FACTOR PHRASES: THE SEMANTICS OF MULTIPLICATIVE MODIFICATION OF
EVENTS, DEGREES, AND NOMINALS, AND THE GRAMMAR OF ARITHMETIC

By

Adam Michael Gobeski

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Linguistics – Doctor of Philosophy

2019

ABSTRACT

FACTOR PHRASES: THE SEMANTICS OF MULTIPLICATIVE MODIFICATION OF
EVENTS, DEGREES, AND NOMINALS, AND THE GRAMMAR OF ARITHMETIC

By

Adam Michael Gobeski

Factor phrases – modifiers such as twice or three times – are objects that show up cross-categorially,

and yet their semantics remains largely neglected, with virtually no work done on them. This

dissertation thus redresses the balance, examining factor phrases in three different domains: the

verbal domain (Floyd walked the dog three times), the adjectival domain (Floyd is three times as

tall as Clyde), and the nominal domain (Floyd has twice Clyde’s wisdom). I argue that while the

verbal form is a type of event counter, as described by Landman (2004), the adjectival and nominal
cases are instead modifiers of type (cid:104)d, d(cid:105): a kind of modifier called a ratio degree. I show how
these ratio degrees interact with degree morphemes such as as and -er, including explaining why,

in English, Floyd is three times as tall as Clyde and Floyd is three times taller than Clyde mean the

same thing, despite the meanings of as and -er not being equivalent. This multiplicative use also

extends into the verbal domain, with sentences such as Floyd walked the dog three times as many

times as Clyde did, and we see that ratio degrees also work successfully in this domain. The use of

ratio degrees is then extended to the nominal domain, with a number of novel observations about

the various kinds of nouns that can occur (mass, count, etc.) and their interactions with factor

phrases. The dissertation concludes with a discussion of basic arithmetic, including a detailed

syntactic and semantic analysis of basic arithmetic phrases such as Three times seven is twenty-one

and Fourteen divided by two is seven – an area that has been almost completely neglected. I show

that these sentences still follow the same basic syntactic rules as the rest of natural language, and

that consequently we can use the tools of natural language to provide a semantics for these basic

arithmetic operations.

Copyright by
ADAM MICHAEL GOBESKI
2019

ACKNOWLEDGEMENTS

“Hey,” I said one day, sitting in the office of my advisor, Marcin Morzycki, “how come you can

say twice as tall but not twice taller?”

“That’s a good question,” he probably replied (it’s been a few years, so the details are being

reconstructed here); “why don’t you go figure it out and report back?”

And so that innocent query led to a path of research which led to this dissertation, as I dis-

covered that this was a virtually unexplored part of the grammar with loads of cool puzzles and

observations that had been left untouched. Many many people have contributed along the way,

both with ideas and suggestions and with moral support. So let me take a moment to acknowledge

them, with the caveat and apology that if I don’t mention a particular name, it has more to do with

the state of my memory than with the size of the contribution they made – particularly given that

this took longer than I had initially anticipated. (Pro tip: move across the country after you finish

your PhD program, not before.)

Marcin Morzycki is one of the best people one could ask for as an advisor; his energy and

enthusiasm for semantic puzzles is infectious, such that even when my own spirits were flagging

about a particularly vexed corner of factor phrases, his genuine interest would often renew my

confidence and my own interest in a given puzzle, and his habit of giving me the occasional in-

tellectual nudge (“You keep trying to do this with X, but have you considered trying it with Y?”)

at just the right moment, or of helping me focus my random thoughts into a coherent picture, is

truly appreciated. I should also mention how much I appreciated his patience, for those meetings

when I’d come in and say, “I haven’t gotten any further on this problem” and he would help me

work through it from a new angle or a reorganization of the facts, rather than simply dismissing me

– and in fact, those were often the meetings where breakthroughs would happen. Consequently,

this thesis would be much poorer without his input and support. And his knack of writing about

things in a clear and entertaining way, while making it apparent why a given puzzle is Cool and

Important, is a rare gift, and one that I’ve done my best to emulate.

iv

The rest of my committee, both past and present, deserve equal praise. Alan Munn matches

a dangerously sharp intellect with a friendly sense of encouragement, such that when he points

out a particular issue or concern it’s because he wants you to properly consider it and therefore

make your work better, and so our conversations were always worthwhile. Cristina Schmitt has

an encyclopedic knowledge of the literature and so can easily say, “I think this person has talked

about times in the verbal context a bit,” thus pointing you in a direction you may not have other-

wise thought to go down, and her questions are always the right kind of challenging, making you

consider some of the broader ramifications of your analysis. Suzanne Wagner benefits from being

both incredibly organized and friendly, such that when you’re encountering a problem either lin-

guistic or bureaucratic, she never makes it seem like you’re bothering her, and she has the ability to

consider things from a larger perspective – always useful when you’re stuck in the gritty details of

a problem. And thanks to Alan Beretta, with whom I enjoyed many intellectual and philosophical

questions over a drink (ginger ale in my case) after lab meetings; I’m sorry that the neurolinguistic

aspects didn’t pan out, but that doesn’t mean I didn’t enjoy the conversations.

I had the pleasure of engaging with a large number of very smart and clever students during

my time in the Linguistics program, including (but by no means limited to) Curt Anderson, Kai

Chen, Alex Clarke, Yan Cong, Shannon Cousins, Karl DeVries, Olga Eremina, Cara Feldscher,

Hannah Forsythe, Josh Herrin, Matt Husband, Joe Jalbert, Greg Johnson, Matt Kanefsky, Tae-

hoon Kim, Ai Kubota (née Matsui), Adam Liter, Kali Morris (née Bybel), Phil Pellino, Gabriel

Roisenberg-Rodrigues, Isaac Sarver, Kay Ann Schlang, Ai Taniguchi, Drew Trotter, and everyone

who participated in the MSU Semantics Group (aka Awkward Time) and listened to me ramble on

about math and related topics. Your feedback, support, and friendship cannot be overstated.

I’d also like to mention a few of my non-linguistics friends who’ve been unflaggingly sup-

portive of me during this process, including Daniel Baker, Tracey and Quinn Canole and kids,

Alex Clark (who’s different from Alex Clarke), Tony Huff, Jason King, Erika Koeppe, CJ O’Hara,

Charlie Wallace, and Paul Wilcox; your friendship has meant the world to me, and I’m incredibly

grateful for it.

v

A special thanks to my family, both by marriage (Dennis Kiley, Trish Kiley, Bridget Kiley,

Ryan and Anna Kiley, and the rest of the extended clan) and by blood (all the Gobeskis and the

Gatzas out there – there are simply too many of you to name!), with a special mention for my

brother Douglas, who has always been willing to try to understand what I’ve been discussing, se-

mantically speaking, and has never stopped driving me forward, and my parents Ann and Keith,

who have been incredibly supportive and understanding, even when some of my projected mile-

stones began to slide a bit. My father in some respects deserves the credit for all of this, as it was

he who, after noting my predilection for taking a semester or two of various different language

courses, said, “You know, there’s a major that deals with language in general...”

And last but of course not least, a special thanks to my wife Brianne, who has always been there

for me and never stopped believing in me, even when I sometimes didn’t share her confidence. You

are incredibly smart, funny, and caring, and I’m extremely lucky to have you in my life.

vi

TABLE OF CONTENTS

LIST OF TABLES .

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ix

CHAPTER 1

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1.1 The basic problem .
1.2 Formal assumptions .

INTRODUCTION .
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CHAPTER 2 BACKGROUND .
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2.1 Numbers .

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2.2 Degrees, gradable adjectives, and comparatives

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2.1.1 The ontology of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
. . . . . . . . . . . . . . . . . . . . 16
2.1.2 The syntax and semantics of numbers
. . . . . . . . . . . . . . . . . . . 23
2.2.1 Degrees and gradable adjectives
. . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 The syntax and semantics of comparatives . . . . . . . . . . . . . . . . . . 26
2.2.3 Numbers as quantities versus numbers as names . . . . . . . . . . . . . . . 36
2.2.4 The nature of degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.1
Plurals in the nominal domain . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.2 Events and plurals in the verbal domain . . . . . . . . . . . . . . . . . . . 48

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2.3 Plurality .

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3.1 Previous research .

CHAPTER 3 FACTOR PHRASES IN THE VERBAL DOMAIN . . . . . . . . . . . . . . 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Doetjes (1997) – Times as classifier
. . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Landman (2004) – Definite times adverbials and degrees
. . . . . . . . . . 54
3.1.3 Landman (2004) – Indefinite times adverbials and groups . . . . . . . . . . 58
3.2 Verbal factor phrases and comparatives . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.1 Additive factor phrases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Scales of measurement vs scales of plurality . . . . . . . . . . . . . . . . . 72
3.2.2
3.2.3 One times or two? .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.4 Multiplicative factor phrases . . . . . . . . . . . . . . . . . . . . . . . . . 80
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
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Less/fewer than .
Fractions
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3.3 Unresolved issues .

3.4 Conclusion .

3.3.1
3.3.2

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CHAPTER 4 FACTOR PHRASES IN THE ADJECTIVAL DOMAIN . . . . . . . . . . . 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1 Previous research .
4.2 The basics of adjectival factor phrases
. 99
4.3 Percentages and ratio degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4 Adjectival factor phrases and ratio degrees . . . . . . . . . . . . . . . . . . . . . . 110
4.5 Ratio degrees and multiplicative verbal comparatives
. . . . . . . . . . . . . . . . 116
4.6 Twice
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

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vii

4.7 Conclusion . .

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5.1 Data .

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CHAPTER 5 FACTOR PHRASES IN THE NOMINAL DOMAIN . . . . . . . . . . . . . 125
. 125
5.1.1 Dimensional and evaluative nouns . . . . . . . . . . . . . . . . . . . . . . 125
5.1.2
Factor phrases and relative clauses . . . . . . . . . . . . . . . . . . . . . . 130
5.1.3 Additional distinctions of the different kinds of nouns . . . . . . . . . . . . 137
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.1 Dimensional nouns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
. 144
5.2.2 Mass nouns and property concepts . . . . . . . . . . . . . . . . . . . . .
5.2.3 Quality readings .
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. 152
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5.3 Conclusion . .

5.2 Analysis .

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Introduction .

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6.2.1 Verb? . .
6.2.2 Noun? . .
6.2.3 Conjunction? .
Preposition? . .
6.2.4

CHAPTER 6 THE SYNTAX AND SEMANTICS OF ARITHMETIC . . . . . . . . . . . 158
. 158
6.1
6.2 The syntactic category of arithmetic terms . . . . . . . . . . . . . . . . . . . . . . 159
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.3 The syntax of multi-morphemic arithmetic terms . . . . . . . . . . . . . . . . . . . 174
6.4 Other related syntactic phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.1 Colloquial versions of multi-morphemic mathematical phrases . . . . . . . 180
. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4.2 Other math prepositions
6.4.3 Mixing mono-morphemic and multi-morphemic math terms
. . . . . . . . 183
6.5 The semantics of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
. . . . . . . . . . . . . . . . . . . . . . 187
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.5.1 The semantics of arithmetic times

6.6 Conclusion .

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CHAPTER 7 CONCLUSION .

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BIBLIOGRAPHY .

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viii

LIST OF TABLES

Table 5.1: Nouns types and their interactions with BE and HAVE . . . . . . . . . . . . . . 137

ix

CHAPTER 1

INTRODUCTION

1.1 The basic problem

Consider the following sentences:

(1)

a. Floyd saw the movie four times.

b. Kirsten is three times as tall as Dennis.

c. Mindy has a hundred times Jack’s intelligence.

d. You’re twice the man your father was.

The sentences in (1) all involve modifiers of either the form N times or (in the case of (1d)) twice.

They all perform a kind of counting operation, either directly in the case of (1a), or via multipli-

cation in the other three cases. What’s more, these modifiers span a wide range of categories: (1a)

modifies a verb phrase, (1b) an adjective, and (1c) and (1d) nominal phrases: a possessive NP and

a relative clause headed by an NP, respectively.

Despite their prevalence in these varied areas of language, these modifiers have been generally

ignored in the semantics literature, to the point where when they do come up, there isn’t even a

common term for them. Bierwisch (1989) calls them factor phrases, but other terms also exist,

including “ratio phrases” (Sassoon 2010a,b), “factor modifiers” (Rett 2008a,b), and (in the case

of the verbal modification version) “cardinality adverbials” (Parsons 1990), “Q-times adverbials”

(Doetjes 1997) or “‘time’ adverbials” (Landman 2004).

This is on the face of it a surprising oversight; while some work has been done on the verbal

version, the other cases are almost completely neglected, despite impacting upon relatively popular

areas of semantics such as degree semantics (which explores the semantics of things such as 6 feet

tall or John is older than Mary). Whether this is due to a lack of interest or a belief that their

1

semantics will be trivial, there is nevertheless a large gap in our understanding not just of factor

phrases themselves, but also consequently the areas in which factor phrases appear.

This dissertation will therefore redress the balance, taking a detailed look at these cross-

categorial modifiers. I will establish that there are two basic versions of times in factor phrases: a

simple event-counting form, as in (1a), and a multiplicative modifier of degrees and related phe-

nomena, as in (1b-d). In addition, there is a related form: arithmetic times, a preposition which

appears in sentences such as (2) but which, as we will see, has a different structure from factor

phrases:

(2) Eight times three is twenty-four.

The dissertation will thus be structured in the following way: the problem has been outlined in

this first chapter, which will conclude with the basic formal assumptions being made throughout

this dissertation. Chapter 2 will then provide background research. Since (as previously noted)

there is little specific background research into factor phrases, this chapter will instead provide a

more general background, providing a look at the literature for some of the key topics that factor

phrases interact with, including the nature of numbers, degrees and related constructions (such

as comparatives and equatives), and plurality in both the nominal and verbal domains. This will

therefore provide the necessary foundation to explore factor phrases properly.

Chapter 3 will cover factor phrases in the verbal domain, as in (1a). As there is some previous

work in this area, I will begin by looking at a couple of the previous proposals in detail: namely,

those found in Doetjes (1997) and Landman (2004). I will then provide novel data demonstrating

that the multiplicative modifier can appear alongside the event-counting form in a sentence as in

(3):

(3) Floyd walked the dog three times as many times as Clyde did.

Chapter 3 will conclude by showing how this multiplicative version interacts with the event-

counting form.

2

Chapter 4 will then discuss the nature of factor phrases in the adjectival domain, as in (1b).

Building on earlier work involving percentages (Gobeski & Morzycki 2017), I will argue that factor
phrases are a form of what I will call a ratio degree: a modifier of type (cid:104)d, d(cid:105), one that modifies
a degree and returns a degree. I will then show that this multiplicative form, despite appearing in

the verbal, adjectival, and nominal domains, has the same basic form in all cases. This is in fact

the natural way to think about such things, but it will have surprising ramifications for other areas

of semantics. In the domain of degree semantics, the treatment of equative phrases (e.g., as tall as

Clyde) will be the most surprising, as they will not be given the standard treatment of providing an

equality meaning but instead will be left uninterpreted altogether. In addition, while the semantics

of the comparative (e.g., taller than Clyde) will be relatively standard, the treatment of factor

phrases will have an impact on the ongoing debate of whether the comparative should merge with

the adjective (tall-er) or the than phrase (-er than Clyde) first, by providing a semantics for factor

phrases that works easily with the -er than Clyde form (sometimes known as the “small DegP”

or “classical” view), but which is incompatible with the tall-er version (sometimes called the “big

DegP” or “alternate” view).

Chapter 5 will cover factor phrases in the nominal domain, as in (1c) and (d). This chapter

will explore, to the best of my knowledge, uncharted waters. I will begin by laying out the data,

showing that a number of surprising facts arise from the interaction of factor phrases and various

types of nouns:

3

(4)

a. Floyd

b. Victoria

?has

has

 is
 three times Clyde’s height.
 *is
 twice Zoe’s beauty.
 *is
 four times the water that cup
 is
 twice the idiot Sarah
 is
 .
 is
 twice the teacher Clara
 is
 *is
 twice the teachers Clara
 *is

*has

*has

*has

*has

has

has

has

 *is
 .
 .

 .

c. This pitcher

has

d. Harry

e. Barbara

f. Barbara

The data in (4) shows a split not just between is and has but also over whether the sentence is

interpreted based on some inherent degree or based on a particular scale imposed upon a noun

such as teacher – but only if the noun is in the singular case. I’ll focus on a couple of these forms

for more detailed analyses, demonstrating that our basic idea for factor phrases of being modifiers
of type (cid:104)d, d(cid:105) remains essentially intact even for readings as varied as these.

Finally, while they aren’t factor phrases per se, I will also take the time in chapter 6 to explore

both the syntax and semantics of basic arithmetic, as in (5) and (6):

(5)

a. Seven times three is twenty-one.

b. Six plus two is eight.

(6)

a. Three subtracted from eleven is eight.

b. Fourteen divided by seven is two.

As factor phrases deal with multiplication (and share the word times with the arithmetic form),

this is a natural move to make. Yet even these simple sentences are curiously unexplored in the

4

literature, with almost no discussion of the structure or meaning of these sentences – with even the

syntactic category of words such as plus and minus unsettled. I will therefore discuss these struc-

tures at some length, arguing that the arithmetic operators are prepositions with both subjects and

objects, while the multi-morphemic forms in (6) are short passive sentences with reduced relative

clauses. And as multiplication shows up in natural language via factor phrases, I will provide a

semantics for arithmetic times which differs from our factor phrases by performing multiplication

not by directly importing the mathematical operation but by using groups as a method of pluralities

of pluralities, thus demonstrating how multiplication can be employed without the need for math-

ematical language as a separate entity from natural language and also bringing arithmetic times in

parallel with verbal times as analyzed by Landman (2004).

Now, before we launch into the discussion of background literature in chapter 2, let’s take a

moment to lay out the formal assumptions being made throughout this dissertation.

1.2 Formal assumptions

Let’s begin by laying out the terminology. This dissertation will be working in a standard

set-theoretic framework, with the following terminological assumptions:

• Propositions are of type t; t represents a truth value from the domain of truth values Dt,
which consists of the set {0, 1}, with 0 being equivalent to false and 1 to true. A proposition
is therefore either true or false.

• Individuals are of type e, taken from the domain of individuals De. Individuals can include
individuals such as Floyd, Clyde, and Greta. Individuals will be represented in functions as

x, y, or z.

• Events are of type s, taken from the domain of events Ds. “Events” here includes similar
things such as states; roughly speaking, we’re talking about sections of time the actions and

states that occur/hold during those sections of time. Events will represented in functions as
e, e(cid:48), etc.

5

• Degrees are of type d, taken from the domain of degrees Dd. Degrees are things that can be
measured along scales, such as 42 or 3-feet. (Degrees will be discussed in further detail in
section 2.2.) Degrees will be represented in functions as d, d(cid:48), etc.

We will limit ourselves to an extensional semantics in this dissertation; an intensional seman-

tics, where we have propositions that hold of specific worlds w (which may or may not be related

to the actual world), is still held to be true but isn’t necessary for our purposes; in other words,

we’ll be ignoring the evaluation world.

In addition, we’re assuming that sentences are built up compositionally; each piece has its own

meaning, and each piece combines to create a complete meaning for a given sentence or phrase.
This can be thought of in terms of functions, represented as an ordered pair (cid:104)a, b(cid:105); the left term (a,
in this case) represents the input of the function as a particular semantic type, while the right term
(b) represents the output. So a function from individuals to truth values would be written as (cid:104)e, t(cid:105).
As in standard mathematics, functions can be nested: thus, a function (cid:104)e,(cid:104)e, t(cid:105)(cid:105) (sometimes written
as (cid:104)e, et(cid:105)) is a function from individuals to a function of individuals to truth values. Using functions
is a way of defining more complex types; therefore, an object of type (cid:104)e, t(cid:105) is considered a property
of individuals to truth values. Types can thus be thought of as saturated or unsaturated functions;
our basic types (e, d, etc.) are saturated, while the more complex types ((cid:104)e, t(cid:105)) are unsaturated.

The particular denotation (i.e., formal logical representation) of a word or phrase will be rep-
resented by placing that word inside special brackets, (cid:126) (cid:127) – so if we want the particular denotation
of a given word (say, “Floyd”), we write that as (cid:126)Floyd(cid:127). (cid:126) (cid:127) can be thought of as an interpretation

function.

Now let’s establish some compositional rules, adapted from Heim & Kratzer (1998):

(7) Function Application:

If α has the form

S

γβ

then (cid:126)α(cid:127) = (cid:126)γ(cid:127)((cid:126)β(cid:127)).

6

(8) Non-Branching Node:

If α has the form

α

β

then (cid:126)α(cid:127) = (cid:126)β(cid:127).

Given that, let’s put some pieces together:

(9)

t

Floyde

(cid:104)e, t(cid:105)

sleeps(cid:104)e,t(cid:105)

(10)

a. (cid:126)Floyd(cid:127) = Floyd
b. (cid:126)sleeps(cid:127) = f : De → {0, 1} such that for all x ∈ De, f (x) = 1 iff x sleeps.
c. (cid:126)sleeps(cid:127)((cid:126)Floyd(cid:127))

= [ f : De → {0, 1} such that for all x ∈ De, f (x) = 1 iff x sleeps](Floyd)
= 1 iff Floyd sleeps.

This states that the sentence Floyd sleeps is true if and only if1 Floyd sleeps.

So far so good, but our denotation for sleeps in (10) is a bit unwieldy. So instead let’s introduce

an idea known as lambda calculus, first created by Church (1936) and then adapted for natural

language by Montague (1970, 1973), Partee (1975), and others.

Lambda calculus is a way of reducing the notation of terms of functions. The standard schema

for writing λ-terms, as noted in Heim & Kratzer (1998), is as in (11):

(11)

[λα : φ.γ]

This can be read as “the (smallest) function which maps every α such that φ to γ”. In this schema, α

is the argument variable, φ is the domain condition (the domain over which the function is defined),

1iff in (10b,c) is the traditional way of abbreviating “if and only if”.

7

and γ is the value description (“the value that the function assigns to the arbitrary assignment

represented by α” (Heim & Kratzer 1998:35)).

So in terms of our sentence in (9), Floyd sleeps, the denotation of sleeps from (10b) can be

written as in (12):

(12)

(cid:126)sleeps(cid:127) = λx : x ∈ De.x sleeps

This states the same thing as in (10b). It’s a little odd to have a sentence present in our function,

so instead of writing “x sleeps” we’ll write it as sleep(x), where sleep is whatever we mean by the

verb sleep, with our individual x applying to sleep:

(13)

(cid:126)sleeps(cid:127) = λx : x ∈ De.sleep(x)

(For the rest of the dissertation I will omit the domain condition unless it’s necessary: variables

will be assumed to be in their respective domains, as defined above, unless stated otherwise.)

This setup has the advantage of allowing us to use the same semantics for other languages; for

instance, if we wanted to provide the semantics of the German version of our sentence in (9),

(14) Floyd
Floyd
‘Floyd sleeps’.

schläft.
sleeps

we can use the same semantics as (13), without needing to change anything:

(15)

(cid:126)schläft(cid:127) = λx : x ∈ De.sleep(x)

So now we can take our denotation for sleeps and apply Floyd to it; Floyd is of type e, which

therefore saturates the λx term, allowing us to replace every x in the function with Floyd:

(16)

(cid:126)sleeps(cid:127)((cid:126)Floyd(cid:127)) = [λx.sleep(x)](Floyd)

= 1 iff sleep(Floyd)

This states that the sentence Floyd sleeps is true iff Floyd applies to the sleep predicate (in other

words, if Floyd sleeps).

We can also create more advanced denotations requiring multiple individuals, as in (18):

8

(17) Floyd loves Clyde.

t

Floyde

(cid:104)e, t(cid:105)

loves(cid:104)e,et(cid:105)

Clydee

(18)

(cid:126)loves(cid:127) = [λyλx.x loves y] = [λyλx.love(y)(x)]

Loves requires two pieces, which are both individuals, and since in our tree we start by combin-

ing loves and Clyde via Function Application, Clyde needs to apply to the y part of loves. When we

have n-place predicates, I will write them as, e.g., love(y)(x) to indicate the order in which they’re

being combined. (This is sometimes written in other works as love(y, x); these are different ways

of writing the same thing.)

Completing the denotation for the sentence in (17) yields (19):

(19)

a. (cid:126)Clyde(cid:127) = Clyde

b. (cid:126)Floyd(cid:127) = Floyd

c. (cid:126)loves Clyde(cid:127) = (cid:126)loves(cid:127)((cid:126)Clyde(cid:127)) = [λyλx.love(y)(x)](Clyde) = λx.love(Clyde)(x)

d. (cid:126)Floyd loves Clyde(cid:127) = [λx.love(Clyde)(x)](Floyd)

= 1 iff love(Clyde)(Floyd)

The final result of (19) states that Floyd loves Clyde is true iff the predicate love applies to Clyde

then Floyd, yielding Floyd loves Clyde.

This will thus be the basis of the formal semantics found in this dissertation, with more ad-

vanced types and functions notated appropriately as they come up.

Now that we’ve established both the basic problem of factor phrases and our formal assump-

tions, let’s begin by exploring the previous literature that will provide the foundation which factor

phrases will build upon.

9

CHAPTER 2

BACKGROUND

As noted in the last chapter, factor phrases themselves haven’t been examined in great detail, even

though they interact with a large number of areas of semantics. This chapter will thus provide

a closer look at a number of those areas that factor phrases will impact: the nature of numbers,

degrees and degree semantics (including gradable adjectives and comparative structures), and plu-

rality, both in the nominal and verbal domains. This will provide the background of the semantics

that the analysis of factor phrases will work with and build upon.

2.1 Numbers

The discussion of numbers is an area with an overt philosophical component: do numbers exist

in the real world, or are they simply a construct of humanity? This question lies behind a good

chunk of the work that’s been done on numbers. So let’s take a moment to discuss the ontological

side of things and how it relates to language, before moving to the (more) purely linguistic side of

things.

2.1.1 The ontology of numbers

The question of whether numbers exist independently of the human mind has been the subject of

philosophical debate since at least the time of ancient Greece. Numbers themselves are of course

a part of language, as evidenced by the fact that we have words for them such as “one”, “three”,

etc. But do numbers exist as part of the real world, or are they merely a construct of the human

intellect? The Pythagoreans believed that not only did numbers exist, but that everything in the

universe was made up from numbers. Plato, in his Republic (Book VII), stated that numbers

were abstract but real, and that they formed the basis of intellectual thinking. Of course, the

Greeks also believed that geometry was in fact the key to unlocking the secrets of the universe;

10

numbers were thus a means to an end, for use with geometry. It should be noted that “number”

here means natural numbers – so when irrational numbers were discovered, according to some

versions (such as Iamblichus, albeit indirectly) the discoverer (sometimes said to be Hippasus of
Metapontum) was either exiled or put to death as a result.1 But generally speaking, the prevailing

view for some time was that numbers did exist independently of people, and that numbers (and

thus, by extension, mathematics) were there to be discovered, rather than invented. This position is

known now as mathematical realism, or sometimes mathematical Platonism – due to its similarity

to Plato’s theories regarding Forms (think the Allegory of the Cave and its discussion of shadows

versus real things), rather than because it was something Plato proposed.

For our purposes, however, the story really begins in 1884, when the philosopher and logician

Gottlob Frege published his book The Foundations of Arithmetic. Frege was developing his own

theory as to what a number actually was, after discounting a variety of other possibilities, including

Oscar Schlömilch’s conjecture that numbers are “the idea of the position of an item in a series”

(Frege 1884:37) and John Stuart Mill’s claim that numbers derive from observations of groupings

(Frege 1884:9-12) – a problematic position if you want to work with big, personally unobserved

numbers (or even 0 and 1). But what makes Frege important from our perspective is that he is es-

sentially the first person to apply language itself to this puzzle. Frege argued that by looking at the

language of mathematics, we could answer this question of whether numbers exist independently

or not; in other words, the language of mathematics and the language of everything else both use

the same tools, structures, and meanings, and so therefore the same rules apply.

To this end, Frege noted that (1) and (2) are equivalent in meaning:

1The ancient Greek texts don’t explicitly state this; Iamblichus, in his Life of Pythagoras,
merely states that someone may have died as a result of discovering “incommensurable quanti-
ties”, and that someone else may have died after discovering the regular dodecahedron. (There
had previously been four perfect solids – the cube, tetrahedron, octahedron, and icosahedron – so
this was seen as something of an affront to the gods. (Phillips 1965)) Iamblichus later mentions
that Hippasus was the one who showed how to inscribe a dodecahedron in a sphere, so there is
circumstantial evidence for this story, but nothing direct.

11

(1)

Jupiter has four moons.2

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

While (1), with its use of an adjectival number, makes no ontological commitments about the

existence of four (since adjectives are modifiers rather than entities in their own right), the same

cannot be said about (2). This sentence looks an awful lot like an identity statement, as in (3):

(3) The creator of Doctor Who is Sydney Newman.

Just as (3) commits to the existence of Sydney Newman as an independent object in the world,

so too does (2) seem to commit to the existence of four as an independent object. Frege (1884:69)

explicitly argues against (2) being a predicate along the lines of The sky is blue, stating that the fact

that you can rewrite (2) as (4) shows that this is an identity is, along the lines of “is the same as”:

(4) The number of Jupiter’s moons is the number four, or 4.

Therefore, since the language of math should work the same as the rest of natural language,

the fact that you can get an identity statement as in (2) proves, according to Frege, that numbers do

indeed exist as independent, real-world objects.

Perhaps unsurprisingly, not everyone bought this argument. After all, it would be rather re-

markable if a question with such a long history could be solved just by pulling out a telescope,

counting up the moons you see through it, and verifying the truth conditions. Consequently, this

“Easy Argument” (as Balcerak Jackson 2013 much later dubbed it) didn’t actually settle the de-

bate, and much has been written both for and against Frege’s position. Much of this writing has

approached the Easy Argument from the linguistic side of things, which is where we come in.

Hofweber (2005), for instance, challenges Frege’s idea by pointing out that the occurrences of

four in (1) and (2) aren’t interchangeable; after all, you can’t reasonably utter (5), even though you

might expect, given (2), that the number of Jupiter’s moons is a reasonable substitute for four:

*Jupiter has the number of Jupiter’s moons moons.

(5)
2When Frege wrote this, only the four Galilean moons – Io, Europa, Ganymede, and Callisto –

had been discovered.

12

Hofweber thus argues that there are in fact three different uses of number words: what he calls

the singular-term use, as in (2); the adjectival/determiner use, as in (1); and the symbolic use, i.e.,

the Arabic numeral 4 (which for Hofweber isn’t necessarily identical to the actual word four).

What he crucially doesn’t want to say is that four is ambiguous among these uses, where there’s

multiple versions of four that differ according to their syntactic and semantic properties, and that’s

what leads to the differences; rather, he wants to derive all these uses from the same base form

of the number word. This base form for Hofweber is the determiner form: numbers are objects
of type (cid:104)(cid:104)e, t(cid:105),(cid:104)(cid:104)e, t(cid:105), t(cid:105)(cid:105), which is the standard type for objects known as quantifying determiners,
such as every, no, and some. Any occurrences that look like they might be adjectival are in fact

determiners. Numbers (with the exception of one) take plural agreement, as in (6), and this is said

to be consistent with treating them as determiners:

 *is

are

 in the yard.

(6) Four dogs

Hofweber extends this from not just sentences such as (1), but to arithmetic sentences such as

(7) and (8):

(7) Two and two are four.

(8) Two and two is four.

The plural version in (7) is straightforward: since two as a determiner takes plurality (not to men-

tion the presence of and, which also indicates plurality), this is the expected result. Hofweber

comes a bit unstuck explaining the singular agreement in (8), however, and is forced to resort to a

mechanism he calls “cognitive type coercion”, where the speaker shifts from a high type, such as
(cid:104)(cid:104)e, t(cid:105),(cid:104)(cid:104)e, t(cid:105), t(cid:105)(cid:105), to a lower type, such as e. This then is what causes the singular agreement. Frus-
tratingly, Hofweber doesn’t actually describe how this type-lowering leads to singular agreement;

he seems to treat it as a result of objects of type e being singular, but this ignores that a) objects of

type e can be plural:

13

(9) The Beatles

are

 *is
 the stars of A Hard Day’s Night,
 *is
 going to the movies.

are

and b) type e entities come out plural when combined using and:

(10) Floyd and Clyde

While this cognitive type coercion would thus explain the difference between the two arithmetic

statements, Hofweber states that isn’t in fact what’s going on in (2); instead, he argues that’s

actually a type of focus construction, rewording (1) to become (2). This is akin to (11):

(11)

a.

I had two bagels.

b. The number of bagels I had is two.

(This explanation doesn’t address the fact that (2) isn’t a direct transformation of (1) – that would

be more like The number of moons Jupiter has is four – but that’s perhaps a minor quibble.)

So for Hofweber, Frege’s Easy Argument doesn’t work because the is in (2) isn’t the is of

identity, but rather an is that’s the simple result of reordering (1).

As one might imagine, this explanation didn’t settle the debate. Balcerak Jackson (2013),

while sympathetic to Hofweber’s aims, points out the various linguistic problems with Hofweber’s

account. In addition to the worries already mentioned above, Balcerak Jackson (2013) notes a

number of other concerns: first, if Hofweber is correct and numbers such as four are just standard

quantificational determiners, then it’s very surprising that no other quantificational determiners can

show up in a sentence like (2):

(12)

a. Jupiter has some moons.

b. *The number of moons of Jupiter is some.

(13)

a. Jupiter has many moons.

b. *The number of moons of Jupiter is many.

14

(14)

a. Jupiter has few moons.

b. *The number of moons of Jupiter is few.

(15)

a. Jupiter has a moon.

b. *The number of moons of Jupiter is a.

Second, moving four around for the purpose of focus shouldn’t change its syntactic category

from determiner to noun phrase – compare with (16), in which soccer remains an NP in all cases:

(16)

a. John likes soccer.

b. What John likes is soccer.

c.

It is soccer that John likes.

Finally, Hofweber’s treatment of the number of as a form of placeholder for four in a focus

construction ignores the fact that extra information can be present that the numeral can’t account

for:

(17)

a. Jupiter most likely has four moons.

b. The most likely number of moons of Jupiter is four.

(18)

a. Jupiter is expected to have four moons.

b. The expected number of moons of Jupiter is four.

If four and the number of really were directly interchangeable, the (b) sentences in (17) and (18)

should therefore be problematic; the fact that they’re not indicates that there’s no one-to-one cor-

respondence here.

Balcerak Jackson points out that the real problem with (1) and (2) –and therefore Frege’s Easy

Argument – is that they’re not actually semantically equivalent; if they were, we should expect that

(19) and (20) should be equally acceptable, but while (19) is fine, (20) is a contradiction:

(19)

Jupiter has four moons. In fact, it has sixty-two moons.

(20) The number of moons of Jupiter is four. #In fact, it’s sixty-two.

15

So then why are we willing to accept (2) as a paraphrase of (1)? One possibility is that, if

the differences between the two sentences aren’t directly relevant to the speaker, then they are

ignored unless subsequent commitments make the differences relevant. This may be related to

what Eklund (2005) calls indifference, where speakers aren’t committed to the truth of part of a

statement. For instance, if (21) is uttered, the commitment is to whether or not the man is happy;

if it subsequently turns out that the man wasn’t drinking water but a different clear liquid, that

doesn’t bear on whether or not he looks happy, and so we don’t necessarily judge the sentence as

completely false because of it.

(21) The man drinking water is happy.

Similarly, the fact that (2) has an “exact” reading, while (1) has an “at least” reading, simply

isn’t relevant for most cases, which is why we’re happy to accept (2) as a paraphrase of (1): the

semantic differences don’t lead to a difference in most cases. However, this lack of equivalence

does matter for the ontological discussion, because there it’s important that the two sentences be

semantically equivalent.

Of course, given the complexity and import of the question, Balcerak Jackson (2013) hasn’t

really settled the debate either – merely Hofweber (2005)’s attempt to solve the puzzle. And so the

philosophical question still remains: do numbers independently exist?

Now, since in this dissertation we’re less interested in the ontological question and more inter-

ested in the practical question of “how do numbers behave in language?”, let’s table the ontological

discussion for now and shift gears to look at the more purely linguistic side of numbers: their syn-

tax and semantics, including what their semantic type actually is and why they seem to show up in

so many places.

2.1.2 The syntax and semantics of numbers

While the ontology of numbers is a large and fascinating topic, a number of linguists have more

prosaically ignored the question, essentially (though not overtly) saying, “Regardless of their on-

16

tology, we still have to deal with numbers in practical terms, so how do we do that?” In other

words, what are the syntax and semantics of numbers?

Syntactically, much of the work has focused on the internal structure of complex numerals,

things such as forty-seven and seventy-four million, three hundred and eighty-four thousand, three

hundred and thirty-eight. How can we build up the structure of these compositionally?

Hurford (1975) does so via a set of phrase structure rules, building up increasingly complex

numerals using a combination of multiplication and addition. English, for instance, uses addition

for numbers less than ten (so that two is essentially one plus one), and multiplication for numbers

larger than ten (so that twenty is a form of “two tens”), with a combination of the two approaches

for more complex numbers (so eighty-one is “eight tens, plus one”). Other languages use different

numbers as their base for multiplication – French, for example, uses twenty as its multiplicative

base, and so expresses eighty-one as quatre-vingt-un, literally “four twenty one”, or “four times

twenty, plus one”. Hurford is interested primarily in explaining how and why different languages

approach large numbers differently, and being able to account for the differences with the same set

of rules.

In a similar vein, Ionin & Matushansky (2006) argue that numerals are NP modifiers of type
(cid:104)(cid:104)e, t(cid:105),(cid:104)e, t(cid:105)(cid:105), which makes them different from purely intersective modifiers. This is a response to,
among others, Landman (2004), who argues for the intersective modifier position: the idea is that
if you take a basic predicate of type (cid:104)e, t(cid:105), such as monkey, and want to modify it to have a certain
number of monkeys, you can just perform predicate modification to do so – and so in order to do
that, the number will also have to be of type (cid:104)e, t(cid:105):

(22)

a. (cid:126)monkeys(cid:127) = λx.monkeys(x)
b. (cid:126)two(cid:127) = λx.|x| = 2
c. (cid:126)two monkeys(cid:127) = λx.monkeys(x) ∧ |x| = 2

But Ionin & Matushansky argue that this story doesn’t work. This is because, for a complex

numeral, the simplest version would lead to a contradiction, as (23) illustrates:

17

(23)

a. (cid:126)two(cid:127) = λx.|x| = 2
b. (cid:126)hundred(cid:127) = λx.|x| = 100
c. (cid:126)two hundred books(cid:127) = λx.books(x) ∧ |x| = 2 ∧ |x| = 100

There’s no way for the number of books to equal both 2 and 100, so that approach is a non-starter.

Instead, Ionin & Matushansky (2006) choose to analyze something such as two hundred books as

a set of objects such that two sets each consist of one hundred (different) books, rather than trying

to create an intersection of books, hundred, and two.

(24)

(cid:104)e, t(cid:105)

(cid:104)et, et(cid:105)
two

(cid:104)e, t(cid:105)

(cid:104)et, et(cid:105)
hundred

(cid:104)e, t(cid:105)
books

(25)

(cid:126)two(cid:127) = λP(cid:104)e,t(cid:105)λx.∃S(cid:104)e,t(cid:105)[Π(S )(x) ∧ |S| = 2 ∧ ∀s ∈ S P(s)]

(25) states that for some individual x, there’s a partition S that x applies to, the cardinality of S is

2, and for objects s that are contained in the partition S , s holds of a property P. (S is a partition

Π of an entity x if it’s a cover (that is, its members all sum up to a plural individual X ((cid:70) C = X))

and its cells do not overlap.3) This is just a technical way of saying that there are two hundred

distinct books (or whatever object you’re counting), and you didn’t count up your set of hundred

books twice in order to say you had two hundred books.

Could you use partitions but still treat numbers intersectively? Well, no; Ionin & Matushansky

point out that while that gets you closer, you can’t exclude the possibility of only counting one

hundred books:

3Note that (24) and (25) have been simplified slightly from their original form for the sake of

readability.

