Counting and Other Forms of Measurement

Home , Intersective modifier

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree

Doctor of Philosophy

in the Graduate School of The Ohio State University

Eric Sndyer

∼6 6

Graduate Program in Philosophy

The Ohio State University

2016

Dissertation Committee:

Professor Stewart Shapiro, Advisor

Professor Michael Glanzberg

Professor Craige Roberts

Professor Kevin Scharp

Professor Neil Tennant c Eric Sndyer, 2016 Abstract

This thesis is about number expressions, their meanings, and certain puzzles within the philosophy of mathematics arising in their connection. I develop novel solutions to those puzzles based on recent work within linguistic semantics concerned with the meanings of various measurement-related constructions. Along the way, I further develop those analyses in important ways. Some of the more philosophically signiﬁcant theses I defend are the following: though number expressions take on a greater variety of interrelated meanings than philosophers have traditionally recognized, all of these can be derived assuming that e.g.

‘four’ is basically a numeral, thus vindicating a certain form of the Substantival Strategy; there is a natural language distinction between numbers and cardinalities (a kind of degree), and the familiar Fregean cardinality-operator most charitably refers to the latter, not the former, thus casting considerable doubt on whether Hume’s Principle can generate numbers in the way assumed by Neo-Fregeans; counting involves measuring the cardinality of collec-

1 1 tions of indivisible objects, or “atoms”; fractions like ‘2 2 ’ in ‘2 2 oranges’ involve a form of 1 counting (at least on one interpretation), so that ‘Mary ate 2 2 oranges’ implies that Mary ate three things, namely two whole oranges and one half-orange; the distinction between sortal and non-sortal concepts, at least when construed linguistically, is not exhausted by the familiar count/mass distinction, but is plausibly exempliﬁed by various other semantic phenomenon previously unrecognized by philosophers; and not all concepts answering ‘how many’-questions are sortal. Some of the more linguistically signiﬁcant theses I defend are as follows: though previous accounts have claimed that various uses of ‘four’ can be derived via type-shifting, none of those accounts actually succeeds in deriving meanings appropriate for all of those uses; there are correspondingly distinct referential uses of ‘four’, one referring

ii to a number, the other referring to a degree of cardinality; so-called individuation/measure- ambiguities witnessed in container phrases like ‘four glasses of water’ are a special case of the same ambiguity arising in connection with atomic predicates more generally; despite the traditional conceptions of cardinality, all scales (ordered sets of degrees) are dense, including cardinality scales. My hope is that this thesis not only contributes novel and plausible solutions to important puzzles within the philosophy of mathematics, it also add to our present understanding of the meanings of number expressions and measurement-related phenomena.

iii Acknowledgements

I would like to sincerely thank the members of this committee for their time, patience, insight, and feedback regarding the material in this thesis. Extra special thanks are due to my advisors, Stewart Shapiro and Craige Roberts, for guiding through this entire process.

It is largely due to their help and encouragement that I am in a position to present this material in the ﬁrst place. And it is probably no accident that this material represents a mixture of their particular specialties and interests – the philosophy of mathematics and linguistic semantics. It has truly been an honor to work with them so closely over the past several years, and any future success I might enjoy as an academic should be attributed chieﬂy to their excellent guidance.

I also owe a large debt of gratitude to the other members of this committee, Michael

Glanzberg, Kevin Scharp, and Neil Tennant. I was fortunate to meet Michael at a conference held at Ohio State some years back, and I remember being amazed that someone of his incredible ability and expertise would not only be willing to listen to my ideas, but would also encourage developing them further and stay in touch regarding their development, especially since Michael was not a member of the OSU faculty. That interaction eventually led to my ﬁrst publication, and I have been very fortunate to maintain a dialog (though from a distance) with him ever since. His seemingly endless encouragement and enthusiasm has been invaluable. My interest in measurement-related phenomena was sparked largely by a seminar I took with Kevin on measurement theory. To his credit, he allowed me to develop a semantics for measurement-related expressions which in all honestly probably failed to engage with the central themes of the seminar, but which ultimately grew into much of the work presented in this thesis. His input on much of my published work is evident, and his amazing ability to engage with just about any form of philosophy and to present it to a

iv general philosophical audience has proved critical to my success in grad school. Finally, the primary focus of this dissertation grew directly out of a seminar I took with Neil on Neo-

Logicism. It was through Neil’s own work that I was introduced to the debates mentioned in Chapters 2 and 3, and his excellent work on Neo-Logicism clearly permeates throughout this thesis.

There are too many other people to thank, and there is no way I could do justice to everyone who has helped inﬂuence the ideas found here. Nevertheless, I want to express my sincere gratitude to the following individuals for helpful discussions, exchanges of ideas, or general enthusiasm concerning the project: Maria Aloni, Chris Barker, Thomas Hofweber,

Chris Kennedy, Dan Lassiter, Øystein Linnebo, Carl Pollard, Susan Rothstein, Richard

Samuels, Greg Scontras, Robert Stalnaker, William Taschek, and Joost Zwarts.

Finally, I would like to thank my good friend and collaborator, Jeﬀerson Barlew. Thanks largely to his friendship and the support of the aforementioned individuals, I can sincerely say that the past several years spent at Ohio State have been the best of life. Thanks again to all of you.

v Vita

2004 ...... B.A. English, Piedmont College

2009 ...... M.A. Philosophy, University of Georgia

2009-present ...... Presidential Fellow, the Ohio State University

Publications

Snyder, Eric. 2013. Binding, genericity, and predicates of personal taste. Inquiry 56, pp.

278-306.

Shapiro, Stewart & Eric Snyder. 2015. Vagueness and context. Inquiry.

Snyder, Eric & Stewart Shapiro. Forthcoming. Frege on the real numbers. To appear in

P. Ebert and M. Rossberg (eds.), Essays on Frege’s Basic Laws of Arithmetic. Oxford

University Press.

Snyder, Eric & Jeﬀerson Barlew. Forthcoming. The universal measurer. To appear in

Proceedings of Sinn und Bedeutung 20.

Snyder, Eric. Forthcoming. Numbers and cardinalities: what’s really wrong with the easy

argument for numbers? Linguistics and Philosophy.

Fields of Study

Major Field: Philosophy

vi Table of Contents

Abstract ...... ii

Acknowledgements ...... iv

Vita ...... vi

1 Introduction ...... 1

2 Frege’s Other Puzzle ...... 16 2.1 Frege’s Other Puzzle...... 16 2.2 Two Strategies of Analysis...... 21 2.2.1 The Substantival Strategy...... 21 2.2.2 The Adjectival Strategy...... 27 2.3 Numerals, Determiners, or Adjectives?...... 33 2.4 The Adjectival Theory...... 39 2.4.1 Diﬀerent Uses of Number Expressions...... 44 2.4.2 The Extended Adjectival Theory...... 48 2.5 The Neo-Substantivalist Strategy...... 51

3 The Easy Argument for Numbers ...... 56 3.1 The Easy Argument for Numbers...... 56 3.1.1 A Parallel Ambiguity?...... 58 3.1.2 Various Uses of Number Expressions...... 59 3.1.3 Speciﬁcational and Quantiﬁcational Uses...... 62 3.1.4 Diagnosing the Easy Argument...... 65 3.2 A Brief Review of The Extended Adjectival Theory...... 68 3.3 Extending the Adjectival Theory...... 71 3.3.1 The Individual Concept Analysis...... 74 3.3.2 The Degree-as-Kind Analysis...... 76 3.3.3 Putting It All Together...... 80 3.3.4 Cardinality and Number...... 84 3.4 What’s Really Wrong with the Easy Argument?...... 85 3.5 Conclusion...... 90

vii 4 The Counting Oranges Puzzle ...... 92 4.1 The Counting Oranges Puzzle...... 92 4.2 Individuating and Measuring...... 99 4.3 The Universal Measurer...... 107 1 4.4 How to Count 2 2 Oranges...... 113 4.4.1 Generalized Cumulative Conjunction...... 115 4.4.2 Fractions...... 117 4.4.3 Putting it All Together...... 119 4.4.4 Two Further Objections...... 121 4.5 Conclusion: Don’t KISS the COP!...... 126

5 Concluding Remarks and Future Directions ...... 130 5.1 Concluding Remarks...... 130 5.2 Frege’s Ontological Thesis...... 134 5.2.1 Evidence for Frege’s Ontological Thesis...... 137 5.2.2 Problems for Frege’s Ontological Thesis...... 138 5.3 Sortal Concepts...... 140 5.3.1 Atomicity and Quantization...... 145 5.3.2 Varieties of the Sortal/Non-Sortal Distinction...... 148 5.3.3 Quantization and ‘How Many’-Questions...... 154 5.4 The Sortal Concept Puzzle...... 156 5.4.1 The Proposed Solution...... 158 5.4.2 Cardinalities are Ratios of Magnitudes?...... 160 References...... 168

viii Chapter 1

Introduction

Traditionally, philosophers of mathematics have asked questions like the following:1 What is mathematics about?; Does mathematics have a subject matter, and if so, what is it?;

What do mathematical statements mean?; What is the nature of mathematical truth?;

How can we know mathematics?; Is observation involved, or is it a purely mental exercise?;

What is a proof?; and What is the logic of mathematics?’. These questions are usually posed with branches of advanced mathematics in mind, e.g. number theory, real analysis, complex analysis, set theory, and the like. Less attention has traditionally been payed to the ordinary applications of mathematics like counting ﬁngers and toes, balancing checkbooks, or comparing heights.

There have been two exceptions to this trend in recent times, however. The ﬁrst comes from Neo-Fregeans like Hale and Wright(2001) and their concern with a philosophically satisfactory characterization of arithmetic, especially in connection with what has come to be known as “Frege’s Constraint”: that the applications of arithmetic ought to be built directly into the very fabric of the theory.2 Since counting is assumed to be the relevant application of simple arithmetic, e.g. equations like ‘4 + 3 = 7’, more attention has been paid to counting and its expression in natural language. The second exception is due largely to Hofweber(2005) in a paper titled ‘Number Determiners, Numbers, and Arithmetic’.

The novelty of Hofweber’s view was that it borrows some fairly sophisticated concepts from

1For an excellent introduction to the philosophy of mathematics, see Shapiro(2000). 2Though there is a growing literature on Frege’s Constraint (see e.g. Snyder and Shapiro(2014)), the idea that the characterization of the natural numbers ought to explain their applicability in counting can be found as early as Tennant(1987).

1 linguistics – generalized quantiﬁers, type-shifting, prosody and focus – to argue that natural

language best supports a controversial metaphysical position, namely nominalism, or the view that numbers do not exist. As a result, both philosophers of language and linguists

have since weighed in on the debate, and they have brought increasingly sophisticated

linguistic analyses to bear on some of the traditional questions mentioned above, e.g. ‘Do

numbers exist?’ and ‘How can we know arithmetic truths?’.

This thesis is located squarely within this latter debate. One of the traditional questions

mentioned above is ‘What do mathematical statements mean?’. Depending on how one

interprets “mathematical statement”, this might include ‘Jupiter has four moons’, ‘The

number of Jupiter’s moons is four’, ‘2 + 3 = 5’, or maybe even ‘Mary’s canoe is four feet

longer than John’s’. To adequately answer the question of what these statements mean, we

need to turn to the science of meaning, or linguistic semantics. Though a number of recent

contributions to the above-mentioned debate do that, the resulting analyses are empirically

inadequate in various ways. This dissertation attempts to ﬁll in some of the more signiﬁcant

gaps, while at the same time contributing to our linguistic knowledge of number expressions

and their meanings. My hope is that by getting the key linguistic facts straight, we will

be in a better position to draw signiﬁcant philosophical conclusions based on our ordinary

number talk.

The thesis is divided into three main chapters, each dealing with a diﬀerent puzzle. The

ﬁrst is known as “Frege’s Other Puzzle”, and is the subject of Chapter 2. Frege(1884)

originally pointed out that number expressions like ‘four’ are used in diﬀerent ways. For

example, they can be used referentially and non-referentially, as Frege’s famous pair of

examples Quantiﬁer and Identity illustrate.

(Quantiﬁer) Jupiter has four moons. (Identity) The number of Jupiter’s moons is four.

In particular, ‘four’ in Quantiﬁer appears to be counting how many moons Jupiter has, while

in Identity it seems to naming a certain number. At the same time, Quantiﬁer and Identity

2 appear to be semantically equivalent, and some have thought that diﬀerent occurrences of an expression in semantically equivalent statements must serve the same semantic function.

If so, then how can ‘four’ serve apparently diﬀerent semantic functions in Quantiﬁer and

Identity?

The solution, I submit, is to recognize that diﬀerent occurrences of the same expression needn’t perform the same semantic function, even in semantically equivalent statements.

After pointing out that number expressions have a wider range of uses than those witnessed in Quantiﬁer and Identity, I extend what Landman(2004) calls “the Adjectival Theory” so that meanings appropriate for all of the various uses noted are either witnesses to the basic meaning of ‘four’, namely that of a numeral, or else are derivable via independently motivated type-shifting principles. More speciﬁcally, I show how meanings appropriate for various occurrences of ‘four’ in (1) can be derived via independently needed type-shifting principles under the assumption that ‘four’ enters the lexicon as a numeral, but only if that numeral occurs within a measure phrase in (1a-f).

(1) a. Jupiter has four moons.

b. The number of Jupiter’s moons is four.

c. Jupiter’s moons are four (in number).

d. No four moons of Jupiter orbit Saturn.

e. Jupiter’s moons number four.

f. Mary drank four ounces of water.

g. Four is my favorite number.

h. The number four is my favorite.

I call the resulting semantics “the Extended Adjectival Theory”.

The philosophical signiﬁcance of the Extended Adjectival Theory is that it reveals the

ﬂaws in the two traditional positions regarding the linguistic relationship between Quantiﬁer and Identity. Substantivalists maintain that ‘four’ is a numeral in both examples, and so the

3 truth of either statement entails platonism, or the view that numbers exist. On the other hand, Adjectivalists argue that ‘four’ functions non-referentially in both examples, and so neither supports platonism. According to the Extended Adjectival Strategy, Substantivalists are right to think that ‘four’ functions as a numeral in both examples, while Adjectivalists are right to think that Identity is not a genuine identity statement equating the referents of two singular terms. At the same time, Adjectivalists are mistaken in maintaining that neither occurrence of ‘four’ functions referentially, and Substantivalists are mistaken to think that the post-copular material in Identity refers to a number. Rather, on the semantics I develop, ‘four’ in both examples occurs within a measure phrase, and that measure phrase takes on diﬀerent semantic functions in those diﬀerent examples, thanks to type-shifting.

More specifically, it functions quantificationally in Quantifier and referentially in Identity.

Crucially, however, it’s referent in Identity is not a number, but a “degree”, or an abstract representation of measurement. In particular, it refers to an abstract representation of how many singular individuals, or “atoms”, constitute a given plurality.

This implies a novel solution to Frege’s Other Puzzle, what I call “the Neo-Substantival

Strategy”. According to it, even though ‘four’ qua numeral serves the same semantic function in both of Frege’s examples, the measure phrases that this numeral occurs within take on different but related meanings in those examples, thanks to type-shifting. Consequently, it does not generally follow that occurrences of the same expression in semantically equivalent statements must serve the same semantic function. In fact, to maintain otherwise is to give up on a commonly accepted form of type-shifting. The original purpose of positing type-shifting principles was to explain how one and the same expression can take on different meanings in different syntactic environments, including e.g. coordinating conjunc- tions like ‘and’, the copula ‘be’, and various kinds of noun phrases like ‘cats’, ‘the cat’, and indeed ‘four cats’.3 What’s more, it should not be surprising if certain of those syntactic environments happen to lead to semantically equivalent statements. Consider the different

3See Partee and Rooth(1983), Partee(1986a), and Partee(1986b).

4 occurrences of ‘green’ in (2), for instance.

(2) a. That is a green car.

b. That is a car which is green.

Uncontroversially, ‘green’ serves diﬀerent semantic functions in (2a) and (2b), despite their apparent equivalence. Nevertheless, it is commonly thought that these diﬀerent functions are related via type-shifting.4 According to the Extended Adjectival Strategy, the various occurrences of ‘four’ in (1) are like the occurrences of ‘green’ in (2) in this respect.

The linguistic significance of the Extended Adjectival Strategy is that it is only extant semantics capable of deriving meanings for the various occurrences of ‘four’ in (1) without having to postulate unnecessary lexical ambiguities. The view that number expressions are polymorphic, i.e. they take on different but related meanings in different syntactic environments, is not new. In fact, many have suggested that at least some of the occurrences

witnessed in (1) are derivable via type-shifting, including e.g. Partee(1986b), Geurts(2006),

and Kennedy(2013). However, each of these theories succeeds only on the assumption that

degrees are identical with numbers. Yet the primary thesis of Chapter 3 is that – following

Scontras(2014) – English distinguishes between these two sorts of things. If so, then the

primary problem with the aforementioned analyses is that they wrongly identify the referent

a numeral like the one in (1g), i.e. a number, with the referent of a measure phrase like

the one in (1b), i.e. a degree. On the other hand, the Extended Adjectival Theory presents

a natural way of deriving meanings appropriate for all of (1a-h), while simultaneously

recognizing the sortal distinction between numbers and degrees.

Chapter 3 is concerned with an argument related to Frege’s Other Puzzle. It is commonly

known as “the Easy Argument for Numbers”, and runs as follows. Suppose Quantiﬁer is

true. Then since Identity is equivalent to Quantiﬁer, the former is also true. But Identity

is an identity statement equating the referents of two singular terms, including the numeral

4See e.g. Partee(2004) and Kennedy(2012).

5 ‘four’. Since numerals name numbers and existential generalization is generally valid for

names, Identity entails the existence of a number, namely four. Consequently, Identity

entails Realism,

(Realism) There is a number, namely four, which is the number of Jupiter’s moons.

and so we have a valid argument for a controversial metaphysical claim – that numbers

exist – which proceeds from a seemingly innocuous empirical observation about Jupiter’s

moons, namely that there are four of them.

Philosophers frequently deny the validity of the Easy Argument in one of two ways.

First, they might claim that our ordinary number talk often takes place within an elaborate

pretense that numbers exist, and Realism follows from Identity only within that pretense.

Alternatively, they might argue that ‘four’ functions non-referentially in Identity, and so

Realism simply does not follow from Identity. In my view, however, the problem with the

Easy Argument is that it equivocates. In particular, Realism trades on two importantly

diﬀerent senses of ‘number’ pointed out by Moltmann(2013a). There is a monadic, arith-

metic sense witnessed in examples like (1g,h), and a relational, cardinality sense witnessed

in examples like (1b,c). I present several semantic contrasts involving these diﬀerent senses

and argue that they can be explained if we recognize a sortal distinction between numbers

qua arithmetic objects and degrees qua representations of cardinality. Consequently, the

problem with the Easy Argument is that it attempts to equate what English distinguishes.

That is, the ﬁrst occurrence of ‘number’ in Realism is witness to the monadic, arithmetic

sense, while the second occurrence is witness to the relational, cardinality sense. Conse-

quently, Realism is true on the assumed Fregean semantic representation only if a certain

number is wrongly identiﬁed with a certain cardinality.

Following Moltmann(2013a) and Felka(2014), I assume that Identity is what is known

as a “speciﬁcational sentence”. It has been widely observed that speciﬁcational sentences

are not synonymous with equative sentences, i.e. ordinary identity statements like ‘Cicero

6 is Tully’. On its face, this appears to be problematic for Substantivalists since they assume that Identity is an ordinary identity statement. However, it seems that what’s really needed for the Easy Argument to go through is that the post-copular material in Identity functions as a singular term, at least given the plausible assumption existential generalization is generally valid for singular terms. And on at least one influential analysis of specificational sentences, one due to Heim(1979) and Romero(2005), post-copular terms of specificational clauses are indeed referential. Consequently, it does not necessarily follow that just because

Identity is a speciﬁcational sentence, Identity does not entail Realism.

Indeed, on the semantics I develop, the problem with the Easy Argument is not that

Identity fails to entail the existence of an abstract object. On the contrary, Identity straight- forwardly entails the existence of a cardinality given the Heim/Romero analysis of speciﬁca- tional sentences. What’s more, cardinalities are a kind of degree on the Extended Adjectival

Theory, and degrees – again following Scontras(2014) – are nominalized properties of measured substances. More speciﬁcally, cardinalities are nominalized properties of pluralities constituted by n-many atoms, where n is a number. Thus, degrees as conceived here presuppose a domain of numbers. As a consequence, it follows as a matter of logic that if Identity is true, then a certain number exists, just as the Easy Argument purports to establish.

However, this conclusion will come as little consolation to those philosophers making the

Argument, namely Neo-Fregeans. The primary epistemological thesis of the Neo-Fregean program, at least as outlined by Hale and Wright(2001), is that our knowledge of simple arithmetic statements like ‘3 + 2 = 5’ is ultimately grounded in an abstraction principle known nowadays as Hume’s Principle (HP).

(HP) ∀F, G. #x[F (x)] = #x[G(x)] ↔ F ≈ G

In English, HP “says” that the number F s is identical to the number of Gs just in case

there is a one-to-one mapping from the F s onto the Gs. Crucially, the Fregean cardinality-

operator ‘#x[φ(x)]’ is said to range over numbers, i.e. the very arithmetic objects Frege

7 (1884) attempted to provide an epistemically unassailable foundation for. Indeed, according to an independently held Neo-Fregean thesis, “Frege’s Constraint”, the cardinality-operator must range over the natural numbers. At the same time, the cardinality-operator is also said to capture the meaning of ‘the number of’ witnessed in Identity, i.e. ‘number’ in the relational, cardinality sense. Ultimately, it’s for this reason that counting Jupiter’s moons implies the existence of an abstract arithmetic object, according to Neo-Fregeans.

As a result, the fact that English sortally distinguishes degrees and numbers is philosophically significant for two reasons. First, it offers a novel diagnosis of what’s wrong with the Easy Argument: Realism attempts to equate what English sortally distinguishes. Sec- ondly, it suggests that Hume’s Principle can ground our basic arithmetic knowledge only by conflating degrees and numbers. Nevertheless, many have reasonably believed that our knowledge of basic arithmetic is ultimately grounded in our ability to count. For example, children often learn to do elementary arithmetic by counting fingers. How is this possible if cardinalities and numbers are not the same thing? I suggest that because cardinalities formed on the basis of counting are structurally isomorphic to the natural numbers (i.e. both form ω-sequences), it follows by a form of structuralism that any knowledge we could glean on the basis of counting would also constitute knowledge of basic arithmetic.

The linguistic signiﬁcance of the observations in Chapter 3 is that ‘number’ in the relational sense is what Scontras(2014) calls a “Degree Noun”. As such, it relates substances

(i.e. pluralities or mass quantities) to degrees. In fact, I argue that ‘number’ is a special case of ‘amount’; whereas the latter can relate either mass quantities or pluralities to various sorts of degrees (e.g. ‘that amount of apples’ can refer to the cardinality of the apples or their collective weight), ‘number’ speciﬁcally relates pluralities to degrees of cardinality.

Recognizing this leads to a fairly straightforward extension of Scontras’ semantics to spec- iﬁcational sentences like Identity, concealed questions, ‘how many’-questions, and related constructions, one based on Scontras’ independently motivated semantics for Degree Nouns.

Chapter 4 is concerned with a puzzle due to Nathan Salmon (1997), what I call “the

8 Counting Oranges Puzzle”. Suppose there are three whole oranges on the table. I take one,

cut it in half, eat one of the halves, and set the other half back on the table. Now consider

the Question:

(Question) How many oranges are on the table? 1 (Answer) There are 2 2 oranges on the table.

1 Intuitively, the correct answer to the Question in this situation is the Answer: 2 2 oranges. However, either a half-orange is an orange or it isn’t. If it is, then there are three oranges

on the table, and if it isn’t, then there are only two. In either case, we get an answer to the

Question which isn’t the intuitively correct Answer.

The Counting Oranges Puzzle is theoretically signiﬁcant because it purports to show

that traditional analyses of cardinality expressions are fundamentally mistaken, including

both the Fregean analysis and the Extended Adjectival Theory. That’s because fractional

cardinalities are incoherent on those analyses – the number of oranges on the table cannot

be more than two but less than three, for instance. Yet this is precisely what the Answer

seems to require. That is, we seem to be counting oranges fractionally.

However, I believe that appearances are deceptive in this case. More speciﬁcally, I

argue that the Answer is ambiguous in a way resembling so-called container phrases like

‘four glasses of water’ in (3):

(3) a. Mary put four glasses of water in soup.

b. Mary put four oranges in the punch.

(3a) is known to be ambiguous between an “individuating interpretation” (Mary set four

glasses ﬁlled with water in the soup) and a “measure interpretation” (Mary poured four

glasses’ worth of water into the soup). Based on a number of independent heuristics I develop

for these sorts of ambiguities, I show that examples like (3b) are similarly ambiguous, as is

the Answer.

9 The fact that individuating/measure-ambiguities (I/M-ambiguities) extend beyond numerically modiﬁed container phrases is of independent linguistic interest. That’s because all previous discussions of these ambiguities revolve around container phrases, thus potentially suggesting that I/M-ambiguities are due to idiosyncratic semantic features of those phrases. However, the fact that ‘four oranges’ gives rise to the same phenomenon shows that this cannot be right. Rather, following Snyder and Barlew(2016), I/M-ambiguities are a feature of atomic predicates more generally, i.e. predicates denoting sets of countable individuals. Consequently, I/M-ambiguities are interestingly similar to the grinding/packaging phenomena made famous by Pelletier(1975), illustrated by (4a,b) respectively.

(4) a. There’s a bunch of horse all over the road.

b. We ordered three waters.

Though ‘horse’ is a prototypical count noun, it functions as a mass noun in (4a), Conversely, though ‘water’ is a prototypical mass noun, it is functioning as a count noun in (4b). Both operations are said to be “universal” insofar as they (purportedly) apply to all nouns of the appropriate sort.5 According to the analysis of Snyder and Barlew, I/M-ambiguities are similarly “universal” in that that all atomic predicates are predicted to give rise to them.

On the individuating interpretation, (3b) is true only if there is a plurality of four individual oranges, each of which Mary put in the punch. On the measure interpretation,

(3b) is true if Mary put a quantity of orange in the punch measuring four oranges’ worth.

These interpretations are truth-conditionally distinct. For example, the latter but not the former would be true if Mary poured four oranges’ worth of prepackaged orange pulp into the punch. The primary thesis of Chapter 4 is that the Answer is similarly I/M-ambiguous, and that its intended interpretation – given Salmon’s context – is an individuating interpretation.

According to the latter, I argue, we are counting three things, namely two whole oranges and one half-orange. More generally, I argue that fractions involve a form of counting, at least

5Hence Pelletier’s labels “Universal Grinder” and “Universal Packager”. Note that the purported “uni- versality” of these operations has frequently been called into question, however.

10 on individuating interpretations. But since three is obviously a whole number, Salmon’s example does not require positing fractional cardinalities after all. In other words, the

Counting Oranges Puzzle presents no particular threat to traditional analyses of cardinality expressions.

Following Snyder and Barlew(2016), I assume that measure interpretations are generally due to a type-shifting principle called “the Universal Measurer” (UM). What this does, in eﬀect, is coerce an atomic predicate into a measure expression such as ‘ounce’. For example, UM takes a phrase like ‘glass of water’ or noun like ‘orange’ and coerces it into a measure phrase denoting quantities of water measuring a certain number of glasses’ worth or a measure noun denoting quantities of orange denoting a certain number of oranges’ worth. As a result, unlike individuating interpretations, measure interpretations do not generally involve counting how many atoms constitute a given plurality. Rather, they involve measuring a certain substance according to what Partee and Borschev(2012) call an “ad hoc” unit of measurement, e.g. how much stuﬀ would be contained in a given glass or (perhaps) how much orange mass is contained in a typical orange.

The theoretical upshot is that the traditional analyses of cardinality expressions are not entirely adequate. That’s because they wrongly predict that examples like (4b) should only have individuating interpretations. Conversely, the solution the Counting Oranges Puzzle proposed by Salmon(1997) and also Kennedy and Stanley(2009) wrongly predicts that

(4b) should only have measure interpretations. The fact is that (4b) can have both interpretations, and which interpretation it receives is a matter of context. Ideally, a semantic theory would explain how these diﬀerent interpretations are related, and this is precisely what UM purports to do.

Notice that this solution to the Counting Oranges Puzzle turns on the Answer receiving an individuating interpretation, as suggested by Salmon’s context when specifying the Puz-

1 zle. Thanks to this, ‘2 2 oranges’ in the Answer is interpreted as counting three things. But what if a measure interpretation were intended instead. For example, consider the following

11 variation:

Context: There are numerous orange slices on the table, each from diﬀerent oranges. After you

ask the Question, I give you the Answer.

(Question) How many oranges are on the table? 1 (Answer) There are (approximately) 2 2 oranges on the table.

1 In this case, ‘2 2 oranges’ in the Answer cannot be interpreted as ‘two whole oranges and a single half-orange’. Rather, it is only reasonable interpreted as ‘two and a half oranges’

worth of orange’. At the same time, the Answer is given in response to a ‘how many’-

question, and cardinalities are standardly assumed to be the sorts of things answering ‘how

many’-questions, just as lengths are standardly assumed to be the sorts of things answering

‘how long’-questions. In short, this variation on the Counting Oranges Puzzle appears to

undermine the traditional view that fractional cardinalities are incoherent.

In Chapter 5, after brieﬂy summarizing the major conclusions in previous chapters,

I sketch two further potential extensions of the semantics developed in previous chapters.

Both are related to Frege’s contention in the Grundgesetze that the cardinal numbers and the

real numbers are ontologically distinct, or what I call “Frege’s Ontological Thesis”. I sketch

various linguistic problems for that thesis, suggesting that they trace back to the problematic

presumption that concepts answering ‘how many’-questions are characteristically countable,

or “sortal”.

This leads to the ﬁrst extension of the present project: the so-called sortal/non-sortal

distinction. I suggest that when construed linguistically, at least, Frege’s distinction between

countable and non-countable concepts corresponds to the natural language distinction be-

tween atomic and non-atomic predicates. As such, it is not exhausted by the familiar

count/mass distinction in nouns, contrary to popular philosophical belief; it is also plausi-

bly reﬂected in the semantic functions of e.g. ‘grain of rice’ and ‘ounce of water’, diﬀerent

interpretations of ‘four glasses of water’, and even in diﬀerent kinds of verbs. One important

12 consequence of this new characterization is that not all predicates acceptably forming ‘how many’-questions are countable. For example, we can ask ‘How many ounces of water are in the bucket?’, yet ‘ounce of water’ is not countable. The linguistic analog is that ‘how many’- questions cannot uniformly ask about the number of atoms constituting a given plurality, contrary to inﬂuential analyses like Rullman(1993).

The second major extension involves a puzzle I call “the Sortal Concept Puzzle”. It concerns traditional theories of cardinality within linguistic semantics, according to which cardinalities uniformly represent the number of atoms making up a plurality. Here’s the puzzle. First, the adjective ‘many’ is standardly assumed to be associated with the dimension of cardinality,6 and so ‘that many F s’ or ‘that’s how many F s’ is standardly taken to refer to a cardinality. Now consider Mary’s utterance in (5b).

(5) a. John: ‘There are 3.86 ounces of water in this beaker’.

b. Mary: ‘That’s how many ounces of water are in this beaker too.’

By hypothesis, ‘that’s how many ounces of water’ refers to a cardinality. Moreover, that cardinality is presumably anaphoric on John’s previous utterance, more speciﬁcally on ‘3.86 ounces of water’. This is puzzling for a couple reasons. First, atoms are typically assumed to be the sorts of things we can individuate, and thus count.7 Yet there is plenty of semantic evidence strongly suggesting that we cannot semantically individuate ounces of water. Secondly, since the ratio of an atom to a plurality of atoms is always 1 to some other whole number n, it stands to reason that if cardinalities uniformly represent the number of atoms constituting a plurality, then cardinalities must also be whole. But then how can

‘3.86 ounces of water’ designate a cardinality?

The solution to the puzzle, I want to suggest, is to deny that cardinalities uniformly represent the number of atoms constituting a plurality. Counting involves numbering atoms, but not all cardinalities are formed on the basis of counting. In other words, cardinalities

6See e.g. Rett(2008) and Solt(2009). 7See e.g. Chierchia(1998) or Rothstein(2010).

13 formed on the basis of counting are a special case. Ultimately, what examples like (5) show is that cardinality scales, i.e. sets of (ordered) degrees of cardinality, need to have the same structure as other sorts of scales, e.g. scales of volume or scales of height. How so? I suggest that cardinality measures, like volume or height measures, are really ratios. In the case of atomic predicates like ‘orange’, for instance, the relevant ratio is that of an atomic orange to a plurality of oranges, at least on individuating interpretations. Not so with non-atomic predicates like ‘ounce of water’, however, thus explaining how the latter but not the former can give rise to fractional cardinalities.

The philosophical signiﬁcance of this suggestion is that, if accepted, cardinality would be another form of measurement, even from a Fregean perspective. According to Frege

(1903), the real numbers are ratios of magnitudes. These constitute the “measurement numbers” and answer questions like ‘How long is that canoe?’ or ‘How fast can John run?’.

If Frege’s Ontological Thesis were correct, these would be distinct from the the natural numbers, or the “counting numbers”, which instead answer ‘how many’-questions like ‘How many oranges are on the table?’. On the view suggested here, however, degrees of all kinds, including cardinalities, can be seen as determined by ratios of magnitudes. Consequently, not only is there no linguistic evidence supporting Frege’s Ontological Thesis, if the present suggestion is correct, then the natural language evidence appears to be squarely against it.

As for linguistics, the signiﬁcance of the present suggestion is that it vindicates a controversial thesis Fox and Hackl(2007) call “the Universal Density of Measurement” (UDM), or the claim that all natural language scales are dense. Linguists have traditionally assumed and largely continue to assume that scales come in two varieties: cardinality scales and everything else.8 The relevant diﬀerence is that whereas cardinality scales are assumed to be discrete, all other scales are assumed to be dense. This makes sense if cardinalities are assumed to represent the number of atoms constituting a plurality. After all, cardinalities would be necessarily whole in that case, and so would be isomorphic to the natural numbers.

8Cf. e.g. Rett(2008) and Solt(2009).

14 On the other hand, scales of length, speed, temperature, etc. are dense (and presumably continuous), and so are isomorphic to the real numbers. Consequently, the problem with this bifurcated conception is that it has no obvious means for explaining how ‘3.86 ounces of water’ in examples like (5b) can refer to a cardinality, or indeed how the Answer can refer to a cardinality on the measure interpretation. Put diﬀerently, these sorts of examples provide a fairly straightforward argument for UDM.

Each chapter of the thesis is intended to be more or less self-contained. Consequently, there is some overlap between chapters. I have tried to minimize this where possible, however. Also, I should mention that knowledge of basic formal semantics is presup- posed throughout, though I try to explain potentially unfamiliar concepts as they occur.

For excellent introductory textbooks on formal semantics done in the truth-conditional, model-theoretic tradition assumed here, I recommend Dowty et al.(1981) or Chierchia and

McConnell-Ginet(1990).

15 Chapter 2

Frege’s Other Puzzle

2.1 Frege’s Other Puzzle

In the Grundlagen, Frege (1884, §57) notes that number expressions such as ‘four’ are used

in two importantly diﬀerent ways.

Since what concerns us here is to deﬁne a concept of number that is useful

for science, we should not be put oﬀ by the attributive form in which number

also appears in our everyday use of language. This can always be avoided. For

example, the proposition ‘Jupiter has four moons’ can be converted into ‘The

number of Jupiter’s moons is four’. Here the ‘is’ should not be taken as a mere

copula ... Here ‘is’ has the sense of ‘is equal to’, ‘is the same as’ ... We thus have

an equation that asserts that the expression ‘the number of Jupiter’s moons’

designates the same object as the word ‘four’.

‘Four’ has an “attributive form”, thus suggesting that number expressions function non-

referentially. But it also appears to function as a singular term, thus suggesting that

number expressions function referentially. The former is witnessed in Quantiﬁer, the latter

in Identity.

(Quantiﬁer) Jupiter has four moons.

(Identity) The number of Jupiter’s moons is four.

Frege proposes “converting” the “attributive form” into a numeral by paraphrasing Quan-

tiﬁer as Identity, giving the latter the logical form in (6).

16 (6)# x[moon-of-Jupiter(x)] = 4

The cardinality-operator here – ‘#x[φ]’ – takes a concept φ and returns the number of objects falling under φ. For example, assuming Quantiﬁer is true, it takes the concept moon-of-Jupiter and returns the number 4. Thus, (6) is an identity statement; it is true if the number of objects falling under moon-of-Jupiter is identical to the number four. There are two key components to Frege’s proposed “conversion”. The ﬁrst is that

Identity and Quantiﬁer are semantically equivalent, or true in the exact same situations.

Secondly, Identity is an identity statement equating the referents of two singular terms, including the numeral ‘four’. Taken as a proposal about natural language, these claims jointly lead to a puzzle. According to Felka (2014), occurrences of the same expression in diﬀerent, semantically equivalent statements must serve the same semantic function.

Presumably, (7a,b) are equivalent precisely because the names involved serve the same semantic function in those examples, namely to refer.

