The Universal Scaleand

The Semantics of Comparison

Alan Clinton Bale

Doctor of Philosophy

Department of Linguistics

McGill University

Montreal,Quebec

2006-04-13

A thesis submitted to McGill University in partial fulfillment of the requirements ofthe degree of Doctor of Philosophy

c© Alan Clinton Bale 2006

DEDICATION

This thesis is dedicated to my beautiful wife, Anne, and our two children, Esme

and Seymour.

ii

ACKNOWLEDGEMENTS

There are many people who have helped me in the last four years. I would like

to take this opportunity to acknowledge their contributions to my education and to

the ideas expressed in this thesis. To begin, I would like to thank my committee

members Brendan Gillon, Bernhard Schwarz, and Junko Shimoyama for guiding me

through this thesis. I am especially indebted to my supervisor, Brendan Gillon.

The countless conversations and meetings I have had with him have provided a

rich and bountiful foundation that helped to cultivate my intellectual curiosity. His

contribution to my education is immeasurable. I would also like to thank all the

others who have either commented on versions of this thesis or who have contributed

to my other works involving comparison. In this respect I owe a great deal of gratitude

to David Barner, Jonathan Bobaljik, Dana Isac, Kyle Johnson, Jim McGilvray, Jon

Nissenbaum, and Charles Reiss. This thesis would have been impossible without

their insightful criticisms and helpful suggestions. I would also like to thank all of

the graduate students at McGill. In particular, I would like to thank my office-mates,

Naoko Tomioka and Heather Newell, both as colleagues and as language consultants.

I would also like to acknowledge Lise Vinette and Andrea De Luca for their guidance

and help in fulling the requirements of this degree. In this respect I would also like

to thank Michel Paradis, Glyne Piggot and Lydia White. Finally, I would like to

express my eternal gratitude to my wife and my two children who had to put up

with my stress and anxiety that come all too naturally with graduate studies. Their

emotional support got me through these last four years.

iii

ABSTRACT

Comparative constructions allow individuals to be compared according to dif-

ferent properties. Such comparisons form two classes, those that permit direct com-

parisons (comparisons of measurements as in Seymour is taller than he is wide) and

those that only allow indirect comparisons (comparisons of relative positions on sep-

arate scales as in Esme is more beautiful than Einstein is intelligent). Traditionally,

these two types of comparisons have been associated with an ambiguity in the inter-

pretations of the comparative and equative morphemes (see, Bartsch & Vennemann,

1972; Kennedy, 1999). In this thesis, I propose that there is no such ambiguity.

The interpretations of the comparative and equative morphemes remain the same

whether they appear in sentences that compare individuals directly or relative to two

separate scales. To develop a unified account, I suggest that all comparisons involve

a scale of universal degrees that are isomorphic to the rational (fractional) numbers

between 0 and 1. All comparative and equative constructions are assigned an inter-

pretation based on a comparison of such degrees. These degrees are associated with

the two individuals being compared. Crucial to a unified treatment, the connection

between individuals and universal degrees involves two steps. First individuals are

mapped to a value on a primary scale that respects the ordering of such individuals

according to the quality under consideration (whether it be height, beauty or intelli-

gence). Second, this value on the primary scale is mapped to a universal degree that

encodes the value’s relative position with respect to other values. It is the ability of

iv

the universal degrees to encode positions on a primary scale that enables compar-

ative and equative morphemes to either compare individuals directly or indirectly.

A direct comparison results if measurements such as seven feet participate in the

gradable property (as in Seven feet is tall). Such participation can sometimes result

in an isomorphism between two primary scales and the ordering of measurements

in a measurement system. When this occurs, comparing positions in the primary

scales is equivalent to comparing measurements. If this type of isomorphism cannot

be established then the sentence yields an indirect comparison.

v

ABREGE

Les constructions comparatives permettent de comparer des individus selon

diffrentes proprietes. De telles comparaisons forment deux classes: celles qui per-

mettent des comparaisons directes (comparaisons de mensurations, telles que dans

Seymour est plus haut que large) et celles qui ne permettent que des comparaisons

indirectes (comparaisons de positions relatives sur des echelles separees, telles que

dans Esme est plus belle que Einstein n’est intelligent). Traditionnellement, ces

deux types de comparaisons ont ete associes a une ambigu ite dans l’interpretation

des morphemes comparatifs et equatifs (voir Bartsch & Vennemann, 1972; Kennedy,

1999). Dans la presente these, je propose qu’une telle ambigu ite n’est pas averee.

L’interpretation des morphemes comparatifs et equatifs demeure la meme, qu’ils

paraissent dans des phrases qui comparent des individus directement ou relativement

a deux echelles separees. De maniere a construire une explication unifiee, je suggere

que toutes les comparaisons impliquent une echelle de degres universels qui sont iso-

morphes aux nombres rationnels (fractionnels) entre 0 et 1. Toutes les constructions

comparatives et equatives reoivent une interpretation basee sur une comparaison de

tels degres. Ces degres sont associes aux deux individus en comparaison. Un fait

essentiel a un traitement unifie est que la connexion entre individus et degres uni-

versels implique deux etapes. Premierement, une valeur sur une echelle primaire est

assignee aux individus, laquelle respecte le rang de tels individus selon la qualite

consideree (que ce soit hauteur, beaute ou intelligence). Deuxiemement, un degre

universel qui encode la position relative de la valeur comparee aux autres valeurs est

vi

assigne a cette valeur sur l’echelle primaire. C’est la capacite des degres universels

d’encoder des positions sur une echelle primaire qui permet aux morphemes com-

paratifs ou equatifs de comparer des individus soit directement, soit indirectement.

Une comparaison directe rsulte si une mesure telle que sept pieds fait partie de la

propriete graduable (telle que sept pieds, c’est haut). Une telle participation peut

parfois resulter en un isomorphisme entre deux echelles primaires et l’ordonnance des

mensurations dans un systeme de mesures. Lorsque cela se produit, la comparaison

des positions sur les echelles primaires equivaut a la comparaison de mensurations.

Si ce type d’isomorphisme ne peut etre etabli, la phrase resulte en une comparaison

indirecte.

vii

TABLE OF CONTENTS

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ABREGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Comparatives, Absolutives and Equatives . . . . . . . . . . . . . . 61.2 What are Gradable Adjectives? . . . . . . . . . . . . . . . . . . . 81.3 Analyzing Comparisons . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Commensurable versus Incommensurable Adjectives . . . . . . . . 16

1.4.1 Incommensurable Adjectives and Indirect Comparisons . . 161.4.2 Commensurable Adjectives and Direct Comparisons . . . . 18

1.5 Model Theoretic Conventions . . . . . . . . . . . . . . . . . . . . 191.6 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Empirical Generalizations and Synopsis . . . . . . . . . . . . . . . . . . . 25

2.1 Empirical Generalizations . . . . . . . . . . . . . . . . . . . . . . 252.1.1 Wheeler’s Generalization . . . . . . . . . . . . . . . . . . . 272.1.2 Comparison Classes, Presuppositions, and Incommensura-

bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1.3 The Case of the Hidden Variable . . . . . . . . . . . . . . . 362.1.4 Direct and Indirect Comparisons . . . . . . . . . . . . . . . 40

2.2 A Brief Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.2.1 Linear Comparisons . . . . . . . . . . . . . . . . . . . . . . 512.2.2 From Clauses to Universal Degrees . . . . . . . . . . . . . . 532.2.3 Direct and Indirect Comparisons . . . . . . . . . . . . . . . 552.2.4 Wheeler’s Generalization and Hidden Variables . . . . . . . 592.2.5 Effects of Comparison Classes . . . . . . . . . . . . . . . . 60

viii

2.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 A Brief Summary of Comparison in Linguistic Theory . . . . . . . . . . . 67

3.1 Theories of Comparison . . . . . . . . . . . . . . . . . . . . . . . . 673.1.1 Conjunctive Comparisons . . . . . . . . . . . . . . . . . . . 693.1.2 Non-conjunctive comparisons . . . . . . . . . . . . . . . . . 903.1.3 Summary of Problems . . . . . . . . . . . . . . . . . . . . . 103

3.2 Apologia: An Explanation of Omissions . . . . . . . . . . . . . . . 1043.2.1 Generalized Quantification Over Sets of Degrees . . . . . . 1053.2.2 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4 A Unified Account of Direct and Indirect Comparison . . . . . . . . . . . 117

4.1 The Primary Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.1.1 Cresswell’s Ontology . . . . . . . . . . . . . . . . . . . . . . 1224.1.2 Questions of Circularity & Redundancy . . . . . . . . . . . 132

4.2 The Universal Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.3 The Interpretation of More and As . . . . . . . . . . . . . . . . . 142

4.3.1 The Interpretation of MORE . . . . . . . . . . . . . . . . . 1434.3.2 The Interpretation of As . . . . . . . . . . . . . . . . . . . 1484.3.3 Interpreting Subordinate Clauses . . . . . . . . . . . . . . . 150

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5 An Account of Indirect Comparison . . . . . . . . . . . . . . . . . . . . . 161

5.1 Controlling Comparison Classes . . . . . . . . . . . . . . . . . . . 1615.2 Interpreting Comparison Classes . . . . . . . . . . . . . . . . . . . 1635.3 Examples of Indirect Comparison . . . . . . . . . . . . . . . . . . 1685.4 Potential Problems with Indirect Comparison . . . . . . . . . . . 189

5.4.1 The Problem of Vagueness. . . . . . . . . . . . . . . . . . . 1905.4.2 Introducing Vagueness. . . . . . . . . . . . . . . . . . . . . 1965.4.3 The Problem of Equivalence Classes. . . . . . . . . . . . . . 1995.4.4 Abandoning the Primary Scale. . . . . . . . . . . . . . . . 202

5.5 Summary of Indirect Comparison . . . . . . . . . . . . . . . . . . 206

6 An Account of Direct Comparison . . . . . . . . . . . . . . . . . . . . . . 209

6.1 Measurements in Language . . . . . . . . . . . . . . . . . . . . . . 2116.2 What are Measurement Systems? . . . . . . . . . . . . . . . . . . 213

6.2.1 Two Adjectives, One Measurement System . . . . . . . . . 215

ix

6.2.2 Measurements of Time and Linear Space: A Well-OrderedSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.3 Two Assumptions about Measurements . . . . . . . . . . . . . . . 2216.3.1 Domain of Measurements is Finite . . . . . . . . . . . . . . 2226.3.2 Measurements as Individuals . . . . . . . . . . . . . . . . . 223

6.4 Explaining Direct Comparisons . . . . . . . . . . . . . . . . . . . 2256.4.1 Short and Low: A Problem for Klein . . . . . . . . . . . . 2426.4.2 Cross Polar Anomaly . . . . . . . . . . . . . . . . . . . . . 251

6.5 Further Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536.5.1 Coercing Direct Comparisons . . . . . . . . . . . . . . . . . 2546.5.2 Comparison Classes and Indirect Comparisons . . . . . . . 256

6.6 Concluding remarks on Direct Comparisons . . . . . . . . . . . . 273

7 Residual Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

7.1 Comparisons with the Same Adjective . . . . . . . . . . . . . . . . 2787.2 The Absolutive Construction . . . . . . . . . . . . . . . . . . . . . 2847.3 Degree Modifiers as Universal Degrees . . . . . . . . . . . . . . . . 287

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

8.1 Direct versus Indirect Comparisons . . . . . . . . . . . . . . . . . 2938.2 Other Empirical Generalizations . . . . . . . . . . . . . . . . . . . 298

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

x

CHAPTER 1Introduction

About four years ago, my daughter was swinging in the park while I chatted

with the other parents, giving a lazy push every few seconds. We were talking about

the sizing labels on children’s clothing. For children such labels are usually listed by

age: 3 months, 6 months, 12 months, 18 months, 2 year, 3 years...etc. My daughter

never fit the right size. She was wearing 6 month clothing at 3 months and 12 month

clothing at 6 months. Most parents at the swings had similar experiences. The

general consensus was that there must be some kind of conspiracy.