18

(26)

a. (cid:126)two(cid:127) = λx.∃S [Π(S )(x) ∧ |S| = 2]
b. (cid:126)hundred(cid:127) = λx.∃S [Π(S )(x) ∧ |S| = 100]
c. (cid:126)two hundred books(cid:127)

= λx.∃S [Π(S )(x) ∧ |S| = 2] ∧ ∃S (cid:48)[Π(S (cid:48))(x) ∧ |S (cid:48)| = 100 ∧ books(x)]

The way (26) is set up, all this says is that there are 100 non-overlapping items and also that there

are 2 non-overlapping items, so in principle you could have a set of 100 books and a set of 2 books

drawn from the set of 100 books, leading to two hundred books coming out true if you only have

100 distinct books. This is decidedly not what we want.

They also explain why the semantic type of numerals can’t be the same as that of determin-
ers ((cid:104)et,(cid:104)et, t(cid:105)(cid:105)), as Hofweber argues: if two is a determiner and hundred is also a number and
therefore a determiner, then the two of them simply can’t combine properly: putting hundred and
books (which is type (cid:104)e, t(cid:105)) gives you an object that’s type (cid:104)et, t(cid:105), and that won’t combine with two
properly, since it needs an input of type (cid:104)e, t(cid:105). Trying to combine two and hundred together first
doesn’t work either, because determiners can’t combine together directly (as evidenced by, e.g.,
*the every book). So the only acceptable type is (cid:104)et, et(cid:105).

However, this line of reasoning requires that hundred and two in fact be the same, which doesn’t

actually seem to be true – *Hundred books isn’t acceptable, after all. Ionin & Matushansky, to their

credit, discuss this, with an argument that semi-lexical cardinals such as hundred and million when

they’re the left-most cardinal have an obligatory indefinite determiner (such as a/an) that the other

cardinals lack – although admittedly even they don’t seem thrilled with this explanation. However,

Zweig (2005) has an answer for that, but in order to dig into that we need to look at Kayne (2005)

first.

Kayne (2005) begins with a simple set of observations. Few seems to behave like an adjective,

with characteristics such as taking comparative -er and superlative -est:

(27)

a. John has fewer books than Bill.

b. John has the fewest books of anyone I know.

19

Few can also combine with the word too, which combines with adjectives, not nouns:

(28)

a. John is too rich.

b. *John has too money.

c. John has too few friends.

And while too can sometimes combine with certain prepositional phrases, few still patterns with

adjectives rather than these PP:

(29)

a. John is too much in love.

b. *John is too much rich.

 much

little

 few friends.

c. *John has too

Conclusion: few is an adjective. But if that’s the case, then why does (30), which expresses the

same sentiment as (27), require the extra phrase number of ?

(30)

a. John has a smaller number of books than Bill.

b. John has the smallest number of books of anyone I know.

Kayne’s argument is that few doesn’t modify books directly, but instead is modifying a covert

noun, which he calls NUMBER. So (27) is in fact as in (31):

(31)

a. John has fewer NUMBER books than Bill.

b. John has the fewest NUMBER books of anyone I know.

Kayne notes that this NUMBER cannot be plural: as one piece of evidence, he points out that

every only appears with singular nouns:

(32)

a. every book

b. *every books

However, every can combine with few, despite what appears to be the presence of a plural noun,

and it can’t simply be the presence of an adjective that allows this:

20

(33)

a. every few books

b. *every good books

But if NUMBER is not marked for plural, (33a) is straightforward:

(34)

every few NUMBER books

Kayne mentions in passing that numbers also combine with every (as in every three books), so

either numbers are singular or they also have a covert NUMBER present.4

This idea of a covert NUMBER is one that Zweig (2005) runs with. Zweig points out that a

number of others (including Corbett 1978 and Hurford 1987) have noted that generally speaking,

the lower numerals behave like adjectives, while higher numerals behave like nouns. For instance,

Zweig points out that in Luganda (a Bantu language spoken in Uganda), lower numerals agree

with the noun they’re modifying, like adjectives do, while higher numerals have their own class

prefixes:

(35)

(36)

mi-
AGRmi-

dumu
jug

emi-
mi-
‘Good jugs’

rungi
good

dumu
jug

a. emi-
mi-
‘Two jugs’

e-
AGRmi-

biri
two

dumu
jug

b. emi-
mi-
’Seven jugs’

mu-
mu-

sanvu
seven

And when a complex numeral shows up, the lower numeral takes the class prefix of the higher

numeral, not the noun:

4Kayne also notes that this NUMBER is count; should a mass noun be required (as in the case

of little), AMOUNT shows up instead:

(i)

a. John has little water.
b. *John has little books.
c. John has little AMOUNT water.

21

(37)

dumu
jug

ama-
emi-
mi-
ma-
‘Twenty jugs’

kumi
ten

a-
AGRma-

biri
two

We also notice a difference between higher and lower numerals in English: the lower numerals

can’t show up in places where we would expect nouns, and, as (40) shows, not all numerals are the

same when it comes to combining them:

(38)

a. hundreds of boys

b. *threes of boys

(39)

a. a/several hundred boys

b. *a/several three boys

(40)

a.

four hundred boys

b. *four three boys

How do we explain the difference? By using covert NUMBER. The (more) adjectival numerals

combine with NUMBER, while the higher, (more) nominal numerals take the place of NUMBER.

As Kayne noted and Zweig makes explicit, numerals combine with every:

(41)

a. every three books

b. every hundred books

c. every three hundred books

So our basic structure for these is as in (42):

(42)

a.

b.

[NP [AP three ] NUMBER ]

[NP [AP three ] hundred ]

Now, while that covers the syntactic side of things, that still leaves the semantic side. We’ve

seen some discussion of what the semantic types of numerals are, but those are by no means the

only positions. There’s one important idea that we haven’t discussed yet: that numerals are in fact

their own semantic type, corresponding to an object called a degree.

22

2.2 Degrees, gradable adjectives, and comparatives

2.2.1 Degrees and gradable adjectives

In order to discuss degrees, we first need to take a slight detour to discuss gradable adjectives,

such as tall or old. As Kamp (1975), Klein (1980), Kennedy (1999, 2001), and others point out,

gradable adjectives refer to properties, and predicates with those properties can be compared to

each other. Take a simple sentence such as (43):

(43) Floyd is taller than Clyde.

How do we express the idea that the height of Floyd is greater than the height of Clyde?

One way is to take the heights of everything, order them along a particular dimension (or scale)

according to increasing height, and then verify that the ordering of the two pertinent heights is the

way we want it. Another way to express this is (44):

(44) The degree of Floyd’s height

 is greater than

exceeds

 the degree of Clyde’s height.

So we use a special object called a degree (which we’ll give its own semantic type, d) to

perform these comparisons along (partially) ordered domains. And since numbers can be ordered,

it’s not the craziest idea to think that numbers and degrees are related somehow, and so degrees

could therefore be a sort of number. But before we can explore that, let’s examine what a degree

actually is and what the relationship is between degrees and gradable adjectives.

Now the classical view of gradable adjectives is that their scales have domains that are partially

ordered in some way, usually through a series of points. There are a couple of ways to define

these adjectives: Kennedy (1999) calls these two approaches the Vague Predicate Analysis and

the Scalar Analysis. In the Vague Predicate Analysis (see, among others, McConnell-Ginet 1973,

Kamp 1975, Klein 1980, and van Benthem 1983), adjectives denote functions from objects to truth
values (e.g., (cid:104)e, t(cid:105)), while in the Scalar Analysis (see, among others, Bartsch & Vennemann 1973,
Seuren 1973, Cresswell 1976, Hellan 1981, von Stechow 1984, and Heim 1985), they denote

23

functions from objects to degrees (e.g., (cid:104)e, d(cid:105)). In both analyses, in (43) the degree of Floyd’s
height is a point on the scale that is above the degree of Clyde’s height – so if we utter (45):

(45) Tony is five feet tall,

then the degree of Tony’s height on the scale is a point equivalent to 5 feet on our tallness scale:

(46)

0

Tony

6

2
µtallness (in ft)

4

The advantage of thinking about degrees as points is that they are easier to conceptualize and

thus easier to manipulate; so in a sentence like (47),

(47) Floyd is three inches taller than Clyde.

all that is necessary to find the degree of Floyd’s height to take the degree of Clyde’s height and

add three inches to it – each degree is a single point, not a range of them that must be further

manipulated.

The disadvantage, though, as Kennedy (1999) points out, is that it becomes very difficult to

explain the effects of negative adjectives, such as the following:

(48)

*Tony is five feet short.

If degrees are simply points on a scale, then it is not immediately clear why (48) is bad. After

all, whether an adjective is positive or negative should have no bearing on the specific points of a

scale.

So instead, Kennedy argues (following from Seuren 1978, 1984 and von Stechow 1984; in

addition, see Schwarzschild & Wilkinson 2002 and Schwarzschild 2002, 2006) that degrees are

not points on a scale but are actually intervals: instead of degrees representing points, they in fact

represent sets of points. The idea here is that if Tony is five feet tall, then he’s also four feet tall,

three feet tall, etc. – his height includes every point between 0 feet and 5 feet.

24

(49)

0

Tony

6

2
µtallness (in ft)

4

Positive and negative degrees differ, therefore, by which subset of the scale they represent: for

a positive degree, for any point contained within the degree, every point below that point is also

contained within the degree. Here’s the formal definition for a positive interval:
(50) POS(S ) = {d ⊆ S|∃p1 ∈ d ∀p2 ∈ S [p2 (cid:22) p1 → p2 ∈ d]}

(50) states that for a degree d that is a proper subset of the scale S , there is a point p1 that is
contained within a degree consisting of every point p2 that is itself contained with a scale such that
if p2 is less than or equal to p1, then p2 is contained within the degree d. In simpler language,
degree d contains every point on a scale that is less than or equal to point p1. So as long as p2 is
less than or equal to p1, p2 is a part of the positive degree.

Negative degree intervals are very similar, but whereas positive degrees go from the bottom of

the scale to a given degree, negative degree intervals go from a given degree to the top of the scale

(which in some cases, such as height, is infinity).
(51) NEG(S ) = {d ⊆ S|∃p1 ∈ d ∀p2 ∈ S [p1 (cid:22) p2 → p2 ∈ d]}

The key thing to note is that for a negative degree, p2 is contained only if it is greater than or equal
to p1.

This means that for our attempted sentence in (48), *Tony is five feet short, the interval would

be as in (52):

(52)

0

Tony

6

2
µtallness (in ft)

4

So the reason (48) is unacceptable is that our interval is from 5 feet to infinity, which isn’t a

meaningful measurement: it has no maximal point on the positively-ordered tallness scale. That

brings us to another piece regarding treating degrees as intervals: how are we sure that we’re

25

choosing the right part of the interval? Imagine a scenario in which Floyd is 6 feet tall and Clyde

is 4 feet tall. We wouldn’t want to say that Floyd and Clyde are the same height, but if Floyd’s

height includes all the heights below 6 feet, what stops us from saying that since Floyd’s height

includes 4 feet, he and Clyde are the same in height? Or, indeed, that since both of their heights

incorporate 2 feet, they’re both the same height for that reason? Obviously this is something of a

silly notion, but the point is that it is possible. So what we mean when we say that Floyd is 6 feet

tall is that his maximal height is 6 feet. So, following on from people such as (among others) von

Stechow (1984) and Rullmann (1995), let’s explicitly include a maximality operator. von Stechow

defines this operator as in (53):
(53) Max(P) is true of d iff P(d) and ¬(∃d(cid:48))[P(d(cid:48)) ∧ d(cid:48) > d]

In other words, if P(d) is a property of degrees that applies to a degree d and there doesn’t exist a
degree d(cid:48) that is larger than our degree d, then max(P) is true.

Now with our maximality operator, our scenario in which Floyd and Clyde are 6 feet and 4

feet respectively doesn’t lead to a circumstance in which we say they’re the same height, because

max will select for the maximal height of each of them. And we can also use this idea (and

the corresponding idea of a minimality operator, min) to make explicit the comparison between

positive and negative degrees. For any given object x on a scale S , the maximal element of the

positive degree is equal to the minimal element of the negative degree.

(54) max(posS (x)) = min(negS (x))

So (43) can now be interpreted properly as meaning that Clyde’s height is a proper subset of

Floyd’s height (since they both start at 0 and overlap):

(55)

tall(Clyde) ⊂ tall(Floyd)

2.2.2 The syntax and semantics of comparatives

Now that we’ve established how scales are constructed and ordered, we can begin to interpret a

comparative sentence such as (43), repeated below for convenience.

26

(43) Floyd is taller than Clyde.

Now, remember how we said there were two approaches to the standard degree story, the

Vague Predicate Analysis and the Scalar Analysis? Here’s where that becomes relevant, because

depending on which analysis you choose, you’ll have two different structures.

The “small DegP” or “classical view” (as Bhatt & Pancheva 2004 call it) of degree clause

structures such as comparatives comes from sources like Selkirk (1970), Bowers (1975), Bresnan

(1973), Jackendoff (1977), Hellan (1981), Heim (2000), and others. In this syntax, the degree

clause and the degree morpheme form a constituent. So in the case of (43), the degree clause

than Clyde combines with the morpheme –er, which then combines further up the tree with the

gradable adjective. The fact that degree morphemes require specific complementary words in the

degree clause (than in the case of comparatives, as in the case of equatives) is evidence for this

constituency. (56) is the structure for the simple, in situ version of the adjective phrase:

(56)

AP(cid:104)e,t(cid:105)

DegP(cid:104)(cid:104)d,et(cid:105),et(cid:105)

Deg(cid:104)d,(cid:104)(cid:104)d,et(cid:105),et(cid:105)(cid:105)

XPd

-er

than Clyde

A(cid:104)d,et(cid:105)
tall

Of course, an argument against this viewpoint, as Bhatt & Pancheva (2004) point out, is that

in general, the degree morpheme and the degree clause cannot appear together – a fact against this

constituency.

(57)

*Floyd is [more than Clyde] tall.

We’ll come back to this in a little bit, but for now let’s look at the semantics for this structure,

which is reasonably straightforward; here’s Kennedy & McNally (2005)’s approach:

27

(58)

a. (cid:126)tall(cid:127) = λdλx[tall(x) = d]
b. (cid:126)-er(cid:127) = λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d (cid:31) d(cid:48) ∧ G(d)(x)]
c. (cid:126)than Clyde(cid:127) = dClyde
d. (cid:126)-er than Clyde(cid:127) = λG(cid:104)d,et(cid:105)λx.∃d[d (cid:31) dClyde ∧ G(d)(x)]
e. (cid:126)tall -er than Clyde(cid:127) = λx.∃d[d (cid:31) dClyde ∧ tall(d)(x)]
f. (cid:126)Floyd is tall -er than Clyde(cid:127) =1 iff ∃d[d (cid:31) dClyde ∧ tall(d)(Floyd)]

5

Our final result in (58f) states that Floyd is taller than Clyde if and only if there exists a degree

that is larger than the degree of Clyde’s tallness, and that degree is the tallness of Floyd – which is

precisely what we want.

(A brief tangent: what’s actually in the than clause? We’ve hedged here and just called it

dClyde, but it’s a bit more complicated than that. The general consensus is that it’s something like

(59)

than opi Clyde is di-tall

with the repeated elements having been deleted from the final full sentence. Based on, among

others, Izvorski (1995), Kennedy (1999), and Wellwood (2014), this clause is said to be a form of

wh-clause, with a null operator moving from the degree variable position in DegP up to SpecCP.

This operator triggers Predicate Abstraction so that the sentence Clyde is tall can still be incorpo-
rated into the denotation as an object of type (cid:104)d, t(cid:105). The than morpheme then converts this into a
unique degree, ιd.

(So the semantics work out to be (roughly) as in (60):

(60)

a. (cid:126)di-tall(cid:127) = λx[max{d : tall(x) = d}]
b. (cid:126)Clyde is di-tall(cid:127) = 1 iff max{d : tall(Clyde) = d}
c. (cid:126)opi Clyde is di-tall(cid:127) = λd[max{d : tall(Clyde) = d}]
d. (cid:126)than(cid:127) = λD(cid:104)d,t(cid:105).ιd[D(d)]
e. (cid:126)than opi Clyde is di-tall(cid:127) = ιd[max{d : tall(Clyde) = d}]

5This has been simplified for the sake of clarity.

28

(Ultimately, we’re left with the unique degree that corresponds to the maximal degree of Clyde’s
tallness.6)

This isn’t the only possible way, however; as people such as Heim (2000) point out, another
approach is by treating the DegP as of type (cid:104)dt, t(cid:105). This therefore makes these look like gener-
alized quantifiers – things such as every monkey or some apples – which are of type (cid:104)et, t(cid:105), and
so consequently we’ll call this form a generalized degree quantifier. (This also helps capture the

intuition that Hofweber (2005) subsequently noticed about the relationship between numbers and
quantifiers.) An object of type (cid:104)dt, t(cid:105) can’t be interpreted in situ, however, so it will need to move
up the tree via a mechanism called Quantifier Raising (or QR) in order to be interpretable. In order

for the degree argument of the gradable adjective to be satisfied, the generalized degree quantifier
will leave behind a trace of type d. Here’s the tree for this:7

(61)

t

DegP(cid:104)dt,t(cid:105)

(cid:104)d, t(cid:105)

Deg(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)

XPd

1

t

-er

than Clyde

DPe

(cid:104)e,t(cid:105)

Floyd

is

AP(cid:104)e,t(cid:105)

d1

A(cid:104)d,et(cid:105)
tall

And here’s how the semantics works:
6This version is of course under the semantics system where gradable adjectives are of type
(cid:104)d, et(cid:105). A version under the alternate system would be similar but require some slight modifi-
cation, as adjectives directly denote degrees in that framework. The ultimate result of a degree
corresponding to (say) Clyde’s maximal height will be the same in both versions, however.

7One thing to note about this structure is the presence of the 1 above Floyd is d1-tall in the tree;
this is the index of -er than Clyde that matches the index on the trace d1, which can be interpreted
by Heim & Kratzer (1998)’s Predicate Abstraction Rule, which essentially states that the 1 creates
a lambda of the same type as the trace (so in this case, type d), thus allowing the moved portion to
combine with the remainder of the sentence.

29

(62)

a. (cid:126)1 Floyd is d1-tall(cid:127) = λd[tall(Floyd) = d
b. (cid:126)-er(cid:127) = λdλF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(x)(d(cid:48)(cid:48))} > d]
c. (cid:126)-er than Clyde(cid:127) = λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(x)(d(cid:48)(cid:48))} > dClyde]
d. (cid:126)-er than Clyde 1 Floyd is d1 tall(cid:127) = 1 iff [max{d(cid:48)(cid:48) : tall(Floyd) = d(cid:48)(cid:48)} > dClyde]

Our final result in (62d) states that Floyd is taller than Clyde is true if and only if the maximal

degree corresponding to Floyd’s tallness is greater than the maximal degree of Clyde’s tallness.

This structure is perhaps a bit unusual to the untrained eye, but there’s nothing unorthodox here.

However, Kennedy (1999), following from others like Abney (1987), Larson (1988), Corver

(1990), and Grimshaw (2005), offers an alternate interpretation. In this version, the AP and the

degree morpheme form a constituent which then combines with the degree clause; this would be

consistent with the non-acceptability of (57). (This version is sometimes referred to as the “big

DegP” structure, as the adjective phrase, by adjoining to the comparative, becomes part of the

degree phrase.) This structure also has the potential advantage that no movement needs to occur in

order for proper interpretation to happen.

(63)

DegP(cid:104)e,t(cid:105)

Deg(cid:48)(cid:104)d,et(cid:105)

XPd

Deg(cid:104)(cid:104)e,d(cid:105),(cid:104)d,(cid:104)e,t(cid:105)(cid:105)(cid:105)

AP(cid:104)e,d(cid:105)

than Clyde

-er

tall

The semantics for the alternate syntax are actually even more straightforward than the classical
view. Under this system, gradable adjectives, as you’ll no doubt recall, are of type (cid:104)e, d(cid:105). So we’ll
need a comparative morpheme that combines with this kind of adjective:

(64)

(cid:126)-er(cid:127) = λG(cid:104)e,d(cid:105)λdλx[max{G(x)} > d]

Essentially, what -er does is compare the degree of an individual x on a particular scale (supplied

by the gradable adjective G(x)) with a degree d supplied by the than phrase, and if the degree

30

associated with x is greater than the degree d, the sentence comes out true. Again, remember that

in this case the output G(x) is in fact a degree, so it’s directly comparable with a degree.

Combining our comparative morpheme with a gradable adjective and the than clause leads to

(65), which states that Floyd is taller than Clyde is true if the maximal degree of Floyd’s tallness

is greater than the maximal degree of Clyde’s tallness:

(65)

a. (cid:126)tall(cid:127) = λx.max{tall(x)}
b. (cid:126)-er tall(cid:127) = λdλx[max{tall(x)} > d]
c. (cid:126)-er tall than Clyde(cid:127) = λx[max{tall(x)} > dClyde]
d. (cid:126)Floyd is -er tall than Clyde(cid:127) =1 iff max{tall(Floyd)} > dClyde

So far so good: this is how it works for a simple comparative. But what about a more compli-

cated form, such as (58)?

(66) Floyd is three inches taller than Clyde.

Now instead of directly comparing one height to another, we’re comparing those two and deter-

mining what the difference is between them. Our standard comparative in (58b)/(64) won’t work,

because there’s no place for the differential to go. So instead we’ll need to modify our comparative

to accommodate this. The version for the big DegP semantics is a bit more complicated, but it’s

still generally the same structure:

(67)

DegP(cid:104)e,t(cid:105)

DPd

three inches

Deg(cid:48)(cid:104)d,et(cid:105)

Deg(cid:48)(cid:104)d,(cid:104)d,et(cid:105)(cid:105)

XPd

Deg(cid:104)(cid:104)e,d(cid:105),(cid:104)d,(cid:104)d,(cid:104)e,t(cid:105)(cid:105)(cid:105)(cid:105)

AP(cid:104)e,d(cid:105)

than Clyde

-er

tall

And the denotation for our differential -er is:

(68)

(cid:126)-erdiff(cid:127) = λG(cid:104)e,d(cid:105)λdλd(cid:48)λx[max{G(x)} − d = d(cid:48)]

31

Nothing too complex; now we’re just saying that the difference between our two degrees is equal to

the difference provided by the degree in SpecDP. The only other move we have to make at this point

is to say that a measure phrase like three inches is directly of type d, where the unit name provides

the unit of measurement and matches the scale (or at least doesn’t conflict with) introduced by the

adjective. This is also a standard move (although see Schwarzschild 2005, 2006 for an argument
that measure phrases are actually properties of degrees and thus type (cid:104)d, t(cid:105)).

The computation is thus fairly standard:

(69)

a. (cid:126)-erdiff tall(cid:127) = λdλd(cid:48)λx[max{tall(x)} − d = d(cid:48)]
b. (cid:126)-erdiff tall than Clyde(cid:127) = λd(cid:48)λx[max{tall(x)} − dClyde = d(cid:48)]
c. (cid:126)3 inches(cid:127) = 3-inches
d. (cid:126)3 inches -erdiff tall than Clyde(cid:127) = λx[max{tall(x)} − dClyde = 3-inches]

In other words, the maximal degree that corresponds to the tallness of an individual x minus the

degree of Clyde’s tallness equals three inches.

The classic viewpoint approaches things similarly; here we’ll also use a different form of -er

that takes in an extra degree argument. This also means that we’ve moved from a > relation to an

= relation, although broadly speaking the > relation still holds for a differential measure phrase

(except in the independently somewhat bizarre 0 case, e.g., Floyd is 0 inches taller than Clyde).

And everything else proceeds more or less the same way as it did in (61) and (62):

32

(70)

(71)

t

DegP(cid:104)dt,t(cid:105)

(cid:104)d, t(cid:105)

DPd

DegP(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)

1

t

three inches

Deg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

XPd

DPe

(cid:104)e,t(cid:105)

-er

than Clyde

Floyd

is

AP(cid:104)e,t(cid:105)

d1

A(cid:104)d,et(cid:105)
tall

a. (cid:126)-erdiff(cid:127) = λdλd(cid:48)λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} − d = d(cid:48)]
b. (cid:126)-erdiff than Clyde(cid:127) = λd(cid:48)λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} − dClyde = d(cid:48)]
c. (cid:126)three inches -erdiff than Clyde(cid:127) = λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} − dClyde = 3-inches]
d. (cid:126)three inches -erdiff than Clyde 1 Floyd is d1 tall(cid:127)

= 1 iff [max{d(cid:48)(cid:48) :tall(Floyd) = d(cid:48)(cid:48)} − dClyde = 3-inches]

The result of the computation states that Floyd is three inches taller than Clyde is true if and only

if the difference between the maximal degree corresponding to Floyd’s tallness and the maximal

degree of Clyde’s tallness is three inches.

So. Why might we prefer a structure that undergoes QR? Well, in addition to making our DegP

look like a form of generalized quantifier, this approach also explains certain scopal effects that

shouldn’t be present if the degree phrase had to be interpreted in situ – no movement should equal

no differences in meaning, but, as Heim (2000) points out, this doesn’t actually seem to be the
case:8

8In (72), p is the paper, longw(p, d) means that x is long to degree d in world w, and Acc(w)
is the set of possible worlds accessible from w (the world argument is suppressed when it’s the
utterance world).

33

(72)

(This draft is 10 pages long.) The paper is required to be exactly five pages longer than

that.

a. required > -er: required [[exactly 5 pages -er than that]1 [the paper be t1-long]

∀w ∈ Acc: max{d : longw(p, d)} = 15 pages
-er > required: [exactly 5 pages -er than that]1 [required [the paper be t1-long]]
max{d : ∀w ∈ Acc: longw(p, d)} = 15 pages

b.

In (72a), the paper is exactly fifteen pages long in every acceptable world. In (72b), the paper

is exactly fifteen pages long in those worlds where it is the shortest (that is, the paper must be at

least fifteen pages long). (The two readings may take a minute to become clear, so don’t worry if

it’s not immediately obvious.) So we in fact seem to be left with two distinct readings, depending

on whether required takes scope over the DegP or not.

A perhaps more striking point is that (72b) gives a reading where the degree morpheme and

the degree clause take scope without the AP. This therefore indicates that DegP is in fact separate

from AP – but if that’s the case, why can’t we say *Floyd is more than Clyde tall? In other words,

why do we have constituency facts that seem to be pointing in opposite directions?

Bhatt & Pancheva (2004) argue that the possibility of both constituencies is the result of move-

ment of the comparative morpheme. The comparative starts as a sister to the AP before moving

countercyclically up to become a sister of the degree clause (the than phrase in a standard com-

parative). However, the morpheme is pronounced in its base position, not in the raised position.

34

(73)

a.

AP

A
tall

DegP

Deg
-er

b.

XP

. . . XP

. . .

AP. . .

DegPi
Deg(cid:48)

Deg
-er

degree clause

DegPi

A
tall

Deg
-er

This movement is therefore the reason why we have constituency facts pointing in both directions:

the pronunciation doesn’t reflect the final position of the comparative, so although we say, e.g.,

taller, the actual semantically-interpreted constituent is ultimately -er than Clyde. This would

mean for our purposes that we want to go with the “classical” QRed structure over the alternate

non-QRed version.

The other remaining piece that we’ll need regarding factor phrases is the equative:

(74) Floyd is as tall as Clyde.

The general consensus regarding equatives is that they differ from (non-differential) compara-

tives only in the type of inequality, being non-strict instead of strict:

(75)

(cid:126)as(cid:127) = λdλF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} ≥ d]

There’s some argument as to whether equatives denote a ≥ relation or just a simple = relation;

the reason for thinking that it’s ≥ is because of sentences such as (76):

(76) You must be as tall as this line to ride the rollercoaster.

35

We don’t want to exclude people from the ride if they exceed the height of the line, so that’s why
we want to go with ≥. Mind, this ignores the fact that this sort of sentence seems to have an at
least reading in non-equative constructions as well:

 four feet

this

 tall to ride the rollercoaster.

(77) You must be

So perhaps we don’t want to automatically assume that equatives have to express a ≥ relation over
=.

2.2.3 Numbers as quantities versus numbers as names

Let’s go back to our “nature of numbers” discussion that we tabled earlier. As you’ll recall,

there was uncertainty about whether Frege’s Easy Argument for the existence of numbers (that

is, whether the fact that a statement such as Jupiter has four moons could be paraphrased as a

seeming identity statement (The number of Jupiter’s moons is four) meant that numbers existed

as genuine objects in the real world) was the right way to think about these things. Part of the

issue, it seems, is that numbers can be used in a wide variety of contexts, but that it’s not clear

that they have a direct bearing on each other. To wit, consider the observation from, among others,

Moltmann (2013) and Snyder (2017) about two different categories of number: there’s a difference

between adjectival uses and monadic, arithmetic uses:

a. What’s the number of children? (At least) four.

b. How many of these numbers are even? Four

(78)

(79)

are

 *is
 is

*are

 .
 .

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

a. What’s the number Mary is researching? (??At least) four.

b. Which one of these numbers is even? Four

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

(one).

36

The sentences in (78) are the adjectival use: they sometimes take plural agreement, they’re

subject to at least readings, and they seem to represent a quantity. By contrast, the sentences in

(79) are the arithmetic use: they don’t take plural agreement or permit at least readings, seeming

to represent something closer to a name than a quantity. And we can further demonstrate this using

the phrase the number N:

(80) What’s the number of children? ??The number four.

(81) What’s the number Mary is researching? The number four.

So we have a distinction between uses here. Snyder (2017), following from work including

Partee (1987), Landman (2004), and Geurts (2006), lists six different environments for numbers to

occur in:

(82)

a. Four is a number.

b. The number four is even.

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

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

e. No four moons of Jupiter orbit Saturn.

f. Jupiter has four moons.

Numeral

Predicative Numeral

Predicative Adjective

Specificational

Intersective Modifier

Quantificational

Snyder argues that all of these uses (and their associated types) are derivable from a basic

individual (type e) via a number of independently motivated type-shifts:

(83)

NOM

IDENT

λxλy.y = x
λP.∩λx.P(x)
λPλQλx.P(x) ∧ Q(x)
ADJUNCT
EXISTENTIAL CLOSURE λPλQ.∃x[P(x) ∧ Q(x)]

(from Partee 1987)

(from Partee 1987)

(from Landman 2004)

(from Partee 1987)

These type-shifts can thus be used to derive all the uses in (82):9

9µ# represents the scale of numbers – the cardinality scale, in other words.

37

(84)

Use

Type

Denotation

Numeral

Predicative Numeral

Predicative Adjective

Specificational

Intersective Modifier

Quantificational

4

λx.x = 4

e
(cid:104)e, t(cid:105)
(cid:104)e, t(cid:105)
d
(cid:104)et, et(cid:105)
(cid:104)et,(cid:104)et, t(cid:105)(cid:105) λPλQ.∃x[µ#(x) = 4 ∧ P(x) ∧ Q(x)]

λx.µ#(x) = 4
∩λx.µ#(x) = 4
λQλx.µ#(x) = 4 ∧ Q(x)

The upshot of this is that the reason not all numbers behave the same across all categories

is because they’ve undergone different type-shifts, depending on their environment.

(This, by

the way, is why Frege’s Easy Argument can’t be right – because it’s trying to conflate the sortal

distinction between numbers (type e) and degrees (type d).) For factor phrases, we’ll be focused

primarily on the specificational degree and the adjectival uses.

2.2.4 The nature of degrees

We should take a moment to discuss the denotation for a degree in (84). Prior to this point we’d

been treating degrees as numbers, sometimes (as in the case of measure phrases) with a unit name

attached. But Snyder (2017) is using a different approach, one that derives from an idea from

people such as Scontras (2014) and Anderson & Morzycki (2015), where degrees are treated as a

special semantic object known as a kind.

A kind is a type of semantic classification, based on certain general properties shared by mem-

bers of a particular term. That said, not every member of the group must have these properties to

still hold: in this sense, kinds represent a sort of idealized or generic form. Consider (85):

(85) Dogs have four legs.

We don’t exclude dogs from the kind “dog” if they happen not to have four legs; rather, as a general

rule we say that dogs have four legs, but we’re willing to allow for exceptions from this generality.

And we don’t have to have particular dogs in mind to truthfully utter this statement; a general

38

mental construct is sufficient. Kinds are frequently represented in English as bare plurals, although

this isn’t always the case; contrast this with (86):

(86) Dogs are hiding in my room.

Here we actually do have to have specific dogs in mind; we’re not saying that a property of

being a dog is that they hide in my room. So we need a way to move back and forth between

kinds and specific instantiations without too much difficulty. Chierchia (1984, 1998), extending
an argument proposed by Carlson (1977), argue that we can do this via the operators ∪ and ∩.
∩ (pronounced “down”) takes us from properties to kinds, while ∪ (pronounced “up”) goes from
kinds to properties. The specifics of how this shift works is beyond the current scope of this
dissertation; for our purposes, it’s sufficient to recognize that the ∩ operator in (84) is moving from
the property of a given measurement to its kind counterpart.

So why are we interested in treating degrees as kinds? Part of it has to do with the language used

when discussing degrees; according to, among others, Anderson & Morzycki (2015), languages

such as German use the same words with kinds, degrees, and manners:

(87)

a. KIND:

einen
a

Hund
dog

so
such
‘a dog of the same kind’

b. MANNER:

getanzt
danced

so
such
‘danced such as that’

c. DEGREE:

bin
am

groß
tall

so
such

Ich
I
‘I am this tall’

39

(88)

a. KIND:

ein
a

dieser
this

Hund
dog

so
such
‘a dog such as this’

wie
wh

b. MANNER:

hat
has

so
such

getanzt
Jan
danced
John
‘John danced the way Mary did’

Maria
Mary

wie
wh

c. DEGREE:

bin
am

so
such

wie
Ich
as
I
‘I am as tall as Peter’

groß
tall

Peter
Peter

In addition, adjectives typically thought of as relating to degrees (e.g., tall, young, cold) can be

modified with manner adverbs:

(89)

a. Ethel is awkwardly tall.

b. The guest speaker is disconcertingly young.

c. This office is bitterly cold.

This would particularly surprising if degrees weren’t related to kinds in some way; after all, num-

bers themselves can’t be modified with manner adverbs, so there has to be something else at work:

(90)

a. #awkwardly 7

b. #disconcertingly 22

c. #bitterly 17,329

Consequently, we need degrees to be more than just numbers. One approach is the degree-as-

kind one mentioned here and realized in (84); another is the version found in Grosu & Landman

(1998) (and, with some modification, in Landman 2004), which argues that degrees are sets of

triples consisting of an individual, its cardinality, and the scale which is being measured along.

40

There’s also the work of Friederike Moltmann (2004, 2007, 2009, 2013, 2017), who argues for

a construction called a trope, which is essentially a particular property that’s associated with a

particular individual or object. Because tropes are specific to particular individuals/objects, no

two tropes are identical: for a crayon, say, the trope of being a particular shade of red is unique

to that crayon, and no other red crayon shares that trope of redness. However, tropes can be

qualitatively identical, thus allowing us to make comparison between tropes. Moltmann argues

therefore that degrees are just a particular instance of number tropes associated with particular

objects. Anderson & Morzycki (2015) are broadly comfortable with the general idea of tropes,

arguing that their version of degree kinds could be integrated with Moltmann’s work, as both are

working along similar lines. (Moltmann herself disagrees, however; see Moltmann 2015b for her

reasons why.)

One additional version of degrees that we should now examine is found in the work of Roger

Schwarzschild, including Schwarzschild (2005, 2008, 2012, 2013). Schwarzschild (2005) notes

that one way to think of the comparative is as a gap from one predicate to another – so in Floyd

is taller than Clyde, the gap extends from Floyd’s height to Clyde’s height, and a measure phrase

measures the size of the gap.

(91) Floyd is 2 inches taller than Clyde.

∃h f∃hctall( f, h f ) ∧ tall(c, hc) ∧ 2-inches([hc → h f ])

Schwarzschild notes that this idea that measure phrases are gap-predicates isn’t a new idea –

McConnell-Ginet (1973) makes this move at one point – but that it’s not one that’s been taken

up much in the intervening years. Schwarzschild (2012, 2013) expand upon this with the idea of

directed scale segments: the idea that these gaps have directionality (rising or falling). Take the

Hindi sentence in (92):

(92) anu
Anu
‘Anu is 2 inches taller than Raj.’

lambii
tall.fem

inc
inch

hai
be.pres.sng

raaj
Raj

se
from

do
2

41

Here the difference in height literally means something like “from Raj to Anu”, so Schwarzschild

argues that we should treat it that way.
(93) Let σ be a directed scale segment where: (cid:104)σ = s, e,≺σ(cid:105):

(94)

start(σ) = s (cid:37) (σ) iff s ≺σ e
end(σ) = s (cid:38) (σ) iff e ≺σ s
a. (cid:126)tall(cid:127)g = λσλx[end(σ) = x∧ ≺σ=height]
b. (cid:126)from Raj(cid:127)g = λP[∃σ[(cid:37) (σ) ∧ start(σ) = Raj ∧ P(σ)]]
c. (cid:126)2 inch(cid:127) = λRλσλx[R(σ)(x) ∧ 2(cid:48)(cid:48)(σ)]
d. (cid:126)2 inch tall(cid:127)g = λσλx[end(σ) = x∧ ≺σ=height ∧ 2(cid:48)(cid:48)(σ)]
e. (cid:126)Anu from Raj 2 inch tall(cid:127) = ∃σ[(cid:37) (σ) ∧ start(σ) = Raj ∧ end(σ) = Anu ∧

≺σ=height ∧ 2(cid:48)(cid:48)(σ)]

So now we’re measuring gaps between individuals (or objects, depending on the sentence). So

what is a degree in this system? Schwarzschild (2013) argues that it’s an ordering of individuals

according to where they fall on a given scale (such as height): a degree is a set of possible indi-

viduals in a possible world, divided into pairs of entities and worlds. This has the advantage of

making the system more flexible by making adjunct differentials such as in by phrases (Floyd is

taller than Clyde by 2 inches) more straightforward.

Regardless of the particular system being used, it’s clear that degrees cannot be treated as primi-

tive objects in the semantics consisting only of a number, with everything else sorted out elsewhere;

instead, degrees are clearly much more complex, incorporating at the very least information not

only about numbers but individuals and perhaps even scales as well.

We now leave the world of numbers, degrees, and comparative semantics (temporarily) behind

us, and turn our attention to plurality.

42

2.3 Plurality

2.3.1 Plurals in the nominal domain

We’ve flirted a bit with plurality so far – most notably in (22) et seq. – but primarily in a somewhat

hand-wavy way, where we just declared that something such as monkeys is represented as mon-

keys(x) without going into much detail. Since factor phrases by their nature deal with plurality,

let’s take a moment to discuss how plurality actually works. The best jumping-off point is going to

be the plurality of nouns, so let’s start there.

The basic sense of the semantics of plurality that we’ll be using derives from Link (1983) and

subsequent papers (including, among others, Link 1991, Link 1984, Landman 1989a, Landman

1989b, and Landman 2000 – and see Szabolcsi 1997 for a useful summary of the literature). The

premise is as follows:

In the nominal domain, we assume that singular count nouns are atomic in nature, not com-

pound; e.g., an apple is an individual entity that isn’t made up of other smaller apples – it’s its

own unique entity. In other words, all the things that make up an apple do not constitute apples

themselves but rather something else, and so the singular apple is therefore atomic in this sense.

(Obviously this sidesteps the problem of what actually constitutes an apple and what it takes to

be considered an apple, but those are questions beyond our current scope. The interested reader

is directed to Link 1984 and chapter 12 of Link 1998 for further discussion.) Note that this only

works for so-called “count” nouns, like apple, monkey, or idea; some nouns, like water, flour, and

space, are what’s called “mass” nouns, in which (linguistically, at least), there’s no clear distinc-

tion between individual units, or atoms. If you put an apple and another apple together you get

apples, but if you put water and more water together you just get water, not waters. Mass nouns are

distinguished by their inability to take plural morphology or have their cardinality measured; *two

water isn’t an acceptable phrase in English, and two waters is only good if you force a count dis-

tinction. (And note that this depends on specific meanings, rather than just being associated with

certain words; space is mass when talking about the physical universe, but count when discussing

43

typography, for instance.) Plurality thus is focused primarily on atomic objects, not on mass ones.