(7) a. John saw Mary at the mall.

b. Mary was seen by John at the mall.

c. John saw a Mary at the mall.

However, these are not equivalent to (7c), where ‘Mary’ occurs as a predicate, even if all three examples happen to be talking about the same Mary.

Assuming Felka is right, the occurrences of ‘four’ in Quantiﬁer and Identity ought to serve the same semantic function given the latter are equivalent. At the same time, there are clear semantic diﬀerences between the “attributive form” and numerals. For example, numerals require singular morphology on the verb, while “attributive” ‘four’ requires the plural.

(8) a. Which one of these three numbers is even? [Let’s see. Three isn’t, and ﬁve isn’t

...] Four {is/??are}.

17 b. How many of these eight numbers are even? [Let’s see. Two is, and six is ...]

Four {??is/are}.

Also, while “attributive” ‘four’ is typically acceptable with modiﬁers like ‘exactly’ and

‘almost’, the numeral ‘four’ is not.

(9) a. {Four/??Almost four} is an even number.

b. {Four/Almost four} children clapped.

And numerals license entailments like (10a), but “attributive” ‘four’ does not.

(10) a. Mary divided four by two yesterday morning Mary divided four by two yesterday

b. Mary divided exactly four numbers by two yesterday morning 2 Mary divided exactly four numbers by two yesterday

These diﬀerences suggest that the numeral ‘four’ and “attributive” ‘four’ serve diﬀerent

semantic functions: while the function of the “attributive form” is to count, the function of

a numeral is to refer to a number.

Thus, we are led to what Hofweber (2005) calls Frege’s Other Puzzle. It relies on four seemingly plausible but jointly inconsistent theses.

Frege’s Other Puzzle

(FOP1) Quantiﬁer and Identity are semantically equivalent.

(FOP2) Diﬀerent occurrences of an expression occurring in semantically equivalent statements

serve the same semantic function.

(FOP3) The different occurrences of ‘four’ in Quantifier and Identity serve different semantic

functions.

(FOP4) The diﬀerent occurrences of ‘four’ in Quantiﬁer and Identity are witnesses to the same

expression, namely ‘four’.

18 FOP1 underlies the legitimacy of Frege’s proposed “conversion”. After all, Quantiﬁer and

Identity could not be paraphrases unless they were semantically equivalent. It is possible, though ultimately unsatisfactory, to deny this premise. One could point out that while the follow up to (11a) is perfectly consistent, the follow up to (11b) seems patently contradictory.

(11) a. Jupiter has four moons. In fact, Jupiter has sixty-two moons.

b. ?? The number of Jupiter’s moons is four. In fact, Jupiter has sixty-two moons.

Call this the Non-Equivalence Strategy. The obvious problem with this strategy is that we can formulate the same puzzle substituting ‘Jupiter has exactly four moons’ for

Quantiﬁer, thus bypassing observations like (11).

It is also possible, though ultimately untenable, to deny FOP4, thus leading to what we might call the Homonym Strategy. According to it, the different occurrences of ‘four’ in Quantifier and Identity are witnesses to altogether different expressions, ones which just happen to be spelled and pronounced alike. In general, we do not expect homonyms like the noun ‘fire’ and the verb ‘fire’ to be acceptably intersubstitutable.

(12) a. The rapid oxidation of combustible materials is ﬁre.

b. Let’s {ﬁre/??the rapid oxidation of combustible materials} John.

Similarly, Hofweber notes that substituting ‘the number of Jupiter’s moons’ for ‘four’ in

Quantiﬁer leads to unacceptability despite Identity appearing to establish their coreferen- tiality.

(13) Jupiter has {four/??the number of Jupiter’s moons} moons.

However, because homonyms are typically spelled and pronounced alike as a matter of historical accident, we do not expect their meanings to be related. Thus, the problem with the Homonym Strategy is that the occurrences of ‘four’ in Quantiﬁer and Identity are clearly semantically related; both tell us something about how many moons belong to Jupiter.

19 Traditionally, philosophers have denied FOP3, i.e. that the diﬀerent occurrences of

‘four’ in Quantifier and Identity serve different semantic functions. Two positions emerge as a result: either both occurrences of ‘four’ function referentially or both function non- referentially. Dummett(1991) calls the former view the Substantival Strategy, the latter the Adjectival Strategy. According to Substantivalists like Frege and more recently Hale and Wright(2001), ‘four’ in both Quantifier and Identity is best understood as a numeral referring to the number four. However, according to Adjectivalists like Hodes

(1984), Hofweber(2005, 2007), and Felka(2014), ‘four’ functions quantiﬁcationally, and thus non-referentially, in both of Frege’s examples.1

I’m going to argue here that both sorts of analyses are linguistically problematic. For example, Substantivalists mistakenly analyze Identity as a genuine identity statement equating the referents of two singular terms, including the numeral ‘four’. As a consequence, they cannot semantically explain diﬀerences like those in (14).

(14) a. The number of Jupiter’s moons is {four/??the number four}.

b. The number Mary likes most is {four/the number four}.

On the other hand, Adjectivalists wrongly analyze ‘four’ in Identity as a quantiﬁcational determiner, similar to ‘no’, ‘some’, ‘most’, or ‘all’. Consequently, they cannot semantically explain diﬀerences like those in (15).

(15) a. Jupiter has {four/no/some/most/all} moons.

b. The number of Jupiter’s moons is {four/??no/??some/??most /??all}.

These and several further problems for Substantivalists and Adjectivalists are the subject

of §2 and §3.

All of this suggests that a diﬀerent analysis of Frege’s examples is in order. One such

analysis is sketched in §4. It begins with what Landman(2004) calls “the Adjectival Theory”

1Another prominent defender of the Adjectival Strategy is Moltmann(2013a), who I will discuss in more detail in Chapter 2.

20 and extends that theory to further uses of number expressions, including Identity, thus resulting in what I call “the Extended Adjectival Theory”. The main attraction of the resulting semantics is that it avoids all of the problems noted for the Substantivalist and

Adjectivalist analyses, and it does so in an elegant manner using minimal semantic resources.

What’s more, it suggests a diﬀerent solution to Frege’s Other Puzzle, namely denying FOP2.

This new solution, which I call “the Neo-Substantivalist Strategy”, is sketched in §5.

Ultimately, the purpose of this chapter is two-fold. First, it is intended to solve an inﬂuential puzzle within the philosophy of mathematics, one which despite having received quite a lot of attention in recent times, has yet to receive a satisfactory solution. The second purpose is to introduce the core linguistic assumptions regarding number expressions adopted throughout the rest of this thesis.

2.2 Two Strategies of Analysis

This section introduces two popular responses to Frege’s original examples, the Substanti- val Strategy and the Adjectival Strategy. I will sketch diﬀerent versions of those Strategies available in the literature and point out some of their problems. I begin with the Substan- tival Strategy.

2.2.1 The Substantival Strategy

According to the Substantival Strategy, the “attributive form” of ‘four’ in Quantiﬁer is really best understood as functioning as a singular term. This underlies Frege’s proposed “conversion” of Quantiﬁer into Identity. According to Frege, the appropriate semantic function of number expressions within an ideal logical language is to refer to numbers. Consequently,

Frege proposes translating apparently non-referential uses like ‘four’ in Quantiﬁer into apparently referential uses like ‘four’ in Identity, which Frege analyzes as an identity statement.

As a result, ‘four’ is best understood as functioning as a numeral in both examples, at least

21 for the purposes of an ideal logical language. However, more recent defenders of the Sub- stantival Strategy like Hale and Wright(2001) argue that Frege’s analysis is basically correct for natural language as well.

One apparent advantage of the Substantival Strategy is that it purports to establish an immediate connection between counting and our knowledge of basic arithmetic. On the

Fregean analysis, repeated in (6), Identity identiﬁes the referents of two singular terms, namely ‘the number of Jupiter’s moons’ and the numeral ‘four’.

(6)# x[moon-of-Jupiter(x)] = 4

Because ‘4’ in (6) refers to the number four, the referent of ‘four’ in Identity is the very same object referenced by numerals in arithmetic statements such as ‘2 + 2 = 4’.

Now, the referents of Frege’s cardinality-operator are ﬁxed by an abstraction principle known nowadays as Hume’s Principle (HP).

(HP) ∀F, G. #x[F (x)] = #x[G(x)] ↔ F ≈ G

Here, “≈” is the relation of equinumerosity: two concepts F and G are equinumerous just in case the F s can be mapped one-to-one onto the Gs. Consequently, HP establishes that the number of two concepts is identical just in case they are equinumerous. But since the number of a concept is just that, a number, HP also ﬁxes the identity of certain arithmetic objects: two cardinal numbers n and n0 will be identical just in case they number

equinumerous concepts.

Where is the connection to counting? Hale(2016) explains:

[The cardinality-operator] makes also a direct connection with applications. A

cardinal number, Frege insists, is what answers a question of the form ‘How

many F s are there? Here F needs to be a concept of the right kind – a concept

which carries with it a principle for distinguishing the objects falling under it

... Frege’s canonical terms for cardinal numbers single them out as numbers

22 belonging to such concepts – concepts of kinds of things which can be counted

– and his deﬁnition embodies the basic principle of counting, telling us when

the answers to the questions ‘How many F s are there?’ [and] ‘How many Gs

are there?’ should be given the same cardinal number. Thus the core principle

– a plausible candidate to be the general principle governing the application of

cardinal numbers in counting – is what has since come to be known as Hume’s

Principle.2

Because Hume’s Principle answers the question ‘Are there as many F s as there are Gs?’, it can answer certain kinds of ‘how many’-questions like ‘How many F s are there?’, namely with answers like ‘Just as many as there are Gs’. As such, Hume’s Principle embodies “the basic principle of counting”, namely establishing one-to-one correspondences between diﬀer- ent collections of objects. And since this also ﬁxes the referents of the cardinality-operator, namely cardinal numbers, Hume’s Principle establishes a direct connection between counting and basic arithmetic.

Despite this apparent advantage, the Substantival Strategy faces two sorts of problems.

The ﬁrst involves the analysis of Quantiﬁer and Identity. According to Substantivalists,

Quantifier is paraphrased as Identity, and so ‘four’ in both statements is best understood as functioning as a singular term. However, there are two ways ‘four’ in Quantifier might be “best understood” as functioning as a numeral. The most obvious is that it just is a numeral. In that case, however, it is hard to see how Substantivalists can explain the numerous semantic differences between numerals and “attributive” uses noted in §1, e.g.

(9).

(9a) {Four/??Almost four} is an even number.

(9b) {Four/Almost four} children clapped.

2Hale (2016, p. 328-329).

23 Moreover, because Identity is an identity statement on the Substantivalist analysis, its truth establishes that ‘the number of Jupiter’s moons’ and the numeral ‘four’ are coreferential singular terms. But since coreferential singular terms are presumably intersubstitutable in non-opaque environments on that analysis, it’s also hard to see how Substantivalists can explain the unacceptability of Hofweber(2005)’s example (13).

(13) Jupiter has {four/??the number of Jupiter’s moons} moons.

In short, the view that ‘four’ in Quantiﬁer is best understood as functioning as a numeral because it is one appears to be subject to direct refutation.

On the other hand, the claim might be that ‘four’ in Quantiﬁer is best understood as a numeral because Quantiﬁer is paraphraseable as Identity, and the use of ‘four’ in Identity should be given some sort of priority in the ultimate analysis. Though this seems like an accurate statement of Frege’s intended strategy, it is not one which is obviously available to more recent Substantivalists. Frege was interested in developing an ideal logical language, not natural language as such. Consequently, he was able to dismiss those aspects of natural language which are apparently inconsistent with his ideal language, like e.g. the fact that

‘four’ can be used non-referentially. Hence his proposed “conversion” of Quantiﬁer into

Identity. However, more recent Substantivalists are concerned with natural language as such, and they claim that the natural language facts support Frege’s original analysis. The problem is that because there is no analogous motivation for prioritizing referential uses in the project of natural language semantics, there is no analogous sense in which non- referential uses are “best understood” as referential uses. Rather, as we will see in §4, there are various referential uses and non-referential uses, and it is an interesting empirical question how their meanings are related.

The second sort of problem for Substantivalists is that Identity is not really an identity statement, i.e. a statement equating the referents of two singular terms. Linguists standardly

24 distinguish between the three kinds of copular sentences listed in (16).3

(16) a. Cicero is Tully. (equative)

b. The most famous Roman orator is Tully. (specificational)

c. Cicero is bald. (predicational)

The labels ‘equative sentence’, ‘speciﬁcational sentence’, and ‘predicational sentence’ are

due to Higgins(1973). 4 Equative sentences are prototypical identity statements like (16a).

They equate the referents of two singular terms. Predicational sentences like (16c) predicate

a property like being bald of the subject. Finally, speciﬁcational sentences like (16b) purport

to “specify” an individual under a certain description.

There are good linguistic reasons for thinking that equative and speciﬁcational sentences

are semantically distinct. For example, Mikkelsen(2005) points out that the gender-neutral

pronoun ‘it’ is acceptable in tag-questions following speciﬁcational sentences but not pred-

icational sentences.

(17) a. Cicero is Tully, isn’t {he/??it}?

b. The most famous Roman orator is Tully, isn’t {he/it}?

c. Cicero is bald, isn’t {he/??it}?

(17c) reveals that speciﬁcational sentences diﬀer from equatives in this respect as well.

That’s plausibly because equative sentences are a special case of predicational sentences.

More speciﬁcally, equatives can be seen as predicating the property of being identical to the individual referred to in the predicate.5 For example, (16a) plausibly ascribes to Cicero the property of being identical to Tully.

Mikkelsen also notes a similar contrast in overt questions like (18a) and (18b).

3See Mikkelsen(2005, 2011) and references therein. 4Higgins originally distinguished four kinds of copular clauses. However, the three-way taxonomy assumed here follows Mikkelsen(2005). 5Cf. Partee(1986b). We will see why in §1.4.2.

25 (18) a. Q: Who is the most famous Roman orator?

A: {He/That/It}’s Tully.

b. Q: What is the most famous Roman orator?

A: {He/??That/??It}’s bald.

The question in (18a) has the form of a specificational sentence – it asks about the identity of a particular individual under a certain description – while the question in (18b) has the form of a predicational sentence – it asks about a general property of an individual under a certain description. As Mikkelsen explains, it is reasonable to expect anaphoric pronouns like ‘he’, ‘that’, and ‘it’ to track certain referential features of their antecedent subjects, e.g. their being a person. What (17) and (18) reveal is that predicational sentences do this but specificational sentences do not. Consequently, Mikkelsen reasonably concludes that predicational sentences are genuinely referential, i.e. their subjects are in fact singular terms, but specificational sentences are not. If so, then specificational sentences cannot be generally synonymous with equative sentences either. After all, the name ‘Cicero’ in (16a) is presumably a genuine singular term.6

The signiﬁcance of this taxonomy of copular sentences for Substantivalists is that Iden- tity is more plausibly a speciﬁcational sentence – prototypically a sentence of the form

Definite Description + be + Name – than an equative sentence – prototypically a sentence of the form Name + be + Name.7 This is problematic since the assumption that ‘four’ is a singular term, and so presumably a numeral, rests on Identity being a genuine identity statement. In other words, the fact that Identity is a specificational sentence appears to undermine the presumption that ‘four’ in Identity, let alone ‘four’ in Quantifier, refers to a number. 6For further arguments that specificational and equative sentences are semantically distinct, see Mikkelsen (2005). 7Cf. Moltmann(2013a) and Felka(2014)).

26 2.2.2 The Adjectival Strategy

According to the Adjectival Strategy, the apparent numeral ‘four’ in Identity is really best understood as functioning “attributively”. Diﬀerent versions of the Adjectival Strategy result depending on how one analyzes “attributive” uses. For example, Hodes(1984) analyzes

Quantiﬁer as (19), or the familiar treatment of cardinality expressions within quantiﬁca- tional logic, where ‘F ’ abbreviates moon-of-Jupiter.

(19) ∃x1, ..., x4. [F (x1)∧... F (x4)∧x1 6= x2 ∧... x3 6= x4]∧[∀y. F (y) → y = x1 ∨... y = x4]

According to the ﬁrst major conjunct, there are at least four moons of Jupiter. According to the second, those are the only such moons. The important point is that (19) could be true even if no numbers existed, i.e. even if the variables in (19) failed to range over numbers.

Hodes proposes that the apparent numeral ‘four’ in Identity is interpreted as having the same non-referential function as it does in Quantifier, thanks to a kind of “coding mechanism”. More generally, arithmetic statements like (20a) can be seen as “code” for purely quantificational formulae like (20b), where F and G are variables over first-order concepts, and “∃n” is what Hodes calls a “cardinality-quantifier”, a kind of shorthand for the equivalent of ‘there are n ...’ as usually rendered in first-order logic.

(20) a. 2+3=5

b. ∀F, G. [∃2x. F (x) ∧ ∃3x. G(x) ∧ ¬∃y. F (y) ∧ G(y)] → [∃5x. F (x) ∨ G(x)]

In English, what this “says” is that if there are at least two F s and at least three non- overlapping Gs, then there are at least ﬁve objects which are either F s or Gs. Again, the important point here is that (20b) could be true even if no numbers existed. Consequently, arithmetic truths like (20a) do not imply an ontology of numbers. Moreover, numerals of all sorts are really “code” for blocks of cardinality-quantiﬁers on Hodes’ view. Accordingly, he calls it Coding Fictionalism.

27 According to Hofweber(2005), the main problem with Coding Fictionalism is that it presupposes an overly naive analysis of the meanings of number expressions in the “attributive” form. That’s because ‘four’ in Quantiﬁer is really a quantiﬁcational determiner, at least according to Hofweber, similar to ‘no’, ‘some’, ‘most’, and ‘all’ in (15a),

(15a) Jupiter has {four/no/some/most/all} moons. and quantificational determiners like ‘most’ are not analyzable as blocks of first-order quan- tifiers like ‘∀’ and ‘∃’ followed by Boolean compounds.8 Hence part of the motivation for the standard analysis of natural language determiners, namely Barwise and Cooper(1981)’s

Generalized Quantifier Theory. According to it, quantiﬁcational determiners are relations between sets of individuals, or predicate extensions. For example, ‘some’ denotes pairs of sets having non-empty intersections, ‘all’ denotes the subset relation, etc.

0 0 0 (21) a. no = {hS, S i : S, S ⊆ U and S ∩ S = ∅} J K b. all = {hS, S0i : S, S0 ⊆ U and S ⊆ S0} J K c. four = {hS, S0i : S, S0 ⊆ U and |S ∩ S0| ≥ 4} J K Generally speaking, quantiﬁcational determiners are not plausibly referential expressions.

No serious semantic theory would suggest that ‘no’, for instance, refers to an individual.

Hofweber’s proposal, similar to Hodes’, is that the apparent numeral ‘four’ in Identity is really a quantificational determiner thanks to a syntactic operation he calls displacement. In essence, Hofweber’s view is that Identity results from syntactically “rearranging” the material in Quantifier. As a consequence, ‘four’ “moves” from its “canonical position” as a determiner in Quantifier to post-copular “singular term position” in Identity. Since the resulting sentence has a similar form to examples like (22), which Hofweber claims to be a genuine identity statement, perhaps this explains the mistaken intuition that Identity is a genuine identity statement.

8Cf. Altham and Tennant(1975) and Barwise and Cooper(1981).

28 (22) The composer of Tannh¨auser is Wagner.

Regardless, because post-copular ‘four’ in Identity results from displacement, it retains its

“canonical”, non-referential semantic status as a quantiﬁcational determiner. Call this the

Displacement Analysis. Regarding arithmetic statements like (20a), Hofweber suggests these are really state-

ments involving “bare determiners”, or determiners occurring without an overt noun com-

plement, similar to ‘four’ and ‘most’ in (23).

(23) A: Where did the children go?

B: Four went outside, but most stayed inside to play.

Hofweber notes that arithmetic formulae like ‘2 + 3 = 5’ are expressed in English in a

couple of ways, either in the singular as with (24a) or in the plural as with (24b).

(24) a. Two and three is ﬁve.

b. Two and three are ﬁve.

Hofweber then claims that the number expressions in (24b), like ‘four’ in (23), are bare

determiners, and so cannot be numerals referring to numbers. Finally, Hofweber suggests

that the singular form given in (24a) has the same semantic interpretation as the plural form

in (24a), despite the fact that (24a) has the surface form of a genuine identity statement.

That is, according to Hofweber, the surface syntax of (24a) contains three numerals, ‘and’,

and the ‘is’ of identity. But since those numerals are interpreted as bare determiners, just

like with Coding Fictionalism, arithmetic truths do not generally entail an ontology of

numbers on the Displacement Analysis.

Though the two proposals are similar in important ways, the primary advantage of the

Displacement Analysis over Coding Fictionalism is that the former is constructed around a

superior semantics for “attributive” uses, namely Generalized Quantiﬁer Theory. However,

29 the Displacement Analysis is subject to two major objections. First, it appears to require a non-compositional semantics, as Hofweber himself seems to recognize:

[The Displacement Analysis] ... does not see the occurrence of number words in

singular term position as arising from [a] semantically variable type, but rather

considers their occurring syntactically as singular terms to be contrary to their

semantic type. Believers in [the Displacement Analysis] will hold that not all

syntactic occurrences of a phrase are closely associated with a corresponding

semantic type. They will hold that the syntactic occurrence of a phrase in a

particular position does not necessarily reﬂect its semantic function.9

The suggestion seems to be that though ‘four’ in identity is a numeral, it nevertheless has the semantic function of a quantiﬁcational determiner. Consequently, Hofweber proposes that there is in general no mapping from syntactic categories to semantic types, something which is usually taken to be a minimal requirement on a compositional semantics.10

Secondly, the Displacement Analysis has been heavily criticized on syntactic grounds, and in particular on the implausibility of Hofweber’s displacement operation. For example, both Moltmann(2013a) and Balcerak-Jackson(2013) point out that the syntactic principles and operations actually needed to “transform” Quantiﬁer into Identity would not be recognized by contemporary syntactic theory, and their empirical legitimacy is highly dubious.11 Without those principles and operations available, however, the only motivation for thinking that ‘four’ functions as a quantiﬁcational determiner in both of Frege’s examples is lost.

On a final version of the Adjectival Strategy, one due to Felka(2014), ‘four’ functions as a quantificational determiner in both Quantifier and Identity thanks to a more familiar

9Hofweber (2005, p. 204). 10For example, Chierchia’s (1998, p. 339) ﬁrst guiding principle of natural language semantics is: “Syn- tactic categories at the relevant level of representation, say LF [i.e. Logical Form], are mapped onto corresponding semantic types (thereby determining for each expression what its denotation is going to be).” 11As Hofweber(2014) appears to concede.

30 syntactic operation, namely ellipsis. According to Felka, Identity has the syntactic structure in (25), where strikethrough represents ellipsis.

(25)[ What the number of Jupiter’s moons is] is [Jupiter has four moons].

Here’s the basic idea. Pre-copular ‘the number of Jupiter’s moons’ in Identity expresses an indirect version of the question ‘What is the number of Jupiter’s moons?’, while post- copular ‘four’ expresses an answer to this question, namely Quantifier, both thanks to ellipsis. Because everything but ‘four’ in the post-copular material is elided, ‘four’ appears to be functioning as a singular term. However, it actually has the same non-referential function it has in Quantifier, namely that of a quantificational determiner. Call this the

Elliptical Analysis. The main attraction of the Elliptical Analysis is that it purports to do what the Dis-

placement Analysis does, namely guarantee that post-copular ‘four’ in Identity is really

functioning “attributively”, but without relying on Hofweber’s dubious syntactic operation

of displacement. It is also subject to two objections, however. First, even if we grant that

post-copular ‘four’ in Identity is really a quantiﬁcational determiner thanks to ellipsis, this

will not guarantee that number expressions never refer to numbers. That’s because the

Elliptical Analysis is based on an independently motivated analysis of speciﬁcational sen-

tences, one which treats them uniformly as expressing question-answer pairs.12 However,

not all purportedly referential uses of number expressions come in the form of speciﬁcational

sentences. Consider again basic arithmetic statements like (24a), for instance.

(24a) Two and three is ﬁve.

Because (24a) not a speciﬁcational sentence, the Elliptical Analysis is simply inapplicable

here. But the philosophical attraction of previous versions of the Adjectival Strategy was

chieﬂy metaphysical: both Coding Fictionalism and the Displacement Analysis purport to

explain how securing appropriate truth-conditions for simple arithmetic statements does

12This is sometimes called “the Question in Disguise Theory”. See Schlenker(2003).

31 not require positing a realm of abstract arithmetic objects. There is no such guarantee with the Elliptical Strategy, however.

Secondly, the Elliptical Analysis makes certain false semantic predictions. Felka’s analysis requires that the pre-copular question expressed by Identity is ‘What is the number of

Jupiter’s moons?’. However, Moltmann(2013a), another prominent defender of the Adjec- tival Strategy, suggests instead that it expresses the question ‘How many moons belong to

Jupiter?’. The obvious advantage of Felka’s suggestion is that it more closely resembles the surface syntactic structure of Identity. After all, it’s easy to see how the ‘what’-question predicted by Felka’s analysis could result by eliding certain parts of the “deeper” syntactic structure of the sentence, but it’s hard to see how the ‘how many’-question Moltmann suggests could arise in a similar manner.

At the same time, the ‘what’-question predicted by Felka’s analysis cannot generally be synonymous with the pre-copular material actually contained in Identity. To see why, consider the contrasts in (26).

(26) a. John knows the number of Jupiter’s moons, namely {four/??even}.

b. John knows what the number of Jupiter’s moons is, namely {four/even}.

c. John knows how many moons Jupiter has, namely {four/??even}.

Here, ‘the number of Jupiter’s moons’ in (26a) patterns like ‘how many moons Jupiter has’ in (26c), or the indirect version of the question ‘How many moons does Jupiter have?’, not

‘what the number of Jupiter’s moons is’ in (26b), or the indirect version of the question

‘What is the number of Jupiter’s moons?’.

In this respect, ‘the number of Jupiter’s moons’ in (26a) resembles ‘the murderer of

Smith’ in (27a), an example due originally to Greenberg(1977).

(27) a. John found out the murderer of Smith.

b. John found out who the murderer of Smith was.

32 The pair here are not synonymous, as both Heim(1979) and Frana(2006) point out. Intu- itively, (27a) would be true only if John found out the exact identity of Smith’s murderer, while (27b) could be true even if John found out some less speciﬁc fact about Smith’s murderer, e.g. that he or she was a used car dealer from Idaho. Based on this observation,

Heim and Frana conclude that ‘the murderer of Smith’ cannot be synonymous with ‘who the murderer of Smith was’. Similarly, ‘the number of Jupiter’s moons’ in (26a) is intuitively true only if John knows the exact identity of the cardinality of Jupiter’s moons, unlike

(26b) which would be true even if John only knows some more general fact about their cardinality, e.g. that they are even or large. The fact that (26b) can be followed up with

‘...namely even’, unlike (26a) and (26c), conﬁrms this intuition. Consequently, ‘the number of Jupiter’s moons’ cannot be synonymous with ‘what the number of Jupiter’s moons is’, contrary to what is required on the Elliptical Analysis.

2.3 Numerals, Determiners, or Adjectives?

We have seen that on previous analyses, it is argued that ‘four’ is best understood as functioning either as a numeral or a quantiﬁcational determiner in both of Frege’s examples.

Others have proposed instead that number expressions are adjectives. Thus, the question

I want to ask here is whether number expressions are numerals, determiners, or adjectives.

Alternatively, do they function as numerals in some cases, determiners or adjectives in others?

I begin by noting that number expressions have a broader range of uses than those witnessed in Quantiﬁer and Identity, a fact noted by e.g. Partee(1986a) and Geurts(2006).

Consider the various occurrences of ‘four’ in (28), for instance.

(28) a. Jupiter has four moons. (Quantiﬁcational)

b. The number of Jupiter’s moons is four. (Speciﬁcational)

c. Jupiter’s moons are four (in number). (Predicative Adjective)

33 d. No four moons of Jupiter orbit Saturn. (Intersective Modiﬁer)

e. Jupiter’s moons number four. (Verbal Complement)

f. Mary drank four ounces of water. (Measurement)

g. Four is my favorite number. (Numeral)

h. The number four is my favorite. (Predicative Numeral)

Some of the labels here are due to Geurts. They are intended to indicate the purported semantic function of ‘four’ in the associated example. For clarity’s sake, I will brieﬂy indicate my reasons for each label here. A more complete explanation will emerge in §1.4.

The simplest case, perhaps, is (28g), where ‘four’ purports to function as a numeral, or a name of a number, similar to ‘John’ in ‘John is my favorite neighbor’. It also appears to be a numeral in (28h). However, whereas the name ‘four’ in (28g) purports to refer directly to a number, ‘four’ in (28h) appears to have the semantic type of a predicate, similar to predicative uses of names such as ‘Mary’ in ‘the Mary to my left’. Similarly, for reasons which will soon be apparent, ‘four’ in (28c) appears to be functioning as an adjective having predicative semantic type, similar to e.g. ‘large’ in ‘Jupiter’s moons are large’. Furthermore,

‘four’ appears to function as a modiﬁer in (28d), similar to ‘large’ in ‘No large moons of

Jupiter orbit Saturn’. Following Geurts, I use the label “Quantiﬁcational” (rather than e.g. “Attributive”) to describe uses of ‘four’ like that in Quantiﬁer, in attempt to remain momentarily neutral. And as previously discussed in §1.2.1, ‘four’ in (28b) occurs within

a speciﬁcational sentence. Hence the label “Speciﬁcational” use. Next, ‘four’ in (28e)

occurs as the complement of ‘number’, used as a verb, similar to ‘measure’ in ‘Jupiter’s

moons measure thousands of miles in diameter’. Finally, again following Geurts, (28f) is a

“Measurement” use of ‘four’ because ‘four’ occurs within a typical measure phrase.

The challenge for Substantivalists, and indeed for Adjectivalists, is to explain how the

various occurrences of ‘four’ in (28) can all be witnesses to one and the same expression.

For instance, we saw in §1 that Numeral uses diﬀer semantically from Quantiﬁcational

34 uses in various ways. Similar comments are in order for Measurement uses like (28f). For example, while names like ‘Mary’ cannot normally occur with modiﬁers like ‘exactly’, do not form comparatives, or answer ‘how’-questions, number expressions can, and so can measure phrases like ‘four ounces’.

(29) a. Jupiter has exactly four moons.

b. Jupiter has four more moons than Saturn.

c. How many moons does Jupiter have? Four (moons).

(30) a. Mary has exactly four ounces of water.

b. Mary has four more ounces of water than John.

c. How much water does Mary have? Four ounces (of water).

Importantly, however, not all occurrences of “bare” number expressions (i.e. number expressions occurring without modiﬁers like ‘exactly’ or determiners like ‘no’) pattern alike.

For example, Moltmann(2013a) points out that ‘four’ is acceptable in (14a), unlike ‘the number four’.

(14a) The number of Jupiter’s moons is {four/??the number four}.

(14b) The number Mary likes most is {four/the number four}.

Contrast this with (14b), where there is no such diﬀerence. Since the numeral ‘four’ and

‘the number four’ plausibly refer to the same number, it is hard to see how Substantivalists can explain this contrast assuming that both occurrences of ‘four’ are numerals.

It is at least as difficult to maintain that the various occurrences of ‘four’ in (28) are quantificational determiners, however. The fact is that number expressions and uncontroversial quantificational determiners pattern quite differently. For instance, Partee(1986a) points out that while ‘four’ can occur immediately after a determiner like ‘no’, prototypical determiners cannot.

35 (31) No {four/??all} moons of Jupiter revolve around Saturn.

Similarly, as Landman(2003) observes, number expressions can occur bare in predicate positions, unlike prototypical determiners.

(32) a. Jupiter’s moons are {four/??no} (in number).

b. The number of Jupiter’s moons is {four/??most}.

Also, number expressions can occur as the complement of verbs, unlike familiar determiners.

(33) Jupiter’s moons number {four/??some}.

Finally, and perhaps most tellingly, number expressions can occur as names, unlike any known determiner.

At this point, it might be thought that number expressions are a special kind of quan- tiﬁcational determiner, one which is diﬀerent from other determiners in the ways just mentioned. Against this, notice that color expressions such as ‘green’, which are standardly assumed to be adjectives, have the exact same range of relevant uses as those noted for

‘four’ above.

(34) a. Jupiter has green moons.

b. The color of Jupiter’s moons is green.

c. Jupiter’s moons are green (in hue).

d. No green moons of Jupiter orbit Saturn.

e. Jupiter’s moons are colored green.

f. Green is my favorite color.

g. The color green is my favorite.

It is hard to maintain that number expressions are a special sort of determiner, one which is unlike all other other uncontroversial determiners in many ways and happens to behave just like an adjective.

36 Another way in which number expressions and color expressions are alike is that both are plausibly intersective when functioning as modiﬁers, meaning that they give rise to entailments like those in (35), unlike non-intersective adjectives such as ‘former’.13

(35) a. Those are green moons Those are moons and they are green (in hue)

b. Those are former senators 2 Those are senators and they are former

c. Those are four moons Those are moons and they are four (in number)

As Partee(1986a) and Geurts(2006) point out, ‘four’ gives rise to the same entailment, cf.

(35c). Also, as Landman(2003, 2004) observes, number expressions can “comingle” with other adjectives. According to Kennedy(2012), the characteristic feature of an adjective is that it can combine with other modiﬁers to form larger phrases. For example, ‘green’ can combine with ‘large’ to form ‘a large green ball’, the latter can be combined with ‘round’ to form ‘a large green round ball’, etc. Now consider (36).

(36) a. We shipped the four ferocious lions to Boston.

b. We shipped the ferocious four lions to Boston.

These examples have different truth-conditions. Intuitively, (36a) would be true if we shipped four lions to Boston, and each of them is ferocious. On the other hand, (36b) would be true if there are two groups of four lions, the first is collectively more ferocious than the second, and we shipped the first group to Boston, even if some of those lions are not themselves ferocious. This suggests that ‘four’ can “co-mingle” with ‘ferocious’ in a way generally characteristic of adjectives.

For these sorts of reasons, Partee(1986a) reasonably concludes that number expressions just are adjectives.14 The trouble with this conclusion, however, is that ‘four’ and ‘green’

13Intersective adjectives are so-called because they denote the intersection of two sets in their attributive form. For example, ‘green’ in ‘green house’ denotes the intersection of the set of green things and the set of houses. For a taxonomy of various adjectives, see Partee(2004). 14Cf. also Schwarzchild(2002).

37 diﬀer from typical intersective adjectives like ‘happy’, ‘large’, ‘beautiful’, etc. in one crucial respect: while number and color expressions can function as names, ‘happy’, ‘large’,

‘beautiful’, etc. cannot.

(37) a. {Four/??Fourness} and ﬁve make nine.

b. {Red/??Redness} and blue make purple.

c. {??Red/Redness} and irritation are to be expected.

d. {??Happy/Happiness} and frustration don’t mix.

In order to use ‘happy’ in the same position as a name, we need to add the suﬃx ‘-ness’,

thus eﬀectively nominalizing the adjective, i.e. turning it into a kind of name. However,

we cannot generally do this with number expressions, and though we can do it with color

terms, the result is not synonymous with the color term itself.

To summarize, it appears as though number expressions cannot be analyzed uniformly

as numerals, determiners, or adjectives. Does it follow that ‘four’ is lexically ambiguous, i.e.

there are multiple expressions which happen to be spelled and pronounced alike, a name,

a determiner, and an adjective? No. One of the primary claims of this chapter is that the

full range of uses in (28) can be accounted for on the assumption that ‘four’ is a numeral in

all of those examples, but only if that numeral occurs within a measure phrase in (28a-f).

(28a) Jupiter has four moons. (Quantiﬁcational)

(28b) The number of Jupiter’s moons is four. (Speciﬁcational)

(28c) Jupiter’s moons are four (in number). (Predicative Adjective)

(28d) No four moons of Jupiter orbit Saturn. (Intersective Modiﬁer)

(28e) Jupiter’s moons number four. (Verbal Complement)

(28f) Mary drank four ounces of water. (Measurement)

38 This is why ‘four’ can occur with modiﬁers like ‘exactly’, answer ‘how’-questions, and form comparatives in these examples. In examples like (28g,h), on the other hand, the numeral

‘four’ does not occur within a measure phrase.

(28g) Four is my favorite number. (Numeral)

(28h) The number four is my favorite. (Predicative Numeral)

This is why (28g,h) instead answer ‘which’-questions like ‘Which number is your favorite?’, cannot occur with modiﬁers like ‘exactly’, etc. In the next section, I will show how on the basis of these assumptions, in conjunction with a limited stock of independently motivated type-shifting principles, it is possible to derive meanings appropriate for all uses of ‘four’ in

(28).

2.4 The Adjectival Theory

The core semantics for number expressions adopted here and throughout this thesis is a version of what Landman(2004) calls the Adjectival Theory. An essential feature of the Theory is that in several of the uses discussed above, ‘four’ is a numeral occurring within a measure phrase. Typical measure phrases include the underlined expressions in examples like (38a-c).15

(38) a. John drank exactly four ounces of water.

1 b. Mary is more than 5 2 feet tall. c. That is a 3.68-pound hammer.