Then, while mulling this fact over, one woman made a statement that particu-

larly caught my attention. She was comparing the differences between her two boys:

the eldest was 2 years old at the time, the youngest 3 months. She said that her

youngest was bigger for a 3 month old than her eldest was for a two year old. I

found this statement surprising. Actually, to tell the truth, at the time I did not find

it surprising at all. It was a perfectly reasonable thing to say especially given the

topic of conversation. But later in the afternoon while I was working in my office,

I realized the linguistic significance of the statement. This statement compared two

individuals with regard to two different types of scales. One individual was being

measured with respect to the largeness of two year olds while the other was being

measured with respect to the largeness of 3 month olds.

1

Having worked on comparisons and the semantics of noun phrases (see, Bale &

Barner, 2004), I was familiar with the treatment of such sentences in the linguistic

literature. Generally they were treated as metaphorical extensions of more basic

forms of comparison such as Seymour is bigger than Esme (see Bierwisch, 1987, and

Cresswell, 1976). However, the woman’s statement about her two boys did not seem

to me to be metaphorical or playful. It struck me that the truth (or falsity) of the

statement at the time was definitive, especially in a society where most parents are

quite familiar with growth charts that allow them to calculate what percentile their

children are in compared to others of like age and gender (whether it be in terms of

height, weight or head-circumference). The knowledge of such growth charts make

statements like the one above easy to evaluate. Likewise for the statements below,

all of which have been uttered either by myself or by other parents since that fateful

park date.

(1) a. Ella is heavier for a girl than Denis is for a boy.

b. Seymour is taller for a one year old than Esme is for a four year old.

c. For a one year old, Seymour is much heavier than he is tall.

d. Charlotte is bigger for a three year old than she was for an infant.

I found it curious how easy it was to understand such sentences. The position of

one individual relative to one scale was being compared to the position of another

individual (or perhaps the same individual at a different stage or according to a

different aspect) relative to a different scale. If the first individual was higher in his

or her respective scale, then the sentence would be true. If the second was higher in

2

his or her respective scale, then the sentence would be false. The data demanded a

systematic semantic account.

Such accounts have been proposed (see Kennedy, 1999; Bartsch & Vennemann,

1972) but they usually involve hypothesizing two interpretations of the comparative

morpheme: one interpretation compares individuals relative to two separate scales

while the other compares measurements of individuals directly without relativization.

Some examples of the more standard type of comparison appear below.

(2) a. Esme is taller than Seymour.

b. This bike seat is higher than that one.

c. The bike seat is higher than Seymour’s legs are long.

d. Seymour is taller than he is wide.

The difference with these sentences is that the individuals are no longer compared

by their respective positions on separate scales. It makes no difference if Seymour is

very tall for a one year old and Esme is short for a four year old, Esme can still be

taller than Seymour. Similarly, it makes no difference if the bike seat is quite low

compared to other bikes seats and Seymour’s legs are quite long in comparison to

the legs of other two year olds: the bike seat can still be higher than Seymour’s legs

are long.

Despite the differences in meaning, there is a huge problem with providing two

different interpretations of the comparative morpheme. It cannot explain why the two

types of comparison always share the same morphology across a variety of languages.

In other words, why would the semantic ambiguity correspond with morphological

uniformity?

3

In this thesis, I propose that there is no semantic ambiguity. The interpreta-

tion of the comparative morpheme remains the same whether it appears in sentences

that compare individuals directly or relative to two separate scales. In developing a

unified account, I make some claims that have fundamental consequences for the na-

ture of the human conceptual system. First, I propose that all properties, whether in

terms of height, weight, beauty, intelligence, talent or any other quality, are measured

by the same mental yardstick: a universal scale. The interpretation of comparative

sentences involves a comparison of two degrees from this scale: this holds regardless

of the nature or compatibility of the adjectives involved in the comparison. Interest-

ingly, this universal scale seems to be best represented by the rational (fractional)

number system. As I argue in detail in chapter 4, the fractional characteristic of this

scale is a necessary component for calculating degree assignments. Let me briefly

explain why.

As mentioned above, I propose that comparative constructions are assigned an

interpretation based upon a comparison of two degrees. These degrees are associated

with the two individuals that are being compared relative to the adjectival properties.

Crucial to a unified treatment of comparison, the assignment of degrees involves two

steps. First individuals are mapped to a value on a primary scale that is unique to the

quality under consideration (whether it be height, beauty or intelligence). Second,

this value on the primary scale is mapped to a value on the fractional, universal

scale that encodes the relative position the individual occupies on the primary scale

with respect to other values. It is the ability of the fractional domain to encode

positions on the primary scale that allows comparative morphemes to either compare

4

individuals directly (if the primary scales are isomorphic to a measurement system)

or indirectly (if the primary scales are not isomorphic to a measurement system).

Another fundamental claim made in this thesis is that gradable adjectives (the

adjectives that appear in comparative sentences) are not predicates (contrary to

Klein, 1980, 1981, 1991) and do not relate individuals to measurements (contrary

to Bartsch & Vennemann, 1972, Kennedy, 1999, and Cresswell, 1976). Rather such

adjectives are interpreted as two-place relations between individuals. In other words,

their meaning encodes the relationship between any two individuals in much the same

way that verbs encode the relationship between subjects and objects. This relational

structure is converted into a scale by calculating the quotient structure (as discussed

in Cresswell, 1976, and proposed in Klein, 1991). This scale constitutes the primary

scale that is eventually embedded into the secondary, fractional scale.

The outline of this thesis is as follows. In the remainder of chapter 1, I discuss

some terms, concepts and definitions that will be relevant for the analysis of com-

parative and equative constructions. In chapter 2, I present four generalizations for

which any theory of comparison must provide an explanation. As I show in chapter

3, most theories do not fulfill this requirement. In chapter 4, I present the theory of

a universal scale. Not only does this theory provide a uniform interpretation of the

comparative morpheme, but it also explains why sentences with standard interpreta-

tions become non-standard with the addition of prepositional phrases. For example,

it explains the difference in interpretation of the two sentences below.

(3) a. Seymour is taller than he is wide.

b. Seymour is taller for a man than he is wide for a man.

5

The first sentence is true if Seymour is five feet tall and four feet wide, whereas the

second is not. No other existing theory provides an adequate explanation of either

of these facts. I discuss these kinds of effects in chapter 6. In the eighth and final

chapter, I provide a summary of the empirical advantages of a universal scale.

1.1 Comparatives, Absolutives and Equatives

The terms comparative construction, equative construction, and absolutive con-

struction are used throughout this thesis. In this section, I will briefly discuss what

each of these terms denote.

The term comparative construction will be used to refer to sentences that contain

a comparative morpheme (more or -er) in conjunction with a gradable adjective (such

as beautiful, intelligent, tall or wide). Some examples appear below.

(4) COMPARATIVE CONSTRUCTIONS

a. Comparatives with ‘MORE’

i. Esme is more beautiful than Morag is.

ii. Esme is more beautiful than I expected her to be.

iii. Esme is a more beautiful woman than Morag is.

iv. Esme saw a more beautiful woman than Morag did.

v. Esme is more beautiful than she is intelligent.

vi. Esme is more beautiful than how beautiful I expected her to be.

(Unacceptable in some English dialects.)

vii. Esme is more beautiful than what I expected. (Unacceptable in some

English dialects.)

6

b. Comparatives with ‘-ER’

i. Seymour is taller than Liam is.

ii. Seymour is taller than I expected him to be.

iii. Seymour is a taller boy than Liam is.

iv. Seymour is taller than the boat is wide.

v. Seymour is taller than how tall the doctor expected him to be.

vi. Seymour is taller than what I expected.

The only difference between the first set of sentences and second set is that in the

first the comparative morpheme is realized as a word whereas in the second it is

realized as an affix. Semantically the two groups of sentences are nearly identical.

Unlike comparative constructions, equative constructions only have one phono-

logical realization of the principal morpheme. Some example appear below.

(5) EQUATIVE CONSTRUCTIONS

a. Seymour is as intelligent as Esme is.

b. Seymour is as intelligent as I suspected.

c. Seymour is as intelligent as Esme is beautiful.

d. Seymour is as tall as the boat is wide.

In these sentences, the equative morpheme as appears in conjunction with a gradable

adjective.

In contrast to comparative and equative constructions, there are some sentences

that relate individuals to gradable adjectives without any comparative morphology.

Such sentence will be called absolutive constructions. Some examples appear below.

7

(6) ABSOLUTIVE CONSTRUCTIONS

a. Seymour is intelligent.

b. Esme is very intelligent.

c. Seymour is four feet tall.

d. Seymour is a tall boy.

e. Esme is quite a beautiful girl.

Notice that the sentences in (6) have no overt morphology connected to the gradable

adjective, nor do they compare two individuals through the gradable property.

My discussion of gradable adjectives and comparisons will be confined to these

three constructions. The results from this limited set appear to extend rather

straightforwardly, but I leave this extension to future work.1

1.2 What are Gradable Adjectives?

Throughout this thesis, adjectives will be classified as gradable based on their

distributional properties, primarily whether the adjective can be combined with a

comparative morpheme or degree modifier. In what follows, I first discuss the syn-

tactic characteristics of gradable adjectives before reviewing some of the common

conceptual properties. As discussed, the conceptual properties are only relevant for

a subset of the gradable adjectives.

Gradable adjectives can be defined by their syntactic environment. They are

the only adjectives that permit so-called degree modifiers such as very, quite, really

and incredibly. They are also the only adjectives that appear in comparative and

equative constructions. For example, beautiful is a prototypical gradable adjective

8

whereas dead is not. Accordingly, the acceptability of the sentences below contrast

sharply.

(7) a. Esme is incredibly beautiful.

b. Esme is more beautiful than than anyone ever thought possible.

c. Esme is as beautiful as Anne is.

(8) a. ? Alan Turing is incredibly dead.

b. ? Alan Turing is deader than I thought.

c. ? Alan Turing is as dead as Einstein is.

The sentences in (7) are acceptable without special circumstances. In contrast, the

sentences in (8) are awkward and require special circumstances or idiosyncratic con-

texts to be interpretable. Gradable adjectives, unlike non-gradable adjectives, are

not strained in such syntactic environments.2

Besides syntactic environments, there are at least two semantic properties that

sometimes serve to differentiate gradable from non-gradable adjectives. For example,

many gradable adjectives demonstrate a certain kind of context dependency that

other adjectives do not. Consider the sentences in (9) and (10).

(9) a. Seymour is (very) tall.

b. Felix is (very) intelligent.

c. That car is (quite) fast.

(10) a. Alan Turing is dead.

b. Anita is pregnant.

c. That car is electric.

9

The sentences in (9) have truth conditions that are highly dependent on context.