Now if we want to take our apple and then take another apple to get a plural object of “apples”,

we’ll need a way to do so. One way to do this might be to posit that plurals are a special sort of

noun – but if that’s the case, they need to be of the same type as singular nouns (since they fit in all

the same places as singular nouns, modulo agreement patterns). So how can we describe objects

that themselves consist of multiple objects (so, sets) but that are treated the same type-theoretically,

more or less, as singular objects?

One way to do this is in mathematical terms with a lattice. Putting things somewhat roughly, a

lattice is a partially ordered set in which any two elements have a unique supremum (the elements’

least upper bound, also known as a join) and a unique infimum (the elements’ greatest lower bound,

also known as a meet). If only a join or a meet is present, then the structure that results is known

as a semi-lattice, and this is what we’ll be using with respect to plurals.

A model for this semi-lattice (which comes from Landman 1989a) as it pertains to plurals is

the following structure:

(95) A = (cid:104)(cid:104)A, +, (cid:118), AT(cid:105), (cid:126) (cid:127)(cid:105), where

(cid:104)A, +, (cid:118), AT(cid:105) is a complete join semi-lattice.

What (95) says is that A is a set partially ordered by (cid:118), + is the join operation that takes any
nonempty subset B ⊆ A and maps it onto an element of A (written as +B) and AT is the set of atoms
(in other words, all the nonplurals) in A. If + applies to sets with two elements, such as {a, b}, then
+{a, b} = {a + b}. Or, putting it in less technical terms (and therefore less precise, but we can live
with that for the moment), the join operation + lets you take all the atoms in A and combine them

into plurals. So if you have a set consisting of the atomic elements Edmund, Baldrick, and Percy
({e, b, p}), you can use + to get {e + b}, {e + p}, {b + p}, as well as the join {e + b + p}. Here’s a
diagram to help visualize things:

44

(96)

e + b + p

e + b

e + p

b + p

e

b

p

Now this works well for combining distinct atomic elements, but what if you want cumulative

reference? In other words, how do you talk about monkeys without introducing a new word every

time the members of the set change? Why is it that monkeys uses the same word to describe three

monkeys, seventeen monkeys, and one thousand and twenty-six monkeys? The answer is by saying

that plurals simply indicate that there are multiple members in a set of a given noun (in this case,

monkeys), without necessarily being concerned with exactly how many members of the set there

are. But in order to include this in our grammar, we need to introduce a pluralization operation, *.

Here’s Landman (1989a:562)’s definition:

(97)

(cid:126)*P(cid:127) = {y ∈ A : ∃X ⊆ (cid:126)P(cid:127).y = +X}

So for a given y that’s an element of the set A, there’s a set X that’s a subset of the predicate P such
that y is the smallest element of X such that ∀y ∈ X : y (cid:118) +B. This means that (cid:126)*P(cid:127) is a complete
join semi-lattice itself, and +(cid:126)P(cid:127) is the maximal element of *P. All of this is a formal way of saying

that the * operator allows you to add new members to a set without changing the predicate P itself:

*P has closure under sum formation, meaning that any sum of parts of *P is itself *P.

This allows us to talk about things like monkeys without worrying about how many elements

actually make up monkeys – so long as all the members are monkeys, there could be two monkeys,

three monkeys, a hundred monkeys, etc. In formal terms:

(98)

(cid:126)monkeys(cid:127) = λx.*monkey(x)

So then what do we do if we want to talk about a specific number of monkeys? Landman

(2004) notes that it’s possible to incorporate numerals into the lattice structure by having them

45

(99)

refer to the cardinality of the plural sets:
a. (cid:126)at least two(cid:127) = λx.|x| ≥ 2
b. (cid:126)at most two(cid:127) = λx.|x| ≤ 2
c. (cid:126)(exactly) two(cid:127) = λx.|x| = 2

(99) works by combining a number (in this case 2) with an expression that provides the car-

dinality of the members of a plural noun. Depending on which version of the numerical you’re

using, the relationship between the number and the noun will be different – so, for instance, (99a)

states that there are at least two of a given noun but allows for the possibility of more members of

the set, while (99c) requires that there be exactly two members in the set.

So, combining this with our denotation for monkeys in (98) via predicate modification gives us

(100):

(100)

(cid:126)two monkeys(cid:127) = λx.*monkey(x) ∧ |x| = 2

(100) states that given some individual x, x is a plural object *monkey and that the cardinality of

the members of the plural object equals two – in other words, we have two monkeys.

One question regarding (99) is the question of how the number actually makes it into that

expression in the first place. Landman (2004) and Snyder (2017) argue that a numeral is part of a

specific measure phrase structure, as in (101):

(101)

Measure Phrase

Numerical Phrase

Measure Term

Numerical Relation

Number

(unit name)

(at least/exactly/etc.)

2

This allows for numbers to be measurable in some way, either with units or, if no unit is present,
as a measure of cardinality. This phrase is type (cid:104)e, t(cid:105), as noted in section 2.2.3, and so is easily
combinable with our plural monkeys.

46

A similar approach is used in Hackl (2000), who’s following from Bresnan (1973) and others.

This version treats numbers not as type e but as type d, and in order to move from a pure degree
to a property of individuals (type (cid:104)e, t(cid:105)), Hackl argues that there is in fact a covert many expression
that the number combines with:

(102)

a. (cid:126)two(cid:127) = 2
b. (cid:126)many(cid:127) = {λdλx.|x| = d}
c. (cid:126)two-many(cid:127) = {λx.|x| = 2}

Why do we need this many? Independently, people, including, among others, Selkirk (1970),

Bresnan (1973), and Solt (2015), have argued that the word more is in fact the phonological re-

alization of many plus the -er comparative. This is because many and few (and much and little)

appear to pattern together:

 many
 people
 many
 people
 people
 many

few

few

few

(103)

a. as

b.

too

c. so

But we get more people patterning with fewer people instead of *manier people – so in order to

match the pattern, more must therefore logically consist of many and -er. And by extending this

approach to numbers, we can make the types fit properly: treating 2 as a degree means that many
allows us to properly combine a degree with an object of type (cid:104)e, t(cid:105). And it’s also worth noting
that many here is parallel to degree adjectives like tall: both take in degrees to provide a type of

measure phrase – but while adjectives operate in their own domain, many provides measure phrases

for the nominal domain. Using many thus makes explicit the composition of expressions like (99).

47

So, whether we go with the Hackl (2000) version or the Landman (2004) version, the end result
is an object of type (cid:104)e, t(cid:105) that can successfully combine with our plural object and thus provide the
cardinality of the members of the plurality. Which is precisely the outcome we want.

Now that we have the basics established, we can turn our attention to the subject of plurality in

the verbal domain.

2.3.2 Events and plurals in the verbal domain

It’s been observed by many people (including, among others, Bach 1986, Krifka 1989, and Laser-

sohn 1995; for a different take, see Schein 1993) that events can be treated similarly to nouns with

regard to plurality. What do we mean when we say “events”? We’re talking, roughly, about spe-

cific periods of time and things that occurred during that time. The amount of time varies with the

specific thing that is being discussed. So, for instance (to adapt from Parsons 1990, who’s working

from a long history that goes back to the second century B.C. and the writings of P¯an. ini, but which
specifically incorporates Davidson 1967), for a sentence as in (104),

(104) Brutus stabbed Caesar.

we have some event e such that e is a stabbing, the subject of which is Brutus and the object of

which is Caesar, and e occurred some time in the past. (105) provides (a version of) the logical

form of this:

(105) ∃e[stab(e) ∧ Subject(e)(Brutus) ∧ Object(e)(Caesar) ∧ e ≺NOW]

Bach (1986), based on Carlson (1981), breaks down events into specific differing kinds:

48

(106)

eventualities

states

non-states

dynamic

sit

stand

static
be tall
love x

processes

walk
drive

events

protracted

build

momentaneous

happenings
recognize

notice

culminations

die

summit

For Bach, the split between events and non-events is similar to the count/mass distinction for

nouns we mentioned earlier; events can be atomic, while processes (on their own) cannot: “the

fusion of two runnings is a running, but no two dyings are a dying” (Bach 1986:10). This split

doesn’t seem to directly hold true, however; some events can be non-atomic, while some processes

can be atomic. Frequently, adding certain pieces such as prepositional phrases and measurements

can make otherwise non-atomic events become atomic.

(107)

a. Jamie ran.

b. Jamie ran a mile.

c. Jamie ran to the ship.

Non-atomic

Atomic

Atomic

Rothstein (2004) argues that the atomicity of events isn’t related a count/mass distinction, but

rather to a telic/atelic distinction (that is, whether an event has a designated end point or not). For

her, a VP “is telic if it denotes a set of countable events, and a set of entities P is countable if

criteria are given for determining what is an atomic entity in P. So a VP is telic if the VP expresses

criteria for individuating atomic events, and it is atelic if this is not the case” (Rothstein 2004:157).

In other words, if you can determine an end point, your event is therefore atomic.

So given that framework, we can work within it to discuss plural events. A sentence like (108a)

denotes a complete atomic event; so too does (108b).

49

(108)

a. Susan stumbled.

b. Susan twisted her ankle.

(108a) indicates that there is a stumbling event, the Agent of which is Susan, while (108b)

indicates that there is a twisting event, the Agent of which is Susan and the Theme of which is

Susan’s ankle. These are two atomic events, but they can be combined as in (109):

(109) Susan stumbled and twisted her ankle.

Here, by combining these two atomic events, we in fact get a plural event consisting both of

the stumbling event and the ankle-twisting event with one Agent, Susan, and (presumably, but not

necessarily), one time frame during which the plural event occurred. This is akin to a nominal

plural like Susan and David, or the apples and the oranges. We can also express a plural event by

having multiple participants, as in (110):

(110) Several people have discovered the secret base.

But what do we do if we want something like the two monkeys, but with regard to events? How

do we express cumulative reference? This, as we’ll see in the next chapter, is where factor phrases

come in.

50

CHAPTER 3

FACTOR PHRASES IN THE VERBAL DOMAIN

Let’s begin by examining factor phrases when they modify verb phrases. We’re concerned, in the

most basic case, with sentences such as (1):

(1) Floyd walked the dog three times.

Roughly speaking, (1) says that there’s an event of dog-walking by Floyd, and that there have

been three distinct occurrences of this event. Now, times here doesn’t (obviously) indicate multi-

plication, the way it seems to in other contexts; instead, it appears to be a form of counting.

How do we go about analyzing this? Fortunately, in this domain we have a decent amount of

earlier work that we can build upon.

3.1 Previous research

Of all the various types of factor phrases, verbal factor phrases are the ones that have been

studied the most extensively. Admittedly, much of the discussion seems like an aside or brief

tangent – for instance, Parsons (1990) spends less than a page on them and only claims that they

indicate that there were multiple occurrences of a certain time or event, with little in the way of

detailed analysis. Two of the most extensive examinations of these come from Doetjes (1997) and

Landman (2004), so we’ll explore their work below in sections 3.1.1 through 3.1.3.

3.1.1 Doetjes (1997) – Times as classifier

Doetjes argues that times is a form of classifier that combines what she calls adnominal quanti-

fiers (i.e., quantifiers that combine directly only with NPs, such as many) with verb phrases. The

inclusion of times is a way of making numbers adverbal, as (2) demonstrates:

51

(2)

a. *John has danced the salsa one/two.

b. John has danced the salsa one time/two times.

This pattern holds in other languages, such as French:

(3)

a
has

a. *Jean
Jean
‘Jean has danced the salsa two times’

dansé
danced

salsa
salsa

deux
two

la
the

a
has

salsa
b. Jean
Jean
salsa
‘Jean has danced the salsa one/two times’

une/deux
one/two

dansé
danced

fois
times

la
the

So in order to make a numeral behave as an adverb, this times classifier must be used. Doetjes

argues that times is a kind of classifier because it behaves similarly to other, more standard clas-

sifiers, such as piece or kilo, which divide out mass nouns into countable portions. One of the

reasons to believe times is a classifier is its morphology. In Dutch, for example, liter ‘liter’ doesn’t

take plural morphology when it combines with an adnominal quantifier such as twee ‘two’, and in

fact we get singular agreement with the verb:

(4) Er

[ zit/*zitten
[ sits/sit

]
]

twee
two

liter
liter

wijn
wine

in
in

de
the

kaasfondue
cheese fondue

There
’There are two liters of wine in the cheese fondue’

Compare that with a standard plural noun, as in (5):

(5) Er

[ *?zit/zitten
[ sits/sit

]
]

There
’There are two glasses of wine in the cheese fondue’

twee
two

glaz-en
glass-pl

wijn
wine

in
in

de
the

kaasfondue
cheese fondue

The Dutch word for ‘times’, maal, behaves similarly to (4), not (5):

(6)

is
is

vorige
last

Jan
Jan
‘Jan went to the cinema three times last week’

[ maal/*malen
[ time/times

week
week

drie
three

naar
to

]
]

de
the

film
movie

gegaan
gone

Curiously, however, if you use a quantifier like vele ‘many’ with maal you get the opposite

pattern:

52

(7)

ons
us

heeft
has

[ malen/*maal
Jan
Jan
[ times/time
‘Jan has deceived us many times’

vele
many

]
]

om
around

de
the

tuin
garden

geleid
led

But then this is similar to the other classifiers:

(8)

heeft
has

gisteren
yesterday

Jan
]
Jan
]
‘Jan drank many litres of wine yesterday’

[ *liter/liters
[ liter/liters

vele
many

wijn
wine

gedronken
drunk

Doetjes argues that this pattern has to do with the nature of the quantifiers: degree quantifiers

such as vele ‘many’, which combine with more than just NPs, require plural morphology, and so

these classifiers take on this morphology and become interpreted as real count nouns, rather than

as classifiers; adnominal quantifiers such as twee have no such requirement, and so the classifiers

remain unmarked for plural. In this sense, maal is no different from liter; it’s the nature of the

quantifier that leads to differences between singular and plural morphology.

However, unlike liter, maal and its equivalents don’t have an overt NP complement. Doetjes

argues that while it’s possible that this is because times selects for a VP instead of an NP, it’s more

likely that there’s a covert noun present. One argument for this is that, in Dutch, the verb moves up

to T, which should be blocked by N keer, ‘N times’, if it were in fact selecting the VP and therefore

dominating it: keer would be an intervening head and thus would block the movement. But that’s

not actually the case:

(9)

ati
ate

vorige
last

Jan
Jan
‘Last week, Jan ate cheese fondue three times’

[VP kaasfondue

week
week

drie
three

keer
time

cheese fondue

ti]

Doetjes points out that further evidence against this idea that times is a classifier that selects

for a VP comes from a comparison with a similar classifier, pieces. Pieces works by essentially

portioning out a mass noun like cheese or a count mass noun like furniture. However, if you

combine pieces with a noun that’s already count then it forces it into a mass interpretation, as in

(10c):

53

(10)

a.

b.

three pieces of cheese

three pieces of furniture

c. #three pieces of cup

Now, if times were the VP counterpart to pieces, then we should expect similar behavior. And

indeed, when times combines with an atelic predicate like ran, we do get a “portioning” reading:

(11)

a. Jack ran.

b. Jack ran three times.

(11b) means that there are three separate events when Jack ran – a portioning reading, just as we’d

expect.

Alas, the pattern doesn’t hold. Doetjes points out that the predicate to buy two kilos of olives

has singular reference – it can’t be divided into more than one event of two-kilos-of-olives-buying.

So if times is like pieces, then when we modify this predicate with a factor phrase, we should get

a portioning of our olive-buying into separate chunks, just as cup was portioned out in (10c). But

that’s not what actually happens; instead we get three separate events, buying two kilos of olives

each time:

(12)

a. John bought two kilos of olives.

b. Last week, John bought two kilos of olives three times.

So it therefore doesn’t seem to be the case that times is a classifier with a VP complement;

instead, it seems to be part of a full noun phrase, just with the NP that times selects remaining

unexpressed. (Unfortunately, Doetjes doesn’t go into any detail about the nature of this covert NP.)

3.1.2 Landman (2004) – Definite times adverbials and degrees

Landman (2004) expands upon both this and Rothstein (1995). He begins by noting that Rothstein

pointed out a difference between definite and indefinite time adverbials when they head a relative

clause:

54

(13)

a.

b.

I opened the door the three times the bell rang.

I opened the door three of the times the bell rang.

c. #I opened the door three times the bell rang.

The definite version in (13a) behaves differently from the indirect version: definites work in

a normal adverbial position, but indefinites are degraded. (According to Landman, judgments on

(13c) aren’t always “completely infelicitous”, varying with speakers and certain cases, although he

doesn’t give an example of a more acceptable indefinite case.) This isn’t the case if an indefinite

occurs in a noun phrase position, however:

(14)

I remember with pleasure seven times (that) I had dinner with him.

Rothstein analyzes all these phrases as prepositional phrases with a null preposition; this would

explain why modification seems to be happening with a noun phrase where we’d expect an adver-

bial phrase - prepositional phrases can occur in adverbial positions. But as Landman points out,

if these are in fact PPs instead of DPs, what’s causing the difference in acceptability in (13)? If

it’s about whether or not you’re in an argument position (which might explain the acceptability of

(14)), then why would (13c) be problematic? After all, if these are PPs then (13c) is just as much

an argument position as (14).

Landman instead argues that these structures are similar to (15), which he and Alexander Grosu

argued in Grosu & Landman (1998) should be interpreted with the head noun internally in the

relative clause, with a free degree variable:

(15)

a. The three books (that) there were – on the table were mine.

b. #Three/many books (that) there were – on the table were mine.

(16) The three books (that) there were (n many books) on the table were mine.

In order to make this work, Landman argues that the type for degrees isn’t a simple number,
but rather a pair-wise relationship between an individual and a number, which he calls (e × n).1
This is defined as a pair consisting of an object and its cardinality:

1Strictly speaking, he calls them (d × n), but this is because he’s using d as the type for indi-

55

(17) DEGD = {(cid:104)e, n(cid:105) : e ∈ D and n ∈ N and |e| = n}

This allows us to recover not just a number from a degree, but also the object associated with a
given degree. This also means that our cardinality function, instead of being of type (cid:104)e, n(cid:105), is now
(cid:104)e, (e × n)(cid:105):

(18)

card = λx. < x,|x| >

So, given these pieces, our relative clause in (16) works as follows:

(19)

thatn there were (zn many books) on the table

a.

{max[λzn.∃x[books(x)∧ < x,|x| >= zn∧ on-the-table(x)]}
= {<sum(λx.*book(x) ∧ on-the-table(x)),|sum(λx.*book(x) ∧ on-the-table(x))| >}

Or, in other words, the singleton set that contains the degree which consists of the sum of all the

books on the table, and its cardinality.

Now the relative clause up to this point is currently type (cid:104)(e × n), t(cid:105), but in order to properly
combine with the head noun books it needs to be (cid:104)e, t(cid:105). Landman argues that the head noun has
something called substance, which allows it to be interpreted both internally and externally with

regards to the relative clause. This substance operation thus selects the first part of the degree pair
and provides the set of first elements of the pair, thus shifting to type (cid:104)e, t(cid:105).

(20)

(21)

substance(CP) = {x :< x,|x| > ∈ CP}
(cid:126)books thatn there were (zn many books) on the table(cid:127)

= sum(λx.*book(x)∧ on-the-table(x))

Landman describes (21) as being inherently definite; this means that an indefinite determiner

such as a or the plural Ø can’t combine with it, but the definite the can. Three is added appositively

(roughly, this means it adds the cardinality statement as a presupposition), and our final result is

(22):
viduals and e as the type for events. For the sake of consistency with the rest of this dissertation,
we’ll adjust Landman’s work slightly to continue to use e as the type for individuals and s as the
type for events.

56

(22)

(cid:126)the three books thatn there were (zn many books) on the table(cid:127)

= sum(λx.*book(x)∧ on-the-table(x))
Presupposition: |sum(λx.*book(x)∧ on-the-table(x))| = 3

As Landman (2004:228) states, “the noun phrase denotes the sum of the books on the table, with

presupposition that there are three.”

This is how it works for a standard noun. The difference between those and time(s), however,

is that times doesn’t need to trigger substance. (Obviously it can, as (14) shows, but it doesn’t need

to.) So what that means is that, for a phrase as in (13a), we have a denotation (seen below in (23))
for a full relative clause that’s not of type (cid:104)s, t(cid:105) (s because we’re dealing with verbs and thus events
now, not nouns and individuals) but rather type (cid:104)(s × n), t(cid:105):

(23)

[DPdegree the three times thatn the bell rang (zn many times)]

=<sum(λe.ring(e) ∧ Th(e) = σ(bell)), 3 >
where |sum(λe.ring(e) ∧ Th(e) = σ(bell))| = 3

As this is still a degree, Landman argues that it can’t appear in an argument position; instead,

it will form an adverbial measure phrase. This means that the entire phrase the three times the bell

rang is the degree part of a measure phrase. So using the same structure we saw in (101) back in

section 2.3.1, we get (24):

(24)

Measure Phrase

Degree Phrase

Measure

Degree Relation

DP[degree]

Ø

Ø

the three times the bell rang

The measure this time won’t be cardinality (because we’re dealing with adverbs, not adjectives)

but rather a similar operation that Landman calls CANTOR. CANTOR is a way of counting indi-

rectly, by mapping every sum of events onto a degree of an isomorphic sum of events – and since

57

isomorphic sums of events have the same cardinality, CANTOR is a counting measure. CANTOR
is of type (cid:104)s, (s × n)(cid:105) and combines with a degree phrase to give (25), which is type (cid:104)s, t(cid:105):

(25)

(cid:126)the three times the bell rang(cid:127)(CANTOR)

= λe[CANTOR(e) = <sum(λe.ring(e) ∧ Th(e) = σ(bell)), 3 >]

And so a full sentence would be as in (26):

(26) The three times the bell rang, Dafna opened the door.
∃e[*open(e)∧ *Ag(e) = Dafna ∧ *Th(e) = σ(door) ∧

CANTOR(e) = <sum(λe.ring(e) ∧ Th(e) = σ(bell)), 3 >]

This states that there’s a sum of door openings by Dafna, and CANTOR maps that sum onto the

degree of the three bell ringings. This isn’t quite right, though; we need a way to map bell ringings

to door openings in a one-to-one correspondence. So let’s constrain CANTOR slightly:
(27) CANTOR is a function of type (cid:104)s, (s × n)(cid:105) such that:

for every e ∈ E : e (cid:27) [CANTOR(e)]1

Now CANTOR “maps every sum of degrees onto the degree of an isomorphic sum of events. In

this way CANTOR is a counting measure, because isomorphic sums of events have, of course, the

same cardinality.” (Landman 2004:232)

So what (26) now states is that the sum of door openings by Dafna and the sum of bell ringings

are isomorphic - so every time the bell rang, Dafna opened the door.

This analysis captures the idea that what event adverbials do is perform a counting operation,

one performed indirectly via CANTOR, and it also captures the definiteness effects that we saw in

(13).

3.1.3 Landman (2004) – Indefinite times adverbials and groups

So definite time adverbials essentially denote a form of degree that counts events indirectly. But

what about indefinite time adverbials? We can’t use the same analysis, because they don’t pattern

58

the same as the definites; Landman argues that while (28a) counts events indirectly via contextually

specified events (i.e., via CANTOR), (28b) counts events directly – there’s no correspondence

between Dafna’s jumping and some other event:

(28)

a. Dafna jumped

b. Dafna jumped

 every time
 many times

three times

the three times

 .

 .

Therefore, what Landman calls the Obvious Analysis would seem to be (29):

(29)

(cid:126)three times(cid:127) = λe.|e| = 3

But Landman points out two problems with this. First, if three times is just a simple (cid:104)s, t(cid:105) modi-
fier, then we should expect similar effects from other (potential) event-denoting NP adverbials. But

that’s not the case: as we’ve seen before, an indefinite times phrase with a relative clause doesn’t

work, and neither does an event NP such as burning – burning only works as part of a PP:

(30)

#I opened the door three times the bell rang.

(31)

a. #They burned the documents three burnings.

b. They burned the documents in three burnings.

Second, when our indefinite factor phrase interacts with scope, the intersective Obvious Anal-

ysis provides the wrong analysis; take (32):

(32) Two girls kissed Dafna three times.

We want this to (potentially) mean something like “there were two girls, and each of them kissed

Dafna three times for a total of six kissing events”, but that’s not what the Obvious Analysis

provides:

(33)

(cid:126)Two girls kissed Dafna three times(cid:127)

= 1 iff ∃e[*kiss(e) ∧ *girl(*Ag(e)) ∧ |*Ag(e)| = 2 ∧ *Th(e) = Dafna ∧ |e| = 3]

59

This states that there are three kissing events, the theme of which is Dafna; the agent of these three

kissing events is two girls. This is a possible interpretation of (32), but it shouldn’t be the only one

(and, one might argue, it’s not the natural interpretation).

So, if the Obvious Analysis doesn’t work, then what should we do instead? We need our factor

phrase to actually take in an event and then modify that event, rather than just joining up to it via

predicate modification. Landman (2004) chooses to do thus by treating times as a classifier, just

as Doetjes (1997) argues. We’ve seen Doetjes’s arguments for this in section 3.1.1; in addition,

Landman notes that, when times is in argument position, it matches the noun agreement pattern,

not the classifier:

(34)

herinner
remember

met
Ik
with
I
‘I remember seven times that I had dinner with him.’

[ keren/#keer
[ times/#time

dat
that

ik
I

me
self

zeven
seven

]
]

hem
him

dineerde.
dined

(=seven dinner events)

Now in order to define how the times classifier actually counts events, Landman chooses to

analyze plural events (and plural individuals, for that matter), in terms of groups. Groups (see,

among others, Link 1984 and Landman (1989a,b, 2000)) are a way of treating plural, non-atomic

sets as atomic. Consider (35):

(35) The committee is meeting tomorrow at 3 pm.

Here we have a noun, committee, which consists of multiple members (since it’s unlikely, unless

you’re Princess Leia, to have a committee of one) but which is treated as a single unit. Note that

the properties of the members of the group don’t transfer to the group itself. As Landman (1989a)

points out, we can say that the Beatles is made up of four musicians who are all rock stars, but

while we can say (36a) and (b), it seems odd to combine both of them together in (c); instead, (d)

is a more natural way of expressing this thought:

60

(36)

a. The Beatles are rock stars.

b. The Beatles is a rock group.

c. #The Beatles is a rock group and are rock stars.

d. The Beatles is a rock group and consists of rock stars.

So we need a way to capture this intuition of singular-but-plural, where the Beatles isn’t simply

the sum of John, Paul, George, and Ringo but instead can also be treated a single unit. Groups
are the way to do this. The standard way to indicate this is with an up arrow, ↑. So now we
can move from pure sums (plurals with a definite attached, such as the – essentially, the join,

or maximal element, of the corresponding semi-lattice) to impure atoms (groups). And we also
have a corresponding operator, ↓, to go the opposite direction, from groups to plurals. So for our
Beatles example, (37a) represents the pure sum of the Beatles as a plurality, while (37b) is the

group version:

(37)

a. σx.*Beatles(x)
b. ↑ (σx.*Beatles(x))

The other thing worth noting about a group is that not all members of the group need be present

in order for the group to count as a group; for instance, Paul McCartney is the only member of

the Beatles to actually play on the song “Yesterday”, but we can still utter (38) without fear of

contradiction:

(38)

“Yesterday” is a song by the Beatles.

Now given this, we can return to Landman (2004)’s approach to classifier times. In addition

to groups in the nominal domain, Landman argues for group events as well. The semantics is the

same; the difference is only whether we’re dealing with type e or type s. And the introduction of
group events means that we’ll need group thematic roles as well. Those are represented as ↑*R,
defined as in (39):

(39)

↑*R(↑ (e)) =↑ (*R(e))

61

So if a plural event takes a plural subject, such as σx.*boy as a plural agent, such that all the boys
perform the event together, then the corresponding group ↑ (σx.*boy) is the group agent for the
corresponding group event.

So what times does is turn non-countable sums into countable atoms by making the sums
groups. Landman devises a special semantic type for this counting operation, (cid:104)↑,(cid:104)α, t(cid:105)(cid:105), where α
is either type e or type s. We then need two operators to move us between counting types and
their non-counting counterparts (i.e., from (cid:104)↑,(cid:104)α, t(cid:105)(cid:105) to (cid:104)α, t(cid:105) and back). Landman defines these
operators as in (40):

(40)

a. Counting operator ↑
Let α be type e or s
If a ∈ EXP(cid:104)α,t(cid:105), then ↑a ∈ EXP(cid:104)↑,(cid:104)α,t(cid:105)(cid:105)
(cid:126)↑a(cid:127)M,g = {↑ (x) : x ∈ (cid:126)a(cid:127)M,g}

b. Counting operator ↓
Let α be type e or s
If a ∈ EXP(cid:104)↑,(cid:104)α,t(cid:105)(cid:105), then ↓a ∈ EXP(cid:104)α,t(cid:105)
(cid:126)↓a(cid:127)M,g = (cid:126)a(cid:127)M,g

These are essentially predicate-modifying versions of the standard group and ungroup operators (↑
and ↓, respectively), allowing us to convert more than just individuals into groups. And so times
takes advantage of this by taking in a counting type and producing the non-counting version:

(41)

(cid:126)classifier time(cid:127) = λz(cid:104)↑,(cid:104)s,t(cid:105)(cid:105).↓z

Now we arrive at the question that Doetjes (1997) raised: if times is indeed a classifier, what is

combining with? Doetjes suggested it was a covert NP; Landman (2004) agrees, arguing it’s just
an identity predicate of type (cid:104)(cid:104)↑,(cid:104)s, t(cid:105)(cid:105),(cid:104)↑,(cid:104)s, t(cid:105)(cid:105)(cid:105):

(42)

(cid:126)ØNP(cid:127) = λz.z

Combining these two, we see that times therefore takes in the countable null predicate and shifts it

to the non-countable version.

62

The step-by-step composition of our ultimate three times denotation involves a level of tech-

nical detail that goes beyond the current scope of this dissertation, including the introduction of

categorial grammar and its use of slashed categories and associated mechanisms in order to re-

solve some of the type mismatches that occur along the way; the interested reader is thus directed

to Landman (2004:244–251) for the specific details, but for now we’ll content ourselves with the

finished denotation of three times, provided in (43):

(43)

(cid:126)three times(cid:127) = λz(cid:104)↑,(cid:104)s,t(cid:105)(cid:105)[↑λe.|e| = 3 ∧ [*↓z](e)]

This states that, for some group event e, the cardinality of which is 3, e applies to the non-countable

version of the event property in z.

The only other piece we need at the moment is the APPLY function: this is used essentially

to resolve type clashes by shifting a predicate from a sum to a group; this is a consequence of

resolving a categorial grammar slash category. So when it’s used in combination with our factor

phrase, the pieces play out as in (44):

(44) Dafna jumped three times.

a. (cid:126)Dafna jumped(cid:127) = λe(cid:48)[jump(e(cid:48)) ∧ Ag(e(cid:48)) =Dafna]
b. (cid:126)APPLY Dafna jumped(cid:127) =↑ λe(cid:48)[jump(e(cid:48)) ∧ Ag(e(cid:48)) =Dafna]
c. (cid:126)three times(cid:127)((cid:126)APPLY Dafna jumped(cid:127)) = λz(cid:104)↑,(cid:104)s,t(cid:105)(cid:105)[↑λe.|e| = 3 ∧ [*↓z](e)]

(↑λe(cid:48)[jump(e(cid:48)) ∧ Ag(e(cid:48)) =Dafna])

=↑ λe.|e| = 3 ∧ [*↓↑λe(cid:48)[jump(e(cid:48)) ∧ Ag(e(cid:48)) =Dafna]](e)
=↑ λe.|e| = 3 ∧ [*λe(cid:48)[jump(e(cid:48)) ∧ Ag(e(cid:48)) =Dafna]](e)
d. ∃ f [↑λe.|e| = 3 ∧ [*λe(cid:48)[jump(e(cid:48)) ∧ Ag(e(cid:48)) =Dafna]](e)]( f )

(via existential closure)

Ultimately we end up with an atomic group event f that corresponds to a sum of events of three

countable atomic events, which are a sum of events of Dafna jumping.

Doing things this way also allows us to get the proper interpretation when we have other nu-

merals involved. Recall (32), repeated for convenience here:

63

(32) Two girls kissed Dafna three times.

This sentence has, of course, multiple readings. In the version where two girls combines after

three times, thus meaning that each girl performed three kissing events (for six total kissings), we

essentially get the same semantics as we saw in (44); three times combines with the VP before two

girls does, meaning it combines with essentially x kissed Dafna:

(45)

a. (cid:126)x kissed Dafna(cid:127) = λe(cid:48)[kiss(e(cid:48)) ∧ Ag(e(cid:48)) = x ∧ Th(e(cid:48)) =Dafna]
b. (cid:126)three times(cid:127)((cid:126)APPLY x kissed Dafna(cid:127)) = λz(cid:104)↑,(cid:104)s,t(cid:105)(cid:105)[↑λe.|e| = 3 ∧ [*↓z](e)]

(↑λe(cid:48)[kiss(e(cid:48)) ∧ Ag(e(cid:48)) = x ∧ Th(e(cid:48)) =Dafna])

=↑ λe.|e| = 3 ∧ [*λe(cid:48)[kiss(e(cid:48)) ∧ Ag(e(cid:48)) = x ∧ Th(e(cid:48)) =Dafna]](e)

c. (cid:126)x kissed Dafna three times(cid:127)

= ∃ f [↑λe.|e| = 3 ∧ [*λe(cid:48)[kiss(e(cid:48)) ∧ Ag(e(cid:48)) = x ∧ Th(e(cid:48)) =Dafna]](e)]( f )

(via existential closure)

Now we apply Landman (2000:52)’s Scope Mechanism, which will make (45c) type (cid:104)e, t(cid:105), and that
gets shifted to an object of type (cid:104)e, st(cid:105). Once that’s done, we can combine it with two girls thusly:

(46)

(cid:126)two girls kissed Dafna three times(cid:127)

= ∃x[*girl(x) ∧ |x| = 2 ∧ ∀a ∈ ATOM(x) :
∃ f [[↑λe.|e| = 3 ∧ [*λe(cid:48)[kiss(e(cid:48)) ∧ Ag(e(cid:48)) = a ∧ Th(e) =Dafna]](e(cid:48))]( f )]]

This states that “there is a sum of two girls and for each of these girls there is an atomic group

event corresponding to a sum of events that consists of three atomic events and that is a sum of

events of that girl kissing Dafna.”

The other version, where three times joins after two girls, is slightly different. Here we start

with two girls kissed Dafna, which has the following denotation:

(47)

(cid:126)two girls kissed Dafna(cid:127)

= λe(cid:48)[*kiss(e(cid:48)) ∧ *girl(*Ag(e(cid:48))) ∧ |*Ag(e(cid:48))| = 2 ∧ *Th(e(cid:48)) =Dafna]

64

A couple things to note: first, since we have more than one girl at this point, we have a plural kissing

event (unlike in (45a)); second, because we have a plural event we also have plural thematic roles,

and we also have a plural subject, of which there are two.

What this means is that we’re going to have a slightly different effect when we combine with

three times:

(48)

(cid:126)three times APPLY two girls kissed Dafna(cid:127)

=↑ λe.|e| = 3∧[*↓↑λe(cid:48)[*kiss(e(cid:48)) ∧ *girl(*Ag(e(cid:48)))∧|*Ag(e(cid:48))| = 2 ∧ *Th(e(cid:48)) =Dafna]](e)
Because the input this time was a sum of events, the ↑↓ doesn’t cancel out the way it did in (44c).
So existential closure leads to (49a), which is equivalent to (49b):

(49)

a. ∃ f [↑λe.|e| = 3 ∧ [*↓↑λe(cid:48)[*kiss(e(cid:48)) ∧ *girl(*Ag(e(cid:48))) ∧ |*Ag(e(cid:48))| = 2 ∧

*Th(e(cid:48)) =Dafna]](e)]( f )
b. ∃e[|e| = 3∧[*↓↑λe(cid:48)[*kiss(e(cid:48)) ∧ *girl(*Ag(e(cid:48)))∧|*Ag(e(cid:48))| = 2 ∧ *Th(e(cid:48)) =Dafna]](e)]

Our final result states that “there is a sum of three atomic group events and each of these atomic

group events corresponds to a sum of events of two girls kissing Dafna.”

Implementing things in this way means we need to revise the definite analysis in section 3.1.2

slightly. Instead of treating, e.g., the three times the bell rang as a degree, we need to treat it

as a full measure phrase. This will make it a counting modifier like our indefinite version. This

essentially means that we take our denotation in (25) and combine it with the null NP gap that we

saw in (42). This (after the appropriate type shifts) provides us with the semantics in (50):

(50)

(cid:126)the three times the bell rang Ø(cid:127)

= λz[↑λe[CANTOR(e) = <sum(λe.ring(e) ∧ Th(e) = σ(bell)), 3 > ∧ [*↓z](e)]]

Ultimately, we’ve seen that verbal factor phrases are ways of counting events, with Landman

arguing that definite versions count them indirectly via the CANTOR mechanism, while indefinite

factor phrases count those events directly, shifting them to atomic group events as necessary for

cardinality to count the proper (group) events.

65

One key point that both Doetjes (1997) and Landman (2004) observe is that the verbal factor

phrase imposes atomicity on events, even on events that otherwise wouldn’t be considered atomic.

This is part of times’s being a classifier: much how mass nouns are separated into measurable

portions by unit names, so times imposes atomicity on plural events via the group mechanism.

Recall (11), repeated below:

(11)

a. Jack ran.

b. Jack ran three times.

Our non-atomic running event in (11) gets turned into a group, which therefore makes it an atom.

This doesn’t always work, however, as we see in (51):

(51)

a. Jack ran five times.

b. ?Jack is running five times.

c. *?Jack was running five times.

(51b) is only really good on a reading that signals intent, i.e. that Jack intends to undertake five runs

(as in at a track meet), while (51c) is really bizarre; we can kind of get the sense that there were

five running events, but the use of the progressive makes things difficult to parcel out appropriately,

as if we’re not quite certain where to draw the line to make the portion. We can see this a bit more

clearly in (52):

(52)

a. Douglas was eating an apple three times.

b. Douglas ate an apple three times.

Here the difference is more apparent: while (52b) can mean that Douglas ate three apples (setting

aside the nonsensical individual reading where he eats the same apple three times), (52a) can mean

that only one apple was involved, and that there were three eating events involving that apple, but

that (at least) two of them did not succeed in finishing off the apple. This tells us that there are

restrictions on what events can combine with a verbal factor phrase: the event has to be replicable

(else we get the nonsensical reading for (52b)), and it has to have some sense of a plausible ending.

66

This is why (52a) seems a bit better than (51c): it’s easier to separate incomplete apple-eatings –

one merely needs the apple not to be gone at the end of the event – than it is to separate nebulous,

undefined running events. After all, what makes Jack’s runnings not just one long running event?

What is it about them that we need to separate them into distinct portions? This is a difficult

question to answer, which is why we find (51c) so strange; there’s not even a potential end point

that we simply have to have not reached, as in the apple case. Perhaps we’re running into the

distinction between telicity and boundedness, as discussed by Depraetere (1995), where there is

sometimes a difference between the telicity of a sentence and whether we can conceive of it having

either a starting or an ending point (its boundedness). Perhaps it’s easier to envision avoiding a

specific endpoint when eating an apple than an open-ended one such as running. And so whether

or not we have a rough sense of the size (in a vaguely temporal sense) of an event makes a difference

on how easily we can turn it into a group event.

But regardless of the specific issues with interactions with other constructions, we have a gen-

eral workable sense of what verbal factor phrases do; they package events into groups and then

measure the cardinality of the total number of groups.