Generally speaking, measure phrases are used to express measurement of various sorts.

One of the important features of Landman’s semantics, and indeed of this thesis, is that cardinality is one such form of measurement.

15See Schwarzchild(2002); Schwarzschild(2005).

39 Landman analyzes measure phrases syntactically as consisting of three components: a numerical relation, a numeral, and a measure term. Numerical relations are modiﬁers

such as ‘exactly’, ‘less than’, ‘almost’, etc. A numeral is a name of a number and includes

1 e.g. ‘four’, ‘5 2 ’, and ‘3.68’. A measure term is an expression referring to a unit of measurement, e.g. ‘ounce’, ‘foot’, ‘mile-per-hour’, and ‘degree Celsius’. Thus, a typical

measure phrase like ‘exactly four feet’ will have the following syntactic structure.

Measure Phrase

Numerical Phrase Measure Term

Numerical Relation Numeral feet

exactly four

This is only “typical” because measure phrases can grammatically occur without overtly speciﬁed numerical relations. Consider ‘Mary is four feet tall’, for instance. Landman assigns ‘four feet’ the following syntactic structure, where ‘∅’ represents an empty node.

Measure Phrase

Numerical Phrase Measure Term

Numerical Relation Numeral feet

∅ four

He assumes that a missing numerical relation is interpreted as ‘exactly’, and so ‘exactly four feet’ and ‘four feet’ are synonymous.

One of the central features of the Adjectival Theory is that like numerical relations, measure terms are not grammatically obligatory. In fact, the view is that this is precisely the case with most of the uses of ‘four’ discussed in §1.3. Most of these include measure

40 phrases missing both a numerical relation and a measure term, thus leading to the following syntactic structure:

Measure Phrase

Numerical Phrase Measure Term

Numerical Relation Numeral ∅

∅ four

In such a case, Landman assumes that the measure term defaults to cardinality, thus explaining why e.g. ‘Jupiter has four moons’ is naturally understood as telling us something about the cardinality of Jupiter’s moons.

On the semantic side, numerical relations are analyzed as relations between numbers.

For example, ‘exactly’ denotes pairs of equal numbers.

Numerical Relation denotation

‘exactly’ λn.λn0. n0 = n

‘at least’ λn.λn0. n0 ≥ n

‘less than’ λn.λn0. n0 < n

0 0 ∅ λn.λn . n = n

Again, numerals are treated as names of numbers, where the relevant domain is often assumed to be the reals.16

Numeral denotation

‘four’ 4

‘4’ 4

‘ﬁve and a half’ 5.5

‘4.38’ 4.38

16See e.g. Krifka(1989), Kennedy(1999), and Scontras(2014).

41 Finally, measure terms like ‘ounce’ denote measure functions, or functions from individuals to numbers. Following e.g. Scontras(2014), these are represented here as function symbols of the form ‘µ’.

Measure Term denotation

‘ounce’ µoz

‘foot’ µft

‘degree Celsius’ µdc

∅ µ# As mentioned, a missing measure term is interpreted by default as a cardinality measure, represented here as ‘µ#’. Semantic composition proceeds as follows. A numeral combines with a numerical relation, thus returning a function from numbers to truth-values. Composing the latter with a measure function named by the measure term then results in a predicate true of individuals having a certain measure. For example, combining the denotations for ‘exactly’ and ‘four’ from above lead to the denotation in (39d), and this composes with the measure function denoted by ‘foot’ to return the denotation for ‘exactly four feet’ in (39e).

(39) a. exactly = λn.λn0. n0 = n J K b. four = 4 J K c. foot = µft J K d. exactly four = λn0. n0 = 4 J K e. exactly four feet = exactly four ◦ foot J K J K J K = λx. µft(x) = 4

The result is a predicate true of individuals measuring exactly four feet (in length).

Similar comments hold for measure phrases missing overt numerical relations or measure

terms, as is the case with e.g. ‘four oranges’. In such a case, the denotations given above

ultimately return a predicate true of individuals having a cardinality of four.

42 0 0 (40) a. ∅ = λn.λn . n = n J K b. four = 4 J K c. ∅ = µ# J K 0 0 d. ∅ four = λn . n = 4 J K e. ∅ four ∅ = ∅ four ◦ ∅ J K J K J K = λx. µ#(x) = 4

More speciﬁcally, the predicate in (40e) is assumed to be true of “pluralities” in the sense of

Link(1983), or groups of singular individuals. These are modeled algebraically as sums in a

semilattice structure like the following, where ‘t’ represents the sum-operation and arrows

represent mereological relations:17

a t b t c

a t b a t c b t c

a b c

The “lowest” elements in the structure here – a, b, and c – are the atoms, and these form

the denotations of singular count nouns such as ‘orange’, ‘horse’, ‘ﬁnger’, etc. Pluralities are sums of atoms – a t b, a t c, b t c, and a t b t c – and are the denotations of plural nouns

such as ‘oranges’, ‘horses’, and ‘ﬁngers’. The cardinality predicate in (40e) is assumed to

be true of those pluralities consisting of exactly four atoms.

This constitutes the basic ingredients of the Adjectival Theory. Its main attraction,

Landman tells us, is its simplicity and elegance:

[T]he simplest assumptions give the correct results by the simplest means: the

measure phrase is built up by applying a numerical relation to a number, forming

a numerical predicate, and composing the numerical predicate with a measure

17For formal deﬁnitions of the relevant structures, see e.g. Link(1983), Krifka(1989), or Landman(2004).

43 ... The power of the adjectival theory is that it provides a simple and elegant

analysis of numerical phrases in the nominal domain.”18

Unlike other theories, the Adjectival Theory provides a uniﬁed account of diﬀerent sorts of numerical phrases using minimal semantic resources, namely a domain of numbers. The elegance of the theory comes from its ability to derive meanings appropriate for the various uses of ‘four’ noted in §1.3. The next two sections show how.

2.4.1 Diﬀerent Uses of Number Expressions

We saw earlier that number expressions have a multiplicity of uses, and that not all of these uses have the same meaning. The question here is how to derive those meanings within the Adjectival Theory. Landman only explicitly discusses four uses, namely Predicative

Adjective, Intersective Modiﬁer, Quantiﬁcational, and Measurement uses.

(28a) Jupiter has four moons. (Quantiﬁcational)

(28c) Jupiter’s moons are four (in number). (Predicative Adjective)

(28d) No four moons of Jupiter orbit Saturn. (Intersective Modiﬁer)

(28f) Mary drank four ounces of water. (Measurement)

I will present Landman’s derivations of meanings appropriate for these uses here and show how similar derivations can be given for the remaining uses in the next section.

According to the Adjectival Theory, the unifying factor in the various occurrences of

‘four’ in (28a,c,d,f) is that they all occur within a measure phrase, thus resulting in a predicate of some sort. The basic idea is that meanings appropriate for (28a,c,d,f) are either witnesses to this predicate, or else are derivable from them via independently motivated type-shifting principles. Let’s begin with Predicative Adjective uses like (28c). As mentioned, if ‘four’ occurs alone within a measure phrase, the result is a predicate true of four-membered pluralities, or (41).

18(Landman, 2004, p. 15).

44 (41) λx. µ#(x) = 4

This is a plausible meaning for the occurrence of ‘four’ in (28c).19 Thus, assuming ‘Jupiter’s moons’ refers to the plurality consisting of moons belonging to Jupiter, (28c) will be true just in case that plurality consists of exactly four atomic moons.

Next consider Intersective Adjective uses like (28d). Recall from above that according to Generalized Quantifier Theory, quantificational determiners like ‘no’ denote relations between sets, or predicate extensions. If so, then ‘four moons of Jupiter’ in (28d) needs to have predicative semantic type, and this could only happen if ‘four’ were functioning as a modifier, similar to ‘green’ in ‘No green moons of Jupiter ...’. Moreover, we have already seen that when functioning as a modifier, ‘four’ is intersective. Now, there is a well- known semantic correspondence holding between predicative and attributive occurrences of intersective adjectives.

(42) a. My car is green.

b. My green car is falling apart.

More speciﬁcally, the attributive form entails the predicative form, so that if (42b) is true, then so is (42a). This is usually explained via type-shifting,20 e.g. through Landman’s

“ADJUNCT”.

(43) λP.λQ.λx. P (x) ∧ Q(x) (ADJUNCT)

This function takes a predicate and returns an intersective modiﬁer. The idea, roughly

speaking, is that the basic meaning of ‘green’, i.e. the meaning we learn, has predicative

semantic type. Thus, when ‘green’ occurs in attributive positions (as with (42b)), a type-

mismatch results. In other words, ‘green’ has the wrong semantic type to occur in attributive

19I assume ‘in number’ has the denotation in (i), so that combining ‘four’ with ‘in number’ results in the same cardinality predicate.

(i) in number = λn.λx. µ#(x) = n J K 20See e.g. Partee(1986a), Partee(2004), and Kennedy(2012).

45 position. This mismatch triggers an application of ADJUNCT, which when applied to the basic predicative meaning of ‘green’, coerces it into a modiﬁer, thus ﬁxing the mismatch.

Furthermore, because the resulting modiﬁer is intersective (i.e. comes in the form of a conjunction), the attributive form is predicted to entail the corresponding predicative form.

Similarly, applying ADJUNCT to the cardinality predicate in (41) returns a modiﬁer true of those four-membered pluralities with property Q.

(44) ADJUNCT(λx. µ#(x) = 4) = λQ.λx. µ#(x) = 4 ∧ Q(x)

This is an appropriate meaning for Intersective Modiﬁer uses like (28d) since combining it

with a predicate like ‘moons of Jupiter’ returns a predicate true of four-membered pluralities

consisting of moons belonging to Jupiter. Consequently, (28d) will be true just in case no

such plurality of moons revolves around Saturn.

Next consider Quantiﬁcational uses like (28a). Following Partee(1986a), Landman

assumes that meanings appropriate for these arise from combining the modiﬁer denotation

given in (44) with Partee’s Existential Closure (EC) given in (45).

(45) λP.λQ.∃x. P (x) ∧ Q(x) (EC)

In eﬀect, this shifts a predicate into an existentially bound generalized quantiﬁer, or second-

order property. For example, combining it with a predicate like ‘green moons’ returns a

second-order property true of some plurality of green moons. As a result, (46) would be

true just in case there is at least one plurality of moons such that each is green and belongs

to Jupiter.

(46) Jupiter has green moons.

Notice that this would not be false if Jupiter also happened to have blue moons and red

moons as well, since in that case there would still be at least one plurality of green moons.

Similarly, applying Existential Closure to a predicate like ‘four moons’ returns an exis-

tentially bound second-order property true of four-membered pluralities of moons, or (47).

46 (47) EC(λx. µ#(x) = 4 ∧ moons(x)) = λQ.∃x. µ#(x) = 4 ∧ moons(x) ∧ Q(x)

Consequently, (28a) (i.e. Quantiﬁer) will be true just in case there is at least one four-

membered plurality of moons, each of which belongs to Jupiter. Notice that this predicts

lower-bounded truth-conditions for Quantiﬁcational uses. In other words, (28a) would be

true even if Jupiter happened to have more than sixty moons,21 as there would still be a

plurality of four such moons in that case.22

Finally, consider Measurement uses like (28f). These can be handled in an exactly

parallel fashion. On the Adjectival Theory, a measure phrase like ‘four ounces’ is predicted

to denote a predicate true of things measuring four ounces, or (48a).

(48) a. four ounces = λx. µoz(x) = 4 J K b. ADJUNCT(λx. µoz(x) = 4) = λQ.λx. µoz(x) = 4 ∧ Q(x))

c. EC(λx. µoz(x) = 4 ∧ water(x)) = λQ.∃x. µoz(x) = 4 ∧ water(x) ∧ Q(x))

Combining this predicate with ADJUNCT, ‘water’, and Existential Closure results in (48c),

or a meaning appropriate for Measurement uses.23 According to the resulting truth-

conditions, (28f) will be true just in case Mary drank at least one quantity of water measur-

ing four ounces. Again, this plausibly predicts lower-bounded truth-conditions: intuitively,

(28f) would not be false if Mary happened to drink a gallon of water, for instance.

21Which it apparently does. According to Balcerak-Jackson(2013), Jupiter actually has between sixty-two and sixty-four moons. 22Why, then, are Quantificational uses so often interpreted as “two-sided”, i.e. as meaning ‘exactly n’? Traditionally, this has been explained as a Gricean implicature (see Horn(1972)). The arguments runs like this. Suppose Quantifier has lower-bounded truth-conditions, so that an utterance of Quantifier proffers the proposition that Jupiter has at least four moons. Now, having more than four moons asymmetrically entails having four moons; for example, having ten moons entails having four moons but not vice versa. Hence, if Jupiter had more than four moons, it would be more informative to say so. Thus, obeying Grice(1989)’s Quantity maxim requires saying so if one is in that epistemic position. So if a speaker does not say that Jupiter has n moons for some number larger than four, assuming (s)he is being cooperative implicates that for all (s)he knows, Jupiter does not have more than four moons. And this implication in conjunction with the proposition proffered jointly imply that Jupiter has exactly four moons. 23I am ignoring the semantic contribution of ‘of’ here for convenience only. For a fully explicit treatment, see Snyder and Barlew(2016).

47 2.4.2 The Extended Adjectival Theory

The purpose of this section is to extend the Adjectival Theory. In particular, I want to show how meanings appropriate for the remaining uses of ‘four’ discussed above can be derived, namely (28b,e,g,h).

(28b) The number of Jupiter’s moons is four. (Speciﬁcational)

(28e) Jupiter’s moons number four. (Verbal Complement)

(28g) Four is my favorite number. (Numeral)

(28h) The number four is my favorite. (Predicative Numeral)

As before, the plan will be to show how these meanings are either already available given existing semantic resources, or else can be derived via independently motivated type-shifting principles. I call the result the Extended Adjectival Theory. Let’s begin with the easiest case, Numeral uses like (28g). Because we are assuming that measure phrases consist of numerals, meanings appropriate for Numeral uses are already available: numerals are names of numbers. Consequently, (28g) will be true just in case the number referred to by ‘four’ has the property of being my favorite number.

Predicative Numeral uses like (28h) can be derived from numerals via two type-shifters and an independently plausible meaning for ‘the’. The ﬁrst type-shifter is Landman’s AD-

JUNCT, which we have seen is needed to capture how intersective modiﬁers are generated from their predicative counterparts. The second type-shifter needed is Partee(1986a)’s

“IDENT”, given in (49).

(49) λx.λy. y = x (IDENT)

What this does, in eﬀect, is lift a singular term into a predicate, one true of individual just in case that individual is identical with the referent of the original singular term. According to Partee(1986b), it is thanks to this type-shifter that equative sentences are really a special

48 case of predicational sentences.24 Assume that the predicational copula has the meaning in

(50b), and suppose the name ‘Tully’ is a singular term (represented here as ‘t’).

(50) a. Cicero is Tully.

b. be = λP.λx. P (x) J K c. IDENT( Tully ) = λy. y = t J K Then since ‘Tully’ occurs in predicate position in (50a), by hypothesis it creates a type-

mismatch. This triggers a use of IDENT, which when applied to the name, creates the

predicate in (50c), thus resolving the mismatch. As a result, (50a) will be true just in case

the referent of ‘Cicero’ is identical to the referent of ‘Tully’.

Now, assume with Sharvy(1980), Link(1983), Chierchia(1998), and others the meaning

of ‘the’ in (51).

(51) ιP = the largest P if there is one (otherwise undeﬁned)

The sense of “largest” here is mereological. With plural nouns like ‘oranges’, for instance,

(51) predicts that ‘the oranges’ will denote a maximal (salient) plurality of oranges, i.e. a

plurality x such that for any salient atomic orange y, y is part of x. On the other hand, (51)

predicts that ‘the orange’ will refer to a uniquely (salient) orange since no single atomic

orange is part of any other.

Finally, assume that ‘number’ in ‘the number four’ is a predicate true of numbers, similar

to ‘number’ in e.g. ‘Four is a number’. We can then derive a meaning for ‘the number four’

by ﬁrst combining the numeral ‘four’ with IDENT, thus resulting in a predicate true of

numbers identical with four.

(52) a. IDENT( four ) = λn. n = 4 J K b. ADJUNCT( number ) = λP.λn. number(n) ∧ P (n) J K 24Put diﬀerently, there are not two copulas – “the ‘is’ of predication” and “the ‘is’ of identity’ – as philosophers often assume. Rather, there is an ‘is’ of predication, with identity statements being a special case.

49 c. the number four = the (ADJUNCT( number ))(IDENT( four )) J K J K J K J K = ιn. [number(n) ∧ n = 4]

Next, we combine ADJUNCT with ‘number’, thus resulting in a predicate true of a number having some property P . Finally, we combine these with ‘the’, ultimately resulting in a singular term referring the unique number identical to four. Thus, the prediction is that

‘the number four’ is a singular term coreferential with the numeral ‘four’.

The last two uses of ‘four’ to consider are Speciﬁcational and Verbal Complement uses.

I want to suggest that these employ the same meaning, though they do so in diﬀerent ways.

The appropriate meaning can be obtained through a ﬁnal type-shifting principle due to

Partee(1986a), one she calls “NOM”.

(53) λP. ∩λx. P (x) (NOM)

As the name suggests, NOM nominalizes a predicate, i.e. turns it into a kind of name. More speciﬁcally, it turns a predicate into a name of a property.25 These are also understood to be kinds, in the sense of Carlson(1977a) and Chierchia(1984, 1998). Technically, kinds are functions from worlds to maximal individuals of the appropriate sort. For example, the kind DOG, or the presumed referent of ‘canine’, is modeled as a function from worlds w to the maximal plurality of dogs in w. Similarly, the kind WATER, or the presumed referent of ‘H20’, is a function from worlds w to the maximal quantity of water in w. In eﬀect, Chierchia’s ∩-operator (pronounced “down”) takes a predicate P and intensionalizes, turning it into a function from worlds to maximal sums of things in the extension of P .

Partee’s NOM codiﬁes Cheirchia’s ∩-operator as a type-shifting principle.

Chapter 2 of this thesis is dedicated to defending the claim that Partee’s NOM supplies an appropriate referent for ‘four’ in Speciﬁcational uses like Identity. As a preview, following

Scontras(2014), I’m going to argue that the result of applying NOM to the cardinality

25As Partee (1986b, p. 122) emphasizes, NOM in subject to certain restrictions: “[NOM] maps properties to entity-correlates if these exist (the Russell property, for instance, will be acceptable as a predicate but will not have an entity-correlate)”.

50 predicate resulting from a measure phrase is what semanticists call a degree, or an abstract representation of measurement.

∩ (54) NOM(λx. µ#(x) = 4) = λx. µ#(x) = 4

Technically, (54) refers to a function from worlds w to the maximal plurality of four-

membered pluralities in w. This provides an appropriate meaning for Verbal Complement

uses like (28e), assuming a meaning for the verb ‘number’ like the one in (55), where d is a

variable ranging over degrees.

(55) number = λd.λx. number(x, d) J K Thus, (28e) will be true if the plurality referred to by ‘Jupiter’s moons’ has a cardinality

of four. The purpose of Chapter 2 is to show how (28b), or Identity, can be given roughly

similar truth-conditions.

To summarize, the Adjectival Theory provides the basic ingredients to derive meanings

appropriate for the various uses of ‘four’ noted above, and it does so in a fairly simple and

elegant manner. Its theoretical signiﬁcance is that it is the only extant theory capable of

deriving meanings appropriate for (28a-h) without having to postulate lexical ambiguities.26

For now, I want to survey the consequences of the Adjectival Theory for Frege’s Other

Puzzle. This is the business of the next and ﬁnal section.

2.5 The Neo-Substantivalist Strategy

I began this chapter with a puzzle concerning two uses of ‘four’ originally noted by Frege.

26This conclusion relies entirely on arguments given in Chapter 2. There are plenty of semantic theories which claim to be capable of deriving meanings appropriate for at least some of the uses of ‘four’ noted in (28), e.g. Partee(1986a), Geurts(2006), and Kennedy(2012). The problem with all of them, however, is that they either directly or indirectly equate numbers with degrees, an assumption I will argue in Chapter 2 is mistaken. For example, Partee’s semantics does this indirectly, suggesting that meanings appropriate for Numeral uses can be obtained by applying NOM to a basic cardinality predicate. Consequently, ‘four’ in Identity is predicted to be synonymous with the numeral ‘four’. However, we will see plenty of examples suggesting that this cannot be right. Geurts’ analysis is an extension of Partee’s, and so inherits the same problem. Finally, Kennedy’s analysis explicitly equates degrees with numbers, and so is subject to the same objection.

51 (Quantiﬁer) Jupiter has four moons.

(Identity) The number of Jupiter’s moons is four.

Quantiﬁer and Identity seem equivalent. Yet ‘four’ appears have diﬀerent semantic functions

in those examples, thus leading to Frege’s Other Puzzle.

(FOP1) Quantiﬁer and Identity are semantically equivalent.

(FOP2) Diﬀerent occurrences of an expression occurring in semantically equivalent statements

serve the same semantic function.

(FOP3) The different occurrences of ‘four’ in Quantifier and Identity serve different semantic

functions.

(FOP4) The diﬀerent occurrences of ‘four’ in Quantiﬁer and Identity are witnesses to the same

expression, namely ‘four’.

We saw that most philosophers deny FOP3. More speciﬁcally, adherents of the Substantival

Strategy argue that ‘four’ is best understood as functioning referentially in both examples,

while defenders of the Adjectival Strategy argue that ‘four’ is functioning non-referentially

in both examples. However, we also saw that both sorts of analyses suﬀer from various

linguistic problems. The question here is whether the Adjectival Theory can solve Frege’s

Other Puzzle while also avoiding those same problems.

As noted in §1.4.1, one of the core commitments of the Adjectival Theory is that ‘four’

in Quantiﬁcational uses occurs within a measure phrase. As such, it is a numeral, just as

Frege and more recent Substantivalists maintain. It is not this numeral whose meaning gets

shifted into a second-order property, but rather the measure phrase as a whole. Furthermore,

I suggested in §1.4.2 that a similar analysis is appropriate for ‘four’ in Identity. More

speciﬁcally, I suggested that ‘four’ in Identity also occurs within a measure phrase, and

that the resulting cardinality predicate can then be shifted into a singular term referring to

a degree. If so, then once again ‘four’ is functioning as a numeral in Identity, just as Frege

52 and recent Substantivalists maintain, as it is not the numeral itself which gets shifted into a singular term, but rather the entire measure phrase.

Consequently, one might view the Extended Adjectival Theory as a somewhat more sophisticated version of the Substantival Strategy. However, there is one critical diﬀerence between the analysis of Identity suggested in §1.4.2 and the one assumed by other Substanti-

valists: whereas the Fregean analysis assumes that Identity is a genuine identity statement

equating the referents of two singular terms, including the numeral ‘four’, the Extended

Adjectival Theory instead views the referent of the post-copular material in Identity as a

nominalized measure phrase referring to a degree. One of the core theses of Chapter 2 is

that English draws a sortal distinction between numbers and degrees, and so these should

not be equated. This explains various semantic contrasts like (14), for instance.

(14a) The number of Jupiter’s moons is {four/??the number four}.

(14b) The number Mary likes most is {four/the number four}.

If so, then one of the critical problems with previous Substantivalist analyses is that not only do they wrongly conflate specificational sentences with equative sentences, they also wrongly equate the actual referent of the post-copular material in Identity – a degree – with the referent of a numeral – a number. Thus, to differentiate the present view from previous

Substantivalists, let’s call the solution to Frege’s Other Puzzle resulting from the Extended

Adjectival Theory the Neo-Substantivalist Strategy. How exactly does the Neo-Substantivalist Strategy solve Frege’s Other Puzzle? First, like previous Substantivalists, the holder of the present view denies FOP3: since ‘four’ occurs within a measure phrase in both Quantiﬁer and Identity, it is functioning as a numeral in both examples. However, unlike Substantivalists and Adjectivalists, the Neo-

Substantivalist also denies FOP2, i.e. that different occurrences of the same expression in equivalent statements must serve the same semantic function. That’s because the same measure phrase occurs in both Quantifier and Identity, yet it serves different semantic

53 functions in those examples. More specifically, it functions quantificationally in Quantifier and referentially in Identity.

As a consequence, the Neo-Substantivalist rejects any purported advantage of viewing number expressions as having a uniform semantic function. For example, number expressions cannot be analyzed uniformly as quantificational determiners. Thus, any hope one might have had for a Adjectivalist-inspired nominalism,27 one springing from the possibility of extending the Adjectivalist analysis to all uses of number expressions, will have to be abandoned. Rather, one of the common themes running throughout this thesis is that English presupposes an ontology of different sorts of abstract objects, including e.g. numbers, degrees, and kinds, and number expressions can refer to different sorts of these abstracta.

It should be emphasized that FOP2 is independently false, and so can be seen as the real culprit behind Frege’s Other Puzzle. After all, it is possible to construct similar “puzzles” using pairs of examples like (56)-(58).

(56) a. Jupiter has green moons.

b. The color of Jupiter’s moons is green.

(57) a. Mary has a four foot canoe.

b. The length of Mary’s canoe is four feet.

(58) a. Four is my favorite number.

b. The number four is my favorite.

Each of the pairs here are intuitively equivalent despite the underlined expressions appearing to serve diﬀerent semantic functions. This is not really puzzling, or at least it shouldn’t be. In essence, maintaining FOP2 amounts to denying the possibility of type-shifting, a

27Nominalism is the metaphysical view that numbers do not exist. It is opposed to platonism, or the view that numbers exist.

54 phenomenon which is uncontroversial from the perspective of linguistic semantics.28,29 Seen this way, that FOP3 is false is an accidental feature of Frege’s Other Puzzle.

I want to end this chapter by noting an important feature of the analyses discussed in

§1.2. Nearly all of them – the Substantivalist analysis of Hale and Wright(2001), the Coding

Fictionalism of Hodes(1984), and the Displacement Analysis of Hofweber(2005) – seem to assume that a fully adequate solution to Frege’s Other Puzzle should not only explain how the diﬀerent uses of ‘four’ in Quantiﬁer and Identity are linguistically related, but also how counting can lead to a priori arithmetic knowledge, i.e. knowledge of statements like ‘2 + 3 =

5’. I have only addressed the linguistic task in this chapter. The second will be taken in up in Chapter 3, where I will argue that even though English presupposes a sortal distinction between cardinalities (the sorts of degrees answering ‘how many’-questions) and numbers

(the sorts of things referred to by numerals), there is a structural isomorphism holding between degrees formed on the basis of counting and the natural numbers, i.e. 1, 2, 3, ...

Thus, adopting a form of structuralism arguably discharges this additional epistemological burden.

28In fact, Chierchia (1998, p. 340) lists the existence of type-shifting operations as one of the three central assumptions guiding linguistic semantics. 29Contrary to (Hofweber, 2005, p.205), who says: “I think that semantic type coercion [i.e. type-shifting] is the second-best attempt to solve Frege’s Other Puzzle, but once we look at the details we can see that it can’t be right.”

55 Chapter 3

The Easy Argument for Numbers

3.1 The Easy Argument for Numbers

This chapter is concerned with a puzzling argument involving Frege(1884)’s pair of examples

from the previous chapter, Quantiﬁer and Identity.

(Quantiﬁer) Jupiter has four moons.

(Identity) The number of Jupiter’s moons is four.

According to Frege, recall, the function of a number expression within an ideal logical

language would be to refer to a number. Thus, he proposes “converting” apparently non-

referential uses of ‘four’ like Quantiﬁer in terms of apparent identity statements like Identity,

where ‘four’ is a numeral referring to the number four.

Neo-Fregeans, those who have since adopted Frege’s analysis, use Frege’s proposed “conversion” to argue for the existence of numbers.1

(Realism) There is a number, namely four, which is the number of Jupiter’s moons.

Suppose Quantiﬁer is true. Since Identity is a paraphrase of Quantiﬁer, Identity must also

be true. But Identity is an identity statement equating the referents of two singular terms,

1Adherents include Hale and Wright(2001), particularly since they use a standard non-free logic. The constructivist logicist account of Tennant(1987, 1997) also incurs ontological commitment to numbers as abstract objects, but via a more detailed logical route using a free logic. It begins on the basis of existentially committing introduction rules for zero and successor that are argued to be meaning-constituting, and furnishes by way of a self-imposed adequacy condition all instances of what Tennant calls “Schema N”: There are exactly n F s if and only if #x[F (x)] = n.

56 including the numeral ‘four’. And names, like all singular terms, can be validly existentially generalized. Generalizing on ‘four’ leads to Realism, or an explicit acknowledgment of the existence of a number. Thus, we have a valid argument for a controversial metaphysical conclusion – that numbers exist – which follows from a seemingly innocuous observation about

Jupiter’s moons – that there are four of them. This is known as the Easy Argument for Numbers.2 The Easy Argument is so-called because it seems far too easy. Whether numbers exist is an important, longstanding philosophical question. It would be surprising if that question could be answered simply by looking through a telescope and counting some moons. This has suggested to many philosophers that the Easy Argument must be invalid. But how?

There are a couple of popular answers available. The ﬁrst is Fictionalism.3 According to it, numbers don’t really exist, yet we often talk as if they do exist as a kind of convenient

fiction. Within that fiction, the singular terms in Identity refer to numbers. But outside that fiction, the singular terms fail to refer, and so Identity is not true. Assuming ‘four’ in

Quantiﬁer is functioning non-referentially, it could be true whether or not we pretend that numbers exist. If so, then Identity cannot be a truth-preserving paraphrase of Quantiﬁer.

Thus, the truth of Quantiﬁer provides no independent support for the truth of Identity, and so the Easy Argument collapses.

The second response is the Adjectival Strategy from the previous chapter. According to it, recall, ‘four’ in Identity is not a singular term. Rather, it has the same non-referential semantic function it has in Quantiﬁer. According to Moltmann(2013a) and Felka(2014), that’s because Identity is a speciﬁcational sentence, and the best analysis of these suggests that they are not identity statements. Rather, they express question-answer pairs, where

‘the number of Jupiter’s moons’ expresses a question about the cardinality of Jupiter’s moons, and post-copular ‘four’ expresses an answer to that question, namely Quantiﬁer.

2This label comes from Balcerak-Jackson(2013). See Hofweber(2007) for discussion. 3See e.g. Hodes(1984) and Yablo(2005).

57 Assuming ‘four’ is non-referential in the latter, the existential inference from Identity to

Realism is thereby blocked.

I would like to oﬀer an entirely diﬀerent diagnosis of what’s wrong with the Easy Argu- ment. In my view, the problem is that it equivocates. Let me explain.

3.1.1 A Parallel Ambiguity?

Moltmann(2013a) observes that ‘number’ is ambiguous between a relational, cardinality sense, and a monadic, arithmetic sense. The former is plausibly witnessed in examples like

(59), the latter in examples like (60).

(59) a. The number of children is {expanding rapidly / ??irrational}.

b. The children are four in number.

c. The children number four.

(60) a. The number Mary is researching is {??expanding rapidly / irrational}.

b. The number four is even.

c. There are lots of numbers between 5 and 100.

Intuitively, ‘number’ in the ﬁrst sense tells us something about how many members a certain group contains. In the second sense, it expresses a property of arithmetic objects, e.g. 1, 2, etc.

There appears to be a parallel ambiguity in ‘four’. In particular, it appears to have an adjectival meaning plausibly witnessed in examples like (61), and an arithmetic meaning plausibly witnessed in examples like (62).

(61) a. What’s the number of children? (Almost) four.

b. How many of these eight numbers are even? [Let’s see. 2 is even and 6 is even

...] Four {??is / are}.

c. The number of children is four. The number of women is the same (??one).

58 (62) a. What’s the number Mary is researching? (??Almost) four.

b. Which one of these three numbers is even? [Let’s see. 1 isn’t, and 3 isn’t ...]

Four {is / ??are}.

c. The number Mary is researching is four. The number John is researching is the

same (one).

Intuitively, the examples in (61) tell us about how many members a certain group contains, while the examples in (62) tell us something about a particular arithmetic object – the number four.

3.1.2 Various Uses of Number Expressions

We saw in the previous chapter that, following e.g. Partee(1986b), Landman(2004), Geurts

(2006), the various uses of ‘four’ in (63) can be seen as taking on diﬀerent but related meanings, thanks to type-shifting.

(63) a. Four is a number. (Numeral)

b. The number four is even. (Predicative Numeral)

c. Jupiter’s moons are four (in number). (Predicative Adjective)

d. The number of Jupiter’s moons is four. (Speciﬁcational)

e. No four moons of Jupiter orbit Saturn. (Intersective Modiﬁer)

f. Jupiter has four moons. (Quantiﬁcational)

More speciﬁcally, these meanings are all derivable from a single lexical item, namely the numeral in (64), but only if it occurs within a measure phrase in (63c-f).

(64) four = 4 J K It’s for this reason that (63a,b) tell us something about a speciﬁc number, while (63c-f) tell us something about the cardinality of Jupiter’s moons.

59 To brieﬂy review, according to the Adjectival Theory, measure phrases are built from three components: a numerical relation, a number, and a measure term. Numerical relations are relations between numbers and include e.g. ‘=’, ‘<’, etc. Measure terms refer to measure functions, or functions from individuals to numbers, and include e.g. a foot measure ‘µft’ and a cardinality measure ‘µ#’. A typical measure phrase such as ‘exactly four feet’ will thus have the following syntactic structure.

Measure Phrase

Numerical Phrase Measure Term

Numerical Relation Number feet

exactly four

However, not all measure phrases will have overtly speciﬁed numerical relations and measure terms. Indeed, this is precisely the situation in examples like (63c,e,f): we have a measure phrase with a null numerical relation and measure term, thus resulting in the following structure:

Measure Phrase

Numerical Phrase Measure Term

Numerical Relation Number ∅

∅ four

And as the following table indicates, null numerical relations and measure terms default to

= and µ#, respectively.

60 numerical relation denotation measure term measure

0 0 exactly λn.λn . n = n foot µft

0 0 less than λn.λn . n < n ounce µoz

0 0 ∅ λn.λn . n = n ∅ µ#

Consequently, the resulting numerical phrase denotes the characteristic function of being identical to the number four. Composing this function with µ# then results in the cardinality predicate in (65), where x ranges over pluralities and µ# measures their cardinality.

(65) λx. µ#(x) = 4

More speciﬁcally, µ# returns the number of singular individuals constituting a plurality, or “atoms” in the sense of Link(1983) or Krifka(1989).

We saw that meanings appropriate for all of the various occurrences of ‘four’ in (63) can be obtained from either the basic numeral in (64), the cardinality predicate in (65), or from combining these with the following independently motivated type-shifting principles.

Name Author Type-Shifter

IDENT Partee(1986b) λx.λy. y = x

NOM Partee(1986b) λP. ∩λx. P (x)

ADJUNCT Landman(2004) λP.λQ.λx. P (x) ∧ Q(x)

EC Partee(1986b) λP.λQ.∃x. P (x) ∧ Q(x)

More exactly, meanings appropriate for the numerals in (63a,b) are derivable from (64) via type-shifting, while those appropriate for the measure phrases in (63c-f) are derivable from the cardinality predicate in (65) via type-shifting. The resulting denotations are summarized in the following table,

61 Use Type Denotation

Numeral e 4

Pred. Numeral he, ti λx. x = 4

Pred. Adjective he, ti λx. µ#(x) = 4

∩ Speciﬁcational d λx. µ#(x) = 4

Int. Modiﬁer hhe, ti, he, tii λP.λx. µ#(x) = 4 ∧ P (x)

Quantiﬁcational hhe, ti, hhe, ti, tii λP.λQ.∃x. µ#(x) = 4 ∧ P (x) ∧ Q(x) and the derivations of these various meanings are summarized in the following map, where arrows indicate directionality in the derivation.

∩ λx.µ#(x) = 4 λx.µ#(x) = 4 λQ.λx.µ#(x) = 4 ∧ Q(x) NOM ADJUNCT EC IDENT λy.y = 4 4 λP.λQ.∃x.µ#(x) = 4 ∧ P (x) ∧ Q(x)

There are two points worth emphasizing here. First, certain instances of ‘four’ in (63) occur within a measure phrase, others do not. Those which do take on a cardinality-related meaning, while those which do not either denote the number four directly or else a property of that number. Hence the apparent ambiguity in ‘four’ noted in §3.1.1. Secondly, notice that the meanings proposed for Numeral and Speciﬁcational uses are diﬀerent. This will play a crucial role in diagnosing what’s wrong with the Easy Argument.

3.1.3 Speciﬁcational and Quantiﬁcational Uses

My next major contention is that both the Neo-Fregean and Adjectivalist analyses are semantically inadequate. The inadequacy of the Neo-Fregean analysis has already been demonstrated by Moltmann(2013a). If Identity were an identity statement equating the referents of ‘the number of Jupiter’s moons’ and the numeral ‘four’, and if the numeral

62 ‘four’ were coreferential with ‘the number four’ as Frege contends, then we would expect

(66a,b) to be equally acceptable, contrary to fact.4

(66) a. The number of Jupiter’s moons is {four / ??the number four}.

b. The number of interest is {four / the number four}.

This indirectly suggests that Identity is not really an identity statement of the familiar sort;

it is in fact a speciﬁcational sentence.