Whether Seymour, Felix and the car are tall, intelligent and fast respectively depends

on who or what constitutes the domain of comparison. For example, Seymour might

be tall for a jockey but he might not be for a regular man. Felix might be intelligent

for an domestic gerbil but maybe not for a household pet. The car might be fast

compared to other street cars but it might not be compared to formula one race cars.

In contrast, the sentences in (10) are not as context sensitive. Alan Turing is just

as dead when compared to current British mathematicians as he is when compared to

his contemporaries. The status of Anita’s pregnancy does not change when speaking

at a maternity ward as opposed to an all-boys, prep school. Similarly the car is

either electric or it is not. Context cannot change its classification as easily as it did

for fast.

Although in general gradable adjectives demonstrate more context dependency

than other adjectives this is not always the case. Words such as wet, open, bent,

awake, full, flat, closed, and straight all permit degree modifiers and also appear

in comparative and equative constructions, yet they do not seem to allow a high

degree of context dependency, at least no more than dead, pregnant and electric (see

Kennedy, 2005, for a discussion).

Another semantic characteristic that is often cited as exclusively applying to

gradable adjectives involves a certain kind of vagueness (see Klein, 1980, 1982; Kamp,

1975). For many gradable adjectives there is a large group of individuals that have the

property specified by the adjective to a certain degree and yet cannot be classified

10

as belonging or not belonging to the adjectival predicate. This property is best

understood with an example.

Consider how one commonly applies the adjective tall. Given any large group of

people, there are those that would be considered tall and those that would considered

not tall. However, there is also a significant portion that would not be considered

tall nor not-tall even though their height is known. These people are generally close

to an average height.

This type of vagueness is not limited to certain special contexts. In fact, in al-

most any context that has a significant portion of individuals to which the adjective

applies, there always seems to be a group of individuals for which the adjective is

neither true nor false. Whether one is talking about tall midgets or tall basketball

players, there are always those that are near the average height for midgets or bas-

ketball players (respectively) and hence are not easily classified as being tall or not

tall (for a midget/basketball player).

In contrast, adjectives like dead and pregnant do not normally demonstrate this

kind of systematic vagueness. For most contexts, such adjectives divide individuals

into two classes, those that have the property and those that do not. This is not

to say that there are never cases of vagueness. For example, someone who demon-

strates a significant reduction in brain activity but still has a beating heart might

be hard to classify as dead or not dead. However, the conditions that induce such

indeterminacy seem relegated to special cases. Vagueness in such adjectives requires

special conditions and does not seem to be systematically induced by the adjective’s

meaning.

11

Even though many gradable adjectives demonstrate this kind of vagueness, this

is not a property shared by all such adjectives. In fact the same words that pattern

with non-gradable adjectives in terms of context dependency also pattern with them

in terms of vagueness, namely wet, open, bent, awake, full, flat, closed, and straight.

Once again, the semantic property accurately defines a subgroup of gradable adjec-

tives but it is not a suitable characterization of all such adjectives that fit the same

syntactic pattern.

1.3 Analyzing Comparisons

For ease of presentation, I would like to develop a theory neutral way of referring

to different aspects of comparative and equative constructions. The terminology

developed in this section will hopefully assist in describing the core set of facts I

believe any theory of gradable adjectives and/or comparatives must explain. As

discussed below, the terms I use are slightly prejudicial since they bias the reader to

an analysis involving degrees and linear orders: an analysis that I ultimately favour.

However, with a bit of terminological stretching, my choice of words should apply

equally well to theories that employ quantification over comparison classes (Wheeler,

1972) or delineators (Klein, 1980, 1982, 1991; Larson 1988).

To begin, I would like to divide comparative and equative constructions into

two parts; the main clause and the subordinate clause. The main clause will contain

either (i) everything to the left of the than (in comparatives) or (ii) everything to the

left of the second instance of as (in equatives). The subordinate clause will contain

everything to the right of these functional items including as and than themselves.

Some examples should clarify this divide.

12

(11) a. I am prouder of Seymour than Anne is.

b. I am prouder of Seymour than I suspect Anne is.

(12) a. I am as proud of Seymour as Anne is.

b. I am as proud of Seymour as I suspect Anne is.

The main clause for the examples in (11a) and (11b) is the phrase I am prouder of

Seymour, whereas the main clause for the examples in (12a) and (12b) is the phrase

I am as proud of Seymour. The subordinate clauses in examples (11a) and (11b)

are the phrases than Anne is and than I suspect Anne is respectively. Similarly the

subordinate clauses in examples (12a) and (12b) are the phrases as Anne is and as I

suspect Anne is. Note that this divide does not reflect an actual syntactic divide: the

elements in the main clause do not form a constituent separate from the subordinate

clause. In terms of constituency, the subordinate clause is embedded within the main

clause. Despite this potential misrepresentation, the divide between the main and

subordinate clause will be conceptually useful.

Given this classification, I propose that any analysis of comparative and equative

constructions must have the following essential properties. First, there must be an

underlying ordering of individuals associated with the gradable adjectives in the main

and subordinate clause. I will use the term ordering to refer to this order whether it

be in terms of a quasi order, partial order or linear order. Second, some aspect of the

main clause must be able to indicate where the subject is situated in the ordering

associated with main clause. Similarly, some aspect of the subordinate clause must

be able to indicate where the embedded, subordinate-clause subject3 is situated in

the ordering associated with the subordinate clause. I will use the terms position

13

of main-clause subject and position of the subordinate-clause subject to denote the

places the subjects occupy in the orderings. As a final ingredient, all theories of

comparison must have a relation that compares the positions associated with the

main and subordinate clauses. I use the term locus of comparison to refer to this

type of relation.

For theories with extents or degrees in their analysis (such as Cresswell, 1976;

Bartsch & Vennemann, 1972; Seuren, 1973, 1978; von Stechow, 1984a&b; Hellan,

1981; Bierwisch, 1987; Heim, 1985, 2000; and Kennedy, 1999), the terminology de-

veloped above extends straightforwardly. For example, the locus of comparison is

often associated with a linear order that relates individuals, degrees or extents. Fur-

thermore, the position of the main-clause and subordinate-clause subjects are often

represented by degrees or extents that express a measurement of the individual in

question relative to a linear order (the ordering) associated with the underlying ad-

jective.

For theories that analyze comparatives in terms of delineators or comparison

classes (Wheeler, 1972; Klein, 1980, 1982, 1991; Larson, 1988) this choice of termi-

nology is a little more opaque. In such theories, the main and subordinate clauses

are not so much associated with measurements as they are associated with a par-

ticular partition of the domain of individuals to which the adjective can apply. For

simplicity, we can imagine that this partition divides a domain into two sets, a posi-

tive set (extension) and a negative set (counter-extension). The main clause conveys

information about whether its subject belongs to one side or the other of a particular

partition (the positive side or negative side). The subordinate clause does the same,

14

however the partition might differ from the one in the main clause. Comparatives

and equatives are evaluated based on a comparison of where the subjects in the main

and subordinate clauses are situated with regard to these possible partitionings. For

example, a sentence such as Seymour is taller than Esme is would be true if there was

one such partition that had Seymour on the positive side and Esme on the negative

side with respect to tallness.

Even though these theories do not have degrees or extents that represent values

of height or width, there is still a ordering of individuals within the adjectival domain.

For example, the adjectives beautiful, intelligent and tall apply to the domain of

people and order this domain in terms of who has more beauty, who has more

intelligence, and who has more height. The partitions associated with the main

and subordinate clauses respect this ordering. For any two individuals a and b, if

a is on the positive side of a partition and b is ordered above a according to the

adjectival ordering, then b must be on the positive side of the same partition. Such

a consistency in the partitions allows this kind of analysis to capture the right truth

conditions for sentences like Seymour is taller than Esme is.

In terms of these types of theories, the term position will refer to the set an indi-

vidual occupies with respect to a partition: whether it be the positive set or negative

set. The position of the main-clause subject and the position of the subordinate-

clause subject will be the positive or negative sides of the partition, the actual value

being determined by which side the individual occupies. The locus of comparison is

the relation that bases the truth or falsity of the sentence on the positions that the

main and subordinate clause subjects occupy with respect to the partitions.

15

Given this terminological massaging, the terms ordering, position, and locus of

comparison should refer to essential aspects of comparison common to all theories.

1.4 Commensurable versus Incommensurable Adjectives

As mentioned above, one of the most interesting aspects of comparatives and

equatives is that they can compare individuals with respect to different orderings.

Widths can be compared to lengths, happiness to sadness, and beauty to intelligence.

Such interpretations arise when there is a contrast between the adjectives in the main

and subordinate clause. There are two types of relationship between such adjectives:

either they can be commensurable or incommensurable. Adjectives are commensu-

rable if they share the same measurement system and yet are not opposites of one

another. For example, tall and wide are commensurable adjectives. In contrast, in-

commensurable adjectives are ones that are either complete opposites of one another

(e.g., tall versus short) or completely unrelated (e.g., beautiful versus intelligent). As

discussed in this section, when adjectives are incompatible or incommensurable, the

comparison is relative to the position of subjects in their respective orderings. When

the adjectives are compatible or commensurable, the comparison is direct: as if one

were comparing measurements. Below, examples of each kind of adjectival relation

are given followed by a discussion of the type of interpretation they license.

1.4.1 Incommensurable Adjectives and Indirect Comparisons

When the choice of adjective differs from the main clause to the subordinate

clause, the most common interpretation involves relativizing the positions of the

main and subordinate clause subjects in their separate orderings. As an example,

consider the sentences in (13) and (14).

16

(13) a. Seymour is as happy as Esme is sad.

b. Marilyn Monroe is more beautiful than Medusa is ugly.

(14) a. Esme is more beautiful than Einstein was intelligent.

b. Esme is as smart as Sidney Crosby is talented.

The sentences in (13) involve adjectives that are polar opposites: happy is the oppo-

site of sad and beautiful the opposite of ugly. Such opposites do not permit a direct

comparison of emotion or pulchritude but rather favour an indirect comparison. The

relative position of the main-clause subject in an ordering with respect to beauty

or happiness is being compared to the relative position of the subordinate-clause

subject in an ordering with respect to sadness or ugliness.

This indirect comparison is best characterized by the set of entailments exhibited

by the sentences in (13). For example, the first sentence entails that if Esme is very

sad then Seymour must be very happy. Similarly, if Esme is somewhat sad then

Seymour must be somewhat happy. In other words, the relative degree of Esme’s

sadness must match the relative degree of Seymour’s happiness.

A similar entailment is maintained by the second sentence where the amount

of beauty Marilyn Monroe exhibits is claimed to exceed the amount of ugliness that

Medusa exhibits. This sentence entails that if Medusa is only somewhat ugly then

Marilyn Monroe must be more than somewhat beautiful (possibly very beautiful).

Following the same logic, if Medusa is only a little ugly then Marilyn Monroe must be

more than a little beautiful. Once again, positions with regard to beauty and ugliness

are being compared relative to their respective orderings rather than directly to each

other.

17

The sentences in (14) also receive this kind of interpretation even though they

do not involve polar opposite adjectives.4 For example, the sentence in (14a) entails

that if Einstein was very intelligent, then Esme must be more than very beautiful.

Likewise, the sentence in (14b) entails that if Sidney Crosby is very talented then

Esme must be very smart. It is the relative positions that matter, not the absolute

value.

In summary, incommensurable adjectives license an indirect comparison of the

main and subordinate clause subjects. These types of comparison compare the po-

sition of the subjects in two separate orderings.