3.2 Verbal factor phrases and comparatives

Now, we’ve seen that there is indeed a available off-the-shelf semantics that we can use for

our event-counting factor phrases. However, Landman, as we’ve seen, is interested in distributive

vs. collective readings for event-counting factor phrases and how they interact scopally with other

predicates; in other words, how we tease apart the multiple readings that can arise from scopal

ambiguities without breaking other things in the system. This is of course a useful goal, but it’s also

somewhat tangential to the point I would now like to make, and the worry is that the complexity

of Landman’s analysis will obfuscate that point. So at this point let’s make the move to simplify

the denotation of the event-counting factor phrase – not with the goal of replacing it, but merely to

package it away for the sake of clarity. The semantics I will outline below should be compatible

with the denotations in Landman (2004) without any significant effort.

67

So here’s how we’ll simplify the denotation of n times: instead of being Landman’s counting
type modifier of (cid:104)(cid:104)↑,(cid:104)s, t(cid:105)(cid:105),(cid:104)↑,(cid:104)s, t(cid:105)(cid:105)(cid:105), n times will be simply a modifier of type (cid:104)st, st(cid:105). This is
simply so that we’ll have less to keep track of. N times will still track the cardinality of an event
e, and it will still attach as a modifier, as in (53)2, but we’ll package up the specifics of our event
in a special predicate, *time(e): this approach will require that the event be both plural and atomic

(since atomicity is a prerequisite for our * operator).

(53)

(cid:104)s, t(cid:105)

(cid:104)s, t(cid:105)

Floyde

(cid:104)e, st(cid:105)

(cid:104)st, st(cid:105)

three times

walk(cid:104)e,(cid:104)e,st(cid:105)(cid:105)

e

the dog

(54)

(cid:126)n times(cid:127) = λe[*time(e) ∧ |e| = n]

So a sentence such as Floyd walked the dog three times works as in (55):

(55)

a. (cid:126)three times(cid:127) = λe[*time(e) ∧ |e| = 3]
b. (cid:126)Floyd walk the dog(cid:127) = λe.walk(the-dog)(Floyd)(e)
c. (cid:126)Floyd walk the dog three times(cid:127) = λe.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = 3

Our final result states that there is a plural walking event e, the Agent of which is Floyd and the

Theme of which is the dog, and that there are three of these events. This is the desired meaning.

Again, just to be clear, the point isn’t to replace Landman (2004)’s denotation because there’s

a problem with it; this is simply a move done for the sake of clarity.

3.2.1 Additive factor phrases

Now let’s examine a slightly more complex sentence like (56).

2This structure assumes existential closure further up the tree.

68

(56) Floyd walked the dog three more times than Clyde (did).

Intuitively, we want (56) to mean that however many times Clyde walked the dog, Floyd walked

the dog that many times plus three additional times. This is what I will refer to as the “additive”

reading.

In order to understand what happens in (56), it’s worth looking at how this construction works

with a more typical noun phrase with a numeral, like five walruses. When put in a sentence similar

to (56), a sentence like (57) results:

(57) Floyd rescued five more walruses than Clyde (did).

The key thing to note about (57) is that the phrase five more walruses isn’t simply a noun

phrase with a comparative; in fact, something more complicated is going on. In order for (57) to

be interpretable, we need to move some pieces. So what we say is that there is a type of generalized

quantifier, similar to every toilet or some monkeys, present in this sentence. Specifically, it’s not

five more walruses, but actually five -er than Clyde, which is our generalized quantifier: one over
degrees and of type (cid:104)(cid:104)d, t(cid:105), t(cid:105) (as noted in, among others, Heim 2000) that cannot be interpreted
in situ; rather, it must move above the subject position, leaving a trace of itself behind. Thus a

sentence like (57) will have a structure as in (58):

(58)

(cid:104)dt, t(cid:105)

5d

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

-er(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

d

1

∃s

than Clyde

Floyd

rescued

d1-many walruses

69

As mentioned earlier, this structure assumes, following Bresnan (1973), Hackl (2000), and

others, that more is composed of many-er, and that it is the 5 -er than Clyde portion that moves up,

leaving the trace d1 behind. This is the standard assumption for how these types of sentences are
interpreted.

Structurally, a sentence with a factor phrase (56) works the same way as one with a standard

numerical noun phrase, giving us the tree in (59):

(59)

(cid:104)dt, t(cid:105)

3d

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

-er(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

t

d

(cid:104)d, t(cid:105)

1

∃s

than Clyde

t

(cid:104)s, t(cid:105)

(cid:104)s, t(cid:105)

(cid:104)s, t(cid:105)

Floyd walked the dog

d1-many times

One thing in particular that’s common to both sentences in (56) and (57) is the presence of

a differential comparative: a comparative that not only compares two degrees, but also provides

a third degree representing the difference between the first two degrees. The standard view of a

non-differential comparative is one as in (60):

(60) Ezekiel walked the dog more than Agnes.

Here we say that the comparative -er simply compares the number of dog-walkings by Ezekiel

with that of the number by Agnes, as in (61):3
(cid:126)-er(cid:127) = λdλ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} > d

(61)

3(60) is technically ambiguous between a cardinality reading and a total amount reading. One
could imagine a scenario in which Ezekiel walks the dog seven times, but each time is only for
ten minutes, while Agnes walks the dog three times for thirty minutes each. Under the cardinality
reading, (60) is true, but under the total amount reading, it is false. However, as this doesn’t directly
bear on the question of factor phrases (which explicitly allow only the cardinality reading), let’s
acknowledge the ambiguity and then quietly draw a veil over the subject.

70

However, the addition of another degree in both (56) and (57) (in the factor phrase and the

MP, respectively) requires the use of a differential comparative morpheme, which incorporates that

additional degree in its denotation as d:

(62)

(cid:126)-er(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − d ≥ d(cid:48)

Treating the differential comparative as separate from the non-differential comparative is not a

new idea by any means – see, among others, Hellan (1981), von Stechow (1984), Rett (2008b), and

Schwarzschild (2008) – but it is worth noting how exactly this differs from our standard compar-

ative; not only is there a third degree present, but we’ve also moved from a strict inequality in the

standard comparative to a non-strict inequality in the differential. This is to ensure that the truth

conditions for a sentence like (63),

(63) Floyd is five inches taller than Clyde,

are accurate: we don’t want it to be the case that if Floyd is 6 feet tall and Clyde is 5 foot 7 inches,

(63) comes out false – but it would if we were using a strict inequality.

The use of a differential comparative in a sentence like (56) is straightforward. Composition-

ally, it will work as in (64):

(64)

a. (cid:126)Floyd walked the dog d1-many times(cid:127)

= 1 iff ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d1

b. (cid:126)-er(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − d ≥ d(cid:48)]
c. (cid:126)than Clyde(cid:127) = dClyde

(in other words, the number of dog-walkings Clyde has performed)4

d. (cid:126)3 -er than Clyde(cid:127) = λ f(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − dClyde ≥ 3]
e. (cid:126)3 -er than Clyde 1 Floyd walked the dog d1-many times(cid:127)

= 1 iff max{d(cid:48)(cid:48) : ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d(cid:48)(cid:48)}−dClyde ≥ 3

4As before, this is simplified for the sake of clarity.

71

Our final result in (64e) states that there is a plural event of walking the dog, the Agent of which

is Floyd, and the degree that corresponds to the number of times Floyd has walked the dog, minus

the number of times Clyde walked the dog, will be at least three. This is indeed the outcome that

we want.

3.2.2 Scales of measurement vs scales of plurality

So far we’ve seen that additive factor phrases are very similar syntactically to standard MPs, and

that while they’re semantically different, they’re not wildly so. But what about when we take a

slightly different word order, as in (65)?

(56) Floyd walked the dog three more times than Clyde (did).

(repeated for convenience)

(65) Floyd walked the dog three times more than Clyde (did).

The slight change in word order in (65) gives rise to an ambiguity not previously noted in

the literature, it turns out. In addition to an additive reading akin to that of (56), we also have a

multiplicative reading. In other words, both (66a) and (66b) are valid interpretations of (65):

(66)

a.

b.

If Clyde walked the dog two times, Floyd walked it five times.

Additive reading

If Clyde walked the dog two times, Floyd walked it six times. Multiplicative reading

Why do we get a different word order in (65) that nevertheless has the same meaning as in

(56)? One possibility is that the word order of MPs and comparatives is slightly variable. Consider

(67):

(67)

a. This book costs five more dollars than that book.

b. This book costs five dollars more than that book.

Here we appear to get both word orders with a different MP (though it should be worth noting

that (67a) isn’t completely acceptable for all speakers), so at first blush one might think this is

72

simply something that MPs can do when they interact with more. However, this idea falls apart
upon closer examination, as seen in (68)5:

(68)

a. John burned five more dollars than Bill.

b. #John burned five dollars more than Bill.

(68) shows us that the structure num more noun is about cardinality, while the structure num

noun more involves measurement. The difference between measurement and cardinality is in a

sense about arbitrariness and artificiality. When we talk about various measurements and the units

on those measurement scales, we’re discussing more abstract concepts. Although we can describe

what each measurement corresponds to (so a meter is the distance light travels in a vacuum in

1

299792458 of a second, and a kilogram was, until May 2019, the mass of a specific item (the
International Prototype of the Kilogram) designed to be exactly one kilogram in mass), that corre-

spondence is arbitrary and can change if so desired – there’s nothing to stop us from deciding that
the IPK is now equivalent to 2 kilograms, for instance.6 And a further feature of its arbitrariness

is the fact that we can assign differing units to similar objects: a USD $5 bill and a USD $1 bill

are the same size and type of paper and only differ in what’s printed on them. They’re also not

necessarily physically divisible; in other words, you can’t cut a dollar in half and have it be worth
50¢.

Now compare this to cardinality, which is all about counting. While we can have three physical

pieces of US currency in our hand, the value of that currency could be wildly different, depending

on the denomination of those pieces of currency. We may have $3 in our hand or $103 – but the

cardinality of the bills themselves will remain 3. Cardinality is arbitrary only in how we decide

what counts as an individual object, and it can’t be subdivided easily: unlike with measurements,

5Many thanks to Alan Munn (p.c.) for pointing this out.
6And in fact, particularly with regards to SI units, this is known to happen: a meter used to
be 1,650,763.73 wavelengths of the orange-red emission line of the krypton-86 atom, for instance,
while the kilogram was redefined as taking the fixed numerical value of the Planck constant h to
be 6.62607015 × 1034 when expressed in the unit joule-seconds – a move done in part because the
IPK was found to actually be losing mass. Or consider the differences between US standard units
and British Imperial units, such as in the size of a pint.

73

you can’t suddenly decide that a monkey is now equivalent to two monkeys, and you can’t attempt

to reinforce your claim by cutting the monkey in half and claiming each piece as a single monkey.

Cardinality maps to a specific scale that only counts objects – it’s the scale that pluralities map to.

This therefore is why (68b) is semantically anomalous; the act of burning requires physical ob-

jects, not measurements, but (68b) provides us with a word order that can only be about measure-

ments, not pluralities – and in fact the reading of (68b) that is marginally acceptable is explicitly

about the value of the money burned, not the bills themselves. Compare with (68a), where the

dollars are physical objects being burned.

So if (68a) is about objects, not values, why then is (67a) acceptable (albeit degraded for

some)? How can we assign a measurement to a structure not designed for it? The answer seems

to be simply a matter of inferring the desired measurement reading from the cardinality structure:

although strictly speaking we’re dealing with a cardinality reading, we’re able to map that number

to its corresponding scale and appropriate degree, thus allowing a measurement reading for (67a).

But what is it about the word order that leads to this difference in meaning? The likely expla-

nation is that the entire measure phrase is treated as a single degree, one that maps to a specific

scale. The (simplified) tree of a sentence like (67b) is as in (69):

(69)

(cid:104)dt, t(cid:105)

d

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

1

5 dollars

-er(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

d

than that book

This book

cost(cid:104)d,et(cid:105)

d1

Here we see that cost is looking for an individual and a degree (compare with the simpler ver-

sion This book costs five dollars), but because five dollars -er than that book is a degree quantifier

it needs to raise up the tree, as we’ve already seen in things like (59).

Compositionally, things behave similarly to the previous QR’ed structure that we saw in (59):

74

(70)

a. (cid:126)cost(cid:127) = λdλx.cost(x) = d

b. (cid:126)This book costs d1(cid:127) =1 iff cost(this-book) = d1
c. (cid:126)5 dollars -er than that book(cid:127) = λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − dthat-book ≥ 5-dollars
d. (cid:126)5 dollars -er than that book 1 this book costs d1(cid:127)

= λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : cost(this-book) = d(cid:48)(cid:48)} − dthat-book ≥ 5-dollars

(70) states that the difference between the price of this book and the price of that book is 5 dollars,

just as we want.

But what about (67a)? Here five more dollars involves a plural object, rather than a degree –

we’re talking about the cardinality of the plural. So in order to obtain the degree reading, we’ll

need a type-shift. The structure of (67a) will be similar to the tree we saw in 69, but with our

type-shift included:

(71)

(cid:104)dt, t(cid:105)

5d

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

1

-er(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

d

than that book

This book

cost
(cid:104)d, et(cid:105)

value

(cid:104)et, d(cid:105)

d

(cid:104)e, t(cid:105)

d1-many dollars

Here we have d1-many dollars, which is of type (cid:104)e, t(cid:105); it can’t combine directly with cost, so
our type-shift operator, value, comes to the rescue. value in this case simply takes in a (plural)

object and returns the value of the cardinality as a degree, as seen in (72):

75

(72)

a. (cid:126)d1-many dollars(cid:127) = λx.*dollar-bill(x) ∧ |x| = d1
b. (cid:126)value(cid:127) = λP(cid:104)e,t(cid:105).value(P)
c. (cid:126)value d1-many dollars(cid:127) =value(λx.*dollar-bill(x) ∧ |x| = d1)
d. (cid:126)cost value d1-many dollars(cid:127) = λx.cost(x) =value(λx.*dollar-bill(x) ∧ |x| = d1)
e. (cid:126)5 -er than that book 1 this book cost value d1-many dollars(cid:127)

= λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : cost(this-book) =value(λx.*dollar-bill(x) ∧ |x| = d(cid:48)(cid:48))}−
dthat-book ≥ 5-dollars

What (72) ultimately says is the same thing as (70); the primary difference is that instead of

getting the cost directly from the appropriate scale, we infer it from the objects and their lexical

entries. We know that dollar bills have value, and we can speak about a dollar in terms of both

the physical object and the abstract value; with this knowledge we can infer the value from the

object, from both the presence of the plural object dollars and the use of cost, which is only

going to be looking for degrees that are on the price scale. Otherwise, the result would be one of

incommensurability, where the verb (or adjective, depending on the sentence) and the unit of the

measure phrase (or shifted DP) each call for dimensions that are incompatible with each other (see
Klein 1991 and Kennedy 1999 for more).7

So now that we’ve seen that measure phrases like 5 dollars denote a degree on a certain scale,
we can now account for why (68b) is problematic. Burned is a verb of type (cid:104)e,(cid:104)e, t(cid:105)(cid:105), but 5 dollars
is of type d – yet even if a type-shift did occur to accommodate the type clash, incommensurability

would result from incompatible dimensions, as 5 dollars is on the price scale, while burned, if it’s

offering a scale at all (as opposed to simply the desire for a tangible object), is looking for one of

burned items. Therefore the type-shift would need to not only shift the type, but also the scale that

5 dollars maps to.

7One question that remains unresolved is what to do with neologisms. We can, for instance,
utter sentences such as This shirt costs 700 pleknars or Paul is 3 sermzans tall without causing any
acceptability failures. Is this because we can automatically provide scales for a neologism in the
lack of evidence to the contrary? Is it because the verb/adjective carries with it the desired scale,
and as long as the provided unit doesn’t lead to incommensurability the sentence will be OK? Or
is it something else?

76

With all of that, we now turn back to the initial example: the difference between (56) and (65),

repeated below for convenience:

(56) Floyd walked the dog three more times than Clyde (did).

(65) Floyd walked the dog three times more than Clyde (did).

There is a problem to deal with in (65), however. If we say that times is in fact a unit name

like dollars or inches, then we suddenly have no way of counting our events, because we’ve made

times part of a measure phrase instead of standing on its own. In order to handle this structure, it

will be helpful to determine just exactly how many different kinds of times we’re dealing with.

3.2.3 One times or two?

If we want to handle the question of how many different times we have floating around, it might

be helpful to turn to the other reading of (65). While we’ve discussed the additive reading, we still

need to account for the multiplicative reading. Why are there two possible interpretations of (65)?

There are two possibilities as to why. The first possibility is that times is in fact ambiguous between

two different meanings: one that marks differential event cardinality (the additive reading) and one

that performs multiplication on a given cardinality (the multiplicative reading). This multiplication

reading would be similar to a sentence like (73):

(73) Percy is two times as tall as Baldrick.

The second possibility is that there is only one times, and that the meaning differences are due

to differing interactions with the rest of the sentence, either as a result of an altered underlying

syntax or something else.

Additional evidence seems to indicate that the first possibility is correct. First, we can in fact

get overt co-occurrence of times in two positions simultaneously:

(74) Floyd walked the dog three times as many times as Clyde (did).

(75)

?Floyd walked the dog three times more times than Clyde (did).

77

If it was in fact movement of times that led to the different effects, we shouldn’t be able to

phonologically realize both the moved element and its trace. This doesn’t, however, discount the

possibility that times occurs naturally in both positions without movement, and that there is in fact

a deletion that typically occurs, much as we saw in the additive form of (65).

More compelling, however, is the use of other words in place of times. The additive, plural

event version has a handful of synonyms that can be substituted without too much fuss:

(76) Floyd walked the dog three times/on three occasions/in three instances.

(77) Floyd walked the dog on three more occasions than Clyde (did).

(78)

?Floyd walked the dog on three occasions more than Clyde (did).

(78) is somewhat strange (for the reasons mentioned above regarding things like five books

more – occasions, it seems, doesn’t lend itself to a dimension as naturally as times does), but to

the extent in which it’s acceptable it can only have an additive reading. This isn’t surprising, as an

 occasions

times

 as Clyde.

effort to replace the multiplicative times with something like occasions fails miserably:

(79)

*Floyd walked the dog three occasions as many

(79), frankly, isn’t very good, and to the extent in which it has meaning, that meaning is only

the additive version, which is consistent with words like occasions being synonymous with only

one meaning of times.

But perhaps the most compelling evidence for two separate meanings for times comes from

foreign language data. Let’s take Japanese as an example:8

(80)

(81)

sanpo-o
walk-ACC

inu-no
dog-GEN
‘I walked the dog three times.’

san-kai
three-times

shi-ta
do-PAST

sanpo-o
furoido-wa
walk-ACC
Floyd-TOP
‘Floyd walked the dog three more times than Clyde.’ (additive)

kuraido-yori
Clyde-than

san-kai
three-times

inu-no
dog-GEN

ooku
many

shi-ta
do-PAST

8This data comes from Ai Taniguchi (p.c.).

78

(82)

furoido-wa
Floyd-TOP
‘Floyd walked the dog three times more than Clyde.’ (multiplicative)

kuraido-yori
Clyde-than

san-bai
three-times

shi-ta
do-PAST

ooku
many

inu-no
dog-GEN

sanpo-o
walk-ACC

Here we see that Japanese in fact has a separate morpheme for each meaning of times: -kai and

-bai. Despite their appearance, these two morphemes are not cognates and in fact are unrelated

to each other (kai on its own means “cycle”, while bai literally translates as “twice”, though its

meaning can be altered by the addition of a numeral, as seen in (82)). And, just as in English,

the additive form -kai is used in both the simple plural event sentence (80) and the slightly more

complex comparative sentence (81).

Given the evidence, it seems therefore safe to assume that times in English is in fact ambiguous

between a simple plural event meaning and a more complex multiplication one. And the co-

occurrence facts mean we can use that to our advantage to describe the additive reading of (65). If

we say that there is also co-occurrence of both our standard plural times and a unit name version,

then we can handle this sentence without too much difficulty. The structure for (65) will be this:

(83)

(cid:104)dt, t(cid:105)

d

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

3 times

-er(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

d

1

∃s

than Clyde

Floyd

d1-many times

walked

the dog

Here we see a slight change from our earlier trees. As noted above, we say that there are two

times present in LF: the standard plural event version, and a related version that acts as a unit name

79

and maps to a scale of completed events (a form of timeline, if you will). The plural event times is

simply deleted at PF, and the rest proceeds as before, with the minor addition of the times unit:

(84)

a. (cid:126)Floyd walked the dog d1-many times(cid:127)

= 1 iff ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d1

b. (cid:126)3 times -er than Clyde(cid:127) = λ f(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − dClyde ≥ 3-times]
c. (cid:126)3 times -er than Clyde 1 Floyd walked the dog d1-many times(cid:127)

= 1 iff max{d(cid:48)(cid:48) : ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d(cid:48)(cid:48)}−dClyde

≥ 3-times

The difference between (84c) and our standard additive version in (64e) is minor:

instead

of saying that Floyd’s dog-walkings minus Clyde’s dog-walkings are greater than or equal to 3,

we now say that the difference between the two dog-walking sets is 3 times – 3 on the scale of

times as counting events. But although the details are slightly different – now we’re comparing

degrees mapped to a specific scale – the outcome is the same: we’re still comparing the number of

Floyd’s dog-walkings to the number of Clyde’s dog-walkings; it’s just that now we’re making the

dimension along which the comparison is made explicit, associated with a distinct scale instead of

simple cardinality. However, as we were implicitly comparing on this completed event timeline

dimension before, the meanings between the two word orders are functionally the same.

Now that we’ve taken care of the additive reading of (65), let’s turn to the multiplicative version.

3.2.4 Multiplicative factor phrases

In order for multiplication to occur, times needs access to the base degree that will be used to carry

out the calculations – in other words, it needs access to the than phrase. However, multiplicative

times is a modifier of the differential comparative morpheme, so it also needs access to that as

well. What this means is that three times needs to join pretty low, before either the comparative

morpheme or the than phrase is closed off. Given that, the syntactic structure for a sentence like

(65) will be as in (85):

80

(65) Floyd walked the dog three times more than Clyde (did).

(85)

(cid:104)dt, t(cid:105)

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

d

1

-er

than Clyde

∃s

three times

Floyd

d1-many times

walked

the dog

You may have noticed that three times and -er aren’t labeled with types in (85). This is because

we have a couple options open to us. One approach is to change the denotation of times to reflect

the different meaning, but to leave everything else the same as in the additive version. This would

cause a problem, however, as we’ve been using a differential comparative so far; thus in order for

our multiplicative factor phrase to work with a differential, we would need the factor phrase to

satisfy two of the degree arguments:

(86)

(87)

(cid:126)-er(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} ≥ d(cid:48) + d
(cid:126)three times(cid:127) = λg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)λdλ f(cid:104)d,t(cid:105).g(d)((3 − 1)d)( f )

= λg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)λdλ f(cid:104)d,t(cid:105).g(d)(2d)( f )

(88)

(cid:126)three times -er(cid:127) = λdλ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} ≥ 2d + d

(1st version)

It’s apparent that there are a number of drawbacks to the approach in (87): first, three times
is of the dizzyingly high type (cid:104)(cid:104)d,(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)(cid:105),(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)(cid:105). This isn’t necessarily a dealbreaker
– for instance, we have independent evidence that there are high types in degree semantics, as

seen in Meier (2003), who provides a denotation for the word too that’s of the rather high type

81

(cid:104)s,(cid:104)(cid:104)s,(cid:104)(cid:104)s, t(cid:105), t(cid:105)(cid:105),(cid:104)(cid:104)d,(cid:104)s, t(cid:105)(cid:105)(cid:105), t(cid:105)(cid:105)(cid:105)9 – but it’s hardly desirable. More worrying is the way this de-
notation has to work to get to the 3 of three times: by subtracting a degree from the factor and

essentially shifting it to a different degree argument.

The biggest issue with (87), however, is that by virtue of combining with a differential compar-

ative, it becomes incompatible with equatives. It’s independently strange to think of an equative as

being differential in some way, as you’re simply saying that one degree is equal to another degree

(or three times another degree, in the case of our target sentence) – there’s no difference involved.

Now, if we were feeling particularly brazen we could attempt to introduce a differential form

of the equative, as in (90):

(89) Floyd walked the dog three times as many times as Clyde (did).

(differential equative attempt)

(90)

a. (cid:126)as(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} ≥ d(cid:48) + d
b. (cid:126)three times as as Clyde 1 Floyd d1-many times walked the dog(cid:127)

=max{d(cid:48)(cid:48) : ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d(cid:48)(cid:48)} ≥
2(dClyde) + dClyde

While that does work, an independent problem has arisen: given that our denotation for as looks

the same as that for -er, we suddenly can’t rule out an additive reading for equatives. In other

words, we should be able to take a sentence like (89) and have it be the case that if Clyde walked

the dog two times, Floyd walked it five. This is simply not an available reading for (89), but under

our differential comparative version we don’t have a way to rule it out. Therefore it seems that

simply changing the denotation of times and leaving the comparative morpheme as a differential

comparative is a nonstarter.

There is a simpler alternative. Since we know independently that more is also ambiguous

between the standard and differential version, we can say that multiplicative factor phrases simply

combine with the standard, non-differential form of more. This would also solve the issue with
equatives, since equatives and standard comparatives only differ by whether they use ≥ or >. The

9For Meier, s is the type of possible worlds, not events.

82

one minor disadvantage is that standard comparatives are strict inequalities: they do not allow an

“equals” reading. This matters because multiplicative times needs an “equals” reading; if we say

that Floyd walked the dog six times and Clyde walked it two times, we don’t want the sentence

Floyd walked the dog three times more than Clyde to come out as false. We can, however, handle

this by simply having times supply its own non-strict inequality in its denotation. This leads us to

(91):

(91)

a. (cid:126)three times(cid:127) = λg(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)λdλ f(cid:104)d,t(cid:105).g(d)( f ) ∧ max{d(cid:48)(cid:48)(cid:48) : f (d(cid:48)(cid:48)(cid:48))} ≥ 3d
b. (cid:126)three times -er(cid:127) = λdλ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} > d ∧ max{d(cid:48)(cid:48)(cid:48) : f (d(cid:48)(cid:48)(cid:48))} ≥ 3d
c. (cid:126)three times -er than Clyde(cid:127)

= λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} > dClyde ∧ max{d(cid:48)(cid:48)(cid:48) : f (d(cid:48)(cid:48)(cid:48))} ≥ 3dClyde
d. (cid:126)three times -er than Clyde 1 Floyd d1-many times walked the dog(cid:127)

= max{d(cid:48)(cid:48) : ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d(cid:48)(cid:48)} > dClyde ∧
max{d(cid:48)(cid:48)(cid:48) : ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d(cid:48)(cid:48)(cid:48)} ≥ 3dClyde

(91d) looks rather complex, but the key thing to note is that d(cid:48)(cid:48) and d(cid:48)(cid:48)(cid:48) are both pointing at the
same degree, the maximal degree associated with Floyd’s dog-walkings. Thus, what (91d) says is

that the maximal degree of Floyd’s dog-walkings is larger than the degree of Clyde’s dog-walkings,

and that it’s also greater than or equal to the degree of Clyde’s dog-walkings multiplied by three –

which is in fact exactly the outcome we want.

So, we’ve now seen that there are in fact two types of factor phrases operating in the verbal

domain: an additive one and a multiplicative one. The multiplicative version is more complex than

the additive one – although this isn’t perhaps too surprising, given that mathematically, multiplica-

tion is more complex than addition – but both have been shown to work in the verbal domain.

3.3 Unresolved issues

The following sections will bring up some smaller issues related to the use of verbal factor

phrases.

83

3.3.1 Less/fewer than

One advantage of having both conjuncts in our denotation of multiplicative factor phrases in (91a)

is that we can successfully explain why sentences like (92) are bad: the first conjunct contradicts

the second:

(92)

(93)

*Floyd walked the dog three times less times than Clyde.
a. (cid:126)less(cid:127) = λdλ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} < d
b. (cid:126)three times less(cid:127) = λdλ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} < d ∧ max{d(cid:48)(cid:48)(cid:48) : f (d(cid:48)(cid:48)(cid:48))} ≥ 3d

(* on multiplication reading)

(93b) states that the maximal degree of f will be both smaller than d and at least as large as 3d, but

those statements can’t be simultaneously true. This result is exactly what we want.

So we’ve seen what happens when we try to use less with multiplicative times, but what do we

do with an additive sentence like (94)?

(94) Floyd walked the dog three less/fewer times than Clyde.

The key difference, of course, is the substitution of less instead of our differential comparative

more. Given our assumptions on how factor phrases work, we need to change something to ac-

commodate the presence of less. One reasonable move might be to simply reverse the direction of

the inequality in the denotation of more to arrive at less – this, after all, is the difference between

less and more in the standard comparative form:

(86)

(95)

(cid:126)-er(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − d ≥ d(cid:48)
(cid:126)less(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − d ≤ d(cid:48)

However, closer examination of this approach reveals its deficiencies. Suppose Clyde walked

the dog five times. According to (94), our natural intuition is that Floyd would have walked the

dogs only two times. But when we run the computation, that’s not what we get:

(96)

(cid:126)3 less than Clyde(cid:127) = λ f(cid:104)d,t(cid:105).max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} − dClyde ≤ 3

84

(96) should hopefully illustrate the issue. Floyd’s degree of dog-walkings will ultimately be
what d(cid:48)(cid:48) measures, but the equation we get in (96) – simplified in (97) – is, while technically
correct, not the intuitive meaning we want.
(97) dFloyd − dClyde ≤ 3
While −3 (the result, of course, of 2 − 5) is indeed less than 3, it’s not really the answer we’re

looking for; a different approach is thus needed.

Another approach might be to decompose less into little and -er (as suggested by, among

others, Bresnan 1973, Rullmann 1995, Heim 2006, and Büring 2007) and then introduce little

into the system, leaving the -er morpheme untouched. However, a problem immediately presents

itself: little is said to (depending on the analysis) negate either the degree or the adjective it

combines with (i.e., “not tall” or “not to degree d”). However, there’s no adjective in the differential

comparatives that we’ve been working with, and it’s not clear what degree would be negated. In

either case, the general result of little is to essentially reverse the direction of the inequality, which

we have already seen is problematic. Now one way around this might be to negate the sign of the
differential operation (in other words, to change the − to a +) but that doesn’t solve the issue either
– we get another result that’s technically correct but not really intuitive:
(98) dFloyd + dClyde ≥ 3

A way to get the intuitive answer to come out is to stipulate that the arguments on either side

of the differential swap places:

(99)

a. (cid:126)less(cid:127) = λdλd(cid:48)λ f(cid:104)d,t(cid:105).d−max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} ≥ d(cid:48)
b. (cid:126)less than Clyde(cid:127) = λd(cid:48)λ f(cid:104)d,t(cid:105).dClyde−max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} ≥ d(cid:48)
c. (cid:126)3 less than Clyde(cid:127) = λ f(cid:104)d,t(cid:105).dClyde−max{d(cid:48)(cid:48) : f (d(cid:48)(cid:48))} ≥ 3
d. (cid:126)3 less than Clyde 1 Floyd d1-many times walked the dog(cid:127)

= 1 iff dClyde−max{d(cid:48)(cid:48) : ∃e.*walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = d(cid:48)(cid:48)} ≥ 3

85

The good thing is that this is intuitively the outcome that we want: 5 − 2 is indeed equal to 3,
and the inequality points in the right direction (so, for instance, if we have something like Floyd

walked the dog three less times than Clyde – in fact, he walked it even less than that, the number of

Floyd’s dog-walkings should go down, and so the resulting difference between Clyde’s and Floyd’s

dog-walkings will be even larger, which the inequality correctly accounts for). The bad thing is

that this approach is currently stipulative, and it’s not clear how one would derive the denotation

in (99a).

3.3.2 Fractions

A natural inclination for factor phrases is to see how they work with fractions, but the results aren’t

nearly as straightforward as one might hope. For starters, fractions don’t seem to work in the

additive cases, as seen in (100):

(100)

a. *Floyd walked the dog one-half more times than Clyde.

b. *Percy went to the store two-thirds more times than Baldrick.

This isn’t actually that surprising. We’ve already established earlier that the additive, plural

event version of times requires complete events to enumerate – much how plural objects require

complete objects as members of their plurality.

This doesn’t however address the fact that mixed fractions are actually OK in the additive

contexts, as seen in (101):

(101) Floyd walked the dog two-and-a-half more times than Clyde.

I have no explanation for why this is good, but then there doesn’t seem to be a clear story on

how mixed fractions work in general – how does two-and-a-half apples work?

What about the multiplicative form of times? Simply replacing the integer in a times phrase

 one-third

three-fourths

 times

 more (times) than

as many times as

 Clyde.

isn’t very good:

(102)

??Floyd walked the dog

86

This, however, is to be expected. As you’ll recall, the denotation for multiplicative times in-

volves two conjuncts: the first one takes in the comparative stuff and spits it back out, while the

actual factoring occurs in the second conjunct. But look what happens when we put a fraction in:

(103)

(cid:126)one-third times(cid:127) = λg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)λdλ f(cid:104)d,t(cid:105).g(d)( f ) ∧ max{d(cid:48)(cid:48)(cid:48) : f (d(cid:48)(cid:48)(cid:48)(cid:48))} ≥ 1
3d

As with multiplicative less in (93b), (103) yields a contradiction: the maximal degree of f can’t

be both larger than d and one-third its size. This is why (102) is bad.

However, certain fractions without times are acceptable in the equative case (though less so in

the comparative):10

(104)

a. Floyd walked the dog

b. Floyd walked the dog

 half
 *half

a third

?a third

 as many times as Clyde.
 more than Clyde.

Given (102) and (103), it’s clear that a different approach is needed to handle the sentences in

(104). One possibility comes via Kennedy & McNally (2005) and Bochnak (2010), who both treat

fractional amounts (specifically, half ) as the midpoint of a closed scale or a complete quantity.
Bochnak (2010) provides the denotation in (105), where mid(S G) is the midpoint of the closed
scale G:

(105)

(cid:126)half(cid:127) = λG(cid:104)d,(cid:104)e,t(cid:105)(cid:105)λx.G(x)(mid(S G))

This midpoint analysis allows Bochnak to both handle closed-scale adjectives such as open

(half open) while also ruling out open-scale adjectives like tall (half tall). Bochnak also extends

his analysis to include plural objects like books by introducing a partitive morpheme between half
10It might be worth noting that as the fractions become less common, they become less accept-

able:

(i)

*Floyd walked the dog

(cid:40) seven-elevenths

(cid:41)

forty-seven-hundredths

as many times as Clyde.

This could simply be a matter of giving up on interpretability judgments as the fractions get too

difficult to easily determine, however.

87

and the prepositional of phrase (following Schwarzschild 2002, 2006) that provides the quantity

of the noun embedded in the PP:

(106)

(cid:126)µPRT (cid:127) = λP(cid:104)e,t(cid:105)λdλx.P(x) ∧ quantity(x) = d

This combines to create a denotation for half of the books as in (107):

(107)

a. (cid:126)of(cid:127)((cid:126)the books(cid:127)) = λx.x ≤ the.books
b. (cid:126)µPRT (cid:127)((cid:126)of the books(cid:127)) = λdλx.x ≤ the.books ∧ quantity(x) = d
c. (cid:126)half(cid:127)((cid:126)µPRT of the books(cid:127))

= λx.[λdλx(cid:48).x(cid:48) ≤ the.books ∧ quantity(x(cid:48)) = d](x)(mid(S o f.the.books))
= λx.x ≤ the.books ∧ quantity(x) =mid(S o f.the.books))

It’s possible that this approach could be used for mixed fractions. One possibility is to say that

a phrase like two-and-a-half apples is actually two apples and a half of an apple, where two apples

is built up normally while half of an apple is built up in Bochnak’s system. (This wouldn’t explain

why we get plural morphology on apples though, so it can’t only be a case of pure elision.) The

downside, however, is that it still can’t account for why (108) is bad:

(108)

*Floyd walked the dog

 half times

half of a time

 more than Clyde.

Now it does seem reasonable to believe that a similar approach to Bochnak’s could be used for

multiplicative cases like (104a) and have it be that dClyde is treated as the scale that half operates
on. This, however, would require a fairly different denotation for, e.g., half as many from how our

current multiplicative factor phrase works. Consequently, this will be left for future research.

3.4 Conclusion

This chapter has provided a detailed look at adverbial factor phrases, beginning with previous

work by Doetjes (1997) and Landman (2004). We’ve seen that these factor phrases are a form of

counting measure, one that counts up events. The interaction of these factor phrases with events and

88

plurals has also been discussed, noting that factor phrases provide a way to discuss the cardinality

of similar plural events, and that it’s times that imposes atomicity on events.

We’ve also seen that times is in fact ambiguous in the verbal domain between the plural event

version and the multiplicative version, which more closely resembles the adjectival form. This

suggests that many languages, including English and Japanese, encode multiplication into the nat-

ural language itself. We’ve also seen the interaction of factor phrases with comparatives, which

provides an insight into how these interact with both standard and differential comparative mor-

phemes; this has also pointed out an ambiguity regarding sentences with factor phrases like five

times more that had previously gone unnoticed.

The discussion of how the multiplicative factor phrases behave leads us neatly into the next

chapter, where we’ll discuss our factor phrases behave in the adjectival domain – what may seem

to be the more natural home for multiplicative factor phrases.

89

CHAPTER 4

FACTOR PHRASES IN THE ADJECTIVAL DOMAIN

We turn our attention now to factor phrases as they behave in the adjectival domain, as in (1):

(1)

a. Floyd is twice as tall as Clyde.

b. Martha is three times older than Sandy.

Instinctively, we know what this means: Floyd’s height is equal to Clyde’s height times 2, or

Martha’s age is the same as Sandy’s age multiplied by three. However, although we have a gen-

eral sense of what these mean, these constructions are surprisingly underexamined (particularly

given the relative popularity of the semantics of degree phrases, which these interact with). Con-

sequently, there’s rather less research about adjectival factor phrases than there is about the verbal

version; however, there is some, so let’s examine that now before we venture into new territory.

4.1 Previous research

For whatever reason, the area of adjectival factor phrases is decidedly underexplored. Much of

the work notes the issue in passing while grappling with other concerns, which means there’s not

that much to draw upon. However, we’re not completely bereft of research; for instance, there’s

some early work by people such as Hellan (1981) and von Stechow (1984), and while it’s not their

primary focus we can nevertheless use it as a starting point.

For the sentence in (2), Hellan (1981) (via von Stechow 1984) assumes the representation in

(3):

ist
is

wie
(2) Hans
Hans
as
‘Hans is twice as tall as Eva.’

doppelt
double

so
such

groß
tall

Eva.
Eva

(3) ∃d1∃d2∃d3[Hans is d1-tall ∧ Eva is d2-tall ∧ d1 = d2 × d3 ∧ d3 = 2]

90

In other words, there are three degrees d1, d2, and d3, such that the degree to which Hans is tall is
equivalent to the degree of Eva’s tallness, multiplied by a factor of two.

von Stechow (1984) approaches things slightly differently, although he’s in the same ballpark:

(4)

a. Ede is twice as fat as Angelika.

the max.d[Angelika is d-tall]λd2[(∃d, d = 2)∃d1[Angelika is d1 × d2-fat]]

b.

However, von Stechow isn’t really interested in factor phrases in and of themselves; instead

he’s more interested in comparing various approaches to how comparatives work, and thus factor

phrases are a means to an end, a way to say, “How does this particular theory handle them?”. The

upshot is that while he mentions factor phrases and provides an initial semantics for them, this is

ultimately more or less in passing, with little elaboration into factor phrases themselves.

Perhaps the most extensive work comes in Bierwisch (1989). Bierwisch takes up Hellan

(1981)/von Stechow (1984)’s semantics, but he also goes beyond them; for instance, he points

out that equative phrases appear to be based on multiplication and differential comparatives on

addition, as evidenced by (5) and (6):

(5)

John is three feet taller than Mary.

(6)

*John is three feet as tall as Mary.

(7) Formal representation of (5) via Bierwisch (1989):

∃x1∃x2∃x3[[JOHN [T ALL x1]] ∧ [MARY [T ALL x2]] ∧ [x3 = 3 f t] ∧ [x1 = x2 + x3]

(Of course, this doesn’t address the fact that, in English at least, comparatives can also take in

multiplication, as we’ve already seen in (1b).)

