Measurements, Numerals and Scales: Essays in Honour of Stephanie Solt 303073322X, 9783030733223

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PALGRAVE STUDIES IN PRAGMATICS, LANGUAGE AND COGNITION

Measurements, Numerals and Scales Essays in Honour of Stephanie Solt Edited by Nicole Gotzner Uli Sauerland

Palgrave Studies in Pragmatics, Language and Cognition

Series Editors Richard Breheny, Psychology and Language Sciences, University College London, London, UK Uli Sauerland, ZAS, Leibniz-Centre General Linguistics, Berlin, Germany

Palgrave Studies in Pragmatics, Language and Cognition is a series of high quality research monographs and edited collections of essays focusing on the human pragmatic capacity and its interaction with natural language semantics and other faculties of mind. A central interest is the interface of pragmatics with the linguistic system(s), with the ‘theory of mind’ capacity and with other mental reasoning and general problem-solving capacities. Work of a social or cultural anthropological kind is included if firmly embedded in a cognitive framework. Given the interdisciplinarity of the focal issues, relevant research comes from linguistics, philosophy of language, theoretical and experimental pragmatics, psychology and child development. The series aims to reflect all kinds of research in the relevant fields - conceptual, analytical and experimental. Series Editorial Board Kent Bach Anne Bezuidenhout Noel Burton-Roberts Robyn Carston François Recanati Deirdre Wilson Sam Glucksberg Francesca Happé

More information about this series at https://link.springer.com/bookseries/14597

Nicole Gotzner · Uli Sauerland Editors

Measurements, Numerals and Scales Essays in Honour of Stephanie Solt

Editors Nicole Gotzner Cognitive Science/Linguistics Department University of Potsdam Potsdam, Brandenburg, Germany

Uli Sauerland ZAS Leibniz-Centre General Linguistics Berlin, Germany

Palgrave Studies in Pragmatics, Language and Cognition ISBN 978-3-030-73322-3 ISBN 978-3-030-73323-0 https://doi.org/10.1007/978-3-030-73323-0

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Palgrave Macmillan imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Stephanie Solt

Acknowledgments

The contributions to this volume stem from colleagues that share a longstanding working relationship and friendship with Stephanie Solt. All contributions have been reviewed anonymously by at least one colleague. We thank those authors who volunteered for this task, and also thank further external reviewers, in particular, Artemis Alexiadou, Heather Burnett, Richard Breheny, William McClure, Diana Mazzarella, and Kazuko Yatsushiro. We are grateful to Henry Salfner for administrative support and the German research council DFG (grants BE 4348/4-1, GO 3378/1-1 and DFG 1412-A5) for financial assistance.

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Contents

Introduction Nicole Gotzner and Uli Sauerland

1

Partitives, Comparatives and Proportional Measurement Alan Bale

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Modified Numerals, Vagueness, and Scale Granularity Anton Benz and Christoph Hesse

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Uncertainty, Quantity and Relevance Inferences from Modified Numerals Chris Cummins Around “Around” Paul Égré Evaluative Intensification and Positive Polarity: Catalan WELL as a Case Study Elena Castroviejo and Berit Gehrke

59 75

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Contents

She is Brilliant! Distinguishing Different Readings of Relative Adjectives Nicole Gotzner and Sybille Kiziltan

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Numerals Denote Degree Quantifiers: Evidence from Child Language Christopher Kennedy and Kristen Syrett

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Number, Numbers and the Mass/Count Distinction in Daakie (Ambrym, Vanuatu) Manfred Krifka

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Representing Measurement: The View from Nominal Polysemy Louise McNally

187

On the Status of Post-nominal Q Superlatives in Romanian Andreea C. Nicolae and Gregory Scontras

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Unknown Numbers Rick Nouwen

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Some Speculative Remarks on the Semantics of Money Phrases Robert Pasternak

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Quantifying the Register of German Quantificational Expressions: A Corpus-Based Study Uli Sauerland

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Domain Restricted Measure Functions and the Extent Readings of Relative Measures Giorgos Spathas

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Amazing-Hodo Eri Tanaka

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A Novel Probabilistic Approach to Linguistic Imprecision Brandon Waldon

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Contents

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Modification of Measure Nouns E. Cameron Wilson

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Index

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List of Figures

Modified Numerals, Vagueness, and Scale Granularity Fig. 1 Fig. 2 Fig. 3 Fig. 4

Fig. 5

Example item from Cummins et al. (2012, p. 146) Median range size and median distance to most likely value Example item from Hesse & Benz (2020) Median range size of the range of values inferred and expected for modified numeral n, i.e. the distance between the modified numeral and the boundary respondents indicated. Reprinted from JML 111, Hesse & Benz, Scalar bounds and expected values of comparatively modified numerals, Copyright [2020], with permission from Elsevier Distance of most likely value relative to modified numeral n. Reprinted from JML 111, Hesse & Benz, Scalar bounds and expected values of comparatively modified numerals, Copyright (2020), with permission from Elsevier

41 42 46

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List of Figures

She is Brilliant! Distinguishing Different Readings of Relative Adjectives Fig. 1

Fig. 2

Fig. 3

Relation of Horn scale mates to underlying measurement scale and their semantic meaning. Ds represents the standard degere and the dashed line indicates that the Horn scales for positive and negative terms are split. The effect of scalar implicature is presented in the second line Semantic meaning of negated expressions. Ds represents the standard degere and the dashed line indicates that the Horn scales for positive and negative terms are split. The different strengthenede meanings of negated positive and negative terms are presented in the second line Density curves across adjective conditions (dashed line represents the mean)

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Numerals Denote Degree Quantifiers: Evidence from Child Language Fig. 1

Experiment 1, count noun scenario for two lemons (>2)

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Quantifying the Register of German Quantificational Expressions: A Corpus-Based Study Fig. 1

SOLT values for the cardinal numbers from 0 to 50

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A Novel Probabilistic Approach to Linguistic Imprecision Fig. 1

Fig. 2

Standard RSA speaker production probabilities given the speaker’s belief that exactly 9/10 townspeople are asleep. Standard RSA permits under-informativeness, but it does not allow for speakers to produce strictly false utterances; hence, it does not capture imprecise production behavior Partial-truth speaker distributions for most-asleep and some-asleep

312 315

List of Figures

Fig. 3

Fig. 4 Fig. 5

Fig. 6

Imprecise speaker production probabilities, using the definition of the imprecise speaker in (8). The analysis correctly predicts that the-asleep can be used imprecisely but erroneously predicts that imprecise production is felicitous for most-asleep and all-asleep Imprecise speaker production probabilities, using the definition of the imprecise speaker in (9) Imprecise listener interpretation probabilities given observation of the utterance the-asleep (using the pragmatic listener defined in 11) Imprecise RSA listener interpretation probabilities given observation of the-asleep (and given the Model 2 setup in 12)

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List of Tables

Modified Numerals, Vagueness, and Scale Granularity Table 1

Modified numerals’ most common boundaries. Reprinted from JML 111, Hesse & Benz, Scalar bounds and expected values of comparatively modified numerals, Copyright [2020], with permission from Elsevier

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She is Brilliant! Distinguishing Different Readings of Relative Adjectives Table 1

Table 2

Table 3

Overview of non-negated adjectives across conditions (the negated forms included the same adjectives preceded by not ) Context and task for the item pretty. Two example statements are given but, in the experiment, participants were presented with all 8 expressions Summary of cumulative link model with sum coding

126

127 128

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List of Tables

Numerals Denote Degree Quantifiers: Evidence from Child Language Table 1 Table 2 Table 3 Table 4

Mean percentage acceptance of second character’s actions Mean percentage acceptance of second character’s actions, “Stuffing the Bear” items removed Results of a follow-up experiment correcting for the item effect Rejection of higher values in the absence of contextually salient constraints on what is allowed

148 149 150 157

Representing Measurement: The View from Nominal Polysemy Table 1 Table 2

Number and proportion of singular vs. plural -th noun tokens in the COCA corpus Summary of -th noun polysemy

193 204

On the Status of Post-nominal Q Superlatives in Romanian Table 1

The distribution of Q and non-Q adjectives post-nominally

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Quantifying the Register of German Quantificational Expressions: A Corpus-Based Study Table 1

Comparison of baseline statistical model with one including stylistic annotations (table generated using texreg, Leifeld 2014), the Intercept represents marking=colloquial in Model 2

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Modification of Measure Nouns Table 1

Table 2

Rate per thousand at which tokens of the four types of nouns are directly preceded by tokens of cardinal numbers Rate per thousand at which tokens of the four types of nouns are directly preceded by tokens of adjectives (context: [*jj]_)

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List of Tables

Table 3

Table 4

Rate per thousand at which five types of nouns are directly preceded by adjectives with overt degree morphology Adjectives that co-occur with measure nouns at rates greater than with nouns overall. Gradable adjectives are in bold

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Introduction Nicole Gotzner and Uli Sauerland

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Stephanie Solt: Her Influence as a Researcher, Colleague and Mentor

The inspiration for this volume stems from our long and fruitful working relationship with Stephanie Solt. Since her Ph.D. in linguistics in 2009, Stephanie has created a rich and intricate body of work that has proven to be deeply insightful and inspiring for many in the field. The contributions in this volume attest to the powerful influence her ideas have N. Gotzner (B) Universität Potsdam, Leibniz-Zentrum Allgemeine Sprachwissenschaft, Berlin, Germany e-mail: [email protected] U. Sauerland Leibniz-Zentrum Allgemeine Sprachwissenschaft, Berlin, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_1

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in three of the central areas of current semantic theory: measurement, numerals and scales. In her dissertation, Stephanie Solt developed a semantic account of quantity expressions. She became a pioneer in integrating insights from the psychology of measurement and numerical cognition with formal semantic analyses. Her first postdoctoral position was in a project on vagueness, approximation and granularity at the ZAS, followed by a position in a project on indefinites at the University of Amsterdam. Since then, Stephanie has acquired an impressive number of grants on scale structure, degree expressions, polarity and situation-specific uses of quantifiers and numerical expressions. With her ground-breaking work in several areas relating to measurement, quantity and scales, Stephanie quickly established herself as a leading researcher in formal and experimental semantics. Both on an academic and a personal level, we could not imagine the field of semantics without Stephanie Solt. She has provided invaluable mentorship to many colleagues and has inspired junior researchers to pursue a career in academia. These are two among many reasons why she deserves the title of a Jedi Master. Her colleagues value her for her insightful feedback and her combination of linguistic theory with experimental investigations, as witnessed in the following quotes. Heather Burnett: ‘What makes Steph’s work really special is how she combines high level experimental work with subtle, well-grounded (almost “no nonsense”) analyses, a combination which always moves linguistic theory forward in a significant way. She is also a very open, generous colleague, and I am so happy to have had the chance to collaborate with her!’ Chris Cummins: ‘Thanks to the largesse of Euro-XPRAG, Stephanie was the very first person to make an academic visit to me, which was when I was a Ph.D. student in Cambridge in 2010. My hosting of that visit wasn’t exactly a success, in that, just before it was due to end, that Icelandic volcano erupted (https://en.wikipedia.org/wiki/2010_erup tions_of_Eyjafjallaj%C3%B6kull) and all the flights were grounded.

Introduction

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Stephanie eventually made the trip back to Berlin overland, by train and ferry, which took about 24 hours door-to-door.’ Nicole Gotzner: ‘I am indebted to Stephanie for training me and for inspiring me to pursue a career in experimental semantics and pragmatics. More than ten years after being her research assistant, Stephanie is still the first person I am seeking out for advice. She is a great source of inspiration to me on many levels.’ Louise McNally: ‘Stephanie is very tough to convince, and almost always has an insightful counterexample up her sleeve, which she generally prefaces with sort of a squint and a very mild-mannered “Yeeeah...well...I don’t know...” before offering what can amount to a definitive blow to an analysis. I always look forward to questions from Stephanie.’ Uli Sauerland: ‘I have learned and continue to learn from Stephanie about linguistics, about leadership and about life in general. Her evident love of studying language and engaging in discussions of it from the oddest angles has many times drawn me in and made me enjoy so much more what I am doing.’ The chapters in this collection address three core topics centered around Stephanie’s work: measurement theory, the meaning of numerical expressions and scales in semantics and pragmatics. We briefly summarize the main findings presented in this volume and how they relate to the core areas of Stephanie’s work in the following.

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Measurement Theory

Solt developed an account of partitive phrases as partial measure functions, restricted by their nominal complement (Solt 2018a). This account is based on data indicating that partitives with ‘many’ force proportional readings while partitives with ‘more’ makes both proportional readings and direct comparisons accessible. In his chapter ‘Partitives, Comparatives and Proportional Measurement,’ Alan Bale provides novel data

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showing that proportional scales are accessible in both partitive and non-partitive constructions, which challenges Solt’s original proposal. He extends this proposal by arguing that partitive phrases always relativize their degree interpretations to an underspecified scalar limit. This novel view has important ramifications for the denotations of mass nouns, which might have a lower bound (as suggested in Chierchia 1998, 2010). Giorgos Spathas’ chapter entitled ‘Domain restricted measure functions and the extent readings of relative measures’ also follows up on Solt’s account of proportional readings with ‘many’ and ‘most’ (Solt 2018a, b). Spathas discusses degree readings of percentages and proposes a unified analysis of percentages in predicational structures. Together with Solt, he argues that the availability of extent readings crucially relies on domain-restricted measure functions. Louise McNally’s chapter on ‘Representing measurement: The view from nominal polysemy’ argues that semantic analyses of gradability and measurement grounded in measurement theory have been very successful, but they also suffer from a high level of abstractness. McNally highlights Solt’s contribution in 2016, where she suggested ways to connect such analyses to facts about the mental representation of quantity and measure. McNally’s paper is in a similar spirit and it suggests an analysis of polysemy data for English nouns ending in -th such as ‘depth’ and ‘warmth.’ McNally argues that the formal scales underlying such nouns mirror cognitively salient aspects of measuring, as evident in the correlation between measurement properties and the polysemy of the corresponding noun. As observed by Solt (2009, 2014a), measure phrases have the same distribution as quantity adjectives like ‘much’ and ‘many,’ which are functional categories. In ‘Modification of measure nouns,’ Cameron Wilson points out that Solt’s analysis of quantity adjectives as degreeoperators enables a unified account of their diverse uses. Wilson presents a corpus analysis on adjectives in measure noun modification. This research indicates that measure nouns are polysemous with a distinction between container nouns vs. unit and temporal nouns. Wilson also points to several other innovations in semantics that were inspired by Solt’s account, for example in the area of quantity superlatives (Wilson 2018; Coppock et al. 2020).

Introduction

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Nicolae and Scontras investigate the distribution and the interpretations of quantity superlatives in Romanian. They also built on Solt’s (2014a) work on quantity adjectives as quantity superlatives are built from quantity adjectives like ‘many.’ ‘On the status of post-nominal Q superlatives in Romanian’ reconciles seemingly contradictory claims in the literature. By showing that both post-nominal quantity and nonquantity superlatives are sensitive to how context sets the comparison class, Nicolae and Scontras offer a unified analysis of post-nominal superlatives in Romanian.

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Numerical Expressions and Approximation

Solt has shown that numbers are anything but boring and several contributions of this volume follow up on her results. The rich variety of grammatical and semantic properties of numerical expressions that Solt has uncovered shows itself with vagueness, modifiability, polarity and social meaning. The vagueness of numerical expressions can either be inherent, especially for round numbers like ‘sixty,’ or it can be managed by explicit approximating modifiers such as ‘around sixty.’ Extending her earlier dissertation work on vague quantification, Solt has made substantial contributions to both types of vagueness. In joint work, she showed that roundness affects the interpretation of modified numerals like ‘more than n’ and ‘less than n’ (Cummins et al. 2012). A major contribution of Solt’s has been to demonstrate in concrete experimental studies cognitive benefits of round numbers to explain the preference for approximation (Solt et al. 2017). The rich study by Solt (2014b) examines the properties of numerals in comparatives. Just one striking discovery of her work is that modified numerals in comparatives frequently lead to polarity items. Among approximating expressions, Solt and Stevens furthermore show that English ‘some,’ as in ‘some 60 years,’ has been wrongly included in this category by Sauerland and Stateva (2007), and instead should be treated by an extension of the indefinite meaning of ‘some’. More recent work of Solt’s addresses (in progress) also the social meaning and

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polarity of numerical expressions, the latter of which we come back to in the following section. The contributions in this volume explore the full richness of terrain laid out by Solt’s work. Three contributions primarily concern modified numerals. Paul Egre in the chapter ‘Around “around”’ follows up on Solt’s (2014b) work on the approximator ‘around.’ He proposes a refined account of the meaning of ‘around’ more aligned with the spatial basis of the word ‘around.’ Two contributions develop theoretical accounts of the empirical results of Cummins et al. (2012) on the upper bound inferred from modified numerals like ‘more than n.’ Chris Cummins’ chapter ‘Uncertainty, quantity and relevance inferences from modified numerals’ focuses on competing considerations in the speaker’s choice of an utterance. Anton Benz and Christoph Hesse discuss novel data that extend the results of Cummins et al. (2012) in their chapter ‘Modified numerals, vagueness, and scale granularity.’ Remarkably, both contributions to this volume agree that the approximate number system plays the central role, while the role of linguistic factors such as roundness is lower. Of the three contributions relating primarily to bare numerals, Chris Kennedy and Kristen Syrett present experimental data from children concerning the interpretation of bare numerals in root modal sentences. In ‘Numerals denote degree quantifiers: Evidence from child language,’ they show that the interpretation assigned are adult like as expected from the theoretical account of Kennedy (2014). In the chapter ‘Unknown numbers,’ Rick Nouwen addresses summative readings of bare numerals as in ‘Sue had three husbands,’ and relates that the adjectival properties of numerical expressions discovered by Solt (2009). Robert Pasternak examines the various uses of money phrases like ‘five dollars’ in the chapter ‘Some speculative remarks on the semantics of money phrases.’ He suggests that all uses of money phrases should be derived from a denotation to an abstract entity of the monetary unit. Two contributions on numerals also take sociolinguistic aspects into account. In ‘Number, Numbers and the Mass/Count Distinction in Daakie (Ambrym, Vanuatu),’ Manfred Krifka examines the dual and paucal in Daakie with particular attention to two sociolinguistic aspects, namely honorific uses of the dual and affiliative uses of the paucal.

Introduction

7

Uli Sauerland’s chapter ‘Quantifying the register of German quantificational expressions: A corpus based study’ investigates the connection between vagueness and formality using corpus evidence and finds some confirmation for a correlation between the two.

4

Scales and Polarity

A number of further chapters address Solt’s work in the area of measurement scales, entailment scales and polarity. In collaborative work, Solt developed an informativity-based account for the polarity sensitivity of different modifiers (Solt 2018c; Solt and Waldon 2019; Solt and Wilson, 2020). Berit Gehrke and Elena Castroviejo relate to this work by focusing on the Catalan modifier ‘ben.’ Their chapter ‘Evaluative intensification and positive polarity: Catalan WELL as a case study’ targets the modifier’s meaning when combined with different adjectives and its behavior in entailment-canceling contexts. They argue that Solt and Wilson’s account (2020) correctly explains why true degree modifiers (e.g., ‘fairly’) behave like positive polarity items. Catalan ‘ben,’ on the other hand, calls for a different analysis as it appears to express degree intensification without manipulating degrees directly. In ‘Amazing hodo,’ Eri Tanaka discusses the polarity sensitivity of the Japanese equative marker ‘hodo,’ following up on joint work with Stephanie Solt (Tanaka et al. 2019, 2020). Tanaka et al. (2019, 2020) showed that this equative marker behaves like an NPI in some contexts but is polarity insensitive in other contexts. Tanaka provides new data, which suggest that ‘hodo’ exhibits the full range of polarity sensitivity, including PPI uses. She argues that the PPI-hood of both ‘amazing’-hodo and amazingly-adverbs should be attributed to their ‘extreme’ semantics, which renders negated propositions involving ‘hodo’ less informative than its alternatives. The chapter ‘She is brilliant! Distinguishing different readings of relative gradable adjectives,’ addresses the interplay between measurement scales and Horn scales in the area of gradable adjectives. Gotzner and Kiziltan follow up on Solt’s (2015) paper on scales in semantics and

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pragmatics and the studies by Gotzner et al. (2018a, b) on scalar implicature and negative strengthening. Gotzner and Kiziltan show that, under negation, the interpretation of weak scalar terms is asymmetric: positive terms (e.g., ‘not large’) are strengthened more than negative ones (e.g., ‘not small’). Strong scalars, on the other hand, receive a middling interpretation independent of polarity. These findings are in line with Horn’s (1989) account of the Division of Pragmatic Labor. Overall, this research indicates that adjectival meaning should be modeled by integrating multiple factors such as the properties of measurement scales, complexity of alternative expressions and factors relating to the social context.

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Concluding Remarks and Wishes

Stephanie Solt has conducted ground-breaking work on measurement, numerical expressions and scales. As shown by the contributions to this volume, she has shaped and tremendously impacted formal and experimental semantics. Dear Stephanie, we would all like to thank you for your amazing contributions to our field, your inspiration, and your friendship. We feel honored to know you and to celebrate your special anniversary with you.

References Chierchia, G. (1998). Plurality of mass nouns and the notion of “semantic parameter”. In Events and grammar (Ed.), Susan Rothstein, 53–104. Kluwer Academic Publishers. Chierchia, G. (2010). Mass nouns, vagueness and semantic variation. Synthese 174, 99–149. Cummins, C., Sauerland, U., & Solt, S. (2012). Granularity and scalar implicature in numerical expressions. Linguistics and Philosophy 35, 135–169. https://doi.org/10.1007/s10988-012-9114-0.

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Coppock, E., Bogal-Allbritten, E., & Nouri-Hosseini, G. (2020). Universals in superlative semantics. Language 96(3), 471–506. Gotzner, N., Solt, S., & Benz, A. (2018a). Scalar diversity, negative strengthening, and adjectival semantics. Frontiers in Psychology, 9. https://doi.org/ 10.3389/fpsyg.2018.01659. Gotzner, N., Solt, S., & Benz, A. (2018b). Adjectival scales and three types of implicature. Semantics and Linguistic Theory 28, 409–432. Horn, L. R. (1989). A natural history of negation. Chicago: University of Chicago Press. Kennedy, C. (2014). A “de-Fregean” semantics for modified and unmodified numerals. Semantics and Pragmatics 8, 1–44. Sauerland, U., & Stateva, P. (2007). Scalar vs. epistemic vagueness: Evidence from approximators. In Masayuki Gibson & Tova Friedman (Eds.), Proceedings of SALT 17, 228–245. Ithaca, NY: CLC Publications, Cornell University. Solt, S. (2009). The semantics of adjectives of quantity. City University of New York. Solt, S. (2011). Vagueness in quantity: Two case studies from a linguistic perspective. In P. Cintula, C. G. Fermüller, & L. Godo (Eds.), Understanding vagueness: Logical, philosophical and linguistic perspectives. College Publications. Solt, S. (2014a). Q-adjectives and the semantics of quantity. Journal of Semantics 32(2), 221–273. Solt, S. (2014b). An alternative theory of imprecision. In T. Snider, S. D’Antonio, & M. Wiegand (Eds.), Semantics and linguistic theory (Vol. 24, pp. 514–533). Solt, S. (2015). Measurement scales in natural language. Language and Linguistics Compass 9(1), 14–32. Solt, S. (2016a). Degree and quantity: Semantics and conceptual representation. Talk presented the workshop “Referential Semantics One Step Further: Incorporating Insights from Conceptual and Distributional Approaches to Meaning,” ESSLLI 28, Bolzano, IT, August 24. Solt, S. (2016b). On measurement and quantification: The case of most and more than half. Language 92 (1), 65–100. Solt, S. (2018a). Proportional comparatives and relative scales. In Proceeding sof Sinn und Bedeutung 21, ed. Robert Trueswell, Chris Cummins, Caroline Heycock, Brian Rabern, & Hannah Rohde, 1123–1140. University of Edinburgh.

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Solt, S. (2018b). Multidimensionality, subjectivity and scales: Experimental evidence. In Elena Castroviejo, Louise McNally, & Galit Weidman Sassoon (Eds.), The semantics of gradability, vagueness, and scale structure: Experimental perspectives, 59–91. Cham: Springer. Solt, S. (2018c). Approximators as a case study of attenuating polarity items. In S. Hucklebridge & M. Nelson (Eds.), NELS 48: Proceedings of the 48th Annual Meeting of the North East Linguistic Society 3, 91–104. GLSA, Amherst, MA. Solt, S., Cummins, C., & Palmovi´c, M. (2017). The preference for approximation. International Review of Pragmatics 9(2): 248–268. https://doi.org/ 10.1163/18773109-00901010. Solt, S., & Waldon, B. (2019). Numerals under negation: Empirical findings. Glossa: A 18 Journal f General Linguistics 4(1): 113, 1–31. Solt, S., & Wilson, C. (2020). M-modifiers, attenuation and polarity sensitivity. Paper presented at Sinn und Bedeutung 25, September 2020. Tanaka, E., Mizutani, K., & Solt, S. (2019). Existential semantics in equatives in Japanese and German. In Proceedings of the 22nd Amsterdam Colloquium, ed. Julian J. Schlöder, Dean McHugh, & Floris Roelofsen, 377–386. Tanaka, E., Mizutani, K., & Solt, S. (2020). Equative hodo and the polarity effects of existential semantics. New Frontiers in Artificial Intelligence: JSAI-isAI International Workshops, JURISIN, AI-Biz, LENLS, KanseiAI Yokohama, Japan, November 10–12, 2019 Revised Selected Papers 341–353. Wilson, E. C. (2018). Amount superlatives and measure phrases: City University of New York dissertation.

Partitives, Comparatives and Proportional Measurement Alan Bale

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Introduction

As the literature on partitives has slowly grown over the last three decades, the puzzles they pose for grammatical theories have only deepened. Beyond traditional distributional problems discussed in the early literature (Jackendoff 1977; Selkirk 1977; Abbott 1996; Reed 1996; Hoeksema 1996; de Hoop 1997), partitive phrases also seem to limit the way that elements in nominal denotations are measured and compared (see the discussions in Partee 1989; Ahn and Sauerland 2015; Ahn and Sauerland 2017; Penka 2018; Solt 2018). In this chapter, I focus on the interactions between partitive constructions and the grammar of measurement. In particular, I review, analyze and re-imagine some of the generalizations and theoretical proposals advanced in Solt (2018). My reasons for focusing on Solt (2018) are twofold. First, Solt (2018) is one of the few researchers who A. Bale(B) Concordia University, Montreal, QC, Canada © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_2

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thoroughly discusses how the grammar of measurement relates to partitive constructions. In particular, she outlines an interesting puzzle that I will dub Solt’s Generalization, namely that partitive constructions force proportional readings when they combine with many and few but crucially not when they combine with more or fewer. Second, Solt’s analysis elegantly accounts for this generalization (along with other well-known partitive facts) via one semantic mechanism, namely functional restriction. In Solt’s theory, functional restriction both presupposes the sub-part relationship and forces proportional interpretations in certain grammatical contexts. However, her analysis is not without its weaknesses. Solt’s theory relies on the hypothesis that proportional scales are derived from restricted (non-proportional) measure functions. Yet, evidence from nonpartitive comparative constructions suggests that there is a much wider variety of proportional scales than those discussed in Solt (2018), including those that cannot be derived from a restricted measure function (Bale and Schwarz 2020). In this chapter, I will explore whether Solt’s theory can be altered while still preserving its account of Solt’s Generalization. I will attempt to keep the core aspects of her theory intact, particularly functional restriction, while making some fundamental changes to how restricted measure functions relate to proportional scales. One of the key features of this reanalysis is that partitive constructions are predicted to “factor out” the effects of most types of proportional measures. As we will see, this prediction has some empirical support. Non-partitive constructions allow for a greater variety of proportional comparisons than their partitive counterparts. In the conclusion of this chapter, I will discuss some of the consequences of adopting this revision. In particular, if this revision is on the right track, then it follows, based on the behavior of mass nouns in comparative constructions, that all mass nouns have minimal parts, or at least minimal “measurable” parts (c.f., Chierchia 1998; Chierchia 2010).

Partitives, Comparatives and Proportional Measurement

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13

Background Assumptions on Decomposition

Within the literature on degrees and numerals, there is much debate over whether measurement is something that is built into the meaning of a variety of lexical items (e.g., adjectives, numerals, comparative morphemes etc.), or whether measurement is associated with a grammatical morpheme that is part of a more complex morpho-syntactic structure. Although this debate is important, the differences between these two perspectives will not be discussed in this chapter. However, for the sake of exposition, I am going to adopt a decompositional point-of-view when it comes to measurement, such as the one advocated in Solt (2018). However, the reader should keep in mind that many of the points I make can be reimagined through a non-decompositional lens. An essential part of the decompositional treatment of measurement is the assumption that certain operators provide the “glue” between degrees and NPs. For example, under such an assumption, phrases like ‘four rabbits’ might be decomposed into something like [[four meas] rabbits], where meas is a phonologically null morpheme that takes a degree argument before modifying the NP rabbits, restricting it to subgroups of rabbits whose cardinal measure is 4. A full DP results when this modified NP combines with the phonologically null existential determiner λ.P.λQ. ∃x.P(x) & Q(x). For the sake of simplicity, I will use ∃ to represent this determiner. Under a decompositional analysis, phrases with numerals are not the only grammatical constructions that require this kind of “glue.” Nominal comparisons critically involve degrees which restrict NPs through some type of grammatical measurement. For example, a sentence such as [More students passed this term than last] is often interpreted using the same comparative operators as adjectives (i.e., more = many/much+er, see the arguments in Bresnan 1973). The comparative operator originates in a degree argument position, connected to the NP via the meas operator. The comparative morpheme moves at LF, leaving a trace degree variable and triggering the creation of a degree predicate. The ‘than’-clause also creates a degree predicate, as shown in (1).