1.4.2 Commensurable Adjectives and Direct Comparisons

In contrast to incommensurable adjectives, there is a small group of adjectives

that allow for a comparison that is not relativized. Such adjectives are often asso-

ciated with extra-linguistic scales of measurement such as meters and inches (tall,

long, wide, high, deep, etc.) or minutes and days (old, long, late, early). Consider

the sentences below.

(15) a. Seymour is taller than he is wide.

b. The ceiling is as high as Seymour is tall.

The entailments exhibited by the sentences above differ from the entailments demon-

strated in the last section. For example, the sentence in (15a) can be true even if

Seymour is very wide but not very tall. In fact, most stout and round men are still

taller than they are wide. This was not possible when the adjectives were incom-

mensurable. Similarly, the sentence in (15b) can be true even when the ceiling is not

very high (such as an eight foot high ceiling) and yet Seymour is very, very tall (such

18

as eight feet tall). Once again, this is contrary to the entailment patterns exhibited

in the last section. In these examples, it seems as if the measurements are being

compared directly without relativization to the adjectival ordering.

The central issue of this thesis will be to account for the differences exhibited by

commensurable and incommensurable adjectives while still maintaining a uniform

interpretation of the comparative and equative morpheme.

1.5 Model Theoretic Conventions

Throughout this thesis, I adopt some fairly standard assumptions and conven-

tions concerning the semantic interpretation function. These conventions are briefly

outlined in this section.

First, I use the double brackets ‘[[ ]]’ to symbolize the interpretation function

that maps sentences, phrases and lexical items to their denotations. Normally, model

theoretic interpretation functions are relativized to a model and normally this rel-

ativization is symbolized by a subscript or superscript as in ‘[[ ]]M ’. To enhance

notational readability I leave this relativization implicit.

Second, I will assume that the interpretation function maps words and phrases

directly to denotations: such denotations are normally represented as set theoretical

entities. Where possible I will use the set theoretical representations directly. Like

Montague, 1973, I assume that logical representations have no real status in natural

language semantics. However, within formal logic, set theoretical denotations are

assigned to logical formulae. Often these formulae are much easier to understand

on an intuitive level than the set theoretical entities themselves. Thus in instances

19

where the set theoretical representation of the denotation is too opaque, I will use

logical formulae as stand-ins.

Third, I use only two rules of semantic composition in this thesis. In the spirit of

Montague, I adopt the standard rule of functional application as represented below:

Functional application:

If [[γ]] is a function from X to Y and [[φ]] is in the domain of X, then[[γφ]] or [[φγ]] = [[γ]]([[φ]]).

This rule can also be represented in terms of the logical stand-ins for the semantic

denotations. For convenience, I will often use this representation.

Functional application with Logical Formulae:

Where [[γ]] has a logical stand-in of the form λαΨ and [[φ]] has a logicalstand-in β that has the same logical type as α, then [[γφ]] has a logicalstand-in of the form Ψ(β/α) (Ψ with β substituted for free instances of α).

The other rule of semantic composition involves intersective modification. This

rule will be used to interpret how adjectives modify nouns when the adjectives are

used attributively.

Intersective Modification:

If [[γ]] and [[φ]] are sets of the same type, then [[γφ]] = [[γ]] ∩ [[φ]].

As with functional application, this compositional rule can be (at least partially)

represented in terms of logical stand-ins.

Intersective Modification with Logical Formulae:

Where [[γ]] has a logical stand-in of the form λαΨ (Ψ of type t) and [[φ]]has a logical stand-in λβΥ (Υ of type t) and where α and β are of the

20

same type, then [[γφ]] (= [[γ]] ∩ [[φ]]) has a logical stand-in of the formλδ (Ψ(δ/α) & Υ(δ/β)).

5

In any interpretive derivations presented in this thesis, I only exploit these two

rules. I will assume that the default rule is functional application. This rule will be

used without comment. In contrast, in instances where intersective modification is

employed, I will comment on the use of the rule to remind the reader of its possible

application.

1.6 Definitions

As certain mathematical structures play a central role in this thesis, it seems

prudent to offer the reader some basic definitions. Below I provide definitions of all

the important mathematical terms used throughout this thesis. I try to keep my

definitions as accessible as possible.

Transitivity: A relation R is transitive if and only if for all x, y, and zin the domain of R, if R(x, y) and R(y, z) then R(x, z).

Asymmetry: A relation R is asymmetric if and only if for all x and yin the domain of R, if R(x, y) then it is not the case that R(y, x).

Anti-symmetry: A relation R is anti-symmetric if and only if for all xand y in the domain of R, if R(x, y) and R(y, x) then x = y.

Reflexivity: A relation R is reflexive if and only if for all x in the domainof R, R(x, x). (Every element is related to itself.)

Connectivity: A relation R is connected if and only if for all x and yin the domain of R, either R(x, y) or R(y, x)).

21

Quasi Order: a Quasi Order is a relation that is transitive and reflexive.(Also called a Pre Order.)

Partial Order: a Partial Order is a relation that is a Quasi Order butis also anti-symmetric (i.e., a relation that is transitive, reflexive, andanti-symmetric). A prototypical example is the subset relation ‘⊆’.

Linear Order: a Linear Order is a relation that is a Partial Order but isalso connected (i.e., a relation that is transitive, reflexive, anti-symmetric,and connected). A prototypical example is the greater-than-or-equal orthe less-than-or-equal relation in the natural number system, ‘≥’ or ‘≤’.

Strict Order: a Strict Order is a relation that is transitive and asym-metric. A prototypical example is the strict subset relation or the strictsuperset relation, ‘⊂’ or ‘⊃’.

Strict Linear Order: a Strict Linear Order is a Strict Order that is alsoconnected (a relation that is transitive, asymmetric, and connected). Aprototypical example is the strictly greater-than relation or the strictlyless-than relation in the natural number system, ‘>’ or ‘<’.

Injection: an injection is a function that maps every element in the do-main to a distinct element in the co-domain. Stated otherwise, a functionf is injective if and only if for any two elements x and y in the domainof f , f(x) does not equal f(y).

Surjection: a surjection is a function where every element in the co-domain is assigned an element in the domain. Stated otherwise, A func-tion f is surjective if and only if for any element x in the co-domain,there is an element y in the domain such that f(y) = x.

Bijection: a bijection is a function that is injective and surjective.

Homomorphism: A homomorphism is a function from one domain intoanother that is structure preserving. For example, consider the domainof sets (S) and the co-domain of natural numbers (N). If f is a functionfrom S to N , and for all x, y ∈ S, (x ⊇ y) → (f(x) ≥ f(y)), thenf is a homomorphism. The function preserves the ordering of sets in

22

terms of the superset relation with the ordering of numbers in terms ofthe greater-than-or-equal relation.

Isomorphism: An isomorphism is a homomorphism that is also a bijec-tion.

23

Notes

1Other constructions include comparisons without adjectives and comparisonswith the morpheme less.

2These sentences are acceptable if the adjective is coerced into meaning somethinglike dead for a longer period of time or more violently killed. The point here is thatgradable adjectives are not coerced in these syntactic contexts. Also note that certainidiomatic expressions allow dead to appear in gradable syntactic environments, suchas dead as a doornail. The acceptability of the sentence is an exception rather thana rule.

3The subject of the adjectival predicate will be call the the subordinate clausesubject.

4The amount of intelligence someone has is not related to their beauty or talent,at least not in the way that happiness is related to sadness.

5 I leave it implicit that δ is the same type as α and β.

24

CHAPTER 2Empirical Generalizations and Synopsis

Before discussing the details of my proposal, I would like to first establish an

empirical foundation. In this chapter, I build this foundation. In section 2.1, I

present some generalizations for which any theory of comparison must provide an

explanation. In section 2.2, I provide a brief synopsis of a theory that can explain

these generalizations. The full presentation of this theory is left to chapter 4.

2.1 Empirical Generalizations

Comparative constructions have long been a subject of linguistic inquiry in mor-

phology, syntax and semantics. As a result there are a vast number of generalizations

to be made concerning these structures, too many to list even in a long thesis. I only

focus on a few of these generalizations, those that I believe to be particularly rel-

evant for understanding the nature of comparison. These generalizations establish

the core set of facts that any theory of comparison must explain and also shape the

theoretical proposal advanced in chapter 4.

I divide my discussion into four sections. The first is called Wheeler’s Gener-

alization (for reasons to be discussed below). This section examines the properties

of transitivity and asymmetry with respect to comparative constructions. Notably,

such properties exist even when the adjective is a nonsense term or a non-gradable

adjective.

25

The second section relates to hidden arguments called comparison classes. In

this section, I discuss three different ways these hidden arguments influence the

interpretation of comparative and absolutive constructions. The first involves the

calculation of standards of comparison, the second presuppositions induced by com-

parison classes, and the third adjectival incommensurability.

The third section presents evidence for the existence of operators and hidden

variables in the subordinate clauses of equatives and comparatives. One finds evi-

dence for such variables when comparing subordinate clauses with known Wh-islands.

Generally, the sentential complement of than or as cannot have a gradable adjective

contained within a Wh-island.

The fourth and final section explores data related to commensurability and

incommensurability. This section describes what are probably the most relevant

generalizations with regard to the semantics of comparison. As mentioned in the

introduction, often comparative and equative sentences compare individuals with

respect to different gradable adjectives. Sometimes such sentences seem to involve

a direct comparison of measurements, such as when one compares Seymour’s width

with his height. However, sometimes they compare individuals relative to their posi-

tions on separate scales, as when one compares Esme’s beauty with her intelligence.

As discussed below, cross-linguistic considerations argue for a uniform treatment of

both types of comparison. However, other factors will suggest that the two types of

comparison should remain grammatically distinct. An adequate theoretical proposal

must be able to satisfy both properties.

26

2.1.1 Wheeler’s Generalization

In this section, I argue that properties such as transitivity and asymmetry are

not exclusively derived from gradable adjectives but are at least partially induced

by the semantics of comparative morphemes. Below, I discuss the transitivity and

asymmetry of comparatives with more and -er. As noted by Wheeler (1972), such

properties persist even when the adjective is unknown or is a nonsense term. I believe

such evidence suggests that the semantics of the comparative itself must be integral

to the transitive and asymmetric properties it induces independent of the adjective.

To begin, a prototypical property of gradable adjectives in comparative con-

structions is that they result in relations that are transitive and asymmetric. Recall

that if a relation is transitive, then for any three individuals (call them a, b and c)

if a is related to b and b to c, then a must also be related to c. Also recall that

if a relation is asymmetric, then for any two individuals (call them a and b) if a is

related to b then b must not be related to a. Both properties are demonstrated by

the sentences below.

(16) a. Seymour is taller than Esme is.

b. Esme is taller than Anne is.

(17) Seymour is taller than Anne is.

(18) Anne is taller than Seymour is.

Transitivity is demonstrated by an entailment relationship between the sentences in

(16) and (17), namely the truth of the sentences in (16) entails the truth of (17).

Furthermore, no matter what names are substituted for Seymour, Esme, and Anne,

27

this entailment will always hold. Asymmetry is demonstrated by an entailment

relation between the sentences in (17) and (18). The truth of one entails the falsity

of the other. Once again, this holds no matter what names are substituted for

Seymour and Anne.

Asymmetry and transitivity are often remarked upon in the linguistic literature,

however there is little discussion about what aspect of comparatives is responsible for

these properties. Should transitivity be attributed to the interpretation of the adjec-

tive itself or rather should it be attributed to the interpretation of the comparative

morpheme?