Bierwisch also points out that in general, equatives of negative adjectives cannot take factor

phrases. This however only applies to what he calls dimensional adjectives like tall and long:

adjectives that are associated with a scale with independently measurable units; by contrast, eval-

uative adjectives – those adjectives which don’t have a scale with independent units, such as good

and pretty – are unaffected and thus are content to combine with factor phrases of both the positive

and negative variety:

91

(8) Dieses

so
as

long
long

wie
as

der
the

Tisch.
table

Brett
board

ist
is

dreimal
three times

This
‘This board is three times as long as the table.’

(9)

Brett
board

?Dieses
This
‘This board is three times as short as the table.’

dreimal
three times

kurz
short

wie
as

der
the

ist
is

so
as

Tisch.
table

Film
film

Buch.
(10) Der
The
book
‘The film is three times as good as the book.’

dreimal
three times

gut
good

wie
as

das
the

ist
is

so
as

Film
film

das
(11) Der
the
The
‘The film is three times as bad as the book.’

dreimal
three times

schlecht
bad

wie
as

so
as

is
is

Buch.
book

The acceptability of (11) versus the unacceptability of (9) may be due to the fact that multiply-

ing degrees of quality is imprecise and thus does not require the precision that multiplying degrees

of dimension does; thus negative adjectives are more likely to lead to an unacceptability judgment

in those cases. An alternate possibility is that evaluative adjectives are essentially closed scales;

that is, while it is theoretically possible to imagine something of infinite length, there is a point

at which, say, perfect beauty is reached (with the definition of perfection varying from individual

to individual) and that thus there is no such thing as infinite beauty, hyperbolic sentences aside.

Therefore, the presence of an end point at both ends of the scale would make negative adjectives

still acceptable. Or perhaps it’s a synthesis of these two ideas: since we don’t have concrete ideas

about units of happiness or beauty, we don’t become concerned when someone tries to use the

oppositely-ordered version; there’s no need to be concerned about what total intelligence or total

stupidity mean, mathematically speaking.

As far as the syntax of factor phrases, Bierwisch (1989) assumes the following structure:

92

(12)

AP

DP

FP

DP

N

F

DEGREE

S

drei
three

mal
times

so
as

wie
as

S
S

A

lang
long

Note that in this structure the factor phrase (FP) is a modifier of the degree phrase (DP), not an

argument. This is also different from how we treated it in the verbal cases, where the factor phrase

modified Deg directly, rather than DegP.

One final point regarding Bierwisch (1989): two of the words he lists as factor phrases are

half and double. Double, although behaving similarly to other adjectival factor phrases in German

(as we saw in (2)), doesn’t really work the same as n times does in English. It doesn’t seem to

combine with adjectival comparatives at all; nor does it work as a verbal FP. Instead, it only works

 as tall as

 Clyde.

with nominal constructions as in (15):

(13)

(14)

*Floyd is double

taller than
*Floyd walked the dog double.

(15) Floyd is double the height of Clyde.

And while there is an adverbial version, doubly, it doesn’t really work any better in these contexts:

93

(16)

a. ?Floyd is doubly tall.

 as tall as

 Clyde.

b. *Floyd is doubly

taller than
c. *Floyd doubly walked the dog.

d. *Floyd is doubly the height of Clyde.

e. ?This box is doubly heavy.

f. ?My car is doubly old.

Doubly doesn’t seem to work with any dimensional adjectives, in fact; it seems to be a modifier

for evaluative adjectives:

(17)

a. We need to be doubly cautious.

b. This linear algebra class is doubly hard.

c. Clyde found himself doubly lost.

d. ?Jack got doubly drunk.

Based on this, it seems clear that while double/doubly share some characteristics with factor

phrases, they differ in other respects as well. Consequently, let’s tentatively place these in a separate

category from other factor phrases and set them aside for the time being.

Half, however, behaves differently depending on the environment. In some cases it can pattern

along the same lines as twice:

 half
 half

twice

twice

 as tall as Clyde.
 taller than Clyde.

(18) Floyd is

(19)

*Floyd is

Kennedy & McNally (2005) note that it can also modify certain adjectives directly, in a manner

similar to words like mostly.

(20) The glass is half full.

94

(21) Her eyes were half closed.

At this point you no doubt recall Bochnak (2010)’s work on half from section 3.3.2, which

argued that half finds the midpoint of either a closed scale or a complete quantity. So the reason

why (20) and (21) are good is that full and closed both have endpoints and so therefore also have

midpoints. A sentence such as (22) is bad, however, because there’s (conceptually) no endpoint

for being tall, and thus no midpoint.

(22)

*Floyd is half tall.

This might therefore go some way toward explaining the difference between (18) and (19).

Half as tall merely asks to find the midpoint of the second as clause (in the case of (18), Clyde’s

height, which is a complete quantity), but half taller runs into a contradiction with the comparative

-er: half finds the midpoint of Clyde’s height, but as that’s not a point that’s greater than Clyde’s

height, -er no longer holds, which makes the sentence unacceptable.

After Bierwisch (1989) there’s not much that’s been done with adjectival factor phrases. Rett

(2008a) mentions in a footnote that factor phrases don’t preserve the evaluativity – whether or not

an adjective exceeds a given standard – of a construction. (23) has evaluativity: the use of the

word short entails that both people are short, and thus the evaluativity isn’t cancellable, as (24)

demonstrates. But that’s no longer true once a factor phrase is included, as in (25):

(23) Sylvester is as short as Jodie.

(24)

#Sylvester is as short as Jodie, although neither is short.

(25) Floyd is twice as short as Clyde, although neither is short.
Based on this, Rett concludes that factor phrases are base-generated in Deg(cid:48), because that’s where
the evaluativity morpheme is located. (It may be worth noting that Rett is working in the big DegP

system, where the comparative morpheme combines with the adjective before the than phrase.)

However, she states that she can’t make any claims beyond that, as she doesn’t know enough about

factor phrases.

95

The other major contributor to the semantics of adjectival factor phrases is Galit W. Sassoon,

who uses factor phrases in a pair of 2010 papers to make claims about negative adjectives and about

how measurement works in language. Sassoon (2010a) argues that, contrary to what Rett (2008a)

noted, factor phrases don’t combine with negative adjectives in equative constructions. She argues

that while the (a) sentences in (26) are fine, the (b) sentences are not acceptable (although it should

be pointed out that despite Sassoon’s claim, the judgment of the (b) case is by no means universally

 tall
 as the sofa.
 short
 as the sofa.

narrow

wide

agreed upon):

(26)

a. The table is twice as

b. #The table is twice as

Sassoon backs up this claim with a table of Google searches for twice as positive adjective vs

twice as negative adjective, and while the number of twice as neg occurrences isn’t zero, they’re

significantly outnumbered by the positive uses in the majority of cases, which, she claims, is

convincing supporting evidence for her claim about negative adjectives and factor phrases. Sassoon

uses this to argue that, similarly to Kennedy (1999)’s position, negative adjectives are ordered in

the reverse direction from positive adjectives, and it’s uncertainty about the zero point for negative

adjectives that leads to the unacceptability. By the “zero point”, Sassoon is discussing the bottom

of the scale: while we might quibble about what it would mean for something to actually be zero

inches tall, we have a general sense of what meaning is being driven toward; by contrast, zero

inches short isn’t just odd but is in fact nonsensical. In practical terms (for our purposes), Sassoon

(2010a) is content to use a form of von Stechow (1984)’s semantics for factor phrases, while

assuming that the factor phrase combines with the equative morpheme as before as combines with

anything else. (Sassoon, for what it’s worth, is also working in the alternate big DegP syntax.)

(27)

(cid:126)Floyd is twice as tall as Clyde(cid:127) = 1 iff ftall((cid:126)Floyd(cid:127)) = 2 × ftall((cid:126)Clyde(cid:127))

96

(28)

Dan

is

tall

as

Sam

twice

as

Sassoon (2010b) is much more interested in including factor phrases as part of a reanalysis

of measure phrases and related phenomena. Sassoon argues that grammar is sensitive to a four-

level distinction found in measurement theory, as described in Krantz et al. (1971). This four-way

distinction consists of the following four measure types: ratio-scale (or extensive) measures,

interval-scale measures, ordinal-scale measures, and nominal-scale measures. Sassoon points

out that measurement theory has been useful in fields ranging from physics to statistics to psychol-

ogy (namely, psychophysics, or how people perceive and represent stimuli such as sound, color,

and weight), so it’s not an unreasonable move to extend it to linguistics.

Nominal-scale measures are the first level, and they refer to objects that are assigned either

similar or different terms, with little in the way of gradation; the terms themselves don’t matter,

just their similarity with other objects. Examples include eye colors (blue, green, brown) and the

set of truth values {0,1}. The fact that we’ve assigned numbers to truth values doesn’t have an

independent effect on what those truth values are; what matters is whether the truth values are the

same or different from each other.

The second level consists of ordinal-scale measures, such as, well, ordinals (1st, 2nd, 3rd...).

You can now make ordering distinctions (such as “1st is better than 2nd”), but you can’t perform

much in the way of mathematics: you can’t say that 1st place is 2nd place plus 3rd place, for

instance.

The third level, interval-scale measures, adds intervals to the mix. Equal differences in values

represent equivalent scales, but the zero point is arbitrary and negative values are permitted. Thus,

while subtraction can be performed, more complex operations such as multiplication and division

97

cannot. (So while 4 AD is two years along from 2 AD, it’s meaningless to say that the year 4 AD

is twice 2 AD.) Examples include certain temperature scales and the numbering of years.

The fourth and final level, ratio-scale measures, finally allow multiplication and division. The

zero point is no longer arbitrary and addition and subtraction can be performed. Ratio-scale mea-

sures include physical measurement scales such as height, age, and mass. What this means is that

when we say something like Colin is 6 feet tall, what we’re doing is multiplying 6 by the measure-

ment called a foot; in other words, the ratio between Colin’s height and our understanding of a foot

is 6.

(29)

(cid:126)Colin is 6 feet tall(cid:127)c = 1 iff ∀T ∈ Tc :

ftall,t((cid:126)Colin(cid:127)t) = 6 × r f t,t

For all total contexts T that are in the set of c’s completions (Tc), the adjective tall as it applies

to Colin (in other words, Colin’s tallness) is equal to 6 times the object representing a foot in a

completion t. Or more simply, Colin’s height is 6 times 1 foot.

Factor phrases are a bit different in that they measure the ratio of comparison between two

given arguments’ degrees related to a given adjective: so Jo is twice as happy as Tegan if the

ratio between Jo’s happiness and Tegan’s happiness is 2; there’s no need to introduce unit names,

because we’re not comparing to units – indeed, happiness has no units to compare to. Now we

can use factor phrases with measures that do include units, but they have to have an understood,

non-arbitrary zero point to use as a basis of comparison: thus William is twice as old as Matt

is understandable because we know what the zero point for age is, but Matt is twice as young

as William doesn’t make sense because we don’t know where we’re measuring from; as Sassoon

(2010a) noted, zero years young is nonsensical.

That’s basically all the research that’s been done on the semantics of adjectival factor phrases;

so let’s now use the pieces we have to forge ahead and provide a more detailed analysis of how

they actually work.

98

4.2 The basics of adjectival factor phrases

Let’s begin with perhaps the most straightforward example: a factor phrase in an equative

construction:

(30) Floyd is three times as tall as Clyde.

We know generally what we want the outcome to be: we want (30) to be true if Floyd’s height

equals Clyde’s height times 3 (so if Clyde is 2 feet tall, Floyd is 6 feet tall, e.g.). Consequently,

our first concern becomes a question of access: where should three times fit into the structure

such that it can have access to both Clyde’s height and Floyd’s height? In the “classical” system,

Bierwisch (1989) argued that it should modify DegP, as we saw in (12). But this runs into a

slight complication. Recall what our semantics for the standard small DegP structure is (we’ll use

Kennedy & McNally 2005’s denotation for the equative morpheme, adapted ever so slightly to
change (cid:23) to ≥):

(31)

a. (cid:126)as(cid:127) = λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d ≥ d(cid:48) ∧ G(d)(x)]
b. (cid:126)as Clyde(cid:127) = dClyde
c. (cid:126)as as Clyde(cid:127) = λG(cid:104)d,et(cid:105)λx.∃d[d ≥ dClyde ∧ G(d)(x)]

The issue is hopefully clear: if the factor phrase combines with DegP, the as Clyde phrase has

already been incorporated into the equative’s semantics and thus cannot be accessed to be modified

(by, say, multiplying it by 3). As you may recall, we’ve been in a similar position before, with

multiplicative factor phrases in the verbal domain. So, as we did in that case, we’ll shift our factor

phrase so that it adjoins low, directly with the equative morpheme:

99

(32)

AP

DegP

DegP

XP

FP

Deg

as Clyde

three times

as

A

tall

This means that three times modifies our equative morpheme, which is of type (cid:104)d,(cid:104)(cid:104)d, et(cid:105),(cid:104)e, t(cid:105)(cid:105)(cid:105)
– so therefore three times itself will be type (cid:104)(cid:104)d,(cid:104)(cid:104)d, et(cid:105),(cid:104)e, t(cid:105)(cid:105)(cid:105),(cid:104)d,(cid:104)(cid:104)d, et(cid:105),(cid:104)e, t(cid:105)(cid:105)(cid:105)(cid:105). This is an
admittedly mind-bogglingly high type, but if we realize it’s just modifying the equative and could
in fact be considered type (cid:104)α, α(cid:105), this potential objection hopefully subsides.
So given that, we should expect a denotation for three times as in (33):
(cid:126)three timesver 1(cid:127) = λH(cid:104)d,(cid:104)(cid:104)d,et(cid:105),(cid:104)e,t(cid:105)(cid:105)(cid:105)λd(cid:48)λG(cid:104)d,et(cid:105)λx.H(3d(cid:48))(G)(x)

(33)

Setting aside the complex high type, this is actually a fairly straightforward denotation: three
times takes in the equative and alters only the degree, such that it’s 3 × d(cid:48) instead of just d(cid:48), which
is precisely how we’d expect three times to behave. Everything else proceeds as expected:

(34)

a. (cid:126)three timesv1 as(cid:127) = λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d ≥ 3d(cid:48) ∧ G(d)(x)]
b. (cid:126)three timesv1 as as Clyde(cid:127) = λG(cid:104)d,et(cid:105)λx.∃d[d ≥ 3dClyde ∧ G(d)(x)]
c. (cid:126)tall(cid:127) = λdλx[tall(x) = d]
d. (cid:126)three timesv1 as as Clyde(cid:127)((cid:126)tall(cid:127)) = λx.∃d[d ≥ 3dClyde ∧ tall(x) = d]
e. (cid:126)Floyd is three times as tall as Clyde(cid:127) = 1 iff ∃d[d ≥ 3dClyde ∧ tall(Floyd) = d]

Our final result states that Floyd is three times as tall as Clyde is true if there is a degree d that is

greater than or equal to Clyde’s height multiplied by a factor of 3, and that degree d is equal to the

(maximal) tallness of Floyd.

So far so good; reasonably clean and (relatively) easy. But the trouble comes when we try the

same thing with the comparative -er morpheme, as in (35):

100

(35) Floyd is three times taller than Clyde.

Again, here’s Kennedy & McNally (2005)’s denotation:

(36)

(cid:126)-er(cid:127) = λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d > d(cid:48) ∧ G(d)(x)]

(37)

And here’s what happens when we combine it with all our other pieces:
a. (cid:126)three timesv1 -er(cid:127) = λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d > 3d(cid:48) ∧ G(d)(x)]
b. (cid:126)three timesv1 -er than Clyde(cid:127) = λG(cid:104)d,et(cid:105)λx.∃d[d > 3dClyde ∧ G(d)(x)]
c. (cid:126)Floyd is three timesv1 taller than Clyde(cid:127) = 1 iff ∃d[d > 3dClyde ∧ tall(Floyd) = d]

Now we’re saying that Floyd is three times taller than Clyde is true if there is a degree d that is

greater than Clyde’s height after it’s multiplied by a factor of 3, and that degree d is equal to the

(maximal) tallness of Floyd. But that’s not actually what we want (35) to mean; we want it to

mean that Floyd is taller than Clyde, and that Floyd’s height equals Clyde’s height times 3. In

other words, if Clyde is 2 feet tall and Floyd is 6 feet tall, we want (35) to come out as true. But

because of the strict inequality in -er, our scenario will be judged as false.

One way around this is to alter three times to more closely resemble the multiplicative factor

phrase we used in the verbal cases. Thus our factor phrase will now look as in (38):

(38)

(cid:126)three timesver 2(cid:127)

= λH(cid:104)d,(cid:104)(cid:104)d,et(cid:105),(cid:104)e,t(cid:105)(cid:105)(cid:105)λd(cid:48)λG(cid:104)d,et(cid:105)λx.H(d(cid:48))(G)(x) ∧ max{d(cid:48)(cid:48) : G(d(cid:48)(cid:48))(x)} ≥ 3d(cid:48)

And the full computation for the comparative is in (39):

101

(39)

a. (cid:126)three timesv2 -er(cid:127)

= λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d > d(cid:48) ∧ G(d)(x)] ∧ max{d(cid:48)(cid:48) : G(d(cid:48)(cid:48))(x)} ≥ 3d(cid:48)

b. (cid:126)three timesv2 -er than Clyde(cid:127)

= λG(cid:104)d,et(cid:105)λx.∃d[d > dClyde ∧ G(d)(x)] ∧ max{d(cid:48)(cid:48) : G(d(cid:48)(cid:48))(x)} ≥ 3dClyde

c. (cid:126)three timesv2 taller than Clyde(cid:127)

= λx.∃d[d > dClyde ∧ tall(x) = d] ∧ max{d(cid:48)(cid:48) :tall(x) = d(cid:48)(cid:48))} ≥ 3dClyde

d. (cid:126)Floyd is three timesv2 taller than Clyde(cid:127)

= 1 iff ∃d[d > dClyde ∧ tall(Floyd) = d] ∧ max{d(cid:48)(cid:48) :tall(Floyd) = d(cid:48)(cid:48))} ≥ 3dClyde

So there’s a degree d that’s larger than the degree of Clyde’s height, and that degree is the (maximal)
height of Floyd, and meanwhile there’s another degree d(cid:48)(cid:48) that also corresponds to the maximal
height of Floyd, and that degree is greater than or equal to Clyde’s height times 3. As with the
verbal version in chapter 3, the key thing to note is that d and d(cid:48)(cid:48) are both pointing at the same
degree: the degree of Floyd’s height. Once we’ve established that, we see that, while there are ad-

mittedly more moving parts, the final result both satisfies the strict inequality of -er and multiplies

the degree of the than phrase by 3 in precisely the way we want it to.

Here’s the equative version, just to verify that nothing has gone wrong with our new factor

phrase denotation:

(40)

a. (cid:126)three timesv2 as(cid:127)

= λd(cid:48)λG(cid:104)d,et(cid:105)λx.∃d[d ≥ d(cid:48) ∧ G(d)(x)] ∧ max{d(cid:48)(cid:48) : G(d(cid:48)(cid:48))(x)} ≥ 3d(cid:48)

b. (cid:126)three timesv2 as tall as Clyde(cid:127)

= λx.∃d[d ≥ dClyde ∧ tall(x) = d] ∧ max{d(cid:48)(cid:48) :tall(x) = d(cid:48)(cid:48))} ≥ 3dClyde

c. (cid:126)Floyd is three timesv2 as tall as Clyde(cid:127)

= 1 iff ∃d[d ≥ dClyde ∧ tall(Floyd) = d] ∧ max{d(cid:48)(cid:48) :tall(Floyd) = d(cid:48)(cid:48))} ≥ 3dClyde

And here we see that this holds if d is greater than or equal to dClyde, which is precisely what we
want; everything else is unchanged from the comparative -er version.

102

So now we have a perhaps slightly-awkward-but-nevertheless-working semantics for factor

phrases in an adjectival context. But we’ve actually dodged an important puzzle here: why do (30)

and (35) mean the same thing? After all, that’s not remotely true for the non-modified version:

(41) Context: Colin is 6 feet tall. Christopher is also 6 feet tall.

a. Colin is as tall as Christopher.

b. #Colin is taller than Christopher.

In order to figure this out, it will be helpful to look at a different kind of multiplicative modifier:

percentages.

4.3 Percentages and ratio degrees

Before we can properly discuss percentages, let’s take a moment to discuss the interaction

(or lack thereof) of comparatives and equatives with differentials. Recall what happens when we

introduce a measure phrase such as three inches into the comparative construction:

(42) Tom is three inches taller than Colin.

As you’re no doubt aware, (42) states that Tom’s height is greater than Colin’s height, and that the

difference in their heights is three inches. The comparative morpheme we’ve been using doesn’t

have a place for a measure phrase to go, so, as we saw in (71) in section 2.2.2, the standard move

is to use a version of -er that has that slot and to just say that the two -ers are ambiguous. This has

some knock-on effects as you’ll recall: the largest is that, rather than continuing to try to interpret
things in situ, we treat DegP as a generalized degree quantifier of type (cid:104)dt, t(cid:105) that QRs up to the top
of the tree. Here’s the syntax and semantics of it again for reference:

103

(43)

t

DegP(cid:104)dt,t(cid:105)

(cid:104)d, t(cid:105)

DPd

DegP(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)

1

t

three inches

Deg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

-er

XPd

DPe

(cid:104)e,t(cid:105)

than Colin

Tom

is

AP(cid:104)e,t(cid:105)

d1

A(cid:104)d,et(cid:105)
tall

(44)

a. (cid:126)-erdiff(cid:127) = λdλd(cid:48)λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} − d = d(cid:48)]
b. (cid:126)three inches -erdiff than Colin(cid:127) = λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} − dColin = 3-inches]
c. (cid:126)three inches -erdiff than Colin 1 Tom is d1 tall(cid:127)

= 1 iff [max{d(cid:48)(cid:48) :tall(Tom) = d(cid:48)(cid:48)} − dColin = 3-inches]

By contrast, equatives cannot combine with a differential measure phrase, no matter how hard

you try:

(45)

a. *Tom is three inches as tall as Colin.

b. *Tom is as tall as Colin by three inches.

What does all this have to do with percentages? Well, consider (46):

(46)

a. Floyd is 60% taller than Clyde.

b. This Honda is 50% more expensive than that Chevy.

Now we have a multiplicative modifier, but it doesn’t behave like three times does. It’s not

saying “take Clyde’s height and multiply it by 60% to get Floyd’s height”; it’s saying, “take Clyde’s

height, multiply it by 60%, and then add that amount to Clyde’s height to get Floyd’s height”. In

other words, it seems to be behaving more like a differential measure phrase than a non-differential

104

multiplier like three times, in the same way that (42) says “take Clyde’s height and add 3 inches to

his height to get Floyd’s height”.

But we don’t want to just say that percentage phrases are differential measure phrases and call

it a day, because we also have to deal with (47):

(47)

a. Clyde is 63% as tall as Floyd.

b. That Chevy is 67% as expensive as this Honda.

We saw in (45) that measure phrases don’t combine with equatives, so therefore it doesn’t look like

we want to call percentage phrases differential MPs; nor do we want to claim that percentages are

also ambiguous between differential MP and non-differential MP versions. (Well, we could, but it

would feel rather arbitrary and wouldn’t be particularly enlightening.) Another strategy is called

for.

Observe other places where percentages show up:

(48)

a. 20% of 200 is 40.

b. Clyde’s height is 75% of Floyd’s height.

In these cases, the percentage appears to modify a degree and then produce another degree – since

phrases such as 75% of Floyd’s height show up where we would expect a degree such as a measure

phrase to appear:

(49) Clyde’s height is 135 centimeters.

In Gobeski & Morzycki (2017), we argued that this was evidence that percentages are in fact
of type (cid:104)d, d(cid:105): they take in a degree, modify it, and then provide the modified degree as a result.
Gobeski & Morzycki (2017) called these objects relational degrees but acknowledged that the

term wasn’t ideal, as the claim isn’t that percentages are creating a relation between two degrees;

instead, percentages describe a degree in terms of another degree. A perhaps less confusing term
would be the slightly clunky ratio degree: a degree that’s in a ratio with another.1 This would
1Of course, maybe the best term to use would be the adjectival form of ratio, but unfortunately

a rational degree means something else entirely.

105

place ratio degrees in the same spirit as Sassoon (2010b): much as she describes measure phrases

as measurements in a ratio with a degree (usually a base unit, such as 1 foot), so ratio degrees

describe the same intuition, but in a ratio that isn’t x : 1.

The natural question then becomes, what degree is the ratio degree modifying? In (48) the

natural conclusion is the of phrase: Floyd’s height and 200 can both be treated as degrees, and this

is what the percentage seeks to modify:

(50)

a. (cid:126)20%(cid:127) = λd.20% × d
b. (cid:126)20% of 200(cid:127) = 20% × 200

Is there such a degree in our comparative sentence? Gobeski & Morzycki (2017) argued that

there was in fact an elided degree present, similar to (51):

(51) The height of the top hat needs to be about 10% of Floyd’s total height. 10% of Floyd’s

height is 18 centimeters.

Thus, a comparative such as (46a) is in fact something like (52):

(52) Floyd is 60% of Clyde’s height taller than Clyde.

And indeed, if we choose to leave of Clyde’s height unelided, it’s perhaps a bit awkward, but it’s

not outright bad.

So given that, let’s put the pieces together. (52) will simply be a straightforward QRed structure,

as in (53), and the semantics will run standardly as in (54):

106

(53)

(54)

t

DegP(cid:104)dt,t(cid:105)

(cid:104)d, t(cid:105)

DPd

(cid:104)d, d(cid:105)

d

60%

of Clyde’s height

DegP(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)

1

t

Deg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

-er

XPd

DPe

(cid:104)e,t(cid:105)

than Clyde

Floyd

is

AP(cid:104)e,t(cid:105)

d1

A(cid:104)d,et(cid:105)
tall

a. (cid:126)60%(cid:127) = λd.60% × d
b. (cid:126)60% of Clyde’s height(cid:127) = 60% × dClyde
c. (cid:126)-erdiff(cid:127) = λdλd(cid:48)λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} = d + d(cid:48)] 2
d. (cid:126)-erdiff than Clyde(cid:127) = λd(cid:48)λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} = dClyde + d(cid:48)
e. (cid:126)60% of Clyde’s height -erdiff than Clyde(cid:127)

= λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} = dClyde + (60% × dClyde)

f. (cid:126)60% of Clyde’s height -erdiff than Clyde 1 Floyd is d1 tall(cid:127)

= 1 iff [max{d(cid:48)(cid:48) :tall(Floyd) = d(cid:48)(cid:48)} = dClyde + (60% × dClyde)]

Ultimately, Floyd is 60% taller than Clyde is true if the maximal degree of Floyd’s height is equal

to the sum of Clyde’s height and 60% of Clyde’s height, and this is indeed the meaning we want.

But now what do we do with equative cases? The same strategy that we employed for compara-

tives won’t work for equatives because, as we saw in (45), equatives don’t combine with differential

measure phrases – and indeed, while our comparative with an unelided degree was a bit awkward

but not outright bad, the same cannot be said if we try to do the same thing with an equative:

(55)

*Clyde is 63% of Floyd’s height as tall as Floyd.

2I’ve altered this slightly to make the addition aspect clear, but mathematically this is identical

to (44a).

107

So how do we get the percentage in the equative case to multiply the as Floyd degree? We

don’t want to have it modify the equative the way we were doing with three times earlier because

not only would we lose the insight that percentages are transformed degrees, but that would also

require completely reworking our comparative case, and all for a solution that would seem arbitrary

at best, a way of brute-forcing all the pieces into fitting together.

Gobeski & Morzycki (2017) argued that part of the solution has to do with the nature of equa-

tives themselves. The general consensus is that equatives differ from comparatives only in whether
the relation they provide is = (or ≥) or >. As the rest of the syntax for equatives appears to be more
or less the same as that of comparatives, this isn’t an unreasonable proposition. But we’ve seen

that equatives don’t quite behave the same as comparatives; you might recall that one of the conse-

quences of comparatives is in their scopal properties. Heim (2000) pointed out that comparatives

can have different meanings when combined with intentional verbs such as required or allowed,

depending on whether the comparative is interpreted above or below the verb; this is meant to

argue against (only) allowing in situ interpretations of DegP. We saw this in (72) of chapter 2, and

a similar version is provided in (56), where the two readings can be distinguished by adding “so

you’re OK”/”so you’re not OK”:

(56) The paper is required to be less long than that.

a. The paper is required to be less long than that, so you’re OK.

b. The paper is required to be less long than that, so you’re not OK.

(You’ve exceeded the maximum allowable length.)

(You’ve met the minimum threshold.)

But while this is true for comparatives, it’s not clear that the same thing applies to equatives.

For instance, in a sentence such as (57),

(57) The paper is required to be exactly as long as that,

it’s not obvious what the two readings would be; (57) states that the requirement has been met, and

that’s about it. It’s not at all apparent that there’s a scopal interaction between required and the

108

equative.

So given that, Gobeski & Morzycki (2017) propose that the equative morpheme as remains

in fact uninterpreted, and so as as Floyd is in fact the same as just as Floyd – which means our

adjective will simply interpret the as Floyd degree more or less directly:

(58)

AP(cid:104)e,t(cid:105)

DegPd

A(cid:104)d,et(cid:105)

Deg

XPd

tall

(59)

as Floyd

as
a. (cid:126)as as Floyd(cid:127) = (cid:126)as Floyd(cid:127) = dFloyd
b. (cid:126)tall(cid:127) = λdλx[tall(x) = d]

c. (cid:126)tall(cid:127)((cid:126)as as Floyd(cid:127)) = λx[tall(x) = dFloyd]

And now with this in mind, a percentage can slot neatly into the tree, and the semantics work

just as we’d expect:

(60)

DPe

Clyde

is

t

(cid:104)e, t(cid:105)

AP(cid:104)e,t(cid:105)

DegPd

(cid:104)d, d(cid:105)

Deg(cid:48)
d

A(cid:104)d,et(cid:105)

tall

63%

Deg

XPd

as

as Floyd

109

(61)

a. (cid:126)as as Floyd(cid:127) = (cid:126)as Floyd(cid:127) = dFloyd
b. (cid:126)63% as as Floyd(cid:127) = 63% × dFloyd
c. (cid:126)tall(cid:127)((cid:126)63% as as Floyd(cid:127)) = λx[tall(x) = 63% × dFloyd]
d. (cid:126)Clyde is 63% as tall as Floyd(cid:127) = 1 iff tall(Clyde) = 63% × dFloyd

Our final result states that Clyde is 63% as tall as Floyd is true if the (maximal) height of Clyde is

Floyd’s height times 63% – which, again, is precisely what we want.

This approach to percentages accomplishes a few things: it allows us to correctly predict that

percentages with comparatives provide a differential meaning, such that percentages below 100%

nevertheless provide a higher degree than the one present in the than phrase; it also predicts that

percentages with equatives provide a multiplicative meaning, such that we end up with a degree

that’s smaller than the degree in the as phrase for percentages below 100%; and this approach

captures the intuition that these multiplicative modifiers are modifying degrees, rather than com-

parative/equative morphemes, which leads to a more satisfying semantics. And while leaving the

equative uninterpreted may seem a bit unorthodox, this may in fact be the same approach that hap-

pens in other languages: Japanese, for example, does much of its work through yori, which is said

to be the (rough) equivalent to than. (See, for instance, Beck et al. 2004 for further description of

how yori behaves.) So this may not be the unorthodox move it may initially appear to be.

All right, we’ve now established that percentage phrases are ratio degrees of type (cid:104)d, d(cid:105), and
we’ve seen how they work. So, armed with this new approach, let’s reexamine adjectival factor

phrases in this light.

4.4 Adjectival factor phrases and ratio degrees

We’ll begin with the equative case. Previously, I’d suggested that (our second version of) factor
phrases were of type (cid:104)(cid:104)d,(cid:104)(cid:104)d, et(cid:105),(cid:104)e, t(cid:105)(cid:105)(cid:105),(cid:104)d,(cid:104)(cid:104)d, et(cid:105),(cid:104)e, t(cid:105)(cid:105)(cid:105)(cid:105), operating in such a way that they were
trying to do an end run around the actual meaning of the equative/comparative morpheme. Now

let’s once more look at our sentence in (30) – but this time we’ll treat our factor phrase three times

110

as a ratio degree:

(30) Floyd is three times as tall as Clyde.

(62)

(cid:126)three timesversion 3(cid:127) = λd.3 × d

Our structure will be more or less the same as it was for the percentage case in (60) – and one

bonus as a result is that we now have the same structure that Bierwisch (1989) argued for in (14):

(63)

t

(cid:104)e, t(cid:105)

DPe

Floyd

is

AP(cid:104)e,t(cid:105)

DegPd

(cid:104)d, d(cid:105)

Deg(cid:48)
d

A(cid:104)d,et(cid:105)

tall

three times

Deg

XPd

as

as Clyde

And as in the percentage case, we’ll leave our equative as uninterpreted: this means that as as

Clyde will be of type d and functionally identical to as Clyde. Thus the semantics will run in the

same way that they did for the percentages:

(64)

a. (cid:126)as as Clyde(cid:127) = (cid:126)as Clyde(cid:127) = dClyde
b. (cid:126)three timesv3(cid:127)((cid:126)as as Clyde(cid:127)) = 3 × dClyde
c. (cid:126)tall(cid:127)((cid:126)three timesv3 as as Clyde(cid:127)) = λx[tall(x) = 3 × dClyde]
d. (cid:126)Floyd is three timesv3 as tall as Clyde(cid:127) = 1 iff tall(Floyd) = 3 × dClyde

So we get a result that says that Floyd’s height is equal to Clyde’s height multiplied by 3, which is

precisely what we want.

So far so good. But now when we move to the comparative version, we’re going to run into a

problem; in the percentage form, the percentage acted as a differential, adding the amount of the

111

multiplied percentage to the degree in the than phrase. But that’s not what we want to happen with

a factor phrase like three times. We saw back in (35) that it should have the same meaning as (30):

(35) Floyd is three times taller than Clyde.

(Clyde is 2 feet tall, Floyd is 6 feet tall)

(30) Floyd is three times as tall as Clyde.

(Clyde is 2 feet tall, Floyd is 6 feet tall)

So a differential reading isn’t going to be what we want. Right?

Maybe it actually is. Consider the case of Mandarin Chinese (Kai Chen, Ying Khong, p.c.):

(65)

bi
compare

Paul
Paul

gao
tall

John
John
’John is four times as tall (lit. ‘three times taller’) as Paul.’

bei
multiples

san
three

Here we have a case where things work exactly in the way we’d predict them to, based on the

percentage data: Mandarin Chinese treats the factor phrase as a differential that it adds to the base

degree, meaning that three times taller is equivalent to four times as tall.

And it turns out that we can actually get this differential effect in English as well. Consider the

(somewhat marginal) sentence in (66):

(66)

?Floyd is point five times taller than Clyde.

The general consensus of an informal poll was that the meaning of (66) is differential: if Clyde

were four feet tall, Floyd would be six feet tall – in other words, we multiply Clyde’s height by
0.5 and then add that to Clyde’s original height in order to get Floyd’s height.3 And while “point

five” is admittedly a bit strange, we can use certain fractions directly to get this same “multiply

then add” effect without issue:

(67) Paul is a third taller than Tony.

(Tony is 3 feet tall, Paul is 4 feet tall)

(68) This first rope is a fifth longer than that second one.

3It should be noted that one respondent couldn’t get a differential meaning for (66) at all;

consequently, they judged the sentence to be outright bad.

(1st rope is 6 feet long, 2nd rope is 5 feet long)

112

And now consider (69):

(69) Floyd is 1.5 times taller than Clyde.

As one of those informally polled independently pointed out, this sentence is ambiguous: it can be

treated as direct multiplication like three times and therefore in fact mean the same thing as (66) –

which is in itself kind of insane – but it can also be treated as a differential, where we add Clyde’s

height times 1.5 to the original degree of Clyde’s height to get Floyd’s height.

And finally, the belief that (30) and (35) mean the same thing is not in fact a universal one in

English, and we can find people who will argue this point, either out of a sense of prescriptiveness

or because they genuinely have a distinction between the meanings of these two sentences. For

instance, here’s a quote from a short paper by statistician Milo Schield: “If B is three times as

much as A, then B is two times more than A – not three times more than A” (Schield 1999:1). Or

similarly, on the website StackExchange a user asked the following question: “Suppose John has

5 sweets. Is there any difference between ... ’Jack has 3 times as many sweets as John’ [and] ‘Jack

has 3 times more sweets than John’? I prefer the first construction and would know unambiguously

that Jack has 15 sweets in this case.” To this, another user replied, “The question has been asked

many times around the web, and there appear to be two schools: one that agrees with you, and one

that thinks both constructions are OK and takes both to mean 15 sweets. I think those people are
nuts, but hey they might be the majority.”4

So what’s ultimately going on here? This differential interpretation can’t simply be an appeal

to pedantry, as even those who are otherwise fine with (30) and (35) being equivalent in meaning

find that the fractional cases don’t follow the rest of the pattern.

I instead propose that the reason why (35) gets a direct multiplicative reading for many (if not

most) people instead of a differential reading is the same reason why (30) gets a direct multiplica-

tive reading: the comparative -er morpheme is also left uninterpreted when a factor phrase such as

4From https://english.stackexchange.com/questions/7894/x-times-as-many-

as-or-x-times-more-than

113

three times is present. However, we can choose to interpret the comparative if we want to, which

is why some people get a differential reading for three times more.

So the multiplicative version of (35) will have a semantics that’s basically the same as we saw

in (64):

(70)

t

(cid:104)e, t(cid:105)

DPe

Floyd

is

AP(cid:104)e,t(cid:105)

DegPd

(cid:104)d, d(cid:105)

Deg(cid:48)
d

A(cid:104)d,et(cid:105)

tall

three times

Deg

XPd

-er

than Clyde

(71)

a. (cid:126)-er than Clyde(cid:127) = (cid:126)than Clyde(cid:127) = dClyde
b. (cid:126)tall(cid:127)((cid:126)three timesv3 -er than Clyde(cid:127)) = λx.tall(x) = 3 × dClyde
c. (cid:126)Floyd is three timesv3 taller than Clyde(cid:127) = 1 iff tall(Floyd) = 3 × dClyde

And if we really want the differential version, we can treat it the same as our percentage version

in (54):

(72) Floyd is three times Clyde’s height taller than Clyde.

114

(73)

t

DegP(cid:104)dt,t(cid:105)

(cid:104)d, t(cid:105)

DegP(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)

1

t

d

Deg(cid:104)d,(cid:104)d,(cid:104)dt,t(cid:105)(cid:105)(cid:105)

XPd

DPe

(cid:104)e,t(cid:105)

DPd

(cid:104)d, d(cid:105)

three times

Clyde’s height

-er

than Clyde

Floyd

is

AP(cid:104)e,t(cid:105)

d1

A(cid:104)d,et(cid:105)
tall

(74)

a. (cid:126)three timesv3 Clyde’s height(cid:127) = 3 × dClyde
b. (cid:126)three timesv3 Clyde’s height -erdiff than Clyde(cid:127)

= λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} = dClyde + (3 × dClyde)

c. (cid:126)three timesv3 Clyde’s height -erdiff than Clyde 1 Floyd is d1 tall(cid:127)

= 1 iff [max{d(cid:48)(cid:48) :tall(Floyd) = d(cid:48)(cid:48)} = dClyde + (3 × dClyde)]

Thus, our final result here states that the maximal degree of Floyd’s height is equal to the degree

of Clyde’s height added to the degree that’s equal to Clyde’s height multiplied by a factor of 3. In

other words, three times taller is equivalent to four times as tall, which (if we’re, say, Milo Schield)

is the result we’re looking for.

One unresolved question has to do with scopal effects. As you’ll recall, one of the reasons

Heim (2000) argued for a movement-based structure for DegP (the “classical”, small DegP view)

was that you can get scopal effects when the comparative interacts with certain verbs and other

things; we saw this in (56). But if the current approach is correct, then when the comparative

morpheme is uninterpreted we shouldn’t expect scopal interactions with a verb like required: it

should always scope over the comparative.

(75) The paper is required to be exactly three times longer than that.

