and through a devoted complex numeral, and (ii) a stronger notion of aggregate than is often employed in theories of countability. The first point dovetails with increasing acceptance in the literature of at least three types of nouns, substance (noncountable), aggregate (non-countable) and individuals (countable). For instance, Bale and Barner (2009) and Deal (2017) both explicitly set out to model those three noun types. Despite increasing recognition of two sorts of non-countable nouns, the data from Czech implicates a more specific semantics than is usually given in the analysis of nouns such as furniture or footwear in English. First, in those models, aggregate nouns are essentially treated as akin to plural nouns but for the inclusion of atoms, technically speaking as atomic join semi-lattices. Second, even though theories such as those of Barner and Snedeker (2005) or Bale and Barner (2009) do recognize aggregate nouns as a distinct class, they analyze them through lexical fiat, i.e., the noncountable status of such nouns is idiosyncratic. For these accounts, extending such a treatment to -í nouns leaves their regularity and lexical semantic 13

An anonymous reviewer suggests that noun meanings derived with -í may be analyzed akin to furniture-nouns in Chierchia (2010), that is, as singular properties; however, it is difficult to see a path forward for using a denotation which is simply all the instantiations of, say, leaves, as it does not speak to the particular conditions of relatedness, discussed in Section 3.2.1, nor explain the restricted distribution. Chierchia (2010, note 11) carefully distinguishes furniture-nouns from “collectives,” noting that single entities may qualify as furniture, as in “That chair is furniture,” but this is at odds with -í nouns which may not felicitously reference atomic parts, but only coherent collections.

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cohesion unexplained. Why should these particular nouns, and not others, receive such distinguished morphological treatment? As shown in Section 3.2.2.2, Aggregate Numerals, complex numerals derived with -oje, apply to nouns derived by -í, pluralia tantum nouns, and also certain morphologically regular plural nouns which designate entities that typically come together in multiples. Assuming that there is some form of semantic selection, then there should be a common denominator among these noun types. Analyzing aggregates as akin to plurals, however, does not provide enough discriminatory power to bring this to the fore. Since morphologically regular plurals and aggregates are selected for by Aggregate Numerals, the common denominator would be individuated lattices, yet, this would overgenerate, as any plural noun should be able to combine with Aggregate Numerals, contrary to fact. Instead, our semantics must distinguish entity types for whom its members canonically co-occur, as is the case for these nouns in Czech. Finally, we turn to considering the nature of nominal flexibility, which we argue requires a more nuanced view. The countability literature has often put forth that the possible interpretation of nouns is unrestrained, in that every countable noun can find a non-countable use, and vice versa, as is most clearly articulated in Pelletier and Schubert (2004) or Borer (2005). What then should be made of Czech’s limited flexibility? Under such a view, an interpretation as a taxonomic plural should be licensed when conceptually possible, yet, the taxonomic plural is systematically absent as an interpretation of ordinary plural nouns in episodic contexts, even though parallel examples in English demonstrate such uses are conceptually possible. The cause for this discrepancy is obvious: Czech has morphology devoted to expressing taxonomic interpretations which can be employed freely. Accordingly, regular plural noun phrases only receive taxonomic interpretations in grammatical contexts that already foster kind interpretations. Similarly, grinding with bare singular nouns is absent in Czech. Again, there is a plausible reason for this: Unlike in English, Czech does not have articles, so bare singular nouns already serve a role in Czech (see also Cheng et al. 2008; Rothstein 2017). The broader implication is that the potential for nominal flexibility in a given language is influenced and constrained by distinctions already expressed within the morphosyntax of nominals in that language. That is, the lexical semantics of nouns manifests structure that is coordinated with what is expressed in the grammatical structure. 3.4

A Formal Treatment of Czech’s Grammatical Number System

We now turn to examining how to integrate Czech’s nominal system within a formal analysis of countability. We develop an analysis by extending a version of the theory of Krifka (1995). The system in Krifka (1995) provides a useful starting point: It develops explicit representations of several ingredients to the

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semantics of countability, such as taxonomic reference and a notion of “natural units” (in contrast to several other theories, discussed in Sections 3.1 and 3.3, committed to parsimonious views of the semantics underpinning countability). As such, it provides a suitably expressive framework to model many of the distinctions present in Czech. Even so, after presenting the basic components of the theory of Krifka (1995), we still must further develop a variety of extensions to the system in Section 3.5 in order to fully account for the data. 3.4.1

Nominal Semantics in Krifka (1995)

Krifka (1995) integrates two lines of research on nominal semantics. On the one hand, building on Link (1983) and Krifka (1989), among others, the domain of objects is structured according to the basic principles of mereology, and as such models nominal meaning as complete semilattice structures lacking the null element. On the other hand, it builds on the work on generics, in which nominal meaning includes both reference to objects and reference to kinds (Carlson 1980; Krifka et al. 1995; Zamparelli 1999; Müller-Reichau 2006). Krifka (1995) proposes a revision to kind-based nominal semantics: Instead of kinds, the broader category of concepts is used, of which kinds are a special subset. We will adopt this distinction for a general framework of nominal meaning, yet as it will not be critical for the purposes of the present discussion, we will often use the terms kind and concept interchangeably. The two levels of nominal meaning are related by a realization relation R between concepts and the instances of the concept at the level of objects. That is, the referential use of dog is tied to the realizations of the concept dog, namely, the instances of a concept. A second relation discussed in Krifka (1995) is a taxonomic relation T holding between kinds/concepts and their sub-kinds or subconcepts, where the sub-kind reading of dogs would correspond to “different types of dogs,” such as beagle or chihuahua. The basic meaning of a noun in this system is given in (41), where variables ranging over kind-level entities are subscripted by k. In prose, (41) describes “the property of being a specimen or subspecies, or an individual sum of specimens or subspecies” (Krifka 1995, p. 399).14 (41)

λyk λiλx½Ri ðx; yk Þ∨Ti ðx; yk Þ

Given (41), an entity may satisfy the predicate dog in two ways. First, it may be an individual dog, or a plural individual composed of dogs, which are objects related to the kind dog by the realization relation R. Second, the predicate may be satisfied on the taxonomic reading, where the entity must be an individual sub-kind of dog (chihuahua) or sum of sub-kinds, which are related to the kind dog by the taxonomic relation T. The remainder of this 14

In (41), i is a variable of type s ranging over possible worlds – in what follows, we will often simplify by extensionalizing the representations.

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section focuses on object-level interpretations, leaving the discussion of taxonomic-level interpretations for Section 3.4.2. Krifka’s account further includes measure functions to model expressions such as two liters or five ounces, and more pertinently for us, a measure function relevant for countable nouns that counts “natural units” – relative to the kind that the noun names – named the “object unit” operator (OU). The OU operator provides a measure, n, of the number of elements which qualify as instances of the kind. Thus, there are two criteria of applicability at work in the semantics of nominals: one which is “qualitative” and represented by the nominal predicate, for example, gold or dog, and a second which is “quantitative,” represented by a “natural unit” measure function. Countable nouns under this account are two-place relations between numbers and entities ðhn; he; t iiÞ, while non-countable nouns are one-place relations ðhe; t iÞ, as shown the contrasting lexical entries for dog and gold in (42a) and (42b), respectively. (42)

a. 〚dog〛≔λnλx½Rðx; DOGÞ ^ OUðDOG; xÞ ¼ n b. 〚gold〛≔λx½Rðx; GOLDÞ

Krifka, partly inspired by classifier languages such as Mandarin, discusses where the natural unit measure function is located in the extended nominal phrase in English. One option is that the OU operator is endemic to the noun, as is the case in (42a), and cardinal numerals are simply an argument of type n as in (43a). Alternately, the OU operator may be supplied by (cardinal) numerals, that is, they include a built-in classifier as in (43b). (43)

a. 〚½threeNum 〛¼ 3 b. 〚½threeNum 〛¼ λyλx½Rðx; yÞ ^ OUðy; xÞ ¼ 3

While Krifka (1995) provides some arguments for including a built-in classifier for numerals in English, these arguments do not transfer over to Czech. One main argument given by Krifka (1995) is that assuming the OU operator as part of nominal meaning does not dispose of the need for a syntactic distinction between countable and non-countable nouns, as there is a ban in English on using singular count nouns as noun phrases. Yet, Czech differs as bare singulars are permitted, as was shown in Section 3.2. Second, the very nature of the complex numerals in Czech discussed here indicate that building the OU operator into the numeral would be conceptually difficult for number terms generally: If the semantics of the numeral stem troj- `three’ included a built-in classifier as specified in (46b), then the morphemes -ice, -oje, and -oji would then necessarily have to eliminate or overwrite the built-in classifier material in order to count groups, collections, or kinds.15 15

A third argument given by Krifka (1995) is that since many nouns can be used as countable or non-countable nouns in English, as in Czech, building the OU operator into the noun would,

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In sum, we will take kinds as the building block of nominal meaning, as argued for in Carlson (1980) or Krifka (1995) and assumed in much other work.16 In addition, we assume that for countable nouns the shift from kindlevel to object-level reference is accompanied by the lexically specified OU operator. That is, countable common nouns are of type hn; he; t ii. Cardinal numerals, in their primary function, simply supply a number value. We make standard assumptions for the nominal syntax. We assume the determiner phrase includes a NumP layer. The num head can be occupied by cardinals 2–4 and 5 and above, which contribute their cardinal value as the number argument and require plural agreement on the noun.17 Compositionally, an NP of type ⟨n,⟨e,t⟩⟩ has its number argument saturated by a cardinal number of type n, where the plural form is required for agreement. Alternately, num can be occupied by non-quantified singular or plural nouns, whereby singular or plural morphology is semantically valued as the number value 1 or 2, respectively.18 In this case, an NP occupies NUM whereby its number argument is saturated by the noun’s morphological number value. This view accords with standard current formal syntactic analysis of Slavic numerals (see Marušič and Nevins 2009; MiechowiczMathiasen 2012; inter alia).19

16

17

18 19

erroneously, disallow this flexibility. We are skeptical that the observed flexibility in English or Czech necessitates abandoning specifying countability as part of nominal meaning, since the facts could be accounted for, arguably more successfully, by analyzing flexibility as resulting from either (i) true ambiguity, as in the case of string which designates pieces of string or the material, or (ii) pragmatic accommodation. See Grimm (2018) for extended argumentation. The analysis we articulate below is in principle compatible with taking properties as basic, as in Krifka (2003), with a slightly different architecture guiding the type shifts, although we do not elaborate here. We note that there is tension in this analysis between the denotation of cardinal numerals as of type ⟨n⟩, that is, simply supplying a number value, and the syntactic modifier status of Czech cardinals 1–4. Yet, we assume, following Geurts (2006) and Rothstein (2017), among others, that a type-shift is generally available between the argumental ⟨n⟩ and predicative/adjectival type ⟨e,t⟩. This is particularly plausible in the case of Slavic numerals, as they have undergone significant grammaticalization whereby mismatches between syntactic and semantic category are not unusual. (See, for instance, Miechowicz-Mathiasen (2012) who discusses the mixed adjectival and nominal properties of lower numerals in Polish). Still, as a reviewer pointed out, it is important to assess how the modifier status of cardinals 1–4 affects the analysis we propose. We provide another, alternate analysis which represents the modifier status of lower cardinals, yet maintains the rest of the semantics that we propose. As this is not central to main thrust of our study, this is discussed in footnotes 22–24 and 26. (We thank Hana Filip and Peter Sutton for discussion and suggesting this line of analysis.). We assume an exclusive view of plurality although nothing in particular hinges on this choice. See Grimm (2013) for discussion. At first sight, that numerals  5 in Slavic languages show singular agreement on the verb when their NP is used in subject position might appear to be an empirical obstacle. Yet, this singular agreement is convincingly argued by Marušič and Nevins (2009) and Marušič et al. (2015), among others, to be a failure of the verb to agree at all, since in Slavic languages, singular (with neuter gender – as in the case of numerals  5) is the default number for a verb, as witnessed by non-argument weather verbs producing singular (neuter) non-agreement as well.

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The general theory of nominal reference outlined so far is able to account for the basics of nominal reference in Czech, as well. The representations in (44) present the different types of nouns discussed in increasing complexity: kindreferring bare singular (44a), non-countable noun (44b), singular countable noun (44c), plural countable noun (44d), a plural genitive NP form, which provides an unsaturated property meaning (44e), and a (genitive plural) noun combining with a simple cardinal numeral pět ‘five’ (44f).20 (44)

a. 〚½N pes〛¼ DOG b. 〚½NP prach〛¼ λx½Rðx; DUSTÞ c. 〚½NumP pes〛¼ λx½Rðx; DOGÞ ^ OUðDOG; xÞ ¼ 1 d. 〚½NumP psi〛¼ λx½Rðx; DOGÞ ^ OU ðDOG; xÞ  2 º e. ⟦[NP psu]⟧ = λnλx[R(x,DOG) ^ OU(DOG,x) = n] 0 º º = ⟦[NP psu]⟧(⟦[Num pět]⟧) f. ⟦[NumP [Num0 pět][NP psu]]⟧ = λx[R(x,DOG) ^ OU(DOG,x) = 5]

Additionally, we will assume, following Krifka (1995), the availability of a shift from a kind to a number-neutral predicate of the kind whereby the OU operator is existentially bound, given in (45). This shift derives a predicate interpretation which can be passed along to the complex numerals for aggregates. (45)

〚½NP psi〛¼ λx9n½Rðx; DOGÞ ^ OUðDOG; xÞ ¼ n

As we now show, this account, with minor modifications, also extends straightforwardly to the taxonomic interpretations of nominals in Czech. 3.4.2

Taxonomic Interpretations and Numerals

To incorporate the taxonomic interpretations found with bare plurals and simple cardinals, discussed in Section 3.2.3, we generalize the nominal semantics to include taxonomic units from the system of Krifka (1995). Accordingly, the base template for nominal meaning in Czech is (46), with the realization

20

In an alternate analysis which represents the modifier status of lower cardinals, numerals 1–4 may be analyzed as of type ⟨⟨n, a⟩, a⟩ where a is a variable over types. The relevant type for simple cardinal constructions is a = ⟨e,t⟩. The lexical entry for dva ‘two’ is given in (i). This also requires a more specific analysis of the plural morpheme, given in (ii), which derives plural nouns as in (iii). Thus, the derivation of dva psi ‘two dogs’ would proceed as in (iv), delivering a parallel representation to (44f). (i) (ii) (iii) (iv)

⟦[Num0 dva]⟧ = λP:⟨n,⟨e,t⟩⟩[P(2)]:⟨e,t⟩ ⟦-PL⟧ = λkλnλx[R(x, k) ^ OU(k, x) = n ^ n  2] ⟦ [N psi]⟧ = ⟦-PL⟧(⟦ [N pes]⟧) = λnλx[R(x,DOG) ^ OU(DOG,x) = n ^ n  2] ⟦[NumP [Num0 dva][N psi]]⟧ = ⟦[Num0 dva]⟧(⟦[NP psi]⟧) = λP[P(2)](λnλx[R(x,DOG) ^ OU(DOG,x) = n]) = λnλx[R(x,DOG) ^ OU(DOG,x) = n ^ n  2](2) = λx[R(x,DOG) ^ OU(DOG,x) = 2 ^ 2  2]

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operator R and the taxonomic operator T, which can be generalized into a single operator, RT, given in (47). (46)

λyk λx½Ri ðx; yk Þ∨Ti ðx; yk Þ

(47)

RTðx; yÞ $ Rðx; yÞ∨Tðx; yÞ

To this we add the analogue for the OU operator for kinds, the KU (‘kind unit’) operator which, for each possible world, when applied to a kind, delivers the number of subspecies of that kind. Both measure functions, OU and KU, can be joined into one operator, OKU (‘object or kind unit’), defined in (48) Krifka (1995, p. 406). (48)

OKUi ðxÞðyÞ ¼ n $ OUi ðxÞðyÞ∨KUi ðxÞðyÞ ¼ n

The different types of nominal reference are given in their generalized form in (49). (49)

a. 〚½N pes〛¼ DOG b. 〚½NP prach〛¼ λx½RT ðx; DUSTÞ c. 〚½NumP pes〛¼ λx½RTðx; DOGÞ ^ OKUðDOG; xÞ ¼ 1 d. 〚½NumP psi〛¼ λx½RTðx; DOGÞ ^ OKUðDOG; xÞ  2 º = λnλx[RT(x, DOG) ^ OKU(DOG, x) = n] e. ⟦[NP psu]⟧ º f. ⟦[NumP [Num0 5][NP psu]]⟧=λx[RT(x, DOG) ^ OKU(DOG, x) = 5]

These representation secure the desired taxonomic interpretations for countable nouns. For instance, psi ‘dogs’ may receive the interpretation, stated in simplified form, λx½Tðx; DOGÞ ^ KUðDOG; xÞ  2, denoting the set of sums of sub-kinds of dog greater than or equal to 2. It is prudent to retain the representation in (49b) for non-countable nouns, where no unit operator is specified, as non-countable nouns may be used to refer to assemblages of stuff which do not cohere to any units. At the same time, it is necessary to allow non-countable nouns to shift to the representations in (49d)–(49e). We assume a rule which converts non-countable noun templates to countable ones, that is, from λyλx½RTðx; yÞ to λyλx½RTðx; yÞ ^ OKUðx; yÞ ¼ n, when contextually licensed, for example, for taxonomic interpretations in generic or otherwise kind-selecting contexts. This then licenses a reading for, e.g., tři paliva ‘three fuels’, namely λx½Tðx; FUELÞ ^ KUðx; FUELÞ ¼ 3. Packaging interpretations are analogously derived. We now turn to the second method of deriving taxonomic interpretations in Czech. We argue that the taxonomic operator T is also found in the meaning of the complex taxonomic numerals derived by -ojí. The lexical entry for the derivational suffix -ojí is given in (51), designating “different kinds.” In prose, -ojí takes a numeral, a kind, and a sub-kind or individual sum of sub-kinds, checks that all the parts of the (sum of ) sub-kind are a sub-kind of the kind specified by the nominal head and that their cardinality is equal to that specified by the numeral. (50)

〚  oj
ı〛¼ λnλkλx½8zðz < x ^ Tðk; zÞÞ ^ jxj ¼ n

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This ensures, for example in (51), that when combining with a numeral, here dv- ‘two’, and a kind denoting root noun, here [N víno] ‘wine’, which happens to be non-countable at the object level, the resulting cardinal phrase designates a sum of sub-kinds of wine whose cardinality is 2.21 (51)

〚dvoj
ı v
ıno〛¼ λx½8zðz < x ^ TðWINE; zÞÞ ^ jxj ¼ 2

A further welcome result is an explanation of the oddity when combining taxonomic numerals with uniquely denoting noun phrases, as in (52a), and with proper names, as shown in (52b): These nouns do not supply the kind terms which -ojí requires. (52)

a. #dv-ojí noha tohoto stolu two-OJI leg this table ‘two kinds of this table’s leg’ b. #dv-ojí Petr Novák two-OJI Petr Novák ‘two kinds of Petr Novák’

Section 3.2.3 showed distributional differences between the uses of taxonomic numeral phrases, which are contextually unrestricted, and taxonomic interpretations of bare plurals or simple cardinal phrases, which are restricted to non-episodic contexts. This receives an explanation on the account provided here, since bare plurals or simple cardinal phrases require appropriate context, e.g., a generic context, to shift from non-countable to countable noun meanings, while the taxonomic numerals do not depend on contextual support. Section 3.2.3 also demonstrated a difference in interpretation depending on whether a simple cardinal/bare plural was used or a taxonomic numeral was used. Taxonomic uses of simple cardinal and bare plural noun phrases refer to specific, well-established and named sub-kinds, such as specific types of varieties of wine (Chardonnay versus Riesling). This corresponds to exactly the sort of entities that would be returned by the KU operator, that is, true ‘kind units’. On the other hand, taxonomic entities referred to with taxonomic numeral phrases tended to be upper-level taxonomic contrasts, such as red versus white wine. In the analysis of taxonomic numeral phrases given here, any taxonomic entities standing in sub-kind/concept relation to the general kind/concept are permitted, thus, it is expected that their interpretation may 21

Here too the alternate analysis of lower cardinals as of generalized type ⟨⟨n, a⟩, a⟩ may be pursued to represent their adjectival status as modifiers. In this instance, the type will be ⟨⟨n, ⟨k, ⟨e,t⟩⟩⟩, ⟨k, ⟨e,t⟩⟩⟩, as shown for the lexical entry of dva in (i). This combines with the entry for -ojí, shown in (ii), ultimately leading to the same result as in (51). (i) ⟦[Num0 dva]⟧ = λP:⟨n, ⟨k, ⟨e,t⟩⟩⟩ [P(2)]:⟨k, ⟨e,t⟩⟩ (ii) ⟦dvojí⟧ = ⟦[Num0 dva]⟧(⟦-ojí⟧) = λkλx[8z(z < x ^ T(k,z)) ^|x| = 2]]

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cover upper-level taxonomic contrasts, and further, will have a tendency to do so if specific sub-kinds are already referenced by simple cardinal phrases. So far, much of Czech’s nominal system has been able to be analyzed using the elements of Krifka’s (1995) system. We now turn to aspects of Czech’s nominal system which require an extension to this basic framework. 3.5

Extending the Framework: Counting Groups and Aggregates

The last section demonstrated that, with some minor modifications, Krifka (1995) can be successfully used to treat the basic cases of nominal semantics as well as taxonomic numerals. Yet, in order to cope with the data arising from group numerals and from aggregates in Czech, more substantial extensions are required. We first enrich the system with groups in the sense of Landman (1989) to account for group numerals, and then with mereotopology in the sense of Grimm (2012) to account for aggregates. 3.5.1

Group Numerals

The system of Krifka (1995) provides a method to analyze certain types of group nouns, such as herd, making uses of measure functions. As shown in (53), the measure function counts groups in the same way it counts atomic objects. (53)

〚three herds of cows〛¼ λx½Rðx; COWÞ ^ HERDðCOW; xÞ ¼ 3

While this analysis is adequate for the data discussed in Krifka (1995), the derived collective numerals in Czech require a different analysis. In particular, counting with the aid of group numerals involves counting both at the level of the number of groups and the number of individuals inside the groups. As was shown in (17), repeated here as (54), -ice assigns a cardinal value to the members of the groups (troj-ice námořníkuº ‘a group of three sailors’), and the groups themselves can then also be counted. (54)

dvě troj-ice námořník-uº two three-ice sailors-GEN.PL ‘two groups of three sailors’

Accordingly, an analysis of these complex numerals will need to contribute a generalized way to deliver a semantics which both groups objects and counts the number of objects in the group. A straightforward solution is to augment the schema of Krifka (1995) with the group shifting operator " from Landman (1989), which maps sums of individuals to a group. Unlike the other complex numerals, -ice is the head of the noun phrase and takes a genitive argument describing the group members, here, as sailors. We analyze -ice, shown in (55a), as first combining with a number, which

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ultimately feeds the OU operator, and then the property provided by the genitive argument. As previously stated, we assume the plural genitive NP provides an unsaturated property meaning. A minimal example is given in (55c), where the application of dv- ‘two’ and then mužuº to -ice results in the set of sum individuals which are ‘groups of two men’.22 (55)

a. 〚  ice〛¼ λnλPλx½" x ^ PðxÞðnÞ ̊ λnλx½Rðx; MANÞ ^ OUðMAN; xÞ ¼ n b. 〚½NP mǔzu〛¼ c. 〚½NumP dvojice mu ̌zu〛¼ ̊ λx½" x ^ Rðx; MANÞ ^ OUðMAN; xÞ ¼ 2

This meaning is again fully countable, designating a singular individual or atom, but in the domain of groups. In particular, the meaning given in (55c) can be itself be pluralized, as well as counted by means of standard cardinal º as given in numbers, where, for instance, the meaning of dvě trojice mužu, (54), consists of the set of two groups of three men. We assume that the simple cardinals which precede the complex cardinals simply occur as modifiers, that is, the meaning of dvě ‘two’ receives a standard analysis as a modifier as λPλx½PðxÞ ^ jxj ¼ 2.23 3.5.2

Deriving and Counting Aggregates

The derived aggregate nouns and complex numerals which count aggregates require further extensions to Krifka (1995), namely the adoption of a theory of aggregate nouns. First, the derived aggregate nouns and complex numerals for aggregates discussed in Section 3.2, unlike the other phenomena discussed here, showed severe restrictions. The derivational suffix -í only applied to a restricted set of nouns, while the complex numerals for aggregates only applied to connected entities, which in turn could be morphologically encoded in the noun, as with -í derived nouns or pluralia tantum, or due to a noun’s meaning which proffers such a connectedness relation, as in the case of schody ‘stairs’, which typically come in multiples. We build on the account of Grimm (2012) to provide a semantics for aggregate nouns in Czech. Grimm (2012) provides a topologicial extension of mereology in part to treat non-countable aggregate terms in English such as (sand or foliage) as well as to treat morphologically recognized collective/ 22

Pursuing the alternate analysis of lower cardinals as of generalized type ⟨⟨n, a⟩, a⟩ in this case results in the type ⟨⟨n, ⟨⟨e,t⟩,⟨e,t⟩⟩⟩, ⟨⟨e,t⟩,⟨e,t⟩⟩⟩, as shown for the lexical entry of dva in (i). This combines with the entry for -ice, shown in (ii), delivering a parallel result. (i) ⟦[Num0 dva]⟧ = λP:⟨n, ⟨⟨e,t⟩,⟨e,t⟩⟩⟩ [P(2)]:⟨⟨e,t⟩,⟨e,t⟩⟩ (ii) ⟦dvojice⟧ = ⟦[Num0 dva]⟧(⟦-ice⟧) = λnλPλx["x ^ P(x)(2)]

23

We also note that this is a point where the analysis of simple cardinals possessing a built-in OU classifier runs aground, since the objects being counted here are not of the type amenable to being counted by a “natural unit” function, that is, there is no kind (or concept) of group of three sailor which is natural to measure via the OU function.

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singulative classes in languages such as Welsh. The principal tool is to add connectedness relations to the standard mereological framework. While topological extension of mereology in Grimm (2012) relates to core topics in countability, such as how to define atoms and substances, we only discuss the portion of the account relevant for the Czech aggregate data. Central to extending mereology with topological relations is including a relationship of connectedness. The intuitive definition of being connected is that two entities are connected if they share a common boundary. Including a definition of connectedness then interacts with the different definitions and axioms of standard mereology, as discussed in detail in Casati and Varzi (1999). We consider first the basic relation C, connected, which is required to be reflexive and symmetrical, given as axioms in (56) and (57). There are some further intuitive interactions with the mereological relations part, , and overlap, O, that Casati and Varzi (1999) note. First, the axiom in (58) ensures that parthood implies connectedness. From (58) the relation in (59) follows, whereby overlap implies connectedness. (56)

Cðx; xÞ ðReflexivityÞ

(57)

Cðx; yÞ ! Cðy; xÞ ðSymmetryÞ

(58)

x  y ! 8zðCðx; zÞ ! Cðz; yÞÞ

(59)

Oðx; yÞ ! Cðx; yÞ

Connectedness can come in a variety of strengths. Two more specific varieties of connectedness relevant for aggregate noun semantics are external connectedness, when two entities touch on their boundaries, notated as Cext ðx; yÞ and proximate connectedness, when two entities are sufficiently near relative to some distance d, C prox ðx; yÞ. We now turn to apply this extended system to aggregates in Czech. 3.5.2.1 Derived Mass Nouns We now give a semantics to the morpheme -í, whereby it applies to a root noun and returns a connected set of individuals, which we will term a cluster individual, described by a root noun. We first define a transitive connection relation in (60). In prose, x and y are transitively connected relative to a property P, a connection relation C, and a set of entities Z, when all members of Z satisfy P and x and y are connected through the sequence of zis in Z. (60)

Transitive Cðx; y; P; C; Z Þ ¼def 8z 2 Z ½PðzÞ ^ ðx ¼ z1 ^ y ¼ zn Þ ^ Cz1 z2 ^ Cz2 z3 . . . ^ Czn1 zn  where Z ¼ fz1 , z2 ; . . . , zn g (Transitively Connected)

Cluster individuals, relative to a property and connection type, are defined in (61) as a set of entities of the same type connected to one another by virtue of

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each being transitively connected to the others (through a set Y relative to the same property and connection relation). (61)

Clusterðx; P; C Þ ¼ 9Z ½x ¼ Z ^ 8z, z0 2 Z 9Y ½TransitiveCðz; z0 ; P; C; Y Þ

With the additional topological machinery, we are now able to give a semantics to -í. We analyze -í nouns as aggregate nouns which refer to cluster individuals, that is, listí ‘foliage’ refers to connected clusters of leaves and combinations thereof. For -í-derived nouns, the relevant connectedness relation is proximate connectedness, that is, all the individual leaves, or alder trees and other examples from Table 3.1, must be spatially close to one another (to a degree relevant for the noun at hand). Importantly, this is a stronger condition than just being a plural individual, and distinguishes the meaning of -í-derived nouns from simple plural meanings. As given in (62), the morpheme -í applies to the root (designating the kind) and returns the connected clusters. As no individual objects are at issue, the OU operator is existentially bound. (62)

〚 
ı〛¼ λkλx9n½Rðx; k Þ ^ x 2 CLUSTER ^ OUðk; xÞ ¼ n

The semantics given here delivers on the nouns’ properties as discussed in Section 3.2. Since the OU operator is existentially bound, it follows that these nouns resist pluralization and counting with basic cardinal numerals, since composition with such elements would fail. Unlike typical non-countable nouns such as water which allow contextual shifts to countable uses by adding a contextually specified measure function which counts units, -í-derived nouns resist contextual shifts to countable uses; since the OU operator is already part of the denotation, it cannot be supplied from context. Finally, the application of -í was seen to have severe lexical restrictions. The given analysis, which results in cluster individuals, is constrained to apply just to those nouns which describe entities which do come in connected sets, which is a much more restricted class of entities than those to which, e.g., plural morphology would legitimately apply. Although the analysis just given is, we argue, sufficient to account for the core facts concerning -í nouns, we note that there is clearly more to be said to give a full account of the special semantics of terms such as lanoví ‘rigging/ ropes’ (< lano ‘rope’) or krajkoví ‘lacework’ (< krajka ‘lace’), and similarly for maliní ‘cluster of raspberry bushes’ (< malina ‘raspberry’). Mere spatial connectedness of a set of ropes is necessary but not sufficient for them to constitute a ship’s rigging, since, for example, the ropes must be of the appropriate types and organized in the fashion required as specified with respect to the function of rigging on a ship. However, we leave such intriguing details to the side on this occasion. 3.5.2.2 Aggregate Numerals Armed with a semantics for -í-derived nouns, we now return to specifying the semantics for the complex numerals for aggregates expressed by the suffix -oje. As the complex numerals count

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-í-derived nouns, they clearly count cluster individuals; however, complex numerals for aggregates cannot count arbitrary cluster individuals, as there may be many such clusters in a given connected set: A cluster individual of n alder trees will of course contain many smaller cluster individuals of alder trees, n1 and so on. Instead, the complex cardinals for aggregates count maximal clusters, defined in (63), which are then disjoint. We then employ maximal clusters, notated MaxCluster, in our semantics for -oje given in (64). (63)

MaxCluster ðx;PÞ¼9C½Cluster ðx;P;CÞ^8yðClusterðy;P;CÞ^Oðy;xÞ$yxÞ

(64)

〚  oje〛¼ λnλPλx½PðxÞ ^9Y ½8zðz < x^ MaxClusterðz;PÞ $ z 2 Y Þ^ jY j ¼ n

-oje first combines with a numeral, similarly to the other complex numerals, and then with a noun, whose denotation is filtered to a derived set containing just the maximal clusters of the specified cardinality. As with group numerals, aggregate numerals combine with the predicate noun denotation, as opposed to taxonomic numerals which select for the kind denotation.24 While in the case of group numerals, we assumed that the genitive NP complement was an unsaturated NP meaning as it entered into the composition with the group numerals, for aggregate numerals there are several cases to consider. They can count derived nouns like listí ‘foliage’, which directly supply clusters, and, in what is no doubt something of a simplification, we will assimilate pluralia tantum nouns under this class, too. Aggregate numerals also count what are in one sense standard countable nouns such as hranolky ‘French fries’, karty ‘cards’ or klíče ‘keys’ but which also regularly occur in connected sets, and thus also supply clusters, although we leave it open here whether this is lexically specified or pragmatically accommodated. The semantics of listí ‘foliage’ and hranolky ‘French fries’ are given as example meanings in (65) and (66), respectively. For hranolky ‘French fries’ and other such predicates, we assume they undergo a shift to the number-neutral predicate before entering into composition with the aggregate numeral, which has the effect of binding off the number argument.25 (65)

24

25

〚½NP list
ı〛¼ λx9n½Rðx; LEAFÞ ^ x 2 CLUSTER ^ OUðLEAF; xÞ ¼ n

The alternate analysis of lower cardinals as of generalized type ⟨⟨n, a⟩, a⟩ for -oje is parallel to that of -ice, namely analyzing the cardinals as of type ⟨⟨n, ⟨⟨e,t⟩,⟨e,t⟩⟩⟩, ⟨⟨e,t⟩,⟨e,t⟩⟩⟩, whereby the composition returns the same result. The assumption that Czech nouns are ambiguous between predicates (he; ti) and kinds (k) receives some additional support from adjectival modification. Taxonomic numerals are not compatible with stage-level adjectives modifying its head noun – ???dvojí včerejší chleba ‘twokinds of yesterday bread’ – but completely grammatical with kind-level adjectives – dvojí tmavý chleba ‘two-kinds of dark bread’. But there is no such restriction with complex numerals for aggregates: dvoje včerejší/elektronické noviny ‘two-sets of yesterday/digital newspapers’.

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116 (66)

Scott Grimm and Mojmír Dočekal 〚½NP hranolky〛¼ λx9n½Rðx; FRYÞ ^ OUðFRY; xÞ ¼ n

We put the different elements together for the case of hranolky ‘French fries’, and abbreviating (66) as FRY(x), producing the meaning in (67): (67)

〚½NumP patery hranolky〛 ¼ λx9n½FRYðxÞ^9Y ½8zðz < x^MaxClusterðz;FRYÞ $ z 2 Y Þ^ jY j ¼ 5

As such, the semantics arrives at a representation of -oje numerals already given in traditional grammatical descriptions of Czech, that they count just objects which are somehow connected. Further, this analysis provides an explanation for the contrast given in (68), where -oje is infelicitously used with a noun which does not designate connected entities. (68)

#Petr viděl na dvorku dv-oje psy. Petr saw on yard two-OJE dogs ‘Peter saw two sets of dogs in the yard.’

Other uses of -oje numerals indicate that it can be used even when a noun’s semantics does not include (maximal) clusters in its denotation. The example in (69) shows an instance where -oje counts entities described by a non-countable noun which has been coerced into a packaging reading, where the packages are themselves complex objects for which the number of objects is further specified.26 (69)

dv-oje vody po šesti two-OJE water DIST six ‘two packages of water, each consisting of six bottles’

Thus, -oje numerals are at once compatible with nouns which provide the appropriate semantic type, cluster individuals, but may also impose such an interpretation on nouns which do not standardly designate it. 3.6

Outlook: Countability from the Perspective of Czech

We have made the case that increased attention to languages with complex nominal morphology is valuable for gaining insight into countability. In this

26

Under strong pragmatic pressure even examples such as (68) can be acceptable, but only if the context provides the connected meaning which is required by the -oje numerals, for example, in a scenario where the two pluralities of dogs are harnessed to two different sleds. (Thanks to an anonymous reviewer for pressing this point.) Yet, there is a palpable difference between nouns which denote connected entities due to their lexical meaning (and consequently are always compatible with the -oje numerals) and other nouns, as witnessed by a simple Google search: dvoje nádobí ‘two sets of dishes’ returns 142 results, many of which are in “out of the blue” contexts, while dvoje psy ‘two sets of dogs’ returns six results, all of them used in a context providing connected meaning, for instance, two sets of dogs, each set owned separately, or housed in different locations.

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section, we conclude by inverting the usual perspective and considering the English nominal system from the perspective of Czech. Czech demonstrates that aggregates, groups and taxonomic kinds can be counted in their own right through particular morphological means. From the perspective of Czech, the nominal system of English underspecifies the contrasts visible in Czech morphology. More nuanced differences arise as well, concerning the overall properties of the different systems in Czech and English and affecting how different interpretations are accessed. In both Czech and English, the plural morpheme delivers sums of objects or sums of taxonomic sub-kinds and therefore, in the terminology of programming languages, is “overloaded” (in the same way that the + operator is often “overloaded” to perform addition on integers and concatenation on strings). Yet, in Czech the presence of overt grammatical means to count taxonomic sub-kinds, namely taxonomic numerals, impinges on the distributional range of the taxonomic plural, which is restricted to non-episodic contexts, and its interpretational range, which is restricted to specific, well-established kind units. In contrast, the taxonomic sub-kind interpretation in English is not grammatically nor interpretationally restricted, but, apparently, only by pragmatic and/or conceptual possibility. Similarly, we have argued that the restricted interpretative possibilities of bare nouns in Czech with respect to countability are both a function of the richer interpretative possibilities overtly expressed in morphology as well as, and probably more critically, a function of the presence of bare singular count nouns. It remains to be seen how well this generalizes typologically. At present, it is clear that there is no positive evidence for extending to Czech theories which claim that there is a derivational relation between “ground” interpretations of nouns and interpretations as individuals – for example, Bale and Barner (2009) assert that ground interpretations are foundational and that “all count nouns are derived from lexical items that denote non-individuated semi-lattices.” Considering English from the perspective of Czech, ground interpretations are a peripheral phenomenon, which only occur in limited contexts, not foundational. In sum, Czech would appear to be a well-regimented language, where clear constraints determine in which morphosyntactic context different nominal interpretations may arise; in contrast, English would appear to be very permissive and less constrained by morphosyntactic contexts and more open to contextual shifts in meaning. These observations connect to what has been repeatedly observed in other areas of grammar: What are hard grammatical constraints in one language are soft statistical constraints in another (Givón 1979; Bresnan et al. 2001). Thus, the oft-noted particular contextual requirements to produce, e.g., ground interpretations could be seen as a soft statistical constraint against secondary interpretations of nouns designating whole individuals. Clearly much work remains in order to gain a fuller understanding of the relation between morphosyntax and lexical semantics in nominals. Ultimately,

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this should aid us in understanding the causal foundations of countability: Why do languages make the countability distinctions that they do? Further investigations will undoubtedly discover more distinctions, and even within Slavic languages there is much more variation than needs to be investigated and understood. REFERENCES Acquaviva, Paolo (2008). Lexical Plurals: A Morphosemantic Approach. Oxford: Oxford University Press. Bach, Emmon (1986). The algebra of events. Linguistics and Philosophy 9.1: 5–16. Bale, Alan, and David Barner (2009). The interpretation of functional heads: Using comparatives to explore the mass/count distinction. Journal of Semantics 26.3: 217–252. Barner, David, and Jesse Snedeker (2005). Quantity judgments and individuation: evidence that mass nouns count. Cognition 97.1: 41–66. Borer, Hagit (2005). Structuring Sense i: In Name Only. Oxford: Oxford University Press. Bresnan, Joan, Shipra Dingare, and Chris Manning (2001). Soft constraints mirror hard constraints: Voice and person in English and Lummi. In Proceedings of the LFG01 Conference, pp. 13–32. Stanford, CA: CSLI. Bunt, Harry C. (1985). Mass Terms and Model-Theoretic Semantics. Cambridge: Cambridge University Press. Carlson, Gregory N. (1980). Reference to Kinds in English. New York, NY and London: Garland Publishing. Casati, Roberto, and Achille C. Varzi (1999). Parts and Places: The Structures of Spatial Representation. Cambridge, MA: MIT Press. Cheng, G., et al. (2008). Searching semantic web objects based on class hierarchies. In WWW-2008 Workshop on Linked Data on the Web, pp. 199–226. Chierchia, Gennaro (1998a). Reference to kinds across languages. Natural Language Semantics 6.4: 339–405. (1998b). Plurality of mass nouns and the notion of “semantic parameter”. In Susan Rothstein (ed.), Events and Grammar: Studies in Linguistics and Philosophy Vol. 7, pp. 53–104. Dordrecht: Kluwer Academic Publishers. (2010). Language, thought, and reality after Chomsky. In Julie Franck and Jean Bricmont (eds.), The Chomsky Notebook, pp. 142–169. New York, NY: Columbia University Press. Czech National Corpus (2018). www.korpus.cz. Dalrymple, Mary, and Suriel Mofu (2012). Plural semantics, reduplication, and numeral modification in indonesian. Journal of Semantics 29.2: 229–261. Dayal, Veneeta (2004). Number marking and (in) definiteness in kind terms. Linguistics and Philosophy 27.4: 393–450. De Belder, Marijke (2013). Collective mass affixes: When derivation restricts functional structure. Lingua 126: 32–50. Deal, Amy Rose (2017). Countability distinctions and semantic variation. Natural Language Semantics 25: 125–171. Dočekal, Mojmír (2012). Atoms, groups and kinds in Czech. Acta Linguistica Hungarica 59.1–2:109–126.