Wheeler (1972) was the first to present evidence against relying on the adjective

alone to derive these properties. As he discussed, even when the adjective is a

nonsense term without a specified meaning, the transitive and asymmetric properties

still persist. For example, consider the following sentences.

(19) a. Seymour is more planket than Esme is.

b. Esme is more planket than Anne is.

(20) Seymour is more planket than Anne is.

(21) Anne is more planket than Seymour is.

Despite the fact that no native speaker knows what planket means or even if it maps

individuals to degrees, all would still agree that the transitive entailment holds. The

truth of the sentences in (19) entails the truth of (20). Similarly, all speakers would

agree that the truth of (20) entails the falsity of (21) and vice versa.

28

Similar evidence can be gathered from coercion facts. For example, when a non-

gradable adjective such as dead is used in a comparative construction, the transitive

and asymmetric properties still persist. Consider the sentences below.

(22) a. Seymour is deader than Esme is.

b. Esme is deader than Anne is.

(23) Seymour is deader than Anne is.

(24) Anne is deader than Seymour is.

The sentences in (22), (23), and (24) are usually coerced into meaning that the

subject died more violently or has been dead for a longer period of time. No matter

how an interpretation is achieved, the result is the same. The truth of the sentences

in (22) entails the truth of (23). Similarly, the truth of (23) entails the falsity of (24)

and vice versa.

Such evidence cannot be explained by deriving transitivity and asymmetry from

the meaning of the adjective since the properties of the adjective are either unknown

or contrary to the relational requirements. It is not the particular adjective that

guarantees transitivity and asymmetry but rather the comparative morpheme itself.6

2.1.2 Comparison Classes, Presuppositions, and Incommensurability

Hidden arguments called comparison classes are more often discussed with re-

gard to absolutive constructions than comparatives. However, as demonstrated in

this section, their effects can be detected in both. Below, I discuss three different

effects of comparison classes on the semantic interpretation of absolutive and com-

parative constructions. First, I address how comparison classes help determine a

29

standard of comparison for many gradable adjectives. I then review how comparison

classes induce a presupposition: namely the presupposition that the subject of the

adjectival phrase must be a member of the comparison class. Finally, I demonstrate

how comparison classes can force incommensurability.

To begin, within the literature on gradable adjectives (in particular see Wheeler,

1972; Bartsch & Vennemann, 1972; Klein, 1980; Ludlow, 1989; Bierwisch, 1987; and

Kennedy, 1999), many have noted that there seems to be a hidden argument in most

absolutive constructions. This argument denotes a set of individuals that relativizes

the application of a gradable adjective when determining a standard of comparison.

For example, consider the sentences below.

(25) The Mars Pathfinder mission is inexpensive. (Kennedy 1999)

a. The Mars Pathfinder mission is inexpensive for a scientific project.

b. The Mars Pathfinder mission is inexpensive for a project involving inter-

planetary exploration.

(26) Fido is large.

a. Fido is large for a dog.

b. Fido is large for a maltese.

The truth or falsity of the sentence in (25) is context dependent. If the expense of

the mission is compared to other scientific projects, then its cost is rather expensive

and hence the sentence would be false. However if the expense is compared to the

cost of other interplanetary explorations, then its cost is relatively cheap and hence

the sentence would be true.

30

A similar kind of context dependency is exhibited in (26). Even if Fido is a

rather big maltese (maltese are very small dogs), he would still be small for a dog.

However, in comparison to the other maltese he would be classified as large. The

truth of the sentence depends on to whom Fido is being compared.

The name comparison class is used in the literature to refer to the contextually

fixed set that determines this kind of ambiguity. Sometimes this set can be overtly

restricted by a for -clause as in (25a), (25b),(26a) and (26b), or by another prepo-

sitional or gerundive adjunct (eg., among these dogs/scientific projects..., between

these dogs/scientific projects..., considering these dogs/scientific projects, etc.). All

of these clauses restrict the comparison class to being a subset of (and in many cases

equal to) the set denoted by the nominal complement of the preposition or gerund.

No matter how the value of the comparison class is determined, whether by overt

restriction, pragmatic influences or a combination of the two, the grammatical effects

remain constant: comparison classes serve to relativize the standard of comparison.7

Not only do comparison classes help determine a standard of comparison, they

also introduce a presupposition: namely the presupposition that the subject of the

adjectival phrase is a member of the comparison class. Consider the following sen-

tences where the comparison class is overtly restricted by a for -clause.

(27) a. Fido is large for a maltese.

b. It is not the case that Fido is large for a maltese.

c. If Fido is large for a maltese, then I’ll put him on a diet.

d. Is it the case that Fido is large for a maltese?

(28) Fido is a maltese.

31

The sentence in (27a) implies the truth of (28) irrespective of whether Fido is in

fact large for a maltese or not. In other words, even in denying (27a) one still

accepts (28). This is a common characteristic of presuppositions. Furthermore, the

implication that (28) is true projects when (27a) is embedded either under negation

(27b), in the antecedent of a conditional (27c), or in a question (27d). These are all

common characteristics of a presupposition. It seems to be a robust empirical result

that Fido is large for a maltese presupposes Fido is a maltese.

Not only is there evidence concerning presuppositions in absolutive construc-

tions, there is also evidence in comparative constructions. Consider the following

sentences.

(29) a. Seymour is smarter for a lawyer than Pat is.

b. It is not the case that Seymour is smarter for a lawyer than Pat is.

c. If Seymour is smarter for a lawyer than Pat is, then I’ll be surprised.

d. Is it the case that Seymour is smarter for a lawyer than Pat is?

(30) a. Seymour is a smarter lawyer than Pat is.

b. It is not the case that Seymour is a smarter lawyer than Pat is.

c. If Seymour is a smarter lawyer than Pat is, then I’ll be surprised.

d. Is it the case that Seymour is a smarter lawyer than Pat is?

For the sentences in (29) the comparison class is overtly restricted so that it only

includes lawyers. The individuals Seymour and Pat serve as subjects in the main

and subordinate clauses. Perhaps not surprisingly given the facts about absolutives,

these sentences presuppose that Seymour and Pat are lawyers.8

32

The same effects are demonstrated by the sentences in (30), although such sen-

tences are a little more difficult to analyze. Like the sentences in (29), the sentences

in (30) presuppose that Pat and Seymour are lawyers. However unlike the sen-

tences in (29) there are no for -clauses that overtly restrict the comparison classes

to lawyers. Instead, these sentences contain modified nouns (nouns modified by an

adjective) that denote the set of lawyers. Some semanticists such as Cresswell (1976)

and Ludlow (1989) have suggested that modified nouns actually serve as comparison

class arguments to the adjectival phrases. Such a thesis would support my claim that

comparison classes induce presuppositions, but unfortunately the evidence suggests

that these nouns cannot be directly associated with comparison classes. For exam-

ple, in the sentences below there are distinct differences between the ones containing

for -clauses and the ones containing modified nouns.

(31) a. Seymour is a six foot tall man.

b. ? Seymour is six feet tall for a man.

c. Is Pat a tall woman? No, Pat is an average sized man.

d. ? Is Pat tall for a woman? No, Pat is an average sized man.

e. Seymour is a large man for someone who eats so little.

f. ? Among those who eat so little, Seymour is large for a man.

Sentences their have their comparison class restricted by a for -clause do not have

the same semantic and pragmatic interpretation as sentences with a modified noun.

However, even though comparison classes cannot be identified with a modified

noun, it seems to be a fact about such classes that they must be a subset of the

33

denotation of the such nouns. This is demonstrated by the sentences below. All the

sentences contain modified nouns as well as a for -clause that restricts the comparison

class.

(32) a. Fido is a large dog for a maltese.

b. Fido is a large dog for a purebred.

c. ?? Fido is a large maltese for a dog.

With the sentence in (32a), the for -clause contains a noun whose denotation is a

subset of the individuals denoted by the modified noun. The standard of comparison

for such a sentence is calculated in terms of this subset. The sentence in (32b)

has a for -clause containing a noun whose denotation shares some members with the

denotation of the modified noun: the set of purebreds contains some dogs as well

as some cats and horses, among many other animals. Interestingly, the standard of

comparison is not determined by this large set but rather only by the set of purebred

dogs. For example, the sentence in (32b) is synonymous with Fido is a large dog

for a purebred dog. The actual comparison class for such a sentence is once again a

subset of individuals denoted by the modified noun.

In contrast to the sentences in (32a) and (32b), the sentence in (32c) has a

for -clause containing a noun whose denotation is a superset of the denotation of

the modified noun. Interestingly, this sentence is judged to be quite odd. This

oddness would be expected if the comparison class were required to be a subset of

the denotation of the modified noun. With such a requirement the comparison class

would be restricted to maltese. Thus, the for -clause would be quite superfluous in

adding the restriction that the comparison class only contain dogs. Such an addition

34

should be as unacceptable as adding for a maltese to the sentence Fido is a large

maltese (??Fido is a large maltese for a maltese).

Given the fact that comparison classes must be subsets of the modified noun,

the sentences in (30) would involve a comparison class that only contains lawyers.

Once again, such sentences demonstrate the effect of comparison classes inducing

presuppositions.

Presuppositions are not the only detectable effect of comparison classes in com-

parative constructions. Such classes can sometimes also force incommensurability.

Recall in the introduction, I discussed how certain pairings of adjectives lead to

indirect comparisons. For example, the sentence Esme is more beautiful than Ein-

stein was intelligent compares Esme’s position relative to an ordering of beauty to

Einstein’s position relative to an ordering of intelligence. Adjectives that led to an

indirect comparison were called incommensurable. Interestingly, overt specification

of comparison classes produce similar kinds of indirect comparisons even though

the adjective remains the same in the main and subordinate clauses. Consider the

sentences below.

(33) Esme is taller than Seymour is.

(34) a. Esme is taller for a woman than Seymour is for a man.

b. Esme is taller for a woman than Seymour is tall for a man.

c. Esme is a taller woman than Seymour is a tall man.

(35) Fido is larger than Rex is.

(36) a. Fido is larger for a maltese than Rex is for a pitbull.

35

b. Fido is larger for a maltese than Rex is large for a pitbull.

c. Fido is a larger maltese than Rex is a large pitbull.

The sentence in (33) is necessarily false if Esme’s height is less than Seymour’s. This

is not the case for the sentences in (34). Such sentences can be true if Esme is very

tall for a woman but Seymour is not very tall for a man, even when Esme’s height

is actually less than Seymour’s. The comparison is not direct but rather involves

comparing the relative position of Esme in a ordering of women based on height to

the relative position of Seymour in a ordering of men based on height.

Similar results are demonstrated by the sentences in (35) and (36). In (35) the

sentence is necessarily false if Rex is larger than Fido. In contrast, the sentences in

(36) are not necessarily false. The sentences can be true if Fido is very large for a

maltese but Rex is not very large for a pitbull, even when Rex is larger than Fido.

In summary, comparison classes effect the meaning of comparative and abso-

lutive constructions in three different ways: first, they relativize the standard of

comparison, second, they introduce presuppositions, and third, they induce incom-

mensurability.