115

Acknowledging that these judgments are slippery even at the best of times, it’s not clear to me that

(75) has multiple meanings. Part of the issue is that there’s a wrinkle; we could choose to interpret

the comparative as we did in (74), meaning that if we do in fact do that then (75) will indeed

have multiple meanings, based on scopal interactions. I have to confess that with a construction as

complex as this, I’m not confident of my judgments (and neither were others who were informally

asked), but to the extent that I can get multiple meanings for (75) I think it is in fact the differential

version that I’m getting, where three times the length is being added to the current length. But

this is far from certain. As a result, this may be a place where sharper judgments may lead to a

reevaluation of things, but as of now that seems a question better left to future research; we’ll have

to be content to leave this unresolved for now.

It’s worth pausing here for a moment to reflect on what’s been achieved here. Instead of go-

ing with the works-but-feels-somewhat-arbitrary solution initially proposed for adjectival factor

phrases, we’ve instead adopted a more elegant solution. The concept of ratio degrees not only

captures the sense that these phrases are manipulating degrees directly – in essence, moving from

one scale to another, related scale – but also does so in a clear and direct manner, and in a way that

doesn’t require a somewhat unusual syntax of combining the factor phrase with the degree mor-

pheme before anything else happens. In addition, the fact that we can use this setup in combination

with the idea of uninterpreted degree morphemes to clearly and deftly capture the idea that, e.g.,

two times taller can mean the same thing as two times as tall and the idea that it can also mean

three times as tall if the situation demands it makes this solution preferred over the initial, arbitrary

approach.

4.5 Ratio degrees and multiplicative verbal comparatives

At this point, a natural question would be “can we use this same approach with the multiplica-

tive factor phrases in the verbal cases”? So in order to explore this, let’s begin by taking another

look at an example sentence. Here’s (74) from chapter 3, renumbered as (76):

116

(76) Floyd walked the dog three times as many times as Clyde (did).

Without worrying about the factor phrase for a moment, let’s just remind ourselves what the struc-

ture of an equative in a verbal construction is. The degree phrase as as Clyde can’t be interpreted

in situ, so it undergoes Quantifier Raising, moving up to the top of the tree and leaving behind a

type d degree:

(77)

(cid:104)dt, t(cid:105)

(cid:104)d,(cid:104)dt, t(cid:105)(cid:105)

d

1

as

as Clyde

∃s

Floyd walked the dog

(cid:104)s, t(cid:105)
many(cid:104)d,st(cid:105)

d1

times

This means that the degree phrase as as Clyde starts as the sister to many before QRing. The

obvious problem with this structure (for our purposes at least) is that there’s not a good place

for a ratio degree to fit into the generalized degree quantifier, and while we could try to merge it

with, say, as Clyde before combining with the degree morpheme as, that feels arbitrary and not

syntactically plausible.

But what would happen if we chose to leave as uninterpreted, the same way we’ve done in the

adjectival cases? That would make the type of as as Clyde the same as just as Clyde (so d), which

can combine with many without issue – in other words, there’d be no need to QR. Consequently,

our factor phrase can combine with as as Clyde in the verbal case the same way as it did in the

adjectival case:

117

(78)

Floyd

(cid:104)s, t(cid:105)

times

walked

d

many(cid:104)d,st(cid:105)

the dog

(cid:104)d, d(cid:105)

d

three times

as

d

as Clyde

And the semantics will still work out the way we want them to:

(79)

a. (cid:126)as as Clyde(cid:127) = (cid:126)as Clyde(cid:127) = dClyde
b. (cid:126)three timesv3(cid:127)((cid:126)as as Clyde(cid:127)) = λd.3 × d ((cid:126)as as Clyde(cid:127)) = 3 × dClyde
c. (cid:126)manys(cid:127) = λdλe[|e| = d]
d. (cid:126)three times as as Clyde(cid:127)((cid:126)manys(cid:127)) = λe[|e| = 3 × dClyde]
e. (cid:126)three times as as Clyde manys times(cid:127) = λe[*time(e) ∧ |e| = 3 × dClyde]
f. (cid:126)Floyd walked the dog(cid:127) = λe.walk(the-dog)(Floyd)(e)

(a many for events)

g. (cid:126)Floyd walked the dog three times as many times as Clyde(cid:127)

=1 iff ∃e.walk(the-dog)(Floyd)(e) ∧ *time(e) ∧ |e| = 3 × dClyde

It looks a bit complicated, but the key takeaway is that the number of Floyd’s dog-walkings is

equal to the number of Clyde’s dog-walkings multiplied by 3. This is indeed the result we want,

and happily we were able to achieve it while still keeping the same denotation for our multiplicative

factor phrase. Thus the idea of factor phrases as ratio degrees still works, even though we’re in a

different domain.

Now with that sorted, let’s move to the problem of twice.

118

4.6 Twice

So far we’ve been dealing with factor phrases of the form n times. But what about the ones

that end in -ce: twice, the less common thrice, and once? In theory, these should behave like their

respective n times counterparts: two times, three times, and one time. But there’s an asymmetry in

the adjectival case (first noted by Gathercole 1981):

(80)

a. Floyd is two times as tall as Clyde.

b. Floyd is twice as tall as Clyde.

(81)

a. Floyd is two times taller than Clyde.

b. *Floyd is twice taller than Clyde.

Gathercole (1981) argues that (81b) is bad because twice is a contraction of two times, and

according to King (1970), contractions are blocked if there’s a gap after where the contraction

would be – and the movement of -er out to join with either the adjective or much/many (to make

more) creates a gap, which means that two times can’t be contracted to twice in comparative cases.

(Equatives are fine because as doesn’t move.)

One of the problems with this analysis (as Kayne 2015 points out) is that it wrongly predicts

(82) should be bad:

(82) He’s taller than his brother.

Another problem is that twice doesn’t actually appear to be a contraction of two times. In fact,

twice actually seems to predate two times in English. According to the Oxford English Dictionary,

twice is attested to in Old English from at least circa 995:

(83)

his
is

twies
twice

cild
child

θat
that
‘That child [Jesus] is born twice.’ 5

acenned.
born

And twice was still being used a hundred years later:

5From Ælfric’s Catholic Homilies

119

geares
years.gen

(84) Ðises
these
‘That light came in these years to the sepulcher of the Lord ... twice.’ 6

Sepulchrum
sepulcher

Dni ...
of.Lord ...

twiges.
twice

com
came

leoht
light

θet
the

to
to

By contrast, while time itself derives from Old English timen/tima/time, the use of two times

dates to Middle English, circa 1450:

(85) The auicion come to hem bi two tymes. 7

The OED suggests that two times derives from many times, but even that only appears to be

attested to in Middle English, not Old English:

lord
lord

god
God

(86) Ure
our
‘Our Lord God Almighty... has many times spiritually made wine from water.’ 8

manitime
many-times

almichti...
almighty...

gostliche.
spiritual

habbeθ
have.pl

watere
water

maked
made

wyn
wine

of
of

Therefore, it seems unlikely that twice is a contraction of two times; it’s more plausible that

twice originated as an adverb, while two times began as a DP and then acquired an adverbial use

over time. So we’ll need a different approach to the asymmetry seen in (80)/(81).

In Gobeski (2011), I suggested that the asymmetry was the result of twice and two times having

slightly different denotations: twice provided the maximal degree that was greater than or equal to

2 times the than/as phrase degree, while two times merely gave you a degree. This was cashed out
as in (87) and (88):9

(87)

(88)

(cid:126)twice(cid:127) = λF(cid:104)d,et(cid:105)λdλx[max{dx : F(dx)(x)} ≥ 2d]
(cid:126)two times(cid:127) = λF(cid:104)d,et(cid:105)λdλx.∃dx[F(dx)(x) ∧ dx ≥ 2d]

The idea is that the conflict comes about when the AP is incorporated into the factor phrase: in

twice, you’re looking for the maximal degree that is smaller than x’s height (or whatever adjective

6From Anglo-Saxon Chronicle, Peterborough Chronicle, c1122
7From The Book of the Knight of La Tour Landry
8From the Kentish Sermons, c1275
9Note that these denotations are in the big DegP system, where the adjective combines with the

degree morpheme than/as first.

120

you’re working with) and saying that that is equal to twice the degree provided by the factor phrase

– but that essentially creates a mathematical limit: you’ll approach x’s degree but never reach it as

you look for the maximal degree smaller than tall(x).

(89)

(cid:126)twice taller(cid:127) = λdλx[max{dx :tall(x) > dx} ≥ 2d]

Two times, by contrast, just wants a degree, so we can just pick one and move on with the rest

of the sentence:

(90)

(cid:126)two times taller(cid:127) = λF(cid:104)d,et(cid:105)λdλx.∃dx[tall(x) > dx ∧ dx ≥ 2d]

It’s kind of a cute solution, but it’s rather specific to this adjectival environment and it doesn’t

address any of the other issues we saw in the previous section, so there’s a bit of a sense of creating

a tool solely to deal with a specific problem. (There’s also the related concern of why we’d let

our brains crash against (89) when we’re happy to perform various rounding operations with other

numerical operators – see, for instance, Anderson 2015 and the use of numerical some.) So let’s

try to find a better solution. One way to do that could be to see where else twice patterns differently

from two times.

In addition to the adjectival asymmetry, Kayne (2015) brings up a number of other distinctions

between once/one time and twice/two times:

(91)

a. They told us about the one time they thought they were really in danger.

b. *They told us about the once they thought they were really in danger.

(92)

a. He’s going to be just a two-time champion.

b. *He’s going to be just a twice champion.

c. *He’s going to be just a two-times champion.

(93)

a. Two times are enough.

b. Two times is enough.

c. *Twice are enough.

d. Twice is enough.

121

We see here in (91b) and (92b) that twice can’t appear with a determiner, while (93c) indicates that

twice is singular, not plural.

What’s driving the difference between twice and two times? Kayne’s explanation is that -ce is

actually a postposition, one that combines with a number and then a silent morpheme TIME. Thus,

the problem sentences in (91)-(93) are bad because of associated reasons: (91b) is bad because

adpositions can’t be the heads of relative clauses, while (92b) and (92c) are bad because of the

restrictions on, respectively, adpositions and plurals in compounds.

Could the presence of an adposition being leading to the adjectival asymmetry? Is there some-

thing about the comparative that disallows an adpositional phrase? This seems implausible, be-

cause we in fact have evidence that, while there is an asymmetry regarding equatives/comparatives

and prepositional phrases, it actually goes in the other direction:

 two times
 two times

a factor of two

a factor of two

 .
 .

(94)

a. Floyd is taller than Clyde by

b. *Floyd is as tall as Clyde by

Part of the reason why (94) is good for comparatives and bad for equatives is that the by phrase

appears to be differential; we get the same effect in (95) as we do with a standard differential

comparative:

(95) Floyd is taller than Clyde by three inches.

But because of this, we can’t just say that comparatives don’t interact with prepositions and leave

it there; that’s not what the evidence points to.

Does it have something to do with the fact that TIME is singular? It’s somewhat difficult to

tell: the sentences in (96) are problematic for independent, redundant reasons:

(96)

a. *Floyd is one time as tall as Clyde.

b. *Floyd is one time taller than Clyde.

122

However, it might be worth noting that, while (97) isn’t what we could call good, it does seem

marginally improved over (96b):

(97)

??Floyd is one times taller than Clyde.

So could the presence of the singular form be the reason? Is there something that’s incompatible

with the comparative? The presence of the singular might indicate that we’re looking for a one-

to-one correspondence, and that might signal that the comparative isn’t going to be of use in this

case, since the two degrees being compared would end up being equal. (97) could then trick us

into accepting it because the plural throws off the scent, so to speak. After all, while (97) is still

weird, (98) is surprisingly acceptable:

(98) Floyd is zero times taller than Clyde.

We may have a tendency to rebel against this sentence, partly because it’s meaningless to have

direct multiplication by zero be the final result for an existing degree. However, if we choose to

interpret the comparative, it becomes reasonably straightforward:

(99)

a. (cid:126)zero timesv3 Clyde’s height(cid:127) = 0 × dClyde = 0
b. (cid:126)zero timesv3 Clyde’s height -erdiff than Clyde(cid:127)

= λF(cid:104)d,t(cid:105)[max{d(cid:48)(cid:48) : F(d(cid:48)(cid:48))} = dClyde + 0

c. (cid:126)zero timesv3 Clyde’s height -erdiff than Clyde 1 Floyd is d1 tall(cid:127)

= 1 iff [max{d(cid:48)(cid:48) :tall(Floyd) = d(cid:48)(cid:48)} = dClyde + 0]

(99c) states that the sentence is true if Floyd’s height is equal to Clyde’s height plus zero, or just

Clyde’s height, and that is in fact the sense that we have for the meaning of (98): essentially, “Floyd

is not taller than Clyde”.

This does slightly beg the question of why (97) isn’t also good if we choose to interpret the

comparative, but this might have more to do with the fact that one times doesn’t seem to show up

as a multiplier in any context; it may just not be good on independent grounds having to do with

redundancy and such.

123

4.7 Conclusion

So a number of things have been accomplished in this chapter. Perhaps the most significant is

that we now have a working semantics for adjectival factor phrases. Perhaps this may not seem

that exciting to some, but the fact of the matter is that prior to this we only had general descriptions

of how they would work, rather than a compositional semantics, and almost no description of the

accompanying puzzles, such as the N times as/N times -er meaning equivalence. Consequently

we’ve filled in a part of degree semantics that had up to this point been embarrassingly neglected.

But more excitingly, as I noted above we’ve accomplished this in a pleasingly direct and sat-

isfying way; the intuition that factor phrases should directly modify degrees is a natural one but

also a difficult one to achieve, as was hopefully made clear by the initial attempts to do so. This

has now been accomplished in a way that is natural but also connects with other multiplicative

degree modifiers such as percentages. A further consequence is that we’ve established that the

equative morpheme can go uninterpreted – a bit of a bold departure, but as noted above, one po-

tentially employed by other languages. The next step of (sometimes) leaving the comparative

-er morpheme also uninterpreted may seem a bit more unusual, but it has the nice consequence

of not only explaining why comparatives and equatives are treated as having identical meanings

when a factor phrase is involved, but also explaining the variation of judgments regarding factor

phrases and comparatives: sometimes we can get a differential meaning by choosing to interpret

the comparative morpheme.

We left off with a bit of discussion about the nature of twice, where we saw that twice and two

times do not behave the same in all contexts; this could be, if Kayne (2015) is correct, due to the

presence of a singular unpronounced TIME that is part of the syntax of twice, and it may be this

singular TIME that leads to the unacceptability of, e.g., *twice taller.

Now let’s move on to the next chapter, where we’ll venture deeper into uncharted territory: the

nature of factor phrases in nominal contexts.

124

CHAPTER 5

FACTOR PHRASES IN THE NOMINAL DOMAIN

Now we’re going to examine how factor phrases behave when they modify DPs, as in (1):

(1)

a. Floyd is twice Clyde’s height.

b. This sack is three times the weight of that sack.

c. Agnes has twice Bertha’s beauty.

d. You’re twice the man (that) your father was.

As we can see, we get these interacting with various types of DPs: dimensional scalar nouns in

(1a) and (b), evaluative nouns in (1c), and even definite NPs with relative clauses in (1d). As

factor phrases themselves haven’t been extensively studied in really any context, it’s unsurprising

to learn that, as far as I can tell, no one has worked on these nominal versions at all. So we’ll

begin in section 5.1 by sorting through the data, while section 5.2 will provide some initial efforts

to analyze some of the categories of the sorted nominals.

5.1 Data

5.1.1 Dimensional and evaluative nouns

Let’s begin by looking at the forms that most closely match the adjectival versions: nouns that

provide a scale in some way.

125

(2)

a. Floyd is twice Clyde’s height.

b. Agnes is three times Bertha’s age.

c. This bench is twice the length of that bench.

d. This box of pennies is forty times the weight of that box of feathers.

e. This kitchen is twice the size of that kitchen.

f. This monitor is twice the resolution of that television.

Here we have nouns that correspond to what Bierwisch (1989) terms dimensional adjectives: height

to tall, age to old, weight to heavy, etc. In other words, these nouns are scales that things can

be independently measured along. This holds even for nouns such as resolution, where we can

still discuss differences in resolution in independent numerical terms (pixel width of 480, 1080,

4K, etc.) even if we’re not always certain what the appropriate corresponding adjective would

be (sharp? clear? high-res?). Consequently, I’ll refer to these nouns as dimensional nouns, in

parallel with dimensional adjectives.

Dimensional nouns fit into a general construction of individual (in a broad sense) + BE; how-

ever, nominalizations of relevant dimensional adjectives don’t work as substitutions for dimen-

sional nouns.

126

(3)

a. *Floyd is twice Clyde’s

shortness

b. *Agnes is three times Bertha’s

c. *This bench is twice the

d. *This box of pennies is forty times the

youngness

 tallness
 .
 oldness
 .
 longness
 of that bench.
 heaviness
 bigness
 of that kitchen.
 sharpness
 of that television.

shortness

smallness

fuzziness

 of that box of feathers.

lightness

e. *This kitchen is twice the

f. ?This monitor is twice the

It’s possible that these are bad because of the inherent sense of ordering present in the nominal-

izations: shortness makes a claim about which way we’re moving along the scale in a way that

height doesn’t. So perhaps these need to be neutral with regards to their ordering. Another pos-

sibility is that the judgments in (3) may be the result of a blocking effect; we have better words

to express tallness and oldness (height and age, respectively), so it’s weird to settle for a nominal-

ization instead. This might be born out by the comparative acceptability of (3f) – resolution isn’t

a particularly common word, so it doesn’t block sharpness or fuzziness as strongly as size blocks

bigness.

The nouns in (2) are all independently measurable: we have judgments about what twice some-

one’s height means because we can measure their height with independent units (inches, centime-

ters, hands, etc.). But when we try to use scalar nouns that aren’t associated with independent

measures, they’re not acceptable:

127

(4)

a. *Agnes is twice Bertha’s beauty.

 Clyde’s intelligence

 .

b. *Floyd is three times

the intelligence of Clyde
c. *This watch is twice the accuracy of that watch.

d. *Gertrude is three times the humility of Florence.

e. *Sam is twice Fitz’s flexibility.

These nouns measure along a scale, but the scales in question don’t have independent units asso-

ciated with them. Their associated adjectives are in Bierwisch (1989) called evaluative adjectives,

so similarly we’ll call these kinds of nouns evaluative nouns.

Evaluative nouns don’t work in the individual + BE construction, and, unsurprisingly, neither

do nominalizations of their associated adjectives:

(5)

a. *Agnes is twice Bertha’s

ugliness

 prettiness
 .
 smartness
 .
 fastness
 of that watch.
 arrogance
 of Florence.

humbleness

slowness

stupidity

b. *Floyd is three times Clyde’s

c. *This watch is twice the

d. *Gertrude is three times the

So if we want to combine evaluative nouns with a factor phrase, we have two options. Option

1 is to compare an evaluative noun as it relates to an individual to that same noun as it relates to a

different individual, as in (6):

(6)

a. Agnes’s beauty is twice Bertha’s beauty.

b. Floyd’s intelligence is three times Clyde’s intelligence.

c. This watch’s accuracy is twice the accuracy of that watch.

d. Gertrude’s humility is three times the humility of Florence.

128

Option 2 is to use HAVE instead of BE:

(7)

a. Agnes has twice Bertha’s beauty.

b. Floyd has three times Clyde’s intelligence.

c. This watch has twice the accuracy of that watch.

d. Gertrude has three times the humility of Florence.

It’s worth noting that while Option 1 works with dimensional nouns...

(8)

a. Floyd’s height is twice Clyde’s height.

b. Agnes’s age is three times Bertha’s age.

c. The length of this bench is twice the length of that bench.

d. The weight of this box of pennies is forty times the weight of that box of feathers.

e. This kitchen’s size is twice the size of that kitchen.

f. This monitor’s resolution is twice the resolution of that television.

...Option 2 doesn’t really work with individuals:

(9)

a. *Floyd has twice Clyde’s height.

b. *Agnes has three times Bertha’s age.

c. *This bench has twice the length of that bench.

d. This box of pennies has forty times the weight of that box of feathers.

e. *This kitchen has twice the size of that kitchen.

f. This monitor has twice the resolution of that television.

Curiously, Option 2 isn’t consistently bad with dimensional nouns, as (9d) and (f) illustrate.

This may be because we already discuss properties such as weight or (in the case of display screens)

resolution in terms of being possessed by objects, in a way that we don’t really describe people as

“possessing” height or age. Corroborating this, possessing weight suddenly becomes strange when

applied to a person rather than an object such as a box:

129

(10)

*George has twice Denny’s weight.

This is likely related to a distinction brought up in Francez & Koontz-Garboden (2015) re-

garding something called property concepts, which is a way of dividing adjectives into different

groups based on what their function is (such as describing a color or a particular dimension like

height). Francez & Koontz-Garboden argue that there are two types of PC lexemes: those that are
adjectivally denoting, and those that are substance denoting.1 They argue that, generally speaking,

substance denoting PC lexemes require possessive constructs (such as HAVE), while adjectivally

denoting PC lexemes do not. We’ll discuss Francez & Koontz-Garboden (2015) further in a lit-

tle bit, but for now it’s sufficient to note that the split we see here is very possibly the adjecti-

val/substance PC split, although the fact that certain words such as resolution can occur with both,

and that weight can shift between the two depending on context, may indicate that the divide isn’t

quite as rigid as Francez & Koontz-Garboden (2015) make it seem.

5.1.2 Factor phrases and relative clauses

We turn now to when factor phrases modify relative clauses headed by an NP. Relative clauses

headed by dimensional nouns are weird at best, both with BE and with HAVE:

(11)

a. *Ivan is three times the weight his father is/was.

b. ?*Ivan has three times the weight his father has.

(12)

a. ?This box is ten times the weight that box is.

b. ?This box has ten times the weight that box has.

(13)

a. ?Floyd is twice the height Clyde is.

b. ?*Floyd has twice the height Clyde has.

(14)

a. ?This building is twice the height that that building is.

b. ?This building has twice the height that that building has.

1Francez & Koontz-Garboden actually allow for the possibility of additional PC types, but these

are the two they focus on.

130

(15)

a. ?Greta is three times the age her sister is.

b. *Greta has three times the age her sister has.

(16)

a. This monitor is twice the resolution that television is.

b. This monitor has twice the resolution that television has.

Again, resolution is OK with HAVE, even though weight doesn’t really work anymore in the same

way it did in (9d) – although the non-person version in (12) seems a bit better than the person

version in (11) (and the same thing goes for height in (14) vs. (13)). And it’s also fine with BE,

suggesting that resolution actually works a little differently from the rest of the dimensional nouns.

Evaluative noun-headed relative clauses still don’t work with BE but are fine with HAVE:

(17)

a. *Agnes is twice the beauty that Bertha is.

(Good for personified beauty, bad for scalar beauty)

b. Agnes has twice the beauty that Bertha has.

(18)

a. *Floyd is three times the intelligence that Clyde is.

b. Floyd has three times the intelligence that Clyde has.

(19)

a. *This watch is twice the accuracy that that watch is.

b. This watch has twice the accuracy that that watch has.

(20)

a. *Gertrude is three times the humility that Florence is.

b. Gertrude has three times the humility that Florence has.

That said, they’re still acceptable in the noun BE noun case, but only if the embedded noun is

possessive:

(21)

a. Agnes’s beauty is twice the beauty that Bertha’s is.

b. Floyd’s intelligence is three times the intelligence that Clyde’s is.

c. This watch’s accuracy is twice the accuracy that that watch’s is.

In other words, the sentences in (21) are simply relativized versions of (6).

131

Somewhat unsurprisingly, non-scale gradable nouns (that is, nouns with degree arguments that

aren’t (typical) names for scales – see, among others, Morzycki 2005, 2009 for more on this topic)

can also be modified by a factor phrase, but only if the NP is the head of a relative clause:

 Clyde’s idiot

the idiot of Clyde

 .

(22)

a. Floyd is twice the idiot that Clyde is.

b. *Floyd is twice

c. *Floyd is twice an idiot.

(OK on an eventive reading, bad on a degree one)

d. *Floyd has twice the idiot that Clyde has.

(23)

a. Your relationship is twice the disaster that the sinking of the Lusitania was.

b. *Your relationship is twice the Lusitania disaster.

(24)

a. Paul is twice the Tim and Eric fan that Tony is.

b. *Paul is twice Tony’s fan.

c. Charlie is twice the Belieber that Alex is.2

d. *Charlie is twice Alex’s Belieber.

Here the factor phrase modifies the degree of idiocy/disaster/fan/etc., rather than ascribing a size

to the individual (in other words, we’re not saying (22a) because Floyd is physically larger than

Clyde). This is in line with these gradable nouns, where, e.g., big idiot refers to the amount of

idiocy, not the size of the physical idiot.

Nouns that don’t have degree arguments encoded in them instead get what I will call quality

readings, where the comparison is something along the lines of twice/three times/etc. as good as;

once again, these only seem to work when the noun in question is the head of a relative clause:

2A Belieber is a fan of Canadian pop singer Justin Bieber.

132

(25)

a. Your cottage is twice the home that her bungalow is.

b. The red plane is three times the plane that the green plane is.

c. This knife is ten times the knife that that one is.

d. Floyd is twice the football player that Clyde is.

(26)

a. *Floyd is twice a football player that Clyde is.

b. ?Floyd is twice the football player.

c. *Floyd is twice a football player.

d. *Floyd is twice all other football players.

e. *Floyd is twice any other player.

f. ?Floyd is twice their best player.

To the extent that (26b) is good, it feels like there’s an unexpressed context that’s still syntactically

present, like we’ve elided the rest of the relative clause. (26f) is more interesting: perhaps the

presence of the superlative is providing a set of individuals, such that the factor phrase is modifying

the member of the set picked out by the superlative; this might therefore be taking the place of the

relative clause. But note that things get slightly odd when different superlatives are used:

(27)

a. ??Floyd is twice their worst player.

b. ??Floyd is twice their oldest player.

c. ??Floyd is twice their baldest player.

The extra flavor of strangeness seems to come from a sense of choosing an odd highest degree to

measure against: it’s not clear what the advantage of being twice as good as their baldest player

could be, but one could in principle use that as the basis for a comparison of quality.

But then that’s a point: note that in all of these cases (not just the superlatives but the standard

cases in (25) as well), one could imagine that we’re trying to obtain a meaning related to something

such as size or baldness – this might be twice the home because it’s physically bigger, or twice

their baldest player might mean twice as bald. But instead the only available reading is the quality

133

reading, and the size meaning (e.g.) is only good to the extent that one thinks a bigger home is a

better home; you can’t select for only a size reading:

(28)

#Your cottage is twice the home that her bungalow is, but her bungalow is nicer.

Similarly, you can’t make (29) be about the number of frames:

(29) The 48 frame-per-second version of The Hobbit: An Unexpected Journey is twice the film

that the 24 frame-per-second version is.3

We could literally take all the frames of each version, print them out, and point out that there are

twice as many as frames in the 48 fps version as in the 24 fps, but that’s still not a possible meaning

for (29); it can only be about its quality as a movie, not its length or frame rate. And in fact you

can switch the arguments around and still have a perfectly acceptable sentence, no contradictions:

(30) The 24 frame-per-second version of The Hobbit: An Unexpected Journey is twice the film

that the 48 frame-per-second version is.

But there is a way to get the amount reading to come out: change is to has:

(31) The 48 frame-per-second version of The Hobbit: An Unexpected Journey has twice the

film that the 24 frame-per-second version has.

Quality readings therefore seem tied to BE + RC, not HAS + RC. HAS + RC therefore requires

either a plural noun or one that can be considered mass – otherwise the result becomes both silly

and possibly slightly horrific as the universal grinder takes over:

3Real life explanation for the curious: Standard film runs at 24 fps, but this is basically a
holdover from the days of projected sound-synched film (24 fps was the minimum rate that still
provided decent sound), so there’s no intrinsic need to stick to that now that we have digital video.
The Hobbit movies were filmed at 48 fps (because director Peter Jackson wanted to, basically), but
people complained that it looked too real and/or like live TV (because video runs at 30 fps (roughly
speaking), so we’re trained to associate faster frame rates with sports, news broadcasts, and soap
operas), so the format has yet to take off.

134

(32)

a. You’re twice the man your father is.

b. You have twice the men your father has.

c. #You have twice the man your father has.

So we have dimensional nouns, evaluative nouns, gradable nouns, and quality reading noun +

RC constructions. Now let’s cover non-abstract nouns. We can divide these into two categories,

mass and count. Non-abstract mass nouns generally pattern the same as evaluative nouns:

(33)

a. *Floyd is three times Clyde’s water.

b. Floyd has three times Clyde’s water.

c. *Floyd is three times the water that Clyde is.

d. Floyd has three times the water (that) Clyde has.

(34)

a. *This beach is twice the sand of that one.

b. This beach has twice the sand of that one.

c. *This beach is twice the sand that that beach is.

d. This beach has twice the sand (that) that beach has.

(35)

a. This park is twice the fun of that park.

b. ?This park has twice the fun of that park.

c. This park is twice the fun that that park is.

d. This park has twice the fun (that) that park has.

Fun seems to pattern with more closely with resolution in that it’s OK in the simple copula form

and the HAVE with a relative clause form, despite the fact that it’s not at all clear how one would

independently measure fun. (Smiles? Hugs? Memories?) Thus it seems that fun is somewhat on

the line between physical and abstract mass entities.

Non-abstract count nouns really only work with HAS + RC, although for plural nouns that

evoke senses of emotion, the plain BE and HAVE cases get a little better:

135

(36)

a. *This farm is twice the horses of that one.

b. ?This farm has twice the horses of that one.

c. *This farm is twice the horses (that) that one is.

d. This farm has twice the horses that that one has.

(37)

a. ?This haunted house is twice the thrills of that one.

b. ?This haunted house has twice the thrills of that one.

c. *This haunted house is twice the thrills that that one is.

d. This haunted house has twice the thrills that that one has.

(38)

a. ?This sequel is twice the laughs of the original.

b. ?This sequel has twice the laughs of the original.

c. *This sequel is twice the laughs that the original is.

d. This sequel has twice the laughs that the original has.

And note that for these emotion nouns, the plural is required: singular versions are universally

bad – a quality reading isn’t possible like we might otherwise expect.

has

 is
 is
 is
 is

has

has

has

 three times the thrill of that one.
 three times the thrill that that one
 is
 three times the laugh of that one.
 is
 three times the laugh that that one

has

has

 .
 .

(39)

a. *This show

b. *This show

(40)

a. *This movie

b. *This movie

We’ve had a lot of examples so far, so here’s a table to gather all the information:

136

Table 5.1: Nouns types and their interactions with BE and HAVE

Dimensional nouns (height)
Evaluative nouns (beauty)
Dimen./eval. noun VERB same noun
Gradable nouns (idiot)
Non-gradable “quality reading” nouns (film)
Concrete mass nouns (water)
Non-gradable count nouns (horses, thrills)
Fun/resolution

BE HAVE BE + RC HAVE + RC
(cid:88)
*
(cid:88)
*
*
*
*/?
(cid:88)

*
(cid:88)
*
*
*
(cid:88)
?
?/(cid:88)

*
*
(cid:88)
(cid:88)
(cid:88)
*
*
(cid:88)

?
(cid:88)
*
*
*
(cid:88)
(cid:88)
(cid:88)

Now we can see a pattern begin to emerge. Direct comparison with the copula is only used

either with dimensional nouns or when a dimensional or evaluative noun is explicitly compared

with that same noun; otherwise we get HAVE for our scalar nouns. Both our abstract mass nouns

(the evaluative ones) and the concrete mass nouns only work with HAVE. Nouns that don’t denote

(obvious) scales require a relative clause construction, although the meaning changes depending

on whether the noun is gradable or not. Non-gradable plural count nouns such as thrills work best

when HAVE is combined with a relative clause.

5.1.3 Additional distinctions of the different kinds of nouns

The dimensional nouns are associated with independent scales: we have agreed-upon units that

we can thus use to compare an attribute of two individuals. Consequently, quibbling about the

particular factor phrase used is acceptable:

(41) A: This box is twice the weight of that one.

B: No it’s not! It’s three times the weight!

By contrast, evaluative nouns don’t have agreed-upon independent units; thus, making a dis-

agreement that’s about the factor phrase seems odd:

137

(42) A: Agnes has twice Bertha’s beauty.

??B: No she doesn’t! She has three times her beauty!

Similarly, it’s also weird to quibble about the factor phrase when either a gradable noun or a

quality reading-evoking noun is used:

(43) A: Floyd is twice the idiot that Clyde is.

??B: No he’s not! He’s three times the idiot!

(44) A: The Boss Baby is twice the film that Tangled is.

??B: No it’s not! It’s three times the film!

It’s worth noting that this effect goes away somewhat if you use large factor phrases (in other

words, you’re exaggerating):

(45) A: Agnes has twice Bertha’s beauty.

B: No she doesn’t! She has a hundred times her beauty!

(46) A: Floyd is twice the semanticist that Clyde is.

B: No he’s not! He’s ten times the semanticist!

But generally speaking, quibbling about the factor phrase is only acceptable if the two speakers

can independently agree on what the subdivision of the scale is, which is usually achieved by

independent units. Note that if we have an agreed-upon scale (such as, say, IQ scores), quibbling

is allowed:

(47) Context: Floyd scored an 70 on his IQ test, while Clyde scored a 210.

A: Clyde has twice the intelligence that Floyd has.

B: No he doesn’t! He has three times the intelligence!

Another distinction is that some of these constructions can’t stand alone without the factor

phrase. Dimensional nouns are fine:

138

(48)

a. Floyd is Clyde’s height.

b. This kitchen is the size of that kitchen.

Evaluative nouns need some extra machinery; as is, the readings can come out odd:

(49)

a. ?Agnes has Bertha’s beauty.

b. ?This watch has the accuracy of that watch.

(49a), for instance, seems to suggest that Agnes is somehow sharing Bertha’s beauty, rather than

saying that have an equal degree of beauty. The noun BE noun construction doesn’t help matters;

there’s still a sense that they’re sharing the same portion of beauty:

(50)

?Agnes’s beauty is Bertha’s beauty.

Instead we need to say something like is equal to or the degree of, but only in certain constructions:

(51)

a. Agnes’s beauty is equal to Bertha’s beauty.

b. ?#The degree of Agnes’s beauty is the degree of Bertha’s beauty.

c. ?#The amount of Agnes’s beauty is the amount of Bertha’s beauty.

d. ?Agnes has the degree of Bertha’s beauty.

It’s actually slightly surprising that (51b) comes out strange – this could be because we don’t

typically think of evaluative nouns as actually being associated with degrees at all. But then the

fact that (51c) is weird is also strange, because we’re fine with (51a).

Gradable and quality reading nouns behave similarly to evaluative nouns; the constructions are

unacceptable without the factor phrase:

(52)

a. #Floyd is the idiot that Clyde is.

b. #The Boss Baby is the movie that Frozen is.

But we don’t have ways to express the meaning that don’t significantly alter the structure of the

sentences for these:

139

(53)

a. *Floyd is equal to the idiot that Clyde is.

b. ?Floyd is the degree of idiot that Clyde is.

c. Floyd is as much of an idiot as Clyde is.

(54)

a. *The Boss Baby is equal to the movie that Frozen is.

b. ?The Boss Baby is the level/degree of movie that Frozen is.

c. The Boss Baby is as good a movie as Frozen is.

Plural count nouns aren’t terrible when they’re made singular, but all of the slight changes seem

a bit off:

(55)

a. ?This ride is the thrill of that ride.

b. ?This ride is equal to the thrill of that ride.

c. ?This ride is the level of thrills of that ride.

d. This ride is

 as thrilling

as much a thrill

 as that ride.

Fun, by contrast, can’t stand alone, but our minor tweaks to get the desired meaning all work

here without issue:

(56)

a. #Disney World is the fun that Disneyland is.

b. Disney World is equal to the fun that Disneyland is.

c. Disney World is the level/degree of fun that Disneyland is.

d. Disney World is as much fun as Disneyland is.

So if we want to keep our denotation of factor phrases as a ratio degree of type (cid:104)d, d(cid:105), then
it can’t typically be a case of just modifying an inherent degree. Instead we’re going to have to

assume a covert degree that’s present somewhere.

140

5.2 Analysis

So now that we’ve sorted out these nouns into various categories, let’s take a closer look at some

of these categories. We’ll focus on dimensional nouns, evaluative nouns, and quality readings;

the other types (concrete mass nouns and gradable nouns) won’t be explored, although broadly

speaking they should be compatible with the analyses for evaluative nouns and dimensional nouns,

respectively.

5.2.1 Dimensional nouns

Let’s start with (57) as our standard example.

(57) Floyd is twice Clyde’s height.

The initial problem offhand with (57) is that we’ve only specified one height, that of Clyde.

We don’t have Floyd’s height available to compare against; instead it appears on face value that

we’re being asked to treat Floyd the individual as nothing more than just his maximal height.

Consequently, it looks like almost immediately this sentence should be a non-starter.

However, we could assume that there’s a measurement operator, one that associates a partic-

ular degree with a corresponding scale and subsequently combines with an individual. In some
respects, this measurement operator acts as a form of adjective, being of type (cid:104)d, et(cid:105) – but instead
of combining with a comparative phrase it combines with twice Clyde’s height:

(58)

Floyde

t

is

(cid:104)e, t(cid:105)

(cid:104)e, t(cid:105)

µ(cid:104)d,et(cid:105)

d

twice(cid:104)d,d(cid:105)

d

Clyde’s height

141

And the operator µ will have a denotation as in (59):

(59)

(cid:126)µ(cid:127) = λdλx[µ(x) = d]

This operator will allow us to shift from a degree to something capable of combining with an

individual. The restriction we’ll need to place on it is that it’s only compatible with dimensional

nouns and count nouns (with the count noun measurement simply being cardinality).

The final semantics will run as in (60):

(60)

a. (cid:126)Clyde’s height(cid:127) = µHT (Clyde)
b. (cid:126)twice(cid:127) = λd.2 × d
c. (cid:126)twice Clyde’s height(cid:127) = 2 × µHT (Clyde)
d. (cid:126)µ twice Clyde’s height(cid:127) = λx[µ(x) = 2 × µHT (Clyde)]
e. (cid:126)Floyd is µ twice Clyde’s height(cid:127) = 1 iff µ(Floyd) = 2 × µHT (Clyde)]

(60e) states that our sentence is true if and only if the measure of Floyd is equal to twice the degree

of Clyde’s height.

This doesn’t work that well – after all, how do we know that µ(Floyd) is measuring his height?

We’ll need a way to be able to pull the scale out from a degree. This in principle shouldn’t be

unreasonable, as degrees should be limited to their own particular scales: 6 feet, for example, is

only on the height/length scale, not the weight scale. In order to do this we’ll give our µ operator

a little more power: we’ll include a scale component that provides the scale of a given degree. So

we’ll define that as in (61):

(61)

scale(d) := the scale along which d is measured

Now our revised µ operator will be:

(62)

(cid:126)µ(cid:127) = λdλx[µscale(d)(x) = d]

This allows µ to ensure that the measure of an individual’s degree is along the proper scale. So the

now slightly altered version of (60e) is in (63):

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(63)

a. (cid:126)µ twice Clyde’s height(cid:127) = λx[µscale(2 × µHT (Clyde))(x) = 2 × µHT (Clyde)]

= λx[µHT (x) = 2 × µHT (Clyde)]

b. (cid:126)Floyd is µ twice Clyde’s height(cid:127) = 1 iff µHT (Floyd) = 2 × µHT (Clyde)]

Now the result states that the measure of Floyd’s height is equal to two times the measure of

Clyde’s height, which is indeed what we want.

An alternate approach is somewhat similar; it could be the case that (57) is underlyingly (64):

(64) Floyd’s height is twice Clyde’s height.

Here the semantics is pretty simple: we’re comparing height directly to height. So if Clyde’s

height is ultimately of type d, and Floyd’s height is also type d, then this will be reasonably

straightforward. Here’s one possible approach:

(65)

a. (cid:126)twice Clyde’s height(cid:127) = 2 × (µHT (Clyde))
b. (cid:126)Floyd’s height(cid:127) = µHT (Floyd)
c. (cid:126)is(cid:127) = λdλd(cid:48)[d(cid:48) = d]
d. (cid:126)Floyd’s height is twice Clyde’s height(cid:127) = 1 iff µHT (Floyd) = 2 × (µHT (Clyde))

Here we have is setting the two degrees equal to each other, with the result being that the sentence

is true if the measure of Floyd’s height is equivalent to the measure of Clyde’s height multiplied

by 2.