The trouble for recent versions of the Adjectival Strategy is interestingly diﬀerent. As

mentioned in Chapter 2, Felka(2014) argues that Identity has the syntactic structure in

(67), where strikethrough represents ellipsis.

(67)[ What the number of Jupiter’s moons is] is [Jupiter has four moons].

Consequently, the question expressed by the pre-copular material is predicted to be ‘What is the number of Jupiter’s moons?’, while the post-copular material answers that question with Quantiﬁer. However, Moltmann suggests instead that it instead answers ‘How many moons does Jupiter have?’. The obvious advantage of Felka’s suggestion is that it more closely resembles the surface syntax of Identity. After all, it’s hard to see how the pre- copular material could express ‘How many moons does Jupiter have?’ as a function of ellipsis.

One signiﬁcant problem for Felka’s analysis, however, is that it has no obvious means of explaining why (68a) and (68b) are not synonymous.

(68) a. John knows the number of Jupiter’s moons, namely {four/??even}.

b. John knows what the number of Jupiter’s moons is, namely {four/even}.

c. John knows how many moons Jupiter has, namely {four/??even}.

Intuitively, ‘John knows the number of Jupiter’s moons’ is true only if John knows the exact cardinality of Jupiter’s moons. But ‘John knows what the number of Jupiter’s moons

4(66a) is Moltmann’s example. (66b) is added here for contrast.

63 is’ would be true even if John only knows some more general fact about their cardinality, e.g. that it is large or even. Thus, it appears that ‘the number of Jupiter’s moons’ cannot generally be synonymous with the question ‘What is the number of Jupiter’s moons?’.

Like Moltmann and Felka, I think Identity is a speciﬁcational sentence. However, I also think that a better kind of analysis for those constructions is available. Following

Romero(2005), I assume instead that pre-copular definites in specificational clauses denote individual concepts, or functions from worlds to individuals.5 On this analysis, post- copular terms of specificational clauses generally function as referential-type expressions, including presumably ‘four’ in Identity. If so, then it simply does not follow that because

Identity is a speciﬁcational sentence, post-copular ‘four’ must have the same non-referential semantic function it has in Quantiﬁer, contra the Adjectival Strategy. On the other hand, it does follow that pre-copular ‘the number of Jupiter’s moons’ cannot be a referential-type expression, contra the Neo-Fregean analysis.

My next major contention is that the post-copular numeral in Identity occurs within a measure phrase, as shown by examples like (69).

(69) a. The number of Jupiter’s moons is almost four.

b. The number of Jupiter’s moons is four more than Saturn’s.

c. How many moons Jupiter has is four.

(70) a. The length of Mary’s canoe is almost four feet.

b. The length of Mary’s canoe is four feet more than John’s.

c. How long Mary’s canoe is is four feet.

According to Lassiter(2011), occurring with modiﬁers like ‘almost’, forming comparatives, and answering questions like ‘How ...?’ are the hallmarks of degree-denoting expressions, where a degree is an abstract representation of measurement. Given the parallels to

5See also Comorovsky(2008).

64 prototypical degree-terms like ‘four feet’,6 I thus assume that the post-copular measure phrase in Identity denotes a degree of cardinality, or a degree representing the number of atoms making up a plurality.

Following the recent work of Scontras(2014), I assume that degrees more generally are nominalized properties, or kinds in the sense of Carlson(1977a) and Chierchia(1984, 1998). Chierchia and Partee(1986a) have already taught us how to derive kinds from properties: we nominalize them via Chierchia’s ∩-operator in (71a), codiﬁed as Partee’s type-shifting operation “NOM” from above.

(71) a. ∩P = the kind corresponding to P

∩ b. NOM(λx. µ#(x) = 4) = λx. µ#(x) = 4

The result is a kind: a function from worlds to maximal pluralities of the corresponding sort. For example, applying NOM to the cardinality predicate in (65) returns (71b), or the nominalized property of being four in number. Technically, this is a function from worlds to the maximal plurality of four-membered pluralities in that world. On present assumptions, this is also a degree, more speciﬁcally a degree of cardinality.

3.1.4 Diagnosing the Easy Argument

According to the analysis just sketched, (66a) and (66b) are both speciﬁcational sentences, and so the post-copular expressions are plausibly functioning as singular terms.

(66a) The number of Jupiter’s moons is {four / ??the number four}.

(66b) The number of interest is {four / the number four}.

Thus, if both occurrences of ‘four’ and the deﬁnite ‘the number four’ were coreferential, then the present analysis would face the same problem faced by Neo-Fregeans: it would have no clear means of semantically explaining the diﬀerence in acceptability witnessed.

6See e.g. Kennedy(1999) or Schwarzchild(2002); Schwarzschild(2005).

65 My next signiﬁcant contention, again following Scontras(2014), is that English draws a

sortal distinction between numbers and degrees, including degrees of cardinality. Numbers

are atomic individuals, similar to John or that chair, while degrees are nominalized prop-

erties formed on the basis of measurement, e.g. weight or cardinality. Since measurement

presupposes a measure, and since measures presuppose numbers, degrees cannot be identi-

cal with numbers. Nevertheless, both are in the referential domain, and we can distinguish

them as diﬀerent sorts of individuals in a manner similar to Chierchia(1998). Accordingly,

the referential domain looks as follows.

Sort Example

A: Atomic Individuals That’s John.

P: Plural Individuals That’s John and Mary.

N ⊆ A: Numbers That’s the number four.

K ⊆ A: Kinds That kind of rope...

D ⊆ K: Degrees That length of rope...

Degrees of cardinality are the relata of the relational, cardinality sense of ‘number’, while

numbers form the extension of the monadic, arithmetic sense. The explanation of examples

like (66) thus runs as follows: The numeral ‘four’ occurs within a measure phrase in (66a) but

not (66b); both post-copular expressions are singular terms, though the measure phrase in

(66a) refers to a degree; degrees are sortally distinct from numbers; hence, the ‘the number

four’ is coreferential with the post-copular material of (66b) but not (66a); put diﬀerently,

‘the number four’ creates a sort-mismatch (it denotes a diﬀerent sort of thing than what’s

denoted by ‘four’) in (66a) but not (66b), thus explaining the diﬀerence in acceptability

witnessed.

If so, then what’s really wrong with the Easy Argument is that it equates what English

sortally distinguishes, namely degrees and numbers.

(Quantiﬁer) Jupiter has four moons.

66 (Identity) The number of Jupiter’s moons is four.

(Realism) There is a number, namely four, which is the number of Jupiter’s moons.

On the intended Neo-Fregean interpretation, Realism is assumed to have the logical form

in (72), where the variable ‘y’ ranges over numbers and ‘4’ refers to the number four.

(72) ∃y. #x[moon-of-Jupiter(x)] = y ∧ y = 4

However, the ﬁrst occurrence of ‘number’ in Realism is the monadic, arithmetic sense,

while the second occurrence is the relational, cardinality sense. Thus, the former is true

of a diﬀerent sort of thing than the relata of the latter. But then (72) is actually false

since nothing is both a degree and a number. Put diﬀerently, Realism can establish what

Neo-Fregeans intend only if numbers are mistakenly assumed to be degrees.

My view is that while the Easy Argument does not establish the existence of numbers in

the way Neo-Fregeans intend, English does nevertheless presuppose an ontology of numbers.

We purport to refer to them, quantify over them, employ them in measurement, etc. What’s

more, the post-copular material of Identity is functioning referentially on the semantics

for speciﬁcational sentences assumed here. Consequently, the truth of Identity plausibly

implies that Jupiter’s moons have a cardinality. And since cardinalities are degrees formed

on the basis of measurement, and since measures themselves relate individuals to numbers,

the truth of Identity also plausibly implies the existence of a number, thus vindicating a

somewhat diﬀerent version of the Easy Argument.7

7Whether this gives us a good reason for concluding that numbers or cardinalities “really” exist in some metaphysically robust sense is complicated by two further, independent questions. The ﬁrst is whether statements of the form ‘There is/are ...’ are ontologically committing. Moltmann(2013b) argues that they are not. So even if Identity entails something like ‘There is a cardinality measuring how many moons Jupiter has’, it would not obviously follow that cardinalities “really” exist. The second and more fundamental question is whether semantics ever provides good evidence for robust metaphysical conclusions. This, I take it, is the central question of what Bach(1986b) calls “Natural Language Metaphysics”. For example, does the fact that our best semantic theories employ events, worlds, degrees, numbers, etc. provide good evidence that such things “really” exist? I will not attempt to answer that question here.

67 However, this admission will come as little consolation to Neo-Fregeans. The purpose of the Easy Argument is not merely to show that numbers exist, but also that our ordinary arithmetic knowledge is ultimately grounded in our ability to count. This is why the directionality of the paraphrase from Quantifier to Identity is theoretically significant: we are counting moons in the former and identifying numbers in the latter, thus suggesting that we come to know the identity of numbers through counting.8 But if what’s actually being identified in Identity are degrees, and if these are not the same sorts of things as numbers, then the fact that Quantifier can be paraphrased as Identity would appear to be an epistemological red herring.

The rest of the chapter is dedicated to motivating the assumptions just sketched. §3.2

brieﬂy reviews the core semantics for number expressions assumed here, namely the Ex-

tended Adjectival Theory. §3.3 extends that semantics to numerical speciﬁcational sentences

like Identity by marrying two independently motivated analyses, namely Romero’s analysis

of speciﬁcational clauses and Scontras’ analysis of “Degree Nouns”. §3.4 surveys the philo-

sophical consequences of the resulting analysis for the Easy Argument and the Neo-Fregean

program more generally.

3.2 A Brief Review of The Extended Adjectival Theory

I sketched the core semantics for number expressions assumed throughout this thesis in the

previous chapter, calling it “the Extended Adjectival Theory”. It was motivated by two

general considerations. First, ‘four’ can occur in a number of environments that prototypical

quantiﬁcational determiners cannot, e.g. in predicate positions, as a nominal modiﬁer, in

post-copular position of speciﬁcational sentences, or as the complement of the verb ‘number’.

(73) a. Jupiter’s moons are {four / ??most} (in number).

b. No {four / ??most} moons of Jupiter orbit Saturn.

8See e.g. Dummett(1991) and Wright(2000).

68 c. The number of Jupiter’s moons is {four / ??most}.

d. Jupiter’s moons number {four / ??most}.

Thus, number expressions more closely resemble color expressions such as ‘green’ in this respect.

(74) a. Jupiter’s moons are green (in hue).

b. No green moons of Jupiter orbit Saturn.

c. The color of Jupiter’s moons is green.

d. Jupiter’s moons are colored green.

Also, ‘four’ can “co-mingle” with other modiﬁers like ‘ferocious’.9

(75) a. We shipped the {four / most} ferocious lions to Boston.

b. We shipped the ferocious {four / ??most} lions to Boston.

Again, this suggests that number expressions function adjectivally when occurring in modi-

ﬁer positions. That’s because ‘four’ occurs within a measure phrase in (73a-d), and because, according to Schwarzschild(2005), measure phrases like ‘four feet’ in ‘four feet tall’ serve the same semantic function as ‘extremely’ in ‘extremely tall’.

The second general consideration is that the various uses of number expressions take on diﬀerent but related meanings via type-shifting, assuming they are lexically numerals. For example, a meaning appropriate for ‘the number four’ can be obtained in the way suggested in (76), supposing with Sharvy(1980) that ‘the’ functions as the maximality-operator in

(76a).

(76) a. ιP = the largest member of P if there is one (else undeﬁned)

b. λx.λy. y = x (IDENT)

c. λP.λQ.λx. P (x) ∧ Q(x) (ADJUNCT)

9At least in environments like (75). See Landman (2004) for discussion.

69 d. the number four = the (ADJUNCT( number )(IDENT( four )) J K J K J K J K = ιx [number(x) ∧ x = 4]

The resulting denotation is the singular term in (76d) referring to the unique number identical to four. Thus, the numeral ‘four’ and ‘the number four’ are coreferential singular terms: both refer to a particular number, namely four.

As mentioned above, the cardinality meaning of occurrences of ‘four’ results from the numeral occurring within a measure phrase on the Extended Adjectival Theory. In cases like (77a), where ‘four’ occurs without with an overt numerical relation or measure term, the resulting measure phrase is predicted to denote the cardinality predicate in (77b).

(77) a. Jupiter’s moons are four (in number).

b. λx. µ#(x) = 4

Suppose that there are four atomic moons in our model, a, b, c, and d. These can be joined to form pluralities of moons, e.g. the plurality consisting of a and b, notated here as a t b, or the plurality consisting of all four moons, i.e. a t b t c t d. The cardinality predicate in

(77b) is true of those pluralities which consist of exactly four atoms, and so (77a) will be true if the plurality denoted by ‘Jupiter’s moons’ is a t b t c t d.

With the cardinality predicate in place, further cardinality-related meanings can be derived via type-shifting. For instance, following Partee(1986b) and Geurts(2006) in assuming that numerical modiﬁers are intersective, Intersective Modiﬁer uses like (78a) can be derived via Landman’s ADJUNCT.

(78) a. No four moons of Jupiter orbit Saturn.

b. ADJUNCT(λx. µ#(x) = 4) = λQ.λx. µ#(x) = 4 ∧ Q(x)

Consequently, (78a) will be true if no plurality consisting of four moons belonging to Jupiter orbits Saturn. Similarly, Quantiﬁcational uses such as (79a) can be derived from (78b) via

Partee’s Existential Closure (EC).

70 (79) a. Jupiter has four moons.

b. λP.λQ.∃x. P (x) ∧ Q(x) (EC)

c. EC( four moons ) = λQ.∃x. µ#(x) = 4 ∧ moons(x) ∧ Q(x) J K As a result, Quantiﬁcational uses, unlike Predicative Adjective and Intersective Modiﬁer

uses, are predicted to have lower-bounded truth-conditions: (79a) will be true just in case

there is at least one plurality of four moons belonging to Jupiter. This would be true even

if Jupiter had sixty-two moons, as there would still be a plurality of four such moons in

that case.10

Summing up, the Extended Adjectival Theory provides an elegant way of semantically

relating nearly all of the occurrences of ‘four’ in (63) noted earlier. What remains of this

chapter is to show that it can likewise deliver a meaning suitable for ‘four’ in Speciﬁcational

uses like Identity. To do that, however, we will ﬁrst need a semantics for speciﬁcational

sentences more generally.

3.3 Extending the Adjectival Theory

Let’s begin by reviewing some of the evidence against the Neo-Fregean analysis. According

to it, Identity asserts that an identity holds between the referents of two singular terms,

including the numeral ‘four’.

(Identity) The number of Jupiter’s moons is four.

There are two kinds of problematic evidence. The ﬁrst, suggested by Moltmann(2013a), is

indirect. If the Fregean analysis were correct, then the truth of Identity would reveal that

‘the number of Jupiter’s moons’ and the numeral ‘four’ are coreferential singular terms.

Moreover, we should expect coreferential singular terms to be acceptably intersubstitutable

10Though Quantiﬁcational uses are traditionally assumed to be lower-bounded, this assumption has been recently challenged. See e.g. Geurts(2006) and Kennedy(2013).

71 on the Neo-Fregean analysis, including e.g. the numeral ‘four’ and ‘the number four’. How- ever, Moltmann shows that these assumptions are jointly inconsistent.

(80) a. John noticed {the number of Jupiter’s moons/??the number four}.

b. {The number of Jupiter’s moons/??The number four} is surprising.

c. John compared {the number of Jupiter’s moons/??the number four} to the

number of Saturn’s moons.

d. Mary counted {the number of Jupiter’s moons/??the number four}.

Clearly, the pairs here are not synonymous. Even if John happened to notice four moons, it

does not follow that he also noticed an abstract arithmetic object. Likewise, if John found

the number of Jupiter’s moons surprising, it does not follow that he found an abstract

arithmetic object surprising.

And there are further semantic diﬀerences. For example, ‘the number of Jupiter’s moons’

is unacceptable with ‘which’-questions like (81a), while ‘the number four’ is unacceptable

with ‘how many’.

(81) a. Which number is {four/??the number of Jupiter’s moons}?

b. How many moons Jupiter has is {four/??the number four}.

Also, Moltmann observes that while the number of Jupiter’s moons can expand and con-

tract, apparently the number four cannot.

(82) {The number of Jupiter’s moons/??The number four} is expanding rapidly.

Similarly, the number of Jupiter’s moons can diﬀer from the number of interest with respect

to various arithmetic properties.

(83) a. the irrational / negative / prime number of interest

b. ?? the irrational / negative / prime number of Jupiter’s moons

72 None of this is to be expected if ‘the number of Jupiter’s moons’ refers to the number four, thus indirectly suggesting that Identity does not have the function of equating two numbers.

The second kind of evidence against the Fregean analysis is more direct. As we saw in the previous chapter, there are good semantic reasons for thinking that speciﬁcational sentences are not synonymous with equative sentences.

(84) a. Cicero is Tully. (Equative)

b. The most famous Roman orator is Tully. (Speciﬁcational)

c. Cicero is bald. (Predicational)

For example, Mikkelsen(2005)’s observation that gender-neutral pronouns are acceptable in tag-questions accompanying the former but not the latter suggests that while predicational sentences have genuinely referential subjects, speciﬁcational sentences do not.11

(85) a. The most famous Roman orator is Tully, isn’t {he / it}?

b. The most famous Roman orator is bald, isn’t {he / ??it}?

c. Cicero is Tully, isn’t {he / ??it}?

But as both Moltmann and Felka(2014) point out, Identity has the structure of a prototyp-

ical speciﬁcational sentence – Deﬁnite Description + be + Name – by Neo-Fregeans’ lights.

Hence, the problem for the Neo-Fregean analysis is that it blurs the distinction between

speciﬁcational sentences like Identity and genuine equatives like (86).

(86) That is the number four.

More speciﬁcally, the Neo-Fregean analysis wrongly analyzes ‘the number of Jupiter’s

moons’ in Identity as a singular term.

11For more evidence, see Mikkelsen(2005, 2011) and Romero(2005).

73 3.3.1 The Individual Concept Analysis

The analysis of speciﬁcational clauses adopted here was suggested originally by Heim(1979)

and later developed by Romero(2005). I call it the Individual Concept Analysis since according to it, pre-copular deﬁnites occurring in speciﬁcational clauses denote individual

concepts. It is motivated primarily by a notable parallel to concealed questions, or deﬁnites occurring in question-embedding verbs such as ‘know’. Consider e.g. ‘the composer

Mary knows’ in (87), which is ambiguous.

Context: Mary is taking a music history quiz. She is to match each composer with one of their

famous works. Unfortunately, Mary is only able to match Wagner with Tannha¨user.

(87) John knows the composer Mary knows.

Here, several questions of the form ‘Who composed what?’ are relevant. On the ﬁrst

interpretation, Mary knows the answer to one of these questions, namely ‘Who composed

Tannha¨user?’, and John knows the same answer. On the second interpretation, John knows

the answer to a certain “meta-question”, namely for which of the questions of the form ‘Who

composed what?’ did Mary know the answer? It was in fact ‘Who composed Tannha¨user?’.

According to Heim, the ﬁrst interpretation arises if the matrix verb (the ﬁrst ‘knows’)

combines with the extension of ‘the composer Mary knows’, while the second interpretation

arises if the verb instead combines with its intension.

Romero observes that speciﬁcational clauses give rise to a similar ambiguity. Consider

(88), for instance.

(88) The composer Mary thought was Schumann was (in fact) {Wagner / the composer

of Tannh¨auser}.

Again, several questions of the form ‘Who composed what?’ are relevant, and Mary thought

the answer to one of these questions was “Schumann”, e.g. ‘Who composed Tannha¨user?’.

With ‘Wagner’, (88) is naturally interpreted as claiming that Mary’s answer was mistaken.

74 But with ‘the composer of Tannha¨user’, (88) is instead understood as answering a certain

“meta-question”, namely for which of the questions of the form ‘Who composed what?’ did Mary believe the answer was “Schumann”? The answer is in fact ‘Who composed

Tannhaüser?’. According to Romero, this shows that specificational subjects denote individual concepts; the first interpretation of (88) then arises if specificational ‘be’ takes the extension of ‘the composer Mary thought was Schumann’ as argument, while the “meta- question” interpretation arises if ‘be’ instead takes its intension as argument.

Thus, speciﬁcational ‘be’ is assumed to have the denotation in (89), where ‘y’ is a variable ranging over individual concepts.12

(89) be = λx. λy . λw. y(w) = x J K hs,ei Consequently, Romero analyses speciﬁcational sentences like (91a) as (91b), where the semantic contribution of pre-copular ‘the composer of Tannha¨user’ is the individual concept in (90).

(90) λw0. ιx[composer-of-Tannha¨user(x, w0)]

(91) a. The composer of Tannh¨auser is Wagner.

b. (91a) = λw. ιx[composer-of-Tannhaüser(x, w)] = w J K According to (90), ‘the composer of Tannhaüser’ in (91a) denotes a function from worlds w to individuals x such that x is Tannhaüser’s unique composer in w. Combining this with the meaning of the specificational copula in (89) ultimately reduces to (91b), or the set of worlds w (a proposition) such that Tannhaüser’s unique composer in w is identical with

Wagner. Romero then shows that deﬁnites occurring in concealed question environments like (87) are also plausibly analyzed as individual concepts, and that the resulting semantics

12At least for non-“meta-question” interpretations of examples like (91a). A higher type (hs, hs, eii) pre- copular argument is required for the latter. Mikkelsen(2011) suggests that these meanings are plausibly predicted to be available by Partee(1986b)’s polymorphic analysis of the copula, appropriately intensionalized.

75 for concealed questions can be easily extended to a plausible semantics for overt questions

like ‘Who composed Tannha¨user?’. The business of the next two sections is to extend her

analysis to numerical speciﬁcational sentences like Identity.

3.3.2 The Degree-as-Kind Analysis

I begin with the observation that ‘the number of (Jupiter’s) moons’ exhibits an ambiguity

similar to the one noted for ‘the composer of Tannh¨auser’. To see this, consider ﬁrst the

parallel ambiguity in concealed questions like (92).

Context: Mary is taking an astronomy quiz. She is to match each planet with a number

representing how many moons belong to that planet. Unfortunately, she is only able

to match Jupiter with 62.

(92) John knows the number of moons Mary knows.

Here, there are several relevant questions of the form ‘How many moons belong to which

planet?’. On the ﬁrst interpretation of (92), Mary knows the answer to one of these ques-

tions, namely ‘How many moons belong to Jupiter?’, and John knows this same answer.

On the second interpretation, John knows the answer to a certain “meta-question”, namely

for which of the questions of the form ‘How many moons belong to which question?’ did

Mary answer “62”? The answer is ‘How many moons belong to Jupiter?’.

Now consider (93).

(93) The number of moons Mary thought was four was (in fact) {sixty-two / the number

of Jupiter’s moons}.

Again, there are several questions of the form ‘How many moons belong to which planet?’

relevant, and Mary thinks that the answer to one of these is “four”. With ‘sixty-two’,

(93) is naturally understood as claiming that Mary’s answer was incorrect: the planet

in question actually has sixty-two moons. With ‘the number of Jupiter’s moons’, (93) is

76 instead naturally interpreted as answering a “meta-question”, namely for which question

of the form ‘How many moons belong to which planet?’ did Mary answer “four”. The

answer is in fact ‘How many moons belong to Jupiter?’. As before, this makes sense if

speciﬁcational ‘be’ takes an intensional object as its pre-copular argument.

Assuming with Moltmann(2013a) and Felka(2014) that Identity is a speciﬁcational

sentence, it thus appears that a treatment al´athe Individual Concept Analysis is in order.

If so, then pre-copular ‘the number of Jupiter’s moons’ should denote an individual concept,

while the post-copular measure phrase should function as a singular term. However, the

latter does not plausibly refer to a number, unlike e.g. post-copular ‘four’ in (94a).

(94) a. The number or interest is (??almost) four.

b. The number of interest is {irrational/??expanding rapidly}.

c. Which number is of interest is four.

For one thing, we saw above that ‘the number of Jupiter’s moons’ is unacceptable with

adjectives like ‘irrational’. Also, while ‘the number of Jupiter’s moons’ is naturally para-

phrased as a ‘how many’-question, the concealed question in (94a) is naturally paraphrased

as a ‘which’-question.

So the question is how the post-copular material of Identity can be a referential expres-

sion if it is not a numeral. The answer, I submit, is that it is actually a type-lowered measure

phrase, one referring to a degree of cardinality. Following Scontras(2014), I understand

degrees as nominalized properties, or kinds. Evidence for this comes from parallels between

nouns like ‘amount’, or what Scontras calls a Degree Noun, and the noun ‘kind’. He points out that (95a) is ambiguous.

Context: Pointing at four 1 lb. Fuji apples in a bowl.

(95) a. John ate that amount of apples every day for a year.

b. John ate that number of apples every day for a year.

77 c. John ate that kind of apple every day for a year.

On the implausible definite interpretation of (95a), John ate the same four apples we are looking at every day for a year. On the plausible existential interpretation, there is some contextually-salient measurement of the apples in the bowl, e.g. their weight or cardinality, and John ate apples measuring the same amount every day for a year. In other words, ‘that amount of apples’ references a degree of weight or cardinality and says that

John ate diﬀerent apples instantiating that degree every day for a year. As a consequence,

(95a) is rightly predicted to have diﬀerent interpretations, depending on which contextually- salient measurement of the apples is at issue. If it is weight, then (95a) would be true if

John ate diﬀerent numbers of apples every day, so long as they collectively weighed four pounds. Conversely, if it is cardinality, then (95a) would be true if John ate exactly four apples every day, regardless of their collective weight.

As with (95a), the only plausible interpretations of (95b) and (95c) are existential.

However, unlike (95a), (95b) can only be paraphrased as ‘John ate exactly four apples every day for a year’. In other words, it is only plausibly true if ‘that number of apples’ references a degree of cardinality, and if John ate diﬀerent groups of apples instantiating that degree every day for a year. For (95c), the plausible interpretation can be paraphrased as ‘John ate Fuji apples every day for a year’. This results if ‘that kind of apple’ references an abstract kind of apple, namely the Fuji kind, and says of it that John ate diﬀerent apples instantiating that kind every day for a year.13 This is one way in which degrees pattern like kinds.

Another way is in so-called degree relatives, as Scontras also shows. Carlson(1977b) originally observed that ‘there’-insertion is acceptable within relative clauses headed by

‘that’ but not those headed by ‘which’.

(96) John ate the apples {∅/that /??which} there were on the table. 13Unlike (95a,b), a deﬁnite interpretation for (95c) is not even remotely plausible. See Scontras(2014) for an explanation.

78 To explain why, Carlson argues that the gap (“ ”) is ﬁlled by a degree variable, and while ‘that’ can bind both degree and individual variables, ‘which’ can only bind individual variables. Heim(1987), based on observations originally from Saﬁr(1982), notices that

‘there’-insertion similarly fails with ‘which’-questions like (97a), but not with ‘how many’- questions like (97b).

(97) a. ?? Which food truck is there in Columbus?

b. How many food trucks are there in Columbus?

c. What kind of food truck is there in Columbus?

As Heim explains, this can be similarly explained if the diﬀerent question words bind diﬀer- ent sorts of variables. For example, while ‘which’ binds an individual variable, ‘how many’ binds a degree variable. As Scontras notes, ‘what kind’ patterns like ‘how many’ in this respect.

Finally, Scontras notes that ‘amount’ and ‘kind’ are both acceptable in degree relatives, as indeed is ‘number’.

(98) a. John ate the amount of apples {∅/that/??which} Mary ate.

b. John ate the number of apples {∅/that/??which} Mary ate.

c. Mary ate the kind of apples {∅/that/??which} Mary ate.

Here, the plausible interpretations are clearly existential: (98b), for instance, is only plausibly interpreted as claiming that the cardinality of apples John ate is equal to that of the apples Mary ate. Moreover, the facts in (98a-c) make sense if ‘the amount/number/kind of apples’ introduce the same sort of variable, and if ‘that’ but not ‘which’ can bind it.

Three conclusions naturally recommend themselves. First, ‘number’, like ‘amount’, is a Degree Noun. Secondly, ‘number’ is a special case of ‘amount’ in that while ‘amount’ relates quantities of a substance (e.g. apples or water) to a set of degrees representing some contextually-determined measure of that substance (e.g. weight or cardinality), ‘number’ is

79 restricted to relating just pluralities speciﬁcally to degrees of cardinality. Third, based on these and other resemblances to canonical kind-denoting expressions, Scontras concludes that degrees just are kinds. Anderson and Morzycki(2013) reach a similar conclusion, but based on diﬀerent semantic considerations. Accordingly, let’s call this view of degrees the

Degree-as-Kind Analysis. The signiﬁcance of the Degree-as-Kind Analysis for present purposes is that it aﬀords

a straightforward way of deriving degrees from a cardinality predicate. Suppose that post-

copular ‘four’ in Identity occurs within a measure phrase, and so denotes a cardinality

predicate. Then since it occurs in post-copular position, the measure phrase creates a type-

mismatch. However, this is readily ﬁxed by an application of Partee(1986b)’s NOM-shifter

from §1.2, thus resulting in the nominalized property in (71b).

∩ (71b) NOM(λx. µ#(x) = 4) = λx. µ#(x) = 4

Technically, this denotes a function from worlds to the maximal plurality of four-membered

pluralities. This is a kind, according to Chierchia(1998). But since it is formed on the basis

of a cardinality measurement, it is also a degree. More generally, nominalized properties

formed on the basis of measurement are degrees, but since not all nominalized properties

are formed this way, degrees are best understood as kinds of a special sort.

3.3.3 Putting It All Together

The goal here is to marry the Individual Concept Analysis with the Degree-as-Kind Analysis

so as to produce plausible truth-conditions for Identity. According to Scontras(2014), the

Degree Noun ‘amount’ denotes a relation between substances (e.g. apples or water) and a set

of degrees representing some contextually-determined measurement of that substance (e.g.

weight or cardinality). On the semantics oﬀered here, ‘number’ in its relational, cardinality

80 sense also relates substances to a set of degrees, only those substances are restricted specif-

ically to pluralities, and those degrees are restricted speciﬁcally to degrees of cardinality.

This is the denotation given in (99a), where d is a degree and n is a number.

∩ (99) a. number = λP.λd.[∃n. d = λx. µ#(x) = n ∧ P (x)] J K ∩ b. number of apples = λd.[∃n. d = λx. µ#(x) = n ∧ apples(x)] J K Since the set of degrees in (99b) are nominalized properties formed speciﬁcally on the basis

of a cardinality measurement, they are degrees of cardinality. Continuing to assume that

‘the’ is a maximality-operator, ‘the number of apples’ thus denotes the largest such degree

measuring the cardinality of some plurality of apples.

∩ (100) the number of apples = ιd [∃n. d = λx. µ#(x) = n ∧ apples(x)] J K For example, ‘the number of apples in the bowl’ in (101) will reference the maximal degree

representing how many apples there are in the bowl.

Context: Pointing at four 1 lb. Fuji apples in a bowl.

(101) John bought the number of apples in the bowl.

And since the largest number measuring the cardinality of the apples in the bowl is four,

(101) will be true on the existential interpretation if John bought exactly four apples.

A similar analysis can be give for Identity. I assume that ‘Jupiter’s moons’ is what

Barker(1998) calls a “prenominal possessive”. Borrowing a suggestion from Barker, I also

assume that the possessive clitic -s has the meaning stated in (102a), where the relation

variable R is free, suggesting that its interpretation is to be provided by context.

(102) a. -s = λP.λx.λy. P (y) ∧ R(x, y) J K b. -s ( moons )( Jupiter ) = λy. moons(y) ∧ R(j, y) J K J K J K

81 As a result, ‘Jupiter’s moons’ is a predicate true of those pluralities of moons bearing R to

Jupiter, e.g. belonging to Jupiter. This can then combine with the meaning of ‘the number of’ from above, thus resulting in (103), where ‘JMs’ is shorthand for the predicate in (102b).

∩ (103) the number of Jupiter’s moons = ιd[∃n. d = λx.µ#(x) = n ∧ JMs(x)] J K As a result, ‘the number of Jupiter’s moons’ will denote the maximal degree representing

the cardinality of moons bearing R to Jupiter.

Now recall the meaning of the speciﬁcational copula on the Individual Concept Analysis,

repeated in (104a). Combining this with the above degree-denoting meaning for the post-

copular measure phrase results in (104b).

(104) a. be = λx.λy .λw.λx. y(w) = x J K ∩ b. is four = λy .λw.y(w) = λz. µ#(z) = 4 J K Finally, this combines with an intensionalized version of ‘the number of Jupiter’s moons’, represented here (following Romero(2005)) as the individual concept in (105), resulting in the truth-conditions for Identity in (106).

(105) λw0. ιd [number-of-Jupiter’s-moons(d, w0)]

(106) Identity = λw. ιd [number-of-Jupiter’s-moons(d, w)] = J K ∩ λz. µ#(z) = 4

According to (105), ‘the number of Jupiter’s moons’ in Identity contributes a function from

worlds w to degrees d (a sort of individual) such that the maximal cardinality of Jupiter’s

moons in w is d. Ultimately, then, Identity will denote the set of worlds w (a proposition)

in which the maximal degree of cardinality of Jupiter’s moons in w is identical to the

nominalized property of being four in number, itself a degree of cardinality. For example,

Identity will be true in the actual world if Jupiter has exactly four moons.

It is relatively straightforward to extend this semantics to concealed questions and overt

‘how many’-questions, following the analyses of Romero(2005). For example, (107a) is

82 analyzed as (107b), where “EPIJohn” are John’s epistemic alternatives and w0 designates the actual world.

(107) a. John knows the number of Jupiter’s moons.

b. ∀w ∈EPIJohn. ιd [number-of-Jupiter’s-moons(d, w)]

= ιd [number-of-Jupiter’s-moons(d, w0)]

In English, (107a) will be true just in case Jupiter has exactly as many moons in all of

John’s epistemic alternatives as it actually has. ‘How many’-questions like (108a) can then

be analyzed as denoting those propositions truly and exhaustively answering that question,

al´a Karttunen(1977).

(108) a. How many moons does Jupiter have?

b. (108a) w = λw0. ιd [number-of-Jupiter’s-moons(d, w0)] J K = ιd [number-of-Jupiter’s-moons(d, w)]

So if Jupiter actually had four moons, then (108a) would map the actual world to the

proposition expressed by Identity. In other words, Identity would be the actually correct

answer to (108a).

One immediate consequence is that the concealed question in (107a) should be para-

phraseable as a ‘how many’-question, unlike Felka(2014)’s Elliptical Analysis which predicts

that the pre-copular subject of Identity should express the question ‘What is the number

of Jupiter’s moons?’.

(67)[ What the number of Jupiter’s moons is] is [Jupiter has four moons].

Consequently, the analysis here naturally predicts the diﬀerence in acceptability seen in

(68).

(68a) John knows the number of Jupiter’s moons, namely {four/??even}.

(68b) John knows what the number of Jupiter’s moons is, namely {four/even}.

83 (68c) John knows how many moons Jupiter has, namely {four/??even}.

Whereas the ‘what’-question in (68b) asks about some general property holding of the cardinality of Jupiter’s moons, the ‘how many’-question in (68c) asks speciﬁcally about its identity.

3.3.4 Cardinality and Number

There is one more important consequence of the analysis just sketched: it aﬀords a straightforward explanation of the Moltmann(2013a) data cited in §3.1, assuming Scontras(2014) is correct that degrees are sortally distinct from numbers. For example, consider again the diﬀerence noted in (66a,b).

(66a) The number of Jupiter’s moons is {four / ??the number four}.

(66b) The number of interest is {four / the number four}.

Post-copular ‘four’ occurs within a measure phrase referring to a degree of cardinality

(henceforth “cardinality”) in (66a) but not (66b). Since ‘the number four’ is coreferential with the numeral, we have a sort-mismatch in (66a) but not (66b). A similar explanation can be given for (81a,b).

(81a) How many moons Jupiter has is {four / ??the number four}.

(81b) Which number is {four / ??the number of Jupiter’s moons}?

This makes sense if ‘which’ binds individual variables, ‘how many’ binds degree variables, and if numbers qua atomic individuals and cardinalities qua degrees are sortally distinct.

But what about examples like (80)?

(80a) John noticed {the number of Jupiter’s moons / ??the number four}.

(80b) {The number of Jupiter’s moons / ??The number four} is surprising.

84 (80c) John compared {the number of Jupiter’s moons / ??the number four} to the number

of Saturn’s moons.

(80d) Mary counted {the number of Jupiter’s moons / ??the number four}.

The thing to notice here is that the deﬁnites can be paraphrased in terms of ‘how many’,

and so they are all plausibly concealed questions.

(109) a. John noticed how many moons of Jupiter there are.

b. How many moons of Jupiter there are is surprising.

c. John compared how many moons of Jupiter there are to how many moons of

Saturn there are.

d. Mary counted how many moons of Jupiter there are.

As such, they denote (individual concepts of) cardinalities, not numbers.

3.4 What’s Really Wrong with the Easy Argument?

How does the semantics just developed help with the Easy Argument?

(Quantiﬁer) Jupiter has four moons.

(Identity) The number of Jupiter’s moons is four.

(Realism) There is a number, namely four, which is the number of Jupiter’s moons.