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Lima, Suzi (2014). The Grammar of Individuation and Counting. PhD Dissertation, University of Massachusetts Amherst. Link, Godehard (1983). The logical analysis of plurals and mass terms: A latticetheoretical approach. In Rainer Bauerle, Christoph Schwarze, and Arnim von Stechow (eds.), Meaning, Use, and Interpretation of Language, pp. 302–323. Berlin: de Gruyter. Marušič, Lanko, and Andrew Ira Nevins (2009). Two types of neuter: Closest-conjunct agreement in the presence of ‘5&ups’. In A. Fisher, E. Kesici, N. Predolac, D. Zec, W. Browne, and A. Cooper (eds.), Annual Workshop on Formal Approaches to Slavic Linguistics, The Second Cornell Meeting 2009, pp. 301–317. Ann Arbor, MI: Michigan Slavic Publications. Marušič, Franc, Andrew I. Nevins, and William Badecker (2015). The grammars of conjunction agreement in Slovenian. Syntax 18.1: 39–77. Mathieu, Éric (2012). Flavors of division. Linguistic Inquiry 43.4: 650–679. Miechowicz-Mathiasen, Katarzyna (2012). Licensing Polish higher numerals: An account of the accusative hypothesis. Current Issues in Generative Linguistics 81, 58–75. Müller-Reichau, Olav (2006). Sorting the World: On the Relevance of the Kind-Level/ Object-Level Distinction to Referential Semantics. PhD Thesis, Universität Leipzig. Payne, John, and Rodney Huddleston (2002). Nouns and noun phrases. In Rodney Huddleston and Geoff Pullum (eds.), Cambridge Grammar of the English Language, pp. 323–524. Cambridge: Cambridge University Press. Pelletier, Francis J. (1979). Non-singular reference: Some preliminaries. In Francis. J. Pelletier (ed.), Mass Terms: Some Philosophical Problems, pp. 1–14. Dordrecht: Reidel. Pelletier, Francis J., and Lenhart Schubert (2004). Mass expressions. In D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume x, pp. 249–335. Dordrecht: Kluwer. Rothstein, Susan (2017). Semantics for Counting and Measuring: Key Topics in Semantics and Pragmatics. Cambridge: Cambridge University Press. Veselovská, Ludmila (2001). Agreement patterns of Czech group nouns and quantifiers. In N. Corver and H. van Riemsdijk (eds.), Semi-Lexical Categories: The Function of Content Words and the Content of Function Words, pp. 273–320. Berlin: Mouton de Gruyter. Wągiel, Marcin (2015). Sums, groups, genders, and Polish numerals. In Gerhild Zybatow, Petr Biskup, Marcel Guhl, Claudia Hurtig, Olav Mueller-Reichau, and Maria Yastrebova (eds.), Slavic Grammar from a Formal Perspective. The 10th Anniversary FDSL Conference, Leipzig 2013, pp. 495–513. Frankfurt am Main: Peter Lang. Wierzbicka, Anna (1988). The Semantics of Grammar. Amsterdam and Philadelphia, PA: John Benjamins. Winter, Yoad (2001). Flexibility Principles in Boolean Semantics. Cambridge, MA: MIT Press. Wisniewski, Edward J., Christopher A. Lamb, and Erica L. Middleton (2003). On the conceptual basis for the count and mass noun distinction. Language and Cognitive Processes 18: 583–624. Zamparelli, Roberto (1999). Layers in the Determiner Phrase. New York, NY: Garland.

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Individuating Matter over Time Manfred Krifka

A body is a special kind of physical object, one that is roughly continuous spatially and rather chunky and that contrasts abruptly with most of its surroundings and is individuated over time by continuity of displacement, distortion, and discoloration. (Quine 1981)

4.1

Meditations over a Cup of Tea

Worked-out models for the semantics of mass nouns come with an assumption that is in need of qualification: that it is possible to identify the same quantity of mass over time.* For example, the ontology proposed in Link (1983) proposes a set of entities E and a set of portions of matter D. The set of portions of matter might be atomic or atom-less, but in any case consists of well-distinguished individuals. Link’s model does not include times, but temporal notions have played an essential role in motivating the proper treatment of masses and objects. For example, Link models the relation between the meaning of the ring and the meaning of the gold that makes up the ring as one of material constitution, where the entities involved may have different properties; for example, the ring may be new, but the gold that makes up the ring may be old. And to say that the stuff that makes up the ring is old makes sense only if it is possible to identify that stuff across time. Identification over time appears easier for objects, like rings, cups, or persons, because there are criteria that allow us to state that something is the same ring, cup or person that was encountered before. But tracking matter across time is tricky. Consider the predicate the cubic centimeter of tea in the center of the cup of tea in front of me. Is it possible to say that this predicate refers to the same portion of matter at 5:05 p.m. and at 5:06 p.m. (assuming that the cup of tea is just sitting in front of me, untouched)? Physics tells us no:

*

I gratefully acknowledge support by the Bundesminsterium für Bildung und Forschung (Projektförderung Zentrum für Allgemeine Sprachwissenschaft, Berlin, Förderkennzeichen 01UG0711) and the valuable critical comments by an anonymous reviewer to an earlier version of this paper.

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the molecules that make up the tea are in constant movement; some will have left the cubic centimeter in the center of the cup, others will have entered it. We may track the whereabouts of the molecules at 5:05, and point to the dissipated aggregate that these molecules form at 5:06 as the referent of the predicate the tea that constituted the cubic centimeter in the center of the cup at 5:05 – hence the same tea as the one referred to before. But this requires that we identify and trace the molecules that together make up the tea. This can be done only by a thought experiment; it would be physically impossible to trace its 3
1022 molecules over a minute. Our naïve understanding of the world around us, which we try to model in natural language semantics, also discourages talking about a portion of tea over time. If we assume an atomistic ontology, then the same problem arises as with what physics tells us: the tea consists of smallest parts that move around in ways that are impossible to track. Reidentification would be humanly impossible, hence any debate whether something is the same portion of tea as before would be moot. And under a non-atomistic ontology we could not even perform the thought experiment alluded to before, as we would have to track not only a very large number, but an infinity of parts. Assuming random movement of the parts, there would be no way to compute the locations of that infinite number of particles after one minute has gone by. In order to refer to the same portion of tea across time, we could trap that cubic centimeter in some container, separating it from its surrounding. In doing this, we make use of the time stability of objects – the container – to create time stability of masses – the tea that it contains. This leads to the question how solid objects (Quine’s bodies) achieve that – why do we consider them as stable, and why do we think that the stuff inside an object can borrow this stability from the object? Alternatively, we could extract that cubic centimeter with a pipette and place it on a plate so that it forms a puddle. The tea of this puddle at 5:06 p.m. would be the same as the tea in the cubic centimeter that we extracted from the center of the tea cup at 5:05 p.m., disregarding evaporation. This is because a puddle is a portion of liquid with a border that is separated from its neighborhood (which may be another liquid – as when we put the tea in a glass filled with oil). Now it is not the walls of a container, but the border of the stuff that lends the stuff some object-like qualities (note that puddle, bubble, drop, blob etc. are count nouns). And this is sufficient to guarantee the identity of the stuff that makes up the puddle or bubble, without the help of any container.1 1

However – as a reviewer pointed out – this is an idealization. Our natural interpretation of puddle and its ilk allows for leaks (small quantities of water that evaporate or seep into the ground) and gains (a drop of rain or some osmotic accretion of fluids) where we would not hesitate to speak of the same puddle.

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Yet another method would be to shock-freeze the cup of tea, turning it to ice. The parts of the cubic centimeter of tea will then stay put in the same location in the middle of the cup. It is still stuff: it is not the content of a container, it is neither puddle nor bubble, we still refer to it by a mass noun, tea ice or perhaps ice tea (not: iced tea!). But now it is solid, and because it is solid, we can trace the material identity by reference to the spatial location inside the cup. Our thought experiments point to an important ontological difference between two kinds of stuff: there are liquids (and gases) that require some container or border in order to be identified across time, and there are solids that do not need such confinements. The borderline between the two kinds is fuzzy. Take a heap of rice; undisturbed, it may behave like ice, and we might be able to refer to the cubic centimeter in its center. But when we put it into a bag, the grains will be shuffled around, and we will lose the ability to refer to that portion of rice, for all practical purposes. In this paper I will sketch a way to enrich the mereological and topological models that have been proposed to deal with the semantics of mass and count terms (cf. Link 1983; Moltmann 1998; Landman 2011; Rothstein 2011; Grimm 2012a, 2012b; Sutton and Filip 2016, among others). I do this by including a temporal dimension that deals with the issue of reidentification of matter over time, something that has not been considered to the degree necessary before (cf. Inwagen 2006). My goal is to reconstruct a part of the ontology that underlies our use of natural language (in the sense of Bach 1986). The current paper is a first exploration of the matter, following largely my own intuition. Subsequent research on identification over time would have to adduce experimental evidence, and explore the assumptions of the underlying theory and check how far it carries in explaining the semantic properties of matter and objects. 4.2

Parts

Let us review the formal underpinnings of proposals for the semantics of mass nouns and count nouns, as far as they are relevant for my proposal. My assumptions are as standard as possible – cf. e.g., Varzi (2007) and Champollion and Krifka (2016). We assume a domain of entities U. It is best to think of the elements of U as quantities of matter without particular form properties; we will construct individuals out of matter later. The mereological part relation v satisfies the following basic principles on the elements u, u0 etc. of U: (1)

a. u v u b. u v u0 ^ u0 v u00 ! u v u00 c. u v u0 ^ u0 v u ! u ¼ u0

Reflexivity Transitivity Antisymmetry

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We define the relations o of overlap and ⊏ of proper part, as follows: (2)

a. u o u0 :, 9u00 ½u00 v u ^ u00 v u0  b. u ⊏ u0 :, u v u0 ^ ¬ u0 v u

Overlap Proper part

In addition to the axioms in (2), the part relation should also satisfy the supplementation principle. We go for the following strong version: (3)

¬ u v u0 ! 9u00 ½u00 v u ^ ¬ u00 o u0  Strong supplementation

This says that if u is not a part of u0 , then there is an u00 that is a part of u and that does not overlap u0 . That is, u00 is a “witness” that u is not a part of u0 . From strong supplementation, weak supplementation follows, which says that if u0 is a proper part of u, then there is another part of u, u00 , that does not overlap with u0 : (4)

u0 ⊏ u ! 9u00 ½u00 v u ^ ¬ u00 o u0 

Weak supplementation

Strong supplementation excludes that there are entities that are distinct but have the same parts. Hence the axioms (3) and (4) lead to extensional mereology (cf. Simon 1987). It is helpful to sharpen one’s intuition with an example. Take the closed subintervals of [0, 5], that is, the sets of real numbers frj a  r  bg ¼ ½a; b, where a, b are real numbers with 0  a < b  5. Form the closure of these subintervals under set union, that is, in addition to intervals like [1, 2] and [3, 4] we also admit for non-intervals like ½1, 2 ∪ ½3, 4. This set of subsets of fr j0  r  1g forms an extensional mereology with the subset relation as part relation. The axioms in (2) hold, and strong supplementation holds as well. For example, take u = [1, 3] and u0 ¼ ½2, 4; u and u0 overlap, and it holds that there is an u00 , for example [1.3, 1.7] that is a part of u but not a part of u0 . There are additional requirements that we may want to impose. For example, we may require that supplements be unique – this is the strong complementation principle: (5)

¬ u v u0 ! 9u00 8u000 ½u000 v u00 $ ½u000 v u ^¬ u000 o u0  Strong complementation

Take again the example of closed subintervals of ½0, 5 closed under set union. This is strongly complemented. Take u ¼ ½1, 3 and form u0 ¼ ½2, 4, the largest closed subinterval u00 of u that does not overlap with u is ½1, 2. Note that ½1, 2 and ½2, 4 do not overlap in this model, as the singleton set f2g is not a closed subinterval (we only admit those intervals ½a; b for which a < b). Strong complementation leads to “scattered” individuals. For example, take u ¼ ½1, 4 and u0 ¼ ½2, 3; the largest individual that is a part of u but not a part of u0 is ½1, 2 ∪ ½3, 4. With strong complementation, we can define the operation of join, or fusion: (6)

u t u0 ≔ ιu00 ½u v u00 ^ u0 v u00 ^ 8u000 ½u000 v u ^ u000 v u0 ! u000 v u00 

Fusion

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The fusion of two entities u, u0 is the smallest entity u00 that has all the parts of u and u0 as parts; strong complementation guarantees that this is indeed a unique entity. For example, ½1, 3 t ½2, 4 is ½1, 4, and ½1, 2 t ½3, 4 is ½1, 2 t ½3, 4. Fusion can be generalized to arbitrary non-empty sets (I understand here P as an arbitrary set, and P(u) as standing for u2P): (7)

tP≔ιu½8u0 ½Pðu0 Þ !u0 v u^8u00 ½8u0 ½Pðu0 Þ! u0 vu00 ! uvu00  Generalized fusion

This is the sum of all entities that fall under the predicate P, the smallest entity u such that all parts of u are P. Generalized fusion is sometimes considered problematic – e.g., is there indeed an object consisting of this apple and the moon? I take it that there are semantic phenomena that suggest that such entities indeed exist. For example, as physical objects, they fall under properties such as attract each other, or have a distance of 395,290 km of each other; the most straightforward way to make sense of such sentences is to assume that their subjects denote existing objects. With a mereological structure with strong complementation, we can define the notion of a complement of an entity u, written  u, as the fusion of all individuals u00 that do not overlap u. For example, we have ½1, 3 ¼ ½0, 1 ∪ ½3, 5. (8)

u ≔ι u0 8u00 ½u00 v u0 $ ¬ u00 o u

Complement

The notion of a complement is not defined for the sum of all individuals. A structure hU; vi, where v is a mereological part relation that satisfies strong supplementation and strong complementation will be called a mereology here (noting that there are different notions of mereologies). 4.3

Atoms

An atom in the mereological sense is an entity that has no proper parts: (9)

ATðuÞ :, ¬ 9u0 ½u0 ⊏u Atom

It follows that two distinct atoms cannot overlap, as otherwise they would have a common proper part. An atomic mereological structure is one in which every entity is composed of atoms; an atomless mereological structure is one in which there are no atoms. (10)

a. hU; vi is atomless :, ¬ 9u2U½ATðuÞ b. hU; vi is atomic :, 8u2U9U0 ⊆U8u0 2U0 ½ATðu0 Þ ^ u ¼ tU0 

There are structures that are neither atomless nor atomic – structures in which some entities consist of atoms and others don’t. The set of closed subintervals of ½0, 5 closed under union is atomless: Each closed subinterval ½a; b with 0  a < b  5 has parts that are smaller, for

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example ½a0 , b0  with a < a0 < b0 < b. The same holds, of course, for entities composed of closed subintervals.2 Atomic mereologies might be finite or infinite; atomless mereologies are necessarily infinite. The issue of atomicity has been discussed in particular with respect to mass nouns like dust, water, or vacuum. There might be atoms of dust (sometimes we can see them dancing in the light). There are atoms of water in the mereological sense (the H2O molecules), but presumably this scientific discovery should not influence the naïve natural language ontology of water, which should be compatible with both the atomic view and the atomless view. As for the atoms of the mass noun vacuum, even modern physics is silent (except perhaps by referring to the Planck length). I take this as indication that the stage for a natural language ontology should be an atomless mereology (cf. also Asher and Vieu 1995). An atomic mereology could be constructed from that by adding an additional axiom, or by constructing atoms from non-atomic entities (cf. Roeper 1997). 4.4

Touching

Mereology has a notion of overlap; in addition, we may want to say that two entities just touch. There are various proposals to enrich mereology by the topological notion of connection, called mereotopology (cf. Asher and Vieu 1995; Roeper 1997; Casati and Varzi 1999; Varzi 2007; cf. Grimm 2012a, 2012b for applications to the mass/count distinction). Normally, connection is considered as a generalization of the mereological relation of overlap: two entities are connected iff they overlap or touch. Here I will take the relation that two entities just touch each other as basic, which is referred to as “external” or “tangential” connection. Consequently, I will use the term haptomereology for the resulting structure. We will use the symbol ∞ if two entities touch. Obviously, touching should be symmetric; if u touches u0 , then u0 touches u. (11)

u ∞ u0 ! u0 ∞ u Symmetry of touching

Touching is not transitive – if John touches Mary and Mary touches Sue, John does not necessarily touch Sue. In the technical sense understood here, touching also is irreflexive – no entity can touch itself.3 As external connection, touch is incompatible with parthood and, consequently, overlap: 2

3

If we allow for singleton sets in our model, then we get an atomic mereology, with singleton sets such as ½3, 3 ¼ f3g as atoms. But notice that the set of closed subintervals of [0; 5] that also contains singletons is not strongly complemented anymore. For example, there is no unique closed interval that is a part of [1; 3] but does not overlap with [2; 4]: [1; 2] now overlaps with [2; 4] in {2}. And there is no maximal [1; x] with x < 2, as the notion of complement requires. Of course, John can touch his forehead, and by this touch himself. This notion of touching can be understood a involving an entity u (John) that has two distinct parts u0 , u00 (the finger, the forehead) such that u0 and u00 touch.

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a. u ∞ u0 ! ¬ u v u0 Touch excludes parthood b. u ∞ u0 ! ¬ u o u0 Touch excludes overlap

We define a haptomereology as a structure hU; v , ∞i, with a mereological part relation v and a corresponding topological touch relation ∞. Take again the example of the set of closed intervals of [0, 5], closed under set union. We say that two elements touch, u ∞ u0 , iff they do not overlap and there is nothing in between u and u0 . The definition is a bit tricky, as u and u0 may be scattered elements that interlock. But the following definition will do: (13)

u ∞ u0 :, ¬ u o u0 ^ 9u00 ,u000 ½u00 v u ^u000 v u0 ^ ¬ 9u∗ ½u00 < u∗ < u000 

Touch

Here, u < u0 stands for 8r 2 u8r0 2 u0 ½r < r0 , that is, the numbers in u are all smaller than the numbers in u0 . Hence (11) states that two non-overlapping entities u, u0 touch iff they contain parts u00 , u000 such that no entity u∗ fits in between them. For example, [1, 2] and [2, 3] touch: they do not overlap, as {2} is not an entity in the structure, and there is no entity u∗ in between. Also, [2, 3] and the scattered entity ½1, 2 ∪ ½4, 5 touch, as they do not overlap, and there is no u∗ between [2, 3] and the part [1, 2]. The entities [1, 2] and [3, 4] do not touch, as there are entities like [2.3, 2.8] that are in between. In general, and not restricted to the specific model at hand, touch and being in-between are complementary notions: If two entities touch, then there is no entity in-between them, and if there is no entity in-between two nonoverlapping entities, then they touch. There are additional notions (cf. Casati and Varzi 1999, Varzi 2007, Grimm 2012a, 2012b). For the current purpose, we define the notion of an interior part of an entity u as an entity u0 that is a part but that is not touched by any entity that touches u. (14)

u0 ≺u :, u0 v u ^ ¬ 9u00 ½u00 ∞ u ^ u00 ∞ u0  Interior part

For example, [2, 3] is an interior part of [1, 4]. In contrast, [1, 2] is a tangential part of [1, 4], as [0, 1] touches both [1, 2] and [1, 4].4

4

Some additional notions that are of lesser relevance for the current paper are the following. The interior INT(u) of an entity is the sum of its interior parts. This is not always defined, e.g., in the non-atomic model above there is no coherent way to define the interior part of [2, 4], as the open interval ]2, 4[ is not an element of the model. In an atomic model that allows for all sub,sets of [0, 5] as elements, the interior par of [2, 4] would be ]2, 4[. For such models we can furthermore define the exterior of an entity u as the interior of the complement of u. In the atomic model, the complement of [2, 4] is ½2, 4 = [0, 2[ ∪ ]4, 5], whose interior is the same set, hence the exterior of [2, 4] is ½2, 4. We can furthermore define the boundary of u as the complement of the join of interior and exterior of u; for example, the boundary of [2, 4] is {2, 4}. We can furthermore define the closure of an entity as the join of the entity of u with its boundary; for example, the closure of [2, 4] is the same entity, [2, 4], and the closure of ½2, 4 is ½0, 2 ∪ ½4, 5.

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4.5

Wholes

As Varzi (2007) and Grimm (2012a, 2012b) observe, mereology is not a theory of the part–whole relation, but just a theory of parts – we cannot define what counts as a “whole” in a mereology. But we can define wholes in a mereotopology or haptomereology, due to the additional notions of connectedness and touching, respectively. More specifically, we can define the notion of a self-connected entity whose parts are connected, in the sense that every two non-overlapping parts that together make up the entity touch each other: (15)

SCðuÞ :, 8u0 , u00 ½u ¼ u0 t u00 ^ ¬ u0 o u00 ! u0 ∞ u00  Self-connection

For example, in our non-atomic model the entity [2, 4] is self-connected. For example, the entity [2, 4] can be seen as the join of [2, 3] and [3, 4], two nonoverlapping entities (as 3 is not an entity); these two entities touch each other. In contrast, the individual ½1, 2 ∩ ½3, 4 is not self-connected, as it consists of two non-overlap parts [1, 2] and [3, 4] that do not touch.5 4.6

Time and Space

So far we have developed a static picture: Entities that stand in the part relation or the touch relation to each other, but do this for eternity. If we want to incorporate the possibility of change, we have to draw time and space into the picture. I will represent time as a structure hT;

6.6.1

mess mass i-object singular count i-object

Portioning Coffee

Landman 1991 argued the need for a shift from mass DP interpretations to singular count objects in the discussion of packaging. I would use the more appropriate term portioning now. Look at the examples in (2) (with the relative clause present to make it difficult to argue for an analysis in terms of generalized conjunction): (2)

a. De koffie in het kopje en de koffie in de pot die in ons laboratorium onderzocht zijn bevatten allebei strychnine. b. The coffee in the cup and the coffee in the pot that were analyzed in our lab both contained strychnine.

Since both means each on a domain of two, it must distribute to 2. This means that the set relative to which distribution takes place must have a cardinality of 2. This means that the subject in (2) should be interpreted as a plural count object that counts as 2. But coffee is a (mess) mass noun, and, intuitively and formally, the coffee in the pot and the coffee in the cup are coffee, mass, and don’t count as anything. We get the reading required for the correct interpretation of both by singular shift. We start with the mess mass interpretations of coffee in the cup and coffee in the pot: coffee in the cup ! coffee in the pot !

With singular shift, we get two interpretations for the coffee in the cup, and two for the coffee in the pot:

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Iceberg Semantics for Count Nouns and Mass Nouns the coffee in the cup ! ! the coffee in the pot ! !

173 mass i-object singular count i-object mass i-object singular count i-object

We take the two singular shifted interpretations, and combining them with the sum-interpretation of and and we get a count i-object that counts as 2 relative to its base: the sum of two portions of coffee: the coffee in the cup and the coffee in the pot !

Let us think about the Iceberg semantics of both as a VP operator in (2). Since in Iceberg semantics definite DPs are interpreted as i-objects, the minimal grammatical assumption we will have to make is that VPs are interpreted as predicates of i-objects. For simplicity and concreteness I assume the following: contain strychnine ! λx.Contains-strychnine(body(x)) Contains-strychnine (z) iff sufficient amount of strychnine is mixed in with z.

As a VP-operator, distributive both maps predicates of i-objects to predicates of i-objects:

Both maps predicate P of i-objects onto a presuppositional predicate of i-objects: λx.8a 2 Dbase(x)(body(x)): P().

The latter predicate is only defined for i-objects x with a body count of 2 relative to the base. Note that already the fact that the semantics is formulated with Dbase(x) is enough to bring in the presupposition that VPs with both (or each) can only apply to count i-objects, since Dbase(x) presupposes that x is count. Given that the presupposition stated itself brings in reference to Dbase(x), we can replace Dbase(x)(body(x)) by (body(x)] ∩ base(x). We apply both to contain strychnine and derive: both contain strychnine !

This we apply to the interpretation given of the coffee in the pot and the coffee in the cup. As argued, the latter interpretation satisfies the presupposition, so we derive:

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Fred Landman 8a 2 (σ(coffee-in-cup) t σ(coffee-in-pot)] ∩ {σ(coffee-in-cup), σ(coffee-in-pot)}: Contains-strychnine (a)

which is equivalent to: Contains-strychnine (σ(coffee-in-cup)) ^ Contains-strychnine (σ(coffee-in-pot)).

6.6.2

Gillon’s Problem

Another case that can straightforwardly be analyzed with singular shift is what Chierchia 1998 calls Gillon’s problem (from Gillon 1992): (3)

a. The curtains and the carpets resemble each other. b. The drapery and the carpeting resemble each other.

The puzzle is that (3a) allows several readings that are unavailable for (3b). (3a) allows the following readings: Reading 1 Curtains resemble other curtains, curtains resemble carpets, carpets resemble curtains, carpets resemble other carpets. Reading 2 Curtains resemble other curtains and carpets resemble other carpets. Reading 3 Curtains resemble carpets and carpets resemble curtains.

(3b), on the other hand, only allows the third reading: Curtains resemble carpets and carpets resemble curtains. Chierchia sees this as a puzzle for the Supremum Argument: If the drapery is the sum of the curtains, and the carpeting is the sum of the carpets, how can you get (3a) and (3b) to be different? The answer, not surprisingly, is by following the lead of Iceberg semantics. We start with two plausible meaning constraints on the predicate resemble each other: Meaning constraints  resemble each other(x) presupposes that x is a strictly plural count i-object (cardinality  2).  resemble each other(x) concerns the resemblance of parts of body(x) that are in base(x). Let curtain be a disjoint set and let curtain  base(drapery) ⊆ *curtain. Let carpet be a disjoint set and let carpet  base(carpeting) ⊆ *carpet. curtains ! carpets ! drapery ! carpeting !



plural count plural count neat mass neat mass

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We look at the three readings of (3a). Reading 1 This reading we derive in the most obvious way: the curtains ! the carpets ! the curtains and the carpets !

Since curtain ∪ carpet is a disjoint set, this is a plural i-object. This means that the presuppositional meaning restriction on resemble each other is satisfied. We derive for (3a): resemble-each-other()

This interpretation compares elements in curtain ∪ carpet with each other. Reading 2 This reading we can get, for instance, with standard Boolean conjunction: λP.P() ^ P()

Since both DP interpretations satisfy the plurality constraint, we derive for (3a): resemble-each-other() ^ resemble-each-other()

This is a conjunction of a statement comparing curtains with curtains and a statement comparing carpets with carpets. Reading 3 This reading we get by applying singular shift and then sum conjunction: the curtains ! "() = the carpets ! "() = the curtains and the carpets !

{σ(*curtain), σ(*carpet)} is a disjoint set with two elements, so the sum is once again a plural i-object. We derive for (3a): resemble-each-other()

This compares the sum of the curtains with the sum of the carpets. We look at (3b). The basic interpretations for the drapery, the carpeting, and the drapery and the carpeting are neat mass i-objects: the drapery ! the carpeting ! the drapery and the carpeting !

neat mass neat mass neat mass

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These i-sets do not satisfy the plurality presupposition of resemble each other, and this means that they cannot be used in a felicitous derivation for (3b). This means that there are no felicitous derivations for (3b) corresponding to the first two derivations of (3a). What we can do, is what we did for Reading 3 of (3a): apply singular shift to the interpretations of the drapery and the carpeting and take their sum: the drapery ! "() = singular count the carpeting ! "() = singular count the drapery and the carpeting !

plural count

In this case the interpretation of the drapery and the carpeting is a plural i-object with card 2. Hence, resemble each other can apply to this, and what gets compared are the elements of the base: {σ(*base(drapery)), σ(*base(carpeting))}>), the drapery and the carpeting. In sum, the presence of distributive and sum readings for sums of count DPs and their absence for sums of mass DPs is expected in Iceberg semantics with singular shift. In both cases, singular shift accounts for the presence of the reading comparing the body of curtains with the body of carpets. 6.7

Compositionality: The Head Principle and the Semantics of Modifiers

Landman 2016 proposes a compositional theory of base information: The base information inherits up in the interpretation of a complex NP from the interpretation of the syntactic head of that NP (and in the interpretation of a DP from the interpretation of the NP). We concentrate on NPs here. Landman 2016 formulates the following Head principle: Head principle for NPs

and let C ! C, H ! H, with C and H i-sets. then: base(C) = (body(C)] ∩ base(H) The base of the complex is the set of all parts of the body of the complex, intersected with the base of the head. This means that the base information is passed up from the head NP to the complex NP, both for modification structures (adjuncts like adjectives and

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numerical phrases) and for complementation structures (in particular the structures for pseudo partitives like measure phrase three liters of wine and classifier phrase three bottles of wine). The Head principle turns Iceberg semantics into a semantic theory in which the distinctions mass/count, mess/neat, and singular/plural apply not only to lexical items, but to NPs and DPs in general, i.e. a compositional theory of these distinctions. What lies at the basis of this are simple inheritance facts: Fact.

If base(H) is disjoint, then base(C) = (body(C)] ∩ base(H) is disjoint.

This means that the interpretation of a complex noun phrase is count if the interpretation of the head is count, and that its interpretation (in non-extreme cases) is mass if the interpretation of the head is mass. Landman 2016 introduces the general schema for the interpretation of NP modifiers. NP modifiers denote functions from i-sets to i-sets, and their semantics fits the following schema:

Thus, the modifier maps the i-set interpretation P of the head NP onto the i-set interpretation of the complex NP, on the condition that the presuppositions specified in presP are satisfied. The base of the complex is given by the Head principle. With this schema, you only need to specify for a given modifier what bodyP is (and the relevant presuppositions). As a simple example we give the semantics of numerical modifiers. Following Landman 2004, I assume that number relations combine with numbers names to form number predicates, and the semantic operation involved is application: at least !  at most !  -!= number relation four ! 4 number at least four ! λn.n  4 at most four ! λn.n  4 - four ! λn.n=4 number predicate

Number predicates denote sets of numbers. From number predicates we derive numerical predicates through composition with card: at least four ! λn.n  4 ∘ card = λZλx. cardZ(x)  4 numerical predicate - four ! λZλx. cardZ(x) = 4

In Iceberg semantics, numerical predicates are turned into numerical modifiers by fitting their interpretation into the above modifier schema:

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As before, the presupposition that P be count derives from the fact that cardbase(P) is only defined if base(P) is disjoint. We combine this interpretation of four with plural noun cyclists (suppressing index wt for readability): cyclists ! with CYCLIST a disjoint set. We derive: four cyclists ! where body = λx.*CYCLIST(x) ^ cardCYCLIST(x) = 4 The set of sums of cyclists that are sums of four objects in CYCLIST base = (body] ∩ CYCLIST. base = (λz.*CYCLIST(z) ^ cardCYCLIST(z) = 4] ∩ CYCLIST = (t(λz.*CYCLIST(z) ^ cardCYCLIST(z) = 4)] ∩ CYCLIST = λx. x v t(λz.*CYCLIST(z) ^ cardCYCLIST(z) = 4) ^ CYCLIST(x) = λx. x v t(CYCLIST) ^ CYCLIST(x) = CYCLIST

The base is identical to the base of the head. This reflects the fact that numerical predicates are quantitative: If there are at least four cyclists, any cyclist is part of a sum of four cyclists, hence (in non-extreme contexts) the numerical modifier does not impose a restriction on the base of the head. 6.8

Excursus: A Complex Modifier

(4)

a. De vier elkaar bespionerende wielrenners The four each other spying-at cyclists ‘The four cyclists (who were) spying on each other.’

DP D de

NP NUM vier

NP AP

NP

elkaar bespionerende wielrenners

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The Dutch example (4a) has the structure in (4b), and I assume this is how the semantic composition takes place. I use a Dutch example because it clearly shows the composition order of semantic operations. I assume the English paraphrase has the same derivation. We are concerned, then with, the semantics of modifier spying on each other. Given the modifier schema given above, we need to find for this an appropriate interpretation bodyP. bodyP is a predicate of individuals which will combine intersectively with the interpretation of the head along the following lines: bodyP = λx. body(P)(x) ^ φ1(x) the set of body(P)-objects that have φ1

Here φ1 is derived from the interpretation of the VP spy on each other. Spy on each other denotes a property of i-objects, and is restricted to plural count iobjects: Meaning constraint spy on each other(x) presupposes that x is a strictly plural count i-object (cardinality  2)

Since we are forming a set out of this, we may as well restrict ourselves explicitly to defined cases: bodyP = λx. body(P)(x) ^ cardbase(P)(x)2 ^ φ2(x) the set of body(P)-objects that are strictly plural relative to base(P) and that have φ2

What is left is to express φ2(x) in terms of predicate spy on each other of iobjects. My proposal is that this involves reference to Dbase(P)(x): φ2(x) = spy on each other()

This means that, in order to evaluate whether a sum x is in bodyP, you need to look at x as a sum base(P) elements. This, again, requires P to be disjoint. Putting all of this together, we get:

bodyP = λx. body(P)(x) ^ cardbase(P)(x)2 ^ spy on each other() The set of body(P)-objects x that are strictly plural sums of base(P)-objects such that spy on each other() holds baseP = (bodyP] ∩ base(P)

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We combine modifier spying on each other with head cyclists: cyclists spying on each other ! body = λx.*CYCLIST(x) ^ cardCYCLIST(x)2 ^ spy on each other() the set of strictly plural cyclist-sums that are spying on each other. base = (bodyP] ∩ CYCLIST the cyclist-parts of the sum of that set.

We have so far not restricted the meaning of spying on each other beyond assuming that it denotes a set of strictly plural base(H)-sums. Let us consider the following two scenarios: Scenario 1 We are in a bicycle race, one minute before the winner crosses the finish line. Four cyclists have escaped from the peloton and have left the peloton at a distance of fifteen minutes, hence without a chance for a position among the first three. The first four are nervous, the others are relaxed and taking their time. a o

b o

o c

o d

oooooooooooooooooooooooooooooooooooo

Scenario 2 The same situation, with the only difference that two of the four got away and will decide the race between them. Five minutes behind them, the other two will fight for the third place. The first four are nervous, the others are relaxed and taking their time. a o

b o

o c

o d

oooooooooooooooooooooooooooooooooooo

There are, of course, many proposals of different strength for the semantics of reciprocals. It has been proposed that proposals of different strength may be relevant simultaneously, and are ranked in context, and that in a context the

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strongest analysis compatible with that context is chosen (see Dalrymple et al. 1994; Winter 2001). I am assuming something less strong here. What is relevant for me here is that in Scenarios 1 and 2, spy on each other must be given a semantics that is strong enough that all the cyclists in CYCLIST – {a,b,c,d} are completely excluded from the denotation. This excludes analyses in which cyclists in CYCLIST–{a,b,c,d} are allowed to be ‘free-loaders’ on the cyclists in {a,b,c,d} (i.e. it excludes sums like atbtx, with x in the peloton). This still allows in semantic analyses of different strength. I will here choose a medium strong analysis to illustrate the Iceberg semantics in these scenarios: Medium strong analysis of reciprocals spy on each other() iff 8a 2 Dbase(P)(x) 9b 2 Dbase(P)(x): a 6¼ b ^ spy on(a,b) every base(P)-part of x spies on some other base(P)-part of x

With this we derive: Scenario 1 cyclists spying on each other !

body = {atb, atc, atd, btc, btd, ctd, atbtc, atbtd, atctd, btctd, atbtctd} base = {a, b, c, d} Scenario 2 cyclists spying on each other !

body = {atb, btc, atbtctd} base = {a, b, c, d} We apply numerical modifier four to cyclists spying on each other and derive: Scenarios 1 and 2 four cyclists spying on each other !

We apply the definite article and derive: Scenarios 1 and 2 the four cyclists spying on each other !

This is, of course, as it should be, a plural count i-object.

PART II MEASURES AND PORTIONS: HOW MASS COUNTS

6.9

Measure Interpretations of Measure Phrases

This section and the next summarize some of the analyses given in Landman 2016 and in Khrizman et al. 2015. We are concerned with the different interpretations of pseudo partitives, in particular, measure phrases like three

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liters of wine and classifier phrases like three drops of wine. We start with measure interpretations, like the most prominent reading of (5): (5)

Pour 20 milliliters of wine in the sauce. wine to the amount of 20 milliliters

I assume, following Landman 2016, that English measure phrases and classifier phrases have the structure given below, in which the measure or classifier is the head of the construction. Following Landman 2004, 2016 and Rothstein 2011, I assume that they differ in compositional interpretation: While for classifiers the semantic composition follows the syntactic structure, I assume that in measure interpretations, number and measure form a semantic unit: Classifier structure: NP NUM three

Measure interpretation: APPLY NP

NP[head]

INTERSECT NP wine

three

wine

liter measure

measure liter

The relevant components of meaning are a number predicate, a measure i-set, and the i-set of the complement: three ! λn.n=3 number predicate liter ! measure i-set wine ! i-set

 The body of the interpretation of measure liter is the measure function literwt, the function that maps at wt an object onto its volume in liters (which is a non-negative real number).  The body of the interpretation of the measure phrase three liters of wine is derived with function composition (details of the semantic composition are in Landman 2016): (number predicate ∘ measure) ∩ complement ∩ WINEwt (λn.n=3 ∘ literwt) three liters of wine ! body = λx. literwt(x)=3 ^ WINEwt(x) Wine to the amount of three liters

The heart of Landman 2016 concerns the derivation of the base. We have set the body(liter) to literwt, which is a set of object-number pairs. In Iceberg semantics, this means that base(liter) must also be a set of object-number pairs,

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in particular a set that generates literwt under t. This is formalized in Landman 2016, and it is there shown that: Given some independent concerns about Iceberg semantics, additive continuous measure functions like literwt cannot be generated under t from a disjoint base. If follows from this that measure liter is by necessity interpreted as a mess mass i-set, and since measure liter is the head of the measure phrase, it follows that the measure interpretation of three liters of wine is a mess mass interpretation. This is good, because Rothstein 2011 has argued that, when measure phrases have a measure interpretation, they pattern with mass NPs (see the discussion in Landman 2016). In context, a natural choice for base leads to the following interpretation of the measure phrase as a mess mass i-set: three liters of wine ! body = λx. literwt(x)=3 ^ WINEwt(x) Wine to the amount of three liters base = λy.y v tWINEwt ^ literwt(y)  mliter,wt Parts of the sum of the wine that measure less than a contextually given small measure value mliter,wt

Fact. three liters of wine on the measure interpretation denotes a mess mass i-set.

6.10

Portion Interpretations of Measure Phrases

Khrizman et al. 2015 and Landman 2016 give a semantics for various classifier interpretations of measure phrases like three liters of wine. I discuss here portion interpretations. The distinction between measure interpretations and portion interpretations has been all but overlooked (but see Partee and Borschev 2011). The reason for this may be that we have been thinking too much in terms of the dichotomy container/measure interpretation and decided too early that: three bottles/liters of wine: If it’s glass, it’s a container and count. If it’s wine, it’s liquid and mass.

But portion readings of wine, or three liters of wine, are wine, but count. Here, I will first introduce portion readings in the context of a grammatical shift from mass to count: patat (french fries) is a mass noun in Dutch. (There is a count noun patat as well, which means ‘single chip’, but that interpretation is not relevant in the present context.) (6)

[Ordering french fries in Amsterdam]: Drie patat, alstublieft, één met, één zonder, en één met satésaus.

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Fred Landman ‘Three [portions of] french fries, please, one with [mayonnaise], one without [sauce] and one with peanut sauce.’