2.1.3 The Case of the Hidden Variable

Perhaps one of the most thoroughly established generalizations within the com-

parative literature is that subordinate clauses contain a variable bound by some

kind of operator. Although this hypothesis is well-known in semantic approaches

to comparatives,9 perhaps the most convincing evidence comes from the syntactic

literature. As first discussed in Ross (1967), Bresnan (1975), and Chomsky (1977),

but also repeated throughout the comparative literature since, there is an interesting

36

parallel between Wh-constructions and subordinate clauses, one that suggests that

the subordinate clause contains an operator that binds a variable in the adjectival

phrase in much the same way that Wh-elements bind variables in their associated

underlying argument positions. Below I review this syntactic evidence, first dis-

cussing the optional presence of overt Wh-constituents in subordinate clauses before

examining the effects of Wh-islands.

The most noticeable commonality between comparative constructions and Wh-

questions is that both allow Wh-elements to appear at the beginning of a sentential

clause, the only difference being that their presence is obligatory in Wh-questions

whereas it is optional in comparatives. Consider the sentences below.

(37) a. John is taller than how high he expected the basket ball rim to be.

b. John is taller than how long that boat is.

c. Seymour is taller than how tall his coach wanted him to be.

d. Seymour is taller than what I expected. (acceptable in at least some

dialects)

In these sentences, the Wh-element how or what can appear immediately to the right

of than. Furthermore, how even appears as a modifier to the adjective that defines

relevant ordering in the subordinate clause. Considering that Wh-elements often

bind variables in their sentential complement, such evidence suggests that the same

kind of variable binding might be present in subordinate clauses.

Although overt connections to Wh-elements are quite significant, they are not

the only link between questions and comparatives. Interestingly, even when there are

no overt Wh-elements in the subordinate clause, such clauses still demonstrate the

37

effects of movement and variable binding. The evidence for this comes from a parallel

between ungrammatical questions and ungrammatical comparative constructions.

Consider the sentences below.

(38) Comparatives and the Complex NP Constraint

a. i. * How tall does Wilt know a boy who is?

ii. How tall does Bill claim to be?

iii. * How tall did Bill make the claim that he is?

b. i. * Wilt is taller than he knows a boy who is (Bresnan, 1975)

ii. Seymour is taller than Bill claims to be.

iii. * Seymour is taller than Bill made the claim he is.

(39) Comparatives and the Coordinate-structure constraint

a. i. * How tall is Bill strong and?

ii. * How tall is Seymour wide and Mary is?

b. i. * Wilt is taller than Bill is [strong and ](Bresnan, 1975)

ii. * Seymour is taller than he is wide and Mary is.

(40) Effects of Overt Complementizers

a. i. How tall does Mindy believe Seymour is?

ii. ?? How tall does Mindy believe that Seymour is?

b. i. Seymour is taller than Mindy believes he is.

ii. ?? Seymour is taller than Mindy believes that he is.

(41) Sentential Subject Island

38

a. i. That Seymour is tall is interesting.

ii. * How tall is that Seymour is interesting?

b. * Seymour is taller than that Morag is is interesting.

(42) Wh-Island

a. i. What did Mindy ask Fred to carry?

ii. * What did Mindy ask whether Fred carried?

b. i. Seymour carried heavier boxes than Mindy asked Fred to carry.

ii. * Seymour carried heavier boxes than Mindy asked whether Fred

carried.

In each example, the sentences in (a) demonstrate violations (and sometimes non-

violations) of well-known constraints on binding by Wh-operators. These violations

are connected to certain syntactic constructions such as Complex Noun Phrases,

Coordinate-Structures, Overt Complementizers, Sentential Subject Islands, and Wh-

Islands. The sentences in (b) contain subordinate clauses with the exact same syn-

tactic constructions as the sentences in (a). Just as the Wh-questions are ill-formed

(or well-formed) with respect to the constraint, so are the comparatives.

This parallelism suggests that subordinate clauses contain a variable bound

by an operator in much the same way that Wh-operators bind a variable in Wh-

questions. An adequate semantic theory should be able to explain the nature of

these variables and the role of the binder in the interpretation of the subordinate

clause.

39

2.1.4 Direct and Indirect Comparisons

As mentioned in the introduction, when adjectives in the main clause of equa-

tives or comparatives differ from those in the subordinate clause there are two types of

comparison. One is indirect in the sense that the comparison relativizes the position

of the subjects according to the adjectival ordering. This is exhibited by adjective

pairs such as happy/sad, beautiful/intelligent, smart/large, where either the adjec-

tives are polar opposites of each other or their qualities are completely disconnected.

The other kind of comparison is direct in the sense that comparisons seem to involve

measurements that are not relativized to the adjectival ordering. This is exhibited

by pairs such as tall/wide, high/deep, or old/long. Generally, these adjectives are re-

lated to the same kind of extra-linguistic units of measurement (inches, yards, days,

etc.).

In this section, I will discuss both kinds of comparison with regard to two con-

flicting sources of data: one that argues in favour of a unified treatment of both kinds

of comparisons and another that argues for disparate treatments. The first involves

cross-linguistic interpretations of comparative and equative constructions while the

second involves morphological differences and ambiguous interpretations of the same

sentence. An adequate semantic theory of comparative and equative constructions

must account for these seemingly contradictory sources of evidence.

Cross-linguistic Evidence.

The most convincing argument that direct and indirect comparisons are the re-

sult of one interpretation stems from the fact that the same morpheme is present in

both kinds of comparison. Although this maybe a coincidental homophony within

40

a single language, the coincidence seems doubtful when it involves a variety of lan-

guages. Below, I first review how to differentiate the two types of comparison in

English. I then demonstrate that these two different methods of comparison exist in

a variety of languages, and in each language both types of comparison are associated

with the same morphemes.

Recall that the difference between direct and indirect comparisons can be charac-

terized by different entailment relations. For instance, consider the sentences below,

some of which are repeated from the introduction.

(43) a. Esme is more beautiful than Marie Curie was intelligent.

b. If Marie Curie was very intelligent, then Esme is more than very beautiful.

(44) a. Esme is as beautiful as Marie Curie was intelligent.

b. If Marie Curie was very intelligent, then Esme is very beautiful.

(45) a. Seymour is taller than he is wide.

b. If Seymour is very wide, then he is more than very tall.

(46) a. Seymour is as tall as he is wide.

b. If Seymour is very wide, then he is also very tall.

The sentences in (43a) and (44a) entail the sentences in (43b) and (44b). These

sentences contain adjectives that are incommensurable, hence the entailment. In

contrast the sentences in (45a) and (46a) contain commensurable adjectives. As a

result, they do not entail the sentences in (45b) and (46b). The sentences in (45a)

and (46a) can be true even when Seymour is relatively short and yet very wide (as

41

long as his height still exceeds his width). Such a circumstance renders the sentences

in (45b) and (46b) false.

It is possible that these two kinds of comparisons with their different entailment

relations are associated with two different loci of comparison. Such an analysis

has already been proposed by Bartsch & Vennemann (1972) and Kennedy (1999).

According to such theories, the shared phonological and morphological forms of the

two types of comparison are somewhat coincidental.10 However there is a problem

with this analysis. Indirect and direct comparisons are available in a variety of

different languages and in each of these languages the two readings are associated

with the same comparative morpheme.

For example, consider the follow set of sentences from Italian, German,(Quebec)

French, and Romanian.

(47) ITALIAN

a. MariaMaria

eis

piumore

bellabeautiful

dithan

quantohow much

MarieMarie

CurieCurie

siais

intelligente.intelligent.

‘Maria is more beautiful than Marie Curie is intelligent.’

b. Seif

MarieMarie

CurieCurie

eis

moltovery

intelligente,intelligent,

allorathen

MariaMaria

eis

moltovery

bella.beautiful.

‘If Marie Curie is very intelligent then Maria is very beautiful.’

c. LaThe

portadoor

eis

piumore

altahigh

dithan

quantohow much

siais

larga.wide.

‘The door is higher than it is wide.’

d. Seif

lathe

portadoor

eis

moltovery

larga,wide

allorathen

lathe

portadoor

eis

moltovery

alta.high.

‘If the door is very wide then the door is very high.’

42

(48) GERMAN

a. EvaEva

istis

schonermore beautiful

alsthan

EinsteinEinstein

intelligentintelligent

war.was.

‘Eva is more beautiful than Einstein was intelligent.’

b. FallsIf

EinsteinEinstein

sehrvery

intelligentintelligent

war,was

dannthen

istis

EvaEva

mehrmore

alsthan

nuronly

sehrvery

schon.beautiful.

‘If Einstein was very intelligent then Eva is more than very beautiful.’

c. DieThe

Turdoor

istis

hoherhigher

alsthan

sieit

breitwide

ist.is.

‘The door is higher than it is wide.’

d. FallsIf

diethe

Turdoor

sehrvery

breitwide

ist,is,

dannthen

istis

sieit

mehrmore

alsthan

nuronly

sehrvery

hoch.high.

‘If the door is very wide, then it is more than very high.’

(49) FRENCH (Quebec)

a. CharlotteCharlotte

estis

plusmore

bellebeautiful

quethan

MarieMarie

CurieCurie

estis

intelligente.intelligent.

‘Charlotte is more beautiful than Marie Curie is intelligent.’

b. SiIf

MarieMarie

CurieCurie

estis

tresvery

intelligente,intelligent,

alorsthen

CharlotteCharlotte

estis

tresvery

belle.beautiful.

‘If Marie Curie is very intelligent, then Charlotte is very beautiful.’

c. LaThe

tabletable

estis

plusmore

longuelong

qu’than

elleit

estis

large.wide.

‘The table is longer than it is wide.’

d. SiIf

lathe

tabletable

estis

tresvery

large,wide,

alorsthen

elleit

estis

tresvery

longue.long.

‘If the table is very wide, then it is very long.’

43

(50) ROMANIAN

a. ElenaElena

eis

maimore

frumoasabeautiful

decitthan

cithow-much

deof

inteligentainteligent

eis

MarieMarie

Curie.Curie.

‘Elena is more beautiful than Marie Curie is intelligent.’

b. DacaIf

MarieMarie

CurieCurie

eis

foartevery

inteligenta,intelligent,

atuncithen

ElenaElena

eis

maimore

multmuch

decitthan

foartevery

frumoasa.beautiful.

‘If Marie Curie is very intelligent, then Elena is more than very beautiful.’

c. MasaTable-the

eis

maimore

lungalong

decitthan

cithow-much

deof

latawide

eis

usa.door-the.

‘The table is longer than the door is wide.’

d. DacaIf

usadoor-the

eis

foartevery

lata,wide,

atuncithen

masatable-the

eis

maimore

multmuch

decitthan

foartevery

lunga.long.

‘If the door is very wide, then the table is more than very long.’

For each language, the sentence in (a) entails the sentence in (b). In contrast, the

sentence in (c) does not entail the sentence in (d). As with the English examples,

sentences with incommensurable adjectives entail certain propositions that sentences

with commensurable adjectives do not. To reiterate, the morphology of comparison

remains the same even when the interpretation differs. (In Italian this morpheme is

expressed by piu, in German -er, in French, plus, and in Romanian mai.)

Even given this limited number of languages, the phonological and morphological

similarities of direct and indirect comparisons seem to require a systematic explana-

tion. There is clearly a common link between the two types of comparison: one that

could possibly be reflected in the interpretation of the comparative morpheme.

44

Evidence of an Ambiguity.

Although the cross linguistic evidence argues for a uniform treatment of the

comparative and equative morphemes, there is still evidence that direct and indirect

comparisons should be kept distinct. Below I review and evaluate such evidence.

The most obvious difference between direct and indirect comparisons is the

contrast in entailment relations, such as the ones I have been using as a diagnostic.