On the Neo-Fregean analysis, Identity expresses an identity statement equating two arith-

metic objects, and since the latter is a paraphrase of Quantiﬁer, counting Jupiter’s moons is

a proof that numbers exist. The ﬁrst thing to notice about this argument is that an exactly

similar Neo-Fregean analysis of examples like (110a) clearly won’t do.

(110) a. The number of interest is {four / the number four}.

b.# x[of-interest(x)] = 4

85 Obviously, I can be interested in the number four without being interested in four things.

What this shows is that the intended relata of Frege’s cardinality-operator are cardinalities, not numbers. The second thing to notice is that the English paraphrase given for (72), stated as Realism,

(72) ∃y. #x[moon-of-Jupiter(x)] = y ∧ y = 4 is awfully convenient. A diﬀerent English paraphrase, one which should be synonymous with Realism on the Neo-Fregean analysis, is given in (111).

(111) ?? There is a number, namely the number of Jupiter’s moons, which is (the number)

four.

Yet (111) is decidedly odd. This points towards what’s really wrong with the Easy Argu- ment: it equivocates.

On the present view, numbers form the extension of ‘number’ in the monadic, arithmetic sense given in (112a), while cardinalities are the relata of ‘number’ in the relational, cardinality sense given in (112b).14

(112) a. number1 = λx. number(x) J K ∩ b. number2 = λP.λd.[∃n. d = λx. µ#(x) = n ∧ P (x)] J K The intended Neo-Fregean conclusion is that numbers, i.e. the very things Frege(1884) in-

tended to provide an epistemically unassailable foundation for, exist. Thus, the existentially

bound variable in (72) had better range over numbers, or the sorts of things forming the

extension of (112a). But the intended relata of Frege’s cardinality-functor are cardinalities,

or the degrees in (112b). The problem is that Realism employs both senses,

14Is there is a principled reason why ‘number’ is ambiguous between these two uses? One possibility, following Moltmann(2013a), is that ‘number’ is actually homonymous (she claims that in German ‘Zahl’ has both uses but ‘Anzahl’ is used only relationally, for instance). Alternatively, English sortally distinguishes numbers and cardinalities, and this is reﬂected in the two senses. In that case, the two senses may not be completely unrelated since cardinalities are formed on the basis of a cardinality measure, the relata of which is a number.

86 ∗ (Realism ) There is a number1, namely four, which is the number2 of Jupiter’s moons.

and so the equivocation is now obvious. Clearly, the existentially bound variable in (72)

can range over cardinalities or degrees, but not both. But if cardinalities and numbers are

sortally distinct and ‘4’ refers to the number four, then (72) is false. Put diﬀerently, Realism

can establish what Neo-Fregeans intend only if cardinalities are conﬂated with numbers.

On the semantics for speciﬁcational sentences assumed here, the post-copular measure

phrase in Identity is functioning as a singular term. Assuming existential generalization

is generally valid for singular terms, the actual truth of Identity should imply (113b),

paraphrased informally as (113a).

(113) a. Jupiter’s moons have a cardinality of four.

0 0 0 ∩ b. ∃d . ιd [number-of-Jupiter’s-moons(d, w0)] = d ∧ d = λz. µ#(z) = 4

0 0 ∩ c. ∃d [∃n. d = λz. µ#(z) = n]

Moreover, (113b) implies (113c) as a matter of logic, presumably. If so, then the actual

truth of Identity plausibly implies the existence of a number, thus vindicating a somewhat

diﬀerent version of the Easy Argument.

However, this new version of the Argument will not suﬃce for Neo-Fregeans’ epistemo-

logical purposes. One of the fundamental assumptions of the Neo-Fregean program is that

our knowledge of Hume’s Principle (HP) grounds our ordinary arithmetic knowledge.

(114) ∀F, G. #x[F (x)] = #x[G(x)] ↔ F ≈ G (HP)

Here, “≈” is the relation of equinumerosity, where two concepts F and G are equinumerous just in case every F can be mapped to a unique G and vice versa. According to Hale and

Wright(2001), because the cardinality-operator ranges over numbers, and since HP lays

down identity conditions for those objects, our knowledge of HP grounds our knowledge of

arithmetic. What’s more, because HP is analytically true, i.e. true in virtue of the meaning

87 of number, it could not be false. If so, then numbers must exist since HP implies their existence.15

Again, the problem with both arguments is that they conﬂate numbers with cardinalities.

If HP is indeed analytic, then it can be so only in virtue of the cardinality sense of ‘number’.

After all, the deﬁnition is given in terms of Frege’s cardinality-operator. However, HP can ground arithmetic knowledge only if the objects generated by the right-to-left direction are numbers. But then numbers would have to be cardinalities in virtue of the cardinality sense of ‘number’, contrary to fact. For the same reason, the assumed analyticity of HP at best establishes the necessary existence of cardinalities, not numbers.

Then why don’t Neo-Fregeans just take cardinalities to be the fundamental objects of arithmetic instead? The problem is that cardinalities are not the right sorts of things to play this role, according to Neo-Fregeans. This suggestion violates another central Neo-Fregean commitment, namely what Wright(2000) calls Frege’s Constraint, which he describes this way:16

What is it to observe Frege’s constraint? To insist that the general principle

governing the application of a type of number be built into their characterization

from the start is in eﬀect just to insist such numbers be characterized by reference

to a principle which explains what kind of entities they apply to – are of – and

what it is for such entities to be associated with same or diﬀerent such numbers...

To view such principles as philosophically and mathematically foundational is

accordingly to view the applications of the sorts of mathematical objects they

concern as belonging to the essence of objects of these sorts.17

Simply put, Frege’s Constraint requires that the applications of the natural numbers be built directly into their characterization. For Wright, counting is the primary application

15At least once HP is suitably restricted. See Heck(1997) and Tennant(1997). 16Though there is a growing literature of Frege’s Constraint, the felt need to explain the applicability of the natural numbers was articulated earlier in Tennant(1987). 17Wright (2000, p. 325.).

88 of the natural numbers.18 Observing Frege’s Constraint thus amounts to requiring that

Frege’s cardinality-operator ranges over the natural numbers. Otherwise, HP would neglect an essential feature of those objects, contrary to what Frege himself requires.

Thus, the problem with the new, valid form of the Easy Argument from the Neo-Fregean perspective is that it fails Frege’s Constraint. That is, though the actual truth of Quantiﬁer plausibly entails an ontology of cardinalities, and thus numbers, it fails to identify the intended relata of Frege’s cardinality-operator – cardinalities – with numbers. Hence, the intended Neo-Fregean epistemological conclusion, namely that counting moons grants us a priori arithmetic knowledge, is thereby blocked. Rather, if anything, it seems that counting moons grants us epistemological access to cardinalities. And cardinalities are a speciﬁc sort

of degree, i.e. a nominalized property of measured substances. Since numbers are the relata

of measures, the truth of e.g. ‘John drank four ounces of water’ also entails their existence on

the Extended Adjectival Theory. But then what is the epistemological connection holding

speciﬁcally between degrees of cardinality and the numbers on which they are based?

To my mind, structuralism presents an attractive answer to this question. Shapiro (1997) describes the position this way:

The subject matter of arithmetic is the natural-number structure, the pattern

common to any system of objects that has a distinguished initial object and a

successor relation that satisﬁes the induction principle. Roughly speaking, the

essence of a natural number is the relations it has with other natural numbers ...

The natural-number structure is exempliﬁed by the von Neumann ﬁnite ordinals,

the Zermelo numerals, the arabic numerals, a sequence of distinct moments of

time, and so forth. The structure is common to all reductions of arithmetic.19

Degrees of cardinality are nominalized cardinality properties, and cardinality properties

hold of pluralities consisting of a certain number of atoms. Because atoms are indivisible

18Cf. also Tennant(1987). 19Shapiro (1997, p. 5-6), author’s emphasis.

89 wholes,20 numbers measuring the cardinality of a plurality will also be whole. Consequently, the natural ordering on degrees of cardinality will look as follows:

∩ λx. µ#(x) = 1

∩ λx. µ#(x) = 2 .

And this, of course, is just the natural number structure. Hence, by structuralists’ lights, all knowledge one could glean about this particular set of degrees would also constitute knowledge about the natural numbers. In this way, structuralism not only provides a natural link between the two senses of ‘number’ discussed here, it might also explain how our arithmetic knowledge is ultimately grounded in counting, just as Neo-Fregeans insist.

3.5 Conclusion

I have argued for several empirical and theoretical claims here. On the empirical side, I’ve argued that number expressions appear to be ambiguous in way resembling ‘number’ because they often occur within measure phrases having no overt measure term. Thus, while

‘number’ is ambiguous between a monadic, arithmetic sense and a relational, cardinality sense, various occurrences of ‘four’ similarly take on diﬀerent arithmetic and cardinality- related meanings depending on whether they occur within a measure phrase. As before, meanings for various uses of ‘four’ can be obtained via type-shifting on the minimal assumption that ‘four’ is lexically a numeral. As a result, we must semantically distinguish between the referents of genuine number-referring expressions like the numerals ‘four’ and ‘4’ and deﬁnites like ‘the number four’, and degree-denoting expressions like the measure phrase

20In the sense of Krifka(1989), for instance, where a P -atom is an individual satisfying P which has no P -relative proper parts.

90 in Identity. Following Scontras(2014), I’ve argued that numbers are sortally distinct from degrees of cardinality, and that this sortal distinction can explain various semantic contrasts observed by Moltmann(2013a).

Furthermore, I’ve argued that this sortal distinction has signiﬁcant philosophical impli- cations. First, it suggests a plausible diagnosis of what’s wrong with the Easy Argument: it equivocates by using both senses of ‘number’, thus conﬂating degrees and numbers. On the semantics developed here, the post-copular measure phrase in Identity is a singular term, and so Identity plausibly entails that Jupiter’s moons have a cardinality. Since cardinalities are formed on the basis of a cardinality measurement, it also plausibly presupposes the existence of a number. But this will come as little consolation to Neo-Fregeans if Frege’s

Constraint requires the cardinality-operator to range over natural numbers. In that case,

Hume’s Principle could ground arithmetic only if natural numbers are conﬂated with cardinalities, a special sort of degree. On the other hand, no such problem arises if Frege’s

Constraint is abandoned and a form of structuralism is adopted instead.

In the next chapter, I will consider a puzzle due to Salmon(1997) which challenges this conclusion. Salmon argues that because we can answer ‘how many’-questions like ‘How

1 many oranges are on the table?’ fractionally with answers like ‘There are 2 2 oranges on the table’, cardinalities needn’t be whole. If so, then the structural equivalence between cardinalities and the natural numbers noted above would be broken, thus undermining the appeal to structuralism in explaining the connection between counting and basic arithmetic knowledge.

91 Chapter 4

The Counting Oranges Puzzle

4.1 The Counting Oranges Puzzle

Here’s a puzzle due to Salmon(1997). Suppose there are three oranges on the table. I take

one of them, cut it in half, eat one of the halves, and set the remaining half back on the

table. Now consider the Question:

(Question) How many oranges are there on the table?

1 (Answer) There are 2 2 oranges on the table.

1 In the scenario described, the intuitively correct answer is the Answer: 2 2 oranges. Now, a half-orange is either an orange or isn’t. If it is, then there are three oranges on table, and

if it isn’t then there are only two. In either case, we get an answer to the Question which

is not the intuitively correct Answer. Call this the Counting Oranges Puzzle (COP). COP is ﬁrst and foremost a puzzle about the meanings of cardinality expressions, i.e.

expressions whose express purpose is to count. Its theoretical signiﬁcance is that it purports

to undermine traditional analyses of cardinality expressions. For example, according to

Frege(1884)’s familiar analysis, ‘one’, ‘two’, etc. are second-order concepts expressible in

terms of ﬁrst-order quantiﬁers and identity. Thus, examples like (115a) are analyzed as

(115b), where F translates ‘is an orange on the table’.

(115) a. There are two oranges on the table.

b. ∃x. ∃y. F (x) ∧ F (y) ∧ x 6= y ∧ ∀z. F (z) → [z = x ∨ z = y]

92 So (115a) will be true just in case there are at least two oranges on the table and those are the only oranges on the table. Put diﬀerently, (115a) will be true if the class of oranges on the table is two-membered.

Now, classes are extensional: they have objects as members and are identiﬁed by which members belong to them. Moreover, cardinal numbers are classes of equinumerous concepts on Frege’s view, where two concepts F and G are equinumerous just in case every F can be mapped to a unique G and vice versa. As a result, cardinal numbers are necessarily whole. There cannot be a concept F such that more than two but fewer than three objects fall under F . But this is precisely what the Answer seems to require; it appears to be counting the number of oranges on the table in terms of a fractional number. Consequently, stating coherent Fregean truth-conditions for the Answer would appear impossible.

We can put the point diﬀerently using Frege’s cardinality-operator ‘#x[φ]’. This maps a concept to a cardinal number, namely the number of objects falling under that concept.

For example, if there are two oranges on the table, then the cardinality-operator will map the concept expressed by ‘orange on the table’ to the number 2. Suppose we analyze the

Answer similarly.

1 (116) a. The number of oranges on the table is 2 2 .

1 b.# x[orange-on-table(x)] = 2 2

Then the Answer would be true only if the number of objects falling under the concept

1 orange-on-table is identical to the number 2 2 . But since there is no such number, (116a), and thus the Answer, is predicted to be false. Why, then, is the Answer the intuitively correct answer to the Question?

According to Salmon, the problem with the Fregean analysis is its extensionality. Since classes are extensional, cardinal numbers must be whole. Thus, the lesson of COP is that counting is inherently intensional. We don’t count classes of objects, but rather pluralities or groups of objects, relative to some counting property. Depending on what that counting

93 property is, we get diﬀerent answers to ‘how many’-questions. Relative to the property of being a whole orange, for instance, the answer to the Question will be ‘two oranges’.

Relative to the property of being an orange, however, the answer to the Question will be the Answer. That’s because we assign diﬀerent “weights” to members of the plurality in question; we assign “weight” 1 to the whole oranges, “weight” .5 to the half-orange, and adding these “weights” together, we get the Answer: 2.5 oranges.

Some have suggested that a similar lesson applies to the traditional analyses of cardinality expressions within linguistic semantics, e.g. the analyses of Barwise and Cooper(1981),

Partee(1986a), or the (Extended) Adjectival Theory as laid out in Chapter 2. On these accounts, count nouns denote individuated, and thus countable, objects. Cardinalities represent how many such objects constitute a collection, and so must again be whole. Thus, it might be thought that these analyses share the same problematic assumption Salmon at- tributes to Frege’s analysis, namely that count nouns like ‘orange’ are extensional, denoting sets of things which are in some intuitive sense indivisible wholes.

But there is a different semantics for count nouns available, one which does not share this assumption. It is due to Krifka(1989). Basing his semantics on classifier languages, 1 Krifka analyzes all English nouns as basically mass, so that all numerical modification amounts to measuring quantities of stuff – quantities of water, quantities of rice, quantities of orange, etc. On Krifka’s semantics, the primary difference between count nouns like ‘orange’ and mass nouns like ‘water’ is that whereas mass nouns denote quantities of stuff directly, count nouns do so indirectly thanks to a certain classifier-like element built into their meanings, which Krifka calls a natural-unit (NU). In essence, when I say that that there are two oranges on the table, I’m saying that there is a quantity of orange on the table measuring two natural-units of orange.

(117) a. orange = λn.λx. orange(x) ∧ NU(orange)(x) = n J K 1 b. 2 2 oranges = λx. orange(x) ∧ NU(orange)(x) = 2.5 J K 1See Chierchia(1998) and Scontras(2014) for relevant discussions.

94 Here, ‘NU’ is a function mapping individuals to real numbers with respect to some property.

Intuitively, cardinality expressions measure how much of a property a certain thing has with respect to a natural unit of that property. As with Salmon’s proposal, counting is thus property-relative, and diﬀerent measurements will result depending on what the property is. In the context of COP, for instance, relative to the property of being a whole orange,

NU will return the real number 2. But relative to the property of being an orange, nothing obviously prevents NU from returning a non-whole number as the correct answer to the

Question, as suggested in (117b).

Consequently, Krifka’s semantics aﬀords a solution to COP which is strikingly similar to Salmon’s solution, and which Kennedy and Stanley(2009) explicitly endorse. After mentioning Krifka’s semantics, they say the following:

This proposal is very similar to the one suggested by Nathan Salmon (1997,

p. 10), who suggests that ‘... numbers are not merely properties of pluralities

simpliciter, but relativized properties’. An advantage of this analysis is that it

provides a semantics for nominals with fractional number terms ... Assuming

that the sorted NU function is not constrained to return whole numbers as

values, a phrase like ‘2.5 oranges’ denotes a property that is true of orange-

stuﬀ whose measure equals 2.5 orange-units. However, this analysis involves a

commitment to the position that count nouns are in some fundamental sense

semantically the same as mass nouns, in that they denote properties of quantities

of stuﬀ, rather than properties of atomic objects.2

The solution on oﬀer is just this. In the context of COP, we are measuring quantities of orange relative to a natural unit of orange. Since measurements are given in terms of real numbers, and since real numbers needn’t be whole, it’s little wonder that the intuitively

2Kennedy and Stanley (2009, p. 619, fn. 16).

95 correct Answer to the Question involves a non-whole number. Let’s call this the Krifka-

Inspired Solution [+ Salmon] (KISS). KISS is not the only available solution to COP. In fact, I believe that it is not really a solution at all. That’s because while KISS correctly predicts one available interpretation of the Answer, it actually predicts the wrong interpretation in the context of Salmon’s puzzle.

I am going to argue that the Answer is ambiguous in a way resembling so-called container phrases like ‘glass of water’. Consider (118a).

(118) a. There are four glasses of water in the soup.

b. There’s a plurality of glasses x s.t. x consists of four individuals, each of which

is ﬁlled with water and is in the soup.

c. There’s a quantity of water x s.t. x measures four glasses worth and x is in the

soup.

This example is due to Rothstein(2009), who argues that it is ambiguous between what she calls an individuating interpretation (II) in (118b) and a measure interpretation (MI) in (118c).3 For the II, suppose that Mary has a strange way of heating up water; she pours it into glasses and places those glasses into boiling soup. For the MI, suppose that Mary wants to make some soup, and that the recipe calls for a certain amount of water, namely four glasses’ worth. Therefore, she ﬁlls a certain glass four times with water, pouring the contents each time into the soup.

One of my primary empirical contentions here is that I/M-ambiguities are not restricted to just container phrases. For example, ‘orange’ is an ordinary count noun, yet ‘four oranges’ is similarly ambiguous.

(119) a. There are four oranges in the punch.

b. There’s a plurality of oranges x s.t. x consists of four individuals, each of which

is in the punch.

3Cf. also Landman(2004), Partee and Borschev(2012), and Scontras(2014).

96 c. There’s a quantity of orange x s.t. x measures four oranges worth and x is in

the punch.

For the II in (119b), suppose that Mary has already made the punch and that she wants to decorate it for the party. She thinks that adding some fruit would do the trick, and so she drops several apples, some pears, and four oranges into the punch. For the MI in (119c), suppose Mary wants to make some punch for the party and the recipe calls for four oranges worth of orange pulp. Accordingly, Mary takes four whole oranges, pulverizes them, and adds the resulting pulp to the rest of the punch.

As preliminary evidence for there being a genuine ambiguity in both cases, consider the fact that there is an asymmetric entailment relation holding generally between IIs and MIs.

For example, if Mary places four glasses ﬁlled with water into the soup, then there must be four glasses’ worth of water in the soup. On the other hand, if she pours an amount of water measuring four glasses’ worth directly from the tap into the soup, it clearly does not follow that she has placed any glasses in it. Similarly, if Mary drops four whole oranges into the punch, then there must be four oranges’ worth of orange in the punch. But if she pours four oranges’ worth of prepackaged orange pulp directly into the punch, it clearly does not follow that she has placed four whole oranges in it. This indirectly suggests that

‘four oranges’, like ‘four glasses of water’, is I/M-ambiguous. I will develop a number of heuristics which demonstrate this more directly in §4.2. For example, Rothstein points out that while only the MI of ‘four glasses of water’ is acceptably paraphrased in terms of ‘-ful’.

Likewise, only the MI of ‘four oranges’ is acceptably paraphrased in terms of ‘worth’.

Here’s the theoretical signiﬁcance of I/M-ambiguities. IIs involve counting glasses qua individuated containers and oranges qua individuated bodies of fruit. MIs, on the other hand, involve measuring quantities of a substance with respect to some non-standardized unit of measurement, or what Partee and Borschev(2012) call an ad hoc measure. The traditional analyses of cardinality expressions naturally predict IIs, while KISS naturally predicts MIs. The trouble is that neither can obviously capture both interpretations. In

97 the case where Mary pours four oranges worth of prepackaged orange pulp directly into the

punch, for instance, the traditional analyses wrongly predict (119a) to be false since there

needn’t be individual oranges in the punch. The problem for KISS is interestingly diﬀerent.

Though it rightly predicts (119a) to be true in this scenario, it appears to wrongly predict

that (119a) should only have a MI, or so we will argue later on. If so, then the problem

with KISS is that it has no clear means of explaining the numerous semantic diﬀerences

between IIs and MIs demonstrated in §4.2.

I will argue that the Answer is also I/M-ambiguous, just like ‘four oranges’. It has an

II paraphraseable as the AnswerI , an MI paraphraseable as the AnswerM , and these are truth-conditionally distinct.

(AnswerI ) There are two oranges on the table, and there is a half-orange on the table.

(AnswerM ) There are two and a half oranges’ worth of orange on the table.

To show this, I will apply the same heuristics mentioned above to the Answer, revealing

it to be I/M-ambiguous. The question thus becomes: How can the Answer come to have

these distinct interpretations?

To answer it, I will ﬁrst need to explain how I/M-ambiguities arise more generally.

This is done in §4.3. On my analysis, following Rothstein(2009) and Scontras(2014),

IIs arise from combining one of the traditional analyses of cardinality expressions with

countable meanings of the relevant expressions, e.g. ‘glass of water’ or ‘orange’. To account

for MIs, we propose a certain type-shifting principle Snyder and Barlew(2016) call “the

Universal Measurer” (UM). This takes a noun like ‘glass’ or ‘orange’ and converts it into

an ad hoc measure on substances of the appropriate sort, e.g. water or orange. As a result,

‘four oranges’ denotes quantities of orange measuring four oranges worth, thus eﬀectively

reproducing Krifka’s semantics. However, because UM does not have an inverse, there is

generally no way to semantically recover the default denotation of ‘orange’ – the set of

individual oranges – from its measure-like denotation resulting from an application of UM.

98 Consequently, there is no way to semantically recover the II from the MI, thus explaining the asymmetric entailment noted earlier.

On my analysis, the AnswerI entails the AnswerM , but not the other way around, thanks to UM. However, the trouble is explaining how the Answer can have an II in the ﬁrst place.

After all, if IIs result from the traditional analyses of cardinality expressions, yet fractional

cardinalities are incoherent on those analyses, then it is hard to see how an II could arise

1 in the ﬁrst place. I will argue that appearances are deceptive here: ‘2 2 ’ in the Answer is not specifying a fractional cardinality of the oranges on the table. Rather, we are actually counting three things on the II of the Answer, namely two whole oranges and a half-orange.

§4.4 is dedicated to defending this claim against certain objections raised by Salmon. I show how combining independently motivated analyses of the component expressions –

‘two’, ‘and’, and ‘a half’ – in a natural way actually predicts that the Answer should have an interpretation paraphraseable as the AnswerI , and more generally that fractions on IIs 2 involve a form of counting. For instance, ‘2 3 oranges’ on the II counts four things – two whole oranges and two more things, namely thirds of an orange.

This in turn aﬀords a neat solution to COP. Though there are strictly speaking only two

oranges on the table in that context, the Answer remains the intuitively correct answer to

the Question because we are individuating the two whole oranges from the half-orange and

counting these three things separately. But since three is a whole number, COP presents

no particular threat to traditional analyses of cardinality expressions.

4.2 Individuating and Measuring

The purpose of this section is to develop four general heuristics for teasing apart individuat-

ing and measure interpretations.4 I begin by applying those heuristics to uncontroversially

I/M-ambiguous container phrases like ‘four glasses of water’, showing how they successfully

4These are developed more completely in Snyder and Barlew(2016).

99 disambiguate between IIs and MIs. I will then apply those same heuristics to the Answer,

revealing it to be similarly ambiguous.

The ﬁrst heuristic has to do with the acceptability of ‘worth’. Rothstein(2009) observes

that the suﬃx ‘-ful’ disambiguates for MIs in container phrases, as revealed by the following

contrast, where (120) is uttered in a measure context and (121) is uttered in an individuating

context.

M-Context: Mary wants to make some soup. The recipe calls for four glasses worth of water.

Accordingly, Mary ﬁlls a certain glass four times with water, pouring the contents

each time into the soup. John says “As I was passing by the kitchen, I noticed there

was some water in the soup. How many glasses of water did you put in the soup?”

(120) Mary: “Four glasses / glassfuls.”

I-Context: Mary wants to heat up some water for coﬀee. Accordingly, she ﬁlls four glasses with

water and places them into the boiling soup. John says “As I was passing by the

kitchen, I noticed some glasses in the soup. How many glasses of water did you put

in the soup?”

(121) Mary: “Four glasses / ??glassfuls.”

As Rothstein explains, the acceptability of ‘glassfuls’ in (120) but not (121) makes sense

if measure contexts impose an MI, individuating contexts impose an II, and ‘-ful’ disam-

biguates for MIs.

Based on this observation, Rothstein proposes that the function of ‘-ful’ is to convert

container nouns like ‘glass’ into ad hoc measures.5 In other words, ‘glassful’ denotes a

measure of a substance like water in terms of a non-standardized glass-unit, e.g. the amount

of water that would ﬁll a certain glass used to pour water into the soup. One of my

contentions here is that ‘-ful’ is a special case of ‘worth’ in this respect. In general, the

5Cf. also Scontras(2014).

100 function of ‘worth’ is to convert a noun into an ad hoc measure. However, unlike ‘-ful’,

‘worth’ is not limited to just container nouns. Rather it can be used with a variety of

nouns, e.g. ‘four oranges’ worth of orange’, ‘four grains’ worth of rice’, or ‘four ounces’

worth of salt’.6

Thus, my ﬁrst heuristic is a natural extension of Rothstein’s: ‘worth’ is generally ac-

ceptable with MIs but not with with IIs, as seen by e.g. the unacceptability resulting from

substituting ‘glasses worth’ for ‘-ful’ in (121). Now consider the following contrast.

1 M-Context: John is on a special diet which requires him to have 2 2 oranges per day. At Smoothie World, smoothies are made with prepackaged orange pulp, and employees know how

much orange pulp a typical orange produces. In order to fulﬁll his dietary require-

ments, John orders an smoothie at Smoothie World. Mary makes John’s smoothie.

John says “I just want to make sure that smoothie will meet my dietary requirements.

How many oranges are in the smoothie?”

(122) Mary: “Two and a half oranges / oranges’ worth.”

I-Context: There are three oranges on the table. Mary takes one of those oranges, cuts it in half,

eats one of the halves, and places the remaining half back on the table. John says “As

I was passing by the kitchen, I noticed there were oranges on the table. How many

oranges are on the table?”

(123) Mary: “Two and a half oranges / ??oranges’ worth.”

As before, the facts in (122) and (123) make sense if the M-Context imposes an MI of the

Answer, the I-Context imposes an II of the Answer, and ‘worth’, like ‘-ful’, disambiguates for

MIs. Note that the I-Context here is just Salmon’s context for COP, thus suggesting that

6 In ‘four oranges worth of orange’, for instance, ‘worth’ converts ‘orange’ into an ad hoc measure on quantities of orange. However, in a context in which various fruits are being washed, it is possible to use ‘four oranges worth of water’ to measure amounts of water in terms of how much would be used to wash a single orange. Likewise with e.g. ‘four glasses worth of (dish) water’.

101 COP imposes an individuating interpretation of the Answer. The additional diagnostics

developed here will conﬁrm this.

The second diagnostic involves pluralization. As Rothstein points out, container phrases

are acceptable with plural pronouns on IIs but not on MIs.

M-Context: Same as (120). Pointing at the soup, Mary says:

(124) a. ?? Those are four glasses of water.

b. That is four glasses of water.

I-Context: Same as (121). Pointing at the soup, Mary says:

(125) a. Those are four glasses of water.

b. That is four glasses of water.

The plural pronoun ‘those’ refers to a plurality consisting of individuated objects, and we

have a plurality of individuated glasses in the I-context but not in the M-context. Hence

the diﬀerence in acceptability between (124a) and (125a). On the other hand, the singular

pronoun ‘that’ is acceptable in both contexts, plausibly for diﬀerent reasons. On the II,

we can think of the four glasses as constituting a single group,7 thus making a referent for

‘that’ available. On the MI, we have a quantity of water measuring four glassfuls, and this

supplies an appropriate referent for ‘that’.

We see a similar contrast with the Answer.

M-Context: Same as (122). Pointing at the smoothie, Mary says:

1 (126) a. ?? Those are 2 2 oranges.

1 b. That is 2 2 oranges.

I-Context: Same as (123). Pointing at the table, Mary says:

7See Landman(2004).

102 1 (127) a. Those are 2 2 oranges.

1 b. That is 2 2 oranges.

Again, this diﬀerence in acceptability makes sense if ‘those’ refers to some salient plurality

consisting of individuated objects. There is such a plurality in the I-Context but not in the

M-Context, so there is something for ‘those’ to refer to in (127) but not in (126). And both

interpretations are acceptable with ‘that’ since there is an appropriate referent available in

both cases.

The third diagnostic involves the acceptability of modiﬁers like ‘approximately’ and

‘roughly’, or what Lasersohn(1999) calls slack regulators. These are generally acceptable with MIs, but not always with IIs, especially when the cardinality of the plurality in

question is small.8

M-Context: Similar to (120), except that Mary pours water straight from the tap into the soup.

She says:

(128) There are approximately four glasses of water in the soup.

I-Context: Same as (121). Mary says:

(129) ?? There are approximately four glasses of water in the soup.

The diﬀerence in acceptability here is plausibly due to an implicature carried by the use of

‘approximately’. In general, ‘approximately’ implicates that the speaker is unsure whether

the amount speciﬁed is the exact amount which actually obtains. For instance, Mary’s

utterance of (128) implicates that she is unsure whether the amount of water poured into the

soup measures exactly four glassfuls.9 This sort of uncertainty is normal with measurement,

8Slack regulators are generally acceptable with larger cardinalities where an exact measure is not so easily determined. For example, if Mary has a very large vat of soup and she has lost track of how many glasses she has placed in it, it is perfectly acceptable for her to say ‘There are approximately twenty glasses of water in the soup’. 9This is plausibly a quantity implicature (see Grice(1989)). Since ‘exactly four glasses of water’ entails ‘approximately four glasses of water’ but not vice versa, the former is strictly speaking more informative.

103 at least to a certain degree of precision. However, if Mary just placed four glasses ﬁlled

with water into the soup, then there would appear to be little room left for uncertainty as

to how many such glasses there are.

Once again, we see the same contrast in the Answer.

M-Context: Same as (122). Pointing at the smoothie, Mary says:

1 (130) There are approximately 2 2 oranges in the smoothie.

I-Context: John hates doing math problems but loves oranges. Mary oﬀers him the following

deal: for each math problem he solves, he will receive half an orange. So far, John

has solved ﬁve math problems. Robin wants to make sure that John completes his

homework, and so asks “How many oranges has John earned so far?”. Mary responds:

1 (131) ?? John has earned approximately 2 2 oranges.

1 Again, the diﬀerence here makes sense if ‘2 2 ’ is functioning as a measure on the MI, but is specifying a cardinality on the II. Whereas it is perfectly sensible for Mary to estimate

when measuring the amount of orange going into a smoothie, if John has solved exactly ﬁve

math problems, then there would appear to be little room for uncertainty as to how many

oranges he has earned up to this point.10

The ﬁnal heuristic given here involves the nouns ‘number’ and ‘amount’. These can be

used to tease apart IIs and MIs directly, as witnessed by (132) and (133).

Context: John and Mary both want to heat some water for coﬀee, and both do it by placing

glasses ﬁlled with water into boiling soup. However, John’s glasses are exactly half the

So if Mary knew there were exactly four glasses of water in the soup, she would have said so (assuming she’s cooperative). Since she hasn’t said that, she must not know whether there are exactly four glasses of water in the soup. 10 1 To be clear, Mary could acceptably utter ‘There are approximately 2 2 oranges on the table’ in the context of COP. After all, Mary may reasonably doubt that she has cut one of the oranges exactly in half. However, this is plausibly due to imprecision in the fraction word ‘half’ (see Lasersohn(1999)), thus prompting our adjusted I-Context for (131). I will return to the meanings of fraction words in §4.2.

104 size of Mary’s. Both place four of their glasses ﬁlled with water into their respective

soups.

(132) a. There are four glasses of water in Mary’s soup, and there are the same number

of glasses in John’s soup. (true)

b. There are four glasses of water in Mary’s soup, and there is the same amount

of water in John’s soup. (false)

Context: Same as for (132), only John places eight of his glasses into his soup.

(133) a. There are four glasses of water in Mary’s soup, and there are the same number

of glasses in John’s soup. (false)

b. There are four glasses of water in Mary’s soup, and there is the same amount

of water in John’s soup. (true)

This makes sense if ‘number’ relates pluralities speciﬁcally to their cardinalities, while

‘amount’ relates substances to their measures more generally, including but not limited

to e.g. volume.11 Thus, (132a) is true since Mary’s glasses and John’s glasses have the

same cardinality, namely four. But since those glasses contain diﬀerent volumes of water,

(132b) is false. We see the reverse in (133). Since IIs involve counting the individuated

members of a plurality but MIs involve measuring a substance according to some ad hoc

measure, these examples directly reveal ‘four glasses of water’ to be I/M-ambiguous.

We see the same pattern with the Answer.

Context: Similar to (123), except that John has oranges which are exactly half the size of Mary’s

oranges. Like Mary, he has three oranges, cuts one in half, eats one of the halves, and

leaves the other half on his table. 11See Scontras(2014) on ‘amount’. Note that ‘amount’ is ambiguous in a way ‘number’ is not. Suppose, for instance, that we are looking at four 1 lb. oranges. As Scontras suggests, ‘that amount of oranges’ can then refer to two sorts of degrees, namely a weight (4 lbs.) or a cardinality (consisting of four singular oranges). On the other hand, ‘that amount of orange’ can only refer to a weight, while ‘that number of oranges’ can only refer to a cardinality.

105 1 (134) a. There are 2 2 oranges on Mary’s table, and there are the same number of oranges on John’s table. (true)

1 b. There are 2 2 oranges on Mary’s table, and there is the same amount of orange on John’s table. (false)

Context: Same as (134), except that John leaves ﬁve of his oranges on his table.

1 (135) a. There are 2 2 oranges on Mary’s table, and there are the same number of oranges on John’s table. (false)

1 b. There are 2 2 oranges on Mary’s table, and there is the same amount of orange on John’s table. (true)

As before, these examples make sense if ‘number’ here relates pluralities of oranges to their

cardinalities, while ‘amount’ relates quantities of orange to their volume. Like with ‘four

glasses of water’, these examples directly reveal the Answer to be I/M-ambiguous.

Altogether, these heuristics reveal two important facts. First, I/M-ambiguities are not

limited to just container phrases like ‘glass of water’. This is theoretically signiﬁcant, as the

literature on I/M-ambiguities has tended to focus exclusively on container phrases, suggest-

ing perhaps that I/M-ambiguities are limited only to them. However, even ordinary count

nouns like ‘orange’ can give rise to IIs and MIs. Secondly, the Answer is I/M-ambiguous,

and COP imposes an individuating interpretation of the Answer. The ﬁrst observation

raises the following empirical question: If I/M-ambiguities are not due speciﬁcally to the

meanings of container nouns like ‘glass’, then how do I/M-ambiguities arise more gener-

ally? I will argue in the next section that these are made possible by a certain type-shifting

principle called “the Universal Measurer”.

106 4.3 The Universal Measurer

We have seen that I/M-ambiguities are not limited to just numerically modiﬁed container phrases like ‘four glasses of water’. We see similar I/M-ambiguities in ‘four oranges’, for instance. The purpose of this section is to provide a general account of I/M-ambiguities.

More speciﬁcally, I want to extend extant analyses of I/M-ambiguities in a natural way to explain how both ‘four glasses of water’ and ‘four oranges’ can have IIs and MIs.

Let’s begin with I/M-ambiguous container phrases like that in (118a).

(118a) There are four glasses of water in the soup.

The question is how to derive both the II and the MI in a compositional manner. Following

Scontras(2014), I assume that ‘glass’ is lexically a monadic predicate true of individual glasses, or (136).