Clearly, patat is used as a count noun here. Khrizman et al. 2015 introduce a contextual property PORTIONc, a function that in a context maps indices onto contextual sets of portions: Contextual portioning Property PORTIONc maps index wt in context c onto a disjoint set of portions: PORTIONc,wt The point about PORTIONc is that it has no inherent semantic content beyond disjointness: What counts as a portion in c is completely determined by context c. This means several things. In the first place, in essence anything can be portioned by PORTIONc: wine into sips of wine, meat into slices of meat, etc. Secondly, the portions introduced by PORTIONc are not required to be of a fixed size; the natural question you will be asked following up on your request in (6) is ‘Small, medium or large?’ Thirdly, the portions introduced by PORTIONc can have semantic coherence within the context, but need not: If you so want, you can in context divide the water into the water in the puddle, the water in the bath, and the water in your glass. Lima 2015 and Khrizman et al. 2015 argue that this operation underlies the semantics of all nouns in the Amazon language Yudja. In the case of (6), portion shift shifts a mass noun into a count noun: Mass: patat ! Singular count: patat ! < base, base > base = λx.PORTIONc,wt(x) ^ PATATwt(x) The set of contextual portions of french fries.

In (7) we combine count shifted patat and measure phrase zeshonderd gram patat: (7)

Drie patat is zeshonderd gram patat. ‘Three [portions of] french fries is six hundred grams of french fries.’

The mass stuff involved is the same, but the perspective is different: The same stuff is presented simultaneously as a mass sum of six hundred grams of patat mass and as a count sum of three portions. The mass/count distinctions are expressed not in the body of the interpretations, but in the respective bases. Allowing such equations is of course exactly what Iceberg semantics is about. The focus of Landman 2016 is on deriving portion readings for pseudopartitives and showing that the compositional semantics of Iceberg semantics derives the relevant readings as count readings.

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The simplest kind of portion classifiers are shape classifiers like hunk, slice, and drop. These are all based on count nouns. We use drop as an example. The count noun interpretation of drop is: drop !

with DROPwt a (contextually) disjoint set

Count noun drop shifts to a function from i-sets into i-sets by filling the obvious intersective body interpretation into the modifier schema: bodyP = λx.body(P)(x) ^ DROPwt(x)

Doing the calculations and applying the result to the interpretation of wine, we derive: drop of wine ! base = λx.WINEwt(x) ^ DROPwt(x)

Given that DROPwt is a disjoint set, and base is a subset of that set, base is also a disjoint set, and hence is a count i-set. Hence we derive that the complex noun phrase drop of wine is count. Other portion classifier interpretations are discussed in Landman 2016. That paper uses PORTIONc to derive portion readings for measure phrases like three liters of wine. This reading is the most prominent reading of the measure phrase in (8): (8)

He drank three liters of wine, one in the morning, one in the afternoon, one in the evening (we found four empty bottles, so on each time occasion he drank one and a third of a bottle).

At the basis of the derivation lies a shift in interpretation for the measure liter: bodyp = λx. PORTIONc,wt(x) ^ body(P)(x) ^ literwt(x)=1 The set of body(P)-portions that measure one liter

This interpretation is intersective, just like that of portion classifier drop above. This means that, from this stage on, the derivation is similar to the derivation of drop of wine above: We fill in bodyP in the modifier schema, apply to wine, and get: liter of wine ! base = λx. PORTIONc,wt(x) ^ WINEwt(x) ^ literwt(x)=1 the set of portions of wine that measure one liter

Again, PORTIONc,wt is a contextually disjoint set, and hence so is base. Consequently, on this shifted interpretation, liter of wine is a count noun phrase. 6.11

When Mass Counts Caveat: Despite appearances, no animals were harmed in the research for this section.

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We discussed in Part I the semantics of counting modifiers and distributive modifiers. We add count comparison here. The semantics of these constructions involves distribution set Dbase(HEAD): Counting: x is three cats if x has three parts that are in Dbase(HEAD) (where base(HEAD) = CATwt). Distribution: each of the cats purrs if all the parts of the cats that are in Dbase(HEAD) purr. Count comparison: most cats purr if the number of parts of the cats that purr that are in Dbase(HEAD) is larger than the number of parts of its relative complement that are in Dbase(HEAD). In all these cases, the presupposition associated with Dbase(HEAD) is that base(HEAD) is disjoint, and hence these interpretations are predicted to be only felicitous for count nouns. The puzzle is that distribution and count comparison are not restricted to count nouns. The recent literature has shown this extensively for neat mass nouns. Rothstein 2011 and Schwarzschild 2009 discuss distributivity for neat mass nouns. Schwarzschild points out that adjective like big only allow distributive interpretations (he calls them stubbornly distributive). Thus, while noisy in (9) allows a distributive and a collective interpretation, big lacks the collective interpretation. (9)

The noisy boys = ✓the boys that are noisy ✓ the noisy group of boys The big chairs = ✓the chairs that are big  the big group of chairs

Rothstein argues that neat mass noun furniture felicitously combines with big, and when it does big has a distributive interpretation: (10)

The big furniture = ✓ the pieces of furniture that are big  the big group of furniture pieces

Barner and Snedeker 2005 discuss cardinality comparison for neat mass nouns, and show experimentally that speakers readily get cardinality comparison readings for neat mass nouns. (They discuss more than. Landman 2011 argues that the same facts hold for most.) (11)

a. Most farm animals are outside in summer. b. Most livestock is outside in summer.

The relevant reading of the examples with the neat mass noun livestock in (11b) compares livestock that is outside in summer with livestock that isn’t outside in summer. Landman 2011 argues that, while (11b) allows measure comparison interpretations in terms of volume or size of biomass, (11b) prominently allows a cardinality comparison interpretation: the same interpretation that is the only interpretation of (11a), with the plural count noun farm animals.

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The obvious question raised by these examples is this: If distribution and count comparison are linked to the distribution set Dbase(HEAD), then how come distribution and count comparison seem to be possible for neat mass nouns, where the base of the head is not disjoint? I will answer this question by arguing that the situation is actually worse than this: Distribution and count comparison are, in Dutch and German, also possible for mess mass nouns. My examples come from Dutch, but the phenomena discussed have been checked with speakers of German as well. In my experience, the facts discussed here do not naturally carry over to English. For instance, native speakers of English do not much like modifying mess mass nouns with adjectives like big, as in big meat. Examples do occur, as in (12), but they are admittedly hard to find: (12)

It’s not that I can’t cook, but I lack experience with preparing big meat and elaborate meals.1

The situation is very different in Dutch. We consider groot (big) and klein (small). Groot and klein in Dutch are just as stubbornly distributive as big and small are in English. But, while examples like (12) are rare in English, they are quite common in Dutch. Searching the web convincingly shows that the Dutch go with Slagerij Franssen. (In this and other examples below, I provide English paraphrases rather than glosses when glosses don’t add anything essential.) (13)

Slagerij Franssen, Maastricht: Tips voor het bereiden van groot vlees. Het bereiden van groot vlees lijkt voor velen een groot probleem. Liever kiest men dan voor een biefstukje of een filet. Echter, groot vlees heeft veel voordelen!2 ‘Butcher shop Franssen, Maastricht: Tips for preparing big meat. Many seem to regard preparing big meat as a big problem. And so they tend to choose a steak or a filet instead. However, big meat has many advantages!’

Vlees (meat) in Dutch is a mess mass noun, like meat in English. We see in (14) that groot and klein can modify mess mass noun vlees, and when they do, they have a distributive interpretation: (14)

1 2

Het grote vlees ligt in de linker vitrine, het kleine vlees in de rechter vitrine. The big meat lies in the left display compartment, the small meat in the right one. Interpretation: The big hunks of meat are in the left compartment, the small hunks of meat in the right one.

Izzy Rose (2009). The Package Deal: My (Not so) Glamorous Transition from Single Gal to Instant Mom, p. 95. New York, NY: Three Rivers Press. www.slagerij-franssen.nl/pages/bereiden-groot-vlees.htm [accessed 2016].

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We might entertain the possibility that vlees, like patat above, or beer in three beer(s), shifts from a mass noun to a count noun. But that is not the case. The facts in (15) and (16) show that there is no such shift. groot vlees stays a mess mass NP, and it does not pattern with true count nouns: (15) shows that groot vlees allows neither modification with numericals nor number agreement typical of count nouns, and (16) shows that groot vlees only allows comparison that fits its mass nature. (15)

#Drie groot vlees #Drie grote vlezen Dutch # ‘Three big meat’ # ‘Three big meats’

(16)

a. ✓Het meeste van het grote vlees is kameel ✓ ‘Most[mass] of the big meat is camel.’ b. #De meeste van het grote vlees zijn kameel. # ‘Most[count] of the big meat are camel.’

We now look at mass comparison. Our first example concerns rijst (rice). Out of the blue, mess mass nouns like rijst do not allow count comparison. Look at the following container and at (17):

not so many very large grains of white rice

very many very small grains of brown rice

Figure 6.2 (17)

De meeste rijst is bruin. Most rice is brown

Out of the blue, the judgement is that (17) is false, which suggests that, out of the blue, the comparison involved is mass comparison in terms of volume, not count cardinality comparison. But if we set up the context carefully we can trigger count readings. (This has been argued for English by Susan Rothstein p.c. and by Peter Sutton, p.c.

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I am not here expressing a judgement on how readily available the reading in question is in English.) The following example is adapted from an example by Peter Sutton. We are playing a game in which we hide small grains of brown rice and very large grains of white rice (to make it not too difficult for the children). Winner is the one who finds the largest number of grains of rice. The numbers and sizes are as in the above picture. Now, as it turns out, Peter is very good at this game. In fact after the game, we take stock and declare: (18)

De meeste rijst is in het bezit van Peter. ‘Most rice is in the possession of Peter.’

In this context it easy to regard (18) as true and felicitous, even if Peter only found small grains. This interpretation, then, involves count comparison. Clearly, in this case, the context has made the grid grain (of) available and allows count comparison in terms of the cardinality of elements in the grid. Grids are a special kind of portion sets. So the suggestion is that count comparison via portions is possible in Dutch for mess mass nouns, when the portioning is made salient in context. We show the same for mess mass noun vlees. The next picture shows the display compartments of our butcher shop: Left compartment: hunks of veal

Right compartment: hunks of baby duck

Figure 6.3 (19)

Het meeste vlees ligt in de rechter vitrine. ‘Most meat lies in the right display compartment.’

As before, the judgement is not surprising (19) is false. Out of the blue, (19) requires mass comparison in terms of volume and count comparison is not natural at all.

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But now we create a bit of context. Tonight you celebrate your Traditional Family Dinner, at which the two Parents eat the Traditional Meal of veal and the twelve Children eat, by Tradition, baby duck. Hence, you have ordered what is in the above display compartments (which is in fact all the veal and duck we have left in the shop). Disaster strikes the butcher shop: The hunks of baby duck were found to be infected with worms. They have to be destroyed, and can’t be sold. I call you with the following message: (20)

Er is een probleem met uw bestelling. Het meeste vlees bleek besmet te zijn met wormen. We moesten het wegdoen, en we hebben geen tijd om vandaag nog een nieuwe bestelling binnen te krijgen. ‘There is a problem with your order. Most meat turned out to be infected with worms. We had to get rid of it and we don’t have time to get a new order in by today.’

As above, in the case of rijst, the judgement is that (20) can readily get a felicitous and true interpretation in this context. But that means that what is made available for the mess mass noun is a reading that involves count comparison in terms of contextual portions, the hunks of meat in the display compartments. We add one more case. We now compare groot vlees and klein vlees: We look at (21), again out of the blue, with contrastive stress on groot: Left compartment: Small hunks of baby duck Big hunks of pork

Right compartment: Exotic meat Small hunks of baby penguin Huge hunks of elephant steak

Figure 6.4

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Het meeste grote vlees ligt in de linker vitrine. ‘Most big meat lies in the left display compartment.’

The observation is that in this case, we don’t need any extra context to get a felicitous and true reading of (21): We easily get a reading that involves count comparison of big hunks of meat. We observe the following. Out of the blue, in (19) we got only a mess mass reading. We got a counting reading in (20) by creating a context that made counting portions salient. We don’t need to set up that special counting context in (21). There is a ready explanation for this in terms of the distributivity involved in the semantics of groot. For count comparison with mess mass nouns to be possible, the semantics must involve (at some level) portion shift, shift to salient portions that can be counted. A portion-counting context is required to make this shift salient. This is what happens in (20). In the case of (21), the semantics of groot involves distribution, which requires a salient disjoint distribution set to be made available. For mess mass nouns like vlees, such a disjoint set is only available via portion shift. Thus, the semantics of the groot vlees already involves portion shift. This means that no further counting context is required to trigger portion shift, and (21) is felicitous and true. 6.12

How Mass Counts

In this last section I will show why in Iceberg semantics distributivity is possible in the mass domain and propose an analysis of how it works there. The account carries over directly to count comparison as well, though, for reasons of space, I will not give the details of the latter here. Let’s deal with the simplest thing first. We observed that although groot is distributive, it can modify mass nouns and does not shift the mass noun it modifies into a count noun: groot meubilair (big furniture) and groot vlees (big meat) are mass NPs. This is easy to account for in Iceberg semantics, because it is exactly what you expect, given the Head principle. Assume a mess mass denotation for vlees: ! , with base(meat) not generated by a disjoint set. groot vlees !

vlees

body is the interpretation of groot vlees. The Head principle derives as the base: (body] ∩ base(meat), which is the part-set of t(body) intersected with base(meat). t(body) is the meat making up the big portions. Obviously, the intersection of its full Boolean part set with base(meat) forms itself a mess mass base for body: Just as the meat is generated by its minimal mess, the big meat is generated by its minimal mess. Hence, arguably, the interpretation of

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groot vlees is just as mess mass as that of vlees. A similar argument applies to neat mass NP groot meubilair. I repeat what I said at the beginning of this section: The analyses given for counting, distribution, and count comparison with count NPs involve the semantic operation Dbase(HEAD). This requires that base(HEAD) is disjoint, and hence that the head is count. The problem, as we see now, is how to account for distribution and counting in cases where the head is not count. The crucial observation is that the operator D defined in Iceberg semantics is actually not itself linked to base(HEAD). Distribution operator

Thus, the central operator is DZ, where Z is presupposed to be a disjoint set. This does mean that any semantics that involves this operator requires a disjoint set for felicitousness, but it does not require this set to be base(HEAD) itself. So here is the big picture. The semantics of modifier groot is based on set bigHEAD, the general form of which is: bigHEAD = λx.body(HEAD)(x) ^ 8a 2 DZ(x): BIGwt(a) presupposition: Z is disjoint

The semantics of count nouns in languages like English and Dutch require the identification of Z with base(HEAD): Count: Z = base(HEAD) bigHEAD = λx. body(HEAD)(x) ^ 8a 2 Dbase(HEAD)(x): BIGwt(a) presupposition: base(HEAD) is disjoint

When we come to mass nouns, what is immediately clear is that the identification of Z with base(HEAD) is impossible, since base(HEAD) is not disjoint. This means that for big to felicitously modify a mass noun, another interpretation for Z must be found. Above, we discussed two kinds of neat mass nouns: number-neutral neat mass nouns and itemized neat mass nouns. Number-neutral neat mass nouns: poultry The i-set interpretation of poultry is based on a disjoint set BIRDwt, and base(poultry) = *BIRDwt, which is not a disjoint set. Z cannot be linked to base(poultry), so, if big poultry and small poultry are to be felicitous, some other contextually relevant set must be found that Z can be linked

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to. Obviously, for number-neutral neat mass nouns there is one disjoint set that is always contextually available, namely ATOMbase(poultry) (= BIRDwt) (and, arguably, the only other contextually relevant disjoint sets are subsets of BIRDwt). So the natural choice (and really the only natural choice) for Z is: Z = ATOMbase(poultry)

This means that we expect that number-neutral neat mass noun poultry allows distributive and count comparison interpretations, but only interpretations that are the ones we find with corresponding count nouns:  thus, x is big poultry entails that the birds making up x are big.  most poultry is outside is true iff the sum of the poultry that is outside has more birds in it than the sum of the poultry that is not outside. Itemized neat mass nouns: kitchenware Landman 2011 argues that for itemized neat mass nouns like kitchenware, what big distributes to is itself context-dependent, varies in context. This means that Z can be linked to different salient disjoint subsets of the base (or even to sets that result from removing overlap by tinkering in context with the ontology, a process that is called pragmagic in Landman 2020). Mess mass nouns: groot vlees In the case of mess mass nouns like rijst (rice), there is a natural salient disjoint set available, namely the natural grid grain. In the case of vlees (meat), there isn’t even a natural grid available. In the latter case, the only way of finding a disjoint set available is through contextual portioning. In Dutch, if in context, PORTIONc makes a disjoint set of portions salient, the semantics allows Z in DZ to pick up λx.PORTIONc,wt(x) ^ body(P)(x), and derive an interpretation of groot vlees as meat that comes in the form of big portions. The same holds for the choice of Z in counting comparison interpretations of most: The same choice of Z as λx.PORTIONc,wt(x) ^ body(P)(x) can be made available in context via portioning and enters into the semantics of the distribution set. Two final comments. Comment one: the analysis has nothing to say about why such portion interpretations are available for Dutch and German mess mass nouns and not in English. The analysis only explains what happens, if and when it happens. Comment two: on the analysis given, the fact that numerical expressions like at least three and distributors like each cannot apply to

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mass nouns, and the generalization that something similar is true in many languages of the world is not accounted for: It follows from the stipulation that in these languages explicit numerical phrases and explicit distributors do make reference to Dbase(HEAD) and not just DZ. The present analysis, so far, has no explanation for why so many languages make this semantic choice. On the other hand, the present analysis allows for the existence of languages that do not make this stipulation. Such languages would allow numerical phrases to apply to prototypical mass nouns, counting portions. As mentioned above, this is exactly what Lima 2014 argues happens in Yudja: Yudja has no lexical mass/ count distinction, all nouns can be counted, and what is counted is contextual disjoint portions: (22)

Txabïu apeta pe. three blood dripped. ‘Three contextually disjoint portions of blood dripped.’

6.13

In Sum

Iceberg semantics is a fruitful semantic framework for studying differences between count nouns and mass nouns, and between different types of mass nouns (like neat mass nouns and mess mass nouns). Iceberg semantics gives a compositional analysis of these semantic distinctions, which means that it is a natural framework for analyzing the complexities and subtleties of mass and count aspects of the interpretations of complex noun phrases (like measure phrases and classifier phrases). The framework provides insight into how and why counting and count comparison is possible in the mass domain. Acknowledgements The story of the Mad Wigmaker in the appendix below forms the ‘base’ of this paper. It was developed in the course of a seminar on the mass/count distinction that I taught at Tel Aviv University in 2001. The body of this paper derives from two separate presentations at two separate workshops on Countability at the Heinrich Heine University Düsseldorf. A version of Part I was presented at the first workshop, organized in 2013 by Hana Filip and Christian Horn, the workshop that the present volume derives from. A version of Part II was given at the second workshop in 2016, organized by Hana Filip. In between the two workshops in Düsseldorf, I presented Iceberg semantics in two lectures at the 7th International School in Cognitive Sciences and Semantics at the University of Latvia in Riga in 2015.

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Since 2016 I have presented versions of this talk at Tel Aviv University, the Hebrew University of Jerusalem, Bar-Ilan University, the University of Vienna, and the Ohio State University/Maribor/Rijeka Philosophy Conference on Number, held in June 2017 in Dubrovnik. Many thanks to the organizers and participants of all these events for helpful comments and stimulating discussion, with special thanks to Hana Filip, Scott Grimm, Daniel ‘ Hyde, Manfred Krifka, Beth Levin, Suzi Lima, Susan Rothstein, Jurgis Škilters, and Peter Sutton. , A first version of this paper was written while I was at the University of Tübingen as a Humboldt Research Award Fellow in 2015–2016. I thank the Alexander von Humboldt Foundation for this wonderful opportunity, and the Linguistics Department at the University of Tübingen, and in particular Gerhard Jäger, Fritz Hamm, and Heike Winhart, for their much appreciated help. Writing the final version of this paper was facilitated by the helpful comments of Hana Filip, Peter Sutton, and Henk Zeevat. This paper has developed through continuous interaction with Susan Rothstein and through reflection on her work. While obviously we disagree on many details, it does represent something of a common enterprise which involves a solid base and a large body of shared ideas. Toujours gai. These acknowledgements were written in 2017. Susan died in 2019. The acknowledgements stay as they are, because We had fun while it lasted.

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Appendix

The Mad Wigmaker

One night, while Pavarotti is asleep, he falls victim to the Mad Wigmaker. The Wigmaker cuts off his hair, and takes it to his secret laboratory. He spreads out the hairs, numbers them, and cuts each of them into five pieces, numbered 1, . . ., 5, top-down. Then he takes piece 2 of hair 1, piece 1 of hair 2, piece 5 of hair 3, piece 4 of hair 4, and piece 3 of hair 5, and by innovative molecular techniques combines these pieces in that order in the normal way for hair. He goes through all of them in this criss-cross way. After this, since he wants the hair to be longer, he takes half of them, and attaches them with the same molecular techniques to the other half. The result he sticks in a wig. A hundred years pass. The wig gets discovered, Scientists do extensive genetic analysis on it. They find the genetic match with Pavarotti, and the wig is put into a museum under special light. Enter two museum attendants. (23)

a. Paola: This is Pavarotti’s hair. As you can see, in this light, Pavarotti’s hair has a magical golden shine. b. Paolo: These are Pavarotti’s hairs. As you can see, in this light, Pavarotti’s hairs have a magical golden shine.

Paola’s statement, I think, is unproblematic. It is Pavarotti’s hair, and presuably, there is a magical golden shine. Paolo’s statement, on the other hand, is problematic. I do not think that it is true that what we have here is Pavarotti’s hairs. I don’t have a problem assuming that what we have is hairs, but they’re not Pavarotti’s hairs. Consequently, the continuation of Paolo’s statement following is, as far as I am concerned, infelicitous. In Iceberg semantics we can argue as follows. In the starting context wt, we can take the mass noun phrase hair of Pavarotti to denote Pavarotti’s hair in situ: HAIRmass,

wt

=

hair + space regions

Within this denotation, isolate a variant, a partition of tHAIRmass, wt of single strands of hair+space regions that contain single strands of hair:

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HAIRstrand-variant, wt = with HAIRstrand-variant, wt disjoint and tHAIRstrand-variant, wt = tHAIRmass,

wt

We can go one step further, and lift the strands out of their spatial context: HAIRstrand, wt = with HAIRstrand, wt disjoint

Strictly speaking, tHAIRstrand, wt 6¼ tHAIRstrand-variant, wt, but, of course, the two are very closely related: in given situation wt you can readily switch between these two count denotations. We now go to Paola and Paolo in context (wt)0 . Pavarotti’s hair has, under the hands of the Mad Wigmaker, undergone a transformation, but the felicity of (23a) suggests that Pavarotti’s hair in wt and Pavarotti’s hair in (wt)0 are cross-temporally identical, i.e. intensionally identical:

where ~ is a relation of cross-temporal identity. But, unlike in wt, in (wt)0 there is no variant HAIRstrand-variant, (wt)0 in HAIRmass, (wt)0 , and this means that the connection that existed between tHAIRmass, wt and tHAIRstrand-variant, wt and tHAIRstrand, wt is broken in (wt)0 . Thus there is no connection in (wt)0 between tHAIRmass, (wt)0 and tHAIRstrand, wt. This is why (23b) is infelicitous. Thus, the Iceberg semantics framework can be used succesfully to analyze supremum puzzles like the one in (23): It allows for natural contexts like wt, where the mass and count supremums are identical; but mass denotations can stay intensionally identical under transformations for longer than corresponding count denotations, and then the supremum identity is lost, as in (wt)0 .

REFERENCES Barner, David, and Jesse Snedeker (2005). Quantity judgements and individuation: evidence that mass nouns count. Cognition 97: 41–66. Chierchia, Genaro (1998). Plurality of mass nouns and the notion of semantic parameter. In Susan Rothstein (ed.), Events and Grammar, pp. 52–103. Berlin: Springer[Kluwer]. Dalrymple, Mary, Makoto Kanazawa, Sam Mchombo, and Stanley Peters (1994). What do reciprocals mean? In Mandy Harvey and Lynne Santelmann (eds.), Proceedings of SALT 4. Ithaca, NY: Cornell University. Gillon, Brendan (1992). Towards a common semantics of English count and mass nouns. Linguistics and Philosophy 15.6: 597–639. Khrizman, Keren, Fred Landman, Suzi Lima, Susan Rothstein, and Brigitta R. Schvarcz (2015). Portion readings are count readings, not measure readings. In Thomas

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Brochhagen, Floris Roelofsen, and Nadine Theiler (eds.), Proceedings of the 20th Amsterdam Colloquium. Amsterdam: ILLC. Landman, Fred (1989). Groups, i & ii. Linguistics and Philosophy 12.5: 559–605, 723–744. (1991). Structures for Semantics. Berlin: Springer [Kluwer]. (1995). Plurality. In Shalom Lappin (ed.), The Handbook of Contemporary Semantic Theory, 1st ed. Oxford: Blackwell. (2000). Events and Plurality. Berlin: Springer [Kluwer]. (2004). Indefinites and the Type of Sets. Oxford: Wiley-Blackwell. (2011). Count nouns – mass nouns – neat nouns – mess nouns. In Michael ‘ Glanzberg, Barbara H. Partee, and Jurgis Škilters (eds.), Formal Semantics and , Pragmatics: Discourse,Context and Models. The Baltic International Yearbook of Cognition, Logic and Communication 6, 2010, http://thebalticyearbook.org/ journals/baltic/issue/current. (2016). Iceberg semantics for count nouns and mass nouns: classifiers, measures and ‘ portions. In Susan Rothstein and Jurgis Škilters (eds.), Number: Cognitive, , Semantic and Cross-linguistic Approaches. The Baltic International Yearbook of Cognition, Logic and Communication 11, 2016, http://dx.doi.org/10.4148/19443676.1107. (2020). Iceberg Semantics for Mass Nouns and Count Nouns. Berlin: Springer. Lima, Suzi (2014). The Grammar of Individuation and Counting. PhD Dissertation, University of Massachusetts Amherst. Link, Godehard (1983). The logical analysis of plurals and mass terms: A latticetheoretic approach. In Rainer Bäuerle, Urs Egli, and Arnim von Stechow (eds.), Meaning, Use and the Interpretation of Language, pp. 303–323. Berlin: de Gruyter. Partee, Barbara, and Vladimir Borschev (2012). Sortal, relational, and functional interpretations of nouns and Russian container constructions. Journal of Semantics 29.4: 445–486. Pelletier, Francis Jeffry, and Leonard Schubert (1989/2002). Mass expressions. In Dov Gabbay and Franz Guenthner (eds.), The Handbook of Philosophical Logic. Dordrecht: Kluwer. Rothstein, Susan (2011). Counting, measuring, and the semantics of classifiers. In ‘ Michael Glanzberg, Barbara H. Partee, and Jurgis Škilters (eds.), Formal , Semantics and Pragmatics: Discourse, Context and Models. The Baltic International Yearbook of Cognition, Logic and Communication 6, 2010, http:// thebalticyearbook.org/journals/baltic/issue/current. (2017). Semantics for Counting and Measuring. Cambridge: Cambridge University Press. Schwarzschild, Roger. (1996). Pluralities. Dordrecht: Kluwer. (2009). Stubborn distributivity, multiparticipant nouns and the count/mass distinction. In Proceedings of NELS 39. Amherst, MA: GLSA. Sharvy, Richard (1980). A more general theory of definite descriptions. Philosophical Review 89.4: 607–624. Winter, Yoad (2001). Plural predication and the strongest meaning hypothesis. Journal of Semantics 18.4: 333–365.

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7

Indexical Inference: Counting and Measuring in Context Alice G. B. ter Meulen

7.1

Introduction

Counting is a form of situated reasoning, in which conclusions may contain new indexical expressions, dependent for their interpretation on the resulting situation, or on recent past or anticipated future situations.* The dynamic, context changing semantics of Discourse Representation Theory (DRT), first proposed by Hans Kamp (cf. Kamp and Reyle 1993) and extended to cover aspectual verbs and adverbs in Smessaert and ter Meulen (2004), serves us in this paper to investigate such novel forms of indexical inference as systematic, interpretive processes of context change. Novel indexical expressions may be introduced in the conclusion of an argument, even when its premises do not necessarily contain any such indexicals overtly. Our investigations terminate the ancient adagio of traditional logical theory that sensitivity of reasoning processes to context should be discarded, eliminated or ignored. Placing indexical inference instead at the core of the situated logic of context change creates interesting new opportunities to investigate the situated aspects of dynamic information processing in natural languages.1 Instead of addressing the traditional linguistic mass/count distinction, as most chapters in this collection do, this chapter analyzes how we give linguistic expression to counting as a cognitive process of interpretation in context. How we describe what we count in contexts interacts with our expectations and subjective assumptions, which substantially constrain how our counting may be continued. These processes underlying counting find linguistic reflection in the semantics of the interaction between aspectual adverbs and phrases with numerical DPs, interpreted as counting members of the denotation of their *

1

This chapter began as a paper at the conference on Countability and Measurement at the University of Düsseldorf in September 2013. It has subsequently been extended and substantially revised after two helpful reviews, gratefully acknowledged by the author. Any errors or unclear propositions that may remain will be corrected in new developments and subsequent publications. In spite of the rather abundant linguistic research on counting, measuring, numerical determiners and their logical behavior in a dynamic semantics, no publications seem to be concerned with the way aspectual information may support indexical inferences, the topic of the current paper.

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noun. Counting is obviously a process with a beginning and an end, described by aspectual verbs, which contribute presuppositions and constraints on continuations of that counting process. Aspectual adverbs, quantifying over the internal structure of events, contribute to the information content of phrases with numerical DPs a polarized scalar structure of temporal alternatives. Their logical function is to create local cohesion in a counting or measuring context by presupposing a backgrounded stage-level event. This account will further outline a semantic explanation of some new observations regarding the variability in the interpretability of English, French and Dutch data. 7.2

Aspectual Modification of VP-Internal Numerical DPs

If a counting event is described statically by reporting its resulting state, as in (1a), we cannot infer anything about what is actually happening, what may have happened in the recent past, nor what may happen in the near future. (1)

a. There are three students here. b. There are still three students here. c. There are already three students here.

However, if spoken without any marked prosody, the aspectual adverbs still and already in (1b) and (1c) appear to contribute essential, intensional information about the way the speaker anticipates the near future or his knowledge of the recent past of the described, factual situation, assuming as given, constant reference to the current location with the indexical adverb here.2 Instead of the indexical here, other prepositional phrases, such as in the classroom may also be used in (1a)–(1c), even though they are not necessarily interpreted as fixed in reference throughout the inference. The focus of this paper is on indexical inference to investigate how indexicals may appear overtly in conclusions of valid inferences, even though the premises do not contain such indexicals. For this reason, our examples will often contain indexical expressions. To do justice to such indexical inference patterns introducing indexical verbal predicates as arrive or leave in their conclusion, the spatio-temporal location of the reasoning agent must be fixed in the context by the indexical 2

See Smessaert and ter Meulen (2004) for a DRT account of the semantics of aspectual adverbs, based systematically on the temporal adverbs since and until and polarity reversals of aspectual verbs. This DRT account argues in detail how it improves over the well-known account of Löbner (1989) and it underlies much of the discussion in this paper. The interested reader is referred to these original sources to review the basics of DRT or of this account of aspectual adverbs, although Section 7.4 will also review some basic notions of this account. The notions of entailment, truth in a model, (in)valid inference, presupposition and continuation are used here in the standard way introduced in most undergraduate courses in semantic theory.

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adverb here as rigid designator.3 In the absence of any other background information or prosodically marked focus, we may conclude from (1b) (i) that students must be leaving, using the present tense epistemic modal, and (ii) that some may have left, satisfying the existential presupposition of still. From these two conclusions we derive furthermore (iii) that at least three students must have been here earlier, crucially reporting (ii) and (iii) in present perfect tense, as they concern the past of the current situation. From (1c) we infer, contrastingly, (i) that students must be arriving and (ii) some may have arrived, and hence (iii) that at most three students must have been here earlier. DRS conditions introduced by such present perfect clauses block access to the referents introduced within their DRS domain. Hence no anaphoric reference with, for example, a subsequent phrase as They are outside to the students that may have left or may have been here earlier is acceptable for (1b). Idem for (1c): no anaphor can pick up the students that may have arrived or may have been here earlier, since anaphora can refer only to the ones that are actually here. Obviously, had (1b) and (1c) been in the simple past tense, instead of the present tense, any conclusions validly derived from them would equally have to be past tense, conforming to the logic of tensed phrases, where premises can introduce reference times, but simple past tense conclusions must always refer to the one last introduced. The well-known focus particles only and even, which have guided the development of so much of focus semantics with focus meaning based on scalar alternatives (e.g. Rooth 1992, Krifka 2007), do not contribute any such information about the past or present against the background of an ongoing event, cf. (2a). They statically introduce higher (only) or lower (even) alternatives to the current, described situation, derived from the counting scale associated with the numerical determiners in the DPs. (2)

a. There are {only/even} three students here. 6¼> Students may be arriving. 6¼> Students may be leaving. b. There are still only three students here. ¼> Students have been arriving. c. ? There are already only three students here. 6¼> Students may be arriving. ¼> Students may be leaving.

With focused numerical DPs, as only three students in (2b), the aspectual adverb still seems to inherit the increasing scale of alternatives associated with

3

The notion of a rigid designator was first introduced by Saul Kripke in his lectures Naming and Necessity (1971, 1980); for a good introduction, see LaPorte (2018).

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only. Accordingly, the interpretation of (2b) introduces a backgrounded event of students having been arriving, instead of leaving, as still by itself would induce, cf. (1b). The aspectual adverb still in composition with a focus particle simply inherits the polarity of the scalar alternatives introduced by that focus particle and reduces to its basic meaning: its presupposed past and current situation are the same, i.e. only three students were here and are here, but more students may arrive, as only fixes the focus on the low end of the alternatives scale. The aspectual adverb already behaves differently as still, cf. (2c).4 Perhaps, (2c), There are already only three students here, may at first sight seem a quite odd way to give information. It may even be hardly interpretable to many English speakers, indicated here by the question mark. Intuitively, already presupposes a past start of the backgrounded event and asserts that the current count of three is higher than expected. But only indicates that three students constitute the low end of the scale of alternatives, i.e. the alternatives would have more than three students here. However, if (2c) is considered interpretable, even if only for sake of argument, it certainly does not support any conclusion on whether students are arriving, as one would expect based on the meaning of already discussed above. The single temporal conclusion to draw validly from (2c) is that the speaker finds it early for the low count of three students to be realized now. In other worlds, the speaker had expected the lower limit of three students to be realized here later, so he had expected more than three students to be here now and for students to be leaving slower than they do. If it is known or given as background information that students are leaving, one may express, perhaps somewhat ironically, one’s surprise (distress, satisfaction or some other mental attitude) that the low end of the scale is reached so soon with (2c), indicating that students must have been leaving rather more quickly than expected. Besides the indexical inferences in (1) and (2), our intuitions on some other invalid indexical inferences provide further intriguing data for a logical theory of indexical inference with aspectual adverbs. Consider the intuitively invalid inferences in (3), to shed more light on the contextual constraints on patterns of indexical inference. (3)

4

a. There are no longer three students here. 6¼> Students may be arriving. 6¼> Students may be leaving. b. There are not yet three students here. 6¼> Students may be arriving. 6¼> Students may be leaving.

English seems more restricted in the composition of polarized numerical DPs in comparison to Dutch, see Section 7.5.

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As a static premise, (3a), There are no longer three students here, does not support the conclusion that students are leaving, and hence there are fewer than three students here, for (3a) may well be true if students keep arriving and hence there are more than three students here. The only, purely temporal conclusion supported by (3a) is that there were three students here and that the number of students that are now here is not three. But (3a) does not warrant any conclusion about whether that number is now less than three or more. The same observations may be made about (3b), as it may be true in either situation of students leaving or arriving. The aspectual adverbs not yet and no longer rely on, respectively, a future start or a past end of the event described by the main verb, requiring their reference time to be fixed either before or after the event. This explains why the background indexical conclusions are not derivable from (3a) to (3b): they belong to a closed chronoscope, a context which is anchored to the past (3a) and the future (3b), respectively, and, hence, inaccessible from the present later or earlier reference time respectively.5 To explain this in more formal detail, we have to appeal to explicit DRS conditions introduced and discussed below. Although in (3a) the current reference time falls after the event, due to the meaning of no longer, it does provide the information that the situation of there being three students has been terminated, completed or ended, either by students leaving or by more of them arriving. The perfect tense claim that there must have been three students here is of course supported, as it is a presupposition of the aspectual adverb no longer.6 From (3b), with not yet three students, we only infer statically that there are fewer than three students here, where the existential context eliminates the situation that there are no students here as lowest scalar alternative. 7.3

Numerical DPs as External Argument

When the numerical DP three students is moved to the Spec of IP position as external argument, but the aspectual adverb remains in situ, i.e. in its position modifying the VP, as in (4), the interpretation is affected as well as the indexical inference patterns it supports. (4)

5

6

½IP ½DP Three studentsi ½VP are not yet ti here: 6¼> Students are arriving. ¼> The three students are arriving.

The notion of a chronoscope is defined in the theory of Dynamic Aspect Trees (DAT) in ter Meulen (2000). It is informally understood to represent information about events that are anchored to the same reference time, i.e. overlap at least at one and the same moment. In DAT theory, perfect tenses are represented in portable stickers, which, though introduced locally at a current node, may be imported onto the root node of the DAT and filtered down to lower, dependent nodes.

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Accommodation of the presupposition of not yet introduces first a group of (at least) three students into the common ground of (4), which then serves as antecedent to bind VP-internal trace of the indefinite subject DP, in virtue of its syntactical position, projected outside the VP scope. Hence from (4) we may infer that the group of three students the DP refers to will be here later by accommodating the presupposition of the aspectual adverb not yet. That conclusion is not supported by the existential context with the aspectual adverb modifying the VP-internal subject DP in (3b). Projection of a numerical DP to a VP-external position out of the syntactic scope of an aspectual adverb in situ has corresponding, compiling semantic effects: (i) accommodation of the presupposition of the aspectual adverb in situ, i.e. in VP-internal position, (ii) the external indefinite subject DP is interpreted as referring to this presupposed plural group and (iii) the VP-internal trace of the DP is itself subsequently interpreted as coreferential to it. Such binding by indefinites in external argument position via projection of presuppositions of VP-internal aspectual adverbs is impossible with non-scalar determiners as any, which, like all negative polarity expressions, must remain in the syntactic scope of a negative, downwards entailing context for their interpretation. Our intuitive judgments of the validity of such inference patterns with conclusions containing novel indexicals that do not overtly occur in the premises provide us with a starting point to develop a semantic theory of indexical inference to account for the validity of such inferences systematically. Indexical expressions for our purposes include, of course, not only the well-known indexicals as the first- and second-person pronouns, but also spatio-temporal expressions here, there, left, right, back, front, behind, now, then, later and earlier, as well as since and until. We investigate below in more detail the indexical character of the aspectual adverbs in composition with numerical DPs. Verbal tenses, even the present tense, are of course always indexical, as their interpretations depend on the current reference time or time of use of the expression, which separates past from future at any point in the process of interpretation. Our human reasoning in natural language is hence always inherently indexical, unless tense and aspect is disregarded as semantically irrelevant, which, regrettably, is so often the case in logical investigations. 7.4

The Dynamic Semantics of Aspectual Adverbs

The DRT semantics of the four basic English aspectual adverbs, first presented in Smessaert and ter Meulen (2004), repeated here in (5) with slight modifications, primarily captures the Boolean interactions between three truthfunctional aspects of their meaning, together constituting their descriptive content. It is based on the indexical interaction between the speech time, the

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current event and its past onset or future ending, using the basic aspectual verbs and the essential temporal adverbs since and until. The stative, factual temporal content of the four basic English aspectual adverbs modally introduces past ( h) or future (h !) polarity reversals (+ and –). D is the domain of entities to which newly introduced referents may be added, and cn the context at stage n, preserved if the aspectual adverb modifies a stative description, otherwise updated to a next stage. (5)

Basic aspectual adverbs (Smessaert and ter Meulen, 2004)   a. D, c0 vp λx ½INFL not yet PðxÞ D0 , c0 ¼ 

0

0 0 0 D,c0∪

&h! &UNTIL    b. D, c0 vp λx ½INFL already PðxÞ D0 , c0 ¼ 

0

 0 0 0 D,c0∪

& h &SINCE   c. D, c0 vp λx ½INFL still PðxÞ D0 , c0 ¼ 

0

 0 0 0 D,c0∪

&h! &UNTIL   0 d. D, c0 vp λx ½INFL no longer PðxÞ D , c0 ¼ 

0

 0 0 0 D,c0∪

& h &SINCE

To aid in understanding, (5a) may be informally paraphrased as follows: the interpretation of a VP containing the aspectual adverb not yet with an intransitive, main verb P in context c0 given domain D of entities, requires 0

0

(i) adding the condition that x is not P at c0 , where P is the semantic property interpreting the syntactic verb P, for simplicity taken to be a one-place predicate here, adding x to the given domain D to make it a larger domain D’, keeping the context c0 fixed, and (ii) imposes a constraint on all continuations of c0 to add that this condition, 0 that x is not P at c0, ends and until it ends it remains the case, i.e. its end is 0 the first, closest endpoint of not- P ðxÞ in c0 ; the underlying model M of interpretation, not indicated here as parameter, may of course contain 0 many earlier and later occurrences of P ðxÞ. Ending a condition that x is not P amounts logically to starting the condition that x is P, as x must either be in a state of not P or in a state of P, as the classical condition of excluded middle requires. Generalizing (5a) to verbs with more arguments is straightforward and not interesting. The other three rules of interpretation of basic aspectual adverbs (5b)–(5d) reverse the polarity of the state, i.e. already and still require x to be P, or they replace END with START or UNTIL with SINCE, with corresponding switches between operators over all (future) continuations or (past) histories of the current context c0. When the conditions in (5) are adopted as formal update instructions to add new conditions to the given, input DRS with domain D at the construction stage c0, they capture the basic temporal inference potential of the four aspectual adverbs in English. The SINCE and UNTIL conditions always serve to constrain the polarity transitions END or START to the last past or the first

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future one, given the current DRS, excluding any possible, interrupting polarity transition for the ongoing event within the current context. Hence these conditions ensure the continuity of the negative polarity in the first condition 0 > for not yet and no longer, where the entire P-event is either 0 future or past, or the positive polarity > for already and still, when the P-event is going on. Obviously, already and still are in this sense veridical, whereas not yet and no longer are anti-veridical. These DRS conditions in (5) create an essentially indexical dependency on the current state of the context c0. On the numerical scale in the examples (1)–(4) discussed above, counting how many students there are, with time passing, the aspectual adverb still is decreasing toward any of its future end points in the context, associated with leaving, whereas for already it is increasing from its past starting point in the context, associated with arriving, as discussed in the previous sections. These formal properties in (5) can explain now in more detail why (6a) below is perfectly fine and coherent: still introduces modally a future end point of the state of there being just three students and only contributes the scalar alternatives of more students arriving, where three is the low end of that numerical scale. (6)

a. (= (2b)) There are still only three students here. b. (= (2c)) There are already only three students here.