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(1)

A. Bale

[-er [than λd. [∃ [d meas students] [passed last term]]] (than-clause) [λd. [∃ [d meas students] [passed this term]]]] (main clause)

The truth conditions of the entire comparative is determined by comparing these two degree predicates (e.g., (1) is true if and only if the maximal value in the main-clause predicate is larger than the maximal value in the thanclause predicate). Similarly, sentences with quantificational terms like many are also routinely analyzed with operators that take degree predicates as arguments (once again on analogy to adjectival expressions). For example, under a decompositional analysis, computing the truth conditions for a sentence like [Many students passed this term] often involves an operator labelled pos (von Stechow 1984; Kennedy 1999). In such an analysis, a measure operator mediates the relationship between a degree argument and a nominal expression within the degree predicate. See the structure in (2).1 (2)

[pos [λd. [∃ [d meas students] [passed this term]]]]

The truth conditions of the sentence is determined by the application of this pos operator to the degree predicate (e.g., the sentence is true if and only if the maximal degree in the degree predicate is greater than a contextually set standard).2 A benefit of this decompositional perspective is that it highlights and isolates the role of measurement in sentences with nominal comparisons. In this chapter, I am going to focus on the role of measurement, in particular Solt’s (2018) hypothesis that the partitive morpheme itself is one of these measurement operators. To facilitate this discussion, I will assume that measurement operators are a part of the “functional” lexicon. In order not to lose the trees for the forest, I will only vaguely discuss the compo-

1 Solt

(2015) argues that there is an additional operator in these types of constructions that captures the differences between many and few on the one hand and much and little on the other. The details are relevant for the overall semantic theory but pull us slightly off topic in terms of the focus of this chapter. For the sake of simplicity, I will not include such operators in the syntactic or semantic representations here, nor will I review the arguments for this additional operator. 2 For now, I will leave it open whether meas is sometimes phonologically realized as many/much (as argued in Bresnan 1973) or whether it is always phonologically null (as argued in Solt 2015).

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sitional details of comparison beyond the contribution of these operators, especially given the variety of possible analyses that can be built on this base.

3

Solt’s Generalization

To understand Solt’s analysis of partitives, it is important to first understand the empirical problems it is trying to tackle. There are a variety of interesting and related problems, however for time and space reasons I am going to concentrate on the interactions between partitives and proportional measurement. As noted by Partee (1989), what counts as being “many” or “few” is often influenced by the presence or absence of a partitive phrase. Consider the contrast between (3a) and (3b).3 (3)

a. b.

Many of the philosophy majors registered for an Ancient Greek course this year. Many philosophy majors registered for an Ancient Greek course this year.

In (3a), many is followed by a partitive phrase whereas in (3b) it is followed by a simple noun phrase. Critical to the discussion at hand, there is a reading available for (3b) that is not available for its counterpart in (3a). For example, consider a context where there are four hundred philosophy majors. Suppose that twelve of the four hundred registered for an Ancient Greek class. (Let’s further suppose that this twelve made up three quarters of the Ancient Greek class.) In this context, it is possible to read (3b) as expressing something true (as long as 12 counts as being “many”) whereas it is very difficult, if not impossible, to read (3a) in the same way (12 when compared to 400 does not count as “many”). Following Partee (1989), the reading which renders the sentence in (3b) true has often been dubbed the cardinal reading, whereas the reading that renders the sentence in (3a) false has often been dubbed the proportional reading.

3 Given

time and space limitations, in this chapter I will only discuss examples with many and more. However, parallel observations hold for few and fewer.

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Solt (2018) observes that this restriction to a “proportional” interpretation disappears when many is replaced with more. For example, consider the sentence in (4). (4)

More of the philosophy majors registered for an Ancient Greek course this year (compared to last).

The comparative in (4) is ambiguous. To illustrate this, consider a context where 12 out of 30 philosophy majors registered for an Ancient Greek course this year whereas 11 out of 20 philosophy majors registered for the same class last year. The sentence in (4) is true under a raw cardinal reading (12 is greater than 11) but is false under a proportional interpretation (just over half of the majors took the course last year whereas only about a third took it this year). The proportional reading can be brought to the fore by appending “relatively speaking” at the beginning of the sentence. These observations lead to the following generalization, which I will call Solt’s Generalization. (5)

Solt’s Generalization: Many and few do not permit a “cardinal interpretation” when combined with a partitive construction, but yet their comparative counterparts, more and fewer, permit both a cardinal and proportional comparisons in identical syntactic contexts.

I put the phrase cardinal interpretation in quotes in (5) since, as we will soon see, such a label can be somewhat misleading.

4

Solt’s 2018 Analysis

An elegant aspect of Solt’s (2018) analysis of partitive constructions is that it accounts for a variety of empirical generalizations through one theoretical conjecture, functional restriction. In essence, Solt hypothesizes that the partitive morpheme (i.e., of ) has an interpretation that critically involves an underspecified partial measure function. This measure function is partial in the sense that its domain (and hence also its range) is restricted by its DP complement. As we will see, this domain restriction

Partitives, Comparatives and Proportional Measurement

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explains why many sometimes lacks a so-called “cardinal reading.” Below, I first present the heart of Solt’s analysis, namely the interpretation of the partitive morpheme, before outlining her account of Solt’s Generalization. Solt (2018) proposes that the partitive morpheme is interpreted as in (6), where x and y are variables that range over entities and d is a variable that ranges over degrees. The symbol μ represents a measure function that maps entities to degrees. (6)

of  = λx.λd.λy.μdim;x (y) = d

A lot of information is contained within this representation so it might be helpful to unpack it a little, separating its compositional implications from some of its truth conditional nuances. Let’s first consider aspects of its functional type. As specified in (6), Solt hypothesizes that the partitive morpheme takes an individual as its first functional argument, and then a degree, before finally yielding a predicate (i.e., it is a function of type e, d, e, t). For example, in a phrase like [two of the philosophy majors], the partitive morpheme’s first argument would be the DP complement [the philosophy majors] and its second argument would be the number term two, thus yielding a type e, t predicate (i.e., the type of the entire phrase two of the philosophy majors). It is an essential part of Solt’s analysis that number terms, and fractional terms for that matter, denote degrees. Since I am concentrating on comparative constructions in this chapter, I will not discuss this further here. A more critical aspect of Solt’s interpretation for the discussion at hand is the presence of a measure function, symbolized by μ, that maps entities to degrees. In Solt’s analysis, this measure function is underspecified, partially indicated by the unspecified dim in the subscript of μ (dim represents the dimension of numerical values that the measure function maps into— i.e., the co-domain of μ). Underspecified measure functions have played an integral role in the analysis of quantification and nominal expressions over the last few decades (see Cresswell 1976; Schwarzschild 2006; Bale and Barner 2009; Wellwood 2015, 2018; Bale and Schwarz 2020, among others). The justification for such underspecification often hinges on how different types of comparisons are influenced by the NP-complement of

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more (e.g., having more water induces a comparison by volume, buying more books induces a comparison by number, making more noise induces a comparison of sound-energy levels, etc.). Sometimes a single expression even allows for multiple types of comparison (e.g., it is possible to evaluate the sentence [This salt-shaker contains more salt than that one] by either weight or volume—in fact one could add either [as measured by weight but not volume] or [as measured by volume but not weight] without contradiction). The innovation in Solt’s analysis is the association of such a measure function with the partitive morpheme itself rather than with a quantificational element such as more or fewer. In almost all theories with an underspecified measure function, the contextual determination of such a function is not completely free. In particular, as discussed in Schwarzschild (2006), it seems to be an empirical fact that whatever measure function ends up being used in a nominal comparison, it must be homomorphic to the part–whole relationship implicitly associated with members of the nominal denotation—often called the Monotonicity Constraint. In other words, within these types of nominal expressions, measures track the part–whole relation in the sense that if an aggregate x is a proper part of an aggregate y, the measure of x must be strictly less than the measure of y. For example, water in statements such as [I have more water than you] can be used to express a comparison in terms of weight or volume, but not a comparison of temperature. This follows from the fact that weight and volume track the part–whole relationship whereas temperature does not: for any two aggregates of water x and y, if x is a proper sub-portion of y then x necessarily weighs less and occupies less volume than y, but it does not follow that x necessarily has a lower temperature than y. Schwarzschild (2006) observes that similar constraints extend to partitive constructions (e.g., [I have more of the water than you] cannot express a comparison of temperature). According to Solt (2018), not only is this underspecified measure function constrained by monotonicity, its inputs are also grammatically restricted, as represented by the x in the subscript of μdim;x . In fact, formally speaking, this is the role that the DP complement serves. A definition of

Partitives, Comparatives and Proportional Measurement

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this restriction is given in (7), where ≤ is the part–whole relation that orders elements in the domain of discourse.4 (7)

Functional Restriction: For all entities x and y, all measure functions μ, and all dimensions dim, μdim;x (y) is defined if and only if y ≤ x. When defined, μdim;x (y) = μdim (y) (i.e., the same value as the unrestricted version of the measure function).

An immediate consequence of the definition in (7) is that the interpretation of a predicate formed from a partitive construction consists of subparts of the denotation of the embedded DP complement. For example, consider the phrase [two of the philosophy majors]. According to Solt (2018), the partitive morpheme would take the interpretation of the embedded DP as its first argument. If we let z be the collection of philosophy majors, then the result would be λd.λy.μdim;z (y) = d, where z restricts the measure function so that it only measures z and its sub-parts. No matter which degree serves as the argument to this partitive phrase, the result will be a set of elements that are either equal to z or are sub-parts of z. If certain reasonable assumptions are made about contextual variability with respect to many, then this restriction will also immediately account for the lack of so-called “cardinal readings” in certain types of partitives. The details of how these assumptions are formally represented can vary and Solt’s (2018) particular implementation need not be discussed thoroughly here. (In fact, such details might unnecessarily complicate our explanation of Solt’s generalization.) The first “reasonable” assumption is that many involves a comparison to some contextually set standard, in much the same way as vague gradable adjectives like tall. This assumption is not new to Solt (2018). There is a vast literature that discusses the parallel between the behavior of many and certain types of gradable adjectives (see Bresnan 1973; Cresswell 1976; Klein 1981; Westerståhl 1985; Büring 1996; de Hoop and Solà 1996; Hackl 2000; Solt 2009; Romero 2015, 2016; Penka 2018; among others). In particular, this literature often compares 4There is more that can be said about the existence and uniqueness of such a partial order. Within a system like the ones proposed by Link (1983), Chierchia (1998), Barker (1998), an entity can participate in only one such “part-whole” relation that orders the domain of discourse. In other types of ontologies, an entity could be allowed to participate in more than one “part-whole” relations (e.g., Gillon 1992; Rothstein 2010).

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the contextual variability in determining what counts as tall (e.g., the fact that tall for a man connotes a greater height than tall for a boy) to the contextual variability in determining what counts as many (e.g., the fact that many among the 200 million American voters connotes a far greater numerical value than many among the 20 tenure-track professors in my department ). Of critical importance, the setting of such standards seems to be influenced by some type of comparison class.5 Kennedy (2007) suggests, at least with respect to adjectives, that comparison classes restrict the domain of measure functions (e.g., a measure function that maps individuals to their heights might be restricted to a certain class of inputs, such as members of the set of boys vs. men). By restricting the domain, the range is also similarly restricted (e.g., to only the heights of boys or only the heights of men). According to Kennedy (2007), the standard can then be computed by taking some relative value with respect to this restricted range (e.g., some norm or average with respect to the range of heights of men vs. some norm or average with respect to the range of heights of boys). Solt (2018) proposes a similar idea for many. The phrase [many of DP] contains a restricted measure function that is part of the partitive morpheme. As implemented in (6), the measure function is restricted to subportions of the partitive complement (e.g., all the possible sub-aggregates of the philosophy majors in [many of the philosophy majors] ). Even when the measure function is one of cardinality (i.e., cardinality of a group), the standard of comparison is computed relative to a restricted range (i.e., the norm or average size of the various possible groups of philosophy majors). Hence, when [many of the philosophy majors] is interpreted with a cardinality measure function, the result is a comparison that is proportional to the size of the referent of the DP complement [the philosophy majors].6 In fact, the truth conditions will essentially be the same for any measure function that is monotonic to the part–whole relation that orders groups, 5The literature on how comparison classes influence a standard is vast, but see in particular the different implementations of this idea for adjectives in Bartsch and Vennemann (1972), Klein (1991), Kennedy and McNally (2005), Kennedy (2007), Bale (2011) and references therein. See the implementation of this idea with respect to many and few in Penka (2018), Romero (2015, 2016) and references therein. 6 Penka (2018) offers an almost identical account of this restriction, however she implements the restriction using focus structures rather than partial measure functions.

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a requirement already necessitated by the Monotonicity Constraint (see the discussions not only in Solt [2018], but also Penka [2018] and Bale and Schwarz [2020]). In contrast to many, comparative constructions with more do not involve standards or norms that are relativized to a comparison class. Rather, they involve a direct comparison of two measurements. Hence, Solt’s analysis predicts that [more of the philosophy majors] will not be constrained in similar ways. In fact, her analysis predicts that such phrases should induce an ambiguity that is relative to the setting of the underspecified measure function. One possible setting for this measure function, so Solt contends, is a proportional measure. It is important to note that Solt (2018) limits proportional measurements to only those that can be derived from restricted measures. The basic components of Solt’s idea are given in (8). (8)

Proportional Measures: For any measure function μ into a nonproportional dimension dim and for any restriction x, there is a proportional measure associated with μ, symbolized as μdim % ;x , such that for any y in the domain of μdim % ;x , μdim % ;x (y) = μdim;x (y) μdim (y) μdim;x (x) , which is equal to μdim (x) if y ≤ x .

As a result, a sentence like [More of the philosophy majors registered for an Ancient Greek course this year (than last)] can be evaluated with a proportional measure based on the ratio of subgroups of philosophy majors taking an Ancient Greek course relative to the total number of philosophy majors in a given year (this year’s cohort vs. last). Although I have not gone over all of the relevant details (and in fact, as Solt notes, the implementation of the remaining details can vary without affecting the analysis) the general gist should be clear. Partitive constructions with more permit both a proportional and direct cardinal measure since the measure function is underspecified. Partitive constructions with many permit both types of measures too, but the cardinal measure appears to be proportional, in terms of the resulting truth conditions, because the standard is set relative to the range of values associated with the restricted domain of the measure function.

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5

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Problem: Varieties of Proportionality

A problem facing Solt’s analysis concerns the variety of proportional interpretations that are available in nominal comparisons. As outlined in Sect. 4, Solt (2018) hypothesizes that proportional measurements are computed via a restricted measure function. Hence, the definition in (8) serves to not only define, but also limit the number of available proportional measures. For the sake of exposition, let’s call the functions used to calculate the measure of the y and x variables in (8) the numerator and denominator functions, respectively. According to this definition, the numerator and denominator functions are the same. Furthermore, the proportional measure is only defined when the entity measured by the numerator function is a sub-aggregate of the entity measured by the denominator function. In partitive constructions, the entity measured in the denominator is the referent of the DP complement to the partitive morpheme (e.g., the referent of [the philosophy majors] in the phrase [many of the philosophy majors] ). In non-partitive constructions, things are a little different. In the absence of a partitive morpheme, Solt introduces a phonologically null meas operator that provides the compositional “glue” to combine degree terms with bare nominal predicates. A version of this null morpheme is given in (9), which is a slightly modified version of the one that appears in Solt (2018). (9)

meas = λd.λP.λy. P(y) & μ(y) = d, where μ is underspecified.

Critically in (9), the measure function is not grammatically restricted, however since it is underspecified, it is possible that it could have the values μdim;z or μdim % ;z for any μ that obeys the Monotonicity Constraint.7 In particular, z could have the same value as the supremum of the nominal complement, in which case this null morpheme would permit the same range of potential readings as its partitive counterpart, but where the entity measured in the denominator is the supremum of the NP rather than the DP complement of the partitive (e.g., the largest group of philosophy majors in the phrase [many philosophy majors] ).

7 For

any partial order ≤, if μdim is monotonic with respect to ≤, then so is μdim;z and μdim % ;z .

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However, as discussed in Bale and Schwarz (2020), there are a variety of proportional readings that cannot be derived from the proportional measure functions defined above. For example, consider the sentences in (10). (10)

a. b. c.

Proportionally speaking, more cooks applied to our program than to yours. Proportionally speaking, more road signs appear on Rte 101 than on Rte 104. Proportionally speaking, more typos were found in the first draft of this book-chapter than in the first draft of my phonology textbook.

Each of the sentences in (10) starts with the qualifier proportionally speaking in order to suppress a pure cardinal reading. However, as discussed in Bale and Schwarz (2020), such a qualifier is by no means necessary. More critical to the discussion at hand, once a cardinal reading is suppressed, each sentence in (10) is true in a context that is inconsistent with the restrictions proposed in (8). For example, (10a) can be true in a situation where “our program” only had 100 applicants whereas “your program” had 200 applicants, but where 50% of the applicants to our program were cooks (50 in total) and 30% of the applicants to your program were cooks (60 in total).8 Such a reading is evaluated based on comparing ratios of cooks who applied to the total number of applicants. Similar readings occur with the other two sentences in (10). The sentence in (10b) can be true in a situation were the proportion of road signs on Rte 101 relative to the length of Rte 101 is greater than the proportion of roads signs on Rte 104 relative to the length of Rte 104. For example, let’s say Rte 101 is 100 mi long and has 1000 road signs (an average of 10 per mile) whereas Rte 104 is 200 mi long and also has 1000 road signs (an average of 5 per mile). Here, the relevant entity being measured in the denominator seems to be the highway routes.

8 Such

a reading is often called a reverse proportional reading (Westerståhl 1985; Herburger 1997) since it appears that the relevant entity in the denominator value is the number of applicants (i.e., the denotation of the VP) rather than the number of cooks (i.e., the denotation of the NP).

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The sentence in (10c) can be true in a situation where 10 typos per page where found in the (30-page) first draft of this book chapter (300 total) but only 2 typos per page were found in the first draft of my 400-page textbook (800 total). Clearly, what is being compared are the proportions of typos in the respective manuscripts. What is interesting for the issue at hand is what these proportions are relativized to (i.e., the denominator value). On the surface, it appears like the relevant proportions are computed relative to the length of the respective documents. Interestingly, these readings disappear when a partitive construction replaces the NPs in (10). The sentences in (11) are not true in the same circumstances as (10). (11)

Proportionally speaking… a. …more of the cooks applied to our program than to yours. b. c.

…more of the road signs appear on Rte 101 than on Rte 104. …more of the typos were found in the first draft of this book-chapter than in the first draft of my phonology textbook.

The proportional readings licensed in (11) are determined by ratios that are limited by their respective nominal complements. For example, (11a) is only true if, of all the contextually relevant cooks, the sub-proportion that applied to my program is greater than the sub-proportion that applied to yours. Similarly, (11b) is only true if, of all the contextually relevant roads signs, the sub-portion that appears on Rte 101 is larger than the sub-portion that appears on Rte 104. Finally, (11c) is only true if, of all the contextually relevant typos, the sub-portion that is contained within the first draft of this book chapter is greater than sub-portion that is contained within the first draft of my textbook. In fact, despite the addition of “proportionally speaking” to the beginning of these sentences, all of these “proportional readings” happen to be truth conditionally equivalent to a cardinal comparison. This data expands on Solt’s Generalization, demonstrating that partitive constructions limit the number of available proportional readings.

Partitives, Comparatives and Proportional Measurement

6

25

Adapting Solt’s Analysis

At this point, the empirical threshold for what constitutes an adequate semantic analysis of partitives should be clear. Not only should such an analysis account for Solt’s Generalization, it also needs to address the data discussed in Sect. 5. In this section I will explore whether we can alter non-essential aspects of Solt’s analysis in order to adapt and expand its empirical coverage. Given the evidence presented in Sect. 5, it is clear that we need to loosen Solt’s constraint on permissible proportional measures. I do so in (12). (12)

Proportional Measures Redux: Given a non-proportional measure μ that maps a domain D into the dimension dim, and given a contextually determined degree d ∗ that is a member of dim, ∗ there is an associated proportional measure μd% (d ∗ > 0) such ∗ that for any x ∈ D, μd% (x) = μ(x) d ∗ (c.f., Bale and Schwarz 2020).

With this expanded ontology, we can easily account for the varieties of proportionality presented in Sect. 5. For example, if d ∗ were set to the number of applicants to our program—let that number be n—we could derive the proportional measure μn#% = λy. μ#n(y) . Similarly, if d ∗ were set to the number of applicants to your program—let that number be m—we could derive the proportional measure μm#% = λy. μ#m(y) . Provided we allow for different measure functions in the main clause and the than-clause via two instances of the measure morpheme in (9)—a standard assumption in the comparative literature9 —these two proportional measures can account for the most prominent reading of (10a) (i.e., [Proportionally speaking, more cooks applied to our program than (how many applied) to yours] ). By identical reasoning, the expanded ontology in (12), in combination with the underspecified measure function in (9), can also account for the proportional readings in (10b) and (10c). We could set d ∗ to the number 9The motivation for this assumption has two sources. First, in adjectival comparisons, sometimes two different measure functions are overtly expressed as in [This table is longer than it is wide]. Second, sometimes there are overt instances of a second many/much morpheme in the than-clause, as in [I have more stuffed animals than how many you think I have].

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of miles that represent the length of Route 101/104 to account for the most prominent reading of (10b)—[Proportionally speaking, more road signs appear on Rte 101 than (how many appear) on Rte 104]. The main clause measure would be μn#% = λy. μ#n(y) while the than-clause measure would be μm#% = λy. μ#m(y) , where n and m are the number of miles that represent the length of Rtes 101 and 104 respectively. Likewise, we could set d ∗ to the number of pages in the first draft of this book chapter/phonology textbook to account for (10c)—[Proportionally speaking, more typos were found in the first draft of this book-chapter than (how many were found) in the first draft of my phonology textbook]. The main clause measure would be μn#% = λy. μ#n(y) while the than-clause measure would be μm#% = λy. μ#m(y) , where n and m are the number of pages in the first draft of the book chapter and phonology textbook respectively. However, with this expansion of the ontology of proportional measures, we have to be careful with how the measure function is restricted in Solt’s interpretation of the partitive morpheme, repeated in (13). (13)

of  = λx.λd.λy.μdim;x (y) = d

In order to account for Solt’s Generalization, along with the additional proportional data in (11), the measure function in partitive constructions still needs to be restricted to only certain types. For example, we would not want to permit any of the proportional functions discussed relative to the examples in (10) to serve as the underspecified measurement in (13). If they could, then one would not expect the contrast we see between proportional comparisons without partitives versus those with—e.g., the contrast between (10) and (11). Empirically speaking, Solt (2018) would have to add some additional constraint or grammatical mechanism that would limit possible denominator values to the maximal value in the range of the restricted measure function. I think there are at least two options. One option is to hypothesize an explicit ambiguity (i.e., a partitive measure that forms a restricted nonproportional measure versus one that forms a proportional measure with a grammatically determined denominator). However, I think such an option runs counter to the spirit of Solt’s analysis and weakens its account of Solt’s Generalization. I will not pursue this option here. A second option is to

Partitives, Comparatives and Proportional Measurement

27

keep a single interpretation for the partitive morpheme but somehow constrain the effects of the underspecified measure functions within the partitive construction. Let me outline one possible way to do this. A critical part of Solt’s characterization of restricted measure functions is that such functions have a well-defined upper limit in their range, namely the degree associated with the measure of the referent that defines the restriction (i.e., if the restriction is set via the entity x then the upper limit is the degree that represents the value of μdim;x (x)). Although not highlighted in Solt (2018), some of the functions also have a well-defined lower limit in the range (especially if the function is restricted by the referent of a count DP). For example, if the measure function is one that measures cardinality of a group—such as μ# , then μ#;x would have a lower limit, namely 1—i.e., the measure of the atoms that are a part of x. An adaptation of Solt’s analysis could take advantage of these two limits by always relativizing the measure of an entity to one of them. One way to achieve this result would be to have a contextually set function called limit with two settings b for bottom and t for top. Thus, limit b would take a measure function and return the least value in the range (if one exists) and limit t would take a measure function and return the greatest value in the range (if one exists). The setting b or t would need to be contextually determined, depending on the needs of the conversation at hand. With such a function, we could redefine Solt’s interpretation of the partitive morpheme as in (14). (14)

μ

(y)

λx.λd.λy. limitdim;x = d , where α is contextually set to b or α (μdim;x )

t and both instances of μdim are the same variable and hence assigned the same value.

To get a more intuitive understanding of (14), let’s consider the interpretation of the phrases [two of the philosophy majors] and [two percent of the philosophy majors], where μdim is set to the cardinality measure μ# . Sup2 . pose that the philosophy majors = z, two = 2, and two percent = 100 Given these values, the denotation of [two of the philosophy majors] would be μ (y) λy. limit#;z(μ ) = 2. The only non-trivial setting for α would be b (since t yields α #;z μ#;z (y) a value less than 1). Thus, the resulting denotation would be λy. limit =2 b (μ#;z ) which equals λy. μ#;z1 (y) = 2, which equals λy.μ#;z (y) = 2 (i.e., the predicate

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that is true of all groups of philosophy majors consisting of two individuals). Likewise, the denotation of [two percent of the philosophy majors] would μ#;z (y) 2 . In this case, the only non-trivial setting for α would = 100 be λy. limit α (μ#;z ) be t (since b yields a value greater than 1). Thus, the resulting denotation μ#;z (y) 2 which equals λy. μ#;z (y) = 2 (i.e., the predicate that = 100 would be λy. limit 100 μ#;z (z) t (μ#;z ) is true of all groups consisting of 2% of the philosophy majors). Let’s now turn to how (14) would be interpreted in comparative constructions. There are two options for setting α and many different possibilities for setting the value of the underspecified measure function. Let’s first consider the possibility where α is set to t and μ is set to the nonproportional measure of cardinality, μ# . In this case, the value in the numerator will only be defined if y is a sub-aggregate of x. When defined, the numerator will be equal to the cardinal measure of y (i.e., μ# (y)). The value of the denominator will be defined as the topmost value in the restricted function μdim;x , which, by definition, will simply be the cardinal measure of x (i.e., μ# (x)). Hence, the entire fraction will represent the fractional value of the cardinal measure of y relative to the cardinal measure of x, where y ≤ x. This yields the attested proportional interpretation. Keeping the measure function the same, what if α were set to b instead? The value of the numerator would not change but the value of the denominator would now be the lowest value in the range of μdim;x . Since the cardinal measure only applies to entities that have atomic parts, it follows that the lowest value in the range of μ#;x will be 1. (It is important to remember that, when determining the lowest/highest value, we only consider those degrees that are output values associated with some input within the restricted domain.) As a result, the overall fraction would be equal to μ#1(y) which is equivalent to μ# (y). In other words, the entire fraction would be equivalent to the non-proportional measure of y. So far, the new interpretation accounts for the same range of readings as the old one did. However, the two interpretations differ when it comes to proportional measures. To account for the contrast between (10) and (11), we need to make sure that the resulting measures in partitive constructions cannot involve proportions relative to some contextually salient value, such as the number of miles in a highway system or pages in a manuscript. The interpretation in (14) does so in a somewhat clever way.

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29

Let’s consider the details of how the fraction in (14) is computed when the measure function is proportional. According to our revised ontological definition of proportional measures in (12), all such measures involve calculating a numerator value using a non-proportional measure and then setting the denominator value to some contextually salient degree. For the sake of clarity, let’s choose a concrete non-proportional measure, such as the cardinality measure, although, in what follows, the choice of measure does not matter. With this expositional choice in place, we can fix the proportional measure relative to some denominator value n, i.e., μ =μn#% =λz. μ#n(z) . Given such a proportional measure function, let’s consider the fraction in (14) when the limit is set to t. In this case, the numerator value would only be defined if y were a sub-aggregate of x. When defined, μdim;x (y) would equal μ(y) which, by assumption, would be μ#n(y) . The denominator value would be the maximal value in the range of μdim;x which by definition would be equal to μdim;x (x), which in turn is equal to μ#n(x) . The entire fraction would then be μ#n(y) / μ#n(x) , which equals μ#n(y) × μ#n(x) , which in turn equals μμ## (y) (x) . In the end, the denominator value in the original proportional measure μn#% factors out. It is critical to note, this process of “factoring out” holds no matter what non-proportional measure is used in μ(y) μ(x) the numerator—i.e. for any non-proportional μ and any n, μ(y) n / n = μ(x) . In other words, the semantics in (14) ends up collapsing the proportional measures so that they are equivalent to their non-proportional counterparts. The same type of “factoring out” happens when the value of the limit is set to b. Once again, let’s use μ# as our non-proportional measure and set the relevant proportional value of μ to λz. μ#n(z) for some arbitrary n. Given this value for μ, the numerator value in (14) would be defined only if y is a sub-aggregate of x. When defined, it would be equal to μ#n(y) . The denominator value would be set to the smallest degree in the range of μdim;x . Given the nature of μ# , this smallest degree would be n1 . Hence, the overall fraction in (14) would be equal to μ#n(y) / n1 which reduces to μ# (y). In other words, the measure ends up being equivalent to the non-proportional measure of y. Thus, comparisons with partitive phrases are predicted to have truth conditions that depend on non-proportional measures even when the underspecified measure function is, in fact, proportional.