However, this approach has some flaws. First, it’s not at all clear how we could get to (57) to

(64) in syntactic terms; what would be the process that would elide both the initial height and the

possessive morpheme? There’s also the worry that these two sentences aren’t exactly equivalent;

there’s an odd sense that what (64) actually states is not that the the measure of Floyd’s height is

twice that of Clyde’s, but rather that Floyd’s height is literally Clyde’s height twice, as in the actual

object Clyde’s height, as if Floyd could somehow physically share the height that Clyde has. While

this isn’t the only reading present, it is a possible reading, and it’s not a reading that (57) seems to

possess. Thus, despite the slight (but necessary) arbitrariness of our µ operator, it still appears to

be our best bet for dealing with these sorts of sentences.

143

5.2.2 Mass nouns and property concepts

Let’s start by taking a quick look at factor phrases and mass nouns. When we utter a sentence such

as Floyd has three times Clyde’s water (or three times the water Clyde has), what we really mean

is something as in (66):

(66) Floyd has three times the amount of

 Clyde’s water

water that Clyde has

 .

So that leads us to a natural question: generally speaking, how do we portion out mass nouns

such as water? How do we discuss an amount of water? As we saw in chapter 2, mass nouns

can’t combine directly with numbers, i.e., *two water is bad. So if we want to have multiple water

portions, we say things as in (67):

(67)

two bottles/glasses/jugs/cups/etc. of water

Generally speaking, all of these nouns are a way of measuring quantity or amount. We don’t

have a way of naturally dividing water into atomic units, so we impose a way of doing it instead.

Chierchia (2010) suggests doing so via the use of partitions. We discussed partitions briefly in

section 2.1.2 when talking about the internal structure of complex numerals; now we’re going to

look at partitions more closely.

Chierchia (2010:120) defines a partition in the following way: a partition Π is any function of

type (cid:104)et, et(cid:105), such that for any property P, Π(P) satisfies the following requirements:

(68)

a. Π(P) ⊆ P+

A partition of P is a total subproperty of P (where P+ represents the positive extension

of P).

b. AT(Π(P)) = Π(P)

If x is a member of a partition of P, no proper part of x is (relative atomicity).

c. ∀x[Π(P)(x) → ∀y[Π(P)(y) → ¬∃z[z ≤ x ∧ z ≤ y]]]

No members of a partition overlap.

144

Words such as quantity or amount simply provide a measure of a portion of a mass noun. So

for instance, the phrase two quantities of water is realized as in (69):

(69)

(cid:126)two quantities of water(cid:127) = ∃x[µAT,Π(water)(x) = 2]

So a quantity of water is just the measure of a portion of water for some individual x.

Now if we want to have the quantity of water, we can just use the (more or less) standard

approach of using the ι-operator, which signifies a unique individual:

(70)

(71)

(cid:126)the(P)(cid:127) = ιP, where ιP = ∪P if ∪P ∈ P

(cid:126)the quantity of water(cid:127) = ιx[µAT,Π(water)(x)]

So (71) states that x is the unique individual that is the measure of a portion of water.

Now we need a way to apply this to an individual; after all, we don’t want a general quantity of

water, but rather a quantity of water that’s possessed by Clyde. Now that we’re adding both HAVE

and possessive morphemes (’s) into the mix, we need a way to talk about possession. In order to

handle this, it will be useful to discuss the idea of property concepts.

Property concepts are a way of describing adjectives (and consequently, related words such as

our scalar nouns). Dixon (1982) described seven broad categories of property concepts that ad-

jectives fall into: dimension (such as tall), age (young), value (good or bad), color (red), physical

(heavy or hard), speed (fast), and human propensity (happy or jealous). As we saw in section 5.1.1

above, Francez & Koontz-Garboden (2015) divide these property concepts into two broad cate-
gories: adjectivally denoting and substance denoting.4 The substance denoting property concepts

are what we’re going to examine now.

Francez & Koontz-Garboden (2015) point out that there are a number of languages that use

possessives when predicating a substance denoting PC lexeme. For instance, a statement such as

He is very clever is more literally translated in a language such as Hausa as (72):

4Francez & Koontz-Garboden (2017) calls substances “qualities”, but I’ll stick with “sub-

stance” to avoid confusion with the quality readings above.

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(72) àkwai
exists
‘He is very clever.’

dà
with

shì
him

way¯o.
cleverness

So therefore we should take seriously the idea that possession is in fact occurring here, and

while we have some cases in English where substance denoting PC lexemes combine with copulas,

this is more a quirk of English; consequently, the sentences in (73) should be treated as (more or

less) equivalent in meaning:

(73)

a. Kim is wise.

b. Kim has wisdom.

And since our evaluative noun constructions involve HAVE, it looks like we’re indeed dealing with

substance denoting property concepts.

We’ve been tossing this term substance around a bit, but what does it actually mean? For

Francez & Koontz-Garboden, a substance is an abstract mass entity, such as wisdom or strength.

This substance is divided up into portions, and those portions can be possessed by individuals –

so much how a person can possess a portion of the substance water (and the substance water is

therefore the set of all portions of water), a person can also possess a portion of, e.g., beauty. This

therefore is working in the same realm as Chierchia (2010) and the idea of partitions.

As a substance is an abstract mass entity, we’ll define it the same way as other mass entities.

Francez & Koontz-Garboden provide the formal definition of a substance as given in (74).

(74) For any portions p, q ∈ A (where A is a non-empty set of portions), p (cid:22) q ⇔ p (cid:116) q = q

This states that for any portions p and q in A, p is a part of q if and only if the mathematical join
(maximal element, roughly) of p and q is q.

We also want substances to be distinct from each other; we don’t want something to be both

wisdom and beauty, just as we wouldn’t want a concrete mass noun to be both water and milk

(say).

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In addition, any substance P that’s a subset of A is ordered by a total preorder ≤, and the
preorder ≤ preserves the mereological part-of relation, so that given a substance P, and two portions
p, q ∈ P, then p (cid:22) q → p ≤ q. It’s the first of these two that will prove crucial to analyzing property
concepts.

So in order to have a portion of a substance, we’ll need a denotation as in (75):

(75)

(cid:126)wisdom(cid:127) = λp.wisdom(p)

In other words, give me a portion p and I’ll give you a portion of wisdom. So far so good. Now

how do we relate that to an individual? Via a possessive relation:

(76) For any individual a and substance P, a has P iff ∃p[P(p) ∧ π(a, p)]

“For any individual a and substance P, a has P if and only if there is a portion p that is part

of P and there’s a possessive relation (π) between a and p.”

Now we can encode (76) into a possessive denotation:

(77)

(cid:126)poss(cid:127) = λPλxλD.∃Dz[P(z) ∧ π(x, z)]

(77) states that given a substance P, an individual x, and a set of portions D, there exists an indi-

vidual z (restricted to elements of D – in other words, a portion) such that z is a portion of P and

there’s a possessive relation between z and an individual x. As portions are related to mass nouns,

we want portions to themselves be individuals – just individuals restricted to the set of portions,

instead of just any individual.

When we put these pieces together for (73b), we get (78d):

(78)

a. (cid:126)wisdom(cid:127) = λp.wisdom(p)
b. (cid:126)has(cid:127) = λPλxλD.∃Dz[P(z) ∧ π(x, z)]
c. (cid:126)has wisdom(cid:127) = λxλD.∃Dz[wisdom(z) ∧ π(x, z)]
d. (cid:126)Kim has wisdom(cid:127) = λD.∃Dz[wisdom(z) ∧ π(Kim, z)]

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Our final result states that Kim has wisdom if and only if there exists in the domain of portions a
portion z that is a portion of wisdom, and Kim is in a possessive relation with that portion z.5

Broadly speaking, Francez & Koontz-Garboden (2015, 2017) and Chierchia (2010) are getting

at the same thing: the idea of portioning off parts of a mass noun, whether that’s a concrete mass

noun such as water or an abstract one such as wisdom. But one thing we need that Francez &

Koontz-Garboden leave largely unexplored is the idea of comparing specific sizes of portions.

They’re comfortable with the idea that there are different sizes of portions – indeed, they observe
that the ≤ ordering relation that substances undergo requires this idea of different sizes. And they’re
able to perform comparative operations using subset relations:

(79)

(80)

(cid:126)more(cid:127) = λαλxλy.α(y) ⊂ α(x)
where α is the type for PC words
(cid:126)more wisdom(cid:127) = λxλy.(cid:126)wisdom(cid:127)(x) ⊂ (cid:126)wisdom(cid:127)(y)
= λxλy.[λuλD.∃zD[wisdom(z) ∧ π(u, z)]](x) ⊂ [λuλD.∃zD[wisdom(z) ∧ π(u, z)]](y)
= λxλy.{D : ∃zD[wisdom(z) ∧ π(x, z)]} ⊂ {D : ∃zD[wisdom(z) ∧ π(y, z)]}

But how do we actually obtain a specific size to compare with? We need a way to measure the

size of the portion, in order to have a factor phrase make a comparison between the sizes of two

different portions. It’s not enough to say that in Floyd has three times Clyde’s intelligence, Clyde’s

intelligence is a subset of Floyd’s intelligence; we need a way to explicitly compare the sizes.

If we look again at (78), we see that the ultimate result is that Kim possesses wisdom. Another

way to express this same idea, however, is via the possessive ’s marker. In other words, Kim’s

wisdom should come out broadly similar to Kim has wisdom; the main difference is that we want

Kim’s wisdom to ultimately be an individual of type e, rather than a full sentence (type t).
In
order to do this we’ll start by making Kim’s wisdom a property of type (cid:104)e, t(cid:105), which can then be
subsequently modified by a null determiner that provides an ι operator, making the whole thing of

type e – this way Kim’s wisdom will be essentially equivalent to the wisdom of Kim. So let’s tweak

5The λD gets existential closure further up the tree.

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our possessive morpheme a bit to be a property: instead of saying that z is a portion that exists,

we’ll just have it look for a portion instead – so a λ operator:

(81)

(cid:126)’s(cid:127) = λPλxλDλzD[P(z) ∧ π(x, z)]

So now Kim’s wisdom will be as in (82):

(82)

(cid:126)Kim’s wisdom(cid:127) = λzD{D : wisdom(z) ∧ π(Kim, z)}

And thus Kim’s wisdom is looking for a portion z that is a portion of wisdom and is possessed by

Kim. Now we add a null determiner, which will provide the ι operator:

(83)

a. (cid:126)Ø(cid:127) = λF(cid:104)e,t(cid:105).ιF
b. (cid:126)Ø Kim’s wisdom(cid:127) = ιzD{D : wisdom(z) ∧ π(Kim, z)}

(83b) thus gives us the unique individual z (which is limited to the domain of portions D) that is a

portion of wisdom and is possessed by Kim.

Now that we’ve gotten this far, it’s time to bring in our factor phrase. One approach is to use
the same ratio degree that we’ve been using. That will be of type (cid:104)d, d(cid:105), however, so we’ll need a
way to have it combine with our individual portion Kim’s wisdom. Consequently, we’ll introduce

a covert degree morpheme that simply provides the measure of the portion. We have independent

evidence that we need a way to discuss the size of a portion – the fact that we can say things like

the amount of Kim’s wisdom is an indication of this, and the meaning of that is the same meaning

that’s needed when a factor phrase combines with, e.g., Kim’s wisdom. Thus, the sentences in (84)

mean the same thing:

(84)

a. Sally has three times Kim’s wisdom.

b. Sally has three times the amount of Kim’s wisdom.

So let’s introduce a way to move from an individual portion to a degree, a piece that’s type (cid:104)p, d(cid:105).
Here we’ll borrow from Chierchia (2010), which will result in (85):

(85)

(cid:126)amt(cid:127) = λx.µD(x)

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amt simply takes an individual portion and thus provides the measure (in the domain of portions)

of that portion. So when it combines with Kim’s wisdom, we get (86):

(86)

(cid:126)amt Ø Kim’s wisdom(cid:127) = µD(ιzD{D : [wisdom(z) ∧ π(Kim, z)]})

This is now a degree that three times can combine with:

(87)

a. (cid:126)three times(cid:127) = λd.3 × d
b. (cid:126)three times amt Ø Kim’s wisdom(cid:127) = 3 × µD(ιzD{D : wisdom(z) ∧ π(Kim, z)})

This states that three times Kim’s wisdom is a degree that is the measure of the unique portion of

wisdom that Kim possesses, multiplied by 3.

Of course, now we have a bit of a concern. We want three times Kim’s wisdom to itself be

a portion of wisdom that Sally possesses. We could of course type-shift back from a degree to a

portion, but now it feels like we’ve made a shift back and forth somewhat unnecessarily.

What if, instead of saying that three times is rigidly of type (cid:104)d, d(cid:105), we made it more flexible?
As Francez & Koontz-Garboden (2017) argue, portions are totally ordered with regard to their

size. Furthermore, Francez & Koontz-Garboden note that there’s a great deal of overlap between

substances and portions and scales and intervals in the degree sense. Substances and scales both are

ordered sets, and if we adopt an interval-based approach to degrees then intervals can also overlap

with each other in the same way that portions of substances can.

Indeed, Francez & Koontz-

Garboden note that if an interval-based approach were used, their use of substances and portions

could be replaced with scales and intervals without too much violence. The reason they opt for

substances has to do with sentences such as (88):

(88) The Taj Mahal has as much beauty as the Stata Center, though their beauties are very

different.

Substances allow for portions of equal size but which are themselves different portions; thus, the

portions are in the same position in the preorder but are not identical. (This is a similar idea to that

150

of tropes, as in Moltmann 2009.) This would be a difficult meaning to achieve if substances were

treated just like scales.

However, the fact that, in terms of ordering, portions behave like intervals means we can ma-

nipulate them in the same way. We can order substances such as beauty, we can assign them units

if we so choose (such as rating a person’s beauty on a scale from 0-10, as at a beauty contest, for

instance), and thus we can also manipulate them mathematically. In other words, we can multiply

portions by factors to get larger portions, just as we can multiply degrees by factors to get larger

degrees.

So now let’s put this into practice. We’ll alter the semantics of our factor phrase to now be (for

lack of a better term) a ratio portion. Thus our factor phrase in this case will be of type (cid:104)p, p(cid:105):

(89)

(cid:126)three timesportion(cid:127) = λp.3 × p

This will combine with our piece in (83b):6

(90)

(cid:126)three times(cid:127)((cid:126)Ø Kim’s wisdom(cid:127)) = λp.3 × p (ιz[wisdom(z) ∧ π(Kim, z)])

= 3 × (ιz[wisdom(z) ∧ π(Kim, z)])

Now we need another possessive morpheme, for has. This one will be slightly different from

our possessive ’s marker; its denotation will be as in (91):

(91)

(cid:126)has(cid:127) = λrλx.∃s[π(x, s) ∧ r ≤ s]

This simply states that, given a portion r, there exists a portion s that’s in a possessive relationship

with an individual x, and the portion r is less than or equal to the portion s that x possesses. This

is in line with Francez & Koontz-Garboden (2017), who note that we have constructions with

multiple possessives floating around, and so we can combine these by creating an explicit ordering

relation between two portions. Thus, when we combine has with 3 times Kim’s wisdom, we get

(92):

6I’m suppressing the variable D representing the set of portions for the sake of readability, as

it’s not crucial to the rest of the discussion.

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(92)

(cid:126)has(cid:127)((cid:126)three times Ø Kim’s wisdom(cid:127)) = λrλx.∃s[π(x, s) ∧ r ≤ s]

= λx.∃s[π(x, s) ∧ [3 × (ιz[wisdom(z) ∧ π(Kim, z)])] ≤ s]

(3 × (ιz[wisdom(z) ∧ π(Kim, z)]))

This looks complex, but all we’re saying is that there’s a portion s that’s possessed by x, and s is

ordered such that it’s at least the same size as the portion which consists of 3 multiplied by the

unique portion that’s possessed by Kim and is a portion of wisdom. And because these portions

are ordered, s must necessarily also be a portion of wisdom, according to the definition of being

a substance and what it means to be ordered as a substance, as laid out by Francez & Koontz-

Garboden (2017:39).

At this point all that’s left is to combine our individual, Sally, with the rest of the sentence.

That’s accomplished as in (93):

(93)

(cid:126)Sally has three times Ø Kim’s wisdom(cid:127)

= 1 iff ∃s[π(Sally, s) ∧ [3 × (ιz[wisdom(z) ∧ π(Kim, z)])] ≤ s]

The sentence is true if there’s a portion s that’s shared by Sally, and that portion is at least 3 times

the size of the unique portion z that’s a portion of wisdom and which is possessed by Sally. And

once again, because z is a portion of wisdom, s must also be a portion of wisdom. In other words,

this works exactly the way we want it to.

5.2.3 Quality readings

What to do with the quality readings that we get in sentences such as (94)?

(94) Floyd is three times the linguist (that) Clyde is.

Here we get a singular count noun as the head of the relative clause instead of a mass noun. Now

one could imagine treating linguist as a single entity and then trying to get a cardinality reading

from that – in other words, that Floyd were somehow two Clyde-linguists (perhaps stacked on top

of each other under a trenchcoat) – but that’s decidedly not a possible reading. Instead, we take

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Clyde’s quality as a linguist and place it on a scale of quality for linguists, ranking linguists based

on how good they are at linguistics. How do we get this reading?

One way might be to say that the phrase the linguist (that) Clyde is moves from being an entity

of type e to a degree. Similarly to what we did with our dimensional nouns in (60), we can argue

that there’s something that takes in an entity and maps it to a scale of quality, resulting in a degree:

(95)

(cid:126)µquality(cid:127) = λx[µquality(x)]

I’m going to set aside the potentially complicated semantics of the relative clause and merely

assume for the sake of clarity that it has a denotation as in (96):

(96)

(cid:126)the linguist Clyde is(cid:127) = [ιx.x is a linguist ∧ x = Clyde]

This then combines with (95) to get (97):

(97)

(cid:126)µquality the linguist Clyde is(cid:127) = [µquality(ιx.x is a linguist ∧ x = Clyde)]

This is now a degree that twice can manipulate. Now we’ll use our µ operator from (62) to make

this combinable with our individual Floyd:

(62)

(98)

(cid:126)µscale(cid:127) = λdλy[µscale(d)(y) = d]
a. (cid:126)twice µquality the linguist Clyde is(cid:127) = 2 × (µquality(ιx.x is a linguist ∧ x = Clyde))
b. (cid:126)µscale twice µquality the linguist Clyde is(cid:127)

= λy[µquality(y) = 2 × (µquality(ιx.x is a linguist ∧ x = Clyde))]

But if we continue down this course, we’re going to run into a couple problems, as (99) will

hopefully make clear:

(99)

(cid:126)Floyd is µscale twice µquality the linguist Clyde is(cid:127)

= 1 iff [µquality(Floyd) = 2 × (µquality(ιx.x is a linguist ∧ x = Clyde))]

The first problem is that (99) merely states that the measure of Floyd in a general quality sense

is better than Clyde as a linguist – in other words, no claims are made that Floyd is better as a

153

linguist, just that Floyd is generally better. This may in fact be true, but it’s not what we want the

meaning to be. A related problem is that not only is Floyd not measured on the linguist quality

scale, but we don’t actually limit x to only linguists in the first place. In other words, we don’t

want (100) to come out acceptable:

(100)

#Star Wars is twice the linguist Clyde is.

This is, of course, because (101) isn’t a valid sentence: movies can’t be linguists.

(101)

#Star Wars is a linguist.

So it seems like part of the issue is our underlying sentence; we need a way to incorporate the

fact that our subject is also a linguist. Roughly speaking, we want the meaning of (94) to be as in

(102):

(102) Floyd as a linguist is twice as good (at being a linguist) as Clyde as a linguist.

In other words, it looks like we’re restricting the individual to a particular aspect of their being.

This phenomenon is sometimes called “restricted individuals” (Landman 1989b), “guises” (Jäger

2003), “qua-sentences” (Szabo 2003; Asher 2006), or “role nouns” (Zobel 2017). The topic of

role nouns is a very rich and complicated area and could be the subject of a dissertation itself;

consequently, it will be beyond the scope of the current work. Let’s instead limit ourselves to a

general discussion of what we want to have happen.

The idea at its core is that individuals can be restricted in some way; Landman (1989b) observes

the following as evidence for this:

(103)

John is thoroughly corrupt, but as a judge he is trustworthy.

The fact that we can state this without it being a contradiction indicates that there are different

aspects to an individual: while the individual John himself may be corrupt, it’s when he’s restricted

to his judge role that he’s trustworthy. Similarly, if we know both (104a) and (b), (c) is not a valid

inference:

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(104)

a. The judge is on strike.

b. The judge is the hangman.

c. #Therefore, the hangman is on strike.

This because, while our judge in (104) may be on strike as a judge, it does not mean that he’s on

strike as a hangman, and so therefore we cannot conclude (104c).

Zobel (2017) argues that roles in fact are in their own domain (Dr) and thus have their own

type (r), separate from individuals of type e. Consequently we get effects such as in (104) because

a predicate such as is on strike is sensitive to roles; a predicate that isn’t sensitive to roles, such as

is tall, will allow the inference to go through:

(105)

a. The judge is tall.

b. The judge is the hangman.

c. Therefore, the hangman is tall.

Since is tall isn’t sensitive to roles, (105c) is a valid inference.

So what we want our factor phrase in (94) to modify is a degree on a quality scale for roles.

This is a measurement of how much an individual matches the ideal properties of the role of being

a linguist. Note that even with nouns that aren’t obviously role nouns (what Zobel 2017 calls a

class noun), the nouns are type-shifted to behave like roles, as in (32a), repeated below:

(32)

a. You’re twice the man your father is.

This statement isn’t measuring how good you and your father are at being physically male; it’s

instead comparing the (positive) qualities we associate with manhood (honor, strong ethics) as a

result of a particular societal role. And if we take a word that doesn’t have strong connections to

being possible as a role, such as person, the result is slightly strange, as if we’re not quite certain

how to interpret the sentence:

(106)

?You’re twice the person your father is.

155

Is this meant to compare qualities of being a person (as far as that’s a meaningful role)? Is it a size

comparison, akin to twice as big? We’re not sure.

And while roles combine with individuals, those individuals don’t need to be people. After all,

we can meaningfully utter sentences such as (107):

(107) The Birth of a Nation as a technical achievement is a landmark in cinema, but as a piece of

entertainment it’s repellent.

And so therefore our sentences in (29)/(30), where we compared the quality of different versions

of the movie The Hobbit: An Unexpected Journey, still can be measured along the quality of a

particular role (in that case, something along the lines of as a piece of entertainment).

So it looks like the factor phrases in these cases are modifying the measure of the quality of a

role as that role pertains to an individual. The specifics of how that will work will be best left at this

point to future research, but the intuition expressed in (102) appears to be the correct interpretation

of (94): a modification of a measure of quality that’s sensitive to specific roles (and consequently

the individuals who can possess those roles) that hold of given individuals.

5.3 Conclusion

In this chapter we’ve ventured into the previously unexamined area of factor phrases in the

nominal domain. We’ve seen that despite the large number of different domains that these can

occur in, there are, broadly speaking, three distinct groups: dimensional nouns and gradable nouns,

which provide a scale that the factor phrase can directly interact with, without the need for a

possessive morpheme; mass nouns, both concrete mass nouns and abstract substances, which are

divided into portions (ordered sets of subparts of the mass noun), the size of which can be modified

by our factor phrase; and quality readings, in which factor phrases modify a measure of quality for

a particular role noun.

We then provided analyses for dimensional nouns (in which our factor phrase modified the

provided scale) and evaluative nouns, which required the use of property concepts (as discussed in

156

Francez & Koontz-Garboden 2015, 2017), where abstract nouns are represented as substances that

can be divided into portions of varying sizes, similar to how concrete mass nouns can be partitioned

into smaller pieces. This necessitated a slight reevaluation of our factor phrase; instead of being
simply a ratio degree from type (cid:104)d, d(cid:105), we made it more type-flexible and thus also possibly of type
(cid:104)p, p(cid:105), where p is the type of portions. As portions can be ordered and sized in the same way as
degrees-as-intervals, this is a natural move.

We also discussed the case of quality readings, where we determined that these are in fact

sensitive to roles (as discussed by Zobel 2017); thus, factor phrases here measure roles as held

by a given individual along a scale of quality, where the determination is how good an individual

is along an ordering of that particular role. We’ve left the specifics of the semantics for future

research, as the task of incorporating role nouns into our semantics is likely to be a complex

endeavour.

But as a whole, we’ve seen that even in a domain as diverse as the nominal domain, with many

different potential interpretations of the NPs, factor phrases can be still operate as ratio degrees,

behaving in the same way as we saw in the adjectival domain and the multiplicative forms in the

verbal domain. There’s a consistency here that’s consequently satisfying, without needing to resort

to several ambiguous forms of factor phrases in order to operate in different areas; instead, our

denotation for factor phrases remains, happily, simple and unchanged.

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CHAPTER 6

THE SYNTAX AND SEMANTICS OF ARITHMETIC

6.1 Introduction

Now we reach another underexamined corner of natural language. The language of mathe-

matics is an area that has been explored in some ways, but not in others. While – perhaps under-

standably – little has been done with mathematical jargon beyond acknowledging its existence, and

while a great deal has been done with numbers themselves, there’s very little in the way of examin-

ing the interface between the two realms: that is, an analysis of basic arithmetic, simple operations

such as addition and subtraction, which are likely parts of natural language, that overlap with the

less natural, more constructed forms of mathematical jargon. To the extent which these construc-

tions are discussed at all, it’s an examination of their semantics, and even then largely only in pass-

ing (as in, for example, Bierwisch 1989, who as we saw notes that comparative measure phrases

such as Hans is 20cm taller than Eva involve addition, while factor phrase constructions such as

Hans is twice as tall as Eva involve multiplication, but has no further discussion of how addition

or multiplication are in fact derived in the system). Meanwhile, there seems to be no discussion of

the syntax and semantics of these arithmetic constructions, leaving a somewhat noticeable gap in

the analysis of related phenomena such as numbers.

This chapter will thus redress the balance, providing a syntactic and semantic analysis of sen-

tences such as in (1) and (2):

(1)

a. Six plus two is eight.

b. Ten minus three is seven.

c. Two times eight is sixteen.

158

(2)

a. Six added to two is eight.

b. Three subtracted from ten is seven.

c. Two multiplied by eight is sixteen.

d. Fifteen divided by three is five.

The aim of this chapter is to provide a detailed examination of these two types of sentences: the

simple versions in (1), with the specific vocabulary terms plus, minus, and times; and the verbal

versions, with the phrases added to, subtracted from, multiplied by, and others.

This chapter will therefore explore these two related constructions, starting with the syntax of

sentences such as (1), focusing on the syntactic category of these arithmetic operators (as even

that is still to be settled) and ultimately arguing that they are prepositions. This will be followed

by an analysis of the structure of the prepositional phrase, as these things combine with numbers,

and what the ultimate syntactic category of the full phrase is. Next I’ll present an analysis of

sentences such as in (2), demonstrating that these are in fact passivized reduced relative clause

constructions, with an appropriate syntax provided. Then we’ll look at the semantics of our mono-

morphemic operators, with a focus on times, and we’ll explore some of the issues concerning

combining arithmetic with natural language.

6.2 The syntactic category of arithmetic terms

As noted, there’s not much in the way of research into mathematics as part of natural language;

there’s a fair amount of research into numbers, as we saw in section 2.1.2, but not as they pertain to

larger arithmetic operations – to the point that it’s not even clear what syntactic category words such

as plus and times (both in the arithmetic sense) even are. There have been a few passing references

– for instance, in an extensive list of English prepositions, Pi (1999) categorizes plus, minus, and

times as clusion prepositions but doesn’t press the issue further, while Ionin & Matushansky (2006)

mention these phrases in an appendix that focuses on what these uses mean for their numeral

theory, rather than how they’re constructed syntactically. However, there doesn’t appear to be any

159

significant prior work that focuses in on these constructions. So let’s begin by determining what

categories these arithmetic terms are. Let’s start with the data in (3):

(3)

a. Six times three is/equals eighteen.

b. Six plus three is/equals nine.

c. Six minus three is/equals three.

Setting aside for the moment the question of what syntactic category numerals are, let’s focus

on the syntactic category of these arithmetic operators. This seems on the surface like it should be

straightforward, but closer examination reveals it’s not actually clear. So let’s start by examining

some of the possibilities, from perhaps the least likely to the most.

6.2.1 Verb?

The idea that these words are verbs is perhaps the least likely possibility, but it’s not completely

outlandish: we know that time, for instance, can be used as a verb, as seen in (4).

(4)

a. At each track meet, Floyd times Clyde during his 100 meter dash.

b. Agatha precisely timed her entrance.

Admittedly, (4a) requires this extra bit of context to work; it’s a bit awkward out-of-the-blue,

but a reasonable context, such as a coach describing tasks, makes this perfectly acceptable. Does

this mean that arithmetic statements are always present tense verbs, and the -s that we see on the

ends of these math terms are simply 3rd-person present tense agreement? After all, we wouldn’t

expect numerals to be expressed as anything other than 3rd-person.

But, of course, the obvious objection that, while this may be true for the verb time, there are no

verbs *plu or *minu to even provide a starting place for the other math terms. And, perhaps more

damningly, (5) isn’t a complete sentence:

(5)

*Two times four.

160

Nor do these words inflect, take modals or adverbs, or any of the other things we would expect a

verb to do.

(6)

a. *Two timed four.1

b. *Two has timed four.

c. *Two may time(s) four.

d. *Two precisely times four.

Thus, perhaps unsurprisingly, the verb possibility fails fairly definitively.

6.2.2 Noun?

Perhaps these arithmetic operators are nouns. After all, consider (7).

(7)

a. Floyd walked the dog two times.

b. Agnes changed her alternator one time.

If the times of multiplication is the same times in (7), we should expect a singular/plural distinction.

But, in English at least, we don’t actually see any morphological differences:

 one

two

 times four

(8)

a.

b. *one time four

However, this might simply mean that these words don’t have any morphological distinction

between singular and plural cases. If this is the case, then we might expect to see differentiation in

other languages – and in fact, this seems to be the case in Brazilian Portuguese (João Mattos, p.c.):

1There’s a colloquial version of this that’s actually good:
(i) Floyd times’d two by four to get eight.
But note that this version of times requires an agent (along the lines of multiply), and that the -s

remains intact here. A version without a permissable agent is bizarre:

???Two times’d four to get eight.

(ii)
This will be discussed further in section 5.

161

(9)

a. Dois

vez-es
time-pl

três
two(m.)
three
Two times three is six.

é
copula

seis.
six

b. Uma

vez-Ø
time-sg

três
three

one(f.)
One times three is three.

é
copula

três.
three

However, this isn’t the case for other languages. Spanish, for instance, not only shows no

alternation between potential singular and plural agreement in its version of times, but the word

that is used, por, is a common Spanish preposition, not a noun.

(10)

por
by

doce.
a. Tres
three
twelve
Three times four is twelve.

cuatro
four

son
are

por
by

b. Uno
one
One times four is four.

cuatro
four

son
are

quatro.
four

And, perhaps more intriguingly, for many speakers of Brazilian Portuguese (9b) is actually some-

what uncommon; they’re more likely to use the “incorrect” version in (11), which patterns the

same as (9a):

(11) Um

vez-es
time-pl

três
three

one(m.)
One times three is three.

é
copula

três.
three

This could be evidence that vezes once began as a noun but is now some other category, akin to the

English version times, and that the vez/vezes distinction for some speakers lives on as a prescriptive

alternation. This could also be evidence of a change in progress, with vezes losing a singular/plural

distinction in this context. But the takeaway is that Brazilian Portuguese isn’t a slam dunk either

for or against the possibility that these terms are nouns.

A perhaps more compelling argument is the independent fact that nouns don’t take bare com-

plements (that is, complements that aren’t headed by a preposition). So if times is a noun in (8), we

162

would need to explain what four is doing. One possibility is that times four is actually a compound

noun – but if that’s the case we should expect it to be modifiable by an adjective, similar to (12):

(12)

a. wooden baseball bat

b. expensive movie theater

c. easy math problem

But any effort to modify times four fails.

(13)

a. *two precise times four

b. *seven rational plus three

So it would appear that these terms are not likely to be nouns either.

6.2.3 Conjunction?

Let’s consider the possibility that these arithmetic words are conjunctions. There may be some

supporting evidence for this; after all, we can get conjunctions in some parts of arithmetic, and

even in the same location:

(14) Four and seven is/makes eleven.

And in fact plus works as a conjunction in some cases:

(15) This book got great reviews, plus it’s on sale.

So could it be that these operators are conjunctions?

Before we continue, we should take the time to finally tackle the question of what syntactic

type numbers are. As noted back in chapter 2, there is some literature about the syntactic category

of numbers that we can use to our advantage. Recall that Zweig (2005) argues that the numbers

one through nine are adjectives, while higher number terms such as hundred or million are nouns.

However, Zweig also argues that one through nine behave like nouns when not combined with

an overt noun, such as hundred or books. And in fact (16) shows that, setting aside the internal

structure, numbers are ultimately DPs:

163

(16)

a. Six plus three is nine.

 That number

Six

 plus

 three

that number

 is nine.

b.

c. Any odd number times two is an even number.

Intriguingly, it also seems to be the case that the whole math constituent is a DP, as seen in (17):

(17)

a. Floyd evaluated six plus three.

b. Floyd evaluated the calculation.

So if numbers are DPs, and the arithmetic phrase is also a DP, then perhaps words like plus

being conjunctions would buy us this fact, similar to something like Floyd and Clyde.

However, the correspondence between plus and and isn’t as close as we might hope. So, for

instance, while it’s not clear if you can stack conjuncts with arithmetic the way that and can:

(18)

a. Floyd, Clyde, and Agnes went to the store.

b. ?Three, six, and nine is/are eighteen.

you definitely can’t stack conjuncts with the equivalent plus:

(19)

a. Three plus six plus nine is eighteen.

 six, plus

plus six,

 nine is eighteen.

b. *Three,

Of course, it may just be the case that when it comes to mathematics, you can only use and

with pairs of numbers. After all, (20) isn’t really any better than (18b):

(20)

?Three and six and nine is/makes eighteen.

A perhaps minor strike against these being conjunctions: although plus can be used to conjoin

full sentences (as we should expect from a standard conjunction), neither minus nor times can.

164

(21)

a. This book got great reviews, plus it’s on sale.

b. *This book is on sale, minus it got bad reviews.

c. *This book got great reviews, times it’s on sale.

While it’s not clear what (21c) would even mean, we could conceive of a meaning for (21b) of

something similar to but – yet this sentence is decidedly unacceptable.

Of course, we could make the case that (21b) is just an extraposed form of (22), in which case

this wouldn’t be an actual CP conj CP use:

(22)

*This book minus the bad reviews is on sale.

And as we can see, the unacceptability of (21b) might be because it’s a version of a sentence that’s

also bad. But even an effort to create a true CP minus CP structure fails pretty miserably, even

though a comparable CP plus CP structure is fine:

(23)

a.

I made a lot of money at the bake sale, plus I had a great time with you.

b. *I made no money at the bake sale, minus I had a great time with you.

So it really doesn’t seem like minus can be used as a conjunction in the same way that plus

can. It’s also worth flagging that even the good version in (21a) is actually a fairly recent usage:

the Oxford English Dictionary’s first citation for this use of plus is only from 1963. This would be

quite surprising if the category of plus was naturally a conjunction. So while there does seem to be

some evidence pointing to these being conjunctions, it’s not particularly compelling, and it doesn’t

pan out in expected ways. So with that in mind, let’s move on to our fourth and final possibility:

prepositions.

6.2.4 Preposition?

Let’s start by noting that in certain contexts, plus and minus seem to work like standard preposi-

tions, as in (24):

165

(24)

a. The committee plus Floyd attended the meeting.

b. The band minus Clyde went to the award ceremony.

Times, by contrast, is restricted in this use to combining with numerals. There do seem to be some

cases where you can combine these with non-numerals to get metalinguistic readings, although the

object of times still needs to be a numeral.

(25) Floyd is basically MacGyver times seven.

This presupposes, of course, that (25) isn’t just the same mathematical phenomenon as the standard

arithmetic cases, extended to a non-numerical entity, which isn’t remotely clear. Could this perhaps

also be a form of our standard factor phrase? That would make it equivalent to (26):

(26)

?Floyd is basically MacGyver by seven times.

This sort of has the same flavor as (25), but it’s not quite the same; we want MacGyver to be

multiplied in some way in order to get to Floyd on an equivalent scale, but the presence of by both

makes this explicitly differential and slightly strange, as if we’ve somehow lost sight of what the

relevant scale to compared along is. So it doesn’t seem like (25) and (26) are paraphrases of each

other, which means we’re left with a possible metalinguistic use of arithmetic times.

So, the potentially problematic case of times aside, we have evidence that, in at least some

cases, these words can behave like prepositions in non-mathematical contexts. And indeed, this

isn’t a new phenomenon; the Oxford English Dictionary lists uses as early as 1802:

166

(27)

a.

It is...calomel, plus an insoluble subnitrate of mercury.2

b. His government was a system of Bashi-Bazoukery plus slave-raiding.3

c. Competitors offer the whole value of the produce minus that daily potatoe [sic].4

d.

It might be supposed...that acetic acid is alcohol minus carbon.5

Operating under the assumption for now that these operators are prepositions, let’s examine them

a little more closely. It doesn’t seem to be the case that these are modifying nouns, the way other

prepositional phrases (as in (28)) do:

(28)

a. The man with a beard

b.

DP

D
the

NP

NP

PP

N
man

P

with

DP

D
a

NP

N

beard

If these were like (28), we should perhaps expect them to be modifying a noun, akin to something

as in (29). But that doesn’t seem to be the case:

(29)

a. The man with a beard laughed.

b. The man laughed.

2from Philosophical transactions of the Royal Society of London, 1802
3from The Pall Mall Gazette, 1 March 1884
4from A sketch of the state of Ireland, past and present (2nd edition) by John Wilson Croker,

1808

5from Domestic Economy by Michael Donovan, 1830

167

(30)

a. Seven plus three equals ten.

b. #Seven equals ten.

What (30) shows is that these operators aren’t simply modifiers; instead, these seem to be

phrases headed by a PP, containing both a prepositional subject and object. This isn’t a million

miles away from a word like ago, which appears to be an intransitive preposition with a subject

(typically a measure phrase). Notably, ago can’t appear without a subject:

(31)

a. Floyd went to Texas three years ago.

b. *Floyd went to Texas ago.

Nor can you strand ago:

(32)

a.

It’s three years ago that Floyd went to Texas.

b. *It’s three years that Floyd went to Texas ago.

c. *It’s ago that Floyd went to Texas three years.

Interestingly, this seems to be similar to our math terms:

(33)

a. *It’s times three that eighteen equals six.

b.

It’s six times three that eighteen equals.

c. ?It’s three that eighteen equals six times.

So if we run with this idea, we should have a tree that looks something like (34):

(34)

PP

P(cid:48)

DP
six

P

times

DP
three

However, a problem arises. Recall (17), which is repeated for convenience below:

168

(17)

a. Floyd evaluated six plus three.

b. Floyd evaluated the calculation.

How do we account for the fact that we seem to be dealing with a DP, not a PP? Let’s re-

turn to Zweig (2005), building on an idea from Kayne (2005) that there are covert NUMBER and

AMOUNT nouns (depending on whether you’re count or mass) present in the structure of adjec-

tives such as few and many. As you’ll recall from back in section 2.1.2, Zweig argues that lower

quantity numerals have a covert NUMBER present, so that three books is actually three NUMBER

books, and that this NUMBER is a singular noun. We can therefore use the presence of NUMBER

to explain why our PP ultimately behaves like a DP: the phrase combines with this covert NUM-

BER to form a DP. This would mean that we treat a phrase such as six plus three as a type of a

complex numeral, albeit not of the same sort as two hundred (and) seven. So our arithmetic phrase

will be more like NUMBER six plus three.