But (6b) requires some coercion or reinterpretation for semantic reasons, as already merely introduces a past starting point of the current state of there being just three students, but no later end point of that state, when more students may be arriving. A coerced interpretation may require an appeal to the speaker’s subjective information regarding the timing of the state, where already indicates that the speaker considers the past start of the state earlier than he expected (cf. Smessaert and ter Meulen 2004 for a systematic account of subjective speaker information in focused aspectual adverbs). Expressing one’s surprise at having just three students earlier than expected conflicts with only introducing a low end of the scale with alternatives of more students arriving later. However, this may yet make sense, if a tacitly assumed backgrounded event requires leaving students as alternatives, rather than arriving students. One’s surprise is then expressed that their actual departure has been so swift as to have just three students left here so soon, earlier than expected. Such coerced interpretations often require special prosodic contours, if spoken, but it would lead us too far astray to investigate such phonological issues in this paper. Often in aspectual interpretation, specific adjustments in the context may coerce a special interpretation, overruling otherwise common assumptions. In this way, we may coerce (6b) into expressing the speaker’s judgment that it is

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relatively early to have reached the state of having just three students here. Already applies to the VP containing the entire polarized DP only three students and the scalar alternatives of only, i.e., having more than three students here, are coerced to be past, rather than future, as now the background contains the information that the students are leaving. This overrules the otherwise conflicting presupposition that students are arriving associated with already, retaining only its information that the start of there being just three students is considered to be early relative to the speaker’s expectations or other attitudes. 7.5

DRT Semantics of Aspectual Adverbs with Numerical DPs

The various steps in the process of interpretation of (1b) are specified in more formal DRT detail in (7). They require first accommodating the backgrounded presupposed information of (7) in the Common Ground (CG) that (i) there is a group X of students here, using capital reference markers for plural objects as is customary in DRT and (ii) and a group Y of students, maybe a smaller group, are leaving, which, as we discussed above, is triggered by still in this context. Although DRT itself does not account for focus structure to determine the available set of alternatives for any focused phrase, in the interpretation of (7), the alternative continuations of the current situation must have fewer than three students here, according to the presuppositions, already accommodated in DRS conditions. The focus or ‘asserted information’ that the number of students is now three is then added as a DRS condition, constituting a disputable claim to the truth of (7). In the DRS updates for (7) X, Y and e are new reference markers, respectively for plural referents and for the leaving event, extending the given domain D to D0 , when the condition they occur in is first incorporated in the DRS. Their reference is fixed by the simultaneous embedding of the entire DRS in the model, as is standardly assumed in DRT (Kamp and Reyle 1993). (7)

j½There are ½FOC still threestudents herejD0 , cnþ1: CG (i) there are students here ¼> 9X ½studentsðXÞ&locðXÞ ¼ lc0  (ii) students are leaving ¼> 9Y, e ½studentsðYÞ&leaveðe, YÞ&e⊃t0  DRS update of CG: D,c0 ∪¼D0 c1 DRS update of focus meaning: Focus structure : λ < n, t0 > ½ n ¼ jXj&n  3&t1 < t0  Focus : D0 , c1 ∪ >¼ D00 , c2 D, cn

The focus structure of (7) is contributed by still three, which requires some discussion of the relation between syntax, semantics and focus structure, surely a topic itself worthy of several dissertations. Syntactically numerical determiners take a noun to form a determiner phrase (DP), which forms a

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constituent, as the usual movement tests support. Semantically, determiners are considered to denote relations between sets, or, equivalently, a property of the intersection of the set denoted by the noun in subject position and the denotation of the verb phrase, assuming Conservativity as a basic principle of DP interpretation.7 Aspectual adverbs semantically modify this relation in (7), considered as a temporally extended process of counting on a numerical scale. Focus structure comes on top of the supposedly compositional interface of syntactic and semantic structure of a phrase. Information structuring is underpinned by a set of context-sensitive, interactive rules of sharing information between users. To determine focus structure at the level of information structure, a phrase, after having been syntactically parsed and semantically evaluated in the model, is divided into (i) its given information, which includes background, common ground and accommodated presuppositions, i.e. all information that is already collectively considered to be true and not in dispute and (ii) the content which is ‘asserted’ as new information, offered as true by the issuer, but considered disputable, until the addressee accepts it as true too. This focus determines which of the focus alternatives is proposed as true, though disputable new information. Typically, focus answers (wh-)questions based on the given information. Accordingly, for (7) still three answers the question of how many students are here, basically reflecting the shared assumption of the scale of the ongoing counting process. The focus structure of (7) is here specified by lambda-abstraction over ordered pairs of numbers, the alternative cardinalities of the set X of students here, and future times, i.e. possible continuations, to provide a set of twodimensional alternatives with fewer than three students at any later time. The focus itself describes the actual state in which there are three students here in the updated, final DRS for (7) with resulting extended domain D00 and updated context c2. We leave further detailed specifications aside since to spell the entire process of interpretation out in required algorithmic detail would lead us too far astray from our current purposes, which is to argue that indexical inferences should be at the core of any investigations into the dynamics of the semantics for natural language. Assuming the lexical semantics of x leave y as describing an event of transition between the state of x’s being at y to x’s not being at y later, as in (8a), we may infer from (1b) using the present perfect tense that some of the students, who were at the location indexically referred to by here, have left that location. Mutatis mutandis, the indexical inference from (1c) is that some of the students who were not here, but are here now, have hence arrived, exploiting (8b) as the lexical semantics of arrive (□! is the operator 7

See van Benthem and ter Meulen (2011), for a comprehensive collection of recent research on generalized quantifier theory and its logical aspects.

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universally quantifying over the continuations, spc refers to the speaker at the context). (8)

a. 〚leaveðe; x; spc Þ〛¼ λ < x, e > ðlocðxÞ ¼ locðspc Þ&h ! ðmoveðe; xÞ ¼> locðxÞ 6¼ locðspc ÞÞÞ b. 〚arriveðe; x; spc Þ〛¼ λ < x, e > ðlocðxÞ 6¼ locðspc Þ&h ! ðmoveðe; xÞ ¼> locðxÞ ¼ locðspc ÞÞÞ

The anti-veridical aspectual adverbs not yet and no longer share the commonground conditions and the focus structure with their respective veridical counterparts already and still, but reverse the focus polarity, i.e. negate the focus information. Accordingly, not yet three will be interpreted against the same backgrounded information as already three, updating the CG with the backgrounded event that students are arriving, but asserts that the actual number of students here is less than three. Ordinarily, no longer three also shares the backgrounded information of departing students with still three but denies that there currently are at least three students here. With no longer, condition (ii) concerning the ongoing leaving of the students may be dropped from the CG without triggering any coercion effect, as at some point the students will all have left. Clearly, it remains true indefinitely in the future that there are no longer three students, however, irrelevant this fact may later be in new contexts. Without the backgrounded ongoing event, only conclusions stated in the perfect tense may be supported regarding the number of students who were here earlier, as in DAT theory perfect tense clauses are represented as portable stickers, supported even by the root node (see notes 5–6). 7.6

Presupposition Projection and Dialogue

This section explores how aspectual adverbs are interpreted if they are part of an interrogative in dialogue. This provides us with some further insights into their function at the level of information structure, albeit in an as yet informal and preliminary way. The presuppositions speakers may impose regarding the backgrounded event project into the common ground in an interrogative dialogue, as part of all information that is tacitly accepted and not in dispute both by the questioner and the addressee. Consider the use of aspectual adverbs in polarity questions, as in (9). (9)

a. Any students still here? CG: there were students here. b. Any students already here? CG: there were no students here.

Either a positive or a negative answer to the polarity questions in (9a)–(9b) would project, respectively, their positive (9a) or negative (9b) existential presupposition regarding the past presence or absence of students to the DRS common ground. A positive answer to (9a) does not by itself affirm the

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questioner’s subjective assessment that, relative to his own expectations, the students that are now here are late to depart, nor for (9b), that the students that are now here are early to arrive. Accordingly, a negative answer to (9a)–(9b) merely denies that there are any students here. In fact, answering negatively to (9a)–(9b) may even be compatible with disagreeing with the backgrounded event of the question that in the actual situation any students are leaving or arriving at all. Since the adverbial aspectual information contained in such polarity questions of (9a)–(9b) is not in focus, it cannot be addressed by any polarity answer. Aspectual adverbs in interrogatives may, however, also be used elliptically by themselves in dialogue in response to an assertion, especially when supported with marked prosody (indicated here in capitals). In such uses, the subjective speaker information regarding the timing of the event described in the statement is actually in focus, as the factual information stated is accepted and taken for granted in the common ground. Consider the dialogue in (10). (10)

A: Three students are taking the test. B: {STILL? / ALREADY?}

The factual, truth-functional information stated by A is relegated to the common ground, as veridical, by B’s response with an aspectual adverb in an elliptical, prosodically highly marked question. B’s veridical interrogative in (10) is soliciting A’s agreement with B’s subjective assessment that the students are respectively late to end, or early to have started taking the test. B indicates accordingly that she had counterfactually expected that the students had ended the test or, respectively, had not yet started the test by now, i.e. the speech time. A positive polarity response from A to B’s question would simply convey that A agrees with B’s subjective assessment as to the timing of the event he described in his first statement. A negative response from A to B’s aspectual question would, however, be much harder to process, as A may not only disagree with B’s subjective view on the timing, but he may also have a further reason for responding negatively, depending on what background information may be available to address what is at issue. Exclamatory prosody on aspectual adverbs always marks the subjective speaker information as in focus, relegating all factual description to the common ground. Agreement in such, often emotionally charged information exchanges is sought by default, whereas disagreement demands further explanation, clarification and dispute to sort out what information is in fact shared in the common ground of the dialogue or conversation. These observations show that aspectual adverbs play an important role at the level of information structure, as they not only convey factual content but also speaker subjective, often emotionally charged content. Whether the former or the latter is in focus, hence in dispute, depends, however, primarily on the

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syntactic structure of the phrases these adverbs occur in. To date, no theory of interrogatives seems to formalize such interactive dialogues and focus structure. The DRT framework must be enriched first with a formal account of focus structure, before justice can be done to these observations. 7.7

Linguistic Variability: English, Dutch and French

It is well known that second language learners often have a hard time grasping the nuances and different usages of the aspectual expressions of the language they are learning. Often, it is in their use of aspectual expressions that they betray that they are non-native speakers. Given the vast amount of research on the linguistic variability of tense and aspect systems and the search for tangible linguistic universals of temporal reference and temporal reasoning (Bittner 2008, 2014a; Dahl 1985, 2000), it may be of interest to note here only briefly some rather striking dissimilarities between English, Dutch and French in the way aspectual information may get expressed in contexts with numerical DPs. There is no other pretense here than sketchily presenting some data which give us some initial indication of the linguistic variability of the syntax of aspectual information in languages otherwise thought of as rather closely related. In Dutch, a Germanic language that characteristically liberally composes morphologically and lexically complex expressions, the numerical DPs describing the counting process, may be modified by prepositions, as in (11) (bold face indicates focus and capitalization indicates high pitch prosody). (11)

NL a. Er zijn drie studenten over. There are three students over. There are three students left. b. Er zijn nog (steeds) drie studenten over. There are still (always) three students over. There are STILL three students left. c. Er zijn maar drie studenten over. There are only three students over. There are only three students left. d. Er zijn nog maar drie studenten over. There are still only three students over. There are still only three students left. e. Er zijn nog steeds maar drie studenten over. There are still always only three students over. There are STILL only three students left.

In (11a) the numerical drie (three) is static, reporting the result of a counting process, and the postverbal preposition over contributes the information that there were more than these three students here at an earlier time, hence

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presupposing that students are leaving. Without this postverbal preposition over in the Dutch examples of (11a)–(11e) it would not be clear whether students are arriving or departing. The English equivalent would have to use the perfect participle describing the leaving overtly, instead of a prepositional phrase. In (11b)–(11e) the aspectual adverbs and the scale of the numerical DPs provide additional information about the context, more liberally composed lexically in Dutch than English permits. In (11b), the aspectual adverb nog together with the postverbal over conveys the backgrounded information that students are leaving and states that the number of students that were here was three and that presently still three remain here. In (11c), the particle maar (lit. but, only) is a negative polarity item, which indicates together with the postverbal over that three is the lower bound on the numerical scale of students here who remain, without any indication of what may happen to them in any continuation of the backgrounded event of departing students. The composition of the aspectual adverb nog with the negative polarity maar in (11d) with the postverbal preposition over would mean that the number of students here has reached the lower bound of three on the counting scale. Note that the other veridical aspectual adverb al (already), instead of the decreasing nog (still) in (11d), would plainly be ungrammatical in this phrase, leaving no room for any coercion in its interpretation. In (11e), again with the postverbal preposition over, the composition of the aspectual adverbs nog and steeds with the polarity item maar means that the speaker had expected that earlier and now fewer than three students would remain here, lexically expressing what would effectively be communicated by still with marked, high pitch prosody in English. The iteration of two aspectual adverbs and a polarity adverb as in (11e) is ungrammatical in English, as it resorts to marked prosody to convey subjective speaker meaning. It is clear from these initial Dutch data on aspectual adverbs’ interaction with numerical DPs that the postverbal preposition over plays a very important role in backgrounding the departure of students. The aspectual adverbs merely contribute the temporal information that the backgrounded event is continuing (nog/still) or has just been started (al/already). Importantly, in Dutch the students here are not expected in any sense to be leaving too, as is so clearly suggested in the English counterparts with just the veridical aspectual adverbs without any preposition. Let us take a brief look at the Dutch numerical DPs with aspectual adverbs in focus (bold face) with phrases that describe events, not states as in (11) above. Consider the data in (12): (12)

a. Er komen al drie studenten aan. There come already three students to. There arrive already three students.

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b. Er komen nog drie studenten aan. There come still three students to. There arrive three more students. c. Er komen maar drie studenten aan. There come only three students to. There arrive only three students. d. *Er komen al maar drie studenten aan. There come already only three students to. *There arrive already only three students. e. Er komen alsmaar drie studenten aan. There come already – s – only three students to. There arrive three students all the time. f. Er komen nog maar drie studenten aan. There come still only three students to. There arrive only three more students.

The basic aspectual adverbs in (12) work as expected, where in (12a) the three students arriving early are the first ones to arrive, but in (12b) the three arriving students are in addition to the ones who may already be there. But the composition of al (already) with the low scalar focus particle maar (only) in (12d) is prima facie unacceptable, unless coerced into a rather forced meaning of it being early that only three students arrive. Another, quantificational kind of adverb, seemingly morphologically composed of these two, alsmaar, meaning continuation or iteration with perhaps a certain additional emphatic meaning, is acceptable here with an iterative interpretation of groups of three students arriving consecutively, cf. (12e). In English the iterative aspect is expressed in a quantificational adverb all the time, denoting a duration of an event within which different groups of three students arrive consecutively. The aspectual adverb nog with maar in (12f) expresses that the three students arriving now are the last ones to arrive. A preliminary conclusion seems justified, on the basis of these rather limited data, that Dutch requires prepositional phrases to contribute background events of departure or arrival, which we considered to be presuppositions of the English aspectual adverbs. The Dutch aspectual adverbs resort to their core temporal meaning of starting or continuing an event. A much more systematic investigation of the syntactic and semantic variability between English and Dutch aspectual adverbs would be required for any linguistic theory to explain such data in a more comprehensive theory of temporal reasoning (cf. also Smessaert and ter Meulen 2004). These observations do provide us with sufficient evidence to conclude that any linguistic universals regarding indexical inference based on aspectual information must be formulated independently of syntactic categories and their compositional semantics, hence at the level of information structure. In French, a Romance language, the static VP il y a encore (there are still) does not convey anything other than that three students are still here,

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temporally presupposing with still they were here earlier. The aspectual adverb encore (still) in (13a) does not support any inference that there were more than three students earlier, and it may, for instance, simply relate to any people, other than students, who were here earlier. Neither does the aspectual adverb déjà (already) support any background information regarding ongoing events, as it merely contributes the information that the starting point of there being three students is past. To adduce any background event of departing or arriving students in French, the lexical verbs rester (remain) or manquer (lack) are required, which carry the requisite presuppositions ((13b)–(13c), respectively) in focus structure with a numerical DP. Note however, that the presupposition of rester (13b) regarding the past departure of students must be reported in the present perfect and the presupposition of manquer in (13c) regarding the ongoing arrival of students is reported in the simple present, as French only has a somewhat convoluted, static periphrastic progressive (etre en train de . . ./be in the process of . . .). (13)

French a. Il {y a/restent/manquent} {encore/déjà} trois étudiants. there is/remain (plur.)/lack (plur.) still/already three students. Three students are {still/already}{here/remaining/lacking}. b. Des étudiants sont partis. Det students are left. Students have left. c. Des étudiants arrivent. Det students arrive. Students are arriving.

Although these limited and preliminary data only illustrate how French may express indexical information lexically in verbs, these variability observations justify saying that, whereas English compactly expresses indexical information in the presuppositions of its aspectual adverbs and exploits marked prosody to add subjective information regarding the speaker’s judgment of timing of the event, Dutch requires an interaction between aspectual adverbs and postverbal prepositions, and French may resort to lexical verbs to express such indexical information. Hence, any linguistic universals regarding indexical inference will have to be formulated at the level of DRSs, after the composition of meaning for any given syntactic input has been completed. It would lead us too far astray from the more limited goals of this paper to attempt to enrich DRT with a theory of focus and formalize these observations further here. 7.8

Conclusions

This paper has first outlined the semantic properties of aspectual adverbs as they modify verb phrases with numerical DPs in English, basing their

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interpretation on the polarities, start and end points and the temporal adverbs since and until together with the insights of information structure from focus theory. The English aspectual adverbs require accommodation of their presuppositions in the background of an ongoing event which ensures the coherence of the current context. The division of background or common ground in dialogue and the scalar alternatives available as focus structure are distinguished from the actual, disputable claim to truth of the focus or the assertion itself. These three levels, syntax, semantic representation and information structure, serve different linguistic goals. The aspectual information English efficiently expresses into adverbs is expressed in Dutch by relations between aspectual adverbs, numerical DPs and postverbal prepositions. French expresses it primarily in its verbal structure where tense inflections are important. These admittedly sketchy variability data, impressionistic as they are, show that, despite this syntactic variation, the concepts of information structure formally integrated within the semantic representation in DRSs should provide the best way to characterize linguistic universals of indexical inference in future research. REFERENCES Benthem, J. van, and A. ter Meulen (eds.) (2011). Handbook of Logic and Language. Amsterdam and Cambridge, MA: Elsevier/North-Holland and MIT Press. Bittner, M. (2008). Aspectual universals of temporal anaphora. In S. Rothstein, (ed.), Theoretical and Crosslinguistic Approaches to the Semantics of Aspect, pp. 349–85. Amsterdam: John Benjamins. (2014a). Temporality: Universals and Variation. Oxford and New York, NY: Wiley-Blackwell. (2014b). Perspectival discourse referents for indexicals. In Hannah Greene (ed.), SULA 7: Proceedings of the Seventh Meeting on the Semantics of Under-Represented Languages in the Americas. Amherst, MA: GLSA. Dahl, Ö. (1985). Tense and Aspect Systems. Oxford: Blackwell, Dahl, Ö. (ed.) (2000). Tense and Aspect in the Languages of Europe. Berlin: Mouton de Gruyter. Kamp, Hans, and Uwe Reyle (1993). From Discourse to Logic: Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory. Dordrecht: Kluwer. Krifka, M. (2007). Basic notions of information structure. In Féry, C., G. Fanselow, and M. Krifka (eds.), The Notions of Information Structure. Interdisciplinary Studies on Information Structure 6, pp. 13–55. Potsdam: Potsdam Universitätsverlag. Kripke, S. (1980). Naming and Necessity. Cambridge, MA: Harvard University Press. LaPorte, J. (2018). Rigid designators. In E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Spring 2018 ed.), https://plato.stanford.edu/archives/spr2018/entries/ rigid-designators/. Löbner, S. (1989). German schon – erst – noch: An integrated analysis. Linguistics and Philosophy 12.2: 167–212.

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Perry, J. (1992). The Problem of the Essential Indexical and Other Essays. Oxford: Oxford University Press. Rooth, M. (1992). A theory of focus interpretation. Natural Language Semantics 1.1: 75–116. Smessaert, H., and A. ter Meulen (2004). Temporal reasoning with aspectual adverbs. Linguistics and Philosophy 27.2: 209–262. ter Meulen, A. (1995). Representing Time in Natural Language. The Dynamic Interpretation of Tense and Aspect. Cambridge, MA: Bradford Books, MIT Press. Paperback edition published with new appendix, 1997. (2000). Chronoscopes: The dynamic representation of facts and events. In J. Higginbotham et al. (eds.), Speaking of Events, pp. 151–168. Oxford and New York, NY: Oxford University Press. (2007). Cohesion in context: The role of aspectual adverbs. In J. Dölling, T. Heyde-Zybatow, and M. Schäfer (eds.), Event Structures in Linguistic Form and Interpretation. Language, Context and Cognition 5, pp. 435–445. Berlin: de Gruyter. (2012). Modeling temporal reasoning: Aspectual interaction in determiners, adverbs and dialogue. In K. Jaszczolt, and L. Filipovic (eds.), Space and Time in Languages and Cultures, Vol. 1. Human Cognitive Processing, pp. 123–134. Amsterdam: John Benjamins. (2013). Temporal reasoning as indexical inference. In K. Jaszczolt and L. de Saussure (eds.), Oxford Studies on Time in Language and Thought, Vol. 1., pp. 37–45. Oxford and New York, NY: Oxford University Press. (2017). Aspectual quantification in DPs. In K. von Heusinger, and P. Schumacher (eds.), Prominence in Pragmatics. Special issue of Journal of Pragmatics.

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Counting and Measuring and Approximation Susan Rothstein

8.1

The Data and the Problem

In previous work (Pires de Oliveira and Rothstein 2011; Rothstein 2017), I have argued extensively that naturally atomic mass nouns, or object mass nouns, allow quantity evaluations and comparisons either in terms of cardinality or along a dimensional scale, while count nouns require comparison in terms of cardinality. This paper focuses on the theoretical issues which emerge as a result of this observation, and in particular the following question: How can we compare two sums in the denotation of an object mass noun in terms of their cardinality if they are not countable? The most obvious answer to this, offered by Barner and Snedeker (2005) and Bale and Barner (2009), is that object mass nouns have the morphosyntax of mass nouns, but the semantics of count nouns and thus the sums in their denotation are countable. However, this claim is based on the incorrect generalization that object mass nouns, like plural count nouns, always require comparison along a cardinal dimension. Grimm and Levin (2012), Gafni and Rothstein (2014) and Rothstein (2017) show that in fact object mass nouns differ from count nouns and do not always require cardinal comparisons. On the contrary, they behave like substance mass nouns where the dimension of comparison is determined by context – for example, Who has more yarn? can be answered in terms of volume, weight or length (as well as variety of kinds), depending on context. This suggests that object mass nouns and count nouns have different semantic interpretations, with object mass nouns patterning with other mass nouns. The challenge then is to give an account of cardinal comparisons which explain why cardinality is (usually) the only possible dimension for comparisons involving count nouns, but only one of a number of possible dimensions for comparisons involving mass nouns, with the choice between the dimensions to be contextually determined. In this chapter, I offer an account which explains this. The outline of the chapter is as follows. In the rest of this section, I shall review the claims in Barner and Snedeker (2005) and Bale and Barner (2009), and present the data which shows that an alternative account is necessary. 217

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Section 8.2 will formulate the issues more precisely and Section 8.3 will give the semantic background for an alternative account. In Sections 8.4 and 8.5, I present my analysis, showing that while comparison in the count domain directly compares cardinal values, cardinal comparisons are also possible using measure operations and what I shall call cardinal scales. Sections 8.6–8.8 discuss approximation and present evidence that estimation, one form of approximation, is semantically a measure operation which uses these cardinal scales. We begin with the data. It is by now well known that there are mass nouns such as furniture, jewelry, garlic, toast, which have minimal, individuable, discrete entities in their denotations, but which do not have the grammatical properties of count nouns. These have variously been called ‘object mass nouns’ (Barner and Snedeker 2005; Bale and Barner 2009), ‘naturally atomic mass nouns’ (Rothstein 2010), ‘neat mass nouns’ (Landman 2011) and ‘fake mass nouns’ (Chierchia 2010). There are two kinds of evidence that these mass nouns denote sets of discrete individuals closed under sum. First, as Rothstein (2010) and Schwarzschild (2011) independently observe, predicates of individuals can modify object mass nouns, distributing over the minimal elements in their denotations. This is illustrated (1a–c), where big furniture is interpreted as ‘big pieces of furniture’, expensive jewelry as ‘expensive pieces of jewelry’, and so on. (1d) is generally considered infelicitous, since the noun mud denotes a substance which is not inherently divided into discrete minimal quantities. (1)

a. b. c. d.

Please carry the big furniture downstairs first. (from Rothstein 2010) The expensive jewelry is on the third floor of the store. I want three heads of (the) big garlic please. #I want the big mud cleared away by hand.

This suggests that the minimal individuals in the denotation of object mass nouns are discrete individuable entities, even though they are not countable. The second kind of evidence is experimental. Barner and Snedeker (2005) show that questions of the form Who has more N? are compared in terms of cardinality when N is a plural count noun as in (2a), and with respect to a noncontinuous dimension when N is a substance mass noun (2b). When N is an object mass nous (2c), comparison is in terms of cardinality, suggesting that object mass nouns denote sets of countable discrete entities: (2)

a. Who has more pens? b. Who has more toothpaste? c. Who has more furniture/footwear?

On the basis of these results, Barner and Snedeker (2005) and Bale and Barner (2009) argue that object mass nouns have the same semantic interpretations as

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plural count nouns. Both object mass nouns and plural count nouns denote, in essence, atomic Boolean semi-lattices, although object mass nouns have the syntactic properties of mass nouns. However, recent research has suggested that the data are more complex than Barner and Snedeker’s results show. Grimm and Levin (2012) show experimentally that a variety of factors may lead to non-cardinal evaluations for object mass nouns. They show that given two sets of equal cardinality but with one set more diverse or heterogeneous that the other, participants unanimously chose the more diverse set as being ‘more’ than the other set. A sofa, two chairs, a coffee table and a bookcase were consistently considered ‘more furniture’ than five chairs.1 Furthermore, for object mass nouns, context plays a role in choosing a dimension of comparison. Grimm and Levin hypothesized that a possible dimension of comparison is how well a collection or sum of objects fulfills a function in a particular context. They then asked for quantity comparison of two sets of different cardinalities in contrasting contexts. One context focused on the normal function of, for example, furniture, and one context neutralized it. For example a collection consisting of a sofa, an easy chair, a coffee table and a bookcase was compared with a collection consisting of a table and four chairs. In the context enhancing functional orientation, the participants were told that the owners of the respective sets were furnishing their (respective) apartments and were asked, ‘Which room has more furniture in it?’ In the context-neutralized situation, participants were told that each collection was bought by a different dealer, and were asked ‘Who bought more furniture?’ Where function was neutralized, 75 percent of participants compared in terms of cardinality, suggesting that in context-neutral situations, cardinal comparisons are indeed the default. However, when function was in effect, only 35 percent of participants did so. This variation was not seen when the noun was count, either a sortal like chairs or a superordinate like vehicle, where quantity evaluation was only in terms of cardinality. Clearly, when the N was an object mass noun, varying 1

A reviewer correctly wonders whether the participants could have been comparing the number of different kinds of furniture, and indeed this reading cannot be ruled out. However, two points must be made. First, in at least some cases, the same number of kinds were compared and greater functionality with fewer pieces were still considered by many to be ‘more’. For example, in one case, participants were asked to who was wearing more jewelry in a gala situation in which Woman A is wearing two gold bracelets, a diamond tiara, and a ruby and emerald necklace (three kinds, four items), while Woman B is wearing three gold rings, a pearl necklace and a silver bracelet (three kinds, five items). The majority of participants chose woman A, commenting that the jewelry was ‘more showy’ or ‘more valuable’. Second, when N is a count noun, ‘more N’ is not usually interpreted as ‘more kinds of N’. Thus, if this reading is available with object mass nouns, this still supports the claim that semantically, object mass nouns are different from syntactically count nouns. Gafni and Rothstein (2014) suggest that comparison of number of kinds is possible with object mass nouns, along with comparison along other continuous dimensions. Examples like (3) and (4) below support this position.

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the dimension of comparison was possible, and context played a crucial role in choice of dimension of comparison. These results were replicated for Hebrew in Gafni and Rothstein (2014). Data based on interviews with informants discussed in Rothstein (2013, 2017) show that a variety of different dimensions of comparison are available for object mass nouns in English, besides success in fulfilling a certain function. Take the examples in (3): (3)

a. John has more furniture than Bill, so he will need the bigger truck. b. John has more pieces of furniture than Bill, so he will need the bigger truck.

(3a) is widely accepted to be true if John has say, a grand piano, a sofa and a large cupboard, while Bill had two folding chairs, a small table and a small floor cushion, showing that ‘more furniture’ can be interpreted as more in terms of volume. Interestingly, (3a) is also accepted as true if the sentence is presented without any context and without any details about what furniture Bill and John have, suggesting that in this context volume is the default dimension of comparison for furniture. In contrast, informants typically react to (3b) by saying ‘it depends on how big the pieces of furniture are’. This shows that more pieces of furniture, which uses the count classifier phrase, requires evaluation in terms of cardinality, while the choice of dimension for evaluating more furniture can be contextually determined. In (3a), the context is deciding who gets the bigger truck, and this makes the dimension of volume salient. The pair in (4) show that the addition of a relative clause can trigger a change in the choice of dimension of comparison, as the typical responses to the questions in each example indicate. (4a) is a neutral context and the evaluation is cardinal. (4b) is a context in which the relative clause makes weight the salient dimension and the dimension of comparison shifts: (4)

Susan got five letters and a postcard in the mail this morning and Fred got three big parcels with books: a. Who got more mail this morning? Susan b. Who has more mail to carry home? Fred

Landman (2011) shows that object mass nouns vary in terms of dimension of comparison using the determiner most, which involves comparison between two subsets of N, the intersection between N and P, and the intersection between N and the complement of P. When N and P are count nouns, comparison is only in terms of cardinality, and both (5a) and (5b) cannot be true simultaneously. When N is an object mass noun, the dimension of comparison can be explicitly varied as in (5c) and (5d). Setting the dimension in terms of volume is impossible for the count noun (5e):

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a. b. c. d. e.

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Most farm animals are chickens. (examples based on Landman 2011) Most farm animals are cows. In terms of number, most livestock is poultry. In terms of volume, most livestock is cattle. #In terms of volume, most farm animals are cattle.

So while it seems that in context-neutral situations, cardinality is the default choice of dimension for comparing object mass nouns, it is clearly the case that contextual parameters can be manipulated to induce comparison along continuous dimensions. This contrast between count nouns and object mass nouns occurs crosslinguistically. Pires de Oliveira and Rothstein (2011) show that in Brazilian Portuguese so-called bare singular count nouns are in fact object mass nouns (see also Rothstein 2017). Predictably, count nouns with count quantifiers require comparison evaluations in terms of cardinality, while ‘bare singular’ object mass nouns allow comparison along contextually determined dimensions. (6a) requires a cardinal answer, while (6b) allows for an answer either in terms of cardinality or an answer which uses a measure phrase such as ten dollars’ worth, two shelvesful and so on. Schvarcz and Rothstein (2017) show that the same contrast occurs in Hungarian, where hány, the count question word, requires a cardinal answer, and mennyi, the mass question word, allows for the same range of variability as the Brazilian Portuguese quanto, as shown in example (7). (6)

a. Quantos livro-s ele compr-ou? how.many book-PL he buy-PST.PRF.3SG ‘How many books did he buy?’ (Pires de Oliveira and Rothstein 2011, p. 52b) b. Quanto livro você compr-ou? How.much book you buy-PST.PRF.3SG ‘What quantity of books did you buy?’

(7)

a. Hány könyv van a táská-d-ban? How.many book is there the bag-POSS.2SG-in ‘How many books are there in your bag?’ (Schvarcz and Rothstein, 2017) (i) Csak Három. (ii) #Három kiló. Only three. Three kilo ‘Only three.’ ‘Three kilos.’ b. Mennyi könyv fér a táská-d-ban? How.much book fit.PRES.3SG the bag-POSS.2SG-in? ‘What quantity of book fits into your bag?’ (i) Három kiló-t (ii) Hárm-at Three kilo-OM three-OM ‘Three kilos.’ ‘Three’

The contrast can further be replicated in Mandarin, where the classifier phrase tài duō běn shū ‘too much classifier book’ compares in terms of number of

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individual volumes, while the bare NP tài duō shū allows for comparison in terms of volume. (8)

nǐ dài tài duō (běn) shū le, xínglǐ huì chāozhòng de you take too much/many Clvolume book PRF baggage will overweight PRT ‘You have taken too many books, your baggage will be overweight.’ (Rothstein, 2017)

These data indicate a systematic contrast between object mass nouns and count nouns, suggesting a difference in denotation. As a consequence, object mass nouns cannot have the semantics of count nouns. This suggests that their denotations are not countable. So, how can we compare two sums in terms of cardinality if they are not countable? 8.2

Formulating the Problem

Let us rephrase the question asked in the last section in more formal terms, in order to understand more precisely what the problem is. We start with the example in (9), which involves comparing sums in the denotation of a count noun, and thus uses the cardinality function. (9)

Mary has more books than John.

(9) is true if the cardinality of the sum of books that Mary has is greater than the cardinality of the sum of books that John has. Frege (1884) showed that cardinality can be analyzed as a property of sets, with cardinal numerals denoting equivalence classes of sets with the same number of members. In the mereological framework of Link (1984), cardinality is a property of entities of type e (singular and plural individuals). The cardinal property of an entity at type e is dependent on the cardinality of the set of its atomic parts, as in (10): (10)

jxj ¼ n $ jfy : yvATOMIC xgj ¼ n ‘The cardinality of a sum x is n if the cardinality of the set of the atomic parts of x is n’

The cardinality of a set X is n if X is a member of the equivalence class denoted by n, thus a sum a has cardinality n if the set of its atomic parts is in equivalence class denoted by n. More(x,y) in (9) compares cardinalities, with (11) as a plausible interpretation of (9). (We assume a second more which compares values on a dimensional scale for examples such as John drank more wine than Mary, about which we will give details later.) (11)

jtðBOOKS∩OWNED BY MARYÞj > jtðBOOKS∩OWNED BY JOHNÞj

Now assume that (12a) also has a cardinal interpretation. If it is interpreted analogously to (9), we would expect the interpretation in (12b):

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a. Mary has more furniture than John. b tðFURNITURE∩OWNED BY MARYÞj > jtðFURNITURE∩OWNED BY JOHNÞj

The problem is that the cardinality function is not defined for the mass domain, since it makes reference to sets of atomic parts, which are assumed not to be grammatically accessible in the mass domain, with different theories giving different explanations for this lack of accessibility (see below). This is the standard explanation for why numerals, which also make use of the cardinality operation in (10), cannot apply to mass nouns. A numeral like three, denoting the λx:jxj ¼ 3, is not defined for mass nouns, since λx:jfy : yvATOMIC xg ¼ nj is not defined. Using the cardinality operation in evaluating sums in the denotation of FURNITURE∩P as in (12b) should also be not defined, ruling out (12b) as the interpretation of (12a). We thus need to ask how cardinality evaluations are possible for (12a). There are several possible solutions to this problem. The first is to argue, like Barner and Snedeker (2005) and Bale and Barner (2009) that the cardinality function does apply in (12), since object mass noun denotations are countable and have the same properties as plural count noun denotations. Modification by numerals is ruled out because numerals are sensitive to count syntax and not to the count semantic properties of the N-denotation. However, as we have seen, this explanation cannot be correct, since it wrongly predicts that object mass nouns ought always to behave as count nouns do in comparison contexts. The second possibility is to assume that the cardinality function in (12) does not count in terms of atomic parts, but in terms of contextually determined discrete parts, and to define a set of such discrete parts for furniture. This is the approach taken in Landman (2016, Chapter 6 in this volume). In simplified terms, Landman assumes a function M that maps the set denoted by an object mass noun (in fact, in appropriate contexts, any mass noun) onto MIN(N), a discrete set of minimal parts of N which can be accessed by the comparative operation. Non-cardinal comparisons may apply directly to pairs of sums in the denotation of furniture without reference to MIN(N). (One might ask why, in this case, cardinality is apparently the default dimension of comparison, since it is presumably the more complex operation; however, we will ignore this question.) Landman assumes that numerals such as three which also make use of the cardinality function do not have access to MIN(N), thus explaining the infelicity of *three furniture(s). So cardinal more can apply to both count nouns and object mass nouns. There is a third possibility for constructing cardinal comparisons with mass nouns which I will explore in the rest of this chapter. This is the possibility that cardinal evaluations of object mass nouns involve a different operation from cardinal evaluations of count nouns. While comparisons of count nouns involve counting and the direct comparison of cardinal values, comparisons of object mass nouns involve measuring and comparing values on what I will call a

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cardinal scale. Cardinal more operates in the count domain, but non-cardinal more operates in the mass domain. Since cardinal measures, measures on a cardinal scale, are a kind of measure value, non-cardinal more can compare these values too. Prima facie support for this approach comes from the pair less/fewer. While comparison in the positive direction uses only more for both mass and count nouns, comparison in the negative direction uses fewer for count nouns and less for mass nouns as in (13). While fewer clearly allows only cardinal comparisons, less allows either cardinal or non-cardinal comparisons in (13b). (13)

a. John has fewer/#less chairs than Mary. b. John has less/#fewer furniture than Mary. c. John has less water/#fewer water than Mary.