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In summary, the interpretation in (14) allows for a greater variety of proportional measures while still restricting the truth conditions of sentences with partitive constructions. The key to such a restriction is the addition of a contextually defined limit, which, in the end, “factors out” the effects of proportional measures.

7

Concluding Thoughts

The main point of this chapter was not only to offer a revision of Solt’s (2018) analysis, but to also further elucidate the empirical landscape by which any theory of partitives should be evaluated. Hopefully, at the very least, the discussion here has clearly outlined the complex relationship between the grammar of measurement and partitive phrases. An adequate theory of partitive constructions cannot merely account for the fact that partitive morphemes establish a part–whole relationship (as in the theories of Ladusaw 1982; Barker 1998; Rothstein 2010, among others), it also needs to explain why there are limitations on how entities in the denotation of partitive phrases are measured and compared. Solt’s theory explains such limitations, and my revision of her theory expands on its empirical coverage. If my revision of Solt’s theory is on the right track, then there are some interesting consequences for measurement as it relates to mass nouns. Up to this point, almost all of my example sentences have focused on DPs formed from count nouns. As discussed in Sect. 6, these count nouns have minimal parts which create a natural lower bound in the range of a cardinal measure function (i.e., the measure of the atomic parts). The existence of such a lower bound plays a critical role in the explanation of why comparative with partitive phrases are ambiguous (i.e., an upper bound setting to the limit yields a proportional comparison whereas a lower bound setting yields a non-proportional comparison). Given this characterization of the data, one wonders whether a similar type of ambiguity exists for non-atomic mass nouns. Consider the sentence in (15) with the mass nouns paint.

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(15)

31

I removed more of the red paint that was on my left boot than the blue paint that was on my right boot.

Let’s suppose we are in a scenario where I spilled some paint on my boots while painting my apartment. As a result, a large amount of red paint splashed onto my left boot and a small amount of blue paint splashed onto my right boot. Let’s further suppose that I removed half of the red paint that was on my left boot, which happened to be a much larger amount of paint than the entire amount of blue paint that was splashed onto my right boot. Let’s also suppose that, at the same time, I removed almost all of the blue paint that was splashed onto my right boot. Clearly, (15) can be false in such a scenario, a reading that comes to the fore when the sentence is preceded by the prompt proportionally speaking. However, (15) can also be true—a judgment that seems to rely on the total amount of removed red paint being greater than the total amount of removed blue paint. The ambiguity that arises in (15) seems to the same as the ones that arise in partitives that contain count nouns. However, according to the theory advanced in Sect. 6, such an ambiguity is derived from the range of the restricted measure function having both an upper and lower bound. In order to account for the sentences in (15), it must be assumed that the restricted measure functions that apply to sub-aggregates of paint also have a lower bound (i.e., minimal, measurable elements). Such a lower bound could arise due to the denotational nature of mass nouns. For example, perhaps Chierchia (1998, 2010) was on the right track when he hypothesized that all nouns have minimal parts, but only some nouns (i.e., count nouns) have stable atomic parts. Alternatively, perhaps the lower bound is set by the nature of the measure function itself. For example, perhaps the measure functions that derive the non-proportional readings in (15) only map sub-aggregates of paint to milliliters, or some other finegrained measure. In other words, perhaps aggregates that contain less than a milliliter of paint are not within the domain of the measure function to begin with. The viability of this treatment of mass nouns (and mass measure functions) is something that deserves a much richer discussion than I am able to give here. I will have to leave this to future research.

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References Abbott, Barbara. 1996. Doing without a partitive constraint. In Partitives, 25–56. Berlin, Germany: Mouton de Gruyter. Ahn, Dorothy, and Uli Sauerland. 2015. The grammar of relative measurement. In Semantics and linguistic theory (SALT), ed. Sarah D’Antonio, Mary Moroney, and Carol-Rose Little, volume 25, 125–142. Ithaca, NY: LSA and CLC Publications. Ahn, Dorothy, and Uli Sauerland. 2017. Measure constructions with relative measures: Towards a syntax of non-conservative construals. The Linguistic Review 34:215–248. Bale, Alan. 2011. Scales and comparison classes. Natural Language Semantics 19:169–190. Bale, Alan, and David Barner. 2009. The interpretation of functional heads: Using comparatives to explore the mass/count distinction. Journal of Semantics 26:217–252. Bale, Alan, and Bernhard Schwarz. 2020. Proportional readings of many and few: The case for an underspecified measure function. Linguistic and Philosophy 43:673–699. Barker, Chris. 1998. Partitives, double genitives and anti-uniqueness. Natural Language and Linguistic Theory 16:679–717. Bartsch, Renate, and Theo Vennemann. 1972. Semantic structures: A study in the relation between semantics and syntax. Frankfurt am Main: Athen aum. Bresnan, Joan. 1973. Syntax of the comparative clause construction in English. Linguistic Inquiry 4:275–343. Büring, Daniel. 1996. A weak theory of strong readings. In Semantics and linguistic theory 6, ed. Teresa Galloway and Justin Spence, volume 6, 17–34. Ithaca, NY: Cornell University, CLC Publications. Chierchia, Gennaro. 1998. Plurality of mass nouns and the notion of “semantic parameter”. In Events and grammar, ed. Susan Rothstein, 53–104. Dordrecht: Kluwer Academic. Chierchia, Gennaro. 2010. Mass nouns, vagueness and semantic variation. Synthese 174:99–149. Cresswell, M. J. 1976. The semantics of degree. In Montague grammar, ed. B. Partee, 261–292. New York: Academic Press. Gillon, B. 1992. Towards a common semantics for English count and mass nouns. Linguistics and Philosophy 15:597–640.

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Hackl, Martin. 2000. Comparative quantifiers. Doctoral Dissertation, Massachusetts Institute of Technology. Herburger, Elena. 1997. Focus and weak noun phrases. Natural Language Semantics 5:53–78. Hoeksema, Jacob. 1996. Floating quantifiers, partitives and distributivity. In Partitives, 57–106. Berlin, Germany: Mouton de Gruyter. de Hoop, Helen. 1997. A semantic reanalysis of the partitive constraint. Lingua 103:151–174. de Hoop, Helen, and Jaume Solà. 1996. Determiners, context sets, and focus. In Proceedings of the Fourteenth West Coast Conference on Formal Linguistics, ed. Jose Camacho, Lina Choueiri, and Maki Watanabe, 155–167. Center for the Study of Language and Information, Stanford Linguistics Association. Jackendoff, Ray. 1977. X  syntax. Cambridge, MA: MIT Press. Kennedy, Christopher. 1999. Projecting the adjective: The syntax and semantics of gradability and comparison. New York: Garland. Kennedy, Christopher. 2007. Vagueness and grammar: The semantics of relative and absolute gradable adjectives. Linguistics and Philosophy 30:1–45. Kennedy, Christopher, and Louise McNally. 2005. Scale structure, degree modification, and the semantics of gradable predicates. Language 81:345–381. Klein, Ewan. 1981. The interpretation of adjectival, nominal, and adverbial comparatives. In Formal methods in the study of language, ed. J. Groenendijk, T. Janssen, and M. Stokhof, 381–398. Amsterdam: Mathematical Center Tracts. Klein, Ewan. 1991. Comparatives. In Semantik/semantics: An international handbook of contemporary research, ed. A. von Stechow and D. Wunderlich, 673– 691. Berlin: Walter de Gruyter. Ladusaw, William. 1982. Semantic constraints on the English partitive construction. In Proceedings of First West Coast Conference on Formal Linguistics, ed. Daniel Flickinger, Marlys Macken, and Nancy Wiegand, 231–242. Link, Godehard. 1983. The logical analysis of plurals and mass terms: A latticetheoretical approach. In Meaning, use and interpretation of language, ed. R. Baeuerle, C. Schwarze, and Arnim von Stechow, 302–323. Berlin: DeGruyter. Partee, Barbara. 1989. Many quantifiers. In ESCOL 1988. Columbus: Ohio State University. Penka, Doris. 2018. One many, many readings. In Proceedings of Sinn und Bedeuting 21, ed. Robert Trusswell, Chris Cummins, Caroline Heycock, Brian Rabern, and Hannah Rohde, 933–950. https://semanticsarchive.net.

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Reed, Ann. 1996. Partitives, existentials, and partitive determiners. In Partitives, 143–178. Berlin, Germany: Mouton de Gruyter. Romero, Maribel. 2015. The conservativity of many. In Proceedings of the 20th Amsterdam Colloquium, ed. Thomas Brochhagen, Floris Roelofsen, and Nadine Theiler, 20–29. Amsterdam: Institute for Logic, Language, and Computation (ILLC), University of Amsterdam. Romero, Maribel. 2016. Pos, -est, and reverse readings of many and most. In Proceedings of 46th Annual Meeting of the North East Linguistic Society (NELS 46), ed. Brandon Prickett and Christopher Hammerly, 141–154. GLSA (Graduate Linguistics Student Association), Department of Linguistics, University of Massachusetts. https://www.createspace.com/6604179. Rothstein, Susan. 2010. Counting and the mass/count distinction. Journal of Semantics 27:343–397. Schwarzschild, Roger. 2006. The role of dimensions in the syntax of noun phrases. Syntax 9:67–110. Selkirk, Elizabeth. 1977. Some remarks on noun phrase structure. In Formal syntax, ed. Peter Culicover, Tom Wasow, and Adrian Akmajian. New York, NY: Academic Press. Solt, Stephanie. 2009. The semantics of adjectives of quantity. Doctoral Dissertation, City University of New York. Solt, Stephanie. 2015. Q-adjectives and the semantics of quantity. Journal of Semantics 32:1–77. Solt, Stephanie. 2018. Proportional comparatives and relative scales. In Proceedings of Sinn und Bedeutung 21, ed. Robert Truswell, Chris Cummins, Caroline Heycock, Brian Rabern, and Hannah Rohde, 1123–1140. Edinburgh: University of Edinburgh. von Stechow, Arnim. 1984. Comparing semantic theories of comparison. Journal of Semantics 3:1–77. Wellwood, Alexis. 2015. On the semantics of comparison across categories. Linguistics and Philosophy 38:67–101. Wellwood, Alexis. 2018. Structure preservation in comparatives. In Proceedings of Semantics and Linguistic Theory (SALT) 28, ed. Katherine Blake, Forrest Davis, Sireemas Maspong, and Brynhildur Stefánsdóttir, 78–99. Linguistic Society of America. Westerståhl, Dag. 1985. Logical constants in quantifier languages. Linguistics and Philosophy 8:387–413.

Modified Numerals, Vagueness, and Scale Granularity Anton Benz and Christoph Hesse

1

Introduction

If journalists write that More than 20,000 people came to the anti-Corona demonstrations in Berlin, then, semantically, the actual number may be any number higher than 20,000. However, the organisers of demonstrations on 1 August 2020 protested that such statements distributed by the media deliberately underestimate the number of protesters, which they believe to be close to a million or even higher. This complaint obviously draws on a pragmatically motivated inference that is triggered by the phrase more than 20,000. It communicates that the true number is probably closer to 20,000 than to 30,000, and, in any case, below 50,000. A. Benz(B) · C. Hesse Leibniz-Centre General Linguistics, Berlin, Germany e-mail: [email protected] C. Hesse e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_3

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More than 50 years ago, Herbert Paul Grice introduced his theory of conversational implicature in the William James Lectures of 1966/67. What is communicated by a speaker not only depends on what is explicitly coded in language but also on inferences made on the basis of assumptions about the speaker’s linguistic behaviour. These assumptions are formulated as constraints, or maxims, that a speaker is supposed to observe. In a normal utterance situation, the speaker is supposed to be cooperative and contributing to a common purpose of the talk exchange. In particular, speakers are supposed to be truthful and as informative as they can be as long as it is relevant to the conversation; furthermore, they are supposed to avoid marked and unordered utterances. Let us consider, for example, (1). If it is relevant how many students passed, and if the speaker knows the number, then, from an utterance of (1-a), the addressee can infer that (1-b) is false, as otherwise a cooperative and knowledgeable speaker would have said that 9 passed as this would have been more informative. Hence, an utterance of (1-a) pragmatically implies that (1-b) is false. (1)

a. b.

Eight students passed the exam. Nine students passed the exam.

Grice distinguishes two types of implying: entailment and implicature. Which of the two applies to (1) depends on what we take the literal meaning of numerals to be: If we assume that numerals have punctual semantics, then uttering (1-a) entails that (1-b) is false. However if we assume numerals have existential semantics, i.e. an at least n reading, then uttering (1-a) literally means at least 8 passed, which does not exclude greater numbers. Numbers greater than 8 are cancelled by contemplating what the speaker could have said instead of (1-a). If more than 8 passed the speaker should have said so. Therefore, by uttering (1-a) instead of (1-b), (1-a) implicates at least 8 passed but not at least 9, hence it must be exactly 8. This pragmatic inference was called a conversational implicature by Grice (1975). A few years after Grice’s William James Lectures, Laurence R. Horn (1972) worked out a general pattern that this and many other examples follow. He observed that if there is a pair S, W of two alternatives, and if A(S) is a sentence that is logically stronger than the sentence A(W ),

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where A(S) results from replacing S by W in A(S), then an utterance of A(W ) conversationally implicates that the logically stronger sentence A(S) is false. The pair S, W forms a so-called Horn scale. For (1), the alternatives are eight and nine, with eight being the weaker alternative (W ), and nine the stronger (S).1 Hence, according to the general pattern, an utterance of (1-a) conversationally implicates that (1-b) is false, and, therefore, that exactly 8 students passed. Let us turn to comparatively modified numerals. In (2) the numerals in (1) are modified by more than. (2)

a. b.

More than 8 students passed the exam. More than 9 students passed the exam.

Clearly, (2-b) is logically stronger than (2-a). If we apply the same reasoning as before, then an utterance of (2-a) should implicate that (2-b) is false, and, therefore, that exactly 9 students passed. This is obviously not communicated by an utterance of (2-a). This led some to assume that comparatively modified numerals do not give rise to implicatures (e.g. Krifka 1999; Fox & Hackl 2006). One reason is that the modification more than indicates that the speaker does not know the exact number. However, as the introductory example, repeated here as (3-a), shows, utterances with modified numerals do invite inferences about upper bounds.2 (3)

a. b. c.

More than 20,000 people came to the anti-Corona demonstrations. More than 30,000 people came to the anti-Corona demonstrations. More than 50,000 people came to the anti-Corona demonstrations.

A journalist writing (3-a) implicates that, most likely, (3-b) and definitely that (3-c) are false. Modified numerals can convey uncertainty about the 1 In

the neo-Gricean tradition (Horn 1972; Gazdar 1979; Levinson 1983; Horn 1989), it was assumed that a numeral n means n-or-more, i.e. at least n. For convenience, we adopt this semantics for numerals. There are, however, strong arguments that tell against it, see, for example (Geurts 2006; Breheny 2008). 2This depends, of course, on additional assumptions about the speaker’s epistemic state. The speaker should either know the true number, or have, at least, a reliable estimate of it.

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exact number, and, at the same time, an implicature about the most likely value and its upper bound. What exactly are the factors that determine the expectations about these values? In the half century since Grice introduced implicature to the world, there ensued sometimes heated debates about their true nature. Implicature from Horn scales had, for example, been considered context-bound inferences driven by considerations of relevance and processing costs (Sperber & Wilson 1986), a special layer of grammaticalised default inferences (Levinson 2000), or even a part of compositional semantics (Chierchia 2004). In this chapter, we argue that implicatures triggered by comparatively modified numerals are driven by number cognition, and, in particular, by the approximate number sense (Dehaene 2011) and the perceived roundness of numerals. In arguing for the role of the number sense, we follow Solt (2016) who argued that the pragmatic differences of the semantically identical quantifiers most and more than half are rooted in differences in the type of measurement scales used for interpretation: respectively, one with a semiordered structure akin to the approximate number sense, the other one with a more restricted ratio-level measurement structure. In the next section, we consider implicature of modified numerals and their properties. In particular, we recount the results of the seminal experimental study by Cummins et al. (2012). We then discuss three models that try to explain their data (Cummins et al. 2012; Cummins 2013; Benz 2015). As a point of departure, we take Cummins et al.’s hypothesis that the distance between modified numerals and upper bounds are monotonically increasing with the roundness level of the numerals. We will argue that this view does not take into account the effect of number cognition. In Sect. 3 we discuss the results of Hesse & Benz (2020), which support our argument.

2

Implicatures from Modified Numerals

Implicatures arising from comparatively modified numerals have some peculiar properties that make them an interesting topic of study. As we have seen, if a journalist writes that more than 20,000 people came to the anti-Corona demonstrations, then this communicates an upper bound on

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39

the number of people demonstrating. Which upper bounds are inferred by readers may depend on expectations about the journalist’s sources. Most likely, it is an official estimate given by police. If one assumes that police estimates are exact up to 10 or 20%, then 30,000 is an absolute upper bound, and the most likely number is somewhere between 21 and 25 thousand. Hence, we see that modified numerals give rise to two types of implicature: about upper bounds and about most likely values. Taking the lead from scalar implicature of unmodified numerals, one would expect that implicatures about upper bounds are generated by negating stronger alternatives that result from replacing the modified numeral by another, larger numeral. In (4), this is done for the number 90. (4)

a. b. c. d.

More than 90 people signed up today. More than 91 people signed up today. More than 92 people signed up today. More than 93 people signed up today.

If the speaker says more than 90, then she may consider 91 to be a possible value. However, it is inconsistent with intuition that she knows that 91 is the exact number. Intuition indicates that she believes that it is possible that (4-b) is true. If the number is increased one by one, as in (4-b) to (4-d), then there seems to be no cut-off point n where the speaker believes that it is possible that more than n becomes false. However, if we increment the number by 10 instead of 1, as in (5), the picture changes. (5)

a. b. c. d.

More than 70 people signed up today. More than 80 people signed up today. More than 90 people signed up today. More than 100 people signed up today.

Here it seems that a speaker who asserts (5-a), implicates with her utterance that (5-d) is false. Cummins et al. (2012, p. 142) predict that (5-c) implicates that (5-d) is false, and that (5-a) implicates that (5-b) is false. They assume that the distance to the upper bound depends on the granularity level of the modified number such that the upper bound must belong to the same granularity level or a coarser one. For example, 100 is predicted to be an upper bound for more than 90 because 100 is on a

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coarser granularity level than 90. However, 91, 92, and 93 are on a finer granularity level, and, hence, cannot be upper bounds for more than 90. For the prediction that 80 is an upper bound for more than 70, one has to consider the notion of granularity level in more detail. Granularity is related to the notion of roundness. Numbers on a coarser granularity level are rounder than numbers on a finer granularity level, except for those numbers that belong to both levels. There is a rich literature on what counts as roundness. With Cummins et al. and others, we rely here on the study by Jansen & Pollmann (2001) that considered frequencies of numerals in a cross-linguistic corpus of newspaper articles. As an indicator of the degree of roundness Jansen & Pollmann considered the frequencies in approximation contexts created by, for example, Dutch ongeveer, German etwa, French environ, and English about. The more frequent a numeral is in such contexts, the rounder it is assumed to be. Simplifying the picture somewhat, Jansen & Pollmann found that the level of roundness depends on the divisibility by numbers 2, 5, 10, 20, 50, and 100. For our purpose, we can say that a number n is rounder than number m, if the largest number among 2, 5, 10, 20, 50, 100 that divides n without remainder does not divide m without remainder. The least round, or non-round numbers are numbers that are divisible by neither 2 nor 5, e.g. 91, 92, 93, etc. With this criterion, we can see that 100 is rounder than 70, 80, and 90, and that 80 is rounder than 70. The criterion of Cummins et al. then says that m qualifies as upper bound for more than n if m is the next larger number that is as round, or rounder than n. This explains why 80 is an upper bound for more than 70 : 80 is rounder than 70 as it is divisible by 20, but 70 is not. A consequence of Cummins et al.’s criterion is that the distance from the modified numeral to its upper bound increases with its roundness level. For example, the average distance of a number between 20 and 200 with highest divisor 10 to the next rounder number is 10, that for numbers with highest divisor 20 is 25, and that for number with highest divisor 50 is 50. Cummins et al. conducted three experiments that seemed to confirm this hypothesis, and the correlated hypothesis that also the distance to the most likely value increases with roundness level. In their experiments, test subjects had to answer a questionnaire on the Amazon Mechanical Turk platform. The items consisted of a short statement containing a modified

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41

numeral followed by a question which asked subjects to estimate the true number. An example is shown in Fig. 1. Information A newspaper reported the following. “[Numerical expression] people attended the public meeting about the new highway construction project.” Question Based on reading this, how many people do you think attended the meeting?

Between tion]

and

people attended [range condi-

people attended [single number condition] Fig. 1 Example item from Cummins et al. (2012, p. 146)

Cummins et al. distinguished two conditions: in the single number condition, subjects were required to provide a natural number for the most likely value; in the range condition, they had to provide a lower and an upper bound (between m and n) for the true number. The experimental data overall support their introspectively motivated claims as can be seen in Fig. 2. The results show that comparatively modified numerals invite inferences which bear similarity to standard scalar implicature. In particular, subjects feel entitled to make guesses about the intervals which contain the true values. What counts as an alternative to a modified number is determined by the number’s roundness level. One caveat, however, is that a particular discourse domain might raise strong expectations about potential values. Take a dialogue about the length of a new movie for instance. If the speaker uses a comparatively modified numeral to describe its runtime, It’s more than 2 hours long, interlocutors’ world knowledge helps them infer an upper bound not much higher than 3 hours. Cummins et al. (2012) therefore selected contexts that would not come with this sort of strong expectation. Hesse & Benz (2020) determined for their scenarios that there were no strong expectations involved through a pre-test.

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'less than n'

40 0

20

'more than n'

-20

Distance of most likely value from n

-50

0

50

'more than n'

Cummins et al. (2012) Hesse & Benz (2020)

'less than n'

-40

Cummins et al. (2012) Hesse & Benz (2020)

-100

Median range size of inferred values

100

A. Benz and C. Hesse

Fine

Medium Coarse

(a) Range size by granularity; variance is MAD.

Fine

Medium Coarse

(b) Distance by granularity; variance is MAD.

Fig. 2 Median range size and median distance to most likely value

Cummins et al. (2012) explained implicatures from modified numerals as a kind of weaker implicature without being too explicit about their exact status. In a theoretical article, Cummins (2013) proposed an optimality theoretic production model, and Benz (2015) worked out a pipelined production model based on the same intuitions as Cummins et al. that also relied on Jansen & Pollmann’s roundness criteria to explain their experimental data. A problem for the evaluation of any theoretical model is the limited data provided by Cummins et al.’s study. For example, in their first experiment they considered only three numbers (93, 100, 110), and in the second six (60, 77, 80, 93, 100, 200), so that they provide data on only five round numbers. To remedy this problem, Hesse & Benz (2020) set out to collect data on all round numbers between 20 and

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200.3 We will discuss the results in more detail in the next section. The theoretical models and the predictions of Cummins et al. were based on a generalisation of the Hornian account of scalar implicature, the key step of which is a negation of stronger scalar alternatives. Before we turn to the experimental study of Hesse & Benz, we want to get some intuitions why the neo-Gricean paradigm may mislead in the case of comparatively modified numerals. Let us return once more to the journalist writing that More than 20,000 people came to the anti-Corona demonstrations in Berlin. The news is based on a police report, and readers interpret the statement with an expectation about accuracy about the police’s estimates.4 It is unlikely that the police was able to get a precise headcount, rather they made their statement based on a perceptual estimate which requires the use of the visual system and number cognition. Number estimates come with a margin of uncertainty. In the case of a large demonstration, this uncertainty may be in the range of a few thousand, in case of numbers between 20 and 200, it may be in the range of 10–20. However, as the police example shows, the actual margin may also be a bit wider. For the sake of argument, let us assume that the accuracy of number estimates is about 10 for numbers between 20 and 200, and let us consider the examples in (5) again. If it is said that more than 90 people got married today, then with an assumed accuracy of 10%, the speaker can only say this with confidence if the actual number is close to 100, and she can switch to more than 100 only if it is at least 110. The same consideration applies to all other numbers, which predicts that the distance to the upper bound is at least 20. Hence, the prediction of Cummins et al. (2012) and Benz (2015) that more than 70 (5-a) implicates at most 80, and more than 90 (5-c) at most 100 is wrong. Now, let us consider more than 70 (5-a) more closely. The upper bound should be at least 90 if accuracy of estimates is 10. With 90, the upper bound is already close to the very salient number 100. Subjects taking part in an 3The aim was to test the models of Cummins and Benz. Already the pre-tests showed that, in particular, Benz (2015) is not consistent with the data. The main experiments were then run as an explorative study. 4The police actually reported first that about 20,000 people took part. This was later corrected to more than 20,000, and then to fewer than 30,000 (see Tagesspiegel 28.08.2020). This just confirms that there is a certain margin of uncertainty in these estimates.

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experiment can also be expected to be influenced by roundness, so that 100 is a strong attractor for their judgements about upper bounds. They may also allow for an additional error margin that further supports the choice of 100 as upper bound. This means that the distance from more than 70 to its upper bound should be widened by the influence of the attractor 100, whereas the distance from more than 80 to its upper bound should not be influenced. For more than 90, the estimates for the upper bounds should anyway spill over the boundary 100. As there is no equally salient attractor nearby, subjects may add a safety margin and end up at a distance which is yet again higher than for 80. Hence, instead of a long interval for 80 and shorter ones for 70 and 90, as predicted by Cummins et al. and Benz, the pattern may just be the reverse. The most likely value should also be influenced by the expected accuracy of number estimates. If the speaker tries to be as informative as possible, and if the accuracy is about 10, then the most likely value of the true number should be about 10 above the modified numeral.5 If the influence of round attractors could be excluded, then one would even predict that the distance of the most likely value to the modified numeral n is independent of the roundness level of n. According to the alternative account that we just sketched out, estimates for upper bounds and most likely values are determined by the expected accuracy of number estimates and the salience of attractors beyond the minimal bound determined by accuracy. So far, this is just a plausible story that may or may not be true. In contrast to the neo-Gricean accounts it has a stronger connection to insights in number cognition. As argued by Hesse & Benz (2020), the accuracy of number estimates may follow the Weber-Fechner law (Fechner 1860) and be rooted in the approximate number sense (Dehaene 2011). In the next section, we consider the results of Hesse & Benz’s study. As we will see, it generally supports the alternative account.

5 Assuming

the speaker has no preference for either rounding or accurate estimates.

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3

45

The Experimental Evidence

Cummins et al. (2012) found that scalar implicatures are available from modified numerals such as fewer/less than n and more than n at specific levels of scale granularity. This finding disproved the assumption generally held by linguists until then that numerical expressions such as more than n, at least n, and fewer/less than n fail to give rise to scalar implictures in embedded declarative contexts. They also found a straight-forward linear correlation between n’s roundness and range, size, and likelihood of values implicated by the modified numeral: the rounder the modified numeral n, (1) the bigger the range of expected values is, and (2) the farther the most likely of those values is expected to be away from n. In Hesse & Benz (2020), we investigated whether these findings could be reproduced and extended to comparatively modified numerals beyond those granularity levels tested by Cummins et al. (2012).

3.1

Methods and Procedure

Experimental items came in the form of 16 short dialogues between speaker A and B as originally used in Cummins et al. (2012). For example: Speaker B’s answer to A’s question contains a modified numeral. Participants had to indicate the lower and upper bound of the range of values they inferred from the modified numeral, as well as the most likely value in that range. For example, for more than 100 signatures we would expect that the range should stretch from 100 as the lower bound to, say, 150 as upper bound, with 110 being the most likely value in that range. These three responses give us a rough impression of the probability distribution over potential values raised by the modified numeral. It also makes our results directly comparable to Cummins et al. (2012) who elicited the same response variables (Fig. 3). Using the concept of scale granularity Cummins et al. (2012) distinguished three degrees of roundness in their study and had two numerals per roundness level: • 77 and 93 as examples of fine scale granularity (unround)

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A: We wanted new signatures on the petition. How many new people signed it? B: [modified numeral] new people signed it. How many new people signed the petition? From to , most likely . Fig. 3 Example item from Hesse & Benz (2020)

• 60 and 80 as examples of medium granularity (medium round) • 100 and 200 as examples of coarse granularity (very round). We were curious whether the predictions made by Cummins et al. (2012) about roundness could be extended to other numerals than the ones they used, and so we systematically tested all multiples of ten in the same range as they had tested, between 20 and 200, which gave us more example numerals for medium granularity: • 93 as an example of fine granularity (unround) • 20, 30, 40, 60, 70, 80, 90, 110, 120, 130, 140, 160, 170, 180, 190 as examples of medium granularity (medium round) • 100, 200 as examples of coarse granularity (very round). We also tested 50 and 150, but since they are divisible by 50 as well as ten, we consider them examples of an intermediate roundness level inbetween what Cummins et al. (2012) called medium granularity and coarse granularity. In direct comparisons to Cummins et al.’s (2012) results we omit them for that reason, but include them where our explanations go beyond Cummins et al. (2012). Every participant saw all 16 dialogues in a 3 granularity/roundness levels × 2 modifiers design. We tested 420 adult American English native speakers on Amazon Mechanical Turk (53% female) for the range of numerals tested by Cummins et al. (2012), and another 200 adult American English native speakers (48% female) for numerals with powers of ten larger than those tested by Cummins et al. (2012).