At this point we should pause to consider our structural options, as we have a couple for a

phrase such as this. One possibility is to treat this like a small clause, similar to how people such

as Stowell (1981), Matushansky (2008), and Fara (2015), among others, treat constructions as in

(35):

(35) They named [SC the king Arthur].

If we assume that our arithmetic NUMBER phrase behaves similarly, we should then have a

small clause structure as in (36):

169

(36)

DP

D
Ø

NP

N

NUMBER

SC

DP

PP

D
Ø

NP

P
plus

N

NUMBER

AdjP

six

P(cid:48)

D
Ø

DP

AdjP

three

NP

N

NUMBER

It’s not completely clear if this is actually the structure we want, however; for one thing, small

clauses are typically (though not always) associated with a verb – either a verb of naming such as

call or name, as in (35), or a form of the copula (either explicit or implicit), as in (37):

(37) We considered [SC Floyd (to be) a fool].

Yet our arithmetic phrase doesn’t obviously involve either of these. To the extent that we can make

this NUMBER overt, it seems to resist combining with a form of the copula:

 *to be

*being

 [PP six plus three]

(38)

the number

Of course, this may simply be an inability to make NUMBER properly overt – we shouldn’t

necessarily assume that the number in (38) is the same as NUMBER.

However, if we do choose to continue down this path a bit, it leads to our second possibility for

the structure of NUMBER six plus three. If NUMBER can indeed be made overt, as in (39),

(39) The number six is my favorite,

then, as suggested by Moltmann (2013, 2015a, 2017), we may in fact be dealing with an appositive

structure, much as in (40):

170

(40)

a. The name “John”

b. The poet Goethe

c. The planet Jupiter

d. The movie “Back to the Future”

But then what precisely is the structure of the appositives in (40)? Moltmann (2015a) suggests

a structure like (41), where the name is contained inside a Quotation Phrase that can consist of any

material:

(41)

DP

D
the

NP

N

planet

QuotP

N

Jupiter

For Moltmann, quotation is used in the manner described by Saka (1998), where the quoted part

isn’t a referential expression but instead a way of “presenting” or mentioning the term, and which

can include not only the type of expression, but also its meaning and even its referent. According

to Saka, quotation is, semantically speaking, a concept or intension, QUOT, which “ambiguously

or indeterminately maps its argument expression X into some linguistic item saliently associated

with X other than the extension of X” (p. 127). For Moltmann, the non-quotational part is a sortal

head, the type of which will help determine what parts of the quotation are salient – so a sortal like

poet will select for different information from one like movie or name. Under this story, our tree

for NUMBER six plus three would be more like (42):

171

(42)

DP

D
Ø

N

NUMBER

NP

DP

QuotP

PP

D
Ø

NP

P
plus

N

NUMBER

AdjP

six

P(cid:48)

D
Ø

DP

AdjP

three

NP

N

NUMBER

Of course, this runs into the same concern that we had with (38): it’s not clear that overt number

is the same as NUMBER. It may therefore be helpful to examine NUMBER in a slightly different

context. One way to do this is to ask whether the six NUMBER that we see in arithmetic contexts

is the same as the phrase the number six – in other words, should we consider six in this arithmetic

context to be a name, à la Moltmann (2015a), or as an adjective, à la Zweig (2005)? Fortunately, as

you’ll recall Snyder (2017) points out, we can distinguish between the two uses in certain contexts:

b. Which one of these three numbers is even? Four

The examples in (43) show the adjectival, cardinality-denoting use, while (44) is the monadic,

name-based use. So what happens when we try to explicitly use the name-based version in our

basic arithmetic phrase? The result isn’t great:

172

(43)

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

b. How many of these eight numbers are even? Four

are
a. What’s the number Mary’s researching? (?Almost) four.

(44)

 *is
 is

*are

 .
 .

(45)

a. Six plus three is nine.

b. ?The number six plus the number three is the number nine.

However, we can make the adjectival version explicit without difficulty:

(46) Six apples plus three apples is nine apples.

This all indicates that Kayne (2005)’s NUMBER is not identical to the overt word number,

and that the version found in these arithmetic contexts is indeed the adjectival version. This may

therefore be an argument against treating NUMBER six plus three as an appositive, as (42) does.

But the confound is that, while it appears that the adjectival cases aren’t identical to the name-based

ones, that doesn’t necessarily have a bearing on NUMBER six plus three, as six plus three isn’t an

adjective either.

One third possibility could be that six plus three is simply a complement of NUMBER, similar

to how in our adjectival case the numeral is the sister of NUMBER. That would lead to the tree in

(47):

(47)

DP

D
Ø

N

NUMBER

NP

PP

six NUMBER plus three NUMBER

This would provide the advantage of being somewhat parallel to the adjectival structure while

being a fairly standard move for a prepositional phrase. The disadvantage is that this would lose

the idea that six plus three is a form of name for a particular numeral.

The key takeaway here is that all these potential structures involve NUMBER and six plus three;

there’s no argument about whether or not NUMBER is present, but merely (if such a word can be

used) how it combines with the arithmetic phrase. It’s perhaps a bit unfortunate that there doesn’t

seem to be a clear winner among the possibilities, but this is perhaps unsurprising when dealing

173

with a covert element. Regrettably, therefore, the specific realization seems to be a case best left

for future research.

Now let’s turn to one other issue that we’ve been avoiding until now: namely, treating six plus

three as a complex numeral is a bit odd. After all, (48) is rather strange:

(48)

?Floyd bought six plus three apples.

(=Floyd bought nine apples.)

But the problem with (48) seems to be more about violating a Gricean norm; given that we

have a much simpler way to express six plus three (i.e., nine), it’s odd to use the more complicated

version. But if we have a reason to use this more complicated version, this objection goes away:

(49)

a. The full ZIP code used by the US Postal Service consists of five plus four digits.

b. The best of 2014 in six books, one movie, one plus four podcasts, and three apps6

So while this structure might seem unwieldy on the surface, this unwieldiness is ultimately not

a syntactic concern, but rather a semantic one, and we’ve seen that that concern can be mitigated.

6.3 The syntax of multi-morphemic arithmetic terms

Having covered single arithmetic words, let’s now look at phrasal versions, as in (50).

(50)

a. Fourteen divided by seven is/equals two.

b. Seven multiplied by two is/equals fourteen.

c. Seven added to two is/equals nine.

d. Two subtracted from seven is/equals five.

e. Two raised to the 4th power is/equals sixteen.

Here now instead of just a single morpheme, we get a verb plus a preposition. These appear to

be standard prepositions, rather than our arithmetic ones, so that analysis won’t work here. Instead

6from

http://www.sandragulland.com/the-best-of-2014-in-six-books-one-

movie-one-plus-four-podcasts-and-three-apps/

174

we need a different approach. One move might be to treat the prepositional phrase as the external

argument of the verb in a reduced relative clause, similar to (51):

(51)

a.

b.

the cookies baked by the grandmother

the trophy won by the team

But, perhaps unsurprisingly (particularly given that we don’t get by in all cases), this prepositional

phrase is not the same by phrase as you get in a standard passive construction. This is demonstrated

in (52):

(52)

a. #Seven divided fourteen to get/equal two.

b. #Two multiplied seven to get/equal fourteen.

c. #Two added seven to get/equal nine.

d. #Seven subtracted two to get/equal five.

e. #The 4th power raised two to get/equal sixteen.

So then if these prepositional phrases aren’t the standard preposition plus postverbal DP that

comes with passives, what are they? They’re an object of the verb. We can tell they’re not adjuncts

because sentences without them are unacceptable:

(53)

a. *Fourteen divided is two.

b. *Seven multiplied is fourteen.

c. *Seven added is nine.

d. *Two subtracted is five.

Of course, just because these prepositional phrases are objects of the verb and not the passivized

by + DP phrase, that doesn’t mean these aren’t passive constructions. For instance, we can in fact

rewrite these mathematical expressions to have what appear to be active forms without issue:

175

(54)

a. We divided fourteen by seven (to get two).

b. Floyd multiplied seven by two (to get fourteen).

c. Clyde added seven to two (to get nine).

d. Agnes subtracted two from seven (to get five).

e. Horace raised two to the 4th power (to get sixteen).

Additionally, we can add an auxiliary verb to make them look like the standard passive:

(55)

a. Fourteen was divided by seven (to get two).

b. Seven was multiplied by two (to get fourteen).

c. Seven was added to two (to get nine).

d. Two was subtracted from seven (to get five).

The data thus points toward these in fact being passive constructions. But if these are indeed

passive sentences, why don’t we get a passive by-phrase? Because in most of these mathematical

cases it’s left unspoken, because it doesn’t add any additional information. These mathematical

sentences aren’t alone in this; as noted by a number of people, including Chomsky (1981), Jaeggli

(1986), Baker et al. (1989), and Collins (2005), there are a number of passives without overt

by-phrases (often called “short passives”) that nevertheless have these by-phrases syntactically

present. We can determine this because there are short passives that have (for example) Agentive

readings, despite there being no Agent phonologically present – and it can’t be the case that the

Theme (the subject of the passive) is coreferential with the Agent. If this were possible, then we

might expect that (56) could mean (57), but as Baker et al. (1989) point out, it cannot:

(56)

John was killed.

(57)

John committed suicide.

So the lack of an overt Agent in (56) does not mean that it’s not syntactically present. Therefore

let’s take a similar tack with the mathematical phrases: they’re short passives without an overt

176

postverbal DP. And indeed, if the need arises, you can make the passive by phrase overt, as seen in

(58).

(58)

a. Six added to forty-seven by people performing the addition properly is fifty-three.

b. Fourteen divided by seven by Clyde is three.

c. 0.1 multiplied by 0.1 by a certain computer processor is

0.010000000707805156707763671875.

So these particular arithmetic phrases are passive constructions. Thus, following from the work

of Collins (2005), we expect a structure for the more-or-less straightforward passive in (55) to be

as in (59):

(59)

TP

DP

Fourteen

T
was

T(cid:48)

PartP

VoiceP

Part

divided

VP

<DP>

V(cid:48)

Voice
(by)

<V>

PP

by seven

Voice(cid:48)

vP

DP

(everyone)

v(cid:48)

v

<PartP>

Here we have the PP by seven, which is generated as an argument of the verb in V, while fourteen

is generated in SpecVP, as expected. Then the verb divide raises up to Part, but then, following

Collins, instead of moving up to v the entire PartP raises up to the specifier position of VoiceP.
Then the DP fourteen raises out of SpecVP and into the subject position as SpecTP.7

7Note that I’ve covertly included the passive marker by in Voice and the word everyone in the

177

Now we can use this to deal with the standard version as in (50). What additional structure is

needed for this version? One possibility could be that this is a reduced relative clause, such as we

saw in (51); is there a way we can prove this to be the case?

As Kayne (1994) points out, reduced relative clauses cannot stack, while non-reduced relatives

can.

(60)

a. The boy who was swimming in the pool who was taken to the hospital

b. *The boy swimming in the pool taken to the hospital

The slight worry here is that it’s not immediately obvious that you can get non-reduced relatives

with numerals in the first place:

(61)

?Twelve which is divided by three is four.

However, with the proper context you can make them more natural, so this worry subsides:

(62) A twelve which is divided by three is four, while a twelve which is divided by two is six.

So, can we stack the reduced relative clauses? A caution before we begin: we need to be

careful to make sure that we’re actually stacking the relative clauses instead of nesting them. In

other words, we don’t want something as in (63):

(63) Twelve [divided by [four divided by two]]

Given this restriction in mind, let’s now consider (64):

(64)

*Fourteen divided by seven divided by two

It looks like our mathematical phrases do indeed resist stackability, which is thus evidence that

they are indeed reduced relative clauses. So, using the structure for relative clauses proposed by

specifier of vP in order to make it clear where the external argument of the vP in the passive would
go, but strictly speaking, based on the actual sentence in (55a) the DP would in fact be a standard
PRO.

178

Kayne (1994), where the head noun raises into SpecCP8, we should expect a (partial) structure for

(50) to be as in (65):

(65)

DP

DP

D
Ø

CP

C(cid:48)

Fourteen

C
Ø

TP

<DP>

T

T(cid:48)

PartP

Part

divided

VoiceP

Voice(cid:48)

VP

Voice

vP

<DP>

V(cid:48)

<V>

PP

by seven

DP
PRO

v(cid:48)

v

<PartP>

In some ways this structure is very straightforward and unsurprising, but this is in fact a pos-

itive: all the pieces come together in a satisfying manner, with nothing arbitrary or awkward to

make it all work - even the presence of our PP and the lack of an overt external argument can

be explained as the object of our verb and an example of a short passive, respectively. Thus we

now have a simple, elegant structure for these multi-morphemic mathematical phrases, without any

further fuss required.

8I use Kayne (1994)’s system as one possible approach to relative clauses, but, as best as I
can see, nothing regarding these particular constructions hinges on the particular relative clause
analysis used.

179

6.4 Other related syntactic phenomena

6.4.1 Colloquial versions of multi-morphemic mathematical phrases

A construction related to our standard multi-morphemic mathematical expressions like multiplied

by is seen in (66).

(66) Seven times’d by two is fourteen.

This sentence has a strong air of casualness to it; it’s not exactly bad, but it does feel very informal.

The other two arithmetic prepositions may also allow this, although their naturalness varies:

(67)

a. ?*Seven plus’d to two is nine.

b. ?Seven minus’d from nine is two.

It’s (67b) that makes things clearer. On some level the order of the numbers doesn’t matter for

addition and multiplication: these are commutative operations, such that (for instance) seven plus

two means the same thing as two plus seven. Subtraction is not commutative, however: seven minus

two does not equal two minus seven. And we can use this to our advantage, because the order of the

numbers (or operands, if you prefer) flips between the arithmetic version and the multi-morphemic

version:

(68) Seven minus two equals five.

(69) Two subtracted from seven equals five.

If we look again at (67b), we see that the order of the numbers reflects the order seen in (69),

not in (68). This thus indicates that the casual versions seen in (66), (68), and (69) are fairly

straightforward replacements of the mathematical verbs, derived from the prepositional forms –

and indeed, we see no functional difference between these derived versions and the standard forms:

(70)

a. Seven multiplied by two is fourteen.

b. Seven times’d by two is fourteen.

180

(71)

a. We multiplied seven by two.

b. We times’d seven by two.

(72)

a. Seven was multiplied by two (to get fourteen).

b. Seven was times’d by two (to get fourteen).

So this is thus a straightforward replacement of one lexical item with another, with no difference

in meaning.

6.4.2 Other math prepositions

There appears to be one other math term you can use that resembles plus and times, although in

some sense it might be considered to be more on the border of natural language and mathematical

jargon:

(73) Fourteen mod(ulo) three is two.

And it seems to be the case that (73) is a related preposition – just a highly specific one. And

indeed, you can use this in certain dialects to refer to things in non-math cases:

(74)

a. This semantics paper was good modulo the syntax.

b. Modulo his racial politics, Woodrow Wilson was a good president.

So an analysis of modulo should proceed along the same lines as plus, minus, and times.

By contrast, we might be inclined (possibly with the aid of a stiff drink) to also treat sentences

as (75) as a similar phenomenon:

(75) Seven rad(ical) nine is twenty-one.

But beyond the surface resemblance things fall apart rather spectacularly. This is no similar

math term to modulo, a semi-obscure preposition that can employed in the right context. This

instead, to the extent it’s good at all, has a flavor of “I’m reading a formula out loud” – and in fact,

we can see that there’s quite a bit omitted from (75) for the sake of brevity; the actual full English

version would be closer to (76):

181

(76) Seven times the square root of nine is twenty-one.

Yet, perhaps curiously, this ends up resembling a different, perhaps more common version of

the same thing. For whatever reason, English lacks a preposition that means divide; we can express

fractions using ordinals (one fourth, two thirds) or a somewhat awkward of construction (three of

five), but these tend to work only for amounts less than one. Seven thirds sounds needlessly clunky,

while ten of five is unacceptable in this context. The closest thing to have to expressing division

using a single preposition is in (77):

(77) Fourteen over seven is two.

But while one might try to argue that over is in fact the preposition that is used to express

division, the argument falls apart when one tries to use real world objects instead of numbers, as

(78) demonstrates:

(78)

#Fourteen apples over seven is two (apples).

This sentence, while syntactically acceptable, doesn’t have an equivalent meaning to fourteen

apples divided by seven is two apples. To the extent it means anything, it’s expressing an odd

locative proposition, not a mathematical one. So it therefore cannot be the case that over represents

division, other than in a similar “reading a formula” manner to (75) or (79):

(79) Six log base ten of one hundred is twelve.

But it’s a curious gap in English for there to be no mono-morphemic term for division, the way

there is for the three basic arithmetic operations. One possible reason might be that division is in

fact expressed via fractions. For instance, (80) and (81) are mathematically equivalent:

(80) Fourteen divided by two is seven.

(81) One-half of fourteen is seven.

Fractions could therefore be the way to divide in natural language, by separating a set or group

into smaller portions of equal size, with the total number of portions determined by the denomina-

tor of the fraction. A syntax and semantics for fractions, complete with a detailed internal structure

182

of phrases such as two-fifths and an explanation as to why the denominators of fractions are in or-

dinal form instead of cardinal form, is consequently well beyond the scope of this chapter, but this

is a clear starting point for future research into division in natural language.

6.4.3 Mixing mono-morphemic and multi-morphemic math terms

For some speakers, sentences as in (82) aren’t actually outright bad:

(82)

a. ?We multiplied seven times two.

b. ?Clyde added seven plus two.

c. ?*Agnes subtracted seven minus two.

Intriguingly, by contrast, the sentences in (82) get better if you stack the prepositions:

(83)

a. We multiplied seven times two times three.

b. Clyde added seven plus two plus eleven.

c. ?Agnes subtracted seven minus two minus three.

It’s unclear as to whether you can stack the standard prepositions, however.9

(84)

a. ?We multiplied seven by two by three.

b. ?Clyde added seven to two to eleven.

c. ?*Agnes subtracted three from two from seven.

While I’m uncertain as to how to analyze (84) (although one possibility could be elided verbs,

with the full version being along the lines of we multiplied seven by [two multiplied by three]), one

way to analyze the sentences in (83) is to treat the object of the verb as a single constituent, along

the lines of (85):

(85)

[VP multiplied [DP seven times two times three ] ]

9That said, it should be noted that, to the extent they liked any of these, at least two speakers

actually preferred stacking the standard prepositions to stacking the math prepositions.

183

This would combine nicely with the idea that these complex mathematical DPs can in fact be

treated as a form of number, as argued in section 3.4, and so its variable acceptability could be

related to how willing a given speaker is to accept these math verbs without a prepositional phrase

object.

Another possibility is that essentially a rescue strategy is employed, where a smaller part of

the calculation is considered to be a single complex number, and then the wrong preposition is

accepted as a form of “I know what you mean” strategy.

(86)

[VP multiplied [DP seven times two ] [PP times [DP three ] ] ]

A speaker’s willingness to allow sloppy preposition use (that is, to sort of question the preposi-

tion used but nevertheless allow it through versus essentially saying, “Hold up, this is the wrong

preposition and I therefore cannot condone this sentence”) could then also explain the variable

judgments.

6.5 The semantics of arithmetic

Now that we’ve covered the syntactic side of things, let’s add the semantics into the mix. We’ll

focus on times, as that’s what we’ve been working with throughout this dissertation. The first

option might be to simply bring in the same semantics for times that we’ve been using for factor

phrases, but it’s clear pretty quickly that that’s not going to work. The perhaps most obvious

problem is that the syntactic structure is different: for our factor phrases, times combines with

the preceding number, and then that’s used to modify another phrase – but arithmetic times is a

preposition, which means it’s going to take an object before it combines with a subject (which is

in itself potentially another difference, since it’s not obvious that in two times, two is the subject of

times). Another issue that might be a problem is that our factor phrases have been ratio modifiers,
being of type (cid:104)d, d(cid:105) or (cid:104)p, p(cid:105) (with p, as you’ll recall, the type for portions) – but while a sentence
such as (87) might involve degrees, it’s perhaps less certain that’s true for (88):

(87) Two times three is six.

184

(88) Two apples times three is six apples.

Given this, it makes sense to treat arithmetic times as being different from the times we see in factor

phrases. As support for this approach, we see that Japanese has a completely different morpheme

for arithmetic times from either the factor phrase times or the verbal event counting times (Ai
Taniguchi, p.c.)10:

(89)

(90)

sanpo-o
walk-ACC

inu-no
dog-GEN
‘I walked the dog three times.’

san-kai
three-times

shi-ta
do-PAST

furoido-wa
Floyd-TOP
‘Floyd is three times as tall as/taller than Clyde’

san-bai
three-times

kuraido
Clyde

ookii
big

yori
from

(91) ni

kakeru
hang

san
three

wa/ikooru
TOP/equal

two
‘two times three is/equals six’

roku
six

And while -bai can be used to perform multiplication (unsurprisingly), it isn’t in an arithmetic

sense, but rather in a slightly roundabout fashion:

(92) ni-no

san-bai
three-MULT.

-wa
-TOP

two-GEN
‘two “three times” (thrice two) is 6’

roku-da
six-COP

Thus it does indeed seem as if arithmetic times is different from the times in factor phrases.

There’s also an additional point that we’ve been kind of dancing around. In previous chapters,

we’ve just been directly importing the mathematical language directly into the semantics (so that
three times is 3×d, for instance). But given the pervasiveness of these basic mathematical functions
in natural language, is there a way we can get math to come out from the tools we need for

independent purposes?

Addition and subtraction seem relatively straightforward. Addition is simply creating and com-

bining pluralities. In essence, if we think of pluralities as essentially sets, then addition is the union

10kakeru, ’hang’, literally means ’hang’, as in ’hang a picture on the wall’.

185

of sets. If we have a set of two apples and another set of three apples, then combining the two sets

gives you five apples:

(93)

a.

b.

c.

two apples: {applea, appleb}
three apples: {applec, appled, applee}
two apples plus three apples: {applea, appleb} ∪ {applec, appled, applee}

= {applea, appleb, applec, appled, applee}

This is of course more or less how plurality works. And note that even when you try to combine

odd sets, it’s still essentially treated as a union – to the point where we’re not even certain how to

express the sum as something that’s not basically just a union:

(94) Two apples plus three oranges is the sum of two apples or three oranges.

Unless, of course, we shift our nominal reference to something less specific, such as fruit:

(95) Two apples plus three oranges is five fruit.

But this is still just a union.

Subtraction works similarly conceptually, except now we’re removing items from a set. In

set theoretic terms, given two sets A and B we’re looking for the relative complement of A in B,

standardly defined as in (96):

(96) B \ A = {x ∈ B | x (cid:60) A}

In other words, we want all the elements of B that aren’t also in A. So if we have a plurality of five

apples and we want to take away two of them, the elements of the set of five apples without the

two apples we remove will have a cardinality of three apples.

186

(97)

a. five apples: {applea, appleb, applec, appled, applee}
b.

two apples: {applea, appleb}
c. five apples minus two apples:

{applea, appleb, applec, appled, applee} \ {applea, appleb}
= {applec, appled, applee}

Multiplication is going to be a trickier story, however; the presence of factor phrases in the lan-

guage suggests that multiplication should be considered part of natural language, but the question

is how to achieve it. The general idea is that multiplication is a way of counting up pluralities of

pluralities, so that we have four fives, for example. One way to achieve this is through the use of

groups. So let’s take a closer look at how to achieve that, and what issues will crop up.

6.5.1 The semantics of arithmetic times

Let’s begin with a basic sentence:

(98) Two times four is eight.

Let’s claim that this will have a (simplified) structure as in (99):

(99)

is

two

times

eight

four

We’ll set aside the unresolved question that we had in section 6.2.4 of what the structure of the full

subject will be and focus in on the subject of the sentence, two times four, which we’ll treat as two

NUMBER times four NUMBER.

Our first decision is what type we want our numbers to be. One move might be to leave the

whole phrase as a degree, but as we saw in (88), it’s not clear we want to do that. The standard

187

thing to say about phrases such as two apples is that they’re of type (cid:104)e, t(cid:105), so let’s make a similar
move for our bare numerals. This will of course mean that we’re dealing with our covert NUMBER
noun as well. NUMBER, in, among others, Solt (2015) and Anderson (2015)11, is given the same

semantics as many in, e.g., Hackl (2000):

(100)

(cid:126)NUMBER(cid:127) = λdλx[|x| = d]

This means that our DP four NUMBER (for instance) will be

(101)

(cid:126)four NUMBER(cid:127) = λx[|x| = 4]

which is in line with what people such as Landman (2004) say is the semantics of these numerals.

Now we need a semantics for times. As we noted above, we can accomplish multiplication in

a non-mathematical-language way via groups. Multiplication is a way of counting up groups of

equal numbers, after all – this is why a (perhaps slightly archaic) way of expressing multiplication

is as in (102):

(102) Three sevens is twenty-one.

In other words, we have three groups of seven (where seven can be thought of as a set of 7 units),

which when flattened out into a single set sums up to 21 total units. This is the intuition behind

having arithmetic times work via groups.

So we need a morpheme that takes in two numbers, each of type (cid:104)e, t(cid:105), and combines them
somehow. One other potential concern before we begin: as it currently stands, our group operator
↑ takes in a plural individual and makes it a singular group atom. However, we’re going to need a
plurality of groups. This isn’t an unusual move; after all, we need this independently with sentences

such as (103):

(103) Three committees met in the auditorium at noon in order to discuss the budget.

11Although they give the NUMBER morpheme a slightly different name: Solt calls it Meas,

while Anderson (2015) calls it Num.

188

Here we have a case where our group is itself now part of a plurality; consequently, we’ll use *↑
to indicate that there is more than one of a particular group. This makes sense; just as an atomic

noun can be made plural, so can an atomic group noun.

With that established, we can now provide a semantics for arithmetic times, which we’ll call

timesar to avoid confusion with factor phrase times:

(104)

(cid:126)timesar(cid:127) = λG(cid:104)e,t(cid:105)λF(cid:104)e,t(cid:105)λx[*↑ (F)(x) ∧ G(x)]

This says that times will take in two objects, both of type (cid:104)e, t(cid:105), with the first object G(x) being the
number of groups we’ll have and the second object F(x) being the content of each group.

Now we can start putting the pieces together; times will combine with four, and then subse-

quently with two to get (105):

(105)

a. (cid:126)timesar four(cid:127) = λF(cid:104)e,t(cid:105)λx[*↑ (F)(x) ∧ |x| = 4]
b. (cid:126)two timesar four(cid:127) = λx[*↑ [λz[|z| = 2]](x) ∧ |x| = 4]

So x is a 4-membered plurality of groups, and each group consists of a 2-some: thus x is a property

of individuals consisting of four 2-somes, which is precisely the desired outcome.

So far so good, but now let’s add a wrinkle: consider what happens when we introduce concrete

nouns that our numbers are modifying into the mix.

(106)

a. Two apples times four is eight apples.

b. Two times four apples is eight apples.

c. *?Two apples times four apples is eight apples.

Curiously, both (106a) and (b) are good, but (c) is weird; in other words, it seems as though

arithmetic times requires that (at least) one of the pieces it takes in be a bare numeral, but it doesn’t
care which one.12

12It should be noted that acceptability judgments for both (106b) and (106c) seem to vary by

speaker; some people find (106b) odd, while others are completely happy with (106c).

189

Let’s start with (106a). Now, the type of two apples is the same as just two – namely, (cid:104)e, t(cid:105).
(This assumes, in line with Kayne 2005 and Zweig 2005, that two apples is underlyingly two

NUMBER apples.) So we’ll have a denotation as in (107):

(107)

(cid:126)two apples(cid:127) = λx.*apple(x) ∧ |x| = 2

Times four will remain unchanged from (105a), so combining that with (107) gives us (108):

(108)

(cid:126)two apples timesar four(cid:127) = λx[*↑ [λz[*apple(z) ∧ |z| = 2]](x) ∧ |x| = 4]

Now we’re saying that x is a 4-membered plurality of groups, and each group consists of a 2-some

of apples; in other words x is a property of individuals consisting of four 2-somes of apples. Again,

this works the way we’d want it to.

However, applying these denotations to (106b) creates an odd effect:

(109)

a. (cid:126)four apples(cid:127) = λx.*apple(x) ∧ |x| = 4
b. (cid:126)two timesar four apples(cid:127) = λx[*↑ [λz[|z| = 2]](x) ∧ [*apple(x) ∧ |x| = 4]]

In this case, (109) states that x is a plurality of groups composed of 2-somes, that x is also

a plurality of apples, and there are four xs. On the surface this seems OK (although it’s perhaps

slightly strange to argue that x is both a 2-some and apples), but it’s not actually obvious that we

want to say that a plurality of groups can also satisfy our plurality of apples. The problem is that

the way the pluralization operator * is defined, plurals are made up of pure atoms; groups, by being

made up from sums, are in fact impure atoms. If we take Link (1983) seriously, then we don’t want

to mix pure and impure atoms. Of course, as we noted above, we independently need a version of

plurality that accepts impure atoms, in order to explain, e.g., (103). So we could get around this by

extending the pluralization operator * to allow groups to be in the extension of plurality. But if we

do that, then we have to contend with (106c):

(110)

(cid:126)two apples timesar four apples(cid:127) = λx[*↑ [λz[*apple(z)∧|z| = 2]](x)∧*apple(x)∧|x| = 4]

Now x is a group of two apples, while x consists of a plurality of apples, and there are four xs.

This actually seems to work; there’s no incoherency about groups of two apples being pluralities

190

of apples. But as we’ve already noted, some speakers find this sentence bad – so in trying to

make (106b) work, we’ve inadvertently also made (106c) work. This is good news if you’re a

speaker who likes (106), but it seems problematic if you like (106b) but not (c). (If you don’t like

either, then you’ve naturally already been content with our semantics for times, no changes to the

* operator necessary.)

A alternate approach could be to say that, instead of the structure that we’ve been assuming,

like that in (111), we actually have a structure as in (112):

(111)

(cid:104)e, t(cid:105)

(cid:104)e, t(cid:105)

(cid:104)et, et(cid:105)

two

times

(cid:104)e, t(cid:105)

(112)

four apples
(cid:104)e, t(cid:105)

(cid:104)e, t(cid:105)

apples(cid:104)e,t(cid:105)

(cid:104)e, t(cid:105)

(cid:104)et, et(cid:105)

two

times

(cid:104)e, t(cid:105)

four

In essence, we’re arguing that two times four is a complex numeral which then modifies apples

via predicate modification. This is akin to the sentences we saw in (49) in section 6.2.4 and would

be consistent with the larger point being made in that section regarding the structure of these

arithmetic phrases. So if that’s the case, then our complex numeral would be what we saw in

(105b), with the full result being as in (113):

(113)

(cid:126)two times four apples(cid:127) = λx[*↑ [λz[|z| = 2]](x) ∧ [|x| = 4] ∧ *apple(x)]

This says that x is composed of 2-somes, that there are four of these 2-somes, and that x is a

plurality of apples. This is completely coherent.

191

There is a slight worry with this approach, however; observe (114):

(114)

a.

two times one apple

b. ?*two times one apples

It sure looks like one apple is meant to be a constituent in (114) – that’s the way the agreement

facts point, after all. If two times one were a complex numeral, then we should expect (114b) to be

good. So if we treat this seriously, then it looks like our solution in (113) won’t actually work.

One way around this might be to postulate that arithmetic times is ambiguous between a version

that makes F(x) a group (as we saw in (104)) and one that makes G(x) a group:

(104)

(115)

(cid:126)timesar(cid:127) = λG(cid:104)e,t(cid:105)λF(cid:104)e,t(cid:105)λx[*↑ (F)(x) ∧ G(x)]
(cid:126)timesar v2(cid:127) = λG(cid:104)e,t(cid:105)λF(cid:104)e,t(cid:105)λx[(F)(x) ∧ *↑ G(x)]

This might not be the most terrible idea; after all, there is something of a sense that it’s the piece

with apples that’s being grouped together. But while this would solve the problem, it’s ultimately

not a very satisfying answer; it doesn’t provide any insight into what’s going on, and it complicates

the lexicon purely to handle a small quirk in a rather specific case.

But let’s look at (114) again. The first thing to note is that the (b) case isn’t actually as bad (to

my ear, at least) as we might expect; it should be as bad as *one apples, but there’s a sense in which,

while (114b) is weird, it’s not as bad as *one apples is. So perhaps there’s a phonological process

at work that turns the plural into the singular because of the presence of one, but semantically this

is indeed a complex numeral.

It’s not an ideal solution, but then there doesn’t seem to be an ideal solution. Either we allow

in sentences we don’t want, such as (106c), and then explain them away, or we use a different

solution, as in (113), and then explain away the syntactic facts. From a naïve point of view, where

we just want the non-bare numeral to become a group, the approach in (104)/(115) almost seems

like the preferred one, even though it’s the most arbitrary.

One other idea could be to use Landman (2004)’s counting operators ↑ and ↓ in this domain. If

↑ applied to our noun, we’d get:

192

(cid:126)↑four apples(cid:127) =↑ λx.*apple(x) ∧ |x| = 4

(116)
And if times included ↓, then we could get the set of groups corresponding to sums of four apples.
Here’s a possible way to implement this, in the manner of Landman (2004):

(117)

(118)

(cid:126)time(cid:127) = λz(cid:104)↑,(cid:104)e,t(cid:105)(cid:105).↓z
(cid:126)time four apples(cid:127) =↓↑ λx.*apple(x) ∧ |x| = 4

= λx.*apple(↓ (x)) ∧ | ↓ (x)| = 4

(119)

(120)

(cid:126)times four apples(cid:127) =*λx.*apple(↓ (x)) ∧ | ↓ (x)| = 4
(cid:126)two times four apples(cid:127) = λy[*λx.*apple(↓ (x)) ∧ | ↓ (x)| = 4](y) ∧ |y| = 2

As we saw in chapter 3, Landman’s implementation is very complex, with a lot of moving parts

and assumptions in order to get things moving, and it will take some work to get this system to

work nicely with the syntax we’ve established thus far (for instance, Landman doesn’t make time

plural until after it’s merged with its complement, which is rather different from how we’ve been

treating times). But one exciting possibility arises from this: as we saw back in chapter 3, the event-

counting times combined with a null NP before combining with the numeral – in other words, that

times potentially has the same syntactic structure as arithmetic times. This could mean that, rather

than arguing for three different versions of times, there may in fact only be two: the multiplicative

factor phrase version, and the event-counting version, which would be the same as the arithmetic

version. (120) provides a possible denotation for how this could ultimately look, and in this case at

least, everything seems to work as desired. We’ll have to leave it at that for now; at this point, fully

pursuing this path is better left for future research – but it could be a very promising path forward.

However, regardless of concerns with the particular implementation of the theory, the larger

point remains clear: we can indeed achieve multiplication in natural language via groups.

In

principle, this should therefore mean that our denotation for factor phrases that we’ve been using

could be modified to use groups instead of importing multiplication directly. One possible version

of this is provided in (121):

(121)

(cid:126)three times(cid:127) = λd. ↑ (d) ∧ | ↑ (d)| = 3

193

This would of course require us to either extend the group operator to encompass degree-type

objects or to create a degree-group operator that would behave similarly to the standard group

operator. This would of course require us to make the type of degrees more ontologically rich –

so this could be an area where degree kinds, for instance, would not only be relevant but in fact

necessary. But hopefully the intuition is still clear: d is converted into a group, and we state that

there are three of these degree groups.

Further research would of course be needed to work out not just the mechanics of our degree-

group operator but also how this new denotation would interact in the adjectival and nominal

environments, to ensure that any wrinkles that may arise as a result of this version can be smoothed

out. But this could be an encouraging direction to have the three versions of times – the event-

counting, ratio degree, and arithmetic forms – come closer together in their semantics, beyond a

general gesture toward the operation of multiplication.

6.6 Conclusion

In this chapter we pointed out the problem that, despite the large body of research into numbers,

the related area of mathematical phrases remained essentially unexplored. Therefore, in order

to address this oversight, we covered two general areas of basic arithmetic – mono-morphemic

words such as times and multi-morphemic phrases such as divided by – and provided arguments

for both the syntactic category of the mono-morphemic terms, as well as how they fit into the

larger syntactic structure, as well as an analysis for the multi-morphemic phrases. We saw that the

mono-morphemic terms are ultimately prepositions with both objects and subjects, with numerals

that modify a covert NUMBER, in keeping with, among others, Kayne (2005) and Zweig (2005).

We also determined that the multi-morphemic phrases are short passives (passives without a by

phrase), combined with a reduced relative clause in order to get the most common form of the

multi-morphemic phrase (e.g., six multiplied by three is eighteen).

We then went on to discuss the semantics of the mono-morphemic terms, with a focus on times.

We established that, while arithmetic times is different from the times found in our multiplicative

194

factor phrases, it’s nevertheless involved in forming groups – which, depending on the particular

implementation, may put it in line with verbal times, either in a general group-forming sense or by

directly borrowing from Landman (2004)’s version.

This collectively represents a step forward for the analysis of mathematical language as lan-

guage, rather than as a shorthand for the various mathematical operators. This makes sense; in-

deed, the order of operations in arithmetic (parentheses, exponents, multiplication/division, addi-

tion/subtraction) is the direct result of imposing an order on the syntactic ambiguity that can arise

by stringing several arithmetic operations together. This demonstrates that language is perhaps

a deeper part of math than we’re perhaps willing to admit, and so it’s therefore useful to have

an understanding of how language treats arithmetic, rather than just blindly replacing words with

symbols. This chapter has thus narrowed the gap that lies between math and language.

195

CHAPTER 7

CONCLUSION

This dissertation has provided a detailed look at the semantics of factor phrases across a variety of

contexts. We’ve seen that there are three basic types of factor phrase: the event-counting version,

found in the verbal domain in sentences such as Floyd walked the dog three times; the multiplica-
tive modifier, a ratio degree of type (cid:104)d, d(cid:105), found in sentences such as Floyd is three times taller
than Clyde and Gloria has twice Sandy’s beauty; and the arithmetic operator, a preposition which

occurs in mathematical statements such as Four times seven is twenty-eight. While the event-

counting version had been explored in some detail in, among others, Doetjes (1997) and Landman

(2004), the other two forms had been almost completely ignored, and certainly hadn’t been ana-

lyzed with any degree of rigor. Consequently, this dissertation marks a large step forward for an

understanding of these modifiers. We’ve seen that ratio degrees can be introduced into the system

straightforwardly, without a need to create high, elaborate semantic types nor a need to assume a

non-standard structure; the one significant change made is that the equative morpheme as is left

uninterpreted. This is a move which not only provides the right semantics but opens the door to

explaining why in English Floyd is three times taller than Clyde means the same thing as Floyd is

three times as tall as Clyde – because speakers choose to leave the comparative -er uninterpreted

as well. However, the comparative can be interpreted if need be, which is why we occasionally

see a variation in meaning, with a differential interpretation of a factor phrase: the comparative

morpheme is being interpreted in these cases.

This dissertation has also made a number of novel observations, from the fact that the multi-

plicative ratio degree can interact with event-counting version, as in Floyd walked the dog three

times as many times as Clyde did, to the observation that the type of nominal phrase modified

by the factor phrase can lead to very different readings. Indeed, the use of factor phrases in the

nominal domain had been more or less completely unexplored, which means that, despite the fact

that there is a good deal of work in the nominal domain, this dissertation represents a journey into

196

uncharted waters. Furthermore, the domain of arithmetic language is one that had also been al-

most completely ignored, and so this dissertation provides the first detailed syntactic and semantic

analysis of basic arithmetic sentences.

This is exciting because not only have we addressed a number of fairly substantial gaps in our

knowledge of areas such as degree semantics and arithmetic language, we’ve also opened doors

into new corners of these areas, ripe for examination. The nature of factor phrases necessarily

means that there’s a lot of ground to cover, and consequently we’ve had to leave some things

merely illuminated, rather than exhaustively explored. As a result, there’s still things that future

research will need to address, including the nature of fractions and the interaction of factor phrases

and role nouns to provide a “quality” reading. But as many of these areas included phenomena not

even observed before, this dissertation still represents a significant advancement, and the fact that

there are such prominent semantic areas still to examine is in itself an exciting prospect.

197

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