Clearly a separation between the two types of comparison is established by this

evidence alone. Since the evidence has already been discussed quite thoroughly in

the previous sections, I will not review the issue here.

A less convincing argument for separating the two types of interpretation rests

on claims that only indirect comparisons licence certain presuppositions. I call this

argument “less convincing” only because I disagree with claim that such presuppo-

sitions exist. However, let me first present the presuppositions that are claimed to

exist before arguing to the contrary.

According to Bartsch & Vennemann (1972), McCawley (1976) and Kennedy

(1999), the sentences below (all of which contain incommensurable adjectives) have

different implications from sentences with commensurable adjectives.

(51) a. The view is more beautiful than the phone call is urgent.

b. Heather is more intelligent than Paul is devious

The first sentence implies that the view is beautiful while the second implies that

Heather is intelligent. In contrast, the following sentences with commensurable ad-

jectives have no such implications.

(52) a. The ceiling is higher than the girl is tall.

45

b. The boy is taller than the bed is long.

The first does not imply that the ceiling is high nor does the second imply that the boy

is tall. The difference in implication between the two sets of sentences is accredited to

a difference in presupposition. Indirect comparisons presuppose that their subjects

have the comparative property to a large extent whereas direct comparisons have no

such presuppositions.

As stated above, I do not agree with their assessment. First, I am not sure what

they mean by presupposition here. As demonstrated below, the sentences in (51) do

not maintain the same implication when they are embedded under negation, within

a question, or in the antecedent of a conditional.

(53) a. If Esme is more intelligent than Paul is devious, then you will win the

bet. But as you will find out, she is not intelligent and yet Paul is quite

devious. In the end, I will win.

b. Is Esme more intelligent than Paul is devious? ... No. While Paul is

quite devious, Esme is not intelligent.

c. It is not the case that Esme is more intelligent than Paul is devious. In

fact, Esme is not intelligent while Paul is quite devious.

If the implication were a presupposition then one would expect the implication to

project in such constructions.

Second, if such implications were a consequence of the meaning of the compara-

tive morpheme then they should be present in a variety of circumstances. However,

implications like those presented above are easily undone. For example, consider the

46

context where Mary is quite stupid. It seems to me that in describing my unflattering

appearance, I could say either of the following sentences.

(54) a. Unfortunately, I’m as beautiful as Mary is intelligent.

b. Unfortunately, Mary is more intelligent than I am beautiful.

Similarly, in the context where the view is quite ugly, I can express my diminutive

intelligence as follows.

(55) a. Unfortunately, I’m as intelligent as the view is beautiful.

b. Unfortunately, the view is more beautiful than I am intelligent.

In such situations the implications are the opposite of what is claimed in the litera-

ture. It seems to me that there might be a pragmatic explanation for the implications

mentioned above. Often indirect comparisons sound more natural when values near

the extremities of a scale are being compared. Perhaps the implications simply re-

flect this type of expectation. However, I will put this issue aside for now. Whatever

the right explanation is for the implication, it is quite obvious that it cannot be due

to the semantics of indirect comparison.

Another, slightly more convincing argument in favour of maintaining a distinc-

tion between indirect and direct comparison involves morphological differences. As

discussed in Bartsch & Vennemann (1972) and Kennedy (1999), in certain construc-

tions -er tends to only allow direct comparisons whereas more tends to only allow

indirect comparisons. For example, consider the sentences below.

(56) a. Seymour is taller than he is wide.

b. Seymour is more tall than he is wide.

47

c. The couch is longer than the doorway is wide.

d. The couch is more long than the doorway is wide.

The sentences in (56b) and (56d) favour an indirect comparison. For instance, (56b)

entails that if Seymour is very wide then he must be more than very tall. Similarly,

(56d) entails that if the doorway is somewhat wide then the couch must be more than

somewhat long. In contrast the sentences (56a) and (56c) have no such entailments.

Although the contrast seems clear with the sentences above, the dichotomy is

not consistent when considering other examples. For instance, comparatives with

-er are interpreted as comparing measurements indirectly when there is a restriction

on comparison classes in both the main and subordinate clauses. Some examples

appear below.

(57) a. Seymour is taller for a man than Esme is for a woman.

b. Seymour is taller for a man than he is wide for a man.

c. Seymour is a taller man than Esme is a tall woman.

The sentences in (57) permit indirect comparisons despite the fact that the adjectives

have -er endings.

Also, indirect comparisons with -er become more acceptable when there is no

potential direct comparison. For instance, the adjectives pretty, clever, funny, rich,

angry, and happy are not associated with any extra-linguistic measuring devices. As

a result, none of the adjectives are commensurable with each other. Yet, as demon-

strated below, indirect comparisons are still available even though the comparatives

involve -er rather than more.

48

(58) a. Let me tell you how pretty Esme is. She’s prettier than Einstein was

clever.

b. If Esme chooses to marry funny but poor Ben over rich but boring Steve,

then there can be only one explanation. Ben must be funnier than Steve

is rich.

c. Although Seymour was both happy and angry, he was still happier than

he was angry.

The sentences in (58) only have an indirect interpretation even though the adjectives

have -er endings.

Furthermore, in certain contexts an indirect comparison with -er seems possible

even when the two adjectives are commensurable. For example, consider the short

paragraph below.

(59) Although Seymour is quite tall, so are the rest of his classmates. In contrast,

Seymour is much wider than the others. Thus, Seymour is wider than he is

tall (...at least when compare to his classmates).

The co-text before the comparative sets up a situation where Seymour’s width and

height are being compared to the rest of his class. In such situations, indirect com-

parisons with -er are possible, although still a little awkward.

Similarly, this kind of contextual priming is also able to coerce the more mor-

pheme into licensing a direct comparison. Consider the paragraph below.

49

(60) In math class, Seymour was given a box and was asked to measure its height

and width. When he measured his box, Seymour discovered that it was more

high than it was wide.

By setting up the measuring task as a direct comparison the more morpheme is able

to license a reading that is normally reserved for -er endings.11

Although such an ambiguity weakens the morphological distinctions between

direct and indirect comparison, it actually supports a semantic distinction between

the two types of comparison. By hypothesizing two interpretations of the compara-

tive morpheme (one indirect and the other direct), one can easily explain how this

kind of ambiguity is possible. Yet, hypothesizing two different interpretations is not

the only way to account for these facts. As I discuss in detail in chapter 4, one

could keep the interpretation of the comparative morpheme constant and derive the

ambiguity by allowing for more than one kind of set to serve as the comparison class

in certain contexts. However, I will forego the details of this possibility here. For

now, it is sufficient to note that evidence suggests that an interpretive distinction

must be made between direct and indirect comparisons.

In summary, the morphological difference between direct and indirect interpreta-

tions is detectable despite the fact that the contrast is inconsistent and contextually

dependent. Furthermore, not only do the two types of comparison demonstrate dif-

ferent entailment relations, there also seems to be a direct versus indirect ambiguity.

50

A Direct and Indirect Summary.

This section presented some conflicting evidence concerning direct and indirect

comparisons. On one hand, the cross linguistic evidence suggested that both com-

parisons should be derived from the same interpretation. On the other, differences

in entailment relations, morphological associations, and ambiguities in interpretation

all suggest that the distinction should have some kind of grammatical explanation.

2.2 A Brief Synopsis

In this section, I outline the basic components of a theory of comparison that can

account for the generalizations discussed above. Crucially, this theory assigns truth

conditions to comparative and equative constructions that rely on a comparison of

two universal degrees. These universal degrees are associated with the main and sub-

ordinate clauses of comparative and equative constructions. Below, I sketch out some

of the details of this theory before addressing how it can explain the generalizations

discussed in section 2.1.

2.2.1 Linear Comparisons

Like Bartsch & Vennemann (1972), Cresswell (1976), and Kennedy (1999),

(among others) I propose that comparative and equative sentences involve a compar-

ison of two degrees through a linear relation: one degree being associated with the

main clause, the other associated with the subordinate clause. Unlike other theories,

I propose that no matter what the nature of the gradable adjectives in the main and

subordinate clauses is, all degrees belong to the same scale: a universal scale that is

isomorphic to the rational (fractional) number system between 0 and 1.

51

To gain a better understanding of such truth conditions, let me establish a few

symbolic conventions as well as a few definitions involving the nature of the universal

scale. First, degrees in the universal scale (the universal degrees) are represented by

a variable d plus a subscript. The subscript is a rational number. For example, d 13,

d 79

and d 2529

are all members of the universal scale. Furthermore, this scale is ordered

according to the fractional subscripts. For example, for all dx and dy in the universal

scale, dx dy if and only if x ≥ y. Also, for all dx and dy in the universal scale,

dx dy if and only if x > y.

With these conventions, the locus of comparison for comparative sentences can

be represented by the strict linear order ‘’, while the locus of comparison for equa-

tive sentences can be represented by the linear order ‘’. All comparative sentences

thus have truth conditions similar to the formula in (61).

(61) dx dy , where dx is the degree associated with the main clause and dy is

the degree associated with the subordinate clause.

Likewise, all equatives sentences have truth conditions similar to (62).

(62) dx dy , where dx is the degree associated with the main clause and dy is

the degree associated with the subordinate clause.

The formulae in (61) and (62) reduce comparative sentences down to a comparison of

two degrees. The key difference between the theory defended in this thesis and other

degree-based accounts concerns how universal degrees are associated with the main

and subordinate clauses. As discussed in the next four sections, comparison classes,

gradable adjectives, and clausal subjects all affect how such clauses are associated

with universal degrees.

52

2.2.2 From Clauses to Universal Degrees

The universal degree associated with the main and subordinate clauses is deter-

mined by the clausal subject and the gradable adjective in each clause. There are

two steps to the determination:

1. The gradable adjective is used to form a primary scale of equivalence classes

(sets of individuals who have the gradable property to a similar extent). This

scale orders equivalence classes according to the gradable property: an equiva-

lence class X is ordered above a class Y if and only if the members of X have

the gradable property to a greater extent than the members of Y .

2. The equivalence class containing the clausal subject is mapped to a universal

degree that encodes the position of the equivalence class with respect to the

ranking in the primary scale. This universal degree is the one used to evaluate

the truth or falsity of the comparison.

In what follows I discuss each aspect of this determination in a little more detail,

although a complete defence is left to chapter 4.

Building off observations by Cresswell (1976) and Klein (1991), I interpret all

gradable adjectives as two-place relations between individuals.12 For example, I in-

terpret the adjective beautiful as the relation x has as much beauty as y and the

adjective intelligent as x has as much intelligence as y. (See chapter 4 for a dis-

cussion of why such relations are not circularly defined.) Such relations have some

interesting mathematical properties: namely, they are connected, transitive and re-

flexive (connected quasi orders). These properties make the relations quite similar

to linear orders: the type of orders that are needed to establish a scale.