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Participants may have one of two readings of comparatively modified numerals: (i) they may interpret them purely semantically, i.e. not infer scalar bounds, or (ii) they pragmatically infer a scalar bound. The ratio between the number of participants with an unbounded interpretation to those with a bounded interpretation tells us which of the two readings is more prevalent. In the case of purely semantic, unbounded interpretation the readings could be paraphrased as (a) fewer than/less than n  any number smaller than n, (b) more than n  any number larger than n. Participants may indicate that they are not inferring a bound by choosing zero as the lower bound for fewer/less than and a very large number as the upper bound for more than. Participants were also able to leave a comment, which some of them used to clarify when they did not infer a boundary. In the case of pragmatic interpretation where participants infer a bound, participants may indicate a bounded interpretation by choosing a nonzero lower bound for fewer/less than and an upper bound for more than which is closer to n. Cummins et al. (2012) used 1000 as the cutoff point for upper bounds on more than. We found that in our experiments, participants with an unbounded interpretation used much larger upper bounds, and participants with bounded interpretation used upper bounds below 500. In our study, we are trying to reproduce the findings made by Cummins et al. (2012). They found a correlation between the roundness of the modified numeral n and (1) the size of the range of expected values, and (2) the distance of the most likely value from n: They found that more round numerals had larger ranges of expected values and that participants expected the most likely value to be farther away from n. We look at the two correlations in turn, comparing the results in Hesse & Benz (2020) and Cummins et al. (2012).

3.2

Range of Expected Values

Similar to Cummins et al. (2012), we found that the overwhelming majority of participants, 81%, have a bounded interpretation of comparatively modified numerals. In addition, Cummins et al. (2012) found a

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'less than n'

60

'more than n'

Median range size of inferred values -40 -20 0 20 40

-50

0

50

'more than n'

-60

Cummins et al. (2012) Hesse & Benz (2020)

'less than n'

-100

Median range size of inferred values

100

linear correlation between a modified numeral’s roundness and the size of the range of potential values that speakers infer from it: the rounder the numeral—or in Cummins et al.’s terms, the coarser the granularity of the measurement scale which the modified numeral applies to—, the farther the extreme end of the range of expected values moves away from the modified numeral, and thus the bigger the range of expected values is.

25

Fine

Medium Coarse

(a) Range size by granularity; variance is MAD.

50

75 100 125 150 Absolute magnitude of n

175

200

(b) Median range sizes for particular modified numerals n; variance is bootstrapped.

Fig. 4 Median range size of the range of values inferred and expected for modified numeral n, i.e. the distance between the modified numeral and the boundary respondents indicated. Reprinted from JML 111, Hesse & Benz, Scalar bounds and expected values of comparatively modified numerals, Copyright [2020], with permission from Elsevier

Looking at range size per roundness level, we find the same linear correlation as Cummins et al. (2012). Figure 4a shows the median range size for each of the three granularity levels (fine, medium, coarse), comparing our results (Hesse & Benz, 2020, white bars) to the range sizes reproduced from Cummins et al. (2012, gray bars). However, if we look at modified numerals individually, in Fig. 4b, median range size does not correlate with modified numerals’ roundness as predicted. For instance, more than 150 has a larger range than more than 100 and more than 200 although 100 and 200 are arguably more round than 150. In fact, the range for more than 100 and more than 200 is hardly bigger than the range for more than 40, 50, 70, 140 or 160, all of which are less round than 100 and 200. Less

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Table 1 Modified numerals’ most common boundaries. Reprinted from JML 111, Hesse & Benz, Scalar bounds and expected values of comparatively modified numerals, Copyright [2020], with permission from Elsevier n

bnd

20 10 30 20 40 30 50 40 60 50 70 50 80 50 90 80 100 80 110 100

Less than n n 120 130 140 150 160 170 180 190 200

bnd

n

100 100 100 100 100 100 150 150 150

20 30 40 50 60 70 80 90 100 110

More than n bnd n 30 50 60 100 100 100 100 100 120 120

120 130 140 150 160 170 180 190 200

bnd 150 150 200 200 200 200 200 200 250

than 170 has the largest range size of all numerals modified by less than, a larger range than for 150, which is more round than 170, and larger than for the rounder 100 and 200. The pattern dictating range size in Fig. 4b only becomes apparent once we look at the boundaries that participants pick. When we look at the most common boundaries of each modified numeral in Table 1, we see that (1) all boundaries are salient, round numbers which are closest to the modified numeral, and we see that (2) consecutive modified numerals often have the same round boundary. So when participants prefer round boundaries in their responses which are closest relative to the modified numeral’s position on the number line, then range size changes accordingly and relative to these preferred boundaries. Although it is not the modified numeral’s roundness that plays a role in controlling range size, roundness does play a role when it comes to which potential boundaries speakers consider: speakers prefer boundaries which are round, and range size changes accordingly. To put it differently, we can say boundaries come from a scale with coarse granularity, which is only populated by the very same round boundaries from which speakers choose (Krifka 2009). Table 1 also suggests that the modified numeral and the preferred boundary may form a Horn scale (1972) where, e.g., less than 170 implicates

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less than 170, but not less than 100, where less than 170 (the weaker alternative) and less than 100 (the stronger alternative) form the pair of the Horn scale. Importantly, though, Cummins et al. (2012) predict that a potential boundary for less than 170 should be of the same granularity or coarser granularity as 170: this would give 160 as potential boundary (granularity 10 and 20) or 150 (granularity 50). The most common boundary 100, however, is of much coarser granularity, three granularity levels more coarse at least (10, 20, 50, 100). However, what we propose is that saliency of potential boundaries follows from their roundness (more round boundaries are more salient) and the modified numeral’s proximity to them. Summarising, we can make three general statements about boundaries and the range of expected values: 1. The boundary of the range of expected values needs to be at least twice the distance of the most likely value away from the modified numeral. 2. The far bound can be quite far away from the modified numeral, but tends to be within the numeral’s order of magnitude (power of ten) or neighbouring order of magnitude. 3. There is a strong preference for round numbers as range boundaries in close proximity to the modified numeral, as long as 1 is not violated. If we think of the round boundaries we observed in the study populating a coarse-grained scale along a mental representation of the number line (Krifka 2009), then it is purely a modified numeral’s relative position on this mental number line and its proximity to potential boundaries that dictates how likely a boundary is to be preferred by speakers.

3.3

Most Likely Value

Based on their results, Cummins et al. (2012) concluded that there is a linear correlation between a modified numeral’s roundness and how far away from the numeral speakers expect the most likely value (among potential values) to be. The median distances of most likely values per granularity—i.e. roundness—level (fine, medium, coarse) from their study are reproduced as gray bars in Fig. 5a. We see the coarser granularity (i.e. the

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'less than n'

Fine

Medium Coarse

(a) Distance by granularity; variance is MAD.

10% 100% 1%

'less than n'

10% 100%

'more than n'

'more than n'

1%

Cummins et al. (2012) Hesse & Benz (2020)

Distance of most likely value from n relative to n's order of magnitude

Distance of most likely value from n -40 -20 0 20 40

more round) the modified numeral, the farther speakers expect the most likely value to be. At first glance our results (the white bars in Fig. 5a) seem to corroborate Cummins et al. (2012): the most likely value is closer for the fine-grained, unround numerals than for medium round numerals. But notice that the median distance and variance for coarse, very round numerals is much closer to the modified numeral than in Cummins et al.’s results. And while our results for the medium level almost perfectly match Cummins et al.’s—despite us having tested many more examples for this level—median distance is the same as in the coarse level. With the exception of unround 93, the distance of the most likely value is ten for both medium granularity and coarse granularity, contrary to the findings by Cummins et al. (2012). For less than 93 the most likely value is 90, for more than 93 100, speaking to the salience and roundness of 90 and 100.

Tens

Hundreds

ThouTens of sands thousands

(b) Distance relative to n’s order of magnitude, i.e. its power of ten.

Fig. 5 Distance of most likely value relative to modified numeral n. Reprinted from JML 111, Hesse & Benz, Scalar bounds and expected values of comparatively modified numerals, Copyright (2020), with permission from Elsevier

In a follow-up experiment we tested 200 adult American English native speakers (48% female) for numerals with powers of ten larger than those investigated by Cummins et al. (2012) in order to see if the most likely value would correlate with modified numerals’ roundness for those orders of magnitude. The results of the follow-up study suggest that the

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distance of the most likely value scales according to the numeral’s order of magnitude: • • • •

when n is in the order of tens, the most likely value’s distance = 10 when n is in the order of hundreds, distance = 10 when n is in the order of thousands, distance = 100 when n is in the order of tens of thousands, distance = 1000.

Expressed as a fraction—distance of most likely/n’s power of ten—this ratio hovers around 10% across all levels of scale granularity (Fig. 5b, with the exception of the smallest granularity (unround level)). This scaling with the magnitude or intensity of a measured variable is a well-known phenomenon in psychophysics. It appears frequently in human perception where small changes in, say, brightness or loudness go unnoticed while large differences are perceived. This limit on the perceptual resolution is called the just noticeable difference. The just noticeable difference obeys the Weber-Fechner law (Fechner 1860) and displays two characteristic effects: (1) when two values of the measured variable are close together, they are perceived to be the same, but are perceived as different when they are further apart (distance effect ), and (2) two differences, e.g. 12 : 6 and 120 : 60, are similarly noticeable because the relative differences, i.e. the ratio 2 : 1 in this example, is the same for both (size effect ). We therefore concluded in Hesse & Benz (2020) that the ratio we see underlying the distance of modified numerals’ most likely value is in fact a Weber fraction. Importantly, this Weber fraction does not make reference to the modified numeral’s roundness but only to its order of magnitude (i.e., power of ten). We therefore concluded that the modified numeral’s roundness is not a necessary or sufficient predictor of the most likely value. It is well-established in the field of number cognition that humans (among other animals) have two cognitive systems for numerosity: a precise number system and an approximate number system (ANS). The approximate number system can only perceive approximate numerosity because the resolution of the ANS is limited by the Weber-Fechner law (Dehaene 2011). From early research in number cognition (Buckley & Gillman 1974) it is well-established that symbolic representation of number does not mean the precise number system is used; the ANS is nev-

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ertheless at play when symbolically represented numbers are of greater magnitude. The acuity of the ANS in literate societies, required for reliable differentiation of two quantities, has been estimated to be between 7 : 8 and 10 : 11 (Pica et al. 2004; Halberda & Feigenson 2008), which is strikingly similar to our estimate of where most participants placed the most likely value. We can see from this that our experimental participants placed the most likely value where the resolution of the approximate number sense would raise intuitions in speakers that a change in quantity is noticeable. In other words, the distance of the most likely value is the just noticeable difference in numerosity scaled properly to a modified numeral’s order of magnitude.

4

Discussion

In the previous section we supplied empirical evidence which supports the alternative account we sketched in the beginning. When we take stock of the findings, it becomes clear that the approximate number sense (ANS) plays a central role in the bounded interpretation of comparatively modified numerals. This is at least true for the discourse domains we and Cummins et al. (2012) used in our experiments and where speakers do not have strong expectations about potential values. So if participants could not use world knowledge to constrain the range of expected values, the fact that the overwhelming majority of them had a bounded interpretation of comparative scalar quantifiers suggests that there must have been another mechanism for boundedness. We saw in the experimental results that speakers select boundaries on the range of potential values by looking at the proximity of the modified numeral to salient, round numbers on the number line. The scale populated by these potential boundaries could have the multi-tiered structure suggested, among others, by Krifka (2009) where each tier corresponds to a specific level of roundness with its own level of granularity: Similar to the corpus results by Jansen & Pollmann (2001), there would be round boundaries for multiples of 10, 20, 50, 100, and so forth, with rounder boundaries being more attractive. Cummins et al. (2012) assumed that boundaries come from the same scale or a coarser scale than the modi-

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fied numeral. However, if we accept this conjecture that boundaries come from a scale structure independent of the modified numeral’s scale, but one that aligns itself with the mental number line, then we can separate the other two observations we made about boundaries from this observation of boundaries’ roundness. Notice how the other two observations about boundaries only make recourse to aspects relevant to the ANS: The range of potential values needs to be at least twice as big as the most likely value is away from the modified numeral. Since we saw independently that the distance of the most likely value follows the Weber-Fechner law and hovers around the ratio of 10% where Pica et al. (2004) and Halberda & Feigenson (2008) place the acuity of the ANS in adult, literate humans, this means that the minimum range size is also driven by the ANS. We recaptured Cummins et al. (2012) observation that boundaries tend to be of the same or coarser granularity than the modified numeral by observing that they tend to fall in the same or neighbouring order of magnitude (power of ten) as the numeral. But crucially, by phrasing the observation this way we capture the role the Weber fraction plays here where the modified numeral’s order of magnitude is the denominator. So in our view, the way we capture observations about the minimum and maximum range size exclusively make reference to the ANS and never to a modified numeral’s roundness. Solt (2016) looked at the different interpretations of more than half and most, proposing that although they may have an equivalent logical form (somewhere between 50 and 100%), more than half implicates a ratio closer to 50% than most. Solt proposes that this is because more than half and most use scales with different structures for interpretation: Most, she argues, does not require the strict structure of an ordinal ratiolevel measurement scale that more than half requires. Instead, most merely requires a semiordered scale where differentiability of values (and thus their ordering) is limited by a tolerance on a sameness relation ∼. Interestingly, Solt argues that the less strict requirements on the scale structure needed for the interpretation of most shows characteristics of the ANS.

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If the involvement of the ANS is a telltale sign that the scale required for interpretation has a tolerant, semiordered structure à la Solt (2016), then it would seem that the interpretation of comparatively modified numerals does require a semiordered scale structure, too (albeit perhaps with more discriminatory power of ∼ than in the case of most ). This goes against an assumption commonly made in degree semantics that comparative scalar quantification presupposes a purely ratio-level scale structure. If this proposal is true, however, it would mean that the ANS not only plays a role in the interpretation of superlative scalar quantifiers (Solt 2016), but also in the interpretation of comparative scalar quantifiers (Hesse & Benz 2020), and that both types of scalar quantifiers require a semiordered scale. The details of this sketch will have to be worked out in future research, but we hope that it offers steps towards a generalisation of comparative and superlative scalar quantifiers at the semantics–pragmatics interface by incorporating insights from number cognition. Another issue that has to be addressed by future research is the influence of the speaker’s knowledge. As Cummins (this volume) points out, if it is common knowledge that the speaker knows the exact number, then a modified numeral may license inferences of much stricter upper bounds. For example, more than 70 may allow the inference that more than 80 is false. The data provided by Cummins et al. (2012) and Hesse & Benz (2020) cannot rule out that strong assumptions about the speaker’s epistemic states may license stronger implicatures.6 Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft (Grant Nrs. BE 4348/4-2 and BE 4348/5-1). Informed consent was obtained for experimentation with human subjects and their privacy rights are observed by anonymising. We would like to thank Stephanie Solt, Uli Suaerland, Chris Cummins, and two anonymous reviewers for their insightful comments, and the editors of this volume. Both authors have contributed equally.

6 In

neither experiment participants were instructed to assume that speakers know the exact, true numbers. For example, (Cummins et al. 2012, p. 166) asked participants only to assume that the speaker is well-informed, telling the truth, and being co-operative in each case.

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Uncertainty, Quantity and Relevance Inferences from Modified Numerals Chris Cummins

1

Introduction

The use of numerically quantified expressions in natural language gives rise to a wide range of subtle interpretations that are not easily accounted for. This subtlety reflects a combination of several factors, including the potential for numerals themselves to vacillate between cardinal and existential senses (Carston 1998; Breheny 2008), the surprisingly complex semantic contribution of ostensibly straightforward numeral modifiers such as at least /most (Geurts and Nouwen 2007), some (Solt and Stevens 2018) and no(t) (Nouwen 2010; Solt and Waldon 2019) and the rich landscape of possible alternative expressions that naturally arises as a consequence of the structure of the number system. In the case of expressions of the form more than n, the complexity of interpretation is apparently due, principally, to this last point. In more C. Cummins (B) University of Edinburg, Edinburg, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_4

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than n, the semantic contribution of the numeral n is firmly punctual (Solt 2014), and that of more than appears broadly to correspond to the mathematical operator > (Geurts et al. 2010). The same appears to be true, mutatis mutandis, for fewer/less than n. Even so, the kinds of interpretation—hence, we assume, pragmatic enrichment—available from more than n seem to vary depending upon the properties of n as a numeral and the overall architecture of the number line in the vicinity of n. For instance, (1) does not implicate the falsity of (2), whereas (3) appears to implicate the falsity of (4). (1) (2) (3) (4)

More than four people attended. More than five people attended. More than forty people attended. More than fifty people attended.

Cummins et al. (2012) propose that the implicatures arising from more than n are constrained by considerations of the roundness, or salience, of the numerals in the informationally stronger alternatives. For instance, (5) entails (3), but uses a numeral that is less round (and, relatedly, more verbose to express), and this is why (3) fails to implicate the negation of (5). (5) More than forty-three people attended. Treating this as scalar implicature, we could think of this as a matter of equal lexicalisation: just as < warm, hot > form a scale but < hot, very hot > do not, we might argue that < more than forty, more than fifty > form a scale but < more than forty, more than forty-three > do not. In the latter case, the hearer might reasonably infer that the complexity of that stronger alternative is the likely explanation for the speaker’s failure to use it, rather than any concern on the speaker’s part as to the truth of the resulting utterance. Cummins et al. (2012) demonstrate the availability of pragmatically inferred upper bounds for more than n by eliciting hearers’ judgements of the range of possible values to which such a phrase could refer, as well as the point value that the hearers consider most likely. They posit

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that these upper bounds are inferred because the hearer is aware that the speaker would likely have selected a stronger alternative—specifically using a round number, defined following Jansen and Pollmann (2001)— if the speaker was confident of the truthfulness of the resulting statement. The role of roundness in this interpretative process is challenged by Hesse and Benz (2020). They adopt a similar elicitation task to Cummins et al. (2012), but use it to survey a wider selection of values, from more than 20 to more than 25,000, across a series of experiments. On the basis of their results, Hesse and Benz (2020) argue that the range of interpretation given by the hearer is principally predicted by the size of the number involved, rather than its roundness. They note more specifically that “respondents estimate the peak of the likelihood distribution is within a fixed distance from the modified numeral relative to its order of magnitude” (ibid.: 14). They propose that this pattern is ultimately rooted in discriminability considerations, and has to do with the acuity of the Approximate Number System (ANS), a system for imprecise quantity representations, which has also been argued to be invoked by most (Solt 2016). Hesse and Benz (2020: 2) suggest that the ANS “bootstraps scalar implicatureof comparatively modified numerals in impoverished contexts”. In this chapter I attempt to explore the interplay between these two potential sources of pragmatic enrichment.

2

Discriminability Versus Scalar Implicature

It is intuitively appealing to think that the range of values felicitously described by an expression such as more than n is connected in some way to the magnitude of n, with larger values of n generally associated with wider ranges of potential values. Cummins et al. (2012) propose one way to approach this, specifically by appeal to the notion of “roundness” suggested—on the basis of corpus data—by Jansen and Pollmann (2001). On Jansen and Pollmann’s view, the property of roundness is exhibited (only) by the first ten multiples of two, five, ten, 20, (potentially 25), 50, 100, 200, (potentially 250) and so on. That is to say, the

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roundness of a number is not merely a matter of its divisibility properties, but of its divisibility properties given its magnitude.1 The crispness of this definition gives rise to potentially counterintuitive predictions: for instance, 90 is predicted to be round on this account, while 110 is not. While there is some gradience in the notion of roundness offered by Jansen and Pollmann (2001), their approach categorically considers 110 as non-round, and on a par with 111 in that respect. Hesse and Benz (2020) adopt a subtly different approach, initially placing roundness on a five-point scale with the most round numbers being those divisible by 100, the next most round being those divisible by 50 and not by 100, and similarly for the other levels; they then suggest a couple of ways of generalising this kind of approach to larger numbers. It is also intuitively appealing to think that discriminability should play a role in the interpretation of more than n. But it is less clear how the ANS could truly “bootstrap” scalar implicature—and it is perhaps worth separating out these two ideas. I will try to do so by appeal to an example. Suppose that a speaker is asked for information about a numerical quantity—the number of people in the large, crowded room in which they are standing—and chooses to answer in the form more than n. If the speaker is reliable, in the sense of adhering to Grice’s maxim of Quality, they should be unwilling to provide information that they consider might be false. Moreover, if they are relying on their ANS, rather than counting the people in the room, their perception of the number is limited by the acuity of this system, which Hesse and Benz (2020: 14) take to be a Weber fraction of around 0.15 in adults—that is to say, “the average adult would notice without counting if a quantity of…100 items had

1 A reviewer queried whether we are in fact talking about divisibility given order of magnitude. The short answer is: not exactly. We typically say that two numbers are of the same order of magnitude if they belong to the same class with respect to a particular logarithmic base, usually 10. Thus, for instance, 16 and 32 are of the same order of magnitude, both lying between 101 and 102 . But on Jansen and Pollmann’s (2001) definition, 16 is round (because it is one of the first ten multiples of two) while 32 is not. Conversely, if we chose, say, base 2 in our definition of order of magnitude, 60 and 80 would be of different orders of magnitude (they fall on opposite sides of 26 ), but definitionally exhibit the same roundness on the basis of the same divisibility properties.

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another 15 items added or more but not if the difference was fewer than 15 items”. Thus, for a speaker to be comfortable saying more than 100 while adhering to Gricean Quality under these circumstances, the true quantity must be at least 115 (more precisely, 118, as 100 is 15% below this), because otherwise the speaker could not be confident in their perception that it exceeds 100. In that case, they would run the risk of making a false statement, misled by the imprecision of their number sense.2 The hearer, alert to this limitation and wishing to estimate the true value, should add 15% (or rather 17.6%) to the stated value in order to compensate for the speaker’s caution. Thus, it would be reasonable for a hearer to suppose that a speaker who says more than 100 under such circumstances must have perceived a quantity that is in fact in excess of 115. In this scenario, could we say that this reasonable and rational hearer has recovered a quantity implicature? No—in fact, considerations of quantity, in the Gricean sense of the word, do not enter into the hearer’s reasoning, as sketched above. It is not relevant whether or not there are stronger alternatives that could have been uttered by the speaker. The only assumption on which the hearer’s inference rests is that the speaker is adhering strictly to Quality. Indeed, assuming that the speaker has a Weber fraction of 0.15, is aware of the limitations of their perception, and is speaking truthfully, the fact that the speaker must have perceived a quantity in excess of 115 is entailed, not implicated, by their utterance of more than 100. Now, suppose instead that a speaker is discussing a quantity of which they have precise knowledge—they counted the people in the room, in the situation described above—but chooses nevertheless to use an expression of the form more than n to express it. In this circumstance, the hearer does not need to consider the speaker’s acuity, in terms of their perception of numerosities. However, they may be entitled to draw an inference about the falsity of alternative, stronger utterances that the speaker could 2 I assume here that discrimination between quantities at one Weber fraction’s distance is sufficiently reliable for a conscientious speaker to rely upon it—that is, if the quantity is more than one Weber fraction above n, the speaker is willing to say more than n. Given the definition of the Weber fraction this too is an oversimplification, although it is not crucial for the current purpose.

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have produced—that is to say, a quantity implicature. This appears to be a completely different kind of inference from that discussed in the preceding paragraphs, and indeed it relies on a disjoint set of assumptions about the situation at hand, and the basis for the speaker’s knowledge. It is certainly striking that the distribution of round numbers appears to align well with the Weber fraction of 0.15. Indeed, if we adopt Jansen and Pollmann’s (2001) definition (ignoring the role of multiples of 2.5), the round numbers between 10 and 100 inclusive are 10, 12, 14, 15, 16, 18, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90 and 100. The percentage increase between consecutive terms within this sequence varies, but the mean difference is 14.6%; and given Jansen and Pollmann’s treatment of powers of 10, this is in fact true of the whole number line thereafter. That is to say, round numbers happen to correspond roughly to numerosities that are just distinguishable from each other, reliably, by adult humans. Is this a coincidence? At one level, yes, apparently it is. Jansen and Pollmann (2001) argue plausibly that the importance of multiples of two and five in our number perception come from the cognitive salience of doubling and halving operations, while the relevance of base 10 appears to stem directly from the peculiarities of human anatomy. If we had eight fingers, on this logic, the relevant multiples would be two, four and eight, and the round numbers between eight and 64 would be 8, 10, 12, 14, 16, 20, 24, 28, 32, 40, 48, 56 and 64: the average difference between these is 19.0%. But it seems implausible on the face of it that the acuity of the ANS is causally related to our having 10 fingers (or rather, using ten for counting) as opposed to eight. In any case, while the average differences may correspond tolerably well with the acuity of the ANS, round numbers are unevenly distributed, with some having appreciably closer neighbours than this. This throws the challenge of reconciling ANS-based interpretations and alternative-based scalar inferences into sharp relief. Consider a case where a speaker utters more than 90. If the hearer assumes the speaker to be relying on ANS-based knowledge, they are entitled to infer that the true quantity is not less than 106 (90/0.85 = 105.88), as otherwise the speaker could not be confident that it was in fact in excess of 90. But if the speaker were to draw a scalar inference based on the speaker’s non-use of the alternative more than 100, they would be inferring that the true

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quantity is not more than 100. In this case, the two inferential paths lead to flatly contradictory results. Based on examples such as this, I would argue that the relationship between these two interpretative options is one of opposition rather than bootstrapping. The kind of scalar inference sketched out above is potentially appropriate if a speaker has precise knowledge, whereas the ANS-based entailment is appropriate if a speaker’s knowledge is mediated by the ANS. If a speaker’s knowledge is of neither of these kinds, then neither kind of reasoning is licensed, as far as the hearer is concerned. In order to delve into this question of interpretative strategy more fully, it is worth addressing the more general question of what kinds of knowledge state a speaker might be in, and under what circumstance a hearer is entitled to make assumptions about this and use it to guide their interpretation. It is also worth considering why a particular numeral should be chosen as an anchor point, and whether that conveys (or “leaks”; Sher and McKenzie 2006) additional information about what the speaker takes to be relevant to the current discourse purpose. I discuss these issues briefly in the following sections.

3

Implicatures from more than n when the Speaker has Uncertain Knowledge

Assuming that the goal of the hearer is ultimately to reconstruct (some approximation of ) the knowledge state of the speaker, it is relevant to consider what knowledge states would license a speaker to produce more than n. Cummins et al. (2012) focus primarily on the case in which a speaker actually has precise knowledge, but opts to present it in this less precise way—in their Experiment 2, the instructions stipulate that the participant should assume the speaker is “well-informed” as well as “telling the truth and being cooperative” (ibid., 152). This is a necessary assumption for the derivation of a fully fledged scalar implicature, on a Gricean view. In the scenario discussed above, to infer that a speaker with precise knowledge who uttered more than 90 meant to convey “not

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more than 100”, the hearer would first have to note that the stronger alternative more than 100 was not uttered and infer that this was because the speaker was not certain that more than 100 was true. But to reach the ultimate conclusion that the speaker takes more than 100 to be false, the hearer would also have to take a second step, namely assuming that the speaker is knowledgeable as to the truth or falsity of the stronger claim. Given this further assumption, we can reasonably conclude that the speaker who does not know more than 100 to be true in fact knows it to be false. If, however, we assume a scenario in which the speaker’s knowledge about the quantity under discussion is imprecise, we cannot take this final “epistemic step”. And it is easy to imagine how a speaker might have uncertain knowledge of this kind—indeed, we have just discussed one scenario in which they would, namely when they are relying on an ANSbased impression of the quantity in question. So what kind of pragmatic enrichments should be available to a hearer in such a situation? Suppose a speaker utters (6) based on an ANS-driven estimate of cardinality, while adhering to Gricean Quality, and assume again that the speaker’s Weber fraction for their ANS is 0.15. (6) There are more than 100 people here. As argued earlier, in order for the speaker to be able to say (6) with confidence, the true cardinality must be at least 118. If it were precisely 118, the speaker should perceive it as lying within the range (100, 136)— they could reliably tell that it is greater than 100 (but not that it is greater than 101), and they could reliably tell that it is less than 136 (but not that it is less than 135). In this circumstance, if we asked the speaker for their perception of the range and of the most likely value, they should (assuming perfect self-knowledge) report (100, 136) and 118, respectively. Let us further suppose that (6) is uttered in a context in which the relevant scale of alternatives has granularity 50—that is to say, the speaker is only choosing among multiples of 50, and more than 150 is an adjacent scalar alternative to more than 100. (I choose 50 arbitrarily to illustrate the idea: granularities of 100, 25, 20 or 10 might reasonably be in play

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instead.) Under this assumption, the speaker would choose to utter (7) in preference to (6) if they were sure that both were true. (7) There are more than 150 people here. Given the ANS, the speaker could only safely utter (7) for a true cardinality above 176. If the true cardinality were 176 (not reliably distinguishable from 150 by the ANS) or fewer, the speaker could not be confident of the truth of (7) and would be obliged to say (6) instead. Thus, across the range of possible scenarios in which (6) is uttered because the speaker is not sure that (7) holds, we find actual cardinalities in the range [118, 176]. Hence, allowing for the limitations of the ANS, the hearer who considers that < more than 100, more than 150 > is the relevant scale should in fact take (6) to implicate “not more than 176”. If asked for the most likely value, their response should reflect the relative probability they attach to the possible values within the range: for instance, if they feel that the probabilities are likely to be monotone decreasing, on the basis that smaller cardinalities are more probable than larger cardinalities, they should choose the lower bound (118), whereas if they feel that the probability distribution is roughly normal within this range they should choose the mid-point (147). Of course, the use of the ANS is just one way in which a speaker might obtain quantity information that is imprecise in character. To generalise the above argument, let us suppose that a (reliable) speaker is confident that the true value of some numerical quantity x lies in the range [a, a + r ], where r ≥ 0. If < more than n, more than m > constitute an informational scale, then the speaker who says x is more than n asserts that a > n and may implicate that a ≤ m, from which it follows that x > n and that x ≤ m + r. To illustrate this idea with a concrete example, suppose that it is common knowledge that there are still 3000 votes to be counted in the US Presidential election in Arizona, and at this point a news ticker reports the latest results as (8).