(136) glass = λx. glass(x) J K However, ‘glass’ takes on a distinctively relational character in container phrases: on the

II, ‘glass’ in ‘glass of water’ roughly means ‘glass containing water’, for instance. I follow

Rothstein(2009) in assuming that a certain type-shifting principle is responsible, one she calls the Construct State Shift (CSS). This is given in (137a), where P is a noun like ‘glass’, Q is a noun like ‘water’, and R is a contextually-supplied relation holding between members of P and Q.(137b) gives the result of applying CSS to the monadic meaning of

‘glass’ in (136) and combining the result with ‘of water’.12

(137) a. λP.λQ.λx.∃y. P (x) ∧ Q(y) ∧ R(x, y) (CSS)

b. CSS( glass of water ) = λx.∃y. glass(x) ∧ water(y) ∧ R(x, y) J K In essence, CSS transforms a monadic noun like ‘glass’ into a relational noun denoting

glasses which bear R to something else. In the case at hand, the relation determined is that

12Following e.g. Rothstein(2009), I assume for expository convenience here that ‘of’ contributes no semantic content. I drop this assumption in §4.4.2, however.

107 of being ﬁlled with, and the “something else” in question is water. Ultimately, then, we get a predicate true of those glasses which are ﬁlled with quantities of water, or (137b).

From here, it is straightforward to derive the II of (118a), repeated in (118b).

(118b) There’s a plurality of glasses x s.t. x consists of four individuals, each of which is

ﬁlled with water and is in the soup.

We simply apply one of the traditional analyses of cardinality expressions. For concreteness, assume the Adjectival Theory from Chapter 2. Then because it occurs without an overt measure term, ‘four’ in ‘four glasses of water’ denotes a cardinality modiﬁer, thus resulting in the denotation of ‘four glasses of water’ in (138).

(138) four glasses of water = λx. µ#(x) = 4 ∧ glasses(x) ∧ water(y) ∧ R(x, y) J K Consequently, (118a) will be true just in case there’s at least one such plurality of glasses

in the soup. And this, of course, is the II of (118a).

For the MI, Scontras proposes the type-shifting operation in (139a) – he calls it “SHIFTC−M ” – where k is a kind in the sense of Chierchia(1998) and ‘ ∪’ is Chierchia’s “up”-operator returning the instances of a kind.

∪ (139) a. λP.λk.λn.λx. k(x) ∧ µP (x) = n (SHIFTC−M )

∪ b. SHIFTC−M ( glass ) = λk.λn.λx. k(x) ∧ µglass(x) = n J K ∪ c. four glasses of water = λx. WATER(x) ∧ µglass(x) = 4 J K

In eﬀect, SHIFTC−M transforms a monadic noun like ‘glass’ into a measure term like

13 ∪ ‘ounce’. In (139b), “µglass” is an ad hoc measure, and “ WATER” in (139c) denotes quantities of water. Hence, according to (139c), ‘four glasses of water’ denotes quantities of

water measuring four ad hoc glass-units. This predicts correct truth-conditions for (118a)

on the MI: it will be true if there are four glasses’ worth of water in the soup.

13See Scontras(2014) for how these diﬀerent kinds of expressions are related.

108 But now consider (140). It too involves an I/M-ambiguous container phrase, namely

‘four boxes of tires’.

Context: John and Mary work in a recycling plant specializing in recycling two items: aluminum

cans and tires. Once the items are ground, they are sifted and the recycled material

is packaged into boxes. John and Mary ﬁnd some unmarked boxes, and there is some

uncertainty as to what they contain. After dumping the contents of the boxes into

two sorted piles, Mary points at one of the piles and says:

(140) That’s four boxes of {tires/tire}.

Obviously, a MI is intended here: Mary is talking about four boxes’ worth of tires, not four

boxes containing tires. Moreover, (140) plausibly receives a “grinding” interpretation, in

the sense of Pelletier(1975). 14 For one thing, Mary is pointing at ground up bits of tire,

and so she could have just as well used the massivized noun ‘tire’ in the scenario described.

Moreover, (140) isn’t false just because Mary failed to point at piles of whole tires, contrary

15 to what is predicted by SHIFTC−M . Happily, there is a fairly simple ﬁx available. Following Snyder and Barlew(2016), I

propose generalizing Scontras’ analysis of MIs via the alternative type-shifting operation in

(141a), called the Universal Measurer (UM).16

(141) a. λP.λQ.λn.λx. γ(Q(x)) ∧ µP (x) = n (UM)

b. γ(P ) = {x| ∃y. P (y) ∧ x v y}

c. four boxes of tires = λx. γ(tires(x)) ∧ µbox(x) = 4 J K 14I will discuss grinding interpretations further in Chapter 5. 15According to Chierchia(1998)’s analysis of mass nouns, applying the ∪-operator to the kind TIRE returns the set consisting of all singular tires plus the pluralities formed from them. Crucially, however, this denotation does not include bits of tire-matter, and so (139a) wrongly predicts that (140) should be false in the context provided. 16In homage to Pelletier(1975)’s famous Universal Grinder and Universal Packager. UM is plausibly the meaning of ‘worth’, though suitably restricted in various ways.

109 The γ-operator in (141b) is a grinding-operation similar to ones proposed by Link(1983),

Rothstein(2010), and Landman(2011). It takes the members of the extension of a predicate

P and returns their parts, represented here as ‘v’. Consider the set of pluralities of tires, for instance. These consist of individual, whole tires. But the latter also have parts – all of the individual bits of rubber and metal constituting them, for instance. Now, let’s assume that γ returns the set of all such parts. Then according to (141c), the result of applying

UM to ‘box’ and combining the result with the denotations of ‘of tires’ and ‘four’ (in that order) is a predicate true of those parts of pluralities of tires which measure four boxes’ worth. Accordingly, Mary’s utterance of (140) would be true if she were pointing at a pile of whole tires or a pile of assorted bits of tire, so long as the tire-matter she is pointing at collectively measures four boxes’ worth.

But what about I/M-ambiguous non-container phrases, e.g. (119a)?

(119a) There are four oranges in the punch.

Can the II and MI for these be derived in a similar manner? The answer is “Yes”, but with a couple important caveats. Let’s begin with the II. Like ‘box’, ‘orange’ is inherently monadic. Thus, we may plausibly assume that ‘orange’ denotes the set of oranges, or (142).

(142) orange = λx. orange(x) J K Unlike ‘box’, however, ‘orange’ is not relational on IIs. In other words, we count oranges directly qua bodies of individuated fruit, not qua containers of orange. Since this is precisely the meaning of ‘orange’ assumed in (142), we can derive IIs for ‘four oranges’ by combining this denotation directly with one of the traditional analyses of cardinality expressions. For example, the Adjectival Theory correctly predicts the following truth-conditions: (119a) will be true just in case there is a plurality of four oranges, each of which is in the punch.

(119b) There’s a plurality of oranges x s.t. x consists of four individuals, each of which is

in the punch.

110 And this is precisely the II repeated in (119b).

In order to derive the MI in (119c), Snyder and Barlew propose “reﬂexivizing” UM.

(119c) There’s a quantity of orange x s.t. x measures four oranges worth and x is in the

punch.

As the paraphrase suggests, it seems that ‘orange’ needs to supply both the ad hoc measure and the substance being measured on MIs: we are measuring four oranges’ worth of orange.

Intuitively, what’s needed then is a way of guaranteeing that the ﬁrst two arguments of UM are identical. We see something similar with relational verbs like ‘bathe’ when occurring without overt objects. Witness (143b), for instance.

(143) a. John bathed the baby.

b. John bathed.

Unlike (143a), (143b) can only be understood reﬂexively, i.e. as claiming that John bathed himself. Similarly, the MI of ‘four oranges’ doesn’t measure four oranges’ worth of just anything, but only of orange.

According to Reinhart and Siloni(2005), the reﬂexivization of ‘bathe’ can be seen as the consequence of applying a certain type-shifting function – call it “VREF” – to the basic relational denotation of ‘bathe’ in (144a).

(144) a. bathe = λx.λy. bathe(x, y) J K b. λR.λx. R(x)(x) (VREF)

c. VREF( bathe ) = λx. bathe(x, x) J K In eﬀect, VREF resets the arguments of ‘bathe’ to be identical, thus reﬂexivizing the verb.

Snyder and Barlew assume that VREF is one instance of a generalized reflexivization principle, one which takes relations of various types and reflexivizes them. Thus, they assume that another particular instance of this generalized reflexivization principle is “REFL” in

(145a), where ‘Q’ is an expression having the same type as UM.

111 (145) a. λQ.λP.λn.λx. Q(P )(P )(n)(x) (REFL)

b. REFL(UM) = λP.λn.λx. γ(P (x)) ∧ µP (x) = n (RUM)

Similar to (144c), applying REFL to UM eﬀectively resets its ﬁrst two arguments to be iden-

tical, thus resulting in (145b), or what is called the Reflexivized Universal Measurer (RUM).

RUM eﬀectively guarantees that the substance being measured, i.e. “γ(P )”, is of the

same kind as the ad hoc measure being employed, i.e. “µP ”. Thus, it does precisely what we need it to do: it allows us to measure quantities of orange in terms of an ad hoc orange-

unit. Applying RUM to the monadic meaning of ‘orange’ from above results in (146a), thus

generating the meaning of ‘four oranges’ in (146b).

(146) a. RUM( orange ) = λn.λx. γ(orange(x)) ∧ µorange(x) = n J K b. four oranges = λx. γ(orange(x)) ∧ µorange(x) = 4 J K Applying the grinding-operation γ to the extension of ‘orange’ returns all parts of all singular

oranges, including the oranges themselves. Thus, according to the resulting denotation in

(146b), ‘four oranges’ will denote those orange-parts measuring four ad hoc orange-units,

and so (119a) will be true just in case there is an amount of orange in the punch measuring

four oranges’ worth. And this, of course, is just the MI paraphrased in (119c).

The resulting analysis reveals the ﬂaws in both the traditional analyses of cardinality

expressions and KISS. The traditional analyses wrongly predict that the MI of (119a) should

be false in the smoothie scenario since there needn’t be four individual oranges in the

smoothie. KISS rightly predicts that (119a) should be true in that same scenario, but

because it appears to predict that (119a) should only have a MI (see §4.5), KISS has no

obvious way of semantically explaining the various semantic contrasts between MIs and IIs

noted in §4.2. In contrast, on the present view ‘four oranges’ does not uniformly count

oranges or measure quantities of orange in terms of ad hoc units of orange. Rather, it can

do both, and which semantic function it performs depends wholly on context. In sum, both

112 the traditional analyses and KISS capture a genuinely available interpretation of e.g. ‘four

oranges’, but a completely adequate semantics would relate those diﬀerent interpretations

in a natural way. This is exactly what UM does.

1 4.4 How to Count 22 Oranges

We saw that the Answer is I/M-ambiguous in §4.2. The primary question here is whether the

individuating and measure interpretations of the Answer are derivable in the same manner

suggested previously for ‘four oranges’. It should be fairly evident how the MI arises.

(AnswerM ) There are two and a half oranges worth of orange on the table.

Just as with the MI of ‘four oranges’, we apply the Reﬂexivized Universal Measurer to

‘orange’, thus shifting it into a reﬂexivized measure term. As a result, truth-conditions for

the Answer witnessed in the AnswerM are predicted: it will be true just in case there is a quantity of orange on the table measuring two and a half oranges worth. This would be true

1 if, for instance, there happened to be 2 2 oranges worth of orange slices, each from possibly diﬀerent oranges, on the table.

However, this is not the interpretation of the Answer naturally suggested by Salmon’s

puzzle. The diagnostics in §4.2 show that Salmon’s context for COP is an individuating

context, and therefore imposes an II of the Answer. IIs involve individuating members of

pluralities, e.g. glasses containing water or boxes containing tires. In the context of COP, the

1 relevant plurality is the one denoted by ‘the 2 2 oranges on the table’. This plurality consists of three individuals, I submit: two whole oranges and one half-orange. Put diﬀerently, ‘half’

in ‘two and a half oranges’ is functioning as a modiﬁer of ‘orange’, so that ‘two and a half

oranges’ is really being interpreted as ‘two oranges and a half-orange’.

This leads to the II of the Answer, or the AnswerI .

(AnswerI ) There are two oranges on the table, and there is a half-orange on the table.

113 1 If so, then ‘2 2 ’ isn’t specifying a fractional cardinality on the II of the Answer after all. Rather, it’s functioning as a complex modiﬁer, similar to ‘large and small apples’ in (147a).

(147) a. There are large and small apples on the table.

b. There are large apples on the table, and there are small apples on the table.

Like the Answer on the II, (147a) can be paraphrased as a conjunction of modiﬁed noun phrases. And just as one would not reasonably conclude from (147b) that there are apples on the table which are both large and small, one would not reasonably conclude from the

AnswerI that there are oranges on the table numbering both two and also one half. If this informal characterization is correct, then COP presents no special threat to the

1 traditional analyses of cardinality expressions. It would be threatening only if ‘2 2 ’ in the Answer speciﬁed a fractional cardinality. But the claim here is that it fails to do so on either interpretation.

Anticipating this sort of response, perhaps, Salmon(1997) argues the Answer I is not a genuine interpretation of the Answer. More speciﬁcally, he challenges our claim that ‘two and a half’ can function as a complex modiﬁer. To quote him:

The numeral ‘2’ occurring in [the Answer] has been separated from its accom-

panying fraction, and now performs as a solo quantiﬁer. The fraction itself has

been severely mutilated. The numeral ‘1’, which appears as the fraction’s nu-

merator in [the Answer], has ascended to the status of anonymous quantiﬁer,

functioning independently both of its former denominator and of the quanti-

ﬁer in the ﬁrst conjunct. At the same time, the word ‘half’ appearing in [the

AnswerI ] has been reassigned, from quantiﬁer position to predicate position. In 1 eﬀect, the mixed number expression ‘2 2 ’, occurring as a unit in [the Answer], has been blown to smithereens, its whole integer now over here, the fraction’s

numerator now over there, the fraction’s denominator someplace else... Even

114 the schoolboy knows that the phrase ‘and a half’ in [the Answer] goes with the

‘two’ and not with the ‘orange’.17

Colorful rhetoric aside, I take Salmon to be making a fairly simple but substantial point

1 here, namely that it is far from obvious how ‘2 2 ’ functions as a complex modiﬁer, as is required if the AnswerI is genuinely available. If it does not, i.e. if “the phrase ‘and a half’ in [the Answer] goes with the ‘two’ and not with the ‘orange”’, then it is hard to see how the AnswerI could arise compositionally from ‘two and a half oranges’. Thus, I interpret

Salmon’s remarks here as posing a serious challenge: compositionally derive the AnswerI from independently plausible meanings of the relevant component expressions.

I will address Salmon’s challenge head on in what follows. I will show that it is possible

to compositionally derive the AnswerI from independently motivated analyses of the component expressions, namely ‘two’, ‘and’, and ‘a half’. I address the semantic contribution of ‘and’ via Krifka(1990)’s analysis of generalized cumulative conjunction in §4.1. The

contribution of ‘half’ is addressed via Ionin et al.(2006)’s analysis of fraction words in §4.2.

I derive the AnswerI by combining these analyses with the Adjectival Theory in §4.3. I end this section in §4.4 by responding to two further objections raised by Salmon.

4.4.1 Generalized Cumulative Conjunction

Let’s begin with the semantic contribution of ‘and’ in ‘two and a half oranges’. As with

‘large and small apples’, I assume that ‘and’ in the Answer is an instance of cumulative conjunction. Consider the following examples from Krifka(1990).

(148) a. John and Mary met at the mall.

b. This is beer and lemonade.

c. That ﬂag is green and white.

17Salmon (1997, p. 6).

115 (148a) cannot mean that John met at the mall and Mary met at the mall, just as (148b) cannot mean that this liquid concoction is beer and it is lemonade. That’s because the various occurrences of ‘and’ here are witnesses to cumulative conjunction, as opposed to propositional conjunction.18 What’s more, because the different instances of ‘and’ coordi- nate expressions of different semantic types, they must themselves take on different types in those different syntactic environments, thus mirroring generalized propositional conjunction on the treatment of Partee and Rooth(1983). Thus, what’s needed is a theory of generalized cumulative conjunction.

Krifka supplies one such theory. Adapting his analysis of coordinated modiﬁers like

‘green and white’ in (148c) and simplifying somewhat, Krifka’s semantics predicts that

(149a) should have the logical form in (149b) or (149c), depending on whether ‘and’ is understood propositionally or cumulatively.

(149) a. There are large and small apples on the table.

b. ∃x. large(apples(x)) ∧ small(apples(x)) ∧ on-table(x)

c. ∃x.∃y.∃z. x = y ⊕ z ∧ large(apples(y)) ∧ small(apples(z)) ∧

on-table(x)

Here, ‘⊕’ is a join-operation on individuals; in general, if a and b are individuals, then ‘a⊕b’

denotes the plurality consisting of a and b. The incoherent truth-conditions given in (149b)

result if ‘and’ is understood as propositional conjunction, in which case (149a) is true if

there are apples on the table which are both large and small. The coherent truth-conditions

given in (149c) result if instead ‘and’ is understood cumulatively, in which case (149a) is

true if there are apples on the table, parts of which are large and parts of which are small.

My primary contention here is that ‘and’ in ‘two and a half oranges’ in the Answer, like

‘and’ in (149a), is only plausibly interpreted cumulatively.

18Also known as “boolean” and “non-boolean” conjunction. See Partee and Rooth(1983) for related discussion.

116 4.4.2 Fractions

To account for the meanings of fraction words such as ‘half’, ‘quarter’, ‘third’, etc., I adopt the analysis of Ionin et al.(2006) (henceforth IM&R). IM&R model numerators as cardinality expressions, while denominators “package” parts of things into individuals which numerators then count: halves, quarters, thirds, etc. For example, ‘two thirds of an orange’ counts certain proper parts of an orange, namely the thirds. Similarly, ‘two thirds of the oranges’ counts certain proper parts of a plurality of oranges, namely the thirds. In either case, fractions like ‘one half’, ‘two thirds’, etc. essentially involve counting parts of a whole.

According to IM&R, ‘third’ has a fairly complicated meaning consisting of three components. I separate those components here for convenience.

(150) third = λP. λx. ∃y. J K i. P (y) ∧ ∀z[P (z) → z v y] ∧

ii. ∃S [Π(S)(y) ∧ |S| = 3 ∧ x ∈ S ∧

iii. ∃µ [µ ∈ M ∧ ∀s1, s2 [(s1 ∈ S ∧ s2 ∈ S) → (µ(s1) = µ(s2))]]]

Ultimately, ‘third’ is a predicate-modiﬁer: it takes a predicate like ‘of an orange’ or ‘of

the oranges’ and returns a property true of the relevant parts. Suppose for concreteness

that ‘third’ modiﬁes ‘of the oranges’, and assume with Ladusaw(1982) that ‘of’ denotes a

mereological relation, al´a(151a).

(151) a. of = λx.λy. y v x J K b. of the oranges = λy. y v ιx[oranges(x)] J K Then ‘of the oranges’ plausibly denotes a predicate true of the parts of some uniquely salient

plurality of oranges, as suggested in (151b). According to (150), ‘third of the oranges’

expresses a property of these parts, and clauses (i)-(iii) determine which property that is.

More speciﬁcally, clause (i) guarantees that there is a maximal part of the oranges,

namely the entire plurality. Clause (ii) guarantees that there is a partitioning Π of that

117 plurality into three countable, non-overlapping parts.19 The non-overlap condition ensures that we do not count the same part twice. Finally, clause (iii) guarantees that the partitioning of the oranges is quantity uniform: the three parts partitioned by Π all measure the same amount, according to some measure µ in the domain of contextually-determined measures M. In the present case, the contextually-relevant measure is most likely cardinality. Hence, ‘third of the oranges’ is a predicate true of parts of the oranges which, taken together, divide them into three non-overlapping parts having the same cardinality. For example, suppose we are talking about a plurality of twelve oranges. Then the property expressed by ‘third of the oranges’ will hold of those subpluralities which divide the total plurality into three evenly numbered parts. In other words, it will hold of precisely those subpluralities of four oranges.

The key element of IM&R’s semantics is the partitioning function Π. In eﬀect, this

“packages” substances into subparts. Because those subparts are non-overlapping and quantity-uniform, we can count them. In this way, fraction words on IM&R’s account bear a striking resemblance to Scontras(2014)’s category of atomizer nouns such as ‘grain’ and ‘quantity’, which he summarizes this way:

Atomizers (e.g. quantity) are partitioning functions; they compose with a sub-

stance noun and return an atomic (i.e. non-overlapping) set of objects, suscep-

tible to counting.20

The semantic function of ‘grain’, for instance, is to partition a substance like rice into countable units, namely grains of rice. Likewise, ‘quantity’ in ‘a large quantity of the oranges’ partitions the oranges into countable subpluralities. Given this striking resemblance,

I hypothesize that fraction words just are atomizer nouns.

19Π(S)(y) makes S the set of elements of the partition of y. The way that Π and S are deﬁned ensures that S includes all of y and that elements of S do not overlap. See IM&R (2006, p. 318) for details. 20Scontras (2014, p. 153).

118 A prediction of the resulting analysis is that we ought to be able to individuate thirds

of an orange, just as we are able to individuate grains of rice or slices of an orange. This

prediction is borne out in individuating-contexts like the following:

I-Context: There is one orange on the table. Mary cuts it into thirds, eats one of them, and places

the remaining two back on the table. John glances at the table and asks “What kind

of fruit is on the table?”

(152) Mary (pointing at the table): “Those are two thirds of an orange.”

2 Of course, given the general analysis of cardinality sketched above, I predict that ‘ 3 of an orange’ ought to also have a measure interpretation. Hence the contrast witnessed in (153).

2 M-Context: John is on a special diet which requires him to have 3 of an orange per day. At Smoothie World, smoothies are made with prepackaged orange pulp, and employees

know how much orange pulp a typical orange produces. In order to fulﬁll his di-

etary requirements, John orders an smoothie at Smoothie World. Mary makes John’s

smoothie. John says “I just want to make sure that smoothie will meet my dietary

requirements. How many oranges are in the smoothie?”

(153) ?? Mary (pointing at the smoothie): “Those are two thirds of an orange.”

As before, this is to be expected if measure contexts force MIs, at least given the availability

of the Universal Measurer.

4.4.3 Putting it All Together

To show how to derive the AnswerI from the analyses just sketched, I will build the meaning of ‘two and a half oranges’ from the component parts. First, I assume the Adjectival

Theory analysis of cardinality modiﬁers, whereby ‘two’ expresses a cardinality property of

pluralities. This is abbreviated in (154a) as “two”. Secondly, I assume Ionin et al.(2006)’s

119 analysis of fraction words, whereby ‘half’ is a modiﬁer expressing a property of parts of countable individuals. This is abbreviated in (154b) as “a-half”, ignoring the semantic contribution of ‘a’ for expository convenience.

(154) a. two = λP.λx. two(P (x)) J K b. a half = λP.λx. a-half(P (x)) J K The important thing to notice is that ‘two’ and ‘a half’ have the same semantic type. As

such, they can be coordinated by ‘and’ via generalized cumulative conjunction.

Adopting the same analysis of ‘large and small apples’ from above then predicts the

denotation for ‘two and a half oranges’ given in (155).

(155) two and a half oranges = λx.∃y.∃z. x = y ⊕ z ∧ two(orange(y)) ∧ J K a-half(orange(z))

In words, ‘two a half oranges’ will be true of those pluralities consisting of two parts,

namely two oranges and a half-orange. Finally, I assume that ‘there is’ denotes existential

quantiﬁcation and that ‘on the table’ is a distributive predicate, meaning that if it applies to a plurality, then it also applies to all parts of that plurality.21 As a result, the

Answer will have the logical form given in (156).

(156) ∃x.∃y.∃z. x = y ⊕ z ∧ two(orange(y)) ∧ a-half(orange(z)) ∧ on-table(x)

Unpacking the meanings given for ‘two’ and ‘a half’ from above, our ﬁnal truth-conditions

for the Answer can be informally stated as follows: it will be true just in case there is a

plurality x consisting of two parts y and z, where y is a sub-plurality consisting of two

individual oranges and z is an individual half-orange, and each member of x is on the table.

The distributivity of ‘on the table’ ensures that each of the things making up the plurality

y ⊕ z satisﬁes the predicate. It’s for this reason that (149a) is only plausibly understood as

21See Link(1983).

120 (149c). It’s also for this reason that the Answer is paraphraseable as the AnswerI , at least given the logical form in (156).

1 (Answer) There are 2 2 oranges on the table.

(AnswerI ) There are two oranges on the table, and there is a half-orange on the table.

I want to emphasize that the semantics for complex fractions developed here is completely

general. For example, (157a) is predicted to have the logical form in (157b).

2 (157) a. There are 2 3 oranges on the table. b. ∃x.∃y.∃z. x = y ⊕ z ∧ two(orange(y)) ∧ two(thirds(orange(z)))

∧ on-table(x)

2 Crucially, the numerator in ‘ 3 ’ is understood as a cardinality modiﬁer here: it counts two things, namely thirds of an orange. Thus, (157a) will be true if there is a plurality y ⊕ z

such that y is a sub-plurality consisting of two whole oranges, z is a sub-plurality consisting

of two individuals, namely thirds of an orange, and each member of y ⊕ z is on the table.

This is, of course, the individuating interpretation of (157a). In other words, it would be

2 1 the AnswerI were COP formulated using ‘2 3 oranges’ rather than ‘2 2 oranges’.

To summarize, I have compositionally derived the II of the Answer, i.e. the AnswerI , by combining independently motivated semantic analyses of the component expressions.

And this, I take it, discharges Salmon’s challenge of deriving the AnswerI in a principled,

motivated way. Still, Salmon presents two further objections to the claim that the AnswerI is a genuine interpretation of the Answer. What remains of this section is dedicated to

presenting those objections and answering them.

4.4.4 Two Further Objections

I have argued that the AnswerI is a predictable consequence of independently motivated

analyses of ‘two’, ‘and’, and ‘a half’. Nevertheless, Salmon maintains that the AnswerI

121 cannot be a genuine interpretation of the Answer, for two additional reasons. Salmon ﬁrst

charges that the the AnswerI gives the wrong interpretation of the Answer in the context

of COP. Clearly, the AnswerI entails both of its conjuncts, including the Answer2.

(Answer2) There are two oranges on the table.

Thus, if the the AnswerI were true in the context of COP, then the Answer2 would be too. However, this is “subject to direct disproof”, Salmon tells us.

For [the AnswerI ] to be a correct answer to [the Question], it would have to

be true. And this would require its ﬁrst conjunct, [the Answer2], to be true. Now for any pair of numbers n and m, if there are exactly n F ’s, ... and

1 also exactly m F ’s, ... then n and m must be exactly equal. But 2 6= 2 2 .

The two alternatives, [the Answer] and the [the Answer2], are not teammates

but competing rivals. Hence, since it entails [the Answer2], if [the AnswerI ] is a correct answer to [the Question], then [the Answer] is not. In a word,

[the Answer] and [the AnswerI ] are incompatible. Therefore, the latter cannot provide the correct analysis of the former.22

The argument is just this: since the the AnswerI entails the Answer2 but the Answer2 is

incompatible with the intuitively correct Answer in the context of COP, the AnswerI cannot be the correct interpretation of the Answer in that context.

To understand the force of Salmon’s objection, it will help to contrast the present anal-

ysis with the Fregean analysis. According to the latter, the Answer2 should be equivalent to (158a) and the Answer to (158b).

(158) a.# x[orange-on-table(x)] = 2

1 b.# x[orange-on-table(x)] = 2 2 22Salmon (1997, p. 7).

122 1 Since ‘2’ and ‘2 2 ’ should refer to cardinal numbers on the Fregean analysis, and since these are distinct numbers, (158a) and (158b) are not equivalent. More importantly, they appear to be incompatible. If the number of oranges on the table is exactly two, then it would seem

1 that the number of oranges on the table cannot also be exactly 2 2 . But since the Answer is true in the context of COP, then the Answer2 must be false in that context. Thus, the

AnswerI cannot be the correct analysis of the Answer in the context of COP since it would entail something false.

In response, I note ﬁrst that the Answer2 is in fact true in the context of COP. Although it would be somewhat pedantic, one can felicitously and truthfully respond to the Question with (159).

(159) Well, strictly-speaking there are (exactly) two oranges on the table. By the way,

there’s also (exactly) one half-orange on the table.

This is to be expected, of course, if COP imposes an II of the Answer, and the latter consists in counting two oranges and one half-orange. Now, according to Salmon’s objection, the

AnswerI cannot be the correct interpretation of the Answer in the context of COP because it entails the Answer2, and the latter is not equivalent to the Answer. But consider the corresponding entailment from (160a) to (160b).

(160) a. There are large and small apples on the table.

b. There are large apples on the table.

Suppose that I want to know what is on the table and you know this. In that situation, if (160a) is true, then it would uncooperative for you to utter (160b). That’s because

(160a) asymmetrically entails (160b), and so (160a) is strictly more informative than (160b).

Obviously, it doesn’t follow from this that (160b) is false, or that (160a) is the wrong answer to my question.

The same holds for an utterance of the Answer2: though true, it would be uncooperative but not false to utter it in the context of COP. However, it simply doesn’t follow from this

123 that the AnswerI is not a genuine interpretation of the Answer in the context of COP. That

would follow only if the Answer2 were false on the II of the Answer, but it clearly isn’t. If anything, Salmon’s observation proves problematic for his own view. The objection assumes

that ‘two’ means ‘exactly two’. If so, then KISS wrongly predicts that the Answer2 should be false in the context of COP since there aren’t exactly two oranges worth of orange on

1 the table, but 2 2 .

Salmon’s second objection resembles the ﬁrst. As before, the AnswerI clearly entails its

second conjunct, namely the Answer1/2.

(Answer1/2) There is one half-orange on the table.

So if the AnswerI is true, then so is the Answer1/2. Now, imagine the following minimal variation of COP: we take one of the two whole oranges remaining on the table and cut it

in half, setting both halves back on the table. So there is now one whole orange and three

half-oranges on the table. This scenario presents a problem, according to Salmon.

Our current proposal would answer [the Question] ... by saying that there are

3 one and three-halves (1 2 ) oranges on the table. But one might answer [the Question] with the same old [Answer], adding now that one of the oranges has

3 1 been cut in half. And indeed, 1 2 = 2 2 . It is especially tempting to count [the Answer] as still a correct answer since two of the three orange-halves on the

table do indeed come from the same orange ... The conﬁguration of the oranges

on the table has changed, but not their number. By contrast, [the AnswerI ] is not in any way a correct description of the new situation. There is not only

one orange-half on the table. Rather, there are exactly three orange-halves on

the table (together with one whole orange). This further demonstrates that [the

Answer] and [the AnswerI ] are not equivalent.

124 Again, the argument is fairly straightforward: the AnswerI entails the Answer1/2, but the latter is false in the new scenario when the Answer is not. Thus, the AnswerI cannot be an available interpretation of the Answer.

My response to this objection is the same as my response to Salmon’s last objection.

The important thing to note is that, once again, the Answer1/2 is not false on either the original version of COP or Salmon’s modiﬁed version. Thus, one can consistently utter

(161) even in Salmon’s modiﬁed COP:

(161) There is one half-orange on the table. In fact, there are three half-oranges on the

table.

Consequently, Salmon’s conclusion is another non-sequitur, and the fact that (161) is true in this new context again proves problematic for KISS – there is not exactly one or three

1 half-oranges worth of orange on the table, but 2 2 . Nevertheless, Salmon’s remarks illustrate an important point. He rightly notes that one can truthfully answer the Question with the Answer in this modiﬁed version of COP. This is not the only correct answer, of course. One could also truthfully utter (162) in that scenario.

3 (162) There are 1 2 oranges on the table.

As before, I predict that (162) should have a MI according to which we are measuring an

1 amount of orange equal to 2 2 oranges worth. It’s for this reason that we can truly answer the Question with the Answer in this scenario. Thus, Salmon’s objection illustrates the importance of recognizing that not just container phrases are I/M-ambiguous. I conclude that not only is the AnswerI a genuine interpretation of the Answer in the context of COP, but that it is in fact the correct interpretation in that context.

125 4.5 Conclusion: Don’t KISS the COP!

We began with a simple puzzle. If we have three whole oranges, cut one of them in half,

discard one of the halves, and set the remaining half back on the table, then the intuitively

correct answer to the Question is the Answer.

(Question) How many oranges are there on the table?

1 (Answer) There are 2 2 oranges on the table.

Either a half-orange is an orange or it isn’t. This purports to be a dilemma since if a half-

orange is an orange, then there are three oranges on the table, and if it isn’t, then there

are only two. In either case, we get an answer to the Question which isn’t the intuitively

correct Answer.

How does the present analysis of I/M-ambiguities help solve the COP? Given the analysis

of fraction words on the individuating interpretation developed here, I am clearly committed

to embracing one horn of the apparent dilemma: a half-orange is deﬁnitely not an orange.

So there are strictly-speaking only two oranges on the table. But there is also a half-orange

on the table. On the II of the Answer, we are individuating two whole oranges and a single

half-orange, and we are counting them separately. Thus, one may truly answer the Question

with the Answer even though there are strictly-speaking only two oranges on the table. On

the measure interpretation of the Answer, we are measuring two and a half oranges worth

of orange. Thus, the Answer would remain true even if we were to halve the remaining two

whole oranges, or cut them and the single half-orange into slices of various sizes, or even if

we pulverized all of the remaining orange into a bunch of scattered orange pulp. Though

this interpretation of the Answer will always be available thanks to the Universal Measurer,

we have seen that it is not the interpretation plausibly intended in the context of COP.

Thus, the fact that the Answer is I/M-ambiguous is signiﬁcant for a couple reasons.

First, COP does not require positing fractional cardinalities. On the II, we are counting

three things, namely two oranges and one half-orange, and on the MI we are measuring the

126 volume of orange on the table. Seen this way, COP presents no particular threat to the

traditional analyses of cardinality expressions. It would threaten those analyses only if it

required positing fractional cardinalities, but it doesn’t. Secondly, the fact that the Answer

is I/M-ambiguous reveals problems for the traditional analyses of cardinality expressions

and for KISS. Without UM in place, the traditional analyses wrongly predict that the

1 Answer is false when there are 2 2 oranges’ worth of orange slices on the table, each from possibly diﬀerent oranges. While KISS rightly predicts the Answer is true in this context,

it appears to wrongly predicts that the Answer should only have a measure interpretation,

even in the context of COP.

Here’s why. According to KISS, the Answer is true just in case the amount of orange-stuﬀ

on the table measures 2.5 oranges’ worth, where ‘2.5’ names a real number. Consequently, an

explanation of the various semantic contrasts noted in §2 is clearly in order. One notable

possibility mimics the analysis of I/M-ambiguities oﬀered here, only in reverse. On this

view, nouns like ‘glass’, ‘box’, and ‘orange’ are all measure terms in the lexicon, and so give

rise to MIs by default. However, these get coerced into countable predicates by something

like the inverse of UM, thus giving rise to IIs.

Scontras(2014) proposes a type-shifter, one he calls “SHIFT M−C ”, which does just that. This is given in (163), where M is a variable ranging over measure terms.

(163) λM.λx.∃k.∃y.M(k)(1)(y)∧filled-with(y)(x) (SHIFTM−C )

What this does, in eﬀect, is coerce ostensible measure terms into countable predicates. For

instance, applying it to ‘liter’ returns containers ﬁlled with one liter of some substance, e.g.

water. Since containers are generally countable, so is ‘liter’ under such a coercion, thus

explaining how ‘four liters’ can come to have IIs in contexts like (164).

Context: Mary has a strange way of heating up water for coﬀee; she pours it into one liter

bottles and places those bottles into boiling soup. John says “As I was passing by the

127 kitchen, I noticed some bottles ﬂoating in the soup. How many liters of water did you

put in the soup?”

(164) Mary: “Four liters / ??liters’ worth.”

On the version of KISS under consideration, SHIFTM−C is taken to be a general type- shifting operation available for all measure terms, thus explaining why e.g. both ‘glass’ and

‘orange’ give rise to IIs.

The most signiﬁcant problem with this suggestion, as far as I can tell, is that it fails to

generalize. For example, though SHIFTM−C can perhaps be made to work for IIs of ‘four glasses of water’, it does not plausibly extend to ordinary count nouns like ‘orange’ or even

canonical measure terms like ‘ounce’. My intuition is that if one were to cut four oranges in

half, scoop out all of the orange matter, replace it with guacamole, and somehow manage

to get the halves back together again, it would be at best disingenuous, and probably just

false, to say that there are four oranges on the table. The problem with typical measure

terms like ‘ounce’ is that they, unlike ‘liter’, do not plausibly allow for IIs, as evidenced by

contrasts like (165).

Context: Mary sets four oranges, four glasses containing water, and four 1 oz. packets of water

on the table. Pointing at the table, she says:

(165) Each of the four {oranges/glasses (of water)/??ounces (of water)} is for John.

As Rothstein(2009) explains, distributive expressions such as ‘each’ presuppose a domain

of individuated, and thus countable, objects. So if something like SHIFTM−C were generally available to shift measure terms into countable predicates, we would expect to see no

diﬀerence here between e.g. ‘glasses (of water)’ and ‘ounces (of water)’, contrary to fact.

In my view, a far more elegant and satisfactory solution holds that the various expres-

sions in question enter the lexicon having countable meanings, and only take on measure

128 meanings when combined with UM. Hence my conclusion: Don’t KISS the COP; have RUM instead!