If both a cardinal and a non-cardinal operation are available in (13b), then three lexical items are necessary for the interpretation of (13): fewer, which denotes a cardinal comparative operation applying to count noun denotations as in (13a), less1, which denotes a non-cardinal comparative operation applying to mass noun denotations in (13c) and the non-cardinal interpretation of (13b), and less2, which denotes a second cardinal comparative operation applying to mass noun denotations and gives the cardinal evaluation of (13b). Obviously a simpler analysis of (13) would suggest two kinds of operations, a direct comparison of cardinalities denoted by fewer and a single interpretation of less which compares measure values. Comparisons of cardinality with object mass nouns, as in (13b), would have to be the result of a measure operation. In preparation for developing the idea that apparent cardinal comparisons may be the result of comparing measure values, the next section will give some background on counting and measuring. 8.3

Semantics for Counting and Measuring

8.3.1

Counting

Counting is counting of discrete, non-overlapping entities, and giving a cardinal value to a sum involves counting its atomic parts.2 In Rothstein (2010, 2011), I argued that counting is a context-dependent operation: We count, in a particular context k, the entities which in that context are considered atomic 2

As Manfred Krifka has pointed out (p.c.), there are contexts in which it is appropriate to count overlapping objects, for example (i):

(i)

How many squares are there in one face of a Rubik’s cube?

Arguably, these are modal contexts, with (i) equivalent to ‘How many squares can you see/find in one face of a Rubik’s cube?’ Note also Krifka’s discussion of terms like outfits, as in ‘You have three shirts and four pairs of pants. How many different outfits can you make? [. . .] You get twelve outfits’ (cited in Krifka 2009a). However, in the normal case, we count non-overlapping objects.

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entities; more precisely, we count, in a particular context k, instances of N which in that context are considered atomic instances of N. The argument that counting is context-dependent is based on the observation in Rothstein (1999, 2010) that with nouns like fence, wall, place and hedge, as well as thing and object, what counts as an atomic entity, i.e. a single wall or fence, is context-dependent. One stretch of fencing can count as one fence or several fences depending on your choice of what counts as ‘one’, itself depending on what counts as an edge or a border. Crucially, this kind of context-dependency must be distinguished from vagueness with respect to boundaries (Kamp 1975, 2013; Chierchia 2010). For example, we may be vague about who falls under the definition of child (being considered a child with respect to income tax deductions does not mean that one is considered a child with respect to reduced train fares), but at the same time be certain about who, given a precise definition, the atomic children will be. Rothstein (2010) argues that our models contain a rich set of contexts K, where for any k 2 K, k is a set of entities which count as single individuable entities in context k. Nouns are derived from abstract root nouns at type < e, t > via two operations MASS and COUNT. MASSðNroot Þ is the identity function on the root noun denotation, and NMASS is of type < e, t >. In contrast, COUNTðNroot Þ is an operation which maps Nroot onto Nk , of type < e  k, t >, the set Nroot intersected with k, which denotes the set of nonoverlapping, indexed entities which count as semantic atoms in context k. Nk denotes a set of ordered pairs < x, k >, where x is an entity in Nroot ∩k, and k is the context. Entities in a count noun denotation are objects at type e  k and count nouns denote properties of indexed individuals. Count nouns, by definition are semantically atomic. An example is given in (14): (14)

〚stonemass 〛¼ MASSðSTONEroot Þ ¼ fx : x 2 STONEroot g 〚stonecount 〛¼ COUNTk ðSTONEroot Þ ¼ f< x, k >: x 2 STONEroot ∩kg

stonemass denotes a set of quantities of stone, while stonecount denotes a set of k-indexed entities each of which counts as one stone in context k. Crucially, as discussed in Rothstein (2010), k may contain overlapping entities; for example, a wall and the bricks that it is made of may all count as individuable entities. The non-overlap condition is a condition on the output of the count operation so that Nk must be a non-overlapping set of entities. This is why counting can only be relative to a particular nominal description (expressed in the adjectival use of numerals).3

3

Rothstein (2010) discusses the fact that in a situation in which there are three cups and two saucers, How many cups and saucers are there? does not usually allow the answer ‘five’. Likewise, How many boys and girls are coming to the party? will usually exact the answer ‘n boys and m girls’. However, in this latter case, it is possible to reinterpret boys and girls as ‘children’ and answer with one numeral.

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I will not make any commitments about the properties of mass noun (and thus root noun) denotations (see Landman 2016, Chapter 6 in this volume for a theory of mass nouns denotations which is compatible with this account). Crucially, however, count nouns make a set of contextually determined atomic entities grammatically salient, while mass nouns do not. Sums in the denotation of count nouns are sums of k-atoms, entities which count as atomic in context k. When the cardinal function in (10) applies to an entity of type e  k, it counts these atoms, or what we call the k-atomic parts. This cardinal function is as defined in (15a). Since the atomic parts of an entity of type e  k are the k-atoms, it follows that counting the atoms is counting the k-atoms, as in (15). For simplicity, we use xk as short for . (15)

a. for xk ,yk 2 DK : yk vkATOM xk iff yk v xk ^8zk ½zk v yk ! zk ¼ yk  b. if xk 2 D  K, jxk j ¼ n $ jfyk : yk vkATOMIC xk gj ¼ n

(For the precise definition of vkATOMIC , see Rothstein 2010.) Numericals like three count the atomic parts which an entity has. Counting presupposes a k-context, since the result of counting may differ depending on the k-context chosen. So a numerical denotes a function from count predicates onto count predicates, counting the number of k-atomic parts. (16)

⟦three
where: M is a dimension (e.g. volume, weight). U is the unit of measurement in the relevant dimension, in terms of which the scale is calibrated (e.g. liter, kilo) N is the real numbers, or the positive real numbers, or a subset of the real numbers, depending on the nature of the measure and the fine-grainedness of the measurements. MEASUREM, U is a function from objects to values in N.

Since the range of values is the set of the real numbers, measuring is inherently continuous (for a proof of this, see Landman Chapter 6 in this volume). In context, we can make it non-continuous by choosing a subset of the real numbers as our contextually relevant range of values. Our choice of values determines the granularity of the scale (Solt 2015). The choice of unit is expressed in a measure head such as liter. Liter has the denotation in (18a). It combines with a numeral to give a measure predicate, which expresses the property of having a particular measure value (18b), and modifies a noun in the standard way (18c): (18)

a. 〚liter〛¼ λn:λx:MEASUREVOLUME, LITER ðxÞ ¼ n b. 〚3 liter〛¼ λx:MEASUREVOLUME, LITER ðxÞ ¼ 3 c. 〚3 liters wine〛¼ λx:WINEðxÞ ^ MEASUREVOLUME, LITER ðxÞ ¼ 3

Measuring focuses on the properties of the quantity as a whole. If a quantity a measures three liters, we know nothing about its internal structure. We only know that, if our measure function is additive, any way we break a into two non-overlapping parts b and c, the following will hold: (19)

If a ¼ b t c and MEASUREVOLUME, LITER, ðaÞ ¼ 3, then MEASUREVOLUME, LITER, ðbÞ ¼ 3  MEASUREVOLUME, LITER, ðcÞ

8.4

Quantity Evaluations

Quantity evaluations are required to interpret expressions such as much, many and more, as well as little, few, fewer and less (though, for reasons of space, we 4

As Manfred Krifka points out (p.c.), extensive measure functions are additive, and in that sense are also bottom-up functions sensitive to structure since Fða þ bÞ ¼ FðaÞ þ FðbÞ. However, unlike counting functions, additive measure functions can yield a value without any commitment to the internal part-of structure of the sum or quantity that they are measuring (see (19)). Note that we are discussing only extensive measures here, and not mechanisms for comparing values of non-extensive measure functions such as temperature, nor degrees of honesty, cleverness or other abstract properties.

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will focus only on more). They require evaluations either in terms of cardinality or along a continuous dimension. In some cases, the word itself indicates what is the appropriate parameter, for example, (how) much and less indicate that comparison is along a continuous dimension, while (how) many and few indicate that the dimension is cardinal: (20)

a. How much/#many wine did you drink? b. How many/#much bottles of wine did you drink?

(21)

a. John drank fewer/#less bottles of wine than Mary. b. John drank less/#fewer wine than Mary.

In other cases, the quantity term is neutral, for example Hebrew kama in (22), which translates here as either ‘how much’ or ‘how many’ (22a)–(22b), and English more (21c). (22)

a. kama bakbukey yayin šatit? KAMA bottles of wine you-drank ‘How many bottles of wine did you drink?’ b. kama yayin šatit? KAMA wine you-drank? ‘How much wine did you drink?’ c. Who listened to more music/more pieces of music?

In these cases, the parameter for comparison is constrained by whether N is mass or count. When N is count, the atomic structure encoded in the denotation makes the atoms salient and comparison must be in terms of cardinality. Thus, as we already saw in Section 8.1, Who has more books/pieces of furniture? requires a comparison of the value of the cardinality function applied to each plural object (or set). If John has five folding chairs, and Mary has a bed, a piano, a table and a chair, then John has more pieces of furniture than Mary, and if she has five books and he has four, then she has more books than he does. When N is count, the truth conditions for more N can be expressed in paraphrases like (23). (23)

Mary has more books than John is true iff the cardinality of the sum of books that Mary has is greater than the cardinality of the sum of books that John has.

In general, the interpretation of more when it applies to an N at type < e  k, t > makes use of MOREj j as in (24). (24)

x is MOREj j than y if j xk j > j yk j

The semantics for three in (16) expresses the fact that cardinalities are only countable in a context k. Similarly, cardinalities are only comparable in a single k-context, since an agreement on what counts as ‘more’ on a cardinal scale presupposes agreement as to what counts as ‘one’. So cardinal MORE,

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like numerals and the denotation of fewer, is defined only for predicates whose denotation is relativized to a context k. (Note that count nouns privilege cardinal MORE, and do not normally allow evaluation along continuous measure dimensions, a fact that we will come back to in Section 8.9.) Note that de facto you don’t always need to count in order to compare cardinalities. If I ask you ‘Are there more books in my kitchen or in my living room?’, you can give a correct answer by estimating, or by calculating, or by indirect counting (e.g. there are two sets of bookshelves in the kitchen and seven of comparable size in the living room.) But the answer always involves a comparison of cardinalities, and the correct answer is always in terms of which counting value is higher in the sequence of natural numbers. In contrast, when N is mass, Who has more N? requires comparing two values on a (single) dimensional scale to see which is higher, as in (25). More is evaluated with respect to a dimensional scale, with (26) analogous to (24): (25)

Mary drank more wine than John is true iff MEASUREVOLUME, U ðwine that Mary drankÞ > MEASUREVOLUME, U ðwine that John drankÞ

(26)

x is MOREM, U than y if MEASUREM, U ðxÞ > MEASUREM, U ðyÞ

Giving a full compositional semantics for more (. . . than) is beyond the scope of this paper. Example (25), for example, expresses a comparison between two quantities, the wine that Mary drank and the wine that John drank, and extracting these objects compositionally from (25) is not trivial. For our purposes, the crucial point is that the same compositional mechanism that is used to interpret (25) will also be used to interpret more in the count context in (23). Whether more applies to a predicate at type < e, t > or at type < e  k, t >, the comparison will make use of a single semantic operation which is given in (27): (27)

For some function f : MOREf ðx; yÞ iff f ðxÞ > f ðyÞ

The difference, then, between more applied to count nouns and more applied to mass nouns is not in the meaning of more which compares the magnitude of the values which are the output of f, but in function f which gives you the values, and thus in the type of the values. When more applies to count nouns, the f function is the counting operation ⎹⎸ defined for a context k. Since the output of j j is a cardinal number, MOREjj yields a comparison of cardinal values, that is entities of type n. When more applies to mass nouns, the f function is a measure operation, MEASUREDIM, U which gives measure values to compare. Note that we cannot reduce comparison of measure values to a comparison of the cardinality of n. This is because in measuring situations, cardinal values are compared relative to a specified unit and a dimension. Comparisons of

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measure values not only presuppose a common dimension, but also a common unit specification to guarantee that the units in each case are commensurate. It is false that 12 inches of yarn < 20 cm of yarn, despite the fact that 12 < 20. We conclude this section with an important point. Comparisons of both counting and measure values with more are expressions of relative value. ‘Who has more N?’ can always be answered on the basis of exact information, but it can also be answered on the basis of information about the relative quantity values for the two sums being compared without precise quantitative information about either of them. If I know that John has five books and Mary has ten books, then I know that Mary has more books. But if Mary has a large pile of books and John has a small pile of books, I also know that the first pile is ‘more books’ than the second pile without knowing how many books are in each pile. Similarly, a bathtub of water is ‘more water’ than the water in my drinking bottle. We can make these judgments because we can compare relative values (i.e. relative positions on the scale) without being able to name the values themselves. While ‘x is more than y’ is true iff f ðxÞ > f ðyÞ, we can make this assertion without necessarily knowing what either f(x) or f(y) is. We can do this by using explicit indirect evidence, for example by comparing the size of containers, or by using our apparently innate capacities for comparing numerosities using the so-called approximate number system (see review in Hyde 2011 and references cited there.) This is different from answering nonrelative quantity judgments such as ‘How much/many N does John have?’, which in principle requires an answer related to a particular value (unless the answer is given in terms of a relative value, e.g. ‘I don’t know, but less than Mary’). Note that the contrast between comparative, or relative, evaluations and absolute evaluations is not the same as the contrast between approximate and non-approximate values – both comparative and absolute evaluations can involve either approximate or non-approximate values, and approximation can but need not involve explicit numerical values, as we will see in Sections 8.6–8.8. The crucial point is that it is possible to give comparative, relative evaluations without any expression of numerical values (for example, ‘There is clearly more rain in London each year than in the Sahara desert’), whereas giving absolute evaluations involves an expression of a numerical value, even if it approximative. 8.5

Cardinal Comparisons of Object Mass Nouns

We can now return to the question we asked in Section 8.1: How is it that cardinal comparisons of object mass noun denotations are possible, but not obligatory? And what exactly is the meaning of statements and questions like those in (28)? As we already saw, in these cases comparison can be along any contextually relevant dimension, including cardinality.

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a. Mary has more furniture than John has. b. Who has more furniture?

The puzzle is how to represent the truth conditions of the examples in (28) so that can we allow for cardinal comparisons with object mass nouns and at the same time allow cardinality to be only one among a set of possible parameters of comparison. We also need to explain the differences between cardinal comparisons with count nouns and with object mass nouns. Interpreting (28a) as involving comparison of measure values, for example on the scale of volume, is straightforward. Since furniture is a mass noun, measure operations apply to sums in the denotation of furniture unproblematically. Example (29) asserts that some quantity of furniture that Mary owns has a higher value on the volume scale calibrated in cubic meters than any quantity of furniture that John owns. (29)

9x ½FURNITUREðxÞ ^ MARY OWNSðxÞ ^ 8y½FURNITUREðyÞ ^ JOHN OWNSðyÞ ! MOREVOL, METER 3 ðx; yÞ

The cardinal readings, however, are more difficult. We cannot by hypothesis directly use cardinal MORE in (24), because MOREj j presupposes a set of atoms and mass nouns by hypothesis do not have atomic parts. If they did, then we should be able to count mass nouns, since MORE⎸⎸ and cardinal numerals make use of the same cardinality operation. Since furniture does not have a set of atomic parts, we cannot assign a cardinality to the sum of furniture that Mary has and the sum of furniture that John has and directly compare the values. Furthermore, since a non-cardinal reading is available, as in (29), we need a cardinal interpretation which shows why both cardinal and non-cardinal interpretations are readily available for object mass nous but not for count mass nouns. Since furniture is a mass noun and the cardinality operation cannot directly apply to it, the operation which results in a cardinal comparison should somehow be derived from the mechanism for measure comparisons in (26). So we need truth conditions for (28) that allow a cardinal comparison derived from the template in (26) and which, importantly, allow equally for cardinal and non-cardinal evaluations. This means that we must be able to represent the meaning of (28a) on its cardinal reading without directly using the cardinality operation in (24). Note that this issue is not how, in context, you decide whether (28a) is true or what the correct answer is in (28b). You may make this decision by counting pieces of furniture if you want. But even if you do, it must be possible to represent the meaning of the sentences in (28) without directly applying the cardinality operator to entities in the denotation of an object mass noun and thus without using MOREj j . I suggest that, in addition to using MORE⎸⎸, cardinalities can be compared by measuring, using values on a cardinality scale.

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A natural way of treating numbers is as objects of type n (Rothstein 2013, 2017). The natural numbers then form a countably infinite set NN, the set of natural numbers, totally ordered by the successor function. Counting uses this order, comparing values in this set and deciding which of two values is ‘more’ depending on their relative positions in the order. Cardinality comparisons using MOREj j compare the size of a and b by assigning a and b values in NN depending on the number of their atomic parts and comparing which of these values is higher with respect to the successor function on NN . However, we can also treat cardinal values as if they are values on a particular kind of measure scale SM , U , the cardinality scale. Sum a has a cardinality value which is more than the cardinality value of sum b if the value assigned to a is higher on the cardinality scale than the value assigned to b. Since the value of the measure function gives a value to the overall sum and does not make reference to the part-of structure of an entity, the semantic operations do not require access to the atomic parts of the sums. How we assign a cardinal measure value to a sum in the denotation of furniture is (semantically) irrelevant to the measure function, just as it is irrelevant how we measure out three kilos of flour (1  3 kilo unit, 6  0.5 kilo units, 1  1 kilo unit + 1  2 kilo unit etc.). We construct a cardinality scale analogous to SM,U defined in (17) from a set of numbers as follows: (30)

A cardinality scale is an order SCARD, j j k ¼< N, N, j jk > where CARD stands for cardinality, k is the context that determines the set of atoms, N is the set of natural numbers, and j jk is the function that maps x onto the values in N depending on the cardinality of the set of parts of x which are also in k, i.e. j x jk ¼ jfy : y v xg∩k j

N is the set of natural numbers, since the values are not continuous. The scale is not assigned a dimension; put differently, the dimension is arbitrary, and any sequence can be used to model the natural numbers (see Wiese 2003 for discussion). We can use the set k, judiciously chosen, to model the units in terms of which the scale is calibrated, since we use it anyway in j jk . The cardinality scale is arbitrary in dimension because its values do not express dimensional properties of individuals. Weight, volume, cost, temperature and so on all express dimensions along which individuals can be ordered, and having a certain value on certain dimensional scale is a property of individuals. Cardinal scales, however, do not represent properties of individuals but of pluralities. If a cardinal function applies to an individual x, it gives one of two values: either 1 (if x is a countable individual) or undefined (if it is not). The cardinal function provides informative orderings (i.e. in principle using the full range of values) when it applies to pluralities and is apparently

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the only ordering which applies exclusively to pluralities. It is because it measures pluralities along the cardinal dimension that, unlike other measure functions, its values are discontinuous. j jk is a function derived from j j which gives a cardinal value to x not by counting the atomic parts of x, but by counting a related set, the intersection of the set of parts of x with k, fy : y v xg∩k. Since k is by definition a set of atoms, j j can apply to the set X∩k (although since X∩k is of type < e, t >, numerical predicates cannot modify it.) This captures the fact that when we do make cardinal comparisons of pluralities in the denotation of object mass nouns, what is considered a singularity may well be contextually determined, as pointed out in Landman (2016). Crucially, at no point do we have to access a set of minimal parts of the sum x; instead, we take the set of all parts of x, intersect it with k and evaluate the cardinality of the resulting set. MORECARD, j jk compares values on a cardinality scale, MEASURECARD, j jk ðxÞ ¼ n. A value on a cardinality scale gives you a number of units on a dimensional scale, but since CARD is an arbitrary dimension, and the units are k-units, the value n gives information only about how many k-objects x consists of, what its cardinality is. Crucially, the cardinality function used by MORECARD, j jk is not the same as the jj operation used in (15), (16) and (24). The cardinality function jj assigns a numeral property to a sum by accessing and counting the ðkÞatomic parts of the sum. The function used in cardinal measure operations, j jk counts indirectly, assigning a value to x by taking the set of parts of x, intersecting this set with k and giving the cardinality of the resulting set, fy : y v xg∩k. Instead of directly counting the k-indexed atomic parts of x, it constructs an alternative set of entities and gives the cardinality of that set. The interpretation of x is more furniture than y, then, will always use measure more, denoting MOREM, U , since furniture is a mass noun. One interpretation, which compares along the dimension of volume, was given in (29), with MOREVOLUME, METER chosen as the relevant instantiation of MOREM, U . However, an alternative instantiation is MORECARD, j jk , which gives the cardinal interpretation in (31): (31)

IF FURNITUREðxÞ ^ FURNITUREðyÞ, x is MORECARD, j jk than y iff MEASURECARD, j jk ðxÞ > MEASURECARD, j jk ðyÞ ¼ jfx0 : x0 v xg∩kj > j fy0 : y0 v yg∩k j

In sum, then, we can distinguish two ways of comparing in terms of cardinality, one of which involves directly involves the cardinality function, and the other of which is a measure operation which makes indirect use of the cardinality function. When we compare in terms of the cardinality function using the assertion jxk j > jyk j, we are essentially counting, since we assign a numerical

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value n to a plurality depending on the number of its atomic parts and then compare these numerical values. However, we can also assign a cardinal value to a sum, using the measure operation MEASURECARD, j jk ðxÞ ¼ n which assigns values on a scale by indirect use of the cardinality function. Cardinality scales thus allow us to use the set of natural numbers as if they were values on a scale. These allow us to assign cardinal values to sums indirectly, without having access to a set of individual atomically indexed parts, and without being able to count them directly. The operation j jk , like the cardinality operation incorporated into the interpretations of numeral adjectives, makes use of the Fregean cardinality operation which assigns a cardinality to sets, and this is as it should be since, ultimately, comparisons involving cardinalities are dependent on comparing the cardinalities of the relevant sets. However, what the relevant sets are and how they are accessed is very different. Numerals are N0 modifiers which denote properties of sums in N0 and which presuppose that these sum are generated by a set of atomic parts which determines the Boolean structure of the sum and whose cardinality can be directly accessed, i.e. for some sum y, the set fx : xvATOM yg. This itself presupposes that the denotation of N’ itself is an atomic Boolean structure. It is in this sense that direct counting is a bottom-up operation, since the part-of structure of the N-denotation must be accessed in order to give the (cardinal) quantity evaluation. In contrast, evaluations using cardinal scales and the j jk operation, which also involve giving quantity evaluations of sums, do not access a grammatically salient set of atomic parts determining the partstructure of the sum. They access a countable set N∩k without any commitment as to how the denotation of N is related to that set, and in fact without any commitment to the denotation of N being Boolean. Furthermore, since the non-overlap condition sets of semantic atoms is a condition on the COUNTk operation (see discussion in Section 8.3 and references there), it is not even guaranteed that N∩k, as used in the j jk operation, is disjoint. This allows for a situation in which, for example, a sofa which splits into two chairs might count as ‘more furniture’ than a sofa which doesn’t split in the same way, without there being any commitment as to how many pieces of furniture are involved. This brings us back to the connection between the use of cardinal scales and the relative nature of comparative operations such as ‘Who has more N?’ Cardinal scales involve measure operations which are top-down operations that assign values to sums without necessarily accessing a part-of structure. Relative evaluations such as comparatives do not necessarily require access to part-of structure. This is because, as we showed at the end of the previous section, Who has more N? can be answered by plotting relative values on a scale for two sums without giving a numerical value to either of the sums. This is done by estimation, or informed approximation, and is what makes it possible to say John has more furniture than Bill on a cardinal interpretation

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without knowing how much furniture either of them has. So while the truth of John has more furniture than Bill may rest on the relative cardinalities of furnitureJOHN ∩k and furnitureBILL ∩k, the possibility of making that assertion, and even of having it judged true, does not rest on the grammatical operation of counting parts of either sum. This is why the following dialogue is infelicitous, where comparison is taken to be on the cardinal dimension: (32)

A: John has more furniture than Bill. B: How much more? A: #Ten / #many. (OK: ten pieces, many pieces)

Although the cardinal comparison is possible, as is shown by the felicitous response, a bare numeral cannot be used by B to respond, because the parts of furniture are not countable. We will discuss approximation as a measure operation in the next section, but first we conclude this section by clarifying a few remaining points. Rothstein (2009) uses a different formulation of the measure operation from that used here. The measure operation is treated as an operation from individuals into pairs consisting of a number and unit, objects in N  U. In contrast to (18a), liter is analyzed as a function from individuals into pairs in (33a), while the cardinal measure function has the form in (33b): (33)

a. 〚liter〛¼ λnλx:MEASUREVOLUME ðxÞ ¼ < n, LITER > λnλx:MEASURECARD ðxÞ ¼ < n, k > b. j jk ¼

While there are some disadvantages of using this representation, it does have the advantage of clarifying the contrast between values of the cardinality operation, which are numbers of type n, and the values of the cardinal measure operation, which are ordered pairs n  k. Note that while pairs of counting values and pairs of measuring values can both be compared, the operations used are different. This means that counting values and measure values cannot be conveniently compared with each other: (34)

a. John has more carpets than Bill has curtains. b. John has more carpeting than Bill has curtaining. c. #John has more books than Bill has carpeting/furniture.

Example (34a) compares two cardinal values and is true if jcarpetsk j > jcurtainsk j. Example (34b) compares two values of on a dimensional scale and is true if MEASUREM, U ðcarpetsÞ > MEASUREM, U ðcurtainsÞ, leaving open what the dimensional scale is. A cardinal scale is one among several possibilities. Example (34c) is less felicitous, since it compares a cardinal value with a measure value. While these can be brought into accord, it is a more complex operation than those used in either (34a) or (34b).

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We conclude this section with one final question: Assuming that we use a measure operation and cardinal scales to give cardinality values to mass noun denotations, can we do without counting entirely? Can we use cardinal scales to give values to sums in the count domain, and thus reduce all counting to measuring? Krifka (1989) already suggested that all counting involves a measure operation, with count nouns incorporating the measure operation into their interpretation. He proposed that all count nouns are functions at type < n, < e, t >>, i.e. functions from numbers into predicates. Transferring Krifka’s formulae into the format in (33a), count noun meanings have the structure in (35). (35)

〚cowcount 〛¼ λnλx:COWðxÞ ^ MEASURECARD ðxÞ ¼< n, NU >

Either the count noun is applied to a number as in (36a), or n is existentially quantified over as in (36b): (36)

a. 〚two cows〛¼ λx:COWðxÞ ^ MEASURECARD ðxÞ ¼< 2, NU > = the set of sums of cows which measure two Natural Units on the cardinality scale. b. 〚cows〛¼ λx:9n COWðxÞ ^ MEASURECARD ðxÞ ¼< n, NU > = the set of sums of cows which measure some Natural Units on the cardinality scale.

As a theory of mass nouns, this account has problems (see Rothstein 2010, 2017 for discussion), but that is not what concerns us here. The point is that all counting can in principle be reduced to giving values on a cardinality scale, if we want to do so. However, I want to argue that in fact we don’t want to do so. Counting and measuring are two different operations with different properties, and we want to keep them separate. Counting gives us values in N, the set of natural numbers which is a set of entities at type n ordered by the successor relation. While counting is relative to a context k, the set of natural numbers is a set of objects at type n, which are not context-dependent. Counting assigns a cardinality to a sum in context k directly by assigning the set of its atomic parts to an equivalence class, and the cardinality of x is the cardinality of the sum of its atomic ðkÞ parts fy : yvkATOM xg. The set of counting values yielded by the cardinality function is inherently discontinuous. Measuring on a cardinality scale is a more complex operation. Scales are triples < N, M, U, MEASUREM, U > and measuring gives a value relative to a specific scale. The range of a measure values is continuous (for any two values n and m on a scale, there will always be a value n0 such that n < n0 < m). Cardinality scales are one kind of scale. They are triples SCARD, j j k ¼ < N, N , j jk >, which use the set of natural numbers (and the natural successor relation) to build the scale, with a ‘dummy’ dimension, CARD and a contextually dependent unit of calibration derived from k. Measuring assigns

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a cardinal value to a sum x on the basis of the cardinality of a related set fy : y v xg∩k, and the atomic parts of x are not directly accessed. Cardinal scales share an essential property with all measure operations: They assign a value to a sum without it being specified how that value is a function of its part-of structure. If counting is a ‘bottom-up’ perspective on giving quantity evaluations which exploits a single Boolean part-of structure, measuring is a top-down perspective, which allows a quantity evaluation of a sum without a commitment to the part-of structure of the sum. Distinguishing counting from measuring with cardinality scales allows us to keep these two perspectives separate. In the next sections, we look at some grammatical expressions of this contrast. 8.6

Approximation

In the previous section, I argued that direct counting and measuring cardinalities are two different operations, and suggested that there are good conceptual reasons to keep these two operations separate. Measuring cardinalities is a topdown operation which does not require access to a set of semantically atomic parts, and is thus naturally appropriate for comparative, i.e. relative, quantity evaluations. It is equally appropriate for interpretations of vague expressions like a lot of N, which can also have a vague cardinal interpretation, and are implicitly comparative. For example, John has a lot of furniture to carry downstairs can be interpreted on a cardinal scale and is true if the value assigned to the relevant sum of John’s furniture is located in an interval on the cardinal scale which is high relative to some standard. Giving a serious account of the semantics of approximatives is beyond the scope of this paper, but in this section, I do want to show that some approximate or vague cardinal quantity evaluations do not naturally involve direct counting and have the grammatical properties of measure operations. Cardinal measure operations do not require directly accessing atoms, and thus are mechanisms for giving cardinality values by processes other than counting. One such process is a particular kind of approximation operation, namely estimation. Estimation operations are conceptually natural candidates for operations which make use of cardinal scales, and, in the cases that we will look at, they also have grammatical properties of measures. Estimation is a form of approximation in which we give a ‘ballpark’ value to a sum based on indirect evidence. Estimated values are naturally quoted in round numbers; they are values ‘in the region of’. (For explanations of why round numbers are natural approximate values, see Krifka 2009b; Solt 2015). Example (37a) is an example of an estimated value on the scale of weight, based on evidence other than direct measuring, while (37b) shows that explicit cardinal values can also be estimated:

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(37)

a. There is about three kilos of flour left in the jar. It holds six kilos and it is about half-full. b. There were about two hundred people at the party.

The direct method of giving a cardinal value is via counting, and in contrast, estimating a cardinal value is gives an overall approximate cardinal value to a sum without counting, and thus without making a commitment about its atomic part-structure. There are a number of approximation operators in English, some of which are illustrated in (38). (38)

a. about 500 yards away, approximately 5 miles from here, roughly/around an hour b. approximately 500 people, maybe 500 people, roughly 500 people

While counting and measure operations give values in terms of a number, all approximation operations explicitly give values in terms of an interval surrounding a number. (37a) is true if some quantity of flour close enough in weight to three kilos is in the jar. Lasersohn (1999) shows that while measure operations give precise values, a value such as three kilos or 3.8 kilos in fact always determines an interval on a scale, the interval surrounding 3 or 3.8 including those values which are near enough that the difference is irrelevant. Lasersohn calls this range of deviation the ‘pragmatic halo’. If I am asked to measure out three kilos of flour, then normally any quantity between 2.99 kilos and 3.01 kilos will be acceptable. In a shop, a professional scale will distinguish between 2.99 and 3 kilos, but will normally not distinguish between 2.9999 and 3.0001. In general, the size of the interval which counts as ‘3 kilos’ depends on the fine-grainedness of the scale and the size of the quantity being measured. Approximation operators such as those in (38) explicitly extend this interval which is normal in a particular context to a wider interval. Thus the statement that ‘you will need approximately three kilos of flour’ may mean that you need anything between 2.7 and 3.3 kilos, with a deviation of up to 10 percent of the total in either direction. There seem to be a number of different approximation operations available, with different operations associated with different operators. Zaroukian (2011) argued that maybe and like are modal operators, while approximately and roughly seem to be non-modal. Estimation is one natural non-modal approximation operation, in which you give an ‘informed guess’ at the quantity value of a sum, as in (37) above. Estimation gives an ‘overall’ quantity based on indirect evidence, and thus, as we said above, makes no commitment to the part-of structure of the sum. Since direct counting does make a commitment about the atomic part-of structure of a sum, while measuring does not, estimation is very plausibly interpreted as an operation which makes use of cardinal scales.

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Note that not all approximate values involve measuring. Some approximation clearly involves counting, as in (39). (39)

a. I counted about 250 people in the room. b. I made a crochet chain of about 70 stitches.

The approximative values here seem to derive from inaccuracy in counting rather than from a top-down approach. I counted the people in the room, and got to a number which is near 250, but I am not sure how accurate my counting was: I may have missed someone or counted some people twice. Interestingly enough, in approximative counting contexts, precise numbers are more acceptable than in contexts in which approximation seems to result from estimation. While the examples in (40) are a little strange, those in (41) are perfectly acceptable. (40)

a. ?The station is about three hundred and eight meters walk from here. b. ?I have about 308 volumes of law reports on my shelf. c. ?There are about eleven/forty-one people in my class.

(41)

a. I counted about three hundred and eight privates on the ground . . .5 b. Reasons for admission to asylums: 1864 to 1889. I counted about 28 I’d qualify for . . .6 c. I counted about 11 seconds that passed between the sound and the notification badge appearing in the upper right hand corner of the screen.7

The examples in (40) are natural estimation or ‘guessing’ contexts, while in (41) the approximation is naturally the result of uncertainty. For example in (41a), the speaker may be uncertain about whether she counted every private, whether she counted someone twice, or whether some of the individuals she counted were not privates after all.8 Approximative values may also be the result of counting and calculation, as in (42): (42)

Ten rows of fifteen chairs is a hundred and fifty people.

I am not, then, making the claim that all approximation operations are measure operations; however, conceptually, estimation is clearly a candidate for a 5 6 7 8

George Smart, Leaves from the Journals of Sir George Smart, edited by H. Bertram Cox and C. L. E. Cox (Cambridge: Cambridge University Press, 2014), p. 203. www.pinterest.com/pin/426645764680043040/, accessed May 26, 2016. https://discussions.apple.com/thread/7296278?start=0&tstart=0, accessed May 26, 2016. This is expressed very beautifully in the following quotation from Yoel Hoffman’s novel Christ of Fish: “I contemplate unimportant things, like the proper way to count birds. It’s hard to count birds when they are flying here and there, so it is better that they should be resting. They can lift their feet or flap their wings, but if they jump from branch to branch, or stand in each others’ places, it is very difficult to know how many there are” (Y. Hoffman, Christ of Fish paragraph 8. Keter: Jerusalem. 1991. My translation).

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measure operation. In the next two sections, I will show that, in at least two cases, estimation operations have grammatical properties associated with measuring. 8.7

Russian Approximative Inversion

Russian has a dedicated grammatical operation to express approximation, the so-called approximative inversion (AI) construction (Mel’čuk 1985; Franks 1995): (43)

a. ona napisala desjat’ knig she wrote ten books ‘She wrote ten books.’ b. ona napisala knig desjat’ she wrote books ten ‘She wrote about ten books.’

In addition to examples like (43), where a count noun and a numerical change position, approximative inversion occurs with explicit measure heads such as gramm/litrov in (44), and inversion with a count noun and classifer such as štuk ‘items’, čelovek ‘people’ (Yadroff and Billings 1998; Mel’čuk 1985; Matushansky 2015). (44)

a. gramm dvesti (muki) grams two hundred flour ‘about two hundred grams of flour’ b. litrov pjat’ (moloka) liters five milk ‘about five liters of milk’

(45)

a. štuk pjat’ starinnyx knig items five antiquarian books ‘approximately five old/antique books’ b. čelovek desjat’ anesteziologov people ten anesthesiologists ‘approximately ten anesthesiologists’

Given the explicit measure heads, it seems clear that (44) is a measure construction. Khrizman and Rothstein (2015) and Khrizman (2016), as well as Matushansky (2015), argue that the constructions in (43b) and (45), where the numerical is preceded by either a classifier or a count noun, are also measure constructions. They bring a number of pieces of evidence that AI phrases have many properties of measure constructions, based on observations which have been made in the literature over the last twenty years. First, like measure phrases, approximate inversion constructions prefer singular agreement:

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rabotalo/?* rabotali v ètom magazine čelovek pjat’ work.sg/ work.pl in this store people five ‘About five people worked in this store.’ (Franks 1995, p. 166)

Second, Yadroff and Billings (1998) show that AI phrases cannot be modified by which clauses but are compatible with that clauses, as in (47), again patterning like measure predicates: (47)

* knig pjat’, čto/* kotorye my kupili včera books.pl five that/* which.pl we bought yesterday ‘approximately five books that we bought yesterday’ (Yadroff and Billings 1998,p. 332)

Thirdly, AI is degraded in contexts which require individuation, such as reciprocal, control and reflexive constructions (Franks 1995; Yadroff and Billings 1998; Stepanov 2001). These constructions require individuation, and are incompatible with measure phrases (no matter what the agreement on the verb is). This is shown for reciprocals (48a), control structures (48b) and reflexives (48c). (48)

a. *studentov pjat’ pomogalo/pomogali drug drugu students five helped.sg/helped.pl each other ‘About five students helped each other.’ (Franks 1995,p. 166) b. *ženščin pjat’ staralos’/staralis’ kupit etu knigu woman five tried.sg/tried.pl to buy this book ‘About five women tried to buy this book.’ (Franks 1995, p. 167) c. *ženščin pjat’ smotrelo’/smotreli’ na sebja woman five looked.sg/looked.pl on self ‘About five women were looking at themselves.’ (Stepanov 2001, p. 119)

Fourthly, Matushansky and Ruys (2015), citing Mel’čuk (1980), note that in AI constructions, animate count nouns are case marked as inanimate, as in (49a), a characteristic of amount predicates, as shown by (49b): (49)

a. ja videl soldata četyre/ *soldat četyrëx I saw soldier.acc.pl four.acc *soldier.acc.pl four.acc =gen.sg (paucal) =nom.inanm/ =gen.pl =gen.anm ‘I saw about four soldiers.’ b. siloj rovno v tri /*trex medvedja /*medvedej strength exactly in three.acc bear.gen.sg (paucal) /bear. =nom.inanm /three.acc gen.pl =gen.anm ‘as strong as exactly three bears’ Mel’čuk (1980), cited in Matushansky and Ruys (2015)

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Khrizman and Rothstein (2015) suggest that the semantic effects of approximate inversion are to shift the measure function from an operation which maps a sum onto a number, representing a point on a scale, into an operation which maps a sum onto a set of intervals I n associated with or focused on a number n. I n is defined in (50a), and the approximate measure function associated with AI inversion is given in (50b). The effect of AI on liter is to shift the exact interpretation in (51a) onto the approximate interpretation in (51b): (50)

a. In is a set of intervals focused on a number n if 8i 2 In, n 2 i. b. MEASDIM, U APPROX ðxÞ ¼ In, ! 9i 2 In : 9m 2 i : MEASDIM, U ðxÞ ¼ m

(51)

a. 〚liter〛 ¼ λnλx:MEASVOL, LITRE ðxÞ ¼ n, b. 〚literAPPROX 〛 ¼ λnλx:MEASVOL, LITRE ðxÞ ¼ In

Litrov pjat’ then denotes the expression in (52a) and litrov pjat’ moloka has the interpretation in (52b), where litrov pjat’ is a modifier and the whole expression litrov pjat’ moloka denotes quantities of milk which measure approximately 5 liters: (52)

a. 〚litrov pjat0 〛 ¼ λx:MEASVOL, LITRE ðxÞ ¼ I5 b. 〚litrov pjat0 moloka 〛 ¼ λx:MILKðxÞ ^ MEASVOL, LITRE ðxÞ ¼ I5

Liter in (51) is explicitly a measure head measure, mapping portions on milk onto a continuous range of values on the domain of volume. Examples headed by count nouns and by classifiers such as štuk and čelovek map sums of individuals onto a non-continuous range of values in the set of natural numbers. But since the AI examples headed by these classifiers and count nouns have the same grammatical properties as measure constructions, it is highly plausible that they should have an analogous semantics making use of the cardinal measure scale. Khrizman (2016) argues for this in detail, showing that štuk and čelovek are measure classifiers also in non-inverted contexts, but here we will discuss only the AI constructions, giving an interpretation for (45b) čelovek desjat’ anesteziologov ‘approximately ten anesthesiologists’. We assume that a lexical measure classifier operating on the cardinal scale is constructed from an expression like (53) and a noun such as čelovek ‘person’ to give a lexical classifier such as (54a), which, in AI constructions, shifts to the approximative interpretation as in (54b). (53)

λnλx:MEASURECARD, j j k ðxÞ ¼ n ¼ λnλx:jfy : y v xg∩kj ¼ n

(54)

čelovek desjat’ anesteziologov (=(45b)) people.pl.gen ten anesthesiologist.pl.gen ‘approximately ten anesthesiologists’ a. čelovek: ¼ λPλnλx:jfy : y v x ^ PðxÞg∩kj ¼ n ðλx:PERSONðxÞÞ ¼ λnλx:jfy : y v x ^ y 2 PERSONg∩kj ¼ n b. čelovekAPPROX: λnλx:jfy : y v x ^ y 2 PERSONg∩kj ¼ In

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c. čelovek desjat’: λx:jfy : y v x ^ y 2 PERSONg∩kj ¼ I10 d. čelovek desjat’ anesteziologov: λx:jfy : y v x ^ y 2 PERSONg∩kj ¼ I10 ^ x 2 ANESTHESIOLOGIST

Khrizman and Rothstein (2015) argue that when the AI construction is headed by a count noun, the noun has shifted to a classifier interpretation, and the resulting numerical + N is a measure modifier which modifies an lexically empty mass noun, denoting ‘stuff’, i.e. anything in the domain of individuals. For details and supporting arguments, see Khrizman and Rothstein (2015) and Khrizman (2016). The structure for knig desjat’ ‘about ten books’ (lit, ‘books ten’) (43) is given in (55a), and the interpretation in (55b). (55)

a. ½½knigAPPROX desjat0 NP books ten b. λx:jfy : y v x ^ y 2 BOOKg∩kj ¼ In ^ x 2 D ¼ λx:jfy : y v x ^ y 2 BOOKg∩kj ¼ In ‘the set of sums whose set of book parts intersecting k has a cardinality in the range of 10.’