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(8) Biden wins Arizona by more than 9000 votes. If we take this report to be reliable, we are in the above situation, where x denotes Biden’s winning margin in Arizona, n = 9000 and r = 6000 (as 3000 votes, in either direction, could influence the winning margin by 6000). Let us assume that m = 10,000, as the next scalar alternative to 9000. Then, per the above formula, the speaker of (8) may be taken to implicate that Biden’s winning margin in Arizona is no greater than 16,000—but not to implicate any stronger claim than that. In short, the nature of the speaker’s knowledge state is potentially relevant to evaluating the predictions of an implicature-based account of the meaning of expressions such as more than n, because a hearer asked to provide an upper bound for the quantity under discussion (predicted to be m + r, in the above terms) needs to consider that knowledge state alongside information about the relevant scalar alternatives. Both Hesse and Benz (2020) and Cummins et al. (2012) collected theoretically useful information about the hearers’ perceptions of the likely range of possible values, in their respective scenarios, but neither elicited hearers’ opinions about the nature of the speakers’ likely knowledge states. However, as the above discussion indicates, it is perfectly feasible for a hearer to have access to information about this—for instance, that the speaker has precise knowledge, or that they have approximate ANS-mediated knowledge, or that they have imprecise knowledge which is constrained by specific contextual circumstances, as in example (8) above.

4

Inferring the QUD from the Quantity Expression

A further important consideration in interpreting more than n is whether we are in a context in which the speaker intends to convey an implicature, or one in which they do not (a “lower-bound” context, in the terms of Breheny et al. [2006]). Cummins (2017) discusses some reasons why a particular numerically quantified utterance might fail to give rise to an implicature, paying particular attention to the possibility that the

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semantic content of the utterance might answer the present question under discussion (QUD) without further enrichment.3 Some explanation of this kind is certainly required to account for why it appears potentially felicitous for a speaker with precise knowledge to produce an utterance such as (1), repeated below. This is made even clearer in (9), also widely judged felicitous, in which the speaker first makes a more than n assertion and then discloses their precise knowledge. (1) More than four people attended. (9) John has more than two children; in fact, he has five. Presented with (9) out of the blue, a hearer might naturally infer that the specific polar question of whether or not John has more than two children is under discussion at the time of utterance, and that this is why the speaker first expresses their knowledge state relative to that reference point. It is an empirical question to what extent (1), or indeed (6), invites the hearer spontaneously to entertain the corresponding inference. However, in either case, the idea that the speaker may be trying to answer a specific QUD is a possible explanation of their behaviour. This competes with the explanation under which the speaker’s utterance is intended to give rise to a quantity implicature, and the hearer may need to adjudicate between these possible explanations in order to arrive at a correct understanding of the speaker’s meaning. To make this a little more concrete: (1) could be uttered by a speaker who is certain that five people attended and considers it possible that the true number is greater, and who wishes to give a concise and maximally informative account of that knowledge. In this case, the hearer is entitled to enrich the semantic meaning of (1) with the ignorance (quantity) implicature that the speaker is unsure whether or not more than five people attended. Alternatively, (1) could be uttered by a speaker who considers that their current conversational task is merely to answer the 3 For simplicity I restrict this discussion to the scenario in which the speaker wants to address a particular QUD, rather than, for instance, wishing to convey a particular kind of argumentative force (cf. Ariel 2004), or being primed to use particular linguistic material (cf. Cummins 2015). The precise nature of the speaker’s motivation for choosing an informationally weaker utterance using more than n is not crucial to the following discussion.

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question of whether more than four people attended, and knows that the answer is yes. In this case, the hearer is not entitled to enrich the semantic meaning with any implicature: the speaker might be certain that the true number is exactly five, or indeed certain that it is not, and can felicitously say (1) in either case. Hence, if we take it that the task of the hearer is to reconstruct the speaker’s knowledge state, an important auxiliary step is establishing which QUD is being answered. For more than n, the value of n appears to enter into this calculation. Comparing more than 100 with more than four, the former seems somewhat more likely to be presenting a general impression of the quantity under discussion, and admit a quantity implicature, while the latter seems somewhat more likely to be directed towards the specific QUD “whether more than four”, the quantity implicature being suppressed as a result.

5

Rationally Reconstructing the Causes of Speaker Meaning

Cummins et al. (2012) suggest in passing that the interpretation of more than n is atypical when dealing with large non-round values of n, as in (10), for the reasons discussed above: that is, they conjecture that the hearer tends to infer that the speaker is addressing the QUD “whether more than n”. (10) More than 93 people attended. However, this is not a clear-cut matter. Given that round numbers are cognitively salient scale points, QUDs involving round numbers (“whether more than 100”) should perhaps be more likely a priori than those involving non-round numbers (“whether more than 93”). Moreover, even if the QUD is of this form, the speaker might nevertheless respond with a more informative utterance that would entail an answer to the QUD (in this case, more than 100 ). Interestingly, doing so appears to restore quantity implicatures that would be unavailable from the

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more precise utterances: considered as an answer to (11), (12) seems to implicate (13), whereas (14) does not. (11) Edinburgh has about 482,000 inhabitants; is Surat bigger than that? (12) Surat has more than 4 million inhabitants. (13) Surat has fewer than 5 million inhabitants. (14) Surat has more than 482,000 inhabitants. From this perspective, it appears that a rational hearer, attempting to reconstruct the knowledge state of the speaker who utters (10) or (14), needs to draw inferences about the QUD as well as considering the availability of alternatives (and perhaps the inferences that those would in turn invite), the likely nature of the speaker’s knowledge (in the sense of whether it comprises exact knowledge, approximate knowledge derived from the ANS, or something else), and the likely distribution of the quantity under discussion in reality. It would be quite possible—indeed, rational—for the hearer to arrive at a series of provisional and contradictory conclusions about the speaker’s likely meaning given different assumptions. (14) would invite a different interpretation assuming a QUD like (11) than assuming a general “how many” QUD, just as (6) would invite different interpretations if it was based on the speaker’s impression of the size of a crowd as opposed to a precise count. We could think of this as a problem of joint inference, in which the hearer attempts to derive both the QUD and the knowledge state from the utterance, and potentially also additional kinds of information such as the grounds for the speaker’s knowledge. Some recent work has extended the Rational Speech Act (RSA) framework in this kind of direction. For instance, Kao et al. (2014) take a similar approach to hyperbole: however, their model posits only three possible QUDs. It is not immediately clear how best to generalise a model of this type to a setting in which there are arbitrarily many possible QUDs. Within a probabilistic setting like this, the question arises of whether an experimental participant, asked about the possible range or most likely

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value given a more than n stimulus, reasons towards the single most likely explanation of the speaker’s behaviour, or synthesises multiple different possible explanations of the speaker’s behaviour. Strikingly, the results obtained by Hesse and Benz (2020), if fully generalisable, hint at the existence of a possible heuristic, in which hearers estimate that the most likely value given an utterance of the form more than n is to be found one ANS Weber fraction in excess of n. The discussion earlier shows that, for many individual cases, this estimate is likely too high (where the speaker’s knowledge is precise) or too low (where the speaker’s knowledge is filtered through the ANS). However, it is not implausible that, in the round, this is a reasonable estimate of where the peak of the probability distribution should fall for more than n, averaging across a representative sample of circumstances in which such expressions are uttered. The extent to which we could truly ascribe such a result to the operation of the ANS per se is somewhat debatable.

6

Conclusion

For a rational hearer, arriving at the correct interpretation of an expression of the form more than n is in principle a complex matter, drawing upon and integrating several different kinds of linguistic and extralinguistic knowledge. In this respect it is perhaps typical of numerically quantified expressions in language, which acquire complexity both from the sophistication of the linguistic system and from the multifaceted nature of the representation of quantity information in human cognition. In grappling with this challenge, theorists continue to make major strides towards understanding the semantic content of quantity expressions and the nature of the pragmatic processes by which speakers and hearers elaborate those meanings.

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References Ariel, Mira (2004). Most. Language, 80(4): 658–706. Breheny, Richard (2008). A new look at the semantics and pragmatics of numerically quantified noun phrases. Journal of Semantics, 25(2): 93–139. Breheny, Richard, Katsos, Napoleon, and Williams, John N. (2006). Are generalised scalar implicatures generated by default? An on-line investigation into the role of context in generating pragmatic inferences. Cognition, 100(3): 434–463. Carston, Robyn (1998). Informativeness, relevance, and scalar implicature. In Robyn Carston and Seiji Uchida (eds.), Relevance Theory: Applications and Implications, 179–236. Amsterdam: Benjamins. Cummins, Chris (2015). Constraints on Numerical Expressions. Oxford: Oxford University Press. Cummins, Chris (2017). Contextual causes of implicature failure. Discourse Processes, 54(3): 207–218. Cummins, Chris, Sauerland, Uli, and Solt, Stephanie (2012). Granularity and scalar implicature in numerical expressions. Linguistics and Philosophy, 35: 135–169. Geurts, Bart, Katsos, Napoleon, Cummins, Chris, Moons, Jonas, and Noordman, Leo (2010). Scalar quantifiers: Logic, acquisition and processing. Language and Cognitive Processes, 25: 130–148. Geurts, Bart, and Nouwen, Rick (2007). At least et al.: The semantics of scalar modifiers. Language, 83(3): 533–559. Hesse, Christoph, and Benz, Anton (2020). Scalar bounds and expected values of comparatively modified numerals. Journal of Memory and Language, 111: 104068. Jansen, Carel J. M., and Pollmann, Mathijs M. W. (2001). On round numbers: Pragmatic aspects of numerical expressions. Journal of Quantitative Linguistics, 8: 187–201. Kao, Justine T., Wu, Jean Y., Bergen, Leon, and Goodman, Noah D. (2014). Nonliteral understanding of number words. Proceedings of the National Academy of Sciences of the United States of America, 33: 12002–12007. Nouwen, Rick (2010). Two kinds of modified numerals. Semantics and Pragmatics, 3(3): 1–41. Sher, Shlomi, and McKenzie, Craig R. M. (2006). Information leakage from logically equivalent frames. Cognition, 101: 467–494.

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Solt, Stephanie (2014). An alternative theory of imprecision. In Todd Snider, Sarah D’Antonio and Mia Weigand (eds.), Proceedings of the 24th Semantics and Linguistic Theory Conference (SALT 24), 514–533. Washington, DC: Linguistics Society of America. Solt, Stephanie (2016). On measurement and quantification: The case of most and more than half. Language, 92: 65–100. Solt, Stephanie, and Stevens, Jon (2018). Some three students: Towards a unified account of ‘some’. In Sireemas Maspong, Brynhildur Stefánsdóttir, Katherine Blake and Forrest Davis (eds.), Proceedings of the 28th Semantics and Linguistic Theory Conference (SALT 28), 345–365. Washington DC: Linguistics Society of America. Solt, Stephanie, and Waldon, Brandon (2019). Numerals under negation: Empirical findings. Glossa: A Journal of General Linguistics, 4(1): 113.

Around ‘‘Around’’ Paul Égré

1

Introduction

Modifiers such as “around”, “about”, “roughly”, “approximately”, are called approximators by Sauerland and Stateva (2007). Applied to numerals, they allow us to select a wider range of values than denoted by the numeral they modify. For example, “John owns around fifty sheep” conveys that the number of sheep owned by John belongs to some unspecified interval centered on the number 50. An interesting problem, addressed by Krifka (2007) and by Solt (2014), concerns the size and properties of this interval. Another concerns the properties of the numbers modified by “around” and other approximators. Starting with the second issue, it appears that “around n” is more natural when n is a round number. For example, “John owns around 47 sheep” sounds odd compared to “John owns around 50 sheep”. At the very least, P. Égré(B) Institut Jean-Nicod (CNRS, ENS, EHESS), PSL University, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_5

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the context must be special in order for “John owns around 47 sheep” to be felicitously uttered. Maybe the speaker is trying to estimate John’s total number of sheep by adding up her best estimates of various subgroups of the total population. More typically, however, the target number n in “around n” will be a round number. The definition of what to count as a round number is still debated (see Ferson et al. 2015), but on a plausible first analysis at least, a round number may be thought of as a multiple of 5 or 10, given some relevant granularity level. Granted, we find various examples such as “the greatest distance between the audience and the stage is about 17 meters”. But this is because meters can be subdivided into smaller commensurate units, such as centimeters, of which the meter is a multiple, unlike sheep, which are indivisible units for the purpose of counting. The notion of granularity interacts with the first issue raised above, namely with the size of the interval denoted by “around”. Relying on Sauerland and Stateva’s and Krifka’s observations about measurement scales, Solt points out that the interval selected by “around n” depends on a contextually given granularity level. “John owns around 50 sheep” could be still be true if John owns 45 sheep. But if John owns 44 sheep, then “John owns around 40 sheep” should be uttered instead. Intuitively, this is explained if the relevant granularity in this case is set in tens of sheep, in which case “around 50” should denote the interval [45, 55]. Consider the following minimal pair: (1)

This piece of meat weighs about 200 grams.

(2)

This emerald weighs about 200 grams.

In a restaurant, one can expect the granularity to be set in steps of 100 grams, or in steps of 50 grams, but steps of 10 grams appear less relevant. Talking about emeralds, whose substance is so valuable, steps of 1 gram or 10 grams appear much more plausible to report the interval within which the stone’s weight might lie. In physics and in statistics, expressions such as 52.25 ± 0.05 cm are commonly used to report uncertainty in measurements (Kirkup and Frenkel 2006). The term “52.25” sets a precision level (hundredths of cm), and the term ±0.05 sets an error level. The latter means that there

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is a certain probability (conventionally set) for the true value to lie in the interval [52.20, 52.30]. In ordinary language, expressions like “around fifty” make explicit neither the size of the relevant interval nor the precision level attached to the target number. It is therefore natural to assume that both are set either contextually or by default. In (Solt 2014), Solt writes proposes the following general semantics for “Approx n”, where Approx is an approximator such as “around”: (3)

[[Approx n]]=[n − u2 , n + u2 ], with u the contextually given granularity of the relevant scale.

For n = 50, and u = 10, this derives the previous observations regarding “John owns around 50 sheep”. Assuming n = 200 and u = 100 g, then we can explain that (1) is true when talking about a steak of 173 g. On the other hand, for an emerald weighing 173 g, (2) appears misleading and even false when u = 10 g. Solt’s semantics for approximators is plausible and it nicely accounts for examples like (1) and (2). In what follows, however, I outline a different proposal, making room for more flexibility about truth and falsity judgments concerning utterances with approximators. An important motivation behind it is related to the soritical character of expressions like “around” and “about”. Assuming u = 10, then Solt’s account predicts “John owns 50 sheep” to be true if John owns 45 sheep, but false if John owns 44 sheep. This rule seems overly rigid. One way around this prediction may be to suppose that in many contexts, there is some uncertainty about the underlying granularity. When u = 10, the utterance is false, but it would be true if u = 20. However, even when the granularity level is salient, the assumption of a sharp cutoff remains disputable. Consider “John owns around 20 sheep”, and assume 10 is the salient granularity level. It is not clear that the speaker said something false if John actually owns 13 sheep, or 27 sheep, maybe the speaker only said something misleading. In a recent paper, Benjamin Spector, Adèle Mortier, Steven Verheyen and I explore weak truth conditions for “around” that just assume that “x is around n” means that x belongs to some interval centered on n, letting the listener make a probabilistic inference about the interval in question.

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Here I briefly review the main motivations for the account, and I explain how it was derived from a first version in which the meaning of “around” is characterized in terms of quantification over intervals.

2

Around vs. Between

Consider again the sentence: (4)

John owns around 50 sheep.

Assuming the granularity is set to u = 10 in the context, Solt’s account predicts the latter sentence to have exactly the truth conditions of: (5)

John owns between 45 and 55 sheep.

However, this equivalence does not seem to do justice to the fact that intuitively “around” is vague and compatible with an open-ended range of values, whereas “between” specifies a sharp interval. If John actually owns 43 sheep, then (5) is false. But (4) does not seem clearly false in that context. That is, “around” appears to be sorites-susceptible, unlike “between”. If “k is around n” is acceptable, then “k − 1 is around n” also appears to be acceptable, although possibly to a lesser degree (see Lassiter and Goodman 2017; Égré et al. 2019). A second observation that can be made to contrast “around” and “between” is that upon hearing (5), a listener with no particular expectation about the number of sheep that John might own should not infer that 50 is a more likely value than any of the other values in the interval [45, 55]. By contrast, when hearing (4), the same listener should infer that the speaker is not indifferent between values over that same interval: intuitively, 50 is a more likely value to the speaker than more remote values like 45 and 55. In Égré et al. (2021), this intuition is called “peakedness”, to indicate that the listener’s posterior probability distribution on the number of sheep owned by John, starting from a uniform prior, should remain uniform with “between”, but should be nonuniform and peaked on the modified value with “around”. Note that peakedness does not prevent the speaker from using “around 50” even when the speaker knows the actual value to

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be 49 or 51. The point is that the listener will infer that values closer to 50 are more probable to the speaker than values more remote from 50. Listener and speaker may even have common knowledge that 50 is not a possible value in some contexts. To derive the vagueness of “around”, we need to assign “around” truth conditions weaker than those assigned to “between”. To derive peakedness, what is needed is an account of the way in which these truth conditions interact with probability assignments on the possible values of interest.

3

A Weak Parametric Semantics for ‘‘Around’’

The truth conditions proposed with my coauthors in Égré et al. (2021) for “around” are as follows: (6)

[[around n]] y = λx.x ∈ [n − y, n + y]

That is, x is around n provided x is at a distance at most y from n. Unlike the fixed parameter u in (3), which is set by the context, here y is an open parameter. This parameter may be thought of as ranging over granularity levels, but it need not. The basic idea is that by uttering “x is around n”, the speaker does not specify an interval set in advance, but lets the hearer infer that x belongs to some unspecified interval centered on n. This interval may be thought of as a numerical halo in the sense of Lasersohn (1999). However, “around” and other approximators are expressions whose function is to create such halos compositionally. Lasersohn talks of pragmatic halos when the unmodified numeral is used with slack, for instance when “fifty” is used to mean “approximately fifty”. For that reason, Lasersohn himself (1999,§.6.2, pg. 545) distinguishes hedges (like “roughly speaking”, see Lakoff 1973) from slack regulators (like “exactly”), and he describes hedges as “expanding the denotation into the halo”. In our approach, “around” is a variety of hedge in that sense, though it is more accurate to say that it creates such a halo, whose radius is left open. In Égré et al. (2021), the truth conditions stated in (6) are used to account for the vagueness of “around”, and for the difference with “between”.

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Setting details aside here, the leading idea, using Bayesian machinery (see Lassiter and Goodman 2017; Sutton 2018), is to associate to the listener a joint probability distribution on the value y relevant to assess the meaning of “around”, and on the value of the variable x of interest (viz. the number of John’s sheep). With minimal additional assumptions, it can be shown that the posterior probability of x being closer to 50, when hearing “x is around 50”, is comparatively higher than the posterior probability of x taking on more remote values, compared to the corresponding prior probabilities. The intuition behind this result is that the closer x is to 50, the more likely it is to be found in an arbitrary interval centered on 50. This derives peakedness. And likewise, assuming uniform priors on x and y, it can be shown that the posterior probability that x = k is very close to the probability that x = k + 1, in agreement with intuitions of sorites-susceptibility. In other words, starting from parametric truth conditions for “around”, we can substantiate the contrast felt between the use of “around” and the use of “between” to report approximate quantities.

4

‘‘Around’’ as an Existential Quantifier

Let us turn to what I now consider as an imperfect but instructive sketch of the truth conditions in (6), which was a step toward them, and from which further lessons may be drawn. The original motivation behind the truth conditions in (6) is the idea that when x is around n, x must belong to some interval or other, centered on x. This existential quantification in (6) is absent, but it can be retrieved. y y Let An = {[n − k, n + k]|0 ≤ k ≤ y}, that is, An is the set of all intervals of radius less than y, centered on n. Then (6) can be restated equivalently as follows: (7)

y

[[around n]] y = λx.∃I ∈ An (x ∈ I ) y

Clearly, if x belongs to some I in An , then x belongs to [n − y, n + y], which is the largest interval in the set; the converse is also immediate. So (6) and (7) are equivalent.

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In a first sketch of those truth conditions, I had originally proposed to set y to n, yielding weak but non-parametric truth conditions of the form. (8)

[[around n]]= λx.∃I ∈ Ann (x ∈ I )

The choice of y = n was justified by the idea that, when “around” applies to cardinalities for which 0 is the lowest meaningful value, the interval [0, 2n] = [n − n, n + n] is the largest symmetric interval centered on n. Such truth conditions predict that “John owns around 50 sheep” is literally true if and only if John owns between 0 and 100 sheep. Since the granularity level u should not exceed n, those truth conditions are logically weaker than those in (3). The main motivation, in fact, was to assign “around” very weak truth conditions so as to secure the role of “around” and other approximators as mechanisms of error minimization (see Channell 1985; Krifka 2007; van Deemter 2009). However, several problems beset the account given in (8). First of all, although the truth conditions become very weak, “around n” indirectly denotes a fixed interval, namely the interval [0, 2n]. But this remains a rigid interval, whose choice can be disputed for the same reasons that motivated the remark that “around” is vague and sorites-susceptible, unlike “between”. Furthermore, the choice of y = n implies, for example, that 1000 is around 2500, but that 2500 is not around 1000. This prediction has an air of plausibility, since judgments of proximity between numbers can be expected to depend on number sizes (Feigenson et al. 2004). However, this asymmetry in judgments may not hold in all cases. Thirdly, when “around” applies to a negative number along a scale that is not a ratio scale (as in “Today’s temperature is around −20◦ C”), the choice of Ann appears inadequate; this can be fixed, but not without ad hoc assumptions. A more fundamental problem, made vivid to me by Benjamin Spector, concerns the interaction of the truth conditions in (8) with probability assignments. To account for the listener’s inference that 50 is the most likely value when the speaker utters “around 50”, S. Verheyen and I originally proposed to think of the listener as assigning probabilities to all the intervals of the form [n − k, n + k] for 0 ≤ k ≤ n. The model in question makes distinct but qualitatively similar predictions to the ones obtained on the basis of (6), and indeed it was the first basis for that approach (see

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Égré et al. 2021, Appendix B, for details). In particular, it can be used to derive that upon hearing “John owns around 50 sheep”, the listener will consider it unlikely, though not literally ruled out, that John owns 20 sheep. Conceptually, however, Spector’s point is that it is more transparent to assign probabilities about meaning when the meaning of “around” is semantically indeterminate, as reflected in (6), than when the meaning is in fact determinate, as in (8). As a result, the account given in (8) lacks the flexibility and simplicity of (6), which is why the latter set of truth conditions was adopted instead.

5

The Accounts Compared

Let us take stock: (3) presents plausible but overly rigid truth conditions for “around”. (8) states weaker truth conditions in terms of existential quantification over intervals, but they do not get rid of the rigidity problem, instead they shift it toward more extreme values. Unlike those accounts, (6) gives weak parametric truth conditions for “around”, which account for the vagueness of “around”, and dispense with the existential quantification over intervals by the same token. Finally, (7) provides a synthesis, by rephrasing (6) in terms of existential quantification still retaining the parameter on the size of the largest interval denoted by “around”. At this point, one may wonder about the choice between the statement of the meaning of “around” in (6) and the reformulation given in (7) in terms of existential quantification over intervals. About this choice, Benjamin Spector in ongoing work shows that further observations about the distribution of approximators can be explained if “around” patterns like “some”. The main observation there concerns the interaction of “around” with negation (González Rodríguez 2008; Spector 2014), and the fact that “around” and related approximators behave as positive polarity items, as evidenced by the infelicity of the negation of (4) (setting aside echoic uses). (9)

*John does not own around 50 sheep.

The contrast between the positive form (4) and its negation (9) is striking. Solt herself proposed two accounts of this contrast in recent work:

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one in Solt (2018) is based on the idea that “not around 50” must be informationally equivalent to “not 50”, and thereby dispreferred due to creating idle syntactic complexity; the second is based on the idea that “not around 50” selects a non-convex interval when truth conditions like (3) are assumed (see Solt and Waldon 2019, building on Gärdenfors’s criterion of naturalness for concepts). Spector’s own account adopts Solt’s first strategy, but it derives Solt’s informational equivalence between “not around 50” and “not 50” by making use of the existential quantification structure postulated in (7). Without going into the details of his argument here, we can see that despite being truth conditionally equivalent with (6), the logical form underlying (7) may therefore be recruited to derive further facts beside those reviewed above concerning the vagueness of “around”.

6

Spatial Locations and Nearness

A referee suggests that not just negations but also questions sound odd with “around” and other approximators, as in the following example, again setting aside echoic uses: (10)

*Does John own around 50 sheep?

The referee observes that this pattern is reminiscent of the distributional properties of epistemic adverbs like “certainly” and “probably” (Bellert 1977). Considering this analogy, “John owns around 50 sheep” may have to be treated like “John owns probably 50 sheep”, with the approximator being “outside the communicated proposition”. This suggestion is intriguing. It has an air of plausibility too, considering that with some approximators, it is possible to extrapose the corresponding modifier. For instance, in English and in other languages it is possible to say both: (11)

a. b.

John owns more or less 50 sheep. John owns 50 sheep, more or less.

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One difficulty I see for an analysis of “around” as a sentential modifier along those lines concerns whether it would work for occurrences of “around” followed by a non-numeral, such as a location name. Consider the following sentence: (12)

There are nice forests around Paris.

The truth conditions in (6) can account for the meaning of (12). Applied to (12) they predict that there are nice forests at some unspecified distance from Paris, and by Bayesian reasoning these nice forests are more likely to be closer to Paris than to be further away. The point here is that “around” does not seem easily reanalyzable as a sentential modifier (it being a preposition in those examples, unlike “roughly” or “probably" which are adverbs). However, it appears that (12) can be negated without difficulty (“there aren’t nice forests around Paris”) or turned into a question (“are there nice forests around Paris?”). So an account of the infelicity of (9) based on the truth conditions in (6) and (7) must still explain the contrast with the negation of (12). The last example raises a further general issue. Arguably, what (12) conveys is that there are nice forests within some distance near Paris, and similarly for (4), it conveys that the number of sheep owned by John is within some interval near 50. The truth conditions for “around” in (6) and (7) do not incorporate the notion of nearness directly, however. Arguably, nearness is derived pragmatically, and could follow from peakedness, namely from the fact that values more remote from the target are less probable than values closer to the target. Whether this is correct is not completely obvious, however, since when the parameter y is allowed to take large values to begin with, then it may be possible for an object far from Paris to count as “around Paris”, or for a number far from 50 to be “around 50”, even if the posterior probability attached to those possibilities is low. More work needs to be done to settle this issue. If the notion of nearness is to be represented explicitly in the truth conditions of “around”, then it would mean that an additional parameter must be incorporated to account for the vagueness of “around”.

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Conclusions

Solt’s truth conditions for “around” put adequate emphasis on granularity considerations but they fail to capture the vagueness of “around”. Just like the suffix “ish” in “noonish” fails to specify a sharp interval of values around noon, “around n” does not select a fixed interval centered on n, even as the granularity level is salient in the context. Here I have outlined an alternative account presented with my coauthors in Égré et al. (2021), and explained some features and limitations of the first sketch behind that model. Part of the sketch remains worthy of consideration, however, namely the idea that the right analysis of “around” includes not just a parametric element, but also a quantificational structure. This also justifies the title of this short piece: not a straight path to “around”, but a more roundabout way, guided and corrected by collaborative effort, and here meant as a small tribute to the wealth of Stephanie Solt’s work on the semantics of imprecision and approximation in natural language. Acknowledgements This research was supported by the following projects: ANR PROBASEM (ANR-19-CE28-0004-01), ANR FrontCog (ANR-17-EURE0017), as well as by ANR-DFG project ColAForm (ANR-16-FRAL-0010), and the Van Gogh Program (42589PM). I am grateful to Benjamin Spector for several discussions on the topic of this paper, and thankful to an anonymous referee for helpful comments.

References I. Bellert. On semantic and distributional properties of sentential adverbs. Linguistic Inquiry, 8(2):337–351, 1977. J. Channell. Vagueness as a conversational strategy. Nottingham Linguistic Circular, pages 3–24, 1985. K. van Deemter. Utility and language generation: The case of vagueness. Journal of Philosophical Logic, 38(6):607, 2009.