53

As Cresswell (1976) and Klein (1991) both observe, a linear order can be cre-

ated from these types of relations in two steps. First, one can collapse individuals

participating in the relation into equivalence classes. These classes contain all the

individuals who have the gradable property to the same extent (people who are

equally beautiful for the adjective beautiful, people who are equally smart for the

adjective intelligent). Second, one can order the equivalence classes with respect to

the original relation. For example, suppose that the underlying relation were x has

as much beauty as y. Equivalence classes can be formed by gathering together all the

people who are equally as beautiful (all the individuals who are reciprocally related

to one another). These equivalence classes can then be ordered in the following way:

an equivalence class X is ranked above a class Y if and only if a member of X has

as much beauty as a member of Y , but not vice versa. As a result of this ordering

and nature of equivalence classes, a class X is ranked above a class Y if and only if

all the members of X are more beautiful than all the members of Y . Hence by the

nature of the construction, a linear order of equivalence classes encodes an ordering

based on the gradable property (in the example above, an ordering of beauty). As

I propose in chapter 4, such linear orders serve as the primary scales for the main

and subordinate clauses. With such primary scales, one can assign a universal de-

gree to the main and subordinate clauses that is sensitive to the gradable property

associated with the adjective. Let me briefly explain how. Due to the isomorphism

between universal degrees and rational numbers, one can develop a way of mapping

equivalence classes to universal degrees that encodes the position of the equivalence

class in the primary scale. For example, consider the following mapping procedure:

54

for each equivalence class X in a primary scale Γ, map X to the universal degree d zw,

where z is equal to one plus the number of equivalence classes ranked below X in Γ

and where w is equal to the number of equivalence classes in Γ. Such a procedure

interacts with the ranking of the equivalence class. For example, if an equivalence

class X were the second highest equivalence class in a ranking of fifteen, then X

would be mapped to d 1415

. If X were the second lowest equivalence class in the same

ranking, then X would be mapped to d 215

. In general, the higher the equivalence

class is in the primary scale, the closer the fractional number is to one: the lower

it is, the closer the fractional number is to zero. With this mapping procedure, a

degree can be assigned to a clause by mapping the equivalence class containing the

clausal subject (with respect to the primary scale) to the relevant universal degree.

If the universal degrees associated the main and subordinate clauses are deter-

mined in such a way, then a comparison of two universal degrees, as in (61) and

(62), amounts to a comparison of the positions that the clausal subjects’ equivalence

classes occupy in their respective rankings. As described in the sections below, these

types of truth conditions account for all the generalizations discussed in section 2.1.

2.2.3 Direct and Indirect Comparisons

One of the principal advantages of using universal degrees to interpret sentences

involving comparison is that the same interpretation of comparative and equative

constructions can be used to account for both direct and indirect comparisons. Below,

I outline how both types of comparison can be derived from the same interpretation,

starting with indirect comparison.

55

Since universal degrees encode the position that an equivalence class occupies in

the primary scale, a comparison of universal degrees can easily account for indirect

comparison. For example, as proposed earlier, a sentence such as Esme is more beau-

tiful than Marie Curie is intelligent has truth conditions equivalent to the following

formula.

(63) dz dw , where dz is the degree associated with the main clause and dw is

the degree associated with the subordinate clause.

The universal degree dz represents the position of the equivalence class containing

Esme in a scale based on the relation x has as much beauty as y. The higher Esme’s

equivalence class is in this scale, the greater the number of people that she would be

more beautiful than. Also, the higher her equivalence class is, the closer z is to one.

(Likewise, the lower her equivalence class is, the closer z is to zero.)

In contrast to dz , the universal degree dw represents the position of the equiv-

alence class containing Marie Curie in a scale based on the relation x has as much

intelligence as y. The higher Marie Curie’s equivalence class is in this scale, the

greater the number of people she would be more intelligent than. Also, the higher

her equivalence class is, the closer w is to one.

According to the truth conditions outlined for ‘’, the formula in (63) is true

if and only if z is greater than y. However, if z were greater than y, then Esme’s

equivalence class would be higher in a scale of beauty than Marie Curie’s is in a scale

of intelligence. These truth conditions seem to accurately reflect the intuitions people

have about the sentence Esme is more beautiful than Marie Curie is intelligent. It

compares relative positions in two separate scales.

56

In contrast to indirect comparisons, direct comparisons are a little more com-

plicated. The key difference between direct and indirect comparisons is that direct

comparisons are associated with linguistically external measurement systems. Fur-

thermore, measurements from these systems seem to act like individuals in the sense

that they can serve as subjects to adjectival phrases. (For example, statements like

Six feet is tall are perfectly grammatical given the proper context.13 ) As I argue

in chapter 6, the same semantics for indirect comparison can be maintain for direct

comparisons if measurements participate in the adjectival relations in the same way

as other individuals. For example, suppose that measurements like five feet and three

feet participate in the relation associated with tall in much the same way Esme or

Seymour do. Someone has as much height as five feet if they are five feet or taller.

Similarly, suppose that the same kind of measurements participate in the relation

associated with wide. Someone has as much width as three feet if they are three feet

wide or even wider.

The participation of the measurements in the adjectival relation can significantly

affect the composition of the primary scale. With measurements in the relation, the

scale created from tall and wide will be composed of equivalence classes that contain

measurements. Furthermore, if the measurement system determines the precision of

comparison (which does not necessarily hold but could hold), then each equivalence

class will contain at least one measurement. In other words, every individual will

be equivalent to one measurement. Also, since no two measurements from the same

system are equivalent, each equivalence class will only contain that one measurement

(from that measurement system). As a consequence, the ordering of equivalence

57

classes based on tall and wide will be isomorphic to the same measurement system.

For example, if the measurement system were in terms of inches and feet then the

ordering of equivalence classes (whether with respect to height or width) would be

isomorphic to a scale of inches and feet. Thus the position of the equivalence classes

that contains a certain measurement (say three feet) will be the same for both primary

scales: the position will be equivalent to the position of the measurement three feet

in a scale of inches and feet.

Given this information, consider a sentence like Seymour is taller than he is wide.

This sentence seems to have truth conditions based on measurements of Seymour’s

height and width. According to my proposal, such a sentence has truth conditions

similar to Esme is more beautiful than Marie Curie is intelligent.

(64) dz dw , where dz is the degree associated with the main clause and dw is

the degree associated with the subordinate clause.

As with indirect comparison, the universal degree dz represents the position of the

equivalence class that contains Seymour in a scale based on height. However, this

equivalence class also contains a measurement. In a situation where the scale of

heights is isomorphic to the measurement system, this measurement determines the

position of the equivalence class in the scale.

In contrast to dz , dw represents the position of the equivalence class that contains

Seymour in a scale based on width. However, as with the scale of heights, this

equivalence class contains a measurement. In a situation where the scale of widths

is also isomorphic to the measurement system, this measurement determines the

position of the equivalence class in the scale.

58

With both primary scales being isomorphic to the same measurement system,

a comparison of universal degrees is equivalent to a comparison of measurements.

Although certain details and assumptions have to be discussed more thoroughly

to make this account complete (see chapter 6), it seems possible to explain direct

comparisons with the same semantic analysis as indirect comparison.

Interestingly, there is some additional support for this kind of explanation of

direct comparisons. Since such comparisons depend on measurements participating

in the adjectival relations, my proposal predicts that direct comparisons should be

impossible if a for -clause excludes such measurements from participating in the re-

lations (see chapter 4 and section 2.2.5 below). For example, for a man restricts

relations to men only: hence it excludes measurements. As a result, the following

sentence should not allow a direct comparison.

(65) Seymour is taller for a man than he is wide for a man.

Empirically this prediction is borne out. The sentence in (65) is false in a situation

where Seymour is five feet tall and four feet wide.

In summary, the same semantic interpretation can be maintained for both direct

and indirect comparisons, yet differences can also be accounted for by the effect of

measurement systems on primary scales.

2.2.4 Wheeler’s Generalization and Hidden Variables

The present account of comparative constructions also provides an account of

Wheeler’s Generalization and island effects in subordinate clauses. Below, I first

address Wheeler’s Generalization before turning to a discussion of operators and

hidden variables.

59

As shown above, the truth conditions of comparative constructions is based

on a comparison of two degrees by the strict linear order ‘’.14 As a strict linear

order, this relation is transitive and asymmetric. Hence, transitive and asymmetric

entailments should follow from the interpretation of the comparative constructions

independent of the nature of the adjectival relation. This fact explains Wheeler’s

observation concerning transitive and asymmetric entailments with nonsense terms.

An explanation of island effects in subordinate clauses is a little more complex.

Although not discussed above, in chapter 4 I describe how a hidden variable and

operator is needed to assign universal degrees to subordinate clauses (similar to

Kennedy, 1999). This operator moves to bind its variable in much the same way

that a Wh-operator binds its variable. Thus, the process of assigning subordinate

clauses universal degrees requires binding/movement to not violate island conditions.

In summary, the sketch given above describes how the current theory could

account for Wheeler’s Generalization and island effects in subordinate clauses. A

more detailed explanation is presented in section 4.

2.2.5 Effects of Comparison Classes

An interesting consequence of the present account of comparative and equative

constructions is that the semantic influence of comparison classes can be accounted

for by one simple hypothesis: namely that comparison classes restrict the relation

associated with the adjective. Below I describe how such a restriction relativizes the

standard of comparison, induces presuppositions, and forces incommensurability.

I begin with the effects on the standard of comparison. In the comparative liter-

ature, such effects are (perhaps surprisingly) not hotly debated. The general opinion

60

is that comparison classes restrict who participates in the ordering of individuals in

terms of the gradable property. The standard for comparison is then derived from

this restricted ordering: it could be calculated by selecting out a member from the

middle of the ordering and then using the measurement of this individual as the

standard, or it could be associated to a mean value with respect to members of the

ordering (see, Bartsch & Vennemann, 1972; Platts, 1979; Bierwisch, 1987; Kennedy,

2005). For example, the set of individuals that participate in the ordering for an

adjective like large for a maltese is the set of all maltese (ordered in terms of their

size). The standard of comparison is the mean or average measurement with respect

to this set. Hence, something is large for a maltese if it is larger than the average or

mean value of largeness for a maltese.

According to my proposal, the relevant ordering relation associated with the

gradable property is the relation (quasi order) assigned to the gradable adjective.

Thus, restricting the members that participate in this relation determines how an

average or mean value would be calculated. This is exactly what I propose compar-

ison classes do, whether overtly specified in a for -clause or inherently specified by

the context. For example, as already mentioned in the previous sections, I interpret

the adjective beautiful as the relation x has as much beauty as y. In addition to this

kind of interpretation, I propose that this relation, when used in a comparative, equa-

tive or absolutive construction, is restricted by the comparison class. For -clauses can

overtly specify what set constitutes this class. Thus, beautiful for a dog is interpreted

as a relation that is restricted only to dogs, while beautiful for a cat is interpreted

as a relation that is restricted only to cats. As a result of this type of restriction,

61

calculations of the mean or middle value would be relativized to the comparison

class. In summary, by interpreting comparison classes as restrictors operating on the

adjectival relation, one can explain the effects that such classes have on standards of

comparison.

Not only does this kind of restriction account for effects on the standard of

comparison, it can also explain presuppositional implications. In accordance with

Stalnaker (1974) (among others), let’s suppose that presuppositions are simply a list

of criteria that need to be met to assign a truth value to a sentence (whether true or

false). With such a supposition and the assumption that comparison classes restrict

adjectival relations, the semantics for comparatives and equatives outlined above re-

quire that the clausal subjects be members of their clause’s comparison class. Recall

that the truth conditions for comparative and equative constructions (under my pro-

posal) crucially rely on associating universal degrees with the main and subordinate

clauses. This association requires a mapping from the clausal subject to an equiva-

lence class that contains the subject. This equivalence class must also be a member

of the primary scale that is based on the adjectival relation. Thus, a well-formed

interpretation of comparative and equative constructions requires equivalence classes