8.8

Cardinality Estimation in Mandarin

Mandarin also has a construction which shows the direct connection between estimation and measuring, as shown in Li and Rothstein (2012). In this construction, sortal or individuating classifiers are used in measure constructions, and can only have an approximative interpretation. In Mandarin, as is well known, counting necessarily involves sortal classifiers, as in (56a), while measuring expressions use measure classifiers (56b): (56)

a. sān gè xuéshēng three Clgeneral student ‘three students’ b. sān bàng ròu three Clpound meat ‘three pounds of meat’

As Cheng and Sybesma show, the particle de can optionally be inserted after a measure classifier, but not after a sortal or individuating classifier as in (57) (Cheng and Sybesma 1998). (57a) is obviously a measure construction, (57b) can only have a measure interpretation, while (57c), where a measure interpretation is impossible, is infelicitous. (57)

a. sān bàng de three Cl-pound DE ‘three pounds of meat’ b. sān wăn de tāng three Cl-bowl DE soup ‘three bowlfuls of soup’

ròu meat

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Susan Rothstein c. sān gè (*de) xuéshēng three Cl (*DE)student ‘three students’

As Li (2011) and Li and Rothstein (2012) show, the measure interpretation of classifiers like wăn ‘bowl’ and píng ‘bottle’ allows but does not require de (58a), while the sortal interpretation does not allow it at all. (58)

a. wǒ-de wèi néng zhuāngxià sān píng (de) jiǔ my DE stomach can hold three Clbottle DE wine ‘My stomach can hold three bottles of wine.’ (from Rothstein 2017) b. wǒ kāi le sān píng (*de) jiǔ yī zhuō yī píng I open PFV three Clbottle DE wine one table one Cl-bottle ‘I opened three bottles of wine, one bottle for one table.’

However, de phrases are possible with sortal classifiers with high round numbers (Tang 2005, Hsieh 2008) and approximative contexts. The following examples are taken from the PKU Corpus via Hsieh 2008: (59)

a. míngtiān de huódòng xūyào yì bǎi zhāng de fànzuōzi. tomorrow mod activity need one hundred Clpiece DE square-table ‘Tomorrow’s activity needs one hundred square tables.’ b. nàbiān hòng le qī bā kē, shí lái kē de júzi shù. there then plant PFV seven eight Clplant, ten Clplant DE mandarin around tree ‘On that side were planted seven or eight, or around ten mandarin trees.’

The de shows that these constructions involve measuring. The measure operation here is approximative. (59a) assesses the quantity of tables needed to be approximately 100 (k-dependent) zhāng- units, while (59b) estimates the overall sum of mandarin trees planted to have a cardinality of approximately ten. Exact figures are not available, since that would be dependent on individuating the atomic parts via counting. And indeed, in these constructions, precise, non-round numbers are infelicitous: Exact figures are infelicitous because they can be acquired only by counting and not by measuring. (60)

a. #wǒmen xūyào yì bǎilíng bā zhāng de fànzuōzi. we need one- hundred- zero- eight Cl DE square-table ‘We need one hundred and eight square tables.’ b. #yì-nián zhòngzhí le yì bǎi sān-shí qī kē de shùmù. one-year plant-PFV one- hundred- thirty- seven Cl DE tree ‘(They) planted one hundred and thirty-seven trees a year.’

So there is clear grammatical evidence of the association between approximation/estimation and measurement, and the dissociation between the

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use of sortal classifiers in counting contexts (where de is disallowed) and the use of sortal classifiers in estimation contexts, where de is obligatory. A detailed analysis of the interpretation of these constructions is dependent on an full semantics for counting and measuring in Mandarin, which is beyond the scope of this paper. However, an outline of the analysis is as follows. Following Li (2011) and Li and Rothstein (2012), we assume that sortal classifiers are expressions of type < k, < e  k, t >> denoting functions from kinds at type k to a set of atomic instantiations of the kind < e  k, t >>. In other words, sortal classifiers express in the syntax the COUNTk operation that applies in the lexicon in English. The only difference in the operation is that sortal classifiers in Mandarin apply to kind-denoting nouns, while COUNTk applies to set-denoting root nouns in English. Sortal classifiers, then, apply to kind denotations, and yield the set of indexed entities hx; ki, where x is in the intersection of ∪ k and k, as in (61a). As above, xk is a variable over ordered pairs hx; ki. π1 ðxk Þ is the first member of the ordered pair x and π2 ðxk Þ is k; the lexical content of the sortal classifier, for example that the entities in ∩ k have a certain shape or are a certain kind of animal, is expressed as a presupposition. (For some contrasts between sortal classifiers and container classifiers, see Rothstein 2017.) The result of applying the classifier kē used with plants and trees to júzi shù ‘mandarin tree’ is given in (61b): (61)

a. λk λxk: π1 ðxk Þ 2 ∪ k∩k ^ π2 ðxk Þ 2 k jπ1 ðxk Þ 2 P b. λxk :π1 ðxk Þ 2 ∪ MANDARIN TREE ∩ k ^ π2 ðxk Þ 2 k jπ1 ðxk Þ 2 PLANT

shí kē júzi shù ‘ten mandarin trees’ on its ordinary sortal interpretation function has the interpretation in (62): (62)

λxk :π1 ðxk Þ 2 ∪ MANDARIN TREE ∩ k ^ π2 ðxk Þ 2 k ^ jxk j ¼ 10 jπ1 ðxk Þ 2 PLANT

When sortal classifiers are used in cardinal measure constructions as in (59), the classifier denotes a different operation. In (61), the sortal classifier applies to k and gives the set of ordered pairs fx : xv∪ k∩kg  k. Sums in this set can then be counted as in (62). As a measure operator, the classifier denotes the approximate version of the cardinal measure operation MEASURECARD, j jk in (63). (63)

λnλx:MEASURECARD, j jk ðxÞ ¼ In ¼ λnλx:jfy : y v xg∩kgj ¼ In

This modifies the set of instantiations of a kind ∪ k, to give the set of plural entities in ∪ k which have approximately ten parts which are also in the set k, and which are a presupposed to be plants.

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(64)

shí ten

kē de: Clplant DE: λx:MEASURECARD, jjkðxÞ ¼ I10 ¼ λx:jfy : y v xg∩kgj ¼ I10 shí kē de júzi shù: ten Clplant DE mandarin trees: λx:x2 ∪ MANDARIN TREE^MEASURECARD,jjk ðxÞ¼I10 ¼ λx:x2 ∪ MANDARIN TREE^ jfy:yvx∩kgj¼I10 ¼ λx:x2 ∪ MANDARIN TREE^ jfy:yvx∩kgj¼I10

jx 2 PLANT jx 2 PLANT jx 2 PLANT jx 2 PLANT jx 2 PLANT

The sortal classifier takes the set of individuals in ∪ k∩ k and indexes them as countable atoms, yielding a set of type he  k; ti, whose members can be directly counted. The measure operation uses the same set ∪ k∩ k in picking out the set of plural individuals in ∪ k which have approximately ten contextually relevant discrete parts. 8.9

Why Must Count Nouns Be Counted?

In the analysis that I have presented, the choice between comparing object mass nouns in terms of cardinality or along some continuous dimensions is a matter of contextual preference. Object mass noun denotations are measured, and measuring on a cardinal scale is one a set of possible choices for a measure dimension. So why is this contextual preference restricted to object mass nouns? As Bale and Barner (2009) point out, with substance mass nouns like mud and flexible mass nouns like stone, the noun denotations always seem to be compared along a continuous dimension, rather than the cardinal dimension, unlike furniture or jewelry. This is particularly puzzling because in Hungarian and Brazilian Portuguese analogous comparisons using ‘bare singular’ object mass nouns such as Who has more stone/book? allow both cardinal and non-cardinal evaluations. Here is a possible explanation. Let us assume that, in principle, cardinal and non-cardinal evaluations are available for all mass nouns, including mud and stone. English has for the most part a system in which a lexical item has either a core mass reading, like mud, or a core count reading, as in the case of book. Count nouns privilege cardinal evaluations because they encode and make salient the (semantically) atomic structure, while substance mass nouns privilege non-continuous dimensions of evaluation, since a set of atomic entities is not usually salient. With nouns like furniture, cardinal and non-cardinal evaluations are available, since the predicate is generated by a set of discrete entities, but atomic structure is not encoded and thus direct counting is not possible. With flexible nouns, since both count and mass items are available, the count noun privileges a cardinal interpretation, while the mass noun privileges a non-cardinal interpretation. In Brazilian Portuguese and Hungarian, all, or almost all, nouns have a mass form, and a subset of these nouns also have a count form. So while count

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lexical items privilege a counting evaluation, mass lexical items are a default and do not privilege any dimension or make any dimension less available. This kind of pragmatic explanation for the absence of cardinal evaluations of nouns like stone predicts that, in the appropriate context, mass nouns in English should also allow cardinal evaluations, and this seems to be correct. Look at the following situation. There are two fields. Field 1 has a small number of moderate size stones in it. Field 2 has a very large number of very small stones, so the volume is less overall. John and Bill are both being punished for some crime or another with community service which involves clearing stone from public terrain. John has four hours of community service and Bill has only one hour to do, so John was assigned to the field which had more stone to pick up. He was told: (65)

John, you take the field with more stone in it.

In this context, it seems plausible that the larger number of stones with the overall smaller volume will count as ‘more stone’ than the smaller number with greater volume. We do not, however, expect non-cardinal evaluations to be available for count noun denotations, since count nouns are of type he  k; ti, and measure predicates apply to noun denotations of type he; ti. Nonetheless, with some plural count nouns, non-cardinal evaluations do seem to be possible. Solt (2008) discusses examples like Who ate more potatoes/scrambled eggs? or Who packed more clothes? Count superordinates also seem to allow noncardinal evaluations. Gafni and Rothstein (2014), testing data in Hebrew, show that, under experimental conditions, some individuals allowed non-cardinal comparisons for the plural count noun taxšitim ‘jewelry’ and others like it. (66)

le-mi yeš yoter taxšit-im to who there.is more jewel.PL ‘Who has more jewelry?’

These cases suggest that quantity evaluations in the count domain may also be more subtle and complex than we have for the most part considered, but this is a topic for a separate paper. Acknowledgments The first version of this chapter was presented at the Countability Workshop held at the Heinrich-Heine University, Düsseldorf in September 2013. Considerably more sophisticated versions were presented at a Workshop on Nominals at the Goethe University, Frankfurt in February 2016, the second Düsseldorf Workshop on Countability in June 2016, the conference Language in Logic and Conversation at Utrecht University in September 2016 and at the

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Ohio State University/Maribor/Rijeka Philosophy Conference in Dubrovnik (June 2017). In addition, versions were presented at department colloquia at ZAS Berlin (February 2016) the Language, Logic and Cognition Centre, Hebrew University of Jerusalem (January 2017), the Linguistics Department, University of Vienna, (February 2017) and the Linguistics Department, Tel Aviv University (March 2013). I would like to thank the audiences at all these conferences for their helpful and insightful comments, which contributed to the steady development of the ideas in this chapter. I would like to thank Manfred Krifka and an anonymous reviewer for comments on the penultimate draft. I thank Fred Landman for continuous discussion about the ideas developed in this chapter, which have been occupying both of us for many years, Roberta Pires de Oliveira and Brigitta Schvarcz for working with me on Brazilian Portuguese and Hungarian respectively, XuPing Li for pulling me into the complex and fascinating world of Mandarin classifiers and Keren Khrizman for insisting that we work together on approximative inversion in Russian long before I understood why I ought to be interested in it. This work was partially supported by ISF grant 1345/13 and by a Humboldt Research Award which enabled me to spend 2015–2016 at Tübingen University thinking intensively about these issues. REFERENCES Bale, Alan, and David Barner (2009). The interpretation of functional heads: Using comparatives to explore the mass/count distinction. Journal of Semantics 26.3: 217–252. Barner, David, and Jesse Snedeker (2005). Quantity judgements and individuation: Evidence that mass nouns count. Cognition 97.1: 41–66. Cheng, Lisa, and Rint, Sybesma (1998). Yi-wang tang, yi-ge tang: Classifiers and massifiers. The Tsing- Journal of Chinese Studies 28.3, 385–412. Chierchia, Gennaro (2010). Mass nouns, vagueness and semantic variation. Synthese, 174.1: 99–149. Franks, Steven (1995). Parameters of Russian Morphosyntax. Oxford: Oxford University Press. Frege, Gottlob (1884). Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Gafni, Chen, and Rothstein, Susan (2014). Who has more N? Context and variety are both relevant. Paper presented at 1st Conference On Cognition Research of the Israeli Society for Cognitive Psychology (ISCOP), Akko, 10–12 February 2014. Grimm, Scott, and Beth Levin (2012). ‘Who Has More Furniture?’ An Exploration of the Bases for Comparison. Presentation at Mass/Count in Linguistics, Philosophy and Cognitive Science Conference, École Normale Supérieure, Paris, France, December 20–21, 2012. Hsieh, Miao-Ling (2008). The Internal Structure of Noun Phrases in Chinese. Books Series in Chinese Linguistics. No. 2. Taipei: Crane Publishing.

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Hyde, Daniel (2011). Two systems of non-symbolic numeral cognition. Frontiers in Human Neuroscience 5: 150, doi:10.3389/fnhum.2011.00150. Kamp, Hans (1975). Two theories about adjectives. In E. Keenan (ed.), Formal Semantics of Natural Languages, pp. 123–155. Cambridge: Cambridge University Press. Reprinted in K. von Heusinger and A. ter Meulen (eds.), The Dynamics of Meaning and Interpretation. Selected Papers of Hans Kamp, pp. 225–261. Leiden: Brill. Khrizman, Keren (2016). Numerous Issues in the Semantics of Numeral Constructions in Russian. PhD Dissertation, Bar-Ilan University. Khrizman, Keren, and Susan Rothstein (2015). Russian approximative inversion as a measure construction. In G. Zybatow, P. Biskup, M. Guhl, C. Hurtig, O. MuellerReichau, and M. Yastrebova (eds.), Slavic Grammar from a Formal Perspective. The 10th Anniversary FDSL Conference, Leipzig 2013, pp. 259–272. Frankfurt am Main: Peter Lang. Krifka, Manfred (1989). Nominal reference, temporal constitution and quantification in event semantics. In Renate Bartsch, Johan van Benthem, and Peter von Emde Boas (eds.), Semantics and Contextual Expression, pp. 75–115. Dordrecht: Foris. (2009a). Counting configurations. In Arndt Riester and Torgrim Solstad (eds.), Proceedings of Sinn & Bedeutung 13, Working Papers of the SFB 732, 5, pp. 309–324. Stuttgart: Universitätsbibliothek der Universität Stuttgart. (2009b). Approximate interpretations of number words: A case for strategic communication. In Erhard Hinrichs and John Nerbonne (eds.), Theory and Evidence in Semantics, pp. 109–132. Stanford, CA: CSLI Publications. Landman, Fred (2011). Count nouns – mass nouns – neat nouns – mess nouns. In ‘ Michael Glanzberg, Barbara H. Partee, and Jurgis Škilters (eds.), Formal , Semantics and Pragmatics: Discourse, Context and Models. The Baltic International Yearbook of Cognition, Logic and Communication 6, 2010, pp. 1–67, http://thebalticyearbook.org/journals/baltic/issue/current. (2016). Iceberg semantics for count nouns and mass nouns: classifiers, measures and ‘ portions. In Susan Rothstein and Jurgis Škilters (eds.), Number: Cognitive, , Semantic and Cross-linguistic Approaches. The Baltic International Yearbook of Cognition, Logic and Communication 11, 2016, http://dx.doi.org/10.4148/19443676.1107. Lasersohn, Peter (1999). Pragmatic halos. Language 75.3: 522–551. Li, XuPing (2011). On the Semantics of Classifiers in Chinese. PhD Dissertation, Bar-Ilan University. Li, XuPing, and Susan Rothstein (2012). Measure readings of Mandarin classifier phrases and the particle de. Language and Linguistics 13.4: 693–741. Link, Godehard (1984). Hydras: On the logic of relative constructions with multiple heads. In Fred Landman and Frank Veltman (eds)., Varieties of Formal Semantics, GRASS 3, pp. 245–257. Dordrecht: Foris. Matushansky, O. (2015). On Russian approximative inversion. In G. Zybatow, P. Biskup, M. Guhl, C. Hurtig, O. Mueller-Reichau, and M. Yastrebova (eds.), Slavic Grammar from a Formal Perspective. The 10th Anniversary FDSL Conference, Leipzig 2013, pp. 303–316. Frankfurt am Main: Peter Lang.

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Matushansky, O. M., and Ruys, E. G. (2015). Measure for measure. In G. Zybatow, P. Biskup, M. Guhl, C. Hurtig, O. Mueller-Reichau, and M. Yastrebova (eds.), Slavic Grammar from a Formal Perspective. The 10th Anniversary FDSL Conference, Leipzig 2013, pp. 317–330. Frankfurt am Main: Peter Lang. Mel’čuk, I. (1980). Animacy in Russian cardinal numerals and adjectives as an inflectional category. Language 56.4: 797–811. (1985). Poverxnostnyj Sintaksis Russkix Čislovyx Vyraženij. (Surface Syntax of Numerical Expressions in Russian). Wiener Slawistischer Almanach, Sonderband 16. Vienna: Institut fur Slavistic der Universitat Wien. Pires de Oliveira R., and Susan Rothstein (2011). Bare singular noun phrases are mass in Brazilian Portuguese. Lingua 121.15: 2153–2175. https://doi.org/10.1016/j .lingua.2011.09.004. Rothstein, Susan (1999). Fine-grained structure in the eventuality domain: The semantics of predicative adjective phrases and be. Natural Language Semantics 7.4: 347–420. (2009). Measuring and counting in Modern Hebrew. Brill’s Annual of Afroasiatic Languages and Linguistics 1: 106–145. (2010). Counting and the mass/count distinction. Journal of Semantics 27.3: 343–397, https://doi.org/10.1093/jos/ffq007. (2011). Counting, measuring, and the semantics of classifiers. In Michael Glanzberg, ‘ Barbara H. Partee, and Jurgis Škilters (eds.), Formal Semantics and Pragmatics: , Discourse, Context and Models. The Baltic International Yearbook of Cognition, Logic and Communication 6, 2010, http://thebalticyearbook.org/journals/baltic/ issue/current. (2013). Counting, measuring and the mass/count distinction. Paper presented at the Düsseldorf Workshop on Countability, September. (2017). Semantics for Counting and Measuring: Key Topics in Semantics and Pragmatics. Cambridge: Cambridge University Press. Schvarcz, Brigitta R., and Susan Rothstein (2017). Hungarian classifier constructions and the mass–count distinction. In Harry van den Hulst and Anikó Lipták (eds.), Approaches to Hungarian: Volume 15: Papers from the 2015 Leiden Conference, pp. 103–208. Amsterdam and Philadelphia, PA: John Benjamins. Schwarzschild, Roger (2011). Stubborn distributivity, multiparticipant nouns and the count/mass distinction. In S. Lima, K. Mullin, and B. Smith (eds.), NELS 39: Proceedings of the 39th Meeting of the North East Linguistic Society, pp. 661–678. Amherst, MA: GLSA. Solt, Stephanie (2008). Cardinality and the many/much distinction. Paper presented at the LSA Annual Meeting Chicago. (2015). Vagueness and imprecision: Empirical foundations. Annual Review of Linguistics 1.1: 107–127. Stepanov, A. (2001). Late adjunction and minimalist phrase structure. Syntax 4.2: 94–125. Tang, Chih-Chen Jane (2005). Nouns or classifiers: A non-movement analysis of classifiers in Chinese. Language and Linguistics 6.3: 431–472. Wiese, Heike (2003). Numbers, Language and the Human Mind. Cambridge: Cambridge University Press.

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The Count/Mass Distinction for Granular Nouns Peter R. Sutton and Hana Filip

9.1

Granular Nouns as a Notional Class

Granular noun is the label we use to refer to a semantic class of nouns denoting entities that consist of relatively small grains, particles or distinguishable pieces (e.g. lentil, rice, pebble, sand, seed, barley, gravel).* These nouns form a disparate notional class that includes ‘naturally’ occurring objects (rice) and also artifacts (sequin(s)). Nouns in this class can be grammatically encoded as mass (e.g. rice, barley, gravel) or count (e.g. pebble(s), lentil(s)). Some languages also have dual life granular nouns (e.g. seed, cp. many seeds, much seed) and pluralia tantum granular nouns (oats).1 When count, granular nouns are bona fide count nouns in that they are straightforwardly felicitous when directly modified by numerical expressions (three pebbles/lentils). However, this does not mean that they pattern distributionally like prototypical count nouns, such as cat. Many granular count nouns are, in terms of distributional frequencies, more often than not used in the plural (at least when they are the head of an NP, thus excluding non-granular-headed compounds, e.g. lentil soup, bean burger). Furthermore, in some contexts at least, plural granular nouns are slightly awkward to use with count quantifiers like many. For example, it is odd to ask when serving dinner, How many lentils would you like?, which suggests that we often do not care or do not focus on specific numbers. When dual life, granulars such as seed are straightforwardly felicitous in both count and mass constructions. However, the mass sense seems to be restricted to collections of grains; hence, in mass constructions, dual life *

1

This research was funded by the German Research Foundation (DFG) CRC991, project C09. We would like to thank the participants of the Coercion across Linguistic Fields (CALF) workshop held at the DGfS annual conference in Saarbrücken, 2017 and the participants of the Workshop on Countability held at Heinrich Heine University Düsseldorf in June 2016 for their helpful comments. In particular, we would like to thank Eleni Gregoromichelaki, Scott Grimm, Fred Landman, Beth Levin and Susan Rothstein for very useful discussion. Some native speakers find the direct numerical modification of oat(s) felicitous, however. For instance, some speakers accept one oat as straightforwardly felicitous, suggesting that for some native speakers of English, oat is a granular count noun akin to lentil.

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granulars tend to resist any kind of ‘grinding’ reading which accesses the part structure of the individual grains. For instance, After the silo explosion, there was seed all across the farmyard does not seem to have an interpretation in which there are bits of individual seeds all over the farmyard. In this way, the count sense of dual life granular nouns is different from that of count nouns which do permit ‘grinding’, modulo context, as in, for example, After the crash, there was motorbike strewn across the road. When mass, granulars are bona fide mass nouns. Also, as argued by Landman (Chapter 6 in this volume), the ‘grains’ in the denotation of granular nouns are not straightforwardly accessible for cardinality comparisons in comparative constructions. If a has two large grains of wild rice and b has three small grains of pudding rice that total less in volume, it is not obvious that ‘b has more rice than a’ has a true reading. Nonetheless, the fact that the denotations of granular mass nouns are made up of grains/granules is highly salient, even if this grain structure is inaccessible to the counting operation and to cardinality comparisons in comparative constructions. Mass granular nouns will be the main focus of this paper. In particular, we introduce a puzzle relating to why expressions like three rices cannot be coerced to mean ‘three grains of rice’. We label this puzzle the accessibility puzzle. 9.2

The Accessibility Puzzle

Concrete mass nouns can generally be coerced into count noun interpretations, such as container, portion or subkind, depending on context, a fact that has garnered much attention in formal semantics and philosophy since at least Pelletier (1975). (In this paper, we will not discuss the portion reading, however; see Landman, Chapter 6 in this volume, for extensive discussion of the portion reading.) Different classes of nouns diverge with respect to the ease with which they can be coerced into a count interpretation, however. For example, water in (1a)–(1b) is easier to coerce into a count noun interpretation than the granular mass noun rice (2a)–(2b), while count interpretations of mud are possible only in highly specialized contexts, such as technological ones, as illustrated in (3a)–(3b). (1)

a. Three waters, please! e.g. three [glasses/bottles of] water. (container or portion) b. I ordered three waters for the party: still, sparkling, and fruit-flavored for the kids. i.e. three [kinds of] water (subkind)

(2)

a. We ordered the main courses with two plain rice, one egg fried rice and a nan, more than enough for the four of us.2 e.g. two [bowls of] plain rice (container or portion)

2

www.derbytelegraph.co.uk/speciality-dishes-star-turn-littleover-s-red/story-20536589-detail/ story.html [accessed October 10, 2016].

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Peter R. Sutton and Hana Filip b. Context: three kinds of rice: Calmati, Texmati, Kasmati These three rices have basmati’s viscosity and cooking style, but smaller individual grains.3 i.e. three [kinds of] rice (subkind)

(3)

Context: yield points of different mud samples before contamination The three muds experienced particle dispersion at the same temperature with different yield points.4 a. e.g. The three [samples of/vials of] mud . . . (container or portion) b. The three [kinds of] mud . . . (subkind)

Chierchia (2010) poses the question, ‘Why each time that we want to count using a mass noun don’t we simply, automatically “apportion” it as needed? What is to prevent us from interpreting “water” as meaning something like “water amount” or “water quantity”?’ His conclusion is that the mass/count distinction must be rooted in the grammar of natural languages, and not only in ‘general cognition’. We agree with Chierchia, but, in addition, we bring to the table a complexity that raises a follow-up question. While it is true that concrete mass nouns can often be coerced into count noun interpretations given the right context, what has been less explored are cases in which mass-to-count shifts are prohibited or heavily restricted, despite cognitive and contextual factors that would prima facie seem to facilitate them. If the mass/count distinction is in the grammar, why are some coerced mass-to-count interpretations less accessible than others? Let us first consider so-called object mass nouns, also known as fake mass nouns, which include furniture, footwear, cutlery, crockery and equipment, among many others. They have played a key role in the development of many recent theories of the mass/count distinction (Chierchia 1998; Barner and Snedeker 2005; Landman 2011, 2016; Sutton and Filip 2016a, 2016b, 2018). As the examples below show, they strongly resist coercion in numerical counting constructions in which either object units (4), basic-level kind units (5a) or superordinate level kind units (5b) are counted: (4)

#I ordered three furnitures from Ikea: one table and two chairs.

(5)

a. #I ordered two furnitures from Ikea: chairs and tables. b. #I ordered two furnitures from Ikea: bedroom and living room furniture.

3 4

Hensperger, Beth, and Julie Kaufmann (2003). The Ultimate Rice Cooker Cookbook, p. 23. Boston, MA: Harvard Common Press. Adekomaya, Olufemi A. (2013). Experimental analysis of the effect of magnesium saltwater influx on the behaviour of drilling fluids. Journal of Petroleum Exploration and Production Technology 3: 61–67.

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At face value, it seems puzzling that counting of object units in the denotation of furniture is prohibited, given that it consists of conceptually and visually salient individuated entities like individual chairs and tables, for instance. However, these are not accessible for grammatical counting, as the oddity of (5a) shows. Such data are well known, but rarely directly addressed. Rothstein (2015) highlights the fact that object mass nouns lack pluralization with a subkind interpretation (but offers no account). Landman (2011) criticizes the account of Chierchia (1998), in which object mass noun denotations are atomic, which would prima facie seem to predict that they should be accessible to direct grammatical counting, but nonetheless, they are not: The problem is that it is not particularly difficult to semantically or contextually pull a set of atoms out of an atomic structure . . . a child can do it. And there, of course, is the problem: the child doesn’t do it. (Landman 2011)

This, among other considerations, motivated Landman to develop a theory that characterizes object mass nouns (his ‘neat mass nouns’) as those mass nouns expressing a concept which overdetermines what counts as one item for counting. In brief, the reason why we do not ‘pull a set of atoms out of an atomic structure’ is that there are many different ways of partitioning the domain into sets of entities each of which is ‘one’ for the purposes of counting, none of which is privileged over others, and such alternative partitions have members that overlap. Therefore, there is no single determinate way of counting, but rather multiple alternative ones in any given situation. Building upon the ideas of Landman, in Sutton and Filip (2016b), we directly address restrictions on grammatical counting of object mass nouns that concern the cardinality of ordinary particular objects, which instantiate basic-level kinds, as in (4). In Sutton and Filip (2018), we address restrictions on grammatical counting of subkinds of object mass nouns (see also Grimm and Levin 2017). In both cases, we argue that the restrictions can be derived from overlap. In simple terms, object mass nouns overdetermine what counts as one particular unit, and also as what counts as one subkind. This overdetermination leads to overlap, and overlap blocks access to a countable set of object units or to a countable enumeration of subkinds. The use of object mass nouns in counting constructions requires an explicit unit extracting expression (e.g. item of) or an explicit kind extracting expression (e.g. kind of). Now, it turns out, as we observe, that some of the countability properties of object mass nouns are shared by granular mass nouns. Both impose one specific restriction on the range of their admissible mass-to-count shifts: namely, while they allow shifts to portions and subkinds (2a)–(2b), they strongly resist shifts to ordinary particular object units (e.g. single grains), when they are, for instance, directly modified by numericals (6a)–(6b) or combined with a distributive determiner, as in sentence (7):

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(6)

a. # Three rices fell off my spoon. b. # Drei Reis sind vom Löffel gefallen. (German) Three rice.sg be.3.aux from.the.dat spoon fall.pst-ptcp

(7)

# I removed each gravel from the sole of my boot.

This shared property of object mass nouns and granular mass nouns has not yet been noticed at all (to the best of our knowledge), which might be due to the fact that when it comes to the mass domain, granular mass nouns have received much less attention than object mass nouns (but see Chierchia 2010, Grimm 2012, Landman, Chapter 6 in this volume, Hnout 2017). To sum up our observations thus far, mass nouns such as mud and blood are hard to coerce into container or portion readings, barring highly specialized contexts. With respect to mass-to-count object unit shifts, it is, of course, unsurprising that there is no mass-to-count object unit shift available for such prototypical mass nouns, because they denote substances lacking conceptually salient individuated units in their denotations. By the same token, it is, however, unclear why object (or fake) mass nouns and granular mass nouns do not sanction a shift into count interpretations that would involve what are conceptually, and possibly also functionally, salient and individuated object units, in their denotations: for example, individual pieces or items of furniture like tables, stools, chairs, grains of rice and kernels of wheat. In order to use them felicitously in numerical counting constructions or with quantifiers that select for count predicates, it is not sufficient that we know what is (taken to be) a single countable object unit, such as individual pieces of furniture like chairs, grains of rice or kernels, but rather it is necessary that such nouns first be combined with an overt classifier-like ‘object unit’ expression, or a ‘unit excerpting’ operator (in the sense of Talmy 1986) that singles out “a single instance of the specified equivalent units” and sets them “in the foreground of attention” (Talmy 1986, p. 12), which can then serve as input into grammatical counting: for example, a piece/an item, as in a piece /an item of furniture, a grain, as in a grain of rice, or a blade, as in a blade of grass. At first blush, then, it looks as though granular mass nouns pattern with object mass nouns in that the perceptually, cognitively or functionally determined units from which their denotations are built are not accessible for count reinterpretation via coercion. However, there are differences, too. While most would agree that there are minimal units in the denotations of both object mass nouns and granular nouns, with, for example, single chairs and tables as the minimal units in the denotation of furniture, and the single grains of rice as the minimal units in the denotation of rice, these entities that count as ‘one’ for counting for object mass nouns are not the only units that can be grammatically counted (Landman 2011). This is not the case for granulars, however.

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The Count/Mass Distinction for Granular Nouns (8)

Tan bought one item of furniture.

(9)

Tan ate one grain of rice.

257

The sentence in (8) can be true when Tan bought a stool, a dressing table and a mirror, which together form a single functional unit which we refer to as a vanity. Although single mirrors, stools and tables may be minimal in the denotation of furniture, the expression one item of furniture can also be used to refer to sums of such minimal entities. A parallel situation, however, does not hold for (9). One cannot use one grain of rice to refer to any sums of grains. It is not entirely clear how granular nouns could be integrated into extant theories of the mass/count distinction. Let us take, as an example, Landman’s (2011) overlap-based proposal, which has influenced much of our own work (Sutton and Filip 2016a, 2016b, 2017, 2018; Filip and Sutton 2017). On this view, as observed above, what makes object mass nouns mass is that the set of entities that count as one contains, in the same context, overlapping entities (e.g. a table and a vanity which subsumes that table). Granular mass nouns are different from object mass nouns, because for granular mass nouns the set of object units that intuitively count as one in their denotation is always disjoint. Nevertheless, such minimal granular object units are not accessible to grammatical counting operations, directly or via mass-to-count coercion. For instance, rice denotes non-overlapping grains, each clearly demarcated and disjoint from the other, and yet rice cannot be used in counting constructions that directly count individual grains of rice. In order to motivate this behavior, we cannot obviously rely on the overlap property which Landman (2011) compellingly uses to motivate why object mass nouns like furniture are not straightforwardly felicitous in grammatical counting construction. We will call the particular restriction on coercing object mass nouns and granular mass nouns to count interpretations that directly count particular object units, as outlined above, the ‘accessibility puzzle’: Why should conceptually and perceptually salient object units in the denotations of object mass nouns and granular mass nouns not be directly accessible by semantic counting operations, nor facilitate the mass-to-count coercion? Specifically, when it comes to granular mass nouns, which are the main focus of this paper, we will address the following questions: (Q1) What is the semantic distinction between count granular nouns (pebble (s), lentil(s)) and mass granular nouns (rice, barley)? (Q2) Why do granular mass nouns (rice) strongly resist a coercion to count readings which involve the object units in their denotation (individual

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grains of rice in the denotation of rice), even if the context makes such object units clearly conceptually and perceptually salient? For granular mass nouns, why are only shifts to count interpretations with implicit containers available, but not shifts to count interpretations based on individual units? In relation to (Q1), we propose a characterization of the semantics for basic predicates for granular nouns which underspecify whether single grains or aggregates of grains are referred to. This representation captures our general knowledge (common across all granular nouns), that, inter alia, they are made up of grains, and typically come in clustered aggregates. Then, using the mechanisms for individuation that we have independently motivated elsewhere, we outline how such basic predicates can be mapped into a count interpretation (such as with the English lentil) or a mass interpretation (such as with the Czech čočka [‘lentil’, mass]). Such mappings rely, crucially, on two key ingredients: the object identifying function, which identifies perceptually or functionally salient entities in a noun’s denotation, and the schema of individuation, which concerns a perspective on these entities relative to a context of utterance. Count granular noun lexical entries feature the object identifying function and a schema of individuation. Mass granular noun lexical entries lack the object identifying function and have a null schema of individuation. In relation to (Q2), we argue that there is a key difference between mass-tocount coercion in which a contextually salient receptacle concept is used (when three rices can be used to mean ‘three bowls of rice’, for instance), and mass-to-count coercion in which single, individuated object units are referred to (were three rices able to mean ‘three grains of rice’, for instance). In the former case, the implicit receptacle concept (e.g. bowl) supplies a means of partitioning a variety of suitable domains (liquids, granulars and substances) into countable units, either as stuff contained in the receptacle or as stuff to the amount that could be contained within the receptacle. In contrast, mass-to-count unit shifts, were they possible, would amount to simply selecting salient entities in a mass noun’s denotation and making them countable (and so, were they possible, would not, strictly, be coercion at all, given the standard view that coercion involves retrieving and using non-lexically specified information from the context). We argue that such a shift would amount to a generalized, mass-to-count shifting operation. The intriguing consequence of this proposal is that it predicts that for languages with a grammaticized lexical mass/count distinction, object unit shifts are excluded. (This will be discussed in detail in Section 9.5.4). In Section 9.3, we lay out the basis for our formal analysis and also give some background on other relevant accounts of the mass/count distinction. In

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Section 9.5, we give an analysis of object unit extracting classifiers such as grain of and classifiers based on receptacle nouns, such as bowl (of ), which are derived from nouns for concrete physical receptacles, and are used as classifier-like concepts of container, contents/portion or measure (Khrizman et al. 2015; Landman 2016). We then characterize what a process of coercion would look like that made use of concepts, such as ⟦bowl of⟧ and ⟦grain of⟧, and argue that the differences between these two cases reveals an answer to the accessibility puzzle. Given that our analysis of NPs and counting constructions, which ties together insights from lexical semantics and compositional semantics, requires a slight enrichment of the standard compositional semantic toolbox, we outline how our account is straightforwardly compatible with representations of counting constructions in Appendix A and with VPs in Appendix B. 9.3

Background

9.3.1

Background: Granular Nouns in the Context of Current Mass/Count Theories

Chierchia (2010) argues that count nouns have ‘stable atoms’ in their denotation. This means that there are entities in their denotation that are atoms in every context (on every admissible precisification of the noun’s denotation such that an admissible precisification is an adjustment of the extension of an expression licensed by permissible language use). Mass nouns lack stable atoms in their denotation, that is, for mass nouns, it is a vague matter what the grammatically countable units in noun denotations are: Namely, there is no entity that is an atom in the denotation of the predicate at all contexts (on every admissible precisification of the noun’s denotation). In this sense, mass nouns have only unstable individuals in their denotation, and, assuming that counting is counting of stable atoms, mass nouns cannot be directly used in counting constructions. For instance, we have the infelicity of #three muds, unless mud first undergoes a shift into a plausible contextually determined count interpretation. Chierchia couches his vagueness-based analysis of the mass/count distinction in supervaluationist terms. How it works is best shown using his paradigm example of a mass noun rice. It is vague in the following way. It is not the case that, across all contexts, for example, a few grains or single grains of rice fall under the denotation of the predicate rice. But this means that such various quantities of rice are all in the vagueness band of rice; they fall in and out of the denotation of rice depending on the context. There may be some context c, in which cups of rice are rice atoms. There may also be some c0 , such that c0 extends c, where sums of a few grains are rice atoms. There may also be

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some c00 such that c00 extends c0 , in which single grains are rice atoms. Most importantly, there is, therefore, no entity that is a rice atom at every context of evaluation of rice. In this sense, the denotation of rice lacks stable atoms, and so is vague. If rice has no stable atoms in its denotation, and counting is counting stable atoms, on Chierchia’s account, then what rice denotes cannot be counted, which motivates its grammatical mass property. Although rice is Chierchia’s paradigm example to motivate his vaguenessbased (supervaluationist) account, granular nouns are in fact problematic for this account. One problem stems from cross- and intralinguistic count/mass variation among granular nouns. For example, lentil(s) is count in English, but čočka (‘lentil’, Czech) is mass. Aware of such data, Chierchia’s response is “[w]hat this suggests is that standardized partitions for the relevant substances are more readily available in such languages/dialects” (Chierchia 2010, p. 140). However, were we to accept this explanation for cross-linguistic variation, such a response would still face a challenge in accounting for intralinguistic variation and dual life granular nouns. For example, the German noun Same(n) (‘seed(s)’), which is count, has a mass counterpart Saat (‘seed’) and, as we have seen, some languages have dual life granular nouns such as the English seed. Even if we accept that standardized partitions are more readily available in some languages (for some nouns) than others, it is less plausible to adopt the position that the same language both does and does not make standardized partitions available across two co-extensional lexical items (Saat/Same(n)), let alone for a single lexical item that admits of both a count and a mass sense (seed). Simply put, one of our worries with Chierchia’s account is that it does not seem to capture what we take to be the most puzzling property of granular mass nouns. As observed above, granular mass nouns have entities in their denotation that intuitively count as one, for example, individual grains, seeds and the like. Nevertheless, such ‘natural’ object units are not accessible to grammatical counting operations, directly or via mass-to-count coercion, even if the context makes them clearly salient and relevant. Chierchia’s account, in attempting to reduce the property of being individuated, and hence countable, to the property of having stable atoms does succeed in accounting for what makes certain nouns which denote notionally granular entities grammatically mass, but it does so at the cost of losing any conceptually privileged status for the grain structure that is a core property of the denotations of granular nouns. One of the key contributions of Landman (2011, 2016) to our understanding of the mass/count distinction is the idea that its motivating property is disjointness. For Landman (2016), count noun concepts specify, relative to context, a disjoint set of entities for counting, their counting bases. In contrast, the counting bases of mass noun concepts (the set of entities that