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P. Égré, D. Ripley, and S. Verheyen. The sorites paradox in psychology. In S. Oms and E. Zardini, editors, The Sorites Paradox, pages 263–286. Cambridge University Press, 2019. P. Égré, B. Spector, A. Mortier, and S. Verheyen. On the optimality of vagueness: “Around”, “between”, and the Gricean maxims. 2021. Under review. Available on arXiv. L. Feigenson, S. Dehaene, and E. Spelke. Core systems of number. Trends in Cognitive Sciences, 8(7):307–314, 2004. S. Ferson, J. O’Rawe, A. Antonenko, J. Siegrist, J. Mickley, C. C. Luhmann, K. Sentz, and A. M. Finkel. Natural language of uncertainty: Numeric hedge words. International Journal of Approximate Reasoning, 57:19–39, 2015. R. González Rodríguez. Sobre los modificadores de aproximación y precisión. In ELUA, pages 111–128, 2008. L. Kirkup and R. B. Frenkel. An introduction to uncertainty in measurement: Using the GUM (Guide to the expression of uncertainty in measurement). Cambridge University Press, 2006. M. Krifka. Approximate interpretation of number words. In G. Bouma, I. Krämer, and J. Zwarts, editors, Cognitive Foundations of Communication, pages 111–126, 2007. G. Lakoff. Hedges: A study in meaning criteria and the logic of fuzzy concepts. Journal of Philosophical Logic, 2(4):458–508, 1973. D. Lassiter and N. D. Goodman. Adjectival vagueness in a Bayesian model of interpretation. Synthese, 194(10):3801–3836, 2017. U. Sauerland and P. Stateva. Scalar vs. epistemic vagueness: Evidence from approximators. In Semantics and Linguistic Theory, volume 17, pages 228–245, 2007. S. Solt. An alternative theory of imprecision. In Semantics and Linguistic Theory, volume 24, pages 514–533, 2014. S. Solt. Approximators as a case study of attenuating polarity items. In S. Hucklebridge and M. Nelson, editors, NELS 48: Proceedings of the 48th Annual Meeting of the North East Linguistic Society, volume 3, pages 91–104, 2018. S. Solt and B. Waldon. Numerals under negation: Empirical findings. Glossa: A Journal of General Linguistics, 4(1), 2019. B. Spector. Global positive polarity items and obligatory exhaustivity. Semantics and Pragmatics, 7:11–1, 2014. P. R. Sutton. Probabilistic approaches to vagueness and semantic competency. Erkenntnis, 83(4):711–740, 2018.

Evaluative Intensification and Positive Polarity: Catalan WELL as a Case Study Elena Castroviejo and Berit Gehrke

1

Introduction

Bolinger (1972, 16) uses the term ‘intensifier’ for ‘any device that scales a quality, whether up or down or somewhere between the two’ and divides them further into boosters (e.g. perfect, terribly), which target the upper part of a scale and are therefore looking up, compromisers (e.g. rather, fairly), which target the middle of a scale, often trying to look both ways at once, and finally diminishers (e.g. indifferent, little) and minimisers (e.g. a bit of, an iota), which target the lower part of a scale, looking down. E. Castroviejo Centro de Investigación Micaela Portilla Ikergunea, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Vitoria-Gasteiz, Spain e-mail: [email protected] B. Gehrke(B) Institut für Slawistik und Hungarologie, Humboldt Universität zu Berlin, Berlin, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_6

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In this context, Bolinger (1972, 120–124) observes that some intensifiers are not acceptable under negation (1), unless under an echo reading or metalinguistic negation, which we set aside for now. (1)

a. *The girl isn’t quite attractive. b. *He’s not rather foolish!

Other intensifiers under negation give rise to a litotes interpretation (2), which we will also set aside in this paper (see Castroviejo and Gehrke 2020, for discussion and account). (2)

He’s not overly bright. ∼ He’s rather stupid.

In a more recent paper, Nouwen (2013) discusses the behaviour of Dutch intensifiers under negation. In this context he distinguishes between L(ow)-, H(igh)-, and M(iddle)-adverbials, depending on which part of the scale they target, and he observes that L- and M-adverbials behave like Positive Polarity Items (PPIs), whereas H-adverbials do not, but give rise to a litotes reading (3). (3)

a.

Jan is niet erg lang. Jan is not badly tall H-adverbial

b. *Jan is niet best lang. Jan is not best tall M-adverbial c. *De handdoek is niet een beetje nat. the towel is not a bit wet L-adverbial He furthermore notes that evaluative subjective H-adverbials also behave like PPIs (4).

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(4)

a. b. c.

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Jan is (*niet) ongelofelijk lang. Jan is not unbelievably tall Het boek is (*niet) redelijk dik. the book is not orderly thick De atleet sprong (*niet) aardig ver. the athlete jumped not nicely far

Finally, Morzycki (2012, 34) shows that what he calls extreme degree modifiers (e.g. downright, flat-out, positively, full-on) resist embedding under NPI-licensing operators, (5). (5)

a. b. c.

Murderers aren’t (??downright) dangerous. Are murderers (??downright) dangerous? If murderers are (??downright) dangerous, you might want to avoid Harold.

What these data show is that intensifiers exhibit varied behaviour in entailment-cancelling environments. Some can be in the scope of NPI-licensing operators and yield a litotes interpretation. Others cannot be in their scope. Subjectivity/evaluativity can also lead to unacceptability in these environments. Building on much work on the ties between scalarity and PPI-hood (see e.g. Israel 1996, 2011), our focus in this paper is to explore the source or sources of the PPI behaviour of such intensifiers. As a case study we present the intensifying use of the Catalan ad-adjectival modifier1 ben ‘(lit.) well ∼ quite’ (ben), illustrated in (6-a), which also behaves like a PPI, in the sense that it cannot appear in the scope of negation, (6-b). (6)

a.

En Pere és ben simpàtic. the Peter is ben nice ‘Peter is rather/quite nice.’ b. *En Pere no és ben simpàtic. the Peter is not ben nice

1Throughout the paper, we will use the term ‘modifier’ in a rather loose sense, as a cover term for certain adverbs, adverbials and attributive adjectives, without making a commitment to the precise mode of composition.

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We make an attempt at a general account, rather than just an account for ben, and our long-term goal is to provide a principled explanation for the infelicity of some intensifiers in entailment-cancelling contexts more generally. We claim that ‘unbleached’ (in the sense of Nouwen 2020, see below) intensifiers whose base is an evaluative adverb do not directly operate on degrees, even if they have degree effects. Moreover, we point out that information-based explanations for the positive polarity behaviour of a given intensifier apply to true degree expressions, but not necessarily to ben and other evaluative adverbs. The structure of the paper is as follows. In Sect. 2, we set the ground on the relationship between evaluation and intensification. With this in mind, Sect. 3 delves into the characterisation of ben, an evaluative modifier that conveys approval at how well an individual instantiates the property denoted by an adjective (A). In particular, in Sect. 3.1 we provide a characterisation of ben on the basis of the interpretations obtained depending on the characteristics of the modified adjective; in Sect. 3.2 we specify and qualify the PPI behaviour of ben, and in Sect. 3.3 we sketch an analysis of intensifying ben. Turning to previous analyses of PPI-hood in M-modifiers, in Sect. 4 we lay out the essentials of an informativity-based account along the lines of Solt (2018), Solt and Waldon (2019) and Solt and Wilson (2020), discuss why it cannot be extended to ben, and point towards an alternative explanation. Section 5 concludes.

2

Intensification, Valence and Evaluativity

Nouwen (2020) notes that many degree modifiers are derived from or related to evaluative adjectives or adverbs. He observes that those that are derived from adjectives or adverbs with a positive valence (e.g. pretty, fairly) tend to express a medium scalar extent (M-modifiers), while those that are derived from negative ones (e.g. terribly, horribly) tend to target high degrees and usually involve a meaning of excess. This is illustrated in (7) (from Nouwen 2020, 4). (7)

a. b.

Scarlett is pretty tall. Scarlett is terribly tall.

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Nouwen attributes this correlation to what he calls the ‘Goldilocks Principles of Evaluation’ (8) (from Nouwen 2020, 2). (8)

The Goldilocks Principles of Evaluation 1. You can have too much of a good thing. 2. You can’t have so much of a bad thing that it becomes good again.

As he states it: Being removed from the sun is good because it prevents the evaporation of water, but once you’re too far, water will freeze and a good thing turns bad. [...] being close to the sun is good because it prevents the freezing of water, but [...] being too close is bad because water will start to evaporate. (Nouwen 2020, 2)

In other words, only the ‘right’ degree of something can be evaluated as good, and this ‘right’ degree is somewhere in the middle of a given scale, whereas degrees that are too high on a given scale give you some sort of excess reading (too much of that degree).2 Given that Catalan ben is related to a positive evaluative adjective with the meaning ‘good’, the expectation is that it behaves like an M-modifier, in the sense that it relates to the middle part of a scale or to the ‘right’ degree of something. In our case study, we will see that this intuition is more or less correct in that we often get a speaker evaluating the degree to which an individual holds the given adjectival property to be sufficient or satisfying. However, we will also see that the notion of an M-modifier or of relating to the middle part of the scale associated with the adjective might be somewhat of an oversimplification, given that ben interacts differently with a given adjective, depending on whether we are dealing 2 Conversely,

one should also expect that it would be evaluated as bad if something is too low on a scale, which would also be too far away from the ideal middle ground, although this is not directly discussed by Nouwen. We would expect that in the case of adjectival meaning, where we additionally have to take into account the standard (e.g. for warm) and thus do not get the whole scale of (e.g. temperature) but just some part of it, this would rather amount to the negation of the property in question (e.g. not enough warmness ∼ not warm); however, in this case we are in the domain of the antonym of the adjective in question, the order of the scale would be the reverse and we would again get an excess reading (e.g. too cold ).

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with an absolute or relative adjective, as well as whether we are dealing with an adjective with a positive or negative valence (in a given context). Nouwen (2020) himself observes that classifying degree modifiers as low-, middle- and high-range alone is not helpful because many such modifiers cannot easily be classified as such, e.g. surprisingly in (9) (from Nouwen 2020, 12). (9)

It is surprisingly warm today.

All that is necessary in this example is that it is warm, but it could be slightly warm or very warm, and in both cases one could say that it is surprisingly warm. In this context he makes a distinction between bleached intensifiers like terribly, very, on the one hand, and unbleached ones like disgustingly, disappointingly, surprisingly, on the other, noting that the latter ‘convey a property expressed by their adjectival stem and modify the degree relative to that property’ (Nouwen 2020, 14). In contrast, bleached adverbs are argued to ‘end up being associated to the scalar extent their initial unbleached version was regularly associated with’ (Nouwen 2020, 14). We believe that this is in principle correct, but we propose to go one step further in this characterisation. In particular, we argue that there is a division between ‘true’ degree modifiers and modifiers that only indirectly relate to degrees. The former (e.g. a bit, slightly, somewhat, rather, quite, pretty, extremely, very) manipulate a degree directly and can be divided into L-, M- and H-modifiers; the latter (e.g. strangely, weirdly, surprisingly, disappointingly), and this is the new part, merely provide an additional comment on the degree of A(djective)-ness. These two different meaning contributions can be informally paraphrased as in (10). (10)

a. b.

True degree modification: x has the property A to a low/medium/high degree Secondary evaluation of a degree: x has property A to a pos(itive)-degree and this (degree) is strange/weird/surprising/disappointing (to me, the speaker)

We will argue that Catalan ben is of this second type. To put it differently,3 , 4 while we take some degree modifiers to manipulate degrees and 3 We are aware that ascribing such properties to degrees themselves is problematic, as already observed

in Morzycki (2008), so the paraphrase in (10b) should be taken as an informal characterisation. fact that the former are bleached whereas the latter are not, might not be a total coincidence: while some of the former ones are already degree-or scale-oriented lexically (e.g. a bit ), many of the 4The

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to therefore add meaning at the at-issue level, we take others to merely add an evaluation by the speaker on that degree, so that this kind of modification is akin to non-restrictive modification by subjective evaluatives (on which see, e.g., Martin 2014; Umbach 2015), which is arguably not at issue. Let us then turn to our case study.

3

The Case Study: BEN

As observed in Castroviejo and Gehrke (2015), Catalan ben ‘well’ can be used as a modifier of participles, as in (11-a), akin to English well, in which case it arguably modifies an event (cf. Gehrke and Castroviejo 2016), but also as a modifier of genuine adjectives, (11-b), in which case the meaning conveyed is intuitively similar to that of an intensifier (ben), something that (Standard) English well cannot do.5 (11)

a. b.

ben embolicat well wrapped ben simpàtic well nice ‘∼ really/pretty nice’

In this section, we present the distribution of ben and characterise its contribution. Specifically, we provide examples of ben with different adjective types and describe the contexts in which they are felicitous, by comparing it with other ad-adjectival modifiers such as molt ‘very’ or completament ‘completely’. In doing so, we situate ben in the broader family of Madverbials, with a twist. After getting acquainted with the semantics of

latter ones are much broader evaluative adverbs, and it would be surprising if they developed into plain degree modifiers in the first place. There is also nothing a priori good or bad about the lexical meaning of e.g. surprising, whereas with the bleached ones in the first group and their literal (i.e., unbleached) counterparts, such as pretty, terribly, fairly, there is. Whether or not evaluative adverbs with a good or bad valence develop into ‘true’ degree modifiers in the end might then rather be a historical accident. From this perspective, ben, with its clear positive valence and not much more lexical content (see, e.g. Umbach 2015, on good ) might be prone to develop into a ‘true’ degree modifier, it just is not there (yet). 5 On the intensifying use of well in some English dialects, see e.g. Paradis (2000).

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ben, we delve into how much of its lexical semantics gives rise to PPI effects.

3.1

Interaction with Scale Structure and Valence

In this section we show that ben can combine with gradable adjectives with various scale structures and valence—albeit with different interpretive effects. We take this fact to be one argument in favour of claiming that ben does not operate on degrees directly. In the subsequent section, the mentioned interpretive effects will be shown to be the result of the lexical semantics of ben. In previous work (e.g. Castroviejo and Gehrke 2015) we have presented ben as an intensifier that can combine with adjectives with whichever scale structure (Kennedy and McNally 2005), including open scale and closed scale on one or two ends. Here, though, we want to pay more attention to the type of standards of these adjectives, as well as the relation denoted by ben when combined with them. We follow Kennedy and McNally (2005) in distinguishing between absolute and relative adjectives. Absolute adjectives are associated with closed scales and their standard is by default the lexically provided end of the given scale: a minimum standard with adjectives like empty, open, visible, a maximum standard with adjectives like full, closed (the former are partial, the latter total adjectives in Rotstein and Winter 2004). Relative adjectives like tall, big, pretty, ugly, on the other hand, are associated with open scales and their standard is contextually derived as being somewhere on that scale associated with them, depending on a comparison class of individuals. We furthermore make a distinction between adjectives that express an a priori neutral valence (in the sense of Nouwen 2020) and can get a positive or negative valence in a given context, such as tall, empty, open, visible, and adjectives that are more prone to having a positive (e.g. pretty, nice) or negative (e.g. ugly, idiotic ) valence. With absolute adjectives that are a priori neutral with respect to their valence, ben conveys that the standard is met, (12). There is a sense of categorical assertion in the use of ben, as though higher degrees are not

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relevant (which is to be expected given that they are often associated with maximum standards), so the degree reached is satisfactory (approved of ). (12)

Absolute adjectives with a neutral valence a. Totally closed scale: Ben buit ‘empty’, obert ‘open’, visible ‘visible’ b. Upper closed scale: Ben recte ‘straight’, net ‘clean’, pla ‘flat’

If we compare the effect of ben with such absolute adjectives with that of regular degree modifiers, we note that it is similar to that of completament ‘completely’, which Lasersohn (1999) analyses as a slack regulator, but different from molt ‘very’, which actually triggers a relativisation of the absolute predicate in order to ‘reset’ the standard (cf. Kennedy and McNally 2005, on the English counterparts). To illustrate more generally, completament cannot modify a relative adjective like alt ‘tall’, and if molt ‘very’ modifies an absolute adjective like buit ‘empty’, the interpretation may be 80% empty rather than completely empty, (13). (13)

a.

b.

La Isona és {molt, *completament} alta. the Isona is very completely tall ‘Isona is {very, *completely} tall.’ La piscina està {molt, completament} buida. the pool is very completely empty ‘The pool is {very, completely} empty.’

In our view, completament is a true degree modifier which combines with adjectives that encode an upper bound—hence its incompatibility with relative adjectives—and asserts that this upper bound is met (more in accordance with Kennedy and McNally 2005 than with Lasersohn 1999 in this respect); ben, on the other hand, conveys that completeness is reached as an inference from the positive evaluation of a degree. In this sense, it makes a secondary evaluation of a degree (recall the proposed distinction in (10)). Despite sharing the ability to modify absolute adjectives, ben differs from completament in that the former can also combine with relative adjectives, (14).

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(14)

Relative adjectives ben alt ‘tall’, bo ‘good, tasty’, petit ‘small’, dolç ‘sweet’, intel.ligent ‘intelligent’

The meaning effect of ben with relative adjectives is harder to grasp, and this is where the valence of the given adjective (in a given context) plays a crucial role. In particular, with adjectives that have a positive valence (in a given context or in general), the meaning effect is like that of sufficiency or satisfaction (e.g. sufficiently tasty/intelligent). In this respect it is also different from, e.g. molt ‘very’, which resets the standard, whereas with ben the standard for a given adjective seems to remain untouched. With adjectives that have a negative valence (in a given context or in general), the meaning effect is closer to that of an H-modifier, or similar to the effect of English really, in the sense that x clearly has the property in question, and this holds for both relative and absolute adjectives with a negative valence, (15). (15)

Negative valence adjectives a. Relative: Ben pesat ‘annoying’, difícil ‘difficult’, trist ‘sad’ b. Absolute: Ben brut ‘dirty’, malalt ‘sick’

Finally, we also observe with a priori neutral absolute adjectives in combination with ben that in a given context they can either get a meaning of sufficiency or completion, which is when the context makes clear that the adjectival property in question is good, or a meaning of excess, when the property in question is bad in that particular context. This is illustrated for moll / humit ‘wet’ in (16). (16)

a.

b.

(Context: At the hairdresser the hair should be wet to provide a good haircut. → Wet hair is good.) Els cabells (ja) estan ben molls. the hairs already are ben wet ‘∼ The hair is sufficiently wet.’ (Context: After taking a shower I want a dry towel, but the only towel I find is somewhat wet. → Wet towels are bad.)

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La tovallola està ben molla! the towel is ben wet ‘∼ The towel is really wet (and I don’t like that).’ This different effect of the valence of a given adjective on the interpretation of ben as expressing some meaning of sufficiency (with positive adjectives) or excess (with negative adjectives) is again reminiscent of Nouwen’s (2020) Goldilocks Principles in (8). In particular, Nouwen also observes that, e.g., in a context where the property associated with a relative adjective is evaluated as negative, a degree modifier like a bit, which is normally not acceptable with relative adjectives, gets the interpretation of excess, as in (17). (17)

?The towel is a bit thick. ∼ The towel is a bit too thick.  *The towel is a bit thick enough.

He argues that this is so because excess creates a lower bound on the otherwise unbounded scale, and given that modifiers like a bit require lower bounds (i.e., minimum standards) their input requirements are met in such excess contexts. With positive adjectives, on the other hand, the meaning effect of ben rather seems to be the opposite, and this is quite independent of whether we are dealing with relative or absolute adjectives: The degree to which x holds the property in question is sufficient or enough. In line with the Goldilocks Principles, this degree will often appear somewhere in the middle of the scale (but still within the range where the standard is met, even without ben). So for a relative adjective the individual has the property in question (the standard is met) and this degree is sufficient (to be called A or for some purpose or function; see below). For an absolute adjective, on the other hand, the standard is still met, but since it is at the end of a scale, the effect is not that of placing the degree in the middle of the scale, which would be expected for an M-modifier; instead, a sense of satisfaction due to completion obtains.

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BEN in Entailment-Cancelling Contexts

Let us move on to describe the acceptability of ben across entailmentcancelling contexts, which is relevant for our analysis of its PPI-hood (Sect. 4.2). First, negation is the worst context, while polar questions and antecedents of conditionals are not ruled out in certain cases.6 In fact, even though we can find examples such as (18-a), which do not sound completely ill-formed in the adequate context,7 ben under negation is typically unacceptable, (18-b). (18)

a.

?La camisa no està ben neta. Encara has de fregar the shirt neg is well clean still have.2sg of scrub una mica més. a little more ‘The shirt is not well clean. You’ll have to scrub it some more.’ b. #La Isona no és ben alta. No pot entrar a l’equip. the Isona neg is well tall neg can enter to the-team ‘Isona is not well tall. She cannot join the team.’

Second, relative adjectives in general as well as absolute adjectives with a negative valence are also unacceptable under other entailment-cancelling contexts (we illustrate with conditionals, (19-a,b)), whereas absolute adjectives with a positive valence are good in these contexts (19-c,d). (19)

6 We

a. #Fa temps que no veiem la Isona, però si és ben alta, does time that neg see.1pl the Isona but if is well tall li her.dat direm que s’apunti a l’equip de bàsquet. tell.fut.1pl that sign.up.subj.3sg to the-team of basketball

have not tested this systematically or experimentally and we do not know whether the effects are strictly semantic or pragmatic in nature, so we will not commit to a particular account at this point. 7 For the sake of brevity, we cannot delve into the conditions that may turn the example into mildly acceptable. Pending further research, we may think that the modifiee having a conventional standard and the context suggesting a sufficiency reading may play a role.

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Intended; ‘We haven’t seen Isona for a long time, but if she is well tall, we will tell her to join the basketball team.’ b. #Si el terra és ben moll, relliscarem. if the floor is well wet slip.fut.1pl Intended: ‘If the floor is well wet, we will slip.’ c. Si els cabells estan ben molls, ja els pots tallar. if the hairs are well wet already them can cut ‘If the hair is well wet, you can go ahead and cut it.’ d. Si la camisa està ben neta, la plegarem i if the shirt is well clean it.acc fold.fut.1pl and la posarem al calaix. it.acc put.fut.1pl in.the drawer ‘If the shirt is well clean, we’ll fold it and put it in the drawer.’ Third, whenever ben is acceptable in such contexts (i.e., with absolute adjectives with a positive valence), the meaning obtained is one of sufficiency, close to enough, as in (19-c) and (19-d). In fact, taking the two last properties together, the context that allows ben in the antecedent of a conditional (as well as in a polar question or in the scope of a modal adverb) is one where the minimal degree that is met corresponds to the degree of the said property that is needed to fulfil some function. Above, to be able to fold the shirt and put it in the drawer, the shirt has to reach total cleanness. It follows that the sentence in (19-b) would be felicitous if slipping would be a goal to fulfil, just like (19-c) is felicitous because having the hair be wet (enough) is the goal to properly cut it. However, notice that, unlike in the case of enough, ben does not admit a sentential complement that encodes the minimal degree that has to be met as part of its argument structure, (20). (20)

a.

b.

La camisa està prou neta per guardar-la al calaix. the shirt is enough clean to keep-it in.the drawer ‘The shirt is clean enough to keep it in the drawer.’ La camisa està ben neta (*per guardar-la al calaix). the shirt is well clean to keep-it in.the drawer

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Even though the minimal degree that is met or exceeded is not grammatically expressible through a complement clause, it should be noted that once the standard is objectively established and, thus, the predicate becomes less vague, the PPI behaviour is weakened in entailment-cancelling contexts.8 In the following section we address how the lexical semantics of ben is responsible for some of the properties described above.

3.3

Lexical Semantics of BEN

Following previous work on the intensifying use of the Catalan adjective bon ‘good’ (Castroviejo and Gehrke 2019), we propose to analyse ben as a subsective modifier that narrows down the set of individuals bearing the property expressed by the modifiee to the ones that are good instances of the property (according to some judge).9 Hence, also following our previous work, we claim that the intensification effects triggered by ben are not the result of direct quantification over or manipulation of degrees, unlike what might be going on with molt ‘very’ and completament ‘completely’ (on which see Kennedy and McNally 2005, a.o.). Instead, we want to exploit the lexical semantics of evaluative good conveying the modal notion of approval to fulfil a function (see Asher 2011, for a denotation of English good ) (the subsective part), to yield the observed intensification, along with the idea that the composition mode of the evaluative adverb itself is like that of non-restrictive modification (it is not at issue). We start from the denotation of the prototypical subsective adjective skillful, like the intensional version in Morzycki (2015) in (21). (21)

[[skillful ]] = λPe,s,t λxe λws .skillful-as(P)(x)(w)

We assume that Catalan non-intersective bon ‘good’ and ben have the same semantics, but a different phonological spell-out because they have a different syntactic distribution: the former is an ad-nominal modifier, whereas the latter is an ad-adjectival modifier. Hence, building on (21)— 8 Cases

such as (18-a) have to be studied in more depth. the sake of brevity, here we do not delve into the role of the judge, so we leave it for future research.

9 For

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and leaving aside intensionality for simplicity—we can provide a subsective analysis of bon/ben along the lines of (22). (22)

[[bon/ben]] = λPe,t λxe .(good-as(P))(x)

In Castroviejo and Gehrke (2019), we argued that, e.g. in una bona dosi d’insulina ‘a good dose of insulin’, intensification is the result of selecting those instances in the extension of insulin that the relevant judge (by default, the speaker) considers good. Given that container/measure nouns like dose have an extension that describes a partial order, (23), the sole criterion to determine goodness is size. (23)

For all a, b, and c in the extension of a noun like dose, a. Reflexive: a is at least as big as itself. b. Antisymmetric: if a is at least as big as b, b cannot be at least as big as a, unless a and b are the same. c. Transitive: if a is at least as big as b, and b is at least as big as c, then a is at least as big as c.

Because of the monotonicity effects of evaluative modifiers, if a is considered a good instance of a dose, for any bigger instance, it will also be considered good. Hence, intensification arises: bigger instances—but not necessarily smaller ones—are considered good. A good dose is one whose size satisfies the speaker because it is considerably big. A problem that arose for our previous proposal, however, was that if the dose gets too big it should not be a good dose anymore (e.g. one can get an insulin shock). We can now say that this is where Nouwen’s (2020) Goldilocks Principles in (8) are at play again so that doses that are too big will not be considered good anymore. We want to argue that the same phenomenon extends to ben where the modifiee is not a noun, but an Adjectival Phrase with the same lexical properties as dose. In particular, ben combines with a gradable adjective (e.g., alt ‘tall’) that has already merged with the type-shifter pos, (24).10 10 Uli Sauerland (p.c.) points out that this analysis predicts the availability of phrases such as ben 180 cm tall, contrary to fact. In the case of Catalan, this could be ruled out on independent grounds. Note that 180 cm alt ‘180 cm tall’ is ungrammatical (without the use of ben). However, we should investigate what the facts are for other languages that have elements similar to ben. Additionally, if

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a. b. c. d.

pos: λgd,e,t λx.∃d[standard(d)(g)(C) ∧ g(d)(x)] (Kennedy and McNally 2005) alt: λdλx.tall(x) ≥ d pos alt: λx.∃d[standard(d)(tall)(C) ∧ tall(x) ≥ d] ben alt ‘well tall’: λx[(good-as([[pos alt]])(x)]

Therefore, in (24-c), the combination of pos and alt ‘tall’ yields the (characteristic function of the) set of individuals who are tall to at least a standard degree. Then, when ben is added into the derivation, those instances that the relevant judge considers good are selected. What does this amount to? Since the extension of ‘pos alt’ constitutes a partial order as in the case of dose above, the effect is to convey that the selected individuals are good instances of tall individuals. Unlike what happened with the ad-nominal modification case, though, where the extension of dose can include good (big) and bad (small) instances of the object, here the modified predicate only contains individuals whose degree of tallness reaches the contextual standard. Thus, in case all of the individuals in the set are good instances of ‘pos alt’, it is possible that the modification by ben does not narrow down the domain of the modifiee. It follows from our analysis that the standard of comparison of alt ‘tall’ is established out of averaging over a comparison class of individuals (i.e., in a between-individuals fashion, as suggested by Toledo and Sassoon 2011), just as it would be with a positive adjective without ben, and then, when this predicate is modified by ben, the criteria to determine good instances follows a subjective procedure, given the evaluative speaker-oriented nature of ben. We furthermore propose that ben can combine with a relative adjective whenever the context satisfies the following assumption: (25)

x is as A as x SHOULD be.

Should and ought to in English are modal verbs that receive a teleological or goal-oriented interpretation; they have been claimed to express weak necessity (von Fintel and Iatridou 2005, 2008) and, more recently, to we were to extend a similar analysis to other evaluative intensifiers such as surprisingly, we should be able to rule out *surprisingly 2 meters tall. We leave this issue for further research and thank Uli for his remark.

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appeal to negotiable ideals, something shared with e.g. the adjective good (Rubinstein 2012, 2014). We claim that this underlying idea of a goal to be fulfilled through a negotiable—and hence subjective—ideal is what guides the judge into deciding which objects in the extension of ‘pos A’ are good instances thereof. For instance, in (26), the speaker may be looking for an individual that is sufficiently tall for a purpose or that simply definitely counts as tall in a particular context. (26)

La Isona és ben alta. the Isona is well tall ‘∼ Isona is really/pretty tall.’

Bear in mind that alt is a relative adjective whose standard is established through contextual means, when it combines with pos. We expect different outcomes depending on the type of standard of the adjective, as pointed out in Sect. 3.1. Remember that we would like to be able to derive that, on the face of a relative adjective with a positive valence—and hence with a contextual standard—, the degree that is reached is sufficient to stand out as a positive measure for A. Regarding negative valence adjectives, we observed an excess effect, and maximum standard adjectives with neutral valence give rise to a completion effect. Does this follow from the proposed analysis? We argue that it does. Let us turn to negative adjectives likes the one exemplified in (27). (27)

a.

b.

El terra està ben moll. the floor is well wet ‘∼ The floor is really wet.’ Aquesta història és ben trista. this story is well sad ‘∼ This story is really sad.’