129 Chapter 5

Concluding Remarks and Future Directions

5.1 Concluding Remarks

I have argued for a number of substantial conclusions in the previous three chapters. In

Chapter 2, I considered a puzzle due to Hofweber(2005), namely Frege’s Other Puzzle:

how can one and the same expression ‘four’ serve apparently opposing semantic functions

in seemingly equivalent statements like Quantiﬁer and Identity?

(Quantiﬁer) Jupiter has four moons.

(Identity) The number of Jupiter’s moons is four.

The solution, I argued, was to recognize that although ‘four’ is a numeral in both examples,

and so refers to a number, it also occurs within a measure phrase, and this measure phrase

serves diﬀerent semantic functions in Frege’s two examples. More speciﬁcally, it functions

quantiﬁcationally in Quantiﬁer and referentially in Identity, both thanks to type-shifting.

I labeled the resulting view “the Neo-Substantivalist Strategy”. As with previous Sub-

stantivalist analyses, ‘four’ is a numeral in both Quantiﬁer and Identity. However, unlike

previous Substantivalists, the post-copular material in Identity does not refer to a number

on the Neo-Substantivalist analysis, despite functioning as a singular term. Rather, it refers

to a cardinality, or a kind of degree. Following Scontras(2014), I argued in Chapter 3 that

English draws a sortal distinction between numbers and degrees. This explains numerous

semantic contrasts like e.g. (166).

(166) a. The number of Jupiter’s moons is {four/??the number four}.

130 b. The number Mary is considering is {four/the number four}.

Both examples are speciﬁcational sentences, and so the post-copular expressions here are all

plausibly singular terms. But since ‘the number four’ and the numeral ‘four’ are presumably

coreferential, we must conclude that the post-copular material in Identity does not refer to

a number. Rather, it is a measure phrase referring to a cardinality.

This observation also aﬀords an interesting diagnosis of what goes wrong in the Easy

Argument for Numbers. According to it, because Identity is an identity statement equating

two singular terms, including the numeral ‘four’, and since existential generalization is

generally valid for singular terms, Identity entails Realism, or the explicit recognition of the

existence of a number.

(Realism) There is a number, namely four, which is the number of Jupiter’s moons.

Since Quantiﬁer entails Identity, it also entails an ontology of numbers. There are two

problematic assumptions with this argument, however. First, Identity is not an identity

statement equating two singular terms. Nevertheless, on an independently plausible analysis

of speciﬁcational sentences, the post-copular material in Identity is a singular term, and so

the truth of Identity does plausibly entail the existence of an abstract object of some sort,

namely a cardinality. The problem, obviously, is that this can establish what Neo-Fregeans

intend it to – the existence of numbers – only if cardinalities are conﬂated with numbers.

This in turn poses an interesting epistemological question. Cardinalities answer ques-

tions like ‘How many ﬁngers am I holding up?’, and many philosophers believe that we can

come to have a priori arithmetic knowledge on the basis of counting. Thus, if I count two

fingers on my left hand, three fingers on my right hand, and then count all the fingers being

held up in total, discovering that they add to ﬁve, the intuition is that I might have thereby

discovered an a priori arithmetic truth, namely that 2 + 3 = 5. But if cardinalities and

numbers are distinct, then these latter numerals cannot be coreferential with ‘the number

of ﬁngers on my left hand’, ‘the number of ﬁngers on my right hand’, etc. So then how can

131 counting ﬁngers lead to arithmetic discovery? The answer, I suggested, was to recognize

that there is a structural equivalence between cardinalities ordered in a natural way – i.e. a

cardinality scale – and the natural numbers. Thus, by adopting a form of structuralism, we

can say that whatever we learn about cardinalities by counting ﬁngers and toes also counts

as knowledge about the natural numbers.

However, this story was threatened by the Counting Oranges Puzzle considered in Chap-

ter 4. If I start with three oranges on the table, cut one in half and leave one of the halves

on the table, then the intuitively correct answer to the Question is the Answer.

(Question) How many oranges are on the table?

1 (Answer) There are 2 2 oranges on the table.

This appears to be problematic for traditional theories of cardinality since fractional cardi-

nalities are incoherent on those analyses – cardinalities represent the number of countable

objects constituting a collection – and yet this is precisely what making sense of the An-

swer seems to require. To put the point diﬀerently, we could rephrase the Answer as ‘The

1 1 number of oranges on the table is 2 2 ’, whereby ‘2 2 ’ refers to a cardinality on the semantics developed in Chapter 3. But if cardinalities can correspond to the rational or real numbers

in this manner, then obviously they cannot be isomorphic to the natural numbers, thus

undermining the earlier appeal to structuralism.

The way out, I argued, is to recognize that the Answer is ambiguous in a way parallel-

ing ‘four glasses of water’. The latter has an “individuating” interpretation according to

which we are counting four glasses, each of which happens to be ﬁlled with water, and a

“measure” interpretation according to which we are describing quantities of water measur-

1 ing four glassfuls. Similarly, I argued that ‘2 2 ’ oranges’ has an individuating interpretation according to which we are counting two whole oranges and a single half-orange, and a

1 measure interpretation according to which we describing quantities of orange measuring 2 2 oranges’ worth. Since an individuating interpretation is intended in the Counting Oranges

132 Puzzle, and since three is a whole number, the Answer poses no special threat to traditional analyses of cardinalities, nor does it threaten the structuralist epistemological connection between cardinalities and natural numbers.

But what if a measure interpretation were intended instead? For example, suppose that rather than starting with three whole oranges, we started instead with an assortment of random orange slices, each from diﬀerent oranges. Mary eats a certain amount of oranges slices, and we ask the Question. In this case, the Answer could be true, meaning that

1 Mary ate 2 2 oranges’ worth of orange. As before, we could state the Answer diﬀerently 1 1 in this case as ‘The number of oranges Mary ate is 2 2 ’, where ‘2 2 ’ again should refer to a 1 cardinality according to the semantics developed in Chapter 3. Except now ‘2 2 ’ cannot be paraphrased as ‘two oranges and a half-orange’. Rather, it looks as if we are dealing with a genuinely non-whole cardinality. Granted, we are not counting oranges in this case, and so this observation poses no obvious threat to the structural connection between cardinalities formed on the basis of counting and the natural numbers. But nothing in the analysis given in the previous chapter prevents the Answer from specifying a fractional cardinality under a measure interpretation.

In fact, I want to suggest in this last chapter that finite cardinalities can be non-whole, and so are not generally isomorphic to the natural numbers. For that to be the case, however, cardinalities cannot uniformly represent the number of countable objects constituting a collection. Rather, the latter are a special sort of cardinality; they are formed on the basis of counting, itself a form of measurement. Thus, one of the central questions here is what a cardinality generally represents if it is not the number of atoms constituting a given plurality. Ultimately, I want to suggest that cardinalities, like lengths, ages, speeds, etc., represent ratios of a certain sort. Given the Fregean analysis of the real numbers as magnitude-ratios, in effect I’m proposing that cardinalities are isomorphic to the reals, with the cardinalities formed on the basis of counting – the correlates of the (finite) cardinal

133 numbers – being “embedded within the reals”, so to speak.1

More generally, I want to use this chapter to sketch some future directions for the present project. There are at least two immediate extensions which should be of interest to philosophers of mathematics. Both are related to Frege(1903)’s claim in the Grundgesetze that the cardinal numbers and the real numbers are ontologically distinct, or what I call

“Frege’s Ontological Thesis” (§5.5.2). The ﬁrst extension concerns the so-called sortal/non- sortal distinction (§5.5.3). The second is a puzzle I call “the Sortal Concept Puzzle” (§5..54).

I want to suggest here that solving the latter has signiﬁcant ramiﬁcations not only for Frege’s

Ontological Thesis, but also for the analysis of cardinality assumed throughout this thesis.

5.2 Frege’s Ontological Thesis

Frege(1903) introduces his conception of the real numbers as ratios of magnitudes in §73 of the Grundgesetze.

What then is the substance of the assertion that number-signs designate quanti-

ties? Let us look at the applications of arithmetic laws in geometry, astronomy,

physics. Here, indeed, numbers occur in relations to magnitudes, such as length,

mass, light intensity, electrical charge... [A] number-sign cannot on its own des-

ignate a length, a force, and so on, but only in connection with the designation

of a measure, a unit, such as meter, gram and so on. What in that case does a

number-sign refer to on its own? Clearly a magnitude-ratio. And this suggests

itself so strongly that we must not be surprised to encounter this insight early.

1Tennant(2010) argues for a similar conclusion, though he arrives at it via interestingly diﬀerent means. According to Tennant’s “logico-genetic account”, the naturals are constructed as purely logical objects in a logicist-appropriate manner via an abstractionist principle Tennant(1987) calls “Schema N”: # x[F (x)] = n iﬀ there are exactly n F s. The reals, on the other hand, are not purely logical objects, but represent points on an oriented geometric line. Ultimately, the reals are thus ratios of magnitudes, just like Frege (1903) suggests. Nevertheless, unlike with Frege, the natural number four and the real number four are the very same number on Tennant’s account. Consequently, the naturals form a subset of the reals, and so are “embedded” within the reals so to speak.

134 If we now understand by “number” the reference to a number-sign, then real

number is the same as magnitude-ratio.

Consider statements like (167) as uttered within a scientiﬁc context.

(167) a. This beaker contains four ounces of water.

b. This rod is 4.63 meters long.

Intuitively, (167a) tells us something about a particular magnitude – the volume of water in this beaker – in comparison to a unit magnitude of that same kind – an ounce of water.

Similarly, (167b) tells us something about the length of this rod in comparison to a unit magnitude of the same kind, namely a meter. These comparisons between magnitudes can be expressed in terms of ratios, e.g. the length of this rod is four times the length of a meter.

On Frege’s analysis, real numbers are ratios of this sort, abstracting away from particular dimensions of measurement.

Frege goes on to claim that the reals qua magnitude ratios are sortally distinct from the cardinal numbers qua classes of equinumerous, countable concepts. This is what we might call Frege’s Ontological Thesis.

Since the cardinal numbers are not ratios, we have to distinguish them from the

positive whole numbers. Therefore, it is not possible to extend the domain of the

cardinal numbers to that of the real numbers; they are simply completely sepa-

rate domains. The cardinal numbers answer the question “How many objects of

a certain kind are there?” while real numbers can be considered as measuring

numbers that state how large a magnitude is in comparison to a unit ... For this

reason ... we distinguish between the cardinal numbers 0 and 1 from the [real]

numbers 0 and 1.2

In essence, because classes of equinumerous concepts and classes of relations between magnitudes are diﬀerent sorts of things, the cardinals and the reals form “completely separate 2Frege (1893, §157).

135 domains”. To safeguard against any potential ontological confusion, Frege even employs diﬀerent fonts for these diﬀerent sorts of objects: numerals referring to cardinals receive strikethrough; those referring to reals do not.

As Simons(1987) points out, Frege’s Ontological Thesis plays a dialectically important role in Frege’s logicist development of the reals.

Frege now (§164) faces the problem of where to obtain the Relations whose ratios

are to form the real numbers, especially as there are not countably but inﬁnitely

many reals, the result of Cantor which Frege cites in passing ... The related

problem for the natural numbers was solved by deﬁning zero using the empty

class, the extension of a necessarily empty concept, and using this necessarily

existing object to start us off on the trip to the first infinite number. The natural

numbers then all exist of necessity, independent of empirical fact. Frege’s trick

here is even more subtle: he makes uses of the availability of the natural numbers

as already deﬁned to show the existence of suitable Relations on which to base

the reals. Hence, the separateness of the naturals from the reals is of strategic

importance for his plan, since without it he would be moving in a circle.3

Since Frege had earlier shown how to generate the naturals (i.e. the finite cardinals) in a logicist-appropriate manner, his plan is to use these same objects to define magnitudes, and eventually the reals. Consequently, the Ontological Thesis guarantees that Frege is not defining the target objects in terms of themselves.4

To avoid circularity, Frege thus needs to provide a theory-external reason for believing the Ontological Thesis. To this end, Frege appeals to the applications of the real numbers within the sciences, and in particular in how they are used to express measurement. Hence the relevance of Frege’s observation that the cardinals and reals answer diﬀerent sorts of

3Simons (1987, p. 33-34.) 4This presupposes an “impredicativist” view of numbers, one according to which classes of numbers should not be deﬁned in terms of themselves.

136 questions. But why think that answering diﬀerent sorts of questions is evidence that the numerals answering those questions refer to diﬀerent sorts of things?

(168) a. Who laughed? John.

b. Which person laughed? John.

After all, ‘who’-questions and ‘which’-questions seem like diﬀerent sorts of questions, yet

‘John’ in (168a,b) can surely refer to the same individual.

5.2.1 Evidence for Frege’s Ontological Thesis

In Frege’s defense, there are some relevant linguistic diﬀerences between ‘how many’- questions and other sorts of ‘how’-questions, diﬀerences which at least initially support the Ontological Thesis. Consider the contrast in (169), for instance.

(169) a. How many cats are on the mat? {Four/??4.38} cats.

b. How much water is in the bowl? {Four/4.38} ounces.

c. How long is that canoe? {Four/4.38} feet.

According to Frege, recall, cardinal numbers are classes of equinumerous concepts, and the possibility of setting up one-to-one correspondences between classes presupposes that we are dealing with whole, individuated objects. Since ‘how many’-questions ask about the cardinality of a concept, it should come as little surprise that concepts like cat-on-the-mat

can answer ‘how many’-questions but those like water-in-the-bowl cannot. For example, we cannot (usually) ask ‘How many waters are in that bowl?’. Furthermore, because car-

dinalities belong to classes consisting of whole, individuated objects, cardinal numbers are

necessarily whole. Thus, it is little wonder that ‘four’ but not ‘4.38’ is acceptable in (169a).

On the other hand, since real numbers are ratios of magnitudes, they needn’t be whole.

Hence the acceptability of both ‘four’ and ‘4.38’ in (169b,c).

137 Secondly, notice that while ‘four’ can occur alone in response to the ‘how many’-question in (170a), it must occur with a measure term in (170b,c).

(170) a. How many cats are on the mat? Four (cats).

b. How much water is in the bowl? Four ??(ounces).

c. How long is that canoe? Four ??(feet).

In this way, number expressions answering ‘how much’ or ‘how long’-questions resemble prototypical relational expressions missing obligatory arguments, e.g. the verb ‘kick’ in

(171).

(171) What did Mary do to John? Kick ??(him).

This resemblance might be taken to show that ‘four’ in (170b,c), unlike ‘four’ in (170a), is relational, just as we would expect if ‘four’ in (170b,c) refers to a real number, and if real numbers but not cardinal numbers are ratios (i.e. a certain kind of relation).

5.2.2 Problems for Frege’s Ontological Thesis

Unfortunately, similar sorts of linguistic considerations prove problematic for Frege’s On- tological Thesis. It is dialectically important for the Thesis that cardinal numbers characteristically answer ‘how many’-questions. However, the view that all concepts acceptably answering ‘how many’-questions have cardinalities is problematic, as it leads to a number of apparent inconsistencies.5

Consider again the examples in (169a,b).

(169a) How many cats are on the mat? {Four/??4.38} cats.

(169b) How much water is in the bowl? {Four/4.38} ounces.

5Many of the problems noted here come from Snyder and Shapiro(2014).

138 Unlike ‘cats’ in (169a), ‘ounces (of water)’ in (169b) acceptably occurs with a non-whole number, thus suggesting that cat-on-the-mat but not ounce-of-water can have a cardinality. However, notice that we can ask about the volume of the water using a ‘how

many’-question instead.

(172) How many ounces of water are in the bowl? {Four/4.38} (ounces).

This is puzzling for a couple reasons. First, because it answers a ‘how many’-question,

ounce-of-water should have a cardinality. But since it also answers a ‘how much’-question, it seems the same concept cannot have a cardinality. Moreover, because cardinal numbers

are necessarily whole, ‘4.38 ounces’ in (172) should be just as odd as ‘4.38 cats’ in (169a),

contrary to fact. On the other hand, if ‘4.38’ in (172) is a real number, as it seems it should

on Frege’s account, then ‘ounces’ in (172) should not be optional, thus resembling ‘ounces’

in (173).

(173) How much water is in the bowl? 4.38 ??(ounces).

Again, however, this not the case.

We can put the point diﬀerently in terms of Frege’s quantiﬁcational analysis of cardinality expressions. According to it, (174a) should have something like the logical form in (174b), at least in response to a ‘how many’-question, where F abbreviates ounce-of-water-in-

the-bowl.

(174) a. There are four ounces of water in the bowl.

b. ∃x1, ...x4. [F (x1) ∧ ... ∧ F (x2) ∧ x1 6= x2 ∧ ... ∧ x3 6= x4] ∧

[∀y. F (y) → y = x1 ∨ ... ∨ y = x4]

Suppose (174a) is true, and that we divide the water into four separate 1 oz. containers.

Since it is possible to map each container to a unique object falling under a concept like

face-on-Mt.-Rushmore, the ﬁrst major conjunct of (174b) is satisﬁed: there are at least four distinct ounces of water in the bowl. However, the second major conjunct of (174b) is

139 clearly false in this circumstance. After all, we could swap half of the water in some of the containers for half of the water in others, in which case we would have diﬀerent resulting quantities of water, each measuring an ounce. In other words, we would have diﬀerent 1 oz. containers of water. What this shows is that there is no unique mapping from objects falling under face-on-Mt.-Rushmore to those falling under ounce-of-water-in-that-bowl in this scenario. Consequently, (174b) is false.

What’s gone wrong? Apparently the presumption that all concepts acceptably answering

‘how many’-questions have cardinalities cannot be maintained given the Fregean conception of cardinal numbers as classes of equinumerous concepts. If so, then the problem for Frege’s

Ontological Thesis is that answering ‘how many’-questions is not suﬃcient for having a cardinality. Yet the primary, theory-external evidence for subscribing to the Thesis is that cardinal numbers characteristically answer ‘how many’-questions.

5.3 Sortal Concepts

As mentioned earlier, Frege maintains that only certain sorts of concepts are capable of being counted, and so have a cardinality. Hale(2016) elaborates:

A cardinal number, Frege insists, is what answers a question of the form: How

many F s are there? Here F needs to be a concept of the right kind – a concept

that carries with it a principle for distinguishing the objects falling under it. It

must be associated not only with what Dummett [1991] has called a criterion

of application, but also with what Frege himself called a criterion of identity.

Frege’s canonical terms for cardinal numbers single them out as numbers be-

longing to such concepts – concepts of kinds of things which can be counted –

and his deﬁnition embodies the basic principle of counting, by telling us when

the answer to the questions: ‘How many F s are there?’ ‘How many Gs are

there?’ should be given the same answer.

140 The sorts of concepts Hale is talking about here are more familiarly known nowadays as sortal concepts. But what is a sortal concept, exactly? In §54 the Grundlagen, Frege(1884) suggests that countable concepts must satisfy two conditions.

For the concept to which number is ascribed does in general delimit what falls

under it in a deﬁnite way. The concept “letter in the word Zahl” delimits the

Z from the a, the a from the h, and so on. The concept “syllable in the word

Zahl” picks out the word as a whole and as indivisible in the sense that the

parts do not now fall under the concept. Not all concepts are so constituted.

We can, e.g. divide up what falls under the concept “red” in a variety of ways,

without the parts thereby ceasing to fall under that concept. To a concept of

this kind no ﬁnite number belongs. The proposition that units are isolated and

indivisible can, accordingly, be formulated as follows: Only a concept which

delimits what falls under it in a deﬁnite way and allows for no arbitrary division

of [what falls under] it into parts can constitute the unit that relates to a ﬁnite

[cardinal] number.

Frege’s ﬁrst condition on countable concepts is what Koslicki(1997) calls Isolation: it “delimits what falls under it in a deﬁnite way”. Take the concept letter-in-the-word-Zahl , for instance. Vagueness aside, we generally have no problem delimiting the Z from the a, the a from the h, etc. According to the second condition, which Koslicki calls Non-Arbitrary

Division, arbitrary parts of an object falling under a countable concept “do not now fall under the concept”. Again, arbitrary parts of the letter Z are not themselves letters in the

word Zahl; the lower half is not, for instance. Contrast this with a non-countable concept

like red or water. Down to a vague minimal threshold, at least, arbitrary parts of a wholly red patch of light are themselves red, just as arbitrary parts of a quantity of water are

themselves quantities of water.

141 Thus, it would appear that for Frege a concept is sortal just in case it satisﬁes both

Isolation and Non-Arbitrary Division. This conclusion is potentially misleading, however.

As Grandy(2007) explains, the term “sortal” arose in the 1950’s and 1960’s through the work of philosophers like Strawson, Quine, and Geach, and it is not entirely clear that they are all intending to talk about the same phenomenon. For one thing, not all of them use the term ‘sortal’; Quine(1960), for instance, talks about those predicates having

“divisive reference”, as opposed to those having “cumulative reference”. To complicate matters further, Grandy mentions that the contemporary sortal/non-sortal distinction is characterized in at least six diﬀerent ways, whereby a sortal F

(S1): Gives a criterion for counting F s.

(S2): Gives a criterion of identity and non-identity among the F s.

(S3): Gives a criterion for the continued existence of an F .

(S4): Answers a question “What is it?” for an F .

(S5): Speciﬁes the essence of an F .

(S6): Does not apply to parts of an F .

Of the six characterizations here, only S2 and S6 sound remotely like Isolation or Non-

Arbitrary Division, and Frege took the latter to jointly imply S1. In any case, the point here is that at present there is no agreed upon characterization of the sortal/non-sortal distinction.

Instead, as Grandy explains, there is a linguistic distinction between certain kinds of nouns, and many philosophers believe that Frege’s distinction between countable and non- countable concepts tracks this linguistic distinction.

In the least interesting (and uncontroversial) sense, an expression is sortal if

and only if it takes numerical modiﬁers. Thus ‘ice cube’ is sortal because the

adjectives associated with it are numerical–we say “two ice cubes”, but ‘water’

142 is not sortal because the usual modiﬁers are mass terms, i.e. we ask for “two

cups of water”. Or in slightly diﬀerent terms, we ask “How many ice cubes

do you want?” but “How much water do you want?” Of course we can often

make sense of ‘how many’ questions about non-sortals if a suitable measure is

understood. Thus “How many coﬀees would you like?” is really understood as

“How many cups ...?” Many words have two distinct though related meanings,

one of which is sortal and one of which is not. For example, we can ask “How

many chickens do you want?” and “How much chicken do you want?” This

linguistic distinction is often called the “count noun/mass noun” distinction by

both philosophers and linguists.

The count/mass distinction is a familiar one. Characteristically, count nouns such as ‘orange’ can occur with numerical modiﬁers like ‘four’, determiners like ‘a(n)’, and adjectives like ‘many’ and ‘several’; mass nouns such as ‘yarn’ cannot.6

(175) a. Mary has four {oranges/??yarns}.

b. Mary has a(n) {orange/??yarn}.

c. Mary has many {oranges/??yarns}.

As Grandy suggests, many philosophers understand the philosophical distinction between sortals and non-sortals, at least when construed linguistically, as just another label for this linguistic distinction.

However, the count/mass distinction is notoriously ﬂexible and seemingly arbitrary in certain respects. As Grandy suggests, there are certain nouns which seem to ﬁt both categories well, at least based on criteria like those in (175). For example, we can say that

Mary has four chickens or a lot of chicken. Or, as Quine points out, (176) is ambiguous between a count interpretation implying that Mary is a pet owner, or a mass interpretation implying that she’s a carnivore.

6See Grimm(2012).

143 (176) Mary had a little lamb.

Furthermore, consider the fact that while ‘coin’ is a count noun, ‘change’ is mass, despite these appearing to be (roughly) co-extensional. Similarly, though ‘hair’ is mass in English, its Italian equivalent ‘capello’ is count. All of this suggests that the count/mass distinction cannot track something metaphysically signiﬁcant, a point made originally by Quine:

The contrast lies in the terms and not in the stuﬀ they name. It is not a question

of scatter. Water is scattered in discrete pools and glassfuls, and red in discrete

objects; still it is just ‘pool’, ‘glassful’, and ‘object’, not ‘water’ or ‘red’, that

divide their reference. Or, consider ‘shoe’, ‘pair of shoes’, and ‘footwear’: all

three range over exactly the same scattered stuﬀ, and diﬀer from one another

solely in that two of them divide their reference diﬀerently and the third not at

all.7

‘Shoe’, ‘pair of shoes’, and ‘footwear’ denote the same portions of material stuff, yet only the former are count and the latter is mass. Quine’s invited conclusion: though the count/mass distinction appears to track a genuine metaphysical distinction, something like the distinction between “things” and “stuff”, ultimately it is just a reflection of the words themselves.

Lastly, consider the phenomena of grinding and packaging, as made famous by Pel- letier(1975). 8 Grinding is illustrated in (177a), where ‘horse’ – a canonical count noun – is coerced into functioning as a mass noun.

(177) a. After the accident, there was horse all over the road.

b. We ordered four waters.

(177b) illustrates packaging, where ‘water’ – a canonical mass noun – is coerced into functioning as a count noun. Minimally, what grinding and packaging show is that whether

7(Quine, 1960, p. 91). 8Though essentially the same observations were made originally by Quine(1960).

144 certain nouns function as count or mass depends on the syntactic context in which they occur. Again, it is hard to see how this could reﬂect something of genuine metaphysical signiﬁcance, though.

As Grandy explains, the seemingly capricious nature of the count/mass distinction has led many philosophers to doubt the philosophical significance of the sortal/non-sortal distinction. However, Grandy himself draws a different conclusion: the sortal/non-sortal distinction, whatever it might be, should not be identified with the count/mass distinction. I agree, though for different reasons. I think that English draws a distinction between predicates satisfying something like Frege’s Isolation and Non-Arbitrary Division and those which do not, and that this distinction amongst predicates is not exhausted by the count/mass distinction.

5.3.1 Atomicity and Quantization

The claim here is that Frege’s distinction between those concepts meeting Isolation and Non-

Arbitrary Division and those which don’t closely tracks the linguistic distinction between atomic and non-atomic predicates. Recognizing this leads to a richer conception of the sortal/non-sortal distinction (linguistically construed) than philosophers have heretofore recognized.

I begin by reviewing certain formal properties of predicates as originally deﬁned by

Krifka(1989). These can be used to characterize diﬀerent classes of expressions, e.g. singular count nouns, mass nouns, plural nouns, measure phrases, etc. The property most relevant for present purposes is Quantization, as deﬁned in (178), where ‘@’ is the relation of proper parthood.

(178) ∀P. QUA(P ) ↔ ∀x, y. P (x) ∧ y @ x → ¬P (y) (Quantization)

A quantized predicate P is such that no proper part of an object satisfying P itself satisﬁes

P . Singular nouns such as ‘orange’ and various sorts of pseudopartitives, e.g. ‘grain of rice’,

145 ‘glass of wine’, and ‘ounce of water’, are all “quantized” in this sense. Vagueness aside, arbitrary parts of an orange are not an orange, just as arbitrary parts of an ounce of water are not themselves an ounce of water.

On the other hand, plural nouns such as ‘oranges’ and mass nouns such ‘water’ are not quantized. After all, pluralities of oranges can consist of further pluralities of oranges, just as a quantity of water can consist of further quantities of water. Thus, these expressions are said to satisfy Cumulativity as deﬁned in (179), where ‘t’ denotes a join-relation.

(179) ∀P. CUM(P ) ↔ ∀x, y. P (x) ∧ P (y) → P (x t y) (Cumulativity)

Two pluralities of oranges make for a larger plurality of oranges, just as two quantities of

water make for a larger quantity of water. Again, however, if x is a grain of rice (or glass

of wine or ounce of water) and so is y, then the result of combining these together is not

another grain of rice (or glass of wine or ounce of water). Thus, the function of an expression

like ‘grain’, ‘glass’, or ‘ounce’ is to quantize an otherwise cumulative expression.9

Krifka distinguishes two sorts of quantized predicates. The ﬁrst – atomic predicates – have the property of Atomicity, as deﬁned in (180b).

(180) a. ∀P.∀x. Atom(P, x) ↔ P (x)∧¬∃y. y @P x (P -atom) b. ∀P. ATM(P ) ↔ ∀x. P (x) → Atom(P, x) (Atomicity)

As emphasized in Chapter 2, the notion of atomicity here is crucially a relative one: a

P -atom is anything satisfying P which has no P -relative proper parts.10 Thus, something

is an atom only with respect to the sort of thing denoted by a predicate. For example,

Rover can be an atomic dog while having other sorts of parts, e.g. legs and a nose. An

atomic predicate is one whose extensions consists solely of P -atoms. As such, atomicity

clearly implies quantization: since P -atoms have no P -relative proper parts, they cannot

have proper parts which also satisfy P .

9Cf. Chierchia(1998). 10Hence the relativization of the parthood relation in (180a) to the predicate in question.

146 Importantly, however, the converse entailment does not generally hold. That is, not all quantized predicates are atomic. To see this, consider a predicate like ‘ounce of water’. Sup- pose we have a semilattice structure like the following, where arrows indicate mereological relations and a, b and c are each quantities of water.

a t b t c

a t b a t c b t c

a b c

Because ‘water’ is cumulative, all seven “nodes” in the structure satisfy the predicate. Now suppose that a, b, and c each measure half an ounce. Then a t b, a t c, and b t c are each an ounce of water, though a t b t c is not, nor are a, b, or c individually. Hence, ‘ounce of water’ is a quantized predicate in this scenario. But it is not atomic. More generally, measure phrases like ‘ounce of water’ have the property of Strict Quantization.

(181) ∀P. SQU(P ) ↔ QUA(P ) ∧ ¬ATM(P ) (Strict Quantization)

In other words, measure phrases are quantized but not atomic.

Intuitively, the difference between atomic predicates like ‘orange’ and strictly quantized predicates like ‘ounce of water’ is that whereas different atomic oranges cannot share orange- relative parts, different ounces of water can share water-relative parts. This easily seen using diagrams like the one above. If we assume the structure given represents ‘orange’, then a, b, and c are the atoms, and a t b, a t c, etc. are pluralities of oranges. In virtue of being

atomic, none of the individual oranges can share parts which are themselves oranges. On

the other hand, if we view the diagram as representing ‘half ounce of water’, then atb, atc,

and btc will each satisfy ‘ounce of water’ despite sharing parts which are themselves water.

Put diﬀerently, the relevant diﬀerence between atomic predicates and strictly quantized

147 predicates appears to be that the former but not the latter satisﬁes discreteness in (182), where ◦ is the relation of overlap.11

(182) ∀P. DIS(P ) ↔ ∀x, y. P (x) ∧ P (y) ∧ x 6= y → ¬x ◦ y (Discreteness)

In other words, atomic predicates consist of non-overlapping individuals, unlike strictly quantized predicates.

Now recall Frege’s two conditions on countable concepts. According to Isolation, a countable concept “delimits what falls under it in a deﬁnite way”, while according to Non-

Arbitrary Division, arbitrary parts of an object falling under a countable concept “do not now fall under the concept”. My proposal is that we assimilate Isolation with Discrete- ness and Non-Arbitrary Division with Quantization. Since Discreteness is what plausibly separates atomic from strictly quantized predicates, the proposal amounts to assimilating

Frege’s distinction amongst concepts with the atomic/non-atomic distinction amongst predicates. Construed linguistically, at least, identifying the sortal/non-sortal distinction with the atomic/non-atomic distinction implies that the former is not restricted to just the distinction between count nouns and mass nouns. Indeed, the next section illustrates several further instances of the “sortal/non-sortal distinction”, thus construed.

5.3.2 Varieties of the Sortal/Non-Sortal Distinction

One instance of the atomic/non-atomic distinction is illustrated in Scontras(2014)’s three- way taxonomy of pseudopartitives like ‘grain of rice’, ‘glass of wine’, and ‘ounce of water’, or what Scontras calls quantizing nouns.

(183) a. atomizer nouns: ‘grain’, ‘drop’, ‘piece’, ‘quantity’

b. container nouns: ‘glass’, ‘bottle’, ‘box’, ‘basket’

c. measure nouns: ‘ounce’, ‘gallon’, ‘foot’, ‘kilowatt’

11Overlap is deﬁned in terms of parthood, so that two things overlap just in case they share at least one part. This deﬁnition is borrowed from Krifka(2008).

148 Each of these nouns combines with ‘of’ plus a substance noun (i.e. a mass or plural noun)

to form either an atomizer phrase (e.g. ‘grain of rice’, ‘quantity of oranges’), a container

phrase (e.g. ‘glass of water’, ‘box of books’), or a measure phrase (e.g. ‘ounce of water’, ‘mile

of elm trees’). Nevertheless, they serve diﬀerent semantic functions. As the label suggests,

atomizer nouns partition a substance into countable individuals, i.e. atoms. Consequently,

numerical atomizer phrases such as ‘four grains of rice’ are usually understood as counting

individuated items of the relevant sort, e.g grains. On the other hand, measure nouns

measure a substance according to a unit named by the noun. Consequently, numerical

measure phrases such as ‘four ounces of water’ are naturally understood as dividing a

substance into quantities measuring that amount, without necessarily dividing the substance

into countable units. Finally, as we saw in Chapter 4, container phrases such as ‘four glasses

of water’ represent a kind of mixed case; the latter counts atomic glasses in some contexts

and measures amounts of water in others.

Evidence for Scontras’ taxonomy comes from what I will call the individuation data. As Rothstein(2009) explains, there are certain expressions which presuppose a domain of

individuated objects, or atoms. These include distributive expressions such as ‘each’, ‘both’,

and presumably ‘which’, as well as plural pronouns like ‘those’. Now consider the contrasts

in (184) (momentarily ignoring container phrases).

Context: Mary places two oranges, two grains of rice, and two 1 oz. packets of water on a table.

Pointing at the table, Mary says:

(184) a. Each of the {oranges/grains of rice/??ounces of water} is for John.

b. Both {oranges/grains of rice/??ounces of water} are for John.

c. Which {orange/grain of rice/??ounce of water} is for John?

d. Those {oranges/grains of rice/??ounces of water} are for John.

149 The contrasts here make sense if atomizer phrases, like ordinary count nouns but unlike mea-

sure phrases, denote sets of atoms. Similarly, Chierchia(1998) observes that certain quan-

tizing nouns are acceptable with determiners like ‘no’ and ‘most’, unlike measure phrases.

Context: Similar to (184), only there are ﬁve of each item mentioned on the table. John asks

where each item came from. Mary responds:

(185) a. I bought {no orange/most of the oranges} from Susan.

b. I got {no grain of rice/most of the grains of rice} from this bag.

c. ?? I poured {no ounce of water/most of the ounces of water} from the tap.

As before, the contrasts here are to be expected if determiners like ‘no’ and ‘most’ quan-

tify over a domain of atomic individuals,12 and if atomizer nouns, unlike measure nouns,

partition a substance into countable atoms.

Next recall the distinction between individuating and measure interpretations of con-

tainer phrases.

I-Context: Mary has a strange way of heating water for coﬀee: she puts it in glasses and then

sets those glasses into boiling soup. After noticing some glasses in the soup, John

asks: “How many glasses of water are in the soup?”. Mary responds with (186a):

M-Context: Mary wants to make some soup. The recipe calls for four glassfuls of water. Accord-

ingly, Mary ﬁlls a certain glass with tap water four times, pouring the contents each

time into the soup. John wants to know the recipe, and so asks “How many glasses

of water are in the soup?”. Mary responds with (186a):

(186) a. There are four glasses of water in the soup.

b. There’s a group of four glasses x s.t. each of x is ﬁlled with water and is in the

soup.

12Cf. also Landman(2004).

150 c. There’s a quantity of water x s.t. x measures four glasses worth and is in the

soup.

In the I-Context, (186a) receives an individuating interpretation paraphrased in (186b). And

in the M-Context, (186a) instead receives the measure interpretation paraphrased in (186c).

On the analysis presented in Chapter 4, following Rothstein(2009) and Scontras(2014),

individuating interpretations were said to arise if ‘glass’ functions as an atomic (though

relational) predicate, while measure interpretations arise instead if ‘glass’ is coerced into a

measure noun, thus resembling e.g. ‘ounce’.

We also saw in Chapter 4 that this sort of ambiguity is not restricted to just con-

tainer phrases. Ordinary count nouns like ‘orange’ also give rise to individuating/measure-

ambiguities.

I-Context: Mary has made punch for the party. She wants to decorate the punch, and she thinks

some ﬂoating fruit would look nice. So she drops several apples, some pears, and four

oranges into the punch. After seeing the fruit ﬂoating in the punch, John asks: “How

many oranges are in the punch?”. Mary responds with (187a):

M-Context: Mary wants to make punch for the party. The recipe calls for four oranges worth of

orange pulp. Accordingly, she takes four whole oranges and pulverizes them, pouring

the resulting pulp into the punch. After tasting the punch, John wants to know the

recipe. He asks: “How many oranges are in the punch?”. Mary responds with (187a):

(187) a. There are four oranges in the punch.

b. There’s a group of four oranges x s.t. each of x is in the punch.

c. There’s a quantity of orange x s.t. x measures four oranges worth and is in the

punch.