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could be candidates for counting) are overlapping (not disjoint) and overlap makes ‘counting go wrong’. Rothstein (2010) emphasizes the importance of context for the delimitation of count noun denotations from mass ones, and to this goal coins the term ‘counting context’. Count nouns denote sets of entity-context pairs (the entity denoted and the context in which it counts as one), which makes them of type he  k; t i. Mass nouns are of type he; t i, that is, they have the standard predicative denotation. This in effect amounts to the claim that the mass/count distinction can be reduced to this typal distinction between mass and count nouns. The main motivating data for the introduction of the counting context into the lexical entries are singular count nouns like fence, wall and rope, for which, as Zucchi and White (1996, 2001), among others, observed, what counts as ‘one’ can vary from occasion of use to occasion of use. One thing that is striking about the mass/count theories of Chierchia’s, Landman’s and Rothstein’s work is that they all integrate some notion of context-sensitivity, albeit each in a slightly different way. Inspired by their proposals in this regard, in Sutton and Filip (2016a), we defended the idea that the mass/count distinction fundamentally relies on the context-sensitive notion of individuation, whereby the relevant context-sensitivity has two sources: one which was inspired by Chierchia’s proposal regarding precisification relative to context, and another which was inspired by Landman’s and Rothstein’s accounts in which context can determine a disjoint set for counting (yielding a count concept) or leave the counting base overlapping (yielding a mass concept). Our main empirical interest in Sutton and Filip (2016a) lay in crossand intralinguistic patterns of count/mass variation. We attempted to motivate the observation that granular (mass/count) nouns, which in English include rice, lentils, beans and collective artifact nouns, including English object mass nouns like furniture and the corresponding Dutch count nouns like meubels, are distinguished by considerable variation in count/mass lexicalization patterns, cross- and intralinguistically. In contrast, prototypical object denoting nouns (cat, ball) are pretty stably count, and substance denoting nouns (mud, blood) are stably mass, at least in languages with a lexical mass/count distinction. For our analysis of granular nouns, we argued that the way granulars can be individuated is sensitive to context. We proposed that count granulars are interpreted relative to a precisification determined by the context, but that mass granulars are interpreted relative to the intersection of all licensed precisifications such that single grains are excluded from the counting base of the noun. Although our account in Sutton and Filip (2016a) could capture more data than alternative accounts, especially with regard to whether notional classes of nouns will display cross- and intralinguistic mass/count variation, it is, arguably, not as parsimonious as it could be, since it relies on two distinct

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mechanisms for individuation. In this paper, we will still defend the view that individuation is context-sensitive, and that the semantics of common nouns must reflect this; however, here, we propose that the semantics of the countability of granular nouns and of other notional classes of nouns can be accounted for with a unified account of individuation. For the account of granular nouns proposed in this paper, of particular interest is the work of Grimm (2012). He enriches the standard mereological semantic toolbox with topological relations so as to be able to articulate, within lexical entries, certain properties which must be taken as basic or unanalyzed in standard mereological accounts. Most importantly for us, Grimm (2012) introduces the notion of cluster, which allows him to differentiate a mereological sum counting as a clustered entity from a mereological sum viewed as one individual entity. For example, the entry for dog in (10) (simplified from Grimm (2012)) states that x is a realization of the concept Dog, and x is a maximally strongly self-connected (MSSC) individual, i.e. an entity for which every part internally overlaps with the whole. (10)

〚dog〛¼ λx½Rðx; DogÞ ^ MSSCðxÞ

In (11) (simplified from Grimm 2012), the entry for the collective noun cacwn (‘hornet’, Welsh), which refers to a swarm or relatively closely grouped collection of hornets specifies the property CLUSTERP, C , which means that x is a cluster entity the parts of which share property P and are transitively connected under some connection relation C. In other words, a set of sum entities that are realizations of swarms of hornets. (11)

9.3.2

〚cacwn〛¼ λx½Rðx; Hornet Þ ^ x 2 CLUSTERP, C 

Formal Background: Extensional Mereology and Frame Semantics

We assume a domain structured as a complete lattice with the bottom element removed, closed under sum t which is an idempotent, commutative and associative relation. Part v and proper-part ⊏ relations are defined as standard: (12)

avb$atb¼b

(13)

a ⊏ b $ a v b ^ ¬ðb v aÞ

The supremum tP of a predicate P and the upward closure of P under sum ∗ P are also given as standard. (14) (15)

tP ¼ the smallest individual x such that 8y 2 P½y v x ∗

P ¼ ftY : Y⊆Pg

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Two entities a, b overlap ða ∘ bÞ: (16)

a ∘ b $ 9x½x v a ^ x v b

We also make use of the property of predicates being overlapping and disjoint. A predicate is overlapping: (17)

8P½OVERLAPðPÞ $ 9x9y½PðxÞ ^ PðyÞ ^ x ∘ y

(18)

8P½DISJOINT ðPÞ $ ¬OVERLAPðPÞ

The formal tool we employ is a form of frame semantics (see Fillmore 1976 for the original proposal). Our version of frame semantics is inspired, in large part, by Type Theory with Records (TTR) (see Cooper 2012 for an introduction an further references), but it is simpler than TTR and stays closer to simply typed semantic theories. The enrichment to frames is, however, necessary. Frames allow us to provide enough detail about lexical information so as to capture subtle differences between the semantics of members of notional noun classes. At the same time, this preserves a standard Montogovian compositional semantics so as to account for compositionality (see Cooper 2012 for details as to why such integration is desirable if not necessary to provide an adequate analysis for at least some natural language data). Frames, on this approach, are representations of (complex) structured concepts. We assume a basic type f for frames (frames replace propositions), along with other more familiar basic types such as e. Frames are sets of fields. Fields are labeled formulae, with labels to the left of the ‘¼’ and formulae to the right. A single frame can have multiple fields (19), and can be recursive in that frames or abstractions over frames can be parts of fields (20). We also allow complex type formation in the usual way. For example, an expression of he; f i can be formed by abstracting over a type e variable within a frame (21). 2 (19)

2 (20)

4

l1

¼

l2

¼ 2

(21)

¼ ¼ ¼

l1 4... ln

λx:4

3 ϕ1 ...5 ϕn ϕ 1 l3 l4

l1

¼

l2

¼

¼ ¼ ϕ 1 l3 l4



3

ϕ2 5 ϕ3 ¼ ¼



3

PðxÞ 5 ϕ2

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  By specifying labels ðl1 Þ, and paths of labels li ; lj . . . within frames we can select (and then modify) specific parts of frames.5 For example, if the expression in (21) is F, then the following equivalences hold: FðyÞ:l1 $ ϕ1  l FðyÞ:l2 $ 3 l4

¼ ¼

PðyÞ ϕ4



FðyÞ:l2 :l3 $ PðyÞ

The move to a frame-based semantics is motivated by our need to represent lexical semantic details in a way that would be, at best, cumbersome for more mainstream formalisms within formal (compositional) semantics. Frames also allow us to represent dependencies between complex semantic structures in a way that would be far more complex to do in more standard formalisms (we give a concrete example below). We should stress, however, that for any frame, an extensionally equivalent predicate logic expression can be provided. For example, the frame in (22), which is equivalent to the frame in (23), can be ‘de-labeled’ and converted into the extensionally equivalent propositional logic formula in (24):6 

2 (22)

(24)

¼

l5

¼ ¼

l1

¼

l5

¼ ¼

6 6 4 l4 2

(23)

l1

6 6 4 l4

l2 ¼ l3 ¼ H ðl1 ðaÞ:l2 ℐðl1 ðbÞ:l3  l2 ¼ λy: l3 ¼ H ðF ðaÞÞ ℐðGðbÞÞ λy:

3 F ðyÞ GðyÞ 7 7 5 3 F ðyÞ GðyÞ 7 7 5

H ðF ðaÞÞ ^ ℐðGðbÞÞ

An immediate expressive advantage of these kinds of frames is that one can easily define functions that add or modify information in frames in ways that it is not simple to do with more mainstream formalisms. For example, we can define a function that modifies and adds a further condition on the sub-frame labeled l1 in (23). The function in (25), applied to the frame in (22), yields the

5 6

The use of paths is also appropriated from TTR (Cooper 2012). We do not give the full details of de-labeling and conversion here. However, as a heuristic, we suggest the following: 1. For all fields in a frame, replace any labels within formulae with the formulae that they label and perform any λ reductions. For the frame in (22), this would yield the frame in (23); 2. For all fields with a type t formulae in the frame which do not contain a variable in the scope of a λ expression, conjoin the formulae. This yields the formula in (24). In other words, the extension of a frame is characterised by the conjunction of the propositional formulae in its fields.

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frame in (26), which is extensionally equivalent to the predicate logic formula in (27). 2

(25)

l1 6 λF :f 4 ll4 5 l6

¼ ¼ ¼ ¼

3 F :l1 7 F :l4 5 F :l5 K ðl1 ðcÞ:l2

¼

λy:



2

l2 ¼ F ðyÞ l3 ¼ GðyÞ H ðl1 ðaÞ:l2 Þ ℐðl1 ðbÞ:l3 Þ K ðl1 ðcÞ:l2 Þ

(26)

6 l1 6 6 λF :f 6 l4 6 4 l5 l6

(27)

H ðF ðaÞÞ ^ ℐðGðbÞÞ ^ K ðF ðcÞÞ

¼ ¼ ¼

3 7 7 7 7 7 5

Importantly, without the label-formula (attribute-value) structure of frames, it is not trivial to define a function that could apply to (24) and give the result in (27) without specifying constants such as F within the function F (which means one would have to define a different function for every argument frame). Frames make simple the representation of modification of rich, structured information in the lexicon of a particular noun while maintaining compositionality, hence we use frames as our representational format. We use this relatively straightforward compositional mechanism in our nominal semantics. Lexical entries, which represent the basic ‘core’ meaning of concepts, contain a property of type he; f i, where e stands for the entity type (any concrete object or stuff ) and f for the frame type. Functions apply to this core property and yield a predicate which specifies the counting base for that concept and a predicate which specifies the extension of that concept. These two predicates, as we will go on to argue, differ cross-linguistically, yielding, for example, either a count concept or a mass concept. Finally, we note that, although notationally different, our adoption of this kind of frame semantics is merely a natural generalization of formalisms already used in state-of-the-art semantics accounts of the mass/count distinction. For example, Landman (2016, 2017) gives the lexical entries of nouns with ordered pairs hbody; basei, which arguably are a simple ‘frames’ with two fields. Landman uses functions to access each of the projections of these pairs in a manner similar to the way we use labels above. The way in which our approaches differ is that we allow for arbitrarily large numbers of fields and for recursion. 9.4

The Mass/Count Distinction for Granular Nouns

There are three main ingredients for our lexical entries: (i) basic predicates (a part of all lexical entries), (ii) the object identifying function (only a part of

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concepts which allow cardinality comparisons in more than constructions) and (iii) schemas of individuation (which are either specific or null). We introduce these three ingredients, and then we show how they interact in lexical entries of ‘concrete’ nouns that are representative of four different notional classes: granulars, prototypical objects, collective artifacts and substances. 9.4.1

Basic Predicates

A number of different theories of the mass/count distinction assume some kind of basic meaning for nouns. For example, Krifka (1989) assumes numberneutral predicates like CAT and MUD, which encode the qualitative application conditions for a nominal concept, while Rothstein (2010) assumes a ‘root’ predicate, which she associates with mass meanings insofar as Proot ¼ Pmass . We, too, assume a basic meaning for nominal concepts. Like Krifka (1989), but unlike Rothstein (2010), we take basic predicates to be number-neutral, rather than notionally mass. Moreover, our adoption of frame semantics (inspired by the work of Fillmore [1975, 1976] and others) enables us to enrich representations of basic predicates with encyclopedic information, including background beliefs/knowledge. At the same time, what stands for us in the foreground is the interface between lexical semantics and compositional semantics, and hence being specific about the parts of the lexical entries that facilitate words to participate in compositional processes (see above, and also, for more details on this type of integration, see Cooper 2012). The mass/count distinction, on our view, turns on properties of counting base predicates, namely whether they specify disjoint or overlapping sets. Counting base predicates are derived from basic predicates via mechanisms of individuation which we will describe in detail below. Let us illustrate what we mean by basic predicates with two examples: wolf and rice. Take, for example, an expression such as λx:wolf ðxÞ, which, following Krifka (1989), is a number-neutral predicate. This, and other such predicates, we propose, can be unpacked into a whole frame which highlights whatever properties (perceptual or essential) there are that specify properties of, in this case, wolves. A suggestion for part of such a frame is given in (28). This frame (along with all other common noun frames, we propose) is separated into three main fields: ‘unit’, ‘collection’ and ‘extn’ (a mnemonic for ‘extension’). The unit field specifies a property which includes some background knowledge-based information (e.g. that wolves howl and growl), as well as some mereotopological information (that wolfunits are maximally strongly self-connected (MSSC, Grimm 2012). The ‘collection’ field specifies information relating to any sums of entities specified in the unit field. The presence of this field in basic predicate

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frames means that all basic predicates are number-neutral. For wolf, this field records information about sums of wolf-units, namely that they typically come in social groups (i.e. packs, represented by TYPðsocial unit ð" ðyÞÞ). The extension field states that the wolf concept applies either to individuals or to sums. 33   boundedness ¼ msscðyÞ type ¼ ent 6 77 6 animacy ¼ animateðyÞ 6 77 6 6 77 6   77 6 6 fur ¼ fur of ðy; zÞ 77 6 6 unit fur ¼ λz: ¼ λy: 6 77 6 texture ¼ fluffy ð z Þ 6 77 6 77 6 6 4 sound ¼ ½sound ¼ sound of ðyÞ 2 fhowl; growl, . . .g 5 7 6 7 6 λx:6 7 ... ... ¼ 7 6 2 3 7 6 7 6 pack ¼ ∗ unitðyÞ ^ TYPðsocial unitð" ðyÞÞ 7 6 7 7 6 collection ¼ λy:6 behaviour ¼ territorial ð " ð y Þ Þ 4 5 7 6 7 6 5 4 ... ¼ ... 2

(28)

2

extension ¼ unitðxÞ∨collectionðxÞ

A substance denoting predicate, such as mud, where ‘substance’ is understood in the sense of Soja et al. (1991), would not specify any bounded discrete entities at all. This is specified in the unit field, along with other properties like inanimacy, sliminess (when wet) etc. Object mass nouns which denote artifacts will, despite being mass, contain a ‘unit’ field. However, we suggest that, because they refer to artifacts, rather than being specified by boundedness conditions such as self-connectedness, units be defined in terms of fulfilling at least one of a bundle of functions that relate to the ability of an item to be used for the purpose or purposes specific to that type of artifact (for example, functions pertaining to furnishing for furniture). A similar proposal is made by Grimm and Levin (2017); however, we do not commit to the claim that the specification of these functions must vary cross-linguistically to account for mass/count variation. Crucially, the satisfaction conditions for fulfilling a function will not specify a disjoint set of entities. For example, a vanity fulfills a furnishing function as much as the table, stool and mirror that are its proper parts do. This overspecification of what counts of a functional unit (in the sense of Landman 2011) will feed into our theory of individuation as input. What is notable about granular nouns such as rice, lentil(s) and gravel is that they refer to things that are conceptually individuated objects (i.e. the individual grains like single lentils or bits of gravel), but also typically come in clustered collections. That is to say that most of our interactions with rice, lentils and gravel is with collections of grains etc. Our modest suggestion, which is very much in concord with Grimm’s (2012) proposal, is that such properties have a bearing on individuation. Let us partially ‘unpack’ a predicate such as λx:lentilðxÞ as an example.

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268

(29)

Peter R. Sutton and Hana Filip 33 2 2   boundedness ¼ msscðyÞ 77 6 ent type ¼ animacy 6 ¼ animateðyÞ 77 6 6 77 6 6 unit ¼ lens shapedðyÞ ¼ λy:6 shape 77 6 77 6 6 4 colour 6 ¼ colour  of ðyÞ 2 forange;brown;...g 5 7 7 6 7 ... ¼ ... λx:6 7 6 2 3 7 6 ∗ grains ¼ unit ð y Þ 7 6 7 6 7 6 collection ¼ λy:4 cluster ¼ TYPðcluster ÞðyÞ 5 7 6 5 4 . .. ¼ ... extension ¼ unitðxÞ∨collectionðxÞ

The three root fields specified in this frame are, just as with wolf, labeled ‘unit’, ‘collection’ and ‘extn’ (extension). The unit field specifies a property that refers to single grains (including perceptual and inanimacy information). The collection field specifies a property that refers to sum of grains and includes prototypical (but defeasible) mereotopological information, such as typically coming in clusters (TYPðcluster Þ).7 The extn field places an underspecified condition on the entities denoted by the whole frame that they be either a single grain (specified by the unit field) or sums of grains (as specified by the collection field). This typicality of being clustered, we propose, sets granulars aside from other concrete nouns. Pre-theoretically, it also means that granulars have three ways in which one might conceptualize their extension. One way is as single grains, which, plausibly, is how lentil in English is conceptualized. Another way is as sums of single grains, which, plausibly is how rice in English is conceptualized. In these two cases, the information that grains come in clusters is understood as a matter of how we typically encounter them. The third and final way is as a clustered entity with hard-wired, as opposed to defeasible, mereotopological restrictions (a granular noun which denotes topologically related clusters of grains, and not, for example, any scattered sums of grains), as is the case for grawn (‘grain’, Welsh) (Grimm 2012). We now specify two (families of ) functions, which, together, form the machinery for our account of individuation. These are the object identifying function O and context-dependent individuation schemas S i . 9.4.2

The Object Identifying Function (O)

We have just outlined how predicates, such as λx:wolf ðxÞ, λx:mud ðxÞ and λx:lentilðxÞ, which are all of type he; f i, can be seen as shorthand for frames which encode bundles of perceptual and/or functional properties. The object identifying function O is of type hef ; ef i; it applies to such basic frames and 7

We assume something akin to Grimm’s (2012) account can be applied to spell out the details of predicates like cluster.

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selects the sub-frame labeled ‘unit’. The semantic effect of O is to focus on a sub-lattice of the denotation of the basic frame, namely those entities that count as units. If λx:PðxÞ:he;f i , then: 

(30)

Oðλx:PðxÞÞ ¼

λx:ðPðxÞ:unitÞ ðxÞ λx:PðxÞ

if P contains a “unit” field otherwise

In other words, O is a function that selects, on perceptual or functional grounds, entities that notionally count as ‘one’ for a given predicate. For example, when O applies to a frame for λx:riceðxÞ, OðriceÞ selects those parts of the frame that specify what counts as ‘one’, namely single grains. For cat, Oðcat Þ will be a set of single cats. For substance denoting nouns, such as mud, we assume that O is the identity function Oðmud Þ ¼ mud. This captures the fact that substance denoting nouns lack any entities that, on perceptual or functional grounds, notionally count as individuals. When applied to the frame for lentil in (29) above, OðlentilÞ will return the frame labeled ‘unit’. As we stated above, this should pick out the set of single lentils, namely: (31)

Oðλx:lentilðxÞÞ ¼ λx:ðlentilðxÞ:unitÞ ðxÞ   0 2 3 1 boundedness ¼ msscðyÞ ent type ¼  B 6 7 C animacy ¼ animateðyÞ B 6 7 C B 6 7 C ¼ λx:Bλy:6 shape ¼ lens shapedðyÞ 7ðxÞC B 6 7 C @ 4 colour ¼ colour of ðyÞ 2 forange; brown; . . .g 5 A ...

¼ ...  boundedness ¼ msscðxÞ

 3 ent type ¼ 7 6  animacy ¼ animateðxÞ 7 6 7 6 ¼ λx:6 shape ¼ lens shapedðyÞ 7 7 6 4 colour ¼ colour of ðxÞ 2 forange; brown; . . .g 5 2

...

¼

...

This set will be disjoint (single lentils are non-overlapping), and thus can form a counting base that is compatible with a grammatical counting operation. In other words, O is a restriction on the lentil frame such that it selects the part of the frame that specifies single lentils. If, on the other hand, a lexical entry only specifies λx:lentilðxÞ (i.e. the whole frame in (29)) for the entities in the counting base, this will apply to any sums of lentils, an overlapping set. Of course, what overlaps are subsets of individual lentils of a given set of lentils, not one single lentil with another single lentil. Such a set of overlapping subsets would not be compatible with a grammatical counting operation, and so would be the counting base for a mass granular noun that refers to lentils such as the Czech čočka. The object identifying function O alone is insufficient as the basis of an account of individuation which underpins grammatical counting, and by the

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same token the grammatical mass/count distinction. This is because the output of applying O to a predicate may or may not yield a disjoint set that is a prerequisite for counting. For example, for furniture, the sum entity that is a nest of tables and the individual tables that make up the nest can all count as one unit with respect to furniture, but this means that the set of perceptual and or functional units in its denotation is overlapping (see Landman 2011; Sutton and Filip 2016b, 2018). Likewise, for count nouns such as fence, the entities that can count as one on perceptual and functional grounds also do not form a disjoint set (fencing around a garden can count as one fence or as many) (Rothstein 2010). Disjoint sets are derived by means of individuation schemas, which, intuitively, represent perspectives on predicates that yields what counts as one in context. 9.4.3

Individuation Schemas

Specific Individuation Schemas S i : To take the previous example, suppose that a sum entity that is a nest of tables and the individual tables are all functional units in the extension of furniture (in the extension of OðfurnitureÞÞ: ft 1 , t 2 , t 3 ; t 1 t t 2 t t 3 g⊆ + OðfurnitureÞ), such that for a formula φ in our representation language, + φ is the extension of φ. Individuation schemas S i 2 S, which are of type hef ; ef i, apply to a predicate and return a predicate that specifies a maximally disjoint subset of the extension of the argument predicate (Landman 2011): If

+ P is the extension of P, then + ðS i ðPÞÞ⊆ max :disjoint

+P

Applied to the functional units in the extension of OðfurnitureÞ, in our example above, there will be some S j and S k such that:      ft1 , t2 , t 3 g⊆ + S j ðOðfurnitureÞÞ and DISJOINT + S j ðOðfurnitureÞÞ

ft1 t t 2 t t 3 g⊆ + ðS k ðOðfurnitureÞÞÞ and DISJOINT ð+ ðS k ðOðfurnitureÞÞÞÞ

In other words, different perspectives on furniture lead to different disjoint sets of what is one ‘unit’ of furniture (see also Landman 2011). The types of O and S i are such that they can be stacked on top of each other, whereby ðS i ðOðPÞÞ is of type he; f i. For predicates like cat that denote individuated objects (‘objects’ in the sense of Soja et al. 1991) in a stable way across all contexts, sets such as ðS i ðOðcat ÞÞ will have a stable extension for all S i 2 S. Put in the simplest terms, what we view as one cat will not differ from context to next.8

8

This is, to some extent, an idealisation. In purely extensional terms, there is a certain amount of vagueness, under- or over-specification with respect to what counts as ‘one’, even for predicates

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In contrast, for predicates like mud that lack any ‘objects’ in their denotation, the number of possible maximally disjoint subsets of mud one can form is virtually unrestricted (just think of all of the ways that some mud could be carved up into disjoint clumps). The Null Individuation Schema S 0: The schema is inspired by Landman’s (2011) idea that, for the denotations of mass nouns, both prototypical like mud and object mass nouns like furniture, there is in a given context a multiplicity of partitions (his variants) available, none of which, however, is privileged over others as providing ‘the’ unique individuation schema suitable for counting. What counts as ‘one’ under one individuation schema overlaps with what counts as ‘one’ under another. Put differently, mass nouns have overlapping counting bases, which motivates why they cannot be counted, i.e. used in counting constructions (e.g. #three muds). In order to capture such observations, we introduce the notion of a null individuation schema, S 0 . When applied to a predicate, the null individuation schema returns the union of the interpretations of that predicate at each of the individuation schemas in S. (32)

+ ðS 0 ðPÞÞ ¼ ∪S i 2S + ðS i ðPÞÞ

The schema S 0 is null, because effectively it amounts to an identity function on P. For example, if + P ¼ fa; b; a t bg, then there are two maximally disjoint subsets of + P, and hence there   are two individuation schemas, S j , S k on P such that, for example, + S j ðPÞ ¼ fa; bg and +ðS k ðPÞÞ ¼ fa t bg. Therefore the following holds: +ðS 0 ðPÞÞ ¼ fa; bg∪fa t bg ¼ fa; b; a t bg ¼ +ðPÞ

Figure 9.1

like cat. For example, cat is vague/over-/underspecifies whether a cat and/or that cat minus its tail count as one. This issue relates to the Problem of the Many (Unger 1980), and we will not address it here.

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Figure 9.2

In summary, as Figures 9.1 and 9.2 help to show, applying O to a basic predicate A potentially restricts the extension of A down to the (perceptually or functionally specified) objects in the extension of A, and the application of an individuation schema S i restricts this further if OðAÞ is not disjoint. However, the application of S 0 leaves any overlap in OðAÞ unresolved. 9.4.4

Lexical Entries

We adopt a tripartite structure for lexical entries, following Sutton and Filip (2016b) and Filip and Sutton (2017), and also inspired by some independent suggestions in Landman (2011, 2016). In frame-theoretic terms, this is given as three fields: (i) one labeled ‘baspred’, which gives the basic predicate; (ii) one labeled ‘cbase’, which specifies a predicate for the set of entities that is input into the counting function (and is a function on the baspred frame); and (iii) one labeled ‘extn’, which gives the extension (and is a function on the cbase frame). We give a schema for this in (33). We use (*) to indicate that the upward closure under mereological sum operator is not always present in a lexical entry (it is when the relevant concept denotes pluralities). 2

(33)

3 baspred ¼ λz:PðzÞ λx:4 cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 5 extn ¼ ð∗Þ cbaseðxÞ

The reason this tripartite structure is required is best demonstrated by plurals. For example, for the singular cat, the extension and the counting base, namely, the set of entities that count as one, are identical: the set of single cats. For the

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plural cats, the counting base is still the set of single cats, but the extension is the upward closure of this set under mereological sum. All (‘concrete’ common) count nouns have counting base predicates and extension predicates that are indexed to an individuation schema S i specified by the context of use (see Sutton and Filip 2016b, Filip and Sutton 2017 for justification). This means that under some one particular perspective on individuating the entities involved, there is some disjoint set that is suitable for counting, even if what counts as one may vary from context to context, as in the case of count nouns like fence (see also Rothstein 2010). All count nouns are then of type hhhe; f i; he; f ii; he; f ii, a function from an individuation schema S i of type hhe; f i; he; f ii to a function from entities to a frame (of type he; f i) that specifies the counting base and the extension. Interpreted in context, and so when an individuation schema is specified, however, this reduces to an expression of type he; f i. Mass nouns, on the other hand, have lexical entries that are saturated with the null individuation schema. This means that they have overlapping counting bases, and hence cannot be counted, i.e. straightforwardly used in grammatical counting operations. All (‘concrete’) common mass nouns are of type he; f i, a function from entities to a frame that specifies the counting base and the extension. The ‘slot’ in mass noun entries for individuation schemas is filled with the null individuation schema. Having sketched the most basic assumptions motivating our lexical entries, we will now give examples for prototypical object denoting count nouns (cat) and granular count nouns (lentil), granular mass nouns (čočka ‘lentil’, mass, Czech), substance mass nouns (mud) and ‘collective artifact’ nouns which include mass nouns (furniture) and count-counterparts of object mass nouns (e.g. huonekalut ‘items of furniture’, Finnish). There are two binary features that can be defined to differentiate these classes. First, [+O]/[O]: The cbase field does/does not contain the object identifying function (O). Second, [+S]/[S]: The cbase field contains a specific individuation schema (so, [+S]) contains the null individuation schema (so, [S]). Some of the possible classes, along with natural language exemplars are given in Table 9.1. The generalizations that hold are among the following: Count nouns are all [+O,+S], mass nouns are all [S]. Where mass nouns are [+O], we expect them to share some properties with other [+O] nouns such as the availability of cardinality comparisons in comparative constructions. (However, see Rothstein, Chapter 8 in this volume, for an in-depth discussion.) For example, if these classifications are right, then we should expect comparative constructions containing both furniture and fencing ([+O,S])-like nouns to have a cardinality comparison reading available (as well as a measure reading), but for [O,S] mass nouns like rice and mud to only have a measure reading straightforwardly available in such constructions.

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Table 9.1 Summary of the semantic categorisation of noun classes Class

Count/Mass

Example

Categorization

Prototypical object Granular

Count Count Mass Mass Count Mass

cat lentil rice furniture huonkalua mud

[+O,+S] [+O,+S] [O,S] [+O,S] [+O,+S] [O,S]

Collective artifact Substance denoting a

‘(item of ) furniture’, Finnish

Prototypical Object Nouns Are [+O,+S] (cat): Nouns denoting prototypical objects, such as cat, chair or house in English, are lexicalized as count nouns, in number marking languages at least. The entry for cat in (34) has a cbase (counting base) field that specifies a predicate, λx:cat ðxÞ, which is shorthand for a basic predicate expression of type he; f i (a function from entities to a frame that specifies, inter alia, perceptual properties and background knowledge about cats). The object identifying function O applies to this predicate and returns the set of individual cats. Since the set of single cats is always disjoint, the set of single cats under any schema of individuation, S i , will be disjoint. This means that our account correctly predicts that nouns like cat will be lexicalized as count nouns, since the counting base set is disjoint across all individuation schemas. The extension of cat is the same set as the counting base set: the set of single cats under S i .9 2

(34)

baspred 〚cat〛S i ¼ λx:4 cbase extn

3 ¼ λz:cat ðzÞ ¼ λy:S i ðOðbaspredÞÞðyÞ 5 ¼ cbaseðxÞ

The counting base for the plural noun cats, as shown in (35) is the same as for cat: the set of single cats under S i . The extension of cats is this set closed under mereological sum: the set of single cats under S i and sums thereof.

9

It is worth bearing in mind that if labels are replaced by the formulae they label, then the expression in (34) is equivalent to the expression in (i). Furthermore, both (34) and (i) are extensionally equivalent to the predicate logic formula in (ii). 2 3 baspred ¼ λz:cat ðzÞ (i) λx:4 cbase ¼ λy:S i ðOðcatÞÞðyÞ 5 extn ¼ S i ðOðcatÞÞðxÞ (ii)

λx:S i ðOðcatÞÞðxÞ

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2

(35)

3 baspred ¼ λz:cat ðzÞ 〚cats〛 ¼ λx:4 cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 5 extn ¼ ∗ cbaseðxÞ Si

Count Granular Nouns Are [+O,+S] (lentil): The semantics for a count granular noun such as lentil looks very similar to the semantics for nouns like cat. As outlined above, the object identifying function O applies to the base predicate λx:lentilðxÞ and returns the set of single lentils, a disjoint set. This means that, ‘viewed’ under any schema of individuation, this set will still be disjoint The extension for lentil is the set of single lentils as shown in (36). The extension for lentils is the set of single lentils closed under sum (37). 2

(36)

3 baspred ¼ λz:lentilðzÞ 4 〚lentil〛 ¼ λx: cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 5 extn ¼ cbaseðxÞ

(37)

3 baspred ¼ λz:lentilðzÞ 4 〚lentils〛 ¼ λx: cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 5 extn ¼ ∗ cbaseðxÞ

Si

2

Si

Mass Granular Nouns Are [O,S] (čočka [‘lentil’, Czech], rice): Since the semantics for a count granular noun such as lentil looks very similar to the semantics for nouns like cat, this prompts the question why we should expect there to be any mass granular nouns such as čočka (‘lentil’, Czech) or rice. Part of our answer lies in the differences there are between frames represented by base predicates such as λx:lentilðxÞ compared with frames represented by base predicates such as λx:cat ðxÞ (see (28) and (29) above). The frames for base predicates detail perceptual and functional properties, but also mereotopological information, and background and experiential knowledge. For predicates such as λx:lentilðxÞ, this will include the fact that we most frequently encounter granular entities in either aggregated or clustered form. Their referents are formed of grains, but often clustered together such that we cannot even clearly perceive each and every granular entity (see the frame for λx:lentilðxÞ in (29)). This is not the case for λx:cat ðxÞ. As a matter of contingent fact, we usually experience cats as single entities, clearly separated from other cats (even when they are in groups). These contingent facts allow us to more easily ‘look past’ the granular structure of granular entities and conceptualize entities, such as lentils, as aggregates of grains such that these aggregates lack clear, bounded edges. This is the case for the English rice. We clearly know, on the conceptual level, that rice is made up of grains, but we nonetheless do not grammatically individuate rice. The same is also true for mass counterparts of lentils such as the Czech čočka (‘lentil’, mass), an entry for which is given in (38). We represent this mismatch between the conceptual level and the level accessible to grammatical counting operations in terms of a distinction

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between the fields in our lexical entries. The conceptual level is represented by the base predicate λx:lentilðxÞ. The level accessible to grammatical counting is provided in the ‘cbase’ field. The counting base predicate for čočka (‘lentil’, mass, Czech), S 0 ðlentilÞ, is equivalent to the frame in (29) under the null individuation schema S 0 . It is this predicate that is the input to grammatical counting operations (see Appendix A); however, this predicate, in the case of mass (granular) nouns, specifies an overlapping set of aggregates of lentils. 2

(38)

3 baspred ¼ λz:lentilðzÞ 〚cocka〛S i ¼ λx:4 cbase ¼ λy:S 0 ðbaspredÞðyÞ 5 extn ¼ ∗ cbaseðxÞ

Thus, we retain the information that nouns like rice and čočka denote grains, but also how this individuated structure does not get passed up to the level accessible to grammatical counting. Substance Denoting Nouns Are [O,S] (mud): Substance denoting nouns do not denote objects in the sense of Soja et al. (1991), because there is nothing in their denotation that can be identified reliably as an individual object. Like mass granulars, the lexical entries for nouns in this class do not include the object identifying function O,. The difference between a noun like mud and a noun like rice is that the former does not refer to objects, even on the notional/conceptual level. However, because substance denoting nouns are [O,S] on our analysis, this means that they pattern with mass granular nouns, such as rice and čočka (‘lentil’, mass, Czech), on the GRAMMATICAL level. Both mass granular nouns and substance denoting nouns are both bona fide mass nouns, governed by [S], and neither admits of cardinality comparisons in comparative constructions (see Landman, Chapter 6 this volume), which is governed by [O]. The cbase field for mud is S 0 ðmud Þ. The predicate mud stands for a frame that specifies the perceptual properties of mud (plus some background knowledge, among others). This frame applies to anything with those qualities. The null individuation schema S 0 applies to mud to form a predicate that applies to anything which is mud under any schema of individuation (under any way of dividing mud up into disjoint subsets). The set specified by S 0 ðmud Þ is therefore overlapping, and so mud is mass. The extension of mud is the upward closure of this set under sum.10

10

However, since the way in which something like mud can be divided up into disjoint subsets is totally unconstrained by the O function, ∗ S 0 ðmudÞ will be coextensional with S 0 ðmud Þ. This is because any entity that counts as mud could be an entity in a disjoint subset of all mud, therefore the union of all disjoint subsets of mud (∗ S 0 ðmud Þ) will be equivalent to the lattice denoted by mud.

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The Count/Mass Distinction for Granular Nouns 2

(39)

baspred 〚mud〛S i ¼ λx:4 cbase extn

¼ ¼ ¼

277

3 λz:mud ðzÞ λy:S 0 ðbaspredÞðyÞ 5 ∗ cbaseðxÞ

Collective Artifact-Denoting Nouns Are [+O,+S] or [+O,S]: Like granulars, the members of this notional class of nouns display widespread variation in their count/mass lexicalization patterns. For example, the English mass noun furniture has a count-counterpart in Finnish huonekalu(t) (‘piece(s) of furniture’, Finnish). Unlike mass granulars, mass collective artifacts have some grammatical properties in common with count nouns insofar as they admit of cardinaility comparisons in ‘more than’ constructions. We represent this via the inclusion of the object identifying function O in the lexical entries of both count and mass collective artifact nouns. We analyse mass/count variation for this notional class of nous in terms of whether the counting base is interpreted relative to a specific individuation schema S i . As we outlined above, the ‘unit’ field for the furniture concept specifies an overlapping set of any entities that fulfill the functional role of pieces of furniture. Hence OðfurnitureÞ will be an overlapping set. This means that under the perspective of an individuation schema S i , we get a disjoint set (although one whose members may vary with context). Under the null individuation schema S 0 , we get an overlapping set, one which is not fit for counting. Lexical entries of count collective artifact nouns can therefore be described in terms of having the features [+O,+S], and mass collective artifact nouns can be described in terms of having the features [+O,S]. Mass collective artifact nouns (object mass nouns) are mass (and so [S]), but do allow for cardinality comparissons in comparative constructions (and so are [+O]). Thus, for collective artifact nouns, cross- and intralinguitic variation can be accounted for purely via whether the noun encodes an argument for the individuation schema that is salient in the context of utterance (as in (41)), or whether it is saturated with the null individuation schema S 0 (as in (40)). Hence, we can account for the mass noun furniture, which has an overlapping counting base, and the plural count noun huonekalut (‘items of furniture’, Finnish), which, under every individuation schema of utterance, species a disjoint counting base. 2

(40)

〚furniture〛S i

¼

baspred λx:4 cbase extn

3 ¼ λz:furnitureðzÞ ¼ λy:S o ðOðbaspredÞÞðyÞ 5 ¼ ∗ cbaseðxÞ

2

(41)

3 baspred ¼ λz:furnitureðzÞ 〚huonekalut〛S i ¼ λx:4 cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 5 extn ¼ ∗ cbaseðxÞ

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This means that the plural count noun huonekalut is semantically related to the mass noun furniture as the substitution of the null individuation schema with the individuation schema specified by the context 〚 ( huonekalut〛S i ¼〚furniture〛S 0 ↦S i ). 9.5

Semantics for Classifier-Like Expressions and Addressing the Accessibility Puzzle

We will now turn to our main accessibility puzzle of why coerced container readings, but not unit readings, are available for mass granular nouns. To this end, we will first give an analysis of container readings and unit readings in measure (pseudo-partitive) constructions with explicit container and unit extracting classifier-like expressions. 9.5.1

Container Classifiers

As pointed out by, among others, Partee and Borschev (2012), Khrizman et al. (2015) and Landman (2016), there at least four different interpretations of measure constructions, such as two bowls of rice, which are formed with nouns like bowl whose inherent meaning is sortal, namely that of a physical receptacle. Inherently sortal nouns like bowl may assume (at least) four relational (classifier-like) meanings when they are used in the measure construction, which we label as follows: (i) a container, (ii) contents, (iii) (free) portion and (iv) measure interpretation. Here we focus on the container reading, that is, two bowls, each of which contains rice. We leave aside measure interpretations on which two bowls of rice, for instance, has the mass interpretation of ‘rice to the measure of two bowlsful’. Measure interpretations arguably have a different syntactic and semantic structure than container interpretations: The receptacle noun (bowl) is interpreted as a measure function which combines with a numeral to form a measure phrase (three bowls of) (Rothstein 2011). On our account, receptacle nouns (bowl) in their relational (classifier-like) interpretation that concerns container readings are interpreted as functions from expressions of type he; f i to expressions of type he; f i. This means that, for example, bowls of rice forms a constituent that is sanctioned in a counting construction such as three bowls of rice. This analysis of container readings of pseudo-partitive NPs is in line with Rothstein (2011), Partee and Borschev (2012) and Khrizman et al. (2015), among others. The entry for the plural sortal noun bowls is given in (42).11 We assume a function REL that shifts sortal, receptacle nouns into relational container 11

We assume that this differs from the entry for bowl only insofar as it has an extension which is the upward closure of the set of single bowls indicated by * applied to the formular in the extn field.