The good instances of negative valence adjectives should be those that stand out, so they unquestionably entail the positive form. Being negative, even a small amount thereof is enough to count as A; take for instance pesat ‘annoying’ or trist ‘sad’: no matter how annoying or sad, even if only a little, is already too much. Interestingly enough, these cases teach us that the sense of approval expressed by good, introduced by ben, is directed

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towards how the subject instantiates the property; it does not necessarily reflect the speaker’s overall attitude, though, since it would be counterintuitive to approve of a high degree of annoying-ness or sadness.11 As in the previous case, we claim that there is a specific context that will license the intensifying interpretation of ben given the particular characteristics of negative valence adjectives, namely the one in (28). (28)

x is more A than x SHOULD be.

In accordance to the semantics of excess, the degree that is reached exceeds a standard that, again, is established depending on some goals. The floor is wetter than it should be (since one small drop is already too much) and the story is sadder than it should be (since a somewhat sad story is already too sad). Finally, we consider the case of absolute adjectives with a positive valence, like the one illustrated in (29). (29)

La casa està ben neta. the house is well clean ‘∼ The house is really clean.’

The contexts that license the intensifying interpretation of ben as a modifier of positive valence absolute adjectives are the ones where the following assumption is met: (30)

x is as A as x CAN be.

Here can flags circumstantial modality: the conventional standard of the recipient acts as the threshold12 that is met and, as in the first case, sufficiency—not excess—is conveyed. Hence, clearly, when the members of the extension of ‘pos A’ reach the conventional standard of the adjective (in the particular context), they can be considered good instances of the property.

11 Of course, diachronically, such cases must have been read as ironic, but this is not the case at

ben’s current stage. 12 See McNally (2011) for a qualification of this idea. For the sake of brevity, we do not delve into this issue.

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In predications where a maximum degree is entailed, selecting those individuals that the speaker considers good instantiations of the predicate, modification with ben may be innocuous from the point of view of degree manipulation. What our analysis predicts is that the entities in the extension of pos net ‘clean’ reach a conventional standard, and the speaker approves of them as instances of the predicate. Hence, there is an additional layer of modification that does not seem to modify the literal meaning of the Adjective Phrase, but which adds a subjective approval. Our tentative characterisation of this effect is what Lasersohn (1999) called slack regulation. That is, while there may not be a semantic effect, because the extension of the Adjective Phrase is not narrowed down, it may be that the use of ben precludes a loose interpretation of the inference that the standard has been met. Taken all this together, we can conclude that the degree effects that arise from modification with ben can be explained independently of degree quantification and merely on the basis of non-intersective modification with an evaluative modifier given specific contexts that guide what good instances amount to, all of them consistent with the expression of a sufficient degree. Let us then move on to the PPI behaviour of (some) intensifiers in general and ben in particular.

4

Intensification and PPI-Hood

In this section we address the question of whether an analysis based on informativity, along the lines of Solt and Wilson (2020), is adequate to explain the PPI behaviour of ben and other M-intensifiers that do not directly operate on degrees. We put forth that precisely not quantifying over degrees directly plays an important role in giving rise to the PPI behaviour and, hence, argue against such an analysis for our particular case; instead we will deliberate about alternative venues. Nevertheless, we are convinced that Solt and Wilson’s account does work for what we have called ‘true’ degree modifiers.

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4.1

Solt and Wilson (2020)

Solt and Wilson (2020) take a closer look at the PPI behaviour of English M-modifiers , such as fairly, somewhat, sort of, and provide a semantic account for these that explains their lexical differences (e.g. in terms of input requirements), while still capturing their PPI behaviour in a uniform way. For example, they treat fairly as a true degree modifier that takes something of type d, e, t as an input (31), given that it has to directly precede a gradable adjective. (31)

a. b.

[[fairly]]w,i = λAd,e,t λdλx.A(x)(d)∧ stdi (A)(d) where i ∈ I is some available choice of interpretation. Alice is fairly tall. = ∃d[HEIGHTw (alice) ≥ d∧ stdi (tall)(d)]

In prose they state that fairly ‘loosens the interpretation of an adjective by introducing a standard equivalent to EVAL at some accessible index i ∈ I ’, where EVAL is taken as a null morpheme on positive adjectives, which introduces a function specifying degrees that satisfy a contextual standard, following Rett (2008) (similar to pos in other accounts). The semantics of somewhat accounts for its broader distribution, since it is treated as a quantifier over degrees, similarly to measure phrases and much; in particular, it is argued that somewhat involves existential quantification over a scalar interval that may be supplied by EVAL (32). (32)

a. b.

[[somewhat ]]w,i = λDd,t ∃D = ∅[D ⊂ D] The room is somewhat dirty. [[somewhat ]]w,i (λd.the room is d-EVALi dirty) = ∃D = ∅[D ⊂ λd.DIRTYw (r oom) ≥ d∧stdi (dirty)(d)]

Finally, for the analysis of sort of, which also has a wider distribution, they follow Anderson (2016) in employing Lasersohn (1999) pragmatic halos, (33). (33)

a. b.

[[...sort of α...]]w,i = 1 iff ∃α ∈ HALOi (α) such that [[...α ...]]w,i = 1 The concert is sort of free. ∃α ∈ HALOi (free) such that [[α (concert)]]w,i = 1

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To account for the PPI behaviour of these modifiers they pursue two different avenues, building on previous work (Solt 2018; Solt and Waldon 2019), and they illustrate these with fairly. In a nutshell, they assume that M-modifiers in combination with an adjective more or less render the same lexical semantics as their unmodified counterparts; all else being equal, the unmodified version should be the only one existing since it is simpler. However, the modified versions give rise to a scalar implicature that is absent from their unmodified counterparts, due to a competition with very A; as a result, in positive contexts (and only in these; see below) the modified versions are more informative than the unmodified ones, precisely because they come with this additional scalar implicature (with informativity formally defined as in Solt 2018). For example, large and fairly large express more or less the same semantics lexically, (34). (34)

a. b.

large i : SIZE ∈ stdi (large) (common ground: i ∈ I ) fairly large i : SIZE ∈ stdi (large) (for some available i ∈ I )

Only the modified version evokes alternatives that are, presumably, of the same form at some level (e.g. MOD(A), although this is not spelled out in their proposal). As can be seen from the following figure (from Solt and Wilson 2020), large and fairly large both give us the same stretch on the size scale, whereas very large is more informative because the part on the scale is necessarily smaller.

The competition with the more informative (because semantically narrower) very large leads to the implicature of fairly large that the narrower meaning contribution of very large gets negated, as illustrated in the following figure (from Solt and Wilson 2020).

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Now, when these expressions are negated we get a scale reversal effect, or rather, negation leads to the complementary part of the scale. In this case, not large and not fairly large still give us exactly the same stretch on the scale (everything below the standard). The semantic contribution of not very large, however, is now less informative than not fairly large, because the stretch on the scale is larger (everything that is not very tall ), as illustrated below (from Solt and Wilson 2020).

Thus, we do not get the scalar implicature that we got in the positive case (i.e. The house is not fairly large does not give rise to the scalar implicature The house is not very large). As a result, the modified version (not fairly large) expresses the same meaning as the unmodified version (not large), and given that the unmodified version is simpler than the modified one (again, see Solt 2018, for a formal definition), it blocks the modified version, and not fairly large is ruled out. An alternative explanation for the PPI behaviour that Solt and Wilson entertain is to assume that implicatures could also be embedded under negation. In this case, we proceed as before with the non-negated versions, and a scalar implicature would still be derived for fairly large, which involves not very large. In this case, we get a medium stretch on the scale of size which excludes both ends on the scale. If this now gets negated, though, we get a discontinuous stretch on the scale, the lower and the upper part, to the exclusion of the middle part, as illustrated in the following (from Solt and Wilson 2020).

This, however, would give rise to a non-convex meaning and thus violate a general constraint on meanings in discourse (building on Solt and Waldon 2019). While we believe that this kind of account might work for some of the degree modifiers under discussion, e.g. for fairly, we do not see that it can be extended to all of them (see also Castroviejo and Gehrke 2020, for an

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earlier attempt to apply such an account to ben). The main problem we see for extending this account to all M-modifiers, is the motivation for the choice of alternatives to ‘MOD A’. In particular if Solt and Wilson (2020) are on the right track in providing fairly different lexical semantics for such modifiers in treating only some of them as true degree modifiers (e.g. fairly) but others as quantifiers over degrees (e.g. somewhat ) or even in terms of pragmatic halos and thus not necessarily involving degrees at all (e.g. sort of ), an expression like very tall (involving another true degree modifier) should be an alternative only to some of these, but not to all of them (unless all that matters is some kind of superficial commonality, but we do not really see how that should work).

4.2

The PPI Behavior of BEN

As we have commented above, in this paper, we have introduced ben as an M-modifier, so we may expect that an analysis along the lines of Solt and Wilson’s (2020) fares well for ben. However, as stated previously, the potential competitors for ‘ben A’ are not clear, because ben is not a degree expression that quantifies over degrees to begin with. It would be very hard to tease apart the degree effects of ben and other positive valence adverbs in terms of which areas on the scale they target. Therefore, we need to consider alternative venues for an explanation. Here we entertain two (possibly interwoven) research lines that could hopefully be extended to other evaluation-based intensifiers. The main idea we put forth is that it is no coincidence that evaluative adverbs that are (used as) intensifiers, too, have a PPI behaviour; specifically, while they may be in different stages of grammaticalisation, as long as their literal meaning does not concern quantifying over degrees, intensification is the result of an inference. Therefore, their literal meaning is an evaluation that amounts to degree intensification, but certain contextual conditions need to be met for the intensification to come about. In one view, the intensification inferred in the case of ben is the result of the specific contexts that license these interpretations. In other words, the use of ben presupposes that the conditions are met for the alignment between evaluation and intensification. That is, in Catalan (and other

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Romance languages) it has been possible for the word for well (i.e., ben) to behave like a degree expression—broadly construed. The motivation for the alignment between evaluation and intensification are the properties of the contexts that, are shown in (25), (28) and (30) above, establish a link between satisfaction/approval and a sufficient degree of a property for some purpose. In a nutshell, for a relative adjective like tall, if the individual in question is as tall as she should be (to fulfil some purpose, including expectations), then we can make sense of the fact that the speaker selects the good instances of tall individuals. Similarly, in the case of negative valence adjectives, if the individual in question is considered to have some property beyond what she should, then we understand what it means that this individual is a good instantiation of the said property. Finally, in the case of (positive valence) absolute adjectives, the reason for approval or satisfaction coincides with reaching the conventional maximum degree of the property. Now, under this view, if we cannot accommodate what selecting good instances of a scalar property amounts to, the expression may be infelicitous. This is, we contend, what happens in entailment-cancelling contexts, whereby it cannot be taken for granted that the individual in question has a sufficient degree of the property (either because it is openly negated or merely hypothesised). Take, for instance, (35). (35)

La Isona no és (#ben) alta. the Isona neg is well tall

To make sense of ‘well tall’, we must accommodate that the speaker considers Isona to be as tall as she should be. This gives us the criteria to select the good instances of individuals who share the property of being ‘pos tall’. However, when the sentence is negated, negation does not affect the presupposition, which projects. Hence, a contradiction arises between what is said and the presupposition that licenses the intensification interpretation of ben. Importantly, ben being a PPI in its degree reading—it does appear in entailment-cancelling contexts when it is a manner modifier—(35) cannot convey that Isona is not as tall as she should be. In general terms, it could be that M-modifiers with evaluative bases need an affirmative context to support their degree interpretation.

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The second venue, which might also be compatible with the first one, would be to make a clear distinction between at-issue and non-at-issue meaning contributions. Let us return to our informal characterisation of the difference between ‘true’ degree modification and secondary evaluation of a degree in (10), repeated in (36). (36)

a. b.

True degree modification: x has the property A to a low/ medium/high degree Secondary evaluation of a degree: x has property A to a pos(itive)-degree and this (degree) is strange/weird/surprising/ disappointing (to me, the speaker)

The second type of modification is more indirect and does not manipulate the standard degree at all, and we argued that this is what we should get with ben. What happens with negation, and what gets negated? It could be that with ben negation will only target the first conjunct in (36-b) but not the second one (the secondary evaluation). Hence, we would get something like x does not have property A to a pos-degree, and this (degree) is good (sufficient/enough/excessive). Why would that be? Let us look at negation with other evaluative modifiers that (possibly more clearly) add secondary content, e.g. expressives like fucking : (37)

a. b.

The fucking dog is (not) sleeping on the couch. The story is (???not) fucking boring.

Negation with the modifier on a predicative element, as in (37-b), is as degraded as our cases with ben (see Morzycki 2012, for discussion), and the explanation might possibly be the same (we return to this below). If the main predication is about something else and the modifier appears in attributive position, as in (37-a), negation is acceptable, but the contribution of the modifier is still not targeted by negation. In particular, whether or not the sentence contains negation the attitude of the speaker towards the dog remains the same: she really does not like the dog. Hence, negation does not target this meaning component. Interestingly, attributive ben in entailment-cancelling contexts is not problematic either, as we can see, e.g., in (38).

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Pots escollir una jugadora (que sigui) ben alta? can.2sg choose a player.fem that is.subj well tall ‘Can you select a well tall player?’

Although we will not have much more to say about this in the present paper, we note that attributive position of the Adjectival Phrase is equivalent to predication with the subjunctive mood in Catalan (‘a well tall player’ is here equivalent to ‘a player that BE well tall’), and this might play a role here as well. So what goes wrong in predicative position then, and why is this similar to (37-b)? In a previous proposal (Castroviejo and Gehrke 2015), building on Liu’s (2012) account of the PPI behaviour of evaluative adverbs like unfortunately, we assumed that there is a clash between the at-issue tier, at which it is asserted that x does not have the property in question to a standard degree, and the non-at-issue tier, which evaluates the standard degree, which is additionally presupposed to hold; this leads to a clash between the meanings at the at-issue-tier and the presupposition of the non-at-issue-tier (39-a). (39-b) shows in how far this is similar to negation with an expressive modifier. (39)

a. *En Pere no és ben simpàtic. the Peter not is well nice (i) ¬(nice(p)) (ii) nice is well ascribed to Peter. b. ??The story isn’t fucking boring. (i) ¬(boring(story)) (ii) The speaker expresses a negative attitude at the story being boring.

In our former account, the meaning contribution of ben was less worked out, as we can see in (39-a), whereas in the current paper we are more precise about it and also shed more light on the possible modal or other presuppositions that come with it. While we do not necessarily think that (39-a) is enough of a formal account, the gist of it might still be on the right track. Finally, it is possible that both venues go hand in hand, and it might even be the case that it also depends on the grammaticalisation path of

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degree modifiers more generally: evaluative ones might start out as those in (36-b) but then possibly grammaticalise into ‘true’ degree modifiers of the sort in (36-a). For these, a PPI account along the lines of Solt (2018), Solt and Waldon (2019), Solt and Wilson (2020) looks highly promising, given that such modifiers should compete with other ‘true’ degree modifiers. It is possible that ben is heading in that direction (an avenue for future research) and the examples where it is actually better under negation, precisely when one negates the enough component, as in (18-a), might point in that direction,13 but it is just not there (yet).

5

Conclusions

This paper has discussed the lexical semantics and positive polarity of Mmodifiers, in light of varied and inspiring work carried out by Stephanie Solt (and collaborators). By looking at Catalan ben ‘well’, which is both a manner adverbial and a modifier of genuine adjectives, we have argued that, unlike somewhat or fairly, prototypical M-modifiers targeting degrees at the middle part of the scale, evaluative modifiers can express degree intensification without directly manipulating degrees. Moreover, we have claimed that the fact that intensification is an indirect effect may be related to their inability to occur under the scope of entailment-cancelling operators. In fact, we have argued that, while an informativity-based account along the lines of Solt and Wilson’s (2020) succeeds at explaining why degree modifiers like somewhat and fairly are PPIs, this line of research does not seem well suited for evaluative adverbs that operate on degrees only indirectly. Instead, we have entertained two alternative avenues, both having to do with the incongruence created by operators that target the at-issue content, but which leave presupposed or conventionally implicated meaning untouched.

13 Among the Romance language varieties, French bien and Latin American bien seem to have a broader distribution than Catalan ben. Whether or not this involves being at a later stage in grammaticalisation and what the differences are in lexical semantics among the different varieties are interesting questions for future research. We refer the reader to Romero (2007) and GutiérrezRexach and González-Rivera (2017), a.o.

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Acknowledgements This research has been partially supported by the project VASTRUD (PGC2018-096870-B-I00) funded by the Ministry of Science and Innovation (MCI) / Spanish Research Agency (AEI) and the European Regional Development Fund (FEDER, EU), the IT1396-19 Research Group (Basque Government), and GIU18/221 (University of the Basque Country, UPV/EHU. We thank Uli Sauerland for comments on an earlier version, and, even more importantly, we thank Stephanie Solt for her inspiring work.

References Anderson, C. (2016). Intensification and Attenuation Across Categories. PhD thesis, Michigan State University. Asher, N. (2011). Lexical Meaning in Context: A Web of Words. Cambridge University Press, Cambridge. Bolinger, D. (1972). Degree Words. Mouton, The Hague. Castroviejo, E. and Gehrke, B. (2015). A good intensifier. In Murata, T., Mineshima, K., and Bekki, D., editors, New Frontiers in Artificial Intelligence (JSAI-isAI 2014 Workshops, LENLS, JURISIN, and GABA, Kanagawa, Japan, October 27–28, 2014, Revised Selected Papers), Lecture Notes in Computer Science, pages 114–129. Springer, Dordrecht. Castroviejo, E. and Gehrke, B. (2019). Intensification and secondary content: A case study of Catalan good. In Gutzmann, D. and Turgay, K., editors, Secondary Content: The Linguistics of Side Issues, pages 107–142. Brill, Leiden. Castroviejo, E. and Gehrke, B. (2020). Evaluative intensification and positive polarity: Catalan well as a case study. Talk at the ZAS Semantics Colloquium, March 2020. von Fintel, K. and Iatridou, S. (2005). What to do if you want to go to Harlem: Anankastic conditionals and related matters. Ms. MIT. Available online at: http://mit.edu/fintel/www/harlem-rutgers.pdf . von Fintel, K. and Iatridou, S. (2008). How to say ought in foreign: The composition of weak necessity modals. In Guéron, J. and Lecarme, J., editors, Time and Modality, pages 115–141. Springer, Dordrecht. Gehrke, B. and Castroviejo, E. (2016). Good manners: On the degree effect of good events. In Bade, N., Berezovskaya, P., and Schöller, A., editors, Proceedings of Sinn und Bedeutung 20, pages 252–269. semanticsarchive.

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Gutiérrez-Rexach, J. and González-Rivera, M. (2017). Adverbial elatives in Caribbean Spanish. In Fernández-Soriano, O., Castroviejo, E., and PérezJiménez, I., editors, Boundaries, Phases and Interfaces: Case Studies in Honor of Violeta Demonte, pages 108–128. John Benjamins, Amsterdam. Israel, M. (1996). Polarity sensitivity as lexical semantics. Linguistics and Philosophy, 19(6):619–666. Israel, M. (2011). The Grammar of Polarity: Pragmatics, Sensitivity, and the Logic of Scales, volume 127. Cambridge University Press, Cambridge. Kennedy, C. and McNally, L. (2005). Scale structure, degree modification, and the semantics of gradable predicates. Language, 81:345–381. Lasersohn, P. (1999). Pragmatic halos. Linguistics and Philosophy, 75:522–571. Liu, M. (2012). Multidimensional Semantics of Evaluative Adverbs. Current Research in the Semantics Pragmatics-Interface (CRiSPI) 26. Brill, Leiden. Martin, F. (2014). Restrictive vs. nonrestrictive modification and evaluative predicates. Lingua, 149A:34–54. McNally, L. (2011). The relative role of property type and scale structure in explaining the behavior of gradable adjectives. In Nouwen, R., van Rooij, R., Sauerland, U., and Schmitz, H.-C., editors, Vagueness in Communication: International Workshop, VIC 2009, held as part of ESSLLI 2009, Bordeaux, France, July 20–24, 2009. Revised Selected Papers, pages 151–168. Springer, Berlin. Morzycki, M. (2008). Adverbial modification in AP: Evaluatives and a little beyond. In Dölling, J. and Heyde-Zybatow, T., editors, Event Structures in Linguistic Form and Interpretation, pages 103–126. Walter de Gruyter, Berlin. Morzycki, M. (2012). Adjectival extremeness: Degree modification and contextually restricted scales. Natural Language and Linguistic Theory, 30(2):567–609. Morzycki, M. (2015). Modification. Key Topics in Semantics and Pragmatics. Cambridge University Press, Cambridge. Nouwen, R. (2013). Best nogal aardige middenmoters: de semantiek van graadadverbia van het middenbereik. Nederlandse Taalkunde, 18(2):204–214. Nouwen, R. (2020). Goldilocks and degree modification. Ms. Utrecht University. Paradis, C. (2000). It’s well weird: Degree modifiers of adjectives revisited: The nineties. Language and Computers, 30:147–160. Rett, J. (2008). Degree Modification in Natural Language. PhD thesis, Rutgers University. Romero, C. (2007). Pour une définition générale de l’intensité dans le langage. Travaux de linguistique, 54(1):57–68.

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Rotstein, C. and Winter, Y. (2004). Total adjectives vs. partial adjectives: Scale structure and higher-order modifiers. Natural Language Semantics, 12:259– 288. Rubinstein, A. (2012). Roots of Modality. PhD thesis, University of Massachusetts, Amherst. Rubinstein, A. (2014). On necessity and comparison. Pacific Philosophical Quarterly, 95(4):512–554. Solt, S. (2018). Approximators as a case study of attenuating polarity items. In Hucklebridge, S. and Nelson, M., editors, NELS 48: Proceedings of the 48th Annual Meeting of the North East Linguistic Society 3, pages 91–104. GLSA, Amherst, MA. Solt, S. and Waldon, B. (2019). Numerals under negation: Empirical findings. Glossa: a journal of general linguistics, 4(1): 113:1–31. Solt, S. and Wilson, C. (2020). M-modifiers, attenuation and polarity sensitivity. Paper presented at Sinn und Bedeutung 25, September 2020. Toledo, A. and Sassoon, G. (2011). Absolute vs. relative adjectives: Variance within vs. between individuals. In Proceedings of SALT 21, pages 135–154. Cornell University, Ithaca, NY. Umbach, C. (2015). Evaluative propositions and subjective judgments. In van Wijnbergen-Huitink, J. and Meier, C., editors, Subjective Meaning, pages 127– 168. De Gruyter, Berlin.

She is Brilliant! Distinguishing Different Readings of Relative Adjectives Nicole Gotzner and Sybille Kiziltan

1

Introduction

1.1

Outline and Research Questions

When someone tells us Stephanie Solt is intelligent, how do we know which degree of smartness the person had in mind? When does intelligent mean “brilliant” and when does it mean “not brilliant”? The This work was supported by the German Research Foundation (DFG) as part of the Xprag.de Initiative (Grant Nr. BE 4348/4–1) as well as an Emmy Noether grant awarded to NG (Grant Nr. GO 3378/1–1). We are grateful to Louise McNally for helpful comments on an earlier version of this manuscript. NG and SK designed the experiment. NG analyzed the data and wrote the article.

N. Gotzner (B) Universität Potsdam, Leibniz-Zentrum Allgemeine Sprachwissenschaft, Berlin, Germany e-mail: [email protected] S. Kiziltan Leibniz-Zentrum Allgemeine Sprachwissenschaft, Berlin, Germany © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 N. Gotzner and U. Sauerland (eds.), Measurements, Numerals and Scales, Palgrave Studies in Pragmatics, Language and Cognition, https://doi.org/10.1007/978-3-030-73323-0_7

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given example is easy since Stephanie Solt is a genius!1 But in ordinary conversation, speakers make use of different alternative expressions to communicate varying degrees to which a property holds. Sometimes the speaker has a stronger alternative in mind such as Stephanie is brilliant and by uttering a statement involving the weaker term, the hearer may conclude that the stronger term does not hold. This line of reasoning is a standard scalar implicature, where a stronger alternative (brilliant ) stands in an asymmetric entailment relationship with a weaker one (intelligent ) (e.g., Horn 1989). Negation is known to reverse entailment relationships and thus cancels scalar inferences. Yet another kind of pragmatic inference is known to occur when gradable adjectives appear under negation. By saying John is not intelligent, the speaker may want to convey that John is “rather stupid,” which is known as negative strengthening (R-based strengthening, Horn 1989). A well-known observation is that negative strengthening displays a polarity asymmetry with positive terms like intelligent being more likely to be strengthened than their corresponding negative antonyms like stupid (see especially Brown and Levinson 1987; Horn 1989). Most previous work has focused on the interpretation of weak adjectives such as intelligent and stupid under negation (e.g., Ruytenbeek et al. 2017; Tessler and Franke 2018; Gotzner and Mazzarella, in revision; Mazzarella and Gotzner, in revision) while their stronger scale-mates (brilliant and idiotic ) have not been systematically explored in relation to corresponding antonymic expressions (but see Gotzner et al. 2018a, b for an investigation of weak and strong terms without their antonyms). The goal of the current work is to examine how participants interpret different non-negated adjectives and negated expressions relative to each other. We focus on the interpretation of different scale-mates and their antonyms within the class of relative gradable adjectives. More specifically, we address the following two research questions: (i) Do negated terms (e.g., not intelligent ) overlap in meaning with their corresponding simple alternative expressions (e.g., stupid )? and (ii) What interpretation does negated stronger terms (e.g., not brilliant ) receive? To investigate these research questions, we compare the interpretation of 1

She is also the best Jedi Master on Earth, if that is not a contradiction.

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non-negated weak and strong antonymic pairs with their corresponding negated forms. We show that participants in our experiment use distinct portions of a scale when interpreting statements involving non-negated adjectives. When the terms appear under negation, participants distinguish positive and negative weak terms (e.g., not large and not small ) as well as positive weak and strong terms (e.g., not large and not gigantic ). Negative weak and strong terms (e.g., not small and not tiny), in turn, both receive a middling interpretation. We discuss different accounts of this polarity asymmetry and conclude that multiple factors relating to the complexity of alternative expressions as well as the social context need to be integrated when modeling the interpretation of negated adjectives. This chapter is organized as follows. We will first introduce basic assumptions about measurement scales and Horn scales and we describe how the two are related in the case of relative gradable adjectives. We then outline two accounts of negative strengthening and we briefly review the experimental literature on the interpretation of negated adjectives. Section 2 presents the results of our experiment and Sect. 3 discusses the implications of our findings.

1.2

Measurement Scales and Horn Scales

Most semantic analyses model the meaning of gradable predicates via so-called measurement scales, which are sets of totally ordered degrees (Bartsch and Venneman 1973; Cresswell 1977; Bierwisch 1989; Kennedy and McNally 2005; Kennedy 2007; see Solt 2015 for an overview). For example, a gradable adjective like large is assumed to express that the degree of size is higher than a standard degree. In its unmodified (positive) form, the standard degree is taken to be a contextdependent threshold based on a relevant comparison class (see especially Kennedy 2007). The standard for relative adjectives is fixed contextually while other adjectives like full have absolute context-invariant standards (Rotstein and Winter 2004; Kennedy and McNally 2005). The type of standard involved in gradable adjectives crucially affects entailment relations between antonymic pairs. The truth condition of the minimum standard adjective open corresponds to a “non-zero degree

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of closedness” while the maximum standard adjective describes the “maximal degree of closedness.” As shown in (1), the negation of an absolute adjective entails the assertion of its antonym (Cruse 1986; Rotstein and Winter 2004). Since a minimal positive degree corresponds to a negative maximal degree on the same measurement scale, the entailments in (1) follow from the truth conditions of minimum and maximum standard adjectives (Kennedy 2001, 2007). Relative adjectives, on the other hand, are not subject to a similar entailment relation, as displayed in (2). Both large and small make use of the same dimension and degrees but they impose reverse orderings. Yet the antonyms allow for a middle ground since the associated standards may be different (Kennedy 2007). (1) The door is not open ⇒ The door is closed (2) The door is not large  The door is small The literature on the semantics of gradable adjectives models their meaning in terms of measurement scales. In pragmatics, a different notion of scale is employed, that is, Horn scales (see Solt 2015 for an overview of the different notions of a scale). Horn scales are postulated to model scalar inferences. To take an example, the relative adjective large is associated with a stronger term gigantic on a Horn scale. Semantically, large is consistent with a stronger term gigantic and small is consistent with tiny, respectively since the semantics only introduces a lower bound (the threshold for what counts as large or small ). The standard assumption in pragmatics is that the Horn scales for positive and negative terms are split due to the entailment condition for alternatives (Horn 1989 but see Hirschberg 1985). This is because the stronger scale-mate asymmetrically entails the weaker one (but the weak term does not entail the strong one). Therefore, when a speaker chooses the weaker term, the hearer may conclude that the stronger one does not apply; thus deriving a scalar implicature. In Fig. 1, we show how different scalar expressions relate to an underlying measurement scale of size. As shown in the second row of Fig. 1, the effect of scalar implicature is that large gets upper-bounded so that degrees denoted by gigantic are not within the range of large. While the two literatures on scales are usually separated, it has recently been shown that the properties of the measurement scale underlying the

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Fig. 1 Relation of Horn scale mates to underlying measurement scale and their semantic meaning. Ds represents the standard degere and the dashed line indicates that the Horn scales for positive and negative terms are split. The effect of scalar implicature is presented in the second line

semantics of adjectives crucially affects the derivation of pragmatic inferences (Gotzner et al. 2018a, b; Leffel et al. 2019). For example, relative adjectives do not serve as good candidates for scalar implicature computation but instead for negative strengthening. Due to vague semantics, scalar alternatives associated with relative adjectives may not be clearly distinguishable and thus fail to trigger scalar implicatures (Leffel et al. 2019). In turn, the opposite pattern is observed for absolute adjectives, which tend to trigger scalar implicatures but not negative strengthening.2 In the following, we define the phenomenon of negative strengthening and discuss why relative adjectives are well suited to trigger this kind of inference.