In the I-Context, Mary’s utterance receives the individuating interpretation paraphrased

in (187b); in the M-Context, it receives the measure interpretation in (187c). As before,

151 the individuating interpretation results if ‘orange’ functions as an atomic predicate, while the measure interpretation results instead if ‘orange’ gets coerced into a (reflexive) measure noun. Like ‘ounce of water’, both ‘glassful of water’ and ‘orange worth of orange’ are strictly quantized predicates since different quantities of water measuring a glassful can share overlapping (water) parts, just as different quantities of orange measuring one ad hoc orange-unit can share overlapping (orange) parts.

Finally, consider the semantic contrast between telic verbs like ‘read a letter’ and atelic verbs like ‘read letters’. Roughly put, the distinction is between those verbs denoting events which have a natural end point (telic) and those which denote events which are ongoing (atelic). Consider (188).

(188) a. John read {a letter/??letters} in an hour.

b. John read {?a letter/letters} for an hour.

The adverbial ‘in an hour’ speciﬁes the culmination of an event, namely the end of an hour-

long interval. Since ‘read a letter’ has a natural end point but ‘read letters’ does not, only

the former is acceptable with ‘in an hour’. In contrast, ‘for an hour’ speciﬁes the duration

of an ongoing event. Thus, assuming John has a normal reading capacity, the letter is

of normal length, John isn’t distracted, etc., it seems rather unlikely that it would take

him longer than an hour to ﬁnish reading it. Not necessarily so, however, if he is reading

multiple letters, thus explaining the diﬀerence in (188b).

There are well-known parallels between the count/mass distinction and the telic/atelic

distinction. For example, like count nouns, telic verbs are countable; and like mass nouns,

atelic verbs are not.

(189) a. John scored a goal four times.

b. ?? John was scoring goals four times.

152 Also, just as ‘grain’ or ‘ounce’ can be used to quantize an otherwise cumulative noun,

‘for’-adverbials like ‘for an hour’ can be used to similarly “bound” an otherwise ongoing process.

(190) a. ??an ounce of book / an ounce of water

b. ??score a goal for an hour / scoring goals for an hour

For these reasons, many semanticists model the telic/atelic distinction similarly to the count/mass distinction. More speciﬁcally, telic verbs are modeled as atomic, denoting sets of atomic events, while atelic verbs are modeled as cumulative, denoting states or processes.13

In sum, the atomic/non-atomic distinction is not restricted to just the count/mass distinction. We plausibly see the same distinction in pseudopartitives and also in verbs. More- over, we do not need to contrast count nouns with mass nouns to recognize the distinction: individuating and measure interpretations of ‘four oranges’ will suﬃce. What this shows is that the countable/non-countable distinction, and thus the “sortal/non-sortal distinction” as understood here, is essentially context-dependent.14 Whereas grinding and packaging show that whether a noun is atomic depends at least on the syntactic environment in which it occurs, individuation/measure-ambiguities show that extralinguistic context must also be taken into account. After all, ‘oranges’ in (187a) is by all accounts a count noun, yet whether ‘four oranges’ is interpreted as counting oranges or measuring a certain amount of orange will ultimately depend on factors like the speakers’ intentions and purposes in making the utterance, e.g. whether it is intended to give decorating advice or a recipe.

One of the more signiﬁcant consequences of the present proposal is that answering ‘how many’-questions is not suﬃcient for being countable. That’s because strictly quantized predicates like ‘ounce of water’ can answer ‘how many’-questions. This raises an interesting new question: What exactly is the connection between countability and acceptably answering ‘how many’-questions? The answer, I want to suggest, is that a predicate acceptably

13See e.g. Bach(1986a), Krifka(1989), and Link(1998). 14Rothstein(2010) comes to a similar conclusion, though via diﬀerent considerations.

153 answers a ‘how many’-question only if it is quantized (in the singular). If so, then since quantization is not alone suﬃcient to distinguish countable from non-countable predicates, neither is acceptably answering ‘how many’-questions.

5.3.3 Quantization and ‘How Many’-Questions

Let’s begin with a distinction Schwarzchild(2002) draws between two sorts of measure phrases. The ﬁrst kind, which he calls pseudopartitives, take the form (Numeral) + Measure Noun + of + Noun, and include e.g. ‘four ounces of water’, ‘3.86 pounds of potatoes’, etc. The second kind – compounds – take the form (Numeral) + Measure Term

1 + Noun and include e.g. ‘2 2 foot shovel’, ‘8 mile bridge’, etc. Some measure phrases only acceptably form pseudopartitives, while others only acceptably form compounds.

(191) a. I need {four ounces of water/six feet of soil/??100 degrees of water/??24 karats

of gold}.

b. I need some {??four ounce water/??six foot soil/100 degree water/24 karat gold}.

To explain this, Schwarzschild proposes the following generalization: a measure phrase is acceptable in pseudopartitives only if it measures a property which is monotonic with respect to the substance named by the noun; a measure phrase is acceptable in compounds only if it measures a property which is non-monotonic with respect to the substance named by the noun.

The notion of monotonicity involved in Schwarzschild’s generalization has to do with tracking part-whole relations. It is deﬁned formally in (192), where “µ” is a variable over

measure functions (functions from individuals to numbers).

(192) ∀µ.∀x, y. x @ y → µ(x) < µ(y) (Monotonicity)

To illustrate, volume is monotonic with respect to water because if x is a proper part of some

quantity of water y, then the volume of x must be less than that of y. Put diﬀerently, larger

154 quantities of water are more voluminous, and vice versa. On the other hand, temperature is non-monotonic with respect to water since having more water does not imply having hotter water, or vice versa. Hence, Schwarzschild’s generalization correctly predicts that ‘ounce’ should be acceptable with ‘water’ in pseudopartitives but not compounds, while ‘degree’ and ‘water’ should be acceptable in compounds but not pseudopartitives.

Now consider the contrasts in (193).

(193) a. How many {ounces of water/feet of soil} are in this bucket?

b. ?? How many {ounce water/feet soil} are in this bucket?

c. ?? How many {degrees of water/karats of gold} are in this bucket?

The generalization appears to be this: ‘how many’-questions are acceptable with a measure

phrases only if the measure named by the measure term is monotonic with respect to the

substance named by the noun. This would explain why ‘how many’-questions can only be

formed with pseudopartitives, but not with all pseudopartitives.

This is theoretically signiﬁcant because, as Champollion and Krifka(2016) point out,

monotonicity implies quantization. Here’s the “proof”. Suppose x is a quantity of water

measuring an ounce, and let y be a proper part of x. Then since µoz is monotonic with respect to water, µoz(y) < µoz(x). Thus, y is not an ounce of water, and so ‘ounce of water’ is quantized. Likewise with monotonic measure phrases more generally. Combining

this with previous observations suggests the following generalization about ‘how many’-

questions: ‘How many F s ...?’ is acceptable only if F is quantized.

The right conclusion to draw here is that just because we can acceptably ask a question

of the form ‘How many F s are there?’, it does not necessarily follow that we can count

F s. I want to exploit this insight to solve a certain puzzle facing traditional theories of

cardinality within linguistic semantics, including the one assumed throughout this thesis. I

call it “the Sortal Concept Puzzle”.

155 5.4 The Sortal Concept Puzzle

In Chapter 3, recall, I developed a semantics for numerical speciﬁcational sentences based on what I dubbed the “Individual Concept Analysis”. One of the purported advantages of the resulting semantics was that it aﬀorded an independently plausible analysis of ‘how many’-questions according to which (194a) denotes a proposition truly and exhaustively answering that question, or (194b).

(194) a. How many moons does Jupiter have?

b. (194a) w = λw0. ιd. [number-of-Jupiter’s-moons(d, w0)] = J K ιd. [number-of-Jupiter’s-moons(d, w)]

More speciﬁcally, (194a) will denote those worlds w0 in which the maximal cardinality of

Jupiter’s moons in w0 is equal to the maximal cardinality of Jupiter’s moons in w, the world

of evaluation. In plain English, (194a) can be paraphrased as: What’s the number n such

that Jupiter has exactly n-many moons? Exactly similar analyses of ‘how many’-questions

have been independently proposed in the literature.15 What’s common to all of these is

that ‘how many’-questions ask about the cardinality of a collection.

Furthermore, following e.g. Link(1983, 1998), Partee(1986a), Landman(2004), Scontras

(2014), and others, I have assumed throughout this thesis that cardinality is a form of

measurement, and in particular one representing the number of atoms belonging to a given

plurality. Moreover, following Scontras, I argued in Chapter 3 that cardinalities are a sort

of degree, or a nominalized property formed on the basis of a cardinality measure. Two

important consequences follow. First, because ‘how many’-questions ask about cardinalities,

answers to ‘how many’-questions indicate the (maximal) number of atoms constituting a

certain plurality. Secondly, because atoms are “individuated wholes” in the sense that they

do not have proper parts (of the relevant sort), numbers measuring the cardinality of a

15See e.g. Rullman(1993).

156 plurality will always be whole. It’s for this reason that the set of cardinalities is isomorphic to the natural numbers, or the counting numbers.

Put differently, on the traditional analysis of cardinality, counting presupposes individuation. In order to count a certain sort of thing, we first need to be able to distinguish units of that sort. According to Chierchia(1998), certain languages like English mark a difference between those expressions which naturally include such a unit (count nouns) and those which do not (mass nouns).

Consider moreover the impossibility of counting with a mass noun (i.e. the un-

grammaticality of phrases like three furniture). For counting we need to indi-

viduate at a level at which to count... For natural language this has to be a

set of atoms. But a mass noun, unlike count ones, does not correspond to a set

of atoms. Hence it doesn’t provide a suitable counting criterion. That is why

to count a mass noun denotation we need a classiﬁer phrase (like piece of or

truckload of ) or a measure phrase (like tons of ). Classiﬁer phrases map mass

noun denotations into sets of atoms. Measure phrases can be ultimately thought

of as functions from objects into numbers...16

According to Chierchia, counting presupposes atomicity. Since atoms are the sorts of things we semantically individuate, they are the sorts of things we can count. Thus, mass nouns cannot be counted because they are not atomic. In order to count a mass substance, we need a different sort of expression which, when combined with the noun, forms a phrase which denotes a set of atoms. Thus, what Chierchia calls a “classifier phrase” is precisely what Scontras(2014) calls an “atomizer phrase”. These differ from measure phrases in that the former but not the latter denote sets of atoms. Consequently, we can count the denotations of classifier phrases but not necessarily those of measure phrases. Very similar views are expressed by e.g. Link(1998) and Rothstein(2010).

16(Chierchia, 1998, p. 347).

157 Now, here’s the trouble. We have seen that ‘how many’-questions can be answered in

terms of non-whole numbers, as evidenced again by (172).

(172) How many ounces of water are in that bowl? {Four/4.38} (ounces).

So if cardinalities uniformly answer ‘how many’-questions, then ‘4.38 ounces’ in (172) ought

to designate a cardinality. Thus, it appears that cardinalities needn’t be whole after all,

seemingly contrary to the notion developed in Chapter 3. Worse yet, it seems that ‘four

(ounces of water)’ in response to (172) must be counting the number of atoms constituting

some plurality of ounces of water. But this predicts that we can individuate ounces of water,

contrary to the individuation data alluded to earlier.

In sum, we have a puzzle very similar to the one facing Frege’s Ontological Thesis. I

call it the Sortal Concept Puzzle.

The Sortal Concept Puzzle

(SCP1) ‘How many’-questions ask about cardinalities.

(SCP2) Cardinalities represent the number of atoms constituting a plurality.

(SCP3) Atomicity implies individuation.

(SCP4) We cannot semantically individuate ounces of water.

We have seen that each of the premises here appears to be independently plausible, yet they

jointly lead to contradiction. What’s gone wrong?

5.4.1 The Proposed Solution

The right solution, I would argue, is to deny SCP2. Here’s why. First, consider (195).

Context: John says ‘Fred is six feet tall’. Mary replies:

(195) a. Sarah is that tall as well.

158 b. That’s how tall Sarah is as well.

Here, both (195a) and (195b) are naturally interpreted as true just in case Sarah is six

feet tall. Naively, that’s because both ‘that tall’ in (195a) and ‘how tall Sarah is’ in (195b)

demonstratively refer to some height, namely Fred’s, which happens to be six feet, and both

examples ascribe to Sarah the property of having that height as well.

Now consider the parallel examples in (196).

Context: John says ‘Fred ate four oranges’. Mary replies:

(196) a. Sarah ate that many oranges as well.

b. That’s how many oranges Sarah ate as well.

As before, both (196a) and (196b) are naturally interpreted as true just in case Sarah ate

four oranges. Again, that’s naively because ‘that many oranges’ in (196a) and ‘how many

oranges Sarah ate’ in (196b) both demonstratively refer to some cardinality, namely how

many oranges Fred ate, which happens to be four, and both examples ascribe to Sarah the

property of having eaten that many oranges as well.

Finally, consider (197).

Context: John says ‘Fred drank 4.38 ounces of water’. Mary replies:

(197) a. Sarah drank that many ounces of water as well.

b. That’s how many ounces of water Sarah drank as well.

As with the previous examples, both (197a) and (197b) are naturally interpreted as true

just in case Sarah drank 4.38 ounces of water. Again, this is naively because ‘that many

ounces of water’ in (197a) and ‘how many ounces of water Sarah drank’ in (197b) both

demonstratively refer to a cardinality, namely how many ounces of water Fred drank, which

happens to be 4.38, and both examples ascribe to Sarah the property of having drank that

many ounces of water as well.

159 The intuition here, following e.g. Rett(2008) and Solt(2009), is that ‘many’ is asso-

ciated with a particular dimension of measurement, namely cardinality. It is opposed to

‘much’, which can be associated with various dimensions, e.g. volume, depth, or weight.

Consequently, both (198a) and (198b) are naturally understood as claiming that Sarah

drank the same amount of water Fred did, namely 4.38 ounces.

Context: John says ‘Fred drank 4.38 ounces of water.’ Mary replies:

(198) a. Sarah drank that much water as well.

b. That’s how much water Sarah drank as well.

This would make sense if ‘that much water’ in (198a) and ‘how much water Sarah drank’

in (198b) both demonstratively referred to the same volume, namely how much water Fred

drank, which happened to be 4.38 ounces, and if both examples attribute to Sarah the

property of having drank the same amount of water.

What this suggests is that ‘4.38 ounces of water’ can refer to diﬀerent sorts of degrees,

namely a cardinality and a volume. However, this would obviously be impossible if SCP2

were true since ‘ounce of water’ is not an atomic predicate. The question I want to ask

next is how ‘4.38 ounces of water’ can refer to diﬀerent degrees given the Degree-as-Kind

Analysis assumed here. The answer, I want to suggest, is that measures of both sorts are

really ratios.

5.4.2 Cardinalities are Ratios of Magnitudes?

Up to now, following e.g. Link(1983), and Partee(1986a), Landman(2004), and Scontras

(2014), I have been assuming that measure terms such as ‘ounce’, ‘foot’, etc. denote measure

functions, or functions from individuals to numbers. On this picture, ‘four feet of water’

denotes quantities of water measuring four feet, as suggested in (199b).

(199) a. foot = λn.λP.λx. P (x) ∧ µft(x) = n J K 160 b. four feet of water = λx. water(x) ∧ µft(x) = 4 J K However, this analysis is arguably too simplistic. As some have pointed out, ‘four feet of water’ could denote quantities of water measuring four feet in diameter or quantities of water measuring four feet in depth. Since these needn’t be the same quantities of water, Lønning

(1987) and Schwarzchild(2002) argue that measure terms have an additional dimension of semantic complexity. For example, ‘foot’ is said to denote a function mapping individuals to some contextually determined unit of measurement, which then gets mapped to a number via what Champollion and Krifka(2016) call a unit function. As a result, ‘four feet of water’ could denote diﬀerent quantities of water in diﬀerent contexts, depending on which unit of measurement is relevant: a foot measured in terms of diameter or a foot measured in terms of depth.

As with measure terms, I have been assuming up to now that cardinality is given in terms of a measure function ‘µ#’ which returns the number of atomic individuals constituting a plurality. However, I now want to suggest that this too is overly simplistic. According to Krifka(1989)’s analysis, recall, a count noun like ‘orange’ is associated with an unpro- nounced classiﬁer-like element he calls a “natural unit” (NU), so that ‘four oranges’ denotes quantities of orange measuring four “natural units” of orange, or (200b).

(200) a. orange = λn.λx. orange(x) ∧ NU(orange)(x) = n J K b. four oranges = λn.λx. orange(x) ∧ NU(orange)(x) = 4 J K The important point here is that measuring cardinality includes an additional layer of semantic complexity on Krifka’s analysis: it can only be determined with respect to a

“natural unit” of some property.

In a nutshell, I want to propose that both sorts of quantized predicates discussed earlier

– atomic predicates and strictly quantized predicates – can have cardinalities, and that these are determined with respect to something like Krifka’s “natural unit”. More speciﬁcally, a “natural unit” of a quantized predicate is a member of its extension. Thus, for an

161 atomic predicate like ‘orange’, for instance, cardinality will be determined with respect to an atomic orange. But for strictly quantized predicates like ‘ounce of water’, a cardinality will be determined with respect to a quantity of water measuring an ounce.

Now, the basic question here is how ‘4.38 ounces of water’ can come to refer to both a degree of volume and also a degree of cardinality. Suppose ﬁrst that we have a syntactic structure like the following.

of water

4.38 ounces

Consequently, ‘4.38 ounces’ is functioning as a typical measure phrase, and so ‘ounce’ supplies the relevant measure on the Adjectival Theory. This gives us something like the following semantic representation, where ‘oz’ is a function from individuals to a contextually- determined unit-function u.

(201) 4.38 ounces of water = λx. water(x) ∧ oz(u)(x) = 4.38 J K Now suppose that we have the following syntactic structure instead.

4.38 ounces of water

In this case, ‘4.38 ounces’ is not occurring as a measure phrase. Rather, ‘4.38’ occurs without an overt measure term, and so ‘4.38’ is specifying a cardinality on the Adjectival

Theory. Given the Krifka-inspired analysis of cardinality just alluded to, suppose this gives us something like the following semantic representation, where ‘ounce-of-water’ in the second conjunct supplies the relevant “natural unit”.

162 (202) 4.38 ounces of water = λx. ounces-of-water(x)∧NU(ounce-of-water)(x) = J K 4.38

The question now is how to interpret the representations in (201) and (202).

It seems to me that a natural suggestion, following the measurement-theoretic treatment

of Sassoon(2010), would be to interpret the measures here as determining ratios. To

illustrate, Sassoon analyzes measure phrases like ‘four meters tall’ as indicated in (203), so

that something is four tall just in case its height is four times the length of a meter.

(203) four meters tall = λx. x is n times as tall as the meter J K Similarly, ‘four feet of water’ will denote those quantities of water whose diameter or depth

measures four times that of a foot of water.

My proposal is that we understand the measures in (201) and (202) similarly. More

speciﬁcally, the second conjunct of (201) determines a ratio between a certain quantity of

water x and contextually-supplied unit of measurement u, and “says” that the measure of

x is 4.38 times that of u. Supposing we are talking about volume, (201) will be true of

those quantities of water whose volume measures 4.38 times that of a single ounce of water.

Similarly, the second conjunct of (202) determines a ratio between a certain quantity of

water x measured in terms of ounces and a particular ounce of water y such that the

cardinality of x is 4.38 times that of y.

From here, it is straightforward to see how ‘4.38 ounces of water’ can denote a degree of

volume or degree of cardinality given the Degree-as-Kind Analysis: we simply nominalize

the properties in (201) and (202), thus resulting in (204a,b) (respectively).

(204) a. ∩λx. water(x) ∧ oz(u)(x) = 4.38

b. ∩λx. ounces-of-water(x) ∧ NU(ounce-of-water)(x) = 4.38

163 Supposing u is a unit of volume, (204a) will refer to a degree of volume. And since (204b)

is formed on the basis of cardinality, it refers to a degree of cardinality. The former is an

appropriate referent for examples involving ‘much’ like (205a,b),

Context: John says ‘Fred drank 4.38 ounces of water.’ Mary replies:

(205) a. Sarah drank that {many ounces of water/much water} as well.

b. That’s {how many ounces of water/how much water} Sarah drank as well.

while the latter would be appropriate for similar examples involving ‘many’.

Admittedly, the solution just sketched to the Sortal Concept Puzzle is very sketchy.

Nevertheless, an immediate advantage is that it purports to explain how both sorts of

quantized predicates can have cardinalities, and also why cardinalities formed on the basis

of counting are isomorphic to the natural numbers. That’s because cardinalities formed on

the basis of counting are a special case; they measure the ratio of a plurality of atoms to a

single atom. Since this will always result in a whole number when represented as a fraction,

cardinalities formed on the basis of counting will be structurally equivalent to the counting

numbers. This might explain contrasts like the following, for instance.

(169a) How many cats are on the mat? {Four/??4.38} cats.

(169b) How much water is in the bowl? {Four/4.38} ounces.

(172) How many ounces of water are in that bowl? {Four/4.38} (ounces).

It also preserves the predominant intuition that counting presupposes individuation. Even

though ‘that many ounces of water’ can refer to a cardinality, because ‘ounce of water’ is

not atomic, it does not follow that we can count ounces of water. Thus, the present proposal

also preserves the individuation data.

Suppose something like this proposed solution to the Sortal Concept Puzzle is right.

What would follow? Philosophically speaking, the headline would appear to be that count-

ing is just another form of measurement, and that the linguistic evidence is strongly against

164 Frege’s Ontological Thesis. Indeed, the emerging picture is that the natural numbers are

“embedded within the reals”, so to speak, just as Tennant(2010) suggests. 17 That is, nat-

ural language presupposes a single domain of numbers, which we might suppose to be the

reals,18 and these are the referents of numerals occurring within measurement constructions

of various sorts, including ‘four oranges’. Still, counting enjoys a special role amongst the

various forms of measurements, semantically speaking. We see this manifested in diﬀerent

ways, e.g. in the contrast between atomizer and measure phrases and individuating and

measure interpretations.

As for linguistics, the present proposal would provide further support for what Fox and

Hackl(2007) call the Universal Density of Measurement (UDM), or the thesis that all natural language measurement involves mapping to dense scales.

Measurements of height, weight, and the like are commonly thought of as map-

pings between objects and dense scales, while measurements of collections and

individuals, as implemented for instance in counting, are assumed to involve

discrete scales. It is also commonly assumed that natural languages make use

of both types of scales and subsequently distinguishes between two types of

measurement ... [However,] natural language semantics treats all measurement

uniformly as mappings from objects (individuals or collections of individuals) to

dense scales, hence the universal density of measurement.19

Fox and Hackl present a variety of data in support of UDM, including data involving

implicatures, deﬁnite descriptions, and questions like (206).

(206) a. Which books did John not read?

b. ?? How many children does John not have?

17See fn. 1, p. 125. 18Cf. Krifka(1989), Kennedy(1999), and Scontras(2014). 19(Fox and Hackl, 2007, p. 537).

165 Nouwen(2008) comes to the same conclusion based on similar sorts observations. However,

these arguments rely crucially on a controversial analysis of certain kinds of implicatures.20

The argument for UDM given here is much simpler and in some ways more direct. On all

semantic theories I am familiar with, ‘many’ is associated with the dimension of cardinality.

Consequently, the most natural assumption to make is that ‘that many ounces of water’

in the examples above refers to a certain sort of degree, namely a cardinality, just as it is

usually assumed that the underlined expressions in (207) demonstratively refer to degrees

of diﬀerent sorts (heights, weights, ages).21

(207) a. [Pointing at a meter stick:] Mary is that tall.

b. [Pointing at a scale:] John was arrested for that amount of cocaine.

c. [Holding up three ﬁngers:] Mary is that much older than John (in years).

Likewise with ‘that much water’ in the examples above, which refers to a particular volume

(let’s say). This implies that degrees of cardinality have the same structure as degrees of

volume, and so scales formed from sets of those degrees will both be dense and continuous.

I want to emphasize that measure phrases are not really needed to make this point. In

fact, the same point can be made using examples like (196), repeated here but with more

context ﬁlled in.

Context: Fred and Sarah are having a contest to see who can eat the most orange slices (each

from randomly diﬀerent oranges). John says ‘Fred ate four oranges’. Mary replies:

(196a) Sarah ate that many oranges as well.

(196b) That’s how many oranges Sarah ate as well.

Clearly, John’s utterance receives a measure interpretation in this scenario: he is not claim-

ing that Fred ate four whole oranges, but rather four oranges’ worth of orange. Likewise

20In particular, they rely crucially on the presence of a so-called exhaustivity-operator. 21See Scontras(2014) for similar examples and relevant discussion.

166 with Sarah’s utterances of (196a,b). Nevertheless, it seems hard to deny that ‘that many oranges’ in (196a) and ‘how many oranges Sarah ate’ in (196b) refers to a cardinality.

167 References

Altham, E. and N. Tennant (1975). Sortal quantiﬁcation. In E. Keenan (Ed.), Formal

Semantics of Natural Language, pp. 46–58. Cambridge University Press.

Anderson, C. and M. Morzycki (2013). Degrees as kinds. Natural Language and Linguistic

Theory 33, 791–828.

Bach, E. (1986a). The algebra of events. Linguistics and Philosophy 9, 5–16.

Bach, E. (1986b). Natural language metaphysics. In R. B. Marcus and et al (Eds.), Logic,

Methodology, and the Philosophy of Science VII, pp. 573–595.

Balcerak-Jackson, B. (2013). Defusing easy arguments for numbers. Linguistics and Phi-

losophy 36, 447–461.

Barker, C. (1998). Partitives, double genitives, and anti-uniqueness. Natural Language and

Linguistic Theory 4, 679–717.

Barwise, J. and R. Cooper (1981). Generalized quantiﬁers and natural language. Linguistics

and Philosophy 1, 413–458.

Carlson, G. (1977a). Amount relatives. Language 53 (3), 520–542.

Carlson, G. (1977b). Amount relatives. Language 53, 520–542.

Champollion, L. and M. Krifka (2016). Mereology. In P. Dekker and M. Aloni (Eds.),

Cambridge Handbook of Semantics. Cambridge University Press.

Chierchia, G. (1984). Topics in the Syntax and Semantics of Inﬁnitives and Gerunds. Ph.

D. thesis, University of Massachusetts at Amherst.

168 Chierchia, G. (1998). Reference to kinds across languages. Natural Language Semantics 6,

339–405.

Chierchia, G. and S. McConnell-Ginet (1990). Meaning and Grammar: an Introduction to

Semantics. MIT Press.

Comorovsky, I. (2008). Constituent questions and the copula of speciﬁcation. In I. Co-

morovsky and K. von Heusinger (Eds.), Existence: Semantics and Syntax, pp. 49–77.

Springer.

Dowty, D., R. Wall, and S. Peters (1981). Introduction to Montague Semantics. D. Reidel.

Dummett, M. (1991). Frege: Philosophy of Mathematics. Duckworth.

Felka, K. (2014). Number words and reference to numbers. Philosophical Studies 168,

261–268.

Fox, D. and M. Hackl (2007). The universal density of measurement. Linguistics and

Philosophy 29, 537–586.

Frana, I. (2006). The de re analysis of concealed questions: A uniﬁed approach to deﬁnite

and indeﬁnite concealed questions. In M. Gibson and J. Howell (Eds.), Proceedings of

SALT 16.

Frege, G. (1884). Die Grundlagen der Arithmetik. Koebner, Breslau. Translated by J.L.

Austin as The Foundations of Arithmetic. Blackwell, Oxford, 1950.

Frege, G. (1903). Grundgesetze der Arithmetik II. Olms.

Geurts, B. (2006). Take ‘ﬁve’. In S. Vogleer and L. Tasmowski (Eds.), Non-Deﬁniteness

and Plurality, pp. 311–329. Benjamins.

Grandy, R. (2007). Sortals. In E. Zalta (Ed.), Standford Encyclopedia of Philosophy.

169 Greenberg, B. (1977). A semantic account of relative clauses with embedded question

interpretations. Ms., University of California, Los Angeles.

Grice, P. (1989). Studies in the Way of Words. Harvard University Press.

Grimm, S. (2012). Number and Individuation. Ph. D. thesis, Stanford University.

Hale, B. (2016). Deﬁnitions of numbers and their applications. In P. Ebert and M. Rossberg

(Eds.), Abstractionism. Oxford University Press.

Hale, B. and C. Wright (2001). The Reason’s Proper Study: Towards a Neo-Fregean Phi-

losophy of Mathematics. Oxford University Press.

Heck, R. (1997). Finitude and hume’s principle. Journal of Philosophical Logic 26, 589–617.

Heim, I. (1979). Concealed questions. In R. Bauerle, U. Egli, and A. von Stechow (Eds.),

Semantics from Diﬀerent Points of View, pp. 51–60. Springer.

Heim, I. (1987). Where does the deﬁniteness restriction apply? evidence from the deﬁ-

niteness of variables. In E. Reuland and A. ter Meulen (Eds.), The Representation of

(In)Deﬁniteness, pp. 21–42. MIT Press.

Higgins, R. (1973). The Pseudo-cleft Construction in English. Garland.

Hodes, H. (1984). Logicism and the ontological commitments of arithmetic. The Journal

of Philosophy 81, 123–149.

Hofweber, T. (2005). Number determiners, numbers, and arithmetic. The Philosophical

Review 114, 179–225.

Hofweber, T. (2007). Innocent statements and their metaphysically loaded counterparts.

Philosopher’s Imprint 7, 1–33.

Hofweber, T. (2014). Extraction, displacement, and focus: a reply to balcerak-jackson.

Linguistics and Philosophy 37, 263–267.

170 Horn, L. (1972). On the Semantic Properties of Logical Operators. Ph. D. thesis, University

of California, Los Angeles.

Ionin, T., O. Matushansky, and E. Ruys (2006). Parts of speech: Towards a uniﬁed seman-

tics for partitives. In NELS 36: Proceedings of the North East Linguistics Society, pp.

357–370.

Karttunen, L. (1977). Syntax and semantics of questions. Linguistics and Philosophy 1,

3–44.

Kennedy, C. (1999). Projecting the Adjective: The Syntax and Semantics of Gradability

and Comparison. Garland.

Kennedy, C. (2012). Adjectives. In G. Russell and D. Graﬀ-Fara (Eds.), Routledge Com-

panion to the Philosophy of Language. Routledge.

Kennedy, C. (2013). A scalar semantics for scalar readings of number words. In I. Capon-

igro and C. Cecchetto (Eds.), From grammar to meaning: the spontaneous logicality of

language, pp. 172–200. Cambridge University Press.

Kennedy, C. and J. Stanley (2009). On average. Mind 118, 586–646.

Koslicki, K. (1997). Isolation and non-arbitrary division: Frege’s two criteria for counting.

Synthese 112, 403–430.

Krifka, M. (1989). Nominal reference, temporal constitution, and quantiﬁcation in event

semantics. In J. von Bentham, R. Bartsch, and P. von Emde Boas (Eds.), Semantics and

Contextual Expressions. Foris.

Krifka, M. (1990). Boolean and non-boolean and. In L. Kalman and L. Polos (Eds.), Papers

from the Second Symposium on Logic and Language. Akademmiai Kiado.

Krifka, M. (2008). Diﬀerent kinds of count nouns and plurals. Ms., ZAS Berlin and Hum-

boldt Universit¨atzu Berlin.

171 Ladusaw, W. A. (1982). Semantic constraints on the english partitive construction. Pro-

ceedings of WCCFL 1 1, 231–242.

Landman, F. (2003). Predicate-argument mismatches and the adjectival thoery of indeﬁ-

nites. In M. Coene and Y. D’hulst (Eds.), NP to DP. John Benjamins.

Landman, F. (2004). Indeﬁnites and the Type of Sets. Blackwell.

Landman, F. (2011). Count nouns – mass nouns – neat nouns – mess nouns. The Baltic

Yearbook of Cognition, Logic, and Communication 6, 1–67.

Lasersohn, P. (1999). Pragmatic halos. Language 75 (3), 522–551.

Lassiter, D. (2011). Measurement and Modality: the Scalar Basis of Modal Semantics. Ph.

D. thesis, Ney York University.

Link, G. (1983). The logical analysis of plurals and mass terms: A lattice-theoretic approach.

In R. B¨auerle,C. Schwarze, and A. von Stechow (Eds.), Meaning, Use, and Interpretation

of Langauge, pp. 303–323.

Link, G. (1998). Algebraic Semantics in Language and Philosophy. SCLI Publications.

Lønning, J. (1987). Mass terms and quantiﬁcation. Linguistics and Philosophy 10, 1–52.

Mikkelsen, L. (2005). Copular Clauses: Speciﬁcation, Predication, and Equation. Ben-

jamins.

Mikkelsen, L. (2011). Copular clauses. In K. von Heusinger, C. Maienborn, and P. Portner

(Eds.), Semantics: An International Handbook of Natural Language Meaning. de Gruyter.

Moltmann, F. (2013a). Reference to numbers in natural language. Philosophical Studies 162,

499–536.

Moltmann, F. (2013b). The semantics of existence. Linguistics and Philosophy 36, 31–63.

172 Nouwen, R. (2008). Upper-bounded no more: the exhaustive interpretation of non-strict

comparison. Natural Language Semantics 16, 271–295.

Partee, B. (1986a). Ambiguous pseudoclefts with unambiguous be. In S. Bergman, J. Choe,

and J. McDonough (Eds.), Proceedings of the Northwestern Linguistics Society 16. GLSA.

Partee, B. (1986b). Noun phrase interpretation and type-shifting principles. In J. Groe-

nendijk, D. de Jongh, and M. Stokhof (Eds.), Studies in Discourse Representation Theory

and the Theory of Generalized Quantiﬁers. Foris.

Partee, B. (2004). Compositionality in Formal Semantics. Blackwell Publishing.

Partee, B. and V. Borschev (2012). Sortal, relational, and functional interpretations of

nouns and russian container constructions. Journal of Semantics 29, 445–486.

Partee, B. and M. Rooth (1983). Generalized conjunction and type ambiguity. In R. Bauerle,

C. Schwarze, and A. von Stechow (Eds.), Meaning, Use, and Interpretation of Language,

pp. 361–383. De Gruyter.

Pelletier, F. J. (1975). Non-singular reference: some preliminaries. Philosophia 5, 451–465.

Quine, W. (1960). Word and Object. MIT Press.

Reinhart, T. and T. Siloni (2005). The lexicon-syntax parameter: Reﬂexivization and other

arity operations. Linguistic Inquiry 36 (3), 389–436.

Rett, J. (2008). Degree Modiﬁcation in Natural Language. Ph. D. thesis, Rutgers University.

Romero, M. (2005). Concealed questions and speciﬁcational subjects. Linguistics and

Philosophy 28 (6), 687–737.

Rothstein, S. (2009). Individuating and measure readings of classiﬁer constructions: Evi-

dence from modern hebrew. Brill’s Annual of Afroasiatic Languages and Linguistics 1,

106–145.

173 Rothstein, S. (2010). Counting, measuring and the semantics of classiﬁers. Baltic Interna-

tional Handbook of Cognition, Logic and Communication 6.

Rullman, H. (1993). Scope ambiguities in how many-questions. Ms. University of Mas-

sachusetts, Amherst.

Safir, F. (1982). Syntactic Chains and the Definiteness Effect. Ph. D. thesis, Massachussetts

Institute of Technology.

Salmon, N. (1997). Wholes, parts, and numbers. Nous Supplement: Philosophical Perspec-

tives 11: Mind, Causation, and World 31, 1–15.

Sassoon, G. (2010). Measurement theory in linguistics. Synthese 174, 151–180.

Schlenker, P. (2003). Clausal equations. Natural Language and Linguistic Theory 21, 157–

214.

Schwarzchild, R. (2002). The grammar of measurement. In Proceedings of Semantics and

Linguistic Theory (SALT).

Schwarzschild, R. (2005). Measure phrases as modiﬁers of adjectives. Recherches Linguis-

tiques de Vincennes 34, 207–228.

Scontras, G. (2014). The Semantics of Measurement. Ph. D. thesis, Harvard University.

Shapiro, S. (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University

Press.

Shapiro, S. (2000). Thinking About Mathematics: the Philosophy of Mathematics. Oxford

University Press.

Sharvy, R. (1980). A more general theory of deﬁnite descriptions. The Philosophical Re-

view 89, 607–624.

174 Simons, P. (1987). Frege’s theory of the real numbers. History and Philosophy of Logic 8,

25–44.

Snyder, E. and J. Barlew (2016). The universal measurer. In Proceedings of Sinn und

Bedeutung 20.

Snyder, E. and S. Shapiro ((accepted) 2014). Frege on the reals. In P. Ebert and M. Rossberg

(Eds.), Essays on Frege’s Basic Laws of Arithmetic. Oxford University Press.

Solt, S. (2009). The Semantics of Adjectives of Quantity. Ph. D. thesis, The City University

of New York.

Tennant, N. (1987). Anti-Realism and Logic: Truth as Eternal. Clarendon Library of Logic

and Philosophy, Oxford University Press.

Tennant, N. (1997). On the necessary existence of numbers. Nous 31, 307–336.

Tennant, N. (2010). What are the reals really? Ms., The Ohio State University.

Wright, C. (2000). Neo-fregean foundations for real analysis: Some reﬂections on frege’s

constraint. Notre Dame Journal of Formal Logic 41, 317–334.

Yablo, S. (2005). The myth of the seven. In M. Kalderon (Ed.), Fictionalist Approaches to

Metaphysics, pp. 90–115. Oxford University Press.

175