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classifiers. Building on some ideas in Rothstein (2011), this function is given in (43). The entry for the container reading of bowl(s) is given in (44). 2

(42)

〚bowls〛S i ¼

(43)

〚REL〛¼

baspred λx:4 cbase extn 2

baspred 6 cbase 6 6 extn λQ:he;f i :λP:he;f i : λx:6 6 extn restr 4 precon

(44)

3 ¼ λz:bowlðzÞ ¼ λy:S i ðOðbaspredÞÞðyÞ 5 ¼ ∗ cbaseðxÞ

¼ ¼ ¼ ¼ ¼

QðxÞ:baspred QðxÞ:cbase QðxÞ:extn 8z:½cbaseðzÞ ^ z v x ! 9v:½PðvÞ:extn ^ containðz; vÞ CUMðλy:P:extnðyÞÞ

Si 〚bowls of〛¼〚REL〛〚bowls〛 ð Þ¼ 2 baspred ¼ λz:bowlðzÞ 6 cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 6 ∗ 6 extn cbaseðxÞ ¼ λQ:he;f i :λP:he;f i : λx:6 6 extn restr ¼ 8z:½cbaseðzÞ ^ z v x 6 4 ! 9v:½PðvÞ:extn ^ containðz; vÞ ¼ CUMðλy:PðyÞ:extnÞ precon

3 7 7 7 7 7 5

3 7 7 7 7 7 7 5

The lexical entry in (44) takes as an argument P an expression of type he; f i, such as〚apples〛S i or〚rice〛. It returns an expression of type he; f i which has a counting base predicate that specifies the set of single bowls under any schema S i . Its extension is this set closed under mereological sum with a restriction that each of the single bowls contains something in the extension of P. Finally, precon ¼ CUMðλy:P:extnðyÞÞ captures the condition, introduced by REL, on the combinatorial properties of container classifiers that nominal terms to which they are applied have a cumulative denotation (Krifka 1998): (45)

CUMðPÞ $ 8x8y½ðPðxÞ ^ PðyÞÞ ! Pðx t yÞ

This condition is straightforwardly satisfied by bare mass and plural terms (a bowl of rice/apples), but not by singular count terms (#a bowl of (an) apple)). The result of combining the function in (44) with〚apples〛S i is given in (46) (with labels replaced with full formulas to aid readablity). (46)

Si ð Þ¼ 〚bowls of apples〛¼〚bowls of〛〚apples〛 2 baspred ¼ λz:bowlðzÞ 6 cbase ¼ λy:S i ðOðbowlÞÞðyÞ 6 ∗ 6 extn S i ðOðbowlÞÞðxÞ ¼ λx:6 6 extn restr ¼ 8z:½S i ðOðbowlÞÞðzÞ ^ z v x 6 4 ! 9v:½∗ S i ðOðappleÞÞðvÞ ^ containðz; vÞ ¼ precon CUMðλy∗ S i ðOðappleÞÞðyÞÞ

3 7 7 7 7 7 7 5

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This yields a set of entities that are single bowls or sums thereof, each of which contains apples such that counting proceeds in terms of how many such bowls there are. The expression in (46) is extensionally equivalent to the predicate logic formula in (47): (47)

λx:½ ∗ S i ðOðbowlÞÞðxÞ ^ 8z:½S i ðOðbowlÞÞðxÞ ^ z v x ! 9v:½∗ S i ðOðappleÞÞðvÞ ^ containðz; vÞ ^ CUMðλy∗ S i ðOðappleÞÞðyÞÞ

The result of combining the function in (44) with rice is given in (48) (with labels replaced with full formulas). (48)

〚bowls of rice〛 ¼ 〚bowls of ðriceÞ〛 ¼ 2 baspred ¼ λz:bowlðzÞ ¼ λy:S i ðOðbowlÞÞðyÞ 6 cbase 6 ∗ S i ðOðbowlÞÞðxÞ 6 extn ¼ λx:6 6 extn restr ¼ 8z:½S i ðOðbowlÞÞ ^ z v x 4 ! 9v:½∗ S 0 ðOðriceÞÞðvÞ ^ containðz; vÞ ¼ precon CUMðλy∗ S 0 ðOðriceÞÞðyÞÞ

3 7 7 7 7 7 5

This yields a set of entities that are single bowls or sums thereof, each of which contains rice such that counting proceeds in terms of how many such bowls there are. 9.5.2

Unit Extracting Classifiers

The intuitive idea that underlies the semantics of unit extracting classifiers, such as grain of, is that they ‘zoom in’ on and make accessible the units that are inherent in a granular mass noun’s denotation, that is, more precisely, units specified in its basic predicate frame. The result is something extensionally equivalent to a count granular expression. Formally, this is achieved by two elements encoded by unit extractors, such as grain of: the object identifying O function which applies to the basic predicate frame and identifies any entities that can count as single grains; and the individuation schema of utterance S i which may select a subset of these entities as those individuated in the context.12 The lexical entry for the unit extracting classifier grain of is given in (49). It applies to an expression P of type he; f i such as rice and returns a set of entities or sums thereof that are identified as objects in the extension of P (via the object identifying function O) that count as individuated under the individuation schema of utterance S i . The set of single entities are specified as the counting base. 12

For example, in relatively rare cases where two grains have grown such that they intermingle with one another, the individuation schema will determine whether it counts as one or two grains.

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2

(49)

3 baspred ¼ PðxÞ:baspred 〚grains of〛¼ λP:λs:λx:4 cbase ¼ λy:sðOðPðxÞ:baspredÞÞðyÞ 5 extn ¼ ∗ cbaseðxÞ

The result of applying rice to grain of is given in (50). This yields the set of single grains of rice and sums thereof (under schema S i ) such that the single grains of rice (under schema S i ) count as one. 2

(50)

9.5.3

Si

〚grains of rice〛

3 baspred ¼ λz:riceðzÞ 4 ¼ λs:λx: cbase ¼ λy:sðOðbaspredÞÞðyÞ 5 ðS i Þ extn ¼ ∗ cbaseðxÞ 2 3 baspred ¼ λz:riceðzÞ 4 ¼ λx: cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 5 extn ¼ ∗ cbaseðxÞ

Addressing the Accessibility Puzzle

Now that we have given an account of the semantics of granular nouns (Section 9.4) and of container and unit extracting expressions (Section 9.5.2), we are in a position to address the accessibility puzzle. Recall that what we dub the accessibility puzzle is as follows, repeated here for convenience: Why should conceptually and perceptually salient object units in the denotations of object mass nouns and granular mass nouns not be directly accessible by semantic counting operations, nor facilitate the mass-to-count coercion? Specifically, when it comes to granular mass nouns the twin data to be explained are: Implicit unit extracting classifiers: Counting constructions, such as three rices, cannot be coerced into unit interpretations, such as three GRAINS OF rice, whereby the relevant units for counting are inherent in the meaning of rice. Implicit container classifiers: Counting constructions, such as three rices, can be coerced into portion readings, e.g. three BOWLS OF rice, whereby the relevant portions are recovered from context. The account of individuation we have presented here has two key mechanisms that work in unison with each other: the object identifying function O, which identifies objects which are possible candidates for being individuated, and the schema of individuation S i , which selects some subset of these as entities to be counted in the context (see Figure 9.1). We propose to derive the restrictions on coercion by using the semantics of of pseudo-partitive constructions (two bowls of rice) and unit extracting

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constructions (two grains of rice) given above as a window on coercion. In particular, we propose that a difference in the semantics of relational (classifier-like) concepts, such as ⟦bowl of⟧, and unit extracting concepts, such as ⟦grain of⟧, can be used to explain why the former can be used as an implicitly provided functor to repair a type clash, whereas the latter cannot. Coerced mass-to-count shifts require the addition of the container concept retrieved from the context which supplies the unit for counting (both the object identification function and the individuation schema in our terms). These kinds of mass-to-count shifts do not exploit any discrete units which are inherent in the lexical structure of nouns, and a part of their core meaning. So when counting bowls of rice, as in three bowls of rice under its container interpretation, the individuation criterion for counting comes from the contextually determined concept ⟦bowl⟧ that is supplied externally to the noun rice in order to resolve the type clash triggered by two when it directly combines with rices and so restores compositionality. In contrast, when counting grains of rice, the semantics of expressions such as grain of is to make available to the grammar those entities that we know of as natural grain-units as part of our knowledge of rice. On our account, this is done via the introduction of the object identifying function O and the schema of individuation S i into the cbase field of the ⟦rice⟧ frame. This creates a predicate that specifies a disjoint counting base. On the level of the grammar, this has the effect that unit extracting classifiers encode a shifting operation from mass to count: ½CL grain of  ½N rice ) ½N grain of ricecount . In summary, when counting bowls of rice, individuation turns on individuating bowls and their contents. When counting grains of rice, individuation turns on making entities (the grains) that are anyway specified as part of the rice concept available to the grammatical counting operation. With this distinction in hand, we may now precisely say how it has an impact on restricting mass-to-count coercion. 9.5.4

Implicit Unit Extracting Classifiers

Explicit unit extracting expressions, on our analysis, operate by introducing the object unit function and a schema of individuation into the counting base field of the argument mass noun such that the mass noun concept shifts from lacking disjoint individuation criteria (being mass) to having disjoint individuation criteria (being count). Implicit unit-extractor concepts would have to perform the same task. In other words, there would have to be some implicit concept available whose sole task would be to shift mass nouns into count nouns, i.e. some general function that shifts [O,S] concepts to [+O,+S] concepts (i.e. the free insertion of the tools for individuation (O and S i ) into

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any mass concept). Crucially, this would apply not only to mass granular nouns like rice but also substance denoting nouns like mud. It is this general function that shifts [O,S] concepts to [+O,+S] concepts that, we propose, is blocked as a coercion mechanism for any language with a grammaticized lexical mass/count distinction. Here is why. Suppose that a language L has lexicalized mass/count distinction. Further suppose that a mass-to-count unit shift was available in L for mass granular nouns. This mass-to-count unit shift, we have argued, would be equivalent to a generalized mass-to-count shift (that could apply to any mass noun in a suitable context). Hence, L would have available a generalized mass-tocount shift. However, the availability of a generalized mass-to-count shifting operation is incompatible with L being a language with a lexicalized mass/count distinction (since any mass noun could be used as a count noun given some salient, disjoint set of entities in the context). Hence, it cannot be the case that L has a mass-to-count unit shift for mass granular nouns if L is a language with a lexicalized mass/count distinction. In other words, a granular unit-shifting function is equivalent to a general function that shifts [O,S] concepts to [+O,+S] concepts, and licensing this shift in a language is incompatible with a lexicalized mass/count distinction in that language. Indeed, arguably, Yudja is a language which has such a function and also lacks a lexicalized mass/count distinction (see Lima 2014 for arguments for the latter claim.) 9.5.5

Implicit Container Classifiers

Explicit container classifiers, such as bowl, contribute the individuation criteria of the receptacle concept (e.g. λx:S i ðOðbowlÞÞðxÞ) and further require that the receptacle contains things/stuff that are in the extension of the argument nominal concept. That means that, unlike explicit unit extracting classifiers (grain of), container classifiers do not simply encode a function that inserts the object unit function O and the relevant contextual schema of individuation S i into the frame for the common noun to which they apply. The fact that unit extracting and container classifier concepts work in these distinct ways has an impact on whether or not they can be retrieved from the context and used to repair a type clash between, for example, a count quantifier or numerical expression and a mass noun. We have just argued that languages with a grammaticized, lexical mass/count distinction cannot have a mass-tocount unit-shifting operation that shifts granular mass noun concepts into count noun concepts by making available the natural granular units to the grammatical counting operation. This means that no implicitly provided concept like 〚grain of〛S i can be used to resolve a type clash between, for example, ⟦three⟧ and ⟦rice(s)⟧. Crucially, our argument for this turned on the fact that a granular

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unit-shifting operation would be equivalent to a generalized mass-to-count shifting operation. The story with implicitly provided container classifier concepts is different. If, for example,〚bowl of〛S i is salient as a container concept in the context, it can be used to resolve a type clash between, for example, ⟦three⟧ and ⟦rice(s)⟧, since the 〚CL bowl of〛S i shift is not equivalent to a generalized shifting operation from mass noun concepts to the equivalent count noun concept. This is for at least two reasons: (i)〚CL bowl of〛S i adds a concept with its own individuation criteria (λx:S i ðOðbowlÞÞðxÞ) and so does not modify the individuation criteria of〚rice〛; (ii)〚CL bowl of〛S i is not primarily a mass-tocount shifting operation – it applies to concepts which have cumulative extensions and this includes plural count concepts as well as mass concepts. Since container classifier concepts such as〚bowl of〛S i are not equivalent to a generalized shifting operation from mass noun concepts to the relevant count noun concept, there is no reason why they should not be, modulo context, employed to resolve a count/mass type clash. In other words, if suitable, salient-in-the-context receptacle concepts are available, expressions like three rices can be used to mean things like three bowls of rice without amounting to licensing a shift that is incompatible with the relevant language having a lexicalized grammatical mass/count distinction. 9.5.6

The Importance of Granular Nouns in Mass/Count Theories

If our analysis is on the right track, then, intriguingly, it opens up the possibility of treating granular nouns and explicit unit extracting classifiers as testing grounds for the mass/count distinction in a way similar to the role assigned by Chierchia (2010) to object mass nouns: What makes fake mass nouns interesting is that they constitute a fairly recurrent type of non-canonical mass nouns, and yet they are subject to micro-variation among closely related languages. For all we know, the phenomenon of fake mass appears to be restricted to number marking languages. It is unclear that classifier languages like Mandarin and number-neutral languages like Dëne display a class of cognitively count nouns with the morphosyntax of mass nouns. In view of this intricate behaviour, fake mass nouns arguably constitute a good testing ground for theories of the mass/ count distinction. (Chierchia 2010, p. 111)

First notice that we can replace ‘fake mass’ in the above quote with ‘granular mass’ and make a parallel point. Extrapolating further, we can put forward a hypothesis regarding the available mass-to-count shifts in particular types of languages: If a language has a grammaticized lexical mass/count distinction as is typical with number marking languages such as English, German and Finnish, and has expressions equivalent to grain (of ), it cannot license implicit mass-to-count unit-shifting operations.

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Classifier languages have been argued to reflect only the object/substance distinction in their lexical nominal system (with countability reflected in the classifier system). On the assumption that the semantics of individuating classifiers is to make uncountable concepts (such as kinds) into countable predicates, classifier languages also cannot permit coerced mass-to-count unit shifts since this, too, given our analysis, would be equivalent to a generalized mass-to-count shifting operation. If a language has a relatively impoverished classifier (or classifier-like) system, and generally lacks other reflexes of a lexicalized mass/count distinction (such as number marking and specialized mass quantifiers), then there is no in-principle reason why there should not be a generally licensed means of grammatically counting the grains in the denotations of granular nouns. However, given this, not only would such a language lack mass granular nouns, it should also license the grammatical counting of all nouns (including substance nouns, given a suitable context). One language that potentially meets these criteria is Yudja. Yudja has only a few classifier-like expressions that are highly restricted in their distribution (Lima 2019), and all notionally mass nouns are countable in Yudja modulo a suitable context (Lima 2014). 9.6

Conclusions and Comparisons

We have provided a detailed account of the lexical semantics of granular (mass/count) nouns couched within a broader theory of the semantics of lexical nouns and the mass/count distinction. We have also given analyses of measure (pseudo-partitive) constructions formed with what are inherently sortal receptacle Ns, such as bowl, under their container (classifier-like) interpretation and with unit extracting classifier expressions like grain of. We have defended an account of countability based on the interaction of three key ingredients: (i) more detailed lexical semantic representations of basic predicates than has been proposed by previous algebraic, mereo(topo) logical analyses of the mass/count distinction so far (which required a more expressive representational format, namely frames); (ii) the object identifying function O and (iii) schemas of individuation S i 2 S in particular contexts and the null individuation schema S 0 . For example, we can account for why part of the concept for mass granulars, such as rice, is that they come in grains while still not being accessible to grammatical counting operations. Although part of the basic predicates for nouns like rice specify a denotation made up of single grains, these single grains are not uniquely specified by the part of the lexicon that is accessed by grammatical counting operations; in our terms, they are not specified in the cbase field of lexical entries for granular nouns. Arguably, our analysis is an improvement over Chierchia’s (2010) theory in which, essentially, the theoretical functions of O and S are merged into one

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supervaluationist theory, the result of which, we argued, was to lose the ability to maintain a conceptually privileged place for the granular nature of granular nouns. This is because, according to Chierchia (2010), mass granular nouns have unstable entities at the ‘bottom’ of their denotations, just like (mass) substance nouns do. In contrast, our frame-based representation can record the fact that mass granular nouns refer to stuff made up of grains, while also encoding why these entities are not available to grammatical counting operations. One open question is whether our answer to the accessibility puzzle could, in principle, be adopted by (modifying) other theories of the mass/count distinction. The crucial ingredients such a theory would need to have are: (i) a distinction between the semantics of container classifier-like expressions such as bowl of and unit extracting expressions such as grain of such that the latter function to make the units inherent in our general knowledge pertaining to a nominal concept available for counting. Without this feature, one cannot use our explanation for why three rices can mean ‘three bowls of rice’, but not ‘three grains of rice’. (ii) a distinction between a function that determines the set of possible entities for counting (our O), and one that identifies the subset of those for counting in a particular context (our S i ). This distinction enables one to explain why mass granular nouns pattern grammatically with substance mass nouns but many languages also have count granular nouns. Taken together, (i) and (ii) also facilitate an explanation of the semantics of unit extracting expressions. For example, if a theory were to encode only the equivalent of/an alternative for our S i and lack the equivalent of our O (as many other context-sensitive theories of the mass/count distinction seem to do, among them Rothstein 2010 and Chierchia 2010), and that theory were to abide by condition (i), then there would be no principled way to restrict the equivalent of S i in that theory to access only the grains in granular nouns, since with only the equivalent of S i and with no equivalent of O sums of grains would, in principle, be accessible to grammatical counting.

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Appendix A

Counting Constructions

Numerical expressions, when used adjectivally as predicate modifiers, are functions of type hef ; ef i on our account. Applied to the entry for a count noun such as〚cats〛S i , they return a function from entities to frames that has the same cbase and extn fields as the property that is the argument of the type hef ; ef i function, but adds a restriction to the extension such that it is a set of sum entities that have a cardinality of n with respect to the counting base property. The lexical entry for three is given in (A1), and the result of composing this with〚cats〛S i is given in (A2): 2

(A1)

(A2)

baspred 6 cbase 〚three〛¼ λP:he;f i :λx:4 extn extn restr

¼ ¼ ¼ ¼

3 PðxÞ:baspred PðxÞ:cbase 7 5 PðxÞ:extn μcard ðx; cbase; 3Þ

〚three cats〛S i ¼ 〚three〛ð〚cats〛S i Þ ¼ 3 2 baspred ¼ λz:cat ðzÞ 6 cbase ¼ λy:S i ðOðbaspredÞÞðyÞ 7 7 λx:6 5 4 extn ¼ ∗ cbaseðxÞ extn restr ¼ μcard ðx; cbase; 3Þ

We assume, following Landman (2011, 2016), that the cardinality function μcard is defined only for properties that specify disjoint sets of individuals. This is what blocks counting of mass nouns which, by assumption, have ovelapping (non-disjoint) counting base sets. An extensionally equivalent proposition to (A2) in predicate logic is given in (A3). (A3)

λx:∗ S i ðOðcat ÞÞðxÞ ^ μcard ðx; λy:S i ðOðcat ÞÞðyÞ; 3Þ

In words, this is the set of sums of cat units under individuation schema S i that have a cardinality of 3 with respect to the predicate for single cat units under individuation schema S i .

287

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Appendix B

Composition with Verb Frames

Given our proposal of a frame-based semantics for NPs including numerical NPs, we should provide some indication of how our NP semantics could combine with a VP. There are, inevitably, many issues that we will not even begin to address in this paper (cumulativity, distributivity and quantifier scope, to name but a few), and it is only within the scope of this paper to provide an outline of how such compositional mechanisms could work. We will adopt a fairly traditional neo-Davidsonian approach with some insights from Partee (1986). Intransitive VPs such as play will be of type he; hv; f ii as opposed to the standard he; hv; t ii (with v as the type for eventualities). We also adopt a version of Partee’s (1986) A shift, which was originally proposed as type hhet i; hhet i; t ii (i.e. shifting a predicate to a GQ). Here, since we assume a frame-based neo-Davidsonian semantics, our A shift is hhef i; hhe; hv; f ii; f ii. Let us take three cats play as an example. The frame semantics for play will just be a frame-based interpretation of a standard neo-Davidsonian representation: 

(B1)

extn 〚play〛¼ λx:λe: agent

¼ ¼

playðeÞ agent ðe; xÞ



In oder to compose with an expression of type he; f i such as〚three cats〛S i in (A2) above, we assume a type shifting operation that converts a predicate NP into a GQ. We call this Af for the frame-based version of Partee’s A shift, the semantics for which we give in (B2). 2

(B2)

extn Af ¼ λP:he;f i :λV :he;hv;f ii :λe:9x:4 agent agent restr

3 ¼ V ðeÞðxÞ:extn ¼ V ðeÞðxÞ:agent 5 ¼ PðxÞ

288

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289

This gives us the ‘vanilla’ derivation for three cats play: (B3)

〚three cats play〛S i ¼ 〚Af 〛〚three ð cats〛S i Þ〚play〛 ð Þ 3 2 extn ¼ V ðeÞðxÞ:extn 7 6 agent ¼ V 6 37 2 ðeÞðxÞ:agent 7 6 baspred ¼ λz:catðzÞ 7 ð 6 Þ ¼ λV:λe:9x: 6 7 7 〚play〛 6 cbase ð O ð baspred Þ Þ ð y Þ ¼ λy:S i 7 6 7 6 agent restr ¼ ∗ 55 4 extn 4 ¼ cbaseðxÞ extn restr ¼ μcard ðx;cbase; 3Þ 3 2 extn ¼ playðeÞ 7 6 agent ¼ agent ð e; x Þ 37 2 6 7 6 baspred ¼ λz:cat ð z Þ 7 ¼ λe:9x:6 7 6 cbase 7 6 ð O ð baspred Þ Þ ð y Þ ¼ λy:S i 77 6 agent restr ¼ 6 55 4 extn 4 ¼ ∗ cbaseðxÞ extn restr ¼ μcard ðx;cbase; 3Þ

Existential closure can then yield an expression of type f which yields an expression extensionally equivalent to the event semantics formula in (B4): (B4)

9e:9x:½playðeÞ^agent ðe;xÞ^ ∗ S i ðOðcat ÞÞðxÞ^μcard ðx;λy:S i ðOðcat ÞÞðyÞ;3Þ

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Index

addition, 117, 145, 156 adjective absolute qualitative, 18 absolute superlative, 17–18, 23 numeral adjective, 234 quantitative, 38 superlative, 14, 23 superlative quantitative, 18, 20–21, 26–27 adverb aspectual, 199–200, 202–210, 212–214, 216 indexical, 200–201 quantificational, 213 temporal, 152–153, 200, 205, 215 aggregate artifactual, 87 derived, 93, 112 al (Dutch), 212–213 already, 9, 12, 200–202, 204–207, 209–210 American Sign Language, 158 approximation, 218, 230, 234–235, 237–240, 244 approximative inversion, 240 atom, 4–7, 57–58, 60, 66, 79, 87, 103, 112–113, 121, 125–126, 129, 138, 140, 163–166, 171, 226, 228, 231–233, 237, 246, 255, 259 atomic, 1–7, 57–58, 60, 64, 71, 76, 86–87, 92, 103, 111, 121, 125–129, 138–139, 152, 154, 161, 163, 165–167, 169, 217–219, 222–226, 228, 231–234, 236–238, 244–246, 255 atomicity, 1, 4, 7, 57, 126, 147, 161–163, 165, 171 atomless, 4, 125, 166 k-atom, 226, 236 semantic, 8, 225, 234 bare singular, 90, 104, 106, 108, 117, 221, 246 base, 9, 161, 164–174, 176–182, 184–187, 191–194, 196, 237

counting, 75, 260–261, 265–266, 269, 271–274, 276–277, 279–280, 282, 285, 287 basic predicate, 258, 265–266, 272, 274, 280, 285 beaucoup (French), 15, 26–27 body (and base), 9, 165, 168–171, 173–174, 176–185, 191–193 Boolean algebra, 6, 162, 165, 219, 234, 237 semantics, 161 boundary of an entity, 113, 127 bounded, 43, 45, 146–147, 152–154, 159, 267, 275 Brazilian Portuguese, 77–78, 221, 246, 248 Bulgarian, 19 Cantonese, 9, 11, 78, 145–150, 152, 154–159 cardinal expression, 86 cardinality comparison, 10, 43, 186, 188, 219, 222–224, 231, 233–234, 237, 246–247, 253, 266, 273, 276 reading, 15, 40, 42, 44, 46, 231 scales, 10, 44, 231–232, 234, 236 Caribbean Creole, 156 cei mai mult, i, 16, 22, 25–27 Chol, 55 classifiers, 9, 11, 52, 55, 59, 62–71, 78, 106, 112, 145–153, 156, 158–159, 161–162, 177, 182–183, 185, 194, 220–221, 240, 242–246, 248, 256, 259, 278, 284–285 container, 148, 279, 281, 283–284, 286 individuating, 243, 285 numeral, 53–54, 62, 64–65, 70 plural, 64 unit-extracting, 255–256, 280–283, 286 closure existential, 8, 33–34, 36–37, 289 of an entity, 127 under sum, 161, 262, 272, 276, 278

292

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Index cluster CLUSTER (formal definition), 114 connected, 91, 114 individual, 9, 113–116, 262, 268 MaxCluster (formal definition), 115 maximal, 115 coercion, 206, 209, 212, 252, 254, 256–260, 281–282 mass-to-count, 254–255, 257–258, 260, 281–284 mechanism, 283 collection, 10, 91, 93, 96, 103, 106, 137, 150–151, 219, 252, 262, 267 collection field, 268 collective artifacts/artifact nouns, 261, 266, 273, 277 meaning/interpretation/reading, 94, 171, 186 morphology, 93 comparison along a continuous dimension, 218–219, 221, 228, 246 cardinal/cardinality, 217, 219, 223–224, 229–231, 233, 235 class, 18, 20–22, 24 count, 9, 186–189, 191, 193–194 dimension of, 217, 219–220, 223 mass, 188–189 measure, 10, 186, 231 non-cardinal, 224, 247 quantity, 219 connected entities, 112, 116, 141 maximally self-connected, 139, 141 maximally strongly self-connected (MSSC), 8, 262 self-connected, 128, 131–133, 136, 139, 141 set (of individuals), 113–115 strongly self-connected, 128, 266 temporally, 130–131 transitively, 113, 262 connectedness connectedness (formal definition), 113 external, 113 proximate, 114 relation, 112, 114 context contextual portioning, 184, 193 counting/count, 7–8, 60, 65, 91, 191, 229, 239, 245, 261 dependence/dependency, 11, 57, 193, 224, 226, 236, 268 continuation(s), 141, 200, 205, 207–209, 212–213 continuous dimension, 218–219, 221, 228–229, 246

293 marker, 145 range of values, 242 count concept, 261, 265, 284 counterparts, 273, 277 count-to-mass shift. See shift domain, 31, 161, 165, 218, 224, 236, 247 mass/count distinction, 1–4, 7, 9–11, 52–55, 57, 68, 74, 79, 126, 199, 254, 257–261, 265–266, 270, 283–286 meaning/interpretation/reading, 6, 52–54, 57–60, 62–66, 68–74, 76–77, 79, 162, 169, 253–254, 256–259 noun plural, 57, 65, 186, 217–219, 223, 247, 277 NP, 17, 162, 166, 192 predicate, 162, 226, 256 unit, 258, 283–284 countability, foundations of, 86, 118 counting and measuring, 10, 12, 56, 145, 158, 199, 224, 226, 236–237, 245 operation, 8, 11, 165, 169, 229, 253, 257, 260, 269, 273, 275, 281–283, 285–286 cumulative predicate, 5, 12, 146, 153, 158 reference/extension, 1–2, 59, 65–68, 284 verb/verb phrase, 151–152, 158 cumulativity, 1, 146, 152, 157, 159, 288 Czech, 3, 9, 11, 19, 28, 85–91, 93–95, 97, 99–100, 102–104, 106–109, 111–113, 115–117, 258, 260, 269, 273, 275–276 de particle (Mandarin), 155, 243–245 de partitive marker (French), 26, 28 de partitive marker (Romanian), 16–17, 30 degree expression, 59, 65, 68 marker/marking/marked, 155–157 modifier, 59 of individuation, 9, 78 operator, 14, 42 quantification, 38, 42 quantifier, 38 déjà (French), 214 derivational morphology, 11, 86–89, 91, 103 determiner distributive, 255 indefinite, 32, 37 non-scalar, 204 null, 36 numerical, 10, 199, 201, 207 overt, 36 proportional, 14

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294

Index

determiner (cont.) quantificational, 14, 24, 26, 31–33, 36–37 strong, 32 weak, 32 diminutive, 11, 145, 154–157 disjoint base, 164, 166, 171, 183 contextual disjointness, 165, 194 counting base, 277, 282 disjointness and overlap, 163 portion, 194 set, 164, 167, 169, 171, 174–175, 178, 184–185, 191–193, 260–261, 267, 270, 273, 275, 277, 287 subset, 193, 270–271, 276 distributive marker, 94 operator, 166 plural reading, 158 scope, 34 stubbornly, 186–187 universal quantifier, 57 distributivity, 3, 162, 186, 191, 288 DRT, 199–200, 204, 207, 211, 214 dual life noun, 252, 260 durative event, 152, 154, 158 interpretation, 152, 154, 158 Dutch, 74, 162, 179, 183, 187–189, 192–193, 200, 202, 211–215, 261 Dynamic Aspect Trees (DAT), 203, 209 English, 2, 4, 20, 23, 26–27, 29–31, 38, 49, 52–59, 61–69, 73–78, 86, 88, 90, 100–101, 103–104, 106, 112, 117, 119, 147, 151–152, 155, 157, 179, 182, 187–188, 192–193, 197, 200, 202, 204–205, 211–214, 220, 228, 238, 245–247, 252, 258, 260–261, 268, 274–275, 277, 284 estimation, 72, 218, 234, 237–239, 243–244 even, 201 extension field, 267 exterior of an entity, 127 fewer, 8, 38, 59, 224, 227, 229 focus particle, 201, 213 French, 3–4, 14–16, 18–19, 21–22, 25–28, 45, 60–61, 74, 91, 97, 115–116, 200, 211, 213–215 Generalized Quantifiers, 14, 23–25, 33, 169, 208 German, 19, 31, 158, 187, 193, 252, 256, 260, 284 grammatical number, 9, 85, 103

granular count, 257–258, 261, 275, 280, 286 mass, 11, 253, 257–258, 261, 269, 275–278, 283, 285–286 noun, 10, 252–253, 256–258, 260–261, 267–268, 276, 281, 283–286 Head Principle, 176–177, 191 Hungarian, 63–64, 66, 221, 246, 248 -í (nominal suffix, Czech), 89, 91, 93, 95, 97, 102–104, 112–114 -ice (numerical suffix, Czech), 91, 94–95, 106, 111 Iceberg Semantics, 9, 161–162, 164–166, 168–169, 171, 173–174, 176–177, 181–182, 184, 191–192, 194, 196 indefinite/indefinites cardinal, 31, 34–36 DP, 8, 36, 39, 204 plural, 39 strong, 31, 34, 37, 39–40 weak, 33, 40, 43 indexical inference, 12, 199–200, 202–204, 208, 213–215 individuable entities, 3, 218, 225 individual/individuals concepts, 135, 139 continuous, 136 domain of, 5, 243 incrementally changing, 136 matter individuals, 139–140 plural, 105, 114, 152, 222, 246 possible, 135–136, 138 scattered, 124, 127, 268 sum/sum of, 105, 109, 111, 136, 242 individuated elements/objects/entities, 2, 56, 148, 150, 158, 255–256, 258, 267, 270 events, 151–152 groups, 24 individuation boosting, 54, 76–78 schema (of ), 258, 266, 268, 270–271, 273–274, 276, 281–283, 285 internal structure, 147, 200, 227 interval (in a scale), 42, 124–125, 130, 132–133, 138, 156, 237–238 i-object count, 170, 172–174, 179, 181 mass/neat mass, 171–173, 175 plural, 170, 172, 175–176 singular, 172 i-set count, 166, 185

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Index mass/neat mass, 166–168, 183 plural, 171 iterative, 145, 151–154, 158, 213 kinds basic-level, 254–255 kind units, 110, 117, 254 kind-denoting/-referring, 30, 108, 245 (taxonomic) subkind, 105, 109, 117, 253–255 lattice atomic join semi-lattice, 87, 103 join semi-lattice, 1, 21–22, 24 non-atomic join semi-lattice, 1 le plus, 15, 18, 20, 25–27 less, 59, 224, 228 Mandarin, 10, 16, 52–53, 62–64, 67–70, 74, 106, 149, 152, 155, 157, 221, 243, 245, 248, 284 many, 11, 14–17, 25, 27, 29, 32–33, 36–46, 59–63, 65, 67, 70, 77, 90, 92, 95, 208, 221–222, 227, 230, 233–235, 252 mass environment, 53, 60 mass/count distinction, 1–4, 7, 9–11, 52–55, 57, 68, 74, 79, 126, 199, 254, 257–261, 265–266, 270, 283–286 mass-to-count shift. See shift meaning/interpretation/reading, 52, 54, 58, 64, 66, 69–73, 76, 79, 100, 172, 183, 258, 266, 278 mass question word, 221 matter over time, 10, 123, 136, 139 measure function, 5, 41–42, 45, 106, 109, 111, 114, 182, 227, 232–233, 235, 242, 278 measuring, 10, 42–43, 75, 85, 102–103, 157, 199, 223, 227, 229, 231, 235–238, 240, 243–244, 246 mereology extensional, 8, 124 haptomereology, 126, 128–129, 135 haptomereology (spatiotemporal), 129–130 mereotopology, 111, 126, 128 more, 10, 16, 20–21, 26, 38, 41, 43–45, 58, 61, 65, 67, 71, 74, 77, 186, 213, 217–218, 220, 222–223, 227–231, 233–235, 239, 246–247, 266 most, 17–24, 26, 28, 68, 70, 186, 189–191, 193, 221 determiner most, 220 no longer, 203, 206, 209 not yet, 206, 209

295 noun count, 1–7, 9–11, 52–53, 55, 58, 60, 65, 71–75, 77, 89, 117, 122–123, 140, 142, 147, 161, 166, 168, 183–186, 188, 191–194, 217, 219–220, 222–223, 225–226, 229, 231, 236, 240–243, 246–247, 252–254, 259–261, 270, 273–274, 277, 282–284, 287 fake mass, 7, 218, 254, 284 granular mass, 253, 255–257, 260, 273, 280–281, 283 mass, 2–7, 9–10, 52, 55, 57–62, 65, 68–69, 71, 74, 79, 87–88, 113, 121, 123, 126, 135, 147, 161–162, 166–167, 172, 183–184, 186–188, 191–192, 194, 196, 217, 219–220, 222–224, 226, 229, 231, 233, 236, 243, 246–247, 253–260, 267, 271, 273, 276–277, 282–285, 287 mess mass, 6, 167–168, 187–191, 193–194 neat mass, 6, 166–167, 186, 192, 194, 218, 255 notional mass, 58, 63, 66–67, 71–73 number-sneutral neat mass, 166, 192 object mass, 3–4, 6–7, 10, 217–218, 220, 222–224, 230–231, 233, 246, 254–257, 261, 271, 273, 277, 281, 284 singular count, 2, 59, 106, 261 substance mass, 217–218, 246, 273, 286 number approximate, 230 cardinal, 15, 89, 94, 107, 112, 229 large, 37, 42, 46, 122, 247, 265 marking/marker/marked, 53–55, 57–58, 63–66, 70, 75, 77–78, 274, 284 MeasP, 46 natural, 138, 226, 229, 232, 234, 236, 242 non-negative real, 182 non-round, 244 precise, 62, 239 real, 124, 226–227 round, 237, 244 small, 46, 60, 149, 247 number argument, 107, 115 number neutral, 63, 67, 166, 266, 284 number system, 85 number value, 107 Ojibwe, 74 plural individual, 105, 114, 152, 222, 246 marking/marker/marked, 52, 55, 57–58, 62, 65–66, 70, 78 morpheme/morphology, 78, 107, 114, 117

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296

Index

plural (cont.) NP, 14, 23, 162 object, 145, 154, 161, 167, 171, 207, 228 taxonomic, 98, 104, 117 pluralia tantum, 85, 97, 99, 104, 112, 115, 252 plurality of events, 152–153 Polish, 19, 86, 95 portion, 113, 121–123, 135, 161–162, 183–185, 189, 191, 193, 253, 256, 259, 278, 281 contextual portioning, 193 meaning/interpretation/reading, 161–162, 183–185, 193, 253, 256, 281 of matter, 121, 135 PORTIONc (formal definition), 184–185, 193 possible individual, 135–136, 138 predicate individual-level, 32 kind-level, 90 measure, 227, 241, 247 number, 177, 182 number-neutral, 266 proportional many, 11, 14–15, 17, 36, 45–46 (see also many) proportional most, 11, 14–15, 17–18, 23–24, 31 (see also most) pseudo-partitive, 26, 278, 281, 285 quantifier entity-restrictor proportional, 30 existential, 33, 36 proportional, 25, 28–29 set-restrictor proportional, 29 universal, 57, 95 vague, 91 quantity evaluation, 10, 217, 219, 234, 237, 247 quantity expression, 10, 36, 39, 53–56, 58–63, 65–70, 72–80 anti-count, 60–61, 65, 68–69, 72 count, 54, 56–61, 63–64, 67–68, 70–72, 75–77, 79 non-count, 54, 56, 58–61, 65, 67, 69, 72, 77, 79 quantity judgment task/test/experiment, 67, 71–72, 78 quantity modification, 14 quantity system, 10, 54–55, 61, 79 quantization, 146, 151–152, 157–158 quantized element, 148 predicate, 5, 8, 12, 146, 148, 153 verb, 151–152

realization operator/relation, 105, 109 reasoning temporal, 211, 213 reduplication, 9, 11, 145–148, 150–159 adjective, 154–157 and duration, 11 classifier, 148, 150 noun, 151 reidentification, 123, 137, 139–140, 142 relation between sets, 17, 24, 29–30, 208 connection/connectedness, 112–114, 262 interior part, 127, 130, 132, 142 locative, 33 mereological, 113, 126 number, 177 part, 123–125, 127–129, 134, 137, 139 precedence, 129 successor, 236 tangential part, 127 taxonomic, 105 topological, 113, 262 touch, 10, 127–130, 133, 139 transitive connection, 113 relative clause, 172, 220 relative superlative reading, 16, 18–19, 23, 25, 31 restrictor, 28–29, 31 Romanian, 14, 16, 19–21, 25–27, 29–31, 37 Russian, 22, 86, 94, 240, 248 scale cardinal, 218, 224, 228, 234–235, 237–238, 242, 246 cardinality, 10, 44, 231–232, 236 counting, 201, 212 dimensional, 217, 222, 227, 229, 232, 235 granularity, 227 lexical, 159 measure(ment), 56, 232, 242 numerical, 206, 208, 212 of alternatives, 201–202 proportion(al), 38, 43, 45 proportional, 43, 45 semantic atom. See atom Serbo-Croatian, 19 shift count-to-mass, 72 mass-to-count, 254–255, 258, 282–284 portion, 162, 184, 191 singular, 9, 172, 174–176 Slovenian, 19 solid(s), 10, 122–123, 131–132, 134–135, 142, 195 sortal concept, 137

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Index spatiotemporal region, 142 still, 9, 12, 200, 202, 208–209, 213 sum individual, 6, 112, 138 summation, 9, 11, 145–146, 148, 151–157, 159 superlative absolute qualitative, 24 marker, 27 qualitative, 18 superlative quantitative modifier, 27 supplementation, 124–125

297 taxonomic numeral, 11, 86, 98, 101, 109–111, 115, 117 Tuvan, 146 unbounded, 146, 152–154, 159 universal grinder, 100 Yudja, 10, 53, 58, 63, 65–68, 71–73, 76–78, 80, 88, 184, 194, 283, 285

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