1.3

Negative Strengthening and its Polarity Asymmetry

Traditional Neo-Gricean accounts model implicature as an interplay between two conversational principles that constrain each other during conversation (Horn 1989; Levinson 2000). The Q-principle (“Say as much as you can, given R”) derives scalar implicatures and the Rprinciple (“Say no more than you must, given Q”) derives other kinds of implicature like negative strengthening, which occurs with negated 2

The patterns for minimum and maximum standard adjectives are complex, see especially Gotzner et al. (2018a, b) and Leffel et al. (2019) for a discussion.

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antonyms. According to Horn (1989), negative strengthening arises when “under the right conditions, a formally contradictory negation notF will convey a contrary assertion G” (Horn 1989, p. 273). For example, the utterance The house is not large (not-F) may be used to implicate that “the house is rather small” (G). The presence of a middle ground between antonymic expressions is a prerequisite for the availability of negative strengthening (Horn 1989), hence this inference occurs with contrary (e.g., large and small ) but not contradictory antonyms (e.g., dead and alive) (see Paradis and Willners 2006 for an experimental investigation). As described in the last section, relative but not absolute adjectives allow for a middle ground between the extension of the positive and negative term, which explains why relative adjectives are more likely to trigger negative strengthening (see Leffel et al. 2019; Gotzner et al. 2018a, b).3 Another well-known observation is that negative strengthening is more likely to occur with negated positive terms (not large) compared to negative ones (not small ) (Bolinger 1972; Ducrot 1973; Horn 1989; Brown and Levinson 1987; Levinson 2000; Blutner and Solstad 2000). Thus, “not small” is interpreted as a middling term (neither large nor small ) rather than licensing the inference to large. The semantic meaning of negated positive and the effect of negative strengthening for negated positive and negative terms are displayed in Fig. 2. The most prominent explanation of the polarity asymmetry relates to the notion of face management. Brown and Levinson (1987) suggested that when one performs a face-threatening act such as criticizing, “there is a good social motive for saying much less than you mean” (Brown and Levinson 1987, p. 264). Therefore, a statement like Your office is not large will often convey that the room is “rather small” (which is less socially desirable). In turn, because there is typically no good social motive that prevents speakers from saying that Your room is large directly, an utterance like Your room is not small will be less likely to convey the affirmation of the antonym. Horn’s “Division of Pragmatic Labor” states that “The use of a marked (relatively complex and/or prolix) 3 In principle, negative strengthening is available for modified absolute adjectives and when contradictory antonyms are coerced into having a middle ground (see especially Horn 1989 for examples).

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Fig. 2 Semantic meaning of negated expressions. Ds represents the standard degere and the dashed line indicates that the Horn scales for positive and negative terms are split. The different strengthenede meanings of negated positive and negative terms are presented in the second line

expression when a corresponding, unmarked (simpler, less ‘effortful’) alternative expression is available tends to be interpreted as conveying a marked message (one which the unmarked alternative would not, or could not have conveyed)” (Horn 1984, p. 22). That is, using the more complex expression instead of an available shorter and/or easier expression tends to signal that the speaker was not in a position to employ the simpler version felicitously. This principle predicts that the more complex expressions should also differ in interpretation from simpler lexical antonyms. Experimental studies by Giora et al. (2005) and Paradis and Willners (2006) provided evidence for a difference between negated terms (e.g., not happy) and their lexical antonyms (e.g., sad ) but in part did not reveal a polarity asymmetry between positive and negative terms (e.g., not happy and not sad were strengthened to a similar degree). Such an asymmetry, in turn, was found in studies by Colston (1999), Fraenkel and Schul (2008), Ruytenbeek et al. (2017), Tessler and Franke (2018) and Gotzner and Mazzarella (2021). Specifically, negated positive terms like not happy were judged to convey a greater degree of negative strengthening than their negated positive terms like not sad . The study by Ruytenbeek et al. (2017) also showed that polarity interacts with morphological markedness, that is, the polarity asymmetry was greater for morphological pairs (containing negative morphemes such as happy and unhappy) than for non-morphological pairs (involving lexical antonyms like happy and sad ). Furthermore, a previous study by Gotzner and Mazzarella (2021) tested the face-management explanation of the

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polarity of negative strengthening, by manipulating the power relation of dialog partners and their social distance. Gotzner and Mazzarella (2021) show that negative strengthening is affected by the interaction of the sociological variables such as gender, power and social distance. More specifically, there was a stronger polarity asymmetry for female than male participants, and different weightings of power and social distance across gender. While these results were not predicted by the traditional account of the polarity asymmetry of negative strengthening (based on Horn 1989; Brown and Levinson 1987), they do suggest that negative strengthening, including the strength of its polarity asymmetry, is modulated by features of the social context and they are thus indicative of the potential relevance of face management in the interpretation of negated antonyms. In a further set of experiments, though, Mazzarella and Gotzner (accepted) showed that the polarity asymmetry of negative strengthening is not reducible to face-management considerations. They manipulated the face-threatening potential of positive and negative adjectives in context and replicated the standard polarity asymmetry even when negative adjectives were not face-threatening. Thus, there seems to be a role for adjectival polarity that is independent of the face-threatening potential of the formulation in context. The differences in experimental findings concerning the polarity asymmetry between earlier and more recent experimental studies may result from methodological differences. Further, apart from the studies by Gotzner and Mazzarella (2021) and Mazzarella and Gotzner (accepted), most previous studies presented test sentences in isolation rather than embedding them within a conversational context. Hence, it is important to verify the findings in an experimental paradigm that makes distinctions between all different readings relevant. Previous experiments have either investigated the interpretation of weak antonymic pairs (e.g., Ruytenbeek et al. 2017; Tessler and Franke 2018; Gotzner and Mazzarella, 2021) or the relation between weak and strong terms (e.g., Doran et al. 2009; van Tiel et al. 2016; Gotzner et al. 2018a, b). However, no previous study has compared weak and strong scalars and their antonyms within the same context.

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Current Experiment

2.1

Goals and Predictions

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The goal of the current experiment is to investigate more specifically how participants interpret negated weak and strong terms and how their interpretation relates to the non-negated expressions. Here, we introduce a grading paradigm to compare all different terms in the same context. Based on Horn’s Division of Pragmatic Labour, we expect negated forms to be weaker than non-negated alternative expressions. Horn’s implicature account further predicts an asymmetry between negated positive and negative terms. Finally, we expect negated stronger terms to receive a middling interpretation (Horn 1989; Israel 2004). Horn (1989, p. 337) first observed that the negation of a strong positive scalar (e.g., brilliant ) is typically interpreted literally and is less likely to license negative strengthening. The explanation proposed by Horn (1989) relies on the observation that the negation of strong positive adjectives is more naturally used in linguistic contexts containing a previous mention of the adjective (which is then explicitly denied) than as an evaluation initiating an exchange, thus there is no functional motivation for negative strengthening for strong scalars (1989, p. 360).

2.2

Methods

2.2.1 Participants We recruited 60 participants with US IP addresses on Mechanical Turk (30 participants across two experimental lists). Participants were screened for native language and only included in the analysis if their self-reported native language was English. 34 men and 26 women participated in the study. Their mean age was 37.15, with a standard deviation of 10.9 (age range 24–71). The experiment lasted about 10–15 minutes and participants were paid $1 in compensation.

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Table 1 Overview of non-negated adjectives across conditions (the negated forms included the same adjectives preceded by not) Item/ Condition Good Happy Hot Warm Intelligent Large Pretty Nonnegated negative strong Nonnegated negative weak Nonnegated positive weak Nonnegated positive strong

Terrible Miserable Cold

Freezing Idiotic

Tiny

Hideous

Bad

Unhappy Cool

Cold

Silly

Small

Ugly

Good

Happy

Warm

Intelligent Large

Hot

Excellent Delighted Scalding Hot

Brilliant

Pretty

Gigantic Gorgeous

2.2.2 Materials and Procedure We created 7 adjective quadruplets involving a weak and a strong positive term4 ; their weak and strong negative antonyms. These adjectives were embedded in a statement, either in their non-negated or negated form. Hence, there were 8 expressions for a given scale/item and the experiment used a 2 Polarity (positive, negative) × 2 Scalar Strength (weak, strong) × 2 Negation (non-negated, negated) design. Table 1 displays all adjectives used in different conditions. We embedded these expressions in a context with a grading scenario (inspired by the best response paradigm of Gotzner and Benz 2018). Due to the reward-based grading task, distinctions between different readings should be more relevant. Table 2 presents an example context for the item pretty. Participants saw all non-negated and negated forms 4 In the literature of antonymy positive and negative terms have been defined based on three notions of polarity; markedness, dimensionality, and evaluativity (see Cruse 1986). The antonymic pairs of the current study were chosen to be consistently positive and negative across different notions of polarity (see Ruytenbeek et al. 2017 for possible mismatches and ways to operationalize the notions of polarity).

She is Brilliant! Distinguishing Different …

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Table 2 Context and task for the item pretty. Two example statements are given but, in the experiment, participants were presented with all 8 expressions Context: The art class is receiving grades for a picture they drew for a final exam Please decide which grade each picture receives based on the teacher’s statement 1 = hideous, 5 = gorgeous The teacher says: Mary’s picture is gorgeous 12345 Anne’s picture is not pretty 12345 ...

presented concurrently in one context involving different subject noun phrases (see Tessler and Franke 2018 for a discussion of the difference between concurrent and isolated presentation). In each context, there was one speaker with full knowledge (e.g., a teacher) making statements about a group of individuals (e.g., students). Participants’ task was to judge which grade a given person (or, in the example, picture) would receive based on the statement. Judgments were made on a 5-point Likert scale anchored at the negative strong adjective (1) and its positive strong antonym (5). Hence, we measure participants’ interpretation of different expressions on the same Likert scale. Our three factors, polarity, scalar strength, and negation, were all withinsubject and within-item. Hence, each participant saw 7 contexts with 8 statements. The resulting overall number of critical observations was 3360 (56 trials for 60 participants). The experiment was programmed in HTML and run via MTurk’s in-built environment. Participants read an instruction explaining the task with an example. They were told to give a rating based on a given statement with 1 representing the worst ratings and 5 the best one. The running example involved an antonym pair not used in the stimulus set (fantastic and terrible).

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Results

The data were analyzed using R (version 3.6). We excluded 7 participants based on their labeling in the strong conditions (if 2 or more strong adjectives were placed at the opposite end of the scale, a participant’s data were excluded from all further analyses). Figure 1 shows density curves for ratings across adjective conditions. The results were analyzed with cumulative link mixed models using the function clmm() in the ordinal package (Christensen 2015). We included the fixed factors Polarity, Negation, and Scalar Strength, their interactions as well as random slopes for participants and random intercepts for item (as the model with the rull random effects structure did not converge). The factors were sum-coded, accessing the role of given factor relative to the grand mean to compare different adjectives in their negated and non-negated variants. The results of the model are presented in Table 3. Critically, we found an interaction of Polarity and Scalar Strength (B=.96, SD=0.13, z=-7.5, p 3]

(8)

a. has to [ exh [ Mookie get three hits]] b. [∃x[hits(x) ∧ #(x) = 3] ∧ ¬∃x[hits(x) ∧ #(x) > 3]]]

A second version of the scopal analysis treats numerals as scope-taking expressions in their own right, specifically as generalized quantifiers over degrees, of which numbers are a special case (Kennedy 2013; 2015; Buccola and Spector 2016; cf. Frege 1980 [1884], Scharten 1997, von Stechow’s (1984, p. 56) treatment of measure phrases, and Solt’s (2015) treatment 3The assumption that exclusion of alternatives is based on asymmetric entailment is a gross but hopefully benign oversimplification, for the purposes of illustrating this type of analysis. There is a rich and nuanced literature on the question of what this relation actually is (see e.g., Gazdar 1977; Hirschberg 1985; Sauerland 2004; Fox 2007; Bar-Lev and Fox 2017), but this does not bear on the issues discussed in this paper.

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of the quantity terms much and many). The numeral three, for example, has the denotation in (9): it is true of a property of degrees if the maximal degree that satisfies it is the number three. (9)

[[three]] = λP.max{n | P(n)} = 3

As shown by (10), this denotation delivers two-sided truth conditions as the default interpretation of simple sentences like Mookie got three hits, but a lower bounded interpretation can be derived by lowering the quantificational meaning of the numeral to a singular term (number-denoting) meaning by the application of Partee’s (1987) be and iota type-shifts (Kennedy 2015). (10)

a. three [Mookie got t hits] b. max{n | ∃x[hits(x) ∧ #(x) = n]} = 3

Crucially, since numerals are quantificational, they may interact with other operators, and it is this interaction that accounts for the difference in meaning between (2) and (3). The lower bounded interpretation in (2) arises when the numeral takes scope over the universal modal have to, as in (11), and the upper bounded interpretation in (3) arises when the numeral takes scope below the modal, as in (12).4 (11)

a. three [has to [Mookie get t hits]] b. max{n | [∃x[hits(x) < ∧#(x) = n]]} = 3

(12)

a. has to [three [Mookie get t hits]] b. [max{n | ∃x[hits(x) ∧ #(x) = n]} = 3]

All other things being equal, the exhaustification and degree quantifier analyses derive identical truth conditions for structures like (13a-b), where Num denotes the number n and is the basis for calculation of the alternatives to S in the former, and where Num denotes the degree quantifier λP.max{m | P(m)} = n in the latter. 4 (11b)

says that three is the maximum n such that in every world in the modal domain (worlds in which Mookie wins the batting title), there’s a group of hits of size n that Mookie gets. There are groups of size three in worlds in which he gets more than three hits, but there are no groups of size three in worlds which he gets fewer than three hits. (11b) thus places a lower bound of three on the number of hits that we find in each world that satisfies the modal claim.

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(13)

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a. [ ... exh [ S ... Num ... ] ... ] b. [ ... Num [ S ... t ... ] ... ]

The two approaches are thus similar both in the general claim that bounding implications are a scopal phenomenon and in their empirical predictions. Where they differ theoretically is in how the meanings are built up. In the exhaustivity analysis, there are three moving parts, each of which has plausible independent justification: a semantics for numerals that delivers lower bounded content, exhaustification, and calculation of alternatives, with the latter two corresponding to the quantity implicature system. In the degree quantifier analysis, all of these pieces are effectively built into the denotation of the numeral. Our goal in this paper is to examine a set of data that allows us to draw a distinction between these three approaches—local analyses, an exhaustification-based scopal analysis, and the degree quantifier analysis—and, we claim, argues in favor of the latter. The crucial facts involve sentences in which a numeral is embedded in the complement of an existential modal, such as (14). (14)

Mookie can make three errors on the last day of the season and still have the best fielding percentage in the league.

The first conjunct in (14) is most naturally understood as imposing an upper bound on the number of errors that Mookie can make, but unlike (1) and (3), it does not impose a lower bound: it allows for the possibility of Mookie making two, one or zero errors. In the degree quantifier analysis, the upper bound reading of (14) is simply a consequence of the lexical and compositional semantics of the expressions in the sentence: it corresponds to a parse in which the numeral takes scope over the modal: (15)

a. three [can [Mookie make t errors]] b. max{n | ♦[∃x[errors(x) ∧ #(x) = n]]} = 3

(15b) says that three is the maximum n such that there is a world in the relevant modal domain in which Mookie makes at least n errors, which rules out the possibility that he makes more than three errors, and so semantically imposes an upper bound. The numeral may also take scope

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below the modal, deriving the truth conditions in (16b), which require merely that there is a world in the modal domain in which Mookie makes exactly three errors. (16)

a. can [three [Mookie make t errors]] b. ♦[max{n | ∃x[errors(x) ∧ #(x) = n]} = 3]

This meaning is quite weak, because it does not rule anything out, but it does seem to be available. (“Mookie can make three errors; in fact he can make as many as he wants!”) On the exhaustification analysis, the two readings likewise correspond to a scopal interaction, in this case between exh and the modal, with wide scope of exh in (17) delivering upper bounds: (17)

a. exh [can [Mookie make three errors]] b. ♦∃x[hits(x) ∧ #(x) = 3] ∧ ¬♦∃x[hits(x) ∧ #(x) > 3]

(18)

a. can [exh [Mookie make three errors]] b. ♦[max{n | ∃x[errors(x) ∧ #(x) = n]} = 3]

The difference between this analysis and the degree quantifier analysis is that it does not rely only on lexical and compositional semantics, but also on alternative calculation: if, for some reason, the alternatives to the prejacent were not computed, the upper bounded truth conditions in (17b) would not be derived. Finally, on the local analysis, there are two possible interpretations of (14), shown in (19a-b). (19)

a. ♦[∃x[errors(x) ∧ #(x) = 3]] b. ♦[∃!x[errors(x) ∧ #(x) = 3]]

Neither (19a) nor (19b) entails an upper bound. (19b) is logically equivalent to (16), and has the same weak truth conditions. (19a) also has weak truth conditions, and only entails that making fewer than three errors is allowed; it does not entail that making a greater number of errors is not allowed. But (19a) can be strengthened to an upper bounded interpretation via reasoning involving the Maxim of Quantity: (19a) (but not (19b)) is asymmetrically entailed by alternative propositions of the same form but with higher values for the numeral. Using the same Quantity reasoning

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that the classic neo-Gricean analysis appeals to in the case of simple sentences like (1), we can generate the implicature that the speaker believes that Mookie cannot make four, five, etc. errors, which derives the upper bound. The crucial difference between the three approaches to upper bounding inferences, then, is that only the degree quantifier analysis derives an upper bounded interpretation for (14) strictly in terms of the lexical and compositional semantics of the expressions involved: both the exhaustification and local analyses must also invoke whatever mechanisms are involved in the calculation of quantity implicatures, which are grammatical in the former, and (we assume) fully pragmatic in the latter.5 The analyses therefore make different predictions about how sentences like (14) will be evaluated if the quantity implicature mechanisms are suppressed or otherwise inactive: the degree quantifier analysis predicts that upper-bounding inferences will be retained, all other things being equal; the other two analyses predict that they will disappear. We can therefore distinguish between the two approaches by examining how sentences like (14) are understood by a population that has competence with quantification but has difficulty with quantity implicatures. In the next section, we describe an experiment involving one such population.

2

Experiment: Upper Bounds and Existential Modals in Child Language

A broad range of acquisition studies support the conclusion that young children systematically have difficulty computing upper bounding implicatures in contexts in which adults virtually automatically generate such meanings (Barner et al. 2010; Chierchia et al. 2001; Huang et al. 2013; Hurewitz et al. 2006; Gualmini et al. 2001; Guasti et al. 2005; Noveck 2001; Papafragou and Musolino 2003; Papafragou 2006; Smith 1980). Although children’s ability to compute scalar implicatures can be improved 5 We

take it that the semantic difference postulated by the local analysis presumes a fully pragmatic theory of implicature calculation, since there would be no reason for the language learner to posit an upper bounded semantics for numerals if upper bounds could be independently derived from exhaustification of the lower bounded semantics.

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when certain conditions are met, e.g., by asking them to assess conversational interactions rather than descriptions of events (Papafragou and Tantalou 2004), by using ad-hoc and non-lexical scales (Barner et al. 2010; Papafragou and Tantalou 2004; Stiller et al. 2015), by introducing relevant stronger alternatives (Skordos and Papafragou 2016), or by training them on the use of conventional terms (Papafragou and Musolino 2003; Guasti et al. 2005), the general conclusion that they differ from adults in their capacity to automatically calculate upper bounding implications for scalar terms is robust. One notable exception to this generalization is the case of numerals. For example, Papafragou and Musolino (2003) found that Greek-speaking five-year-olds who were assigned to a condition in which they were asked to evaluate sentences with dio “two” in contexts in which a lower bound reading is true but an upper bounded reading is false rejected the sentences on average 65% of the time. (Specifically six of the 10 children in the condition rejected the sentences on three or four of the four trials.) In contrast, children accepted sentences with arxizo “start” and meriki “some” over 80% of the time in contexts in which a sentence with a stronger scalar alternative (the Greek equivalents of finish and all) would have held true, while adults routinely rejected such sentences in these contexts. This pattern is reminiscent of the findings from a statement evaluation task conducted by Noveck (2001) in French, a pattern that was replicated by Guasti et al. (2005) in Italian. Further studies have replicated this difference between numerals and other scalar terms in child language—with the former having upper bounded interpretations and the latter lacking them—using different kinds of methodologies (see e.g., Huang et al. 2013; Hurewitz et al. 2006). The asymmetry between upper bounded interpretations of numerals vs. other scalar terms in child language is now viewed by many as a central argument for a fully semantic account of the former. If this is right, then we already have reason to reject an exhaustification-based account to upper bounding inferences, since it relies on the implicature system—though Barner and Bachrach (2010) propose a different interpretation of the asymmetry that is consistent with such an account. We return to Barner and Bachrach’s proposal in Sect. 3.1 below; in this section, we describe an experiment designed to test the predictions of the local and degree

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quantifier analyses of numerals which assumes that children’s difficulty with implicature extends to all scalar terms, including numerals. The inspiration for our experiment comes from a study by Musolino (2004). In Musolino’s experiment, four- and five-year-old children were exposed to scenarios in which a character had to perform an action with multiple objects, and were told that the character would win a prize under certain conditions. These conditions were described using sentences with the numeral two, including sentences in which two was embedded under an existential modal, as in (20). (20)

Goofy said that the Troll could miss two hoops and still win the coin.

Musolino was interested specifically in whether children correctly understand (20) as not imposing a lower bound, and he found that indeed, in scenarios in which the Troll missed one hoop, children who were told (20) said that it should get the prize more than 80% of the time. Musolino did not test for children’s judgments about sentences like (20) in scenarios in which the upper bound was exceeded, however (e.g., scenarios in which the Troll missed three or more hoops); our experiment was designed to introduce this condition, since it is in precisely such scenarios that the predictions of the three semantic analyses of numerals discussed in the previous section come apart. The local analysis (as well as the exhaustification analysis) can only derive an upper bounded interpretation of sentences like (20) values by implicature, and so predicts that children (of the relevant age) should fail to reject such sentences as descriptions of scenarios in which the upper bound is surpassed. In contrast, the degree quantifier analysis derives an upper bounded interpretation without invoking the implicature system, by scoping the numeral above the modal, and so predicts that children should reject such sentences as descriptions of scenarios in which the upper bound is surpassed.6 Participants 32 children (19 boys, 13 girls; range: 4;0–5;8; Mean 4;9, Median: 4;10) and 32 adults participated in Experiment 1, 16 participants per condition. Children were recruited at area preschools. Adults were undergraduates who earned course credit in a linguistics course in exchange 6 Our experimental stimuli, as well as the results of a separate experiment demonstrating that children

and adults assign the same range of interpretations to sentences in which numerals are embedded under universal root modals, are available at https://semanticsarchive.net/Archive/2E3Y2FjO.

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for their participation. All participants were native speakers of English. Data from two additional adults were excluded due to native speaker status. Materials and procedure The experimental task was a variant of the Truth Value Judgment Task (Crain and Thornton 1998). An experimenter told the participant a series of stories using animated images presented on a computer screen. Each story had the same structure. One character provided instructions to another character. The second character attempted to comply by performing an action. At the end of each story, a puppet, played by a second experimenter (or the experimenter, in the case of adult participants), briefly recalled the story plot and asked whether what the second character did was okay, reminding the participant of the first character’s instructions. The participant’s task was to respond “yes” or “no” (verbally in the case of children, or on a response sheet in the case of adults) and occasionally provide a justification. The experimental session began with two training items. The test session that followed consisted of six test items and four control items, all with the same basic structure. Each test item involved instructions from the second character that featured an existential modal and a numerical expression. Half of these involved numerals plus plural count nouns (e.g., two books/carrots/lemons), and half involved measure phrases plus mass nouns in a pseudopartitive constructions (e.g., two feet of water); there was no significant difference between the two types of nominals. Within each group of three test items, there was a scenario in which the second character performed an action that involved the exact amount required by the first character (=2), another in which the action involved an excess of the amount (always one more, in the case of the numeral/count noun examples) (>2), and another in which the action involved less than the amount (always one less, in the case of the measure phrase/mass noun examples) (2)

Results Recall that the dependent measure is the percentage of times the participants said that the second character’s actions were “okay” when the quantity was less than, equal to, or greater than two (i.e., the percentage of “yes” responses to the question “Is what so-and-so did okay?”). Since

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Mean percentage acceptance of second character’s actions

adults children

2 100.0% 3.1% 100.0% 21.9%

all participants, regardless of age group or condition, readily accepted the items in which the target number was met (the =2 items), we focus our attention on a comparison of the 2 items. The results are presented in Table 1. A McNemar’s test looking at overall results found a highly significant difference between the 2 items for adults ( p 2 items. The exceptional item involved a story about filling a bear with stuffing. In this story, one character is showing another character how to make a toy bear. The first says “you want your bear to be cuddly, but not too stiff, so you are allowed to use two inches of stuffing,” which was measured by putting it into a container marked by a red line. Of the seven acceptances from children in the upper bound condition, six came from this one item, and the lone acceptance from adults in this condition also came from this item. We therefore concluded that this item was producing deviant responses and should be removed from the evaluation of all conditions. In the upper bound condition, this left us with sixteen responses to count noun items and eight responses to the other mass noun item. The revised results are shown in Table 2. A Wilcoxon test comparing children’s responses to the adjusted 2 items revealed a highly significant difference in the same direction as the adults, with children more likely to say “yes” for

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Table 2 Mean percentage acceptance of second character’s actions, ‘‘Stuffing the Bear’’ items removed

adults children

QUANTITY 2 87.5% 100.0% 0.0% 41.7% 100.0% 4.2%

the 2 items (W = 55, z = 2.78, p = .005, two-tailed). In addition, children did not differ from adults in their rates of rejection of the >2 items (U A = 276, z = .24, p = .81, two-tailed). To provide additional confirmation that the initial numbers reflected the influence of a faulty item, we also ran a follow-up experiment on children that replicated all items in the original experiment except for the “stuffing the bear” item, replacing it with a new >2 mass noun item. The follow-up featured the exact same images, except that the bear scenario was changed so that one character was showing another how to fill a container with stuffing in order to ship building materials. The first says to the second, “You want the building materials to be really protected, but you still want enough room for the building materials to go in and not be cushioned too much, so you’re allowed to use 2 inches of stuffing.” In the >2 condition, the second character ends up filling the stuffing past the two-inch line on the container. 15 children (8 boys, 7 girls; range 3;11-6;0; Mean 4;11; Median 4;11) participated. This time, there was a clear and significant difference between the >2 and all D : (a) is entailed by all the other propositions. a. ¬ ∀x. ∈ { a, b, c } → x came. b. ¬ ∀x ∈ { a, b } → x came. c. ¬ ∀x ∈ { a } → x came. …etc. d. EXH O (¬ zen’in [came]) = ¬ all came ∧ ¬(¬ all came) = ¬ all came ∧ all came

contradiction

Contrastive topic wa reverses the situation: Contrastive-topic-marked zen’in is only allowed under the scope of negation. Hara (2006) formulates a pragmatic contribution of contrastive topic wa as follows:

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(22) Let F be the focus-marked element, B the background, R the restriction. The interpretation of CON(B)(F) a. Asserts: B(F) b. Presuppose: ∃F’[F’∈ R ∧ [B(F’) ⇒ B(F)]∧ [B(F)  B(F’)]] c. Implicates: ♦¬ B(F’) Under this formulation, the alternatives to zen’in are scalar ones, namely, some people, most people, all people, and the infelicity of (18a) is due to a presupposition failure: (23) (18a): ‘All-ct came.’ a. Assertion: All people came. b. Alternatives: {Some people came, Most people came, All people...} c. (a) entails all the alternatives in (b). d. There is no F’ such that [F’∈ R ∧ [come (F’) ⇒ ∀x. come (x)]∧ [∀x. come (x)  come (F’)]] e. No implicature produced.

(18b), in contrast, may produce implicature only when negation takes scope over the universal quantifier3 : (24) (18b): ‘All-ct didn’t come.’ a. Assertion: ¬ ∀x. come (x) b. Alternatives: {¬ ∃x. come (x), ¬ ∀x. come(x), …} c. ¬ ∀x. come (x) is entailed by ¬ ∃x. x came d. Implicates: ♦¬ (¬∃x. x came) = ♦ ∃x. x came

If we take (18b) as a narrow-scope-negation proposition, it fails to satisfy the presupposition, as (23) does. In sum, PPI-hood comes from (covert) exhaustification by EXH O by creating a contradictory implicature to an assertion in the case of wider scope negation. An overt CT wa lets negation take scope above a universal quantifier to successfully get an implicature.

3 When

negation takes a narrower scope than ∀, it results in no implicature, as in (23).

Amazing-Hodo

3.2

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Exhaustification by Even-EXH and the PPI-Hood of ‘Amazingly’-Hodo

Inspired by the PPI-hood derived from exhaustification, I propose that ‘amazingly’-hodo induces alternatives that have to be checked by an EXH operator, which leads to the PPI-hood of this expression. I also propose that the EXH-operator at work in ‘amazingly’-hodo is not an only-type but an even-type, as defined in (25). (25) EXH E = λp. p ∧ ∀q ∈ ALT(p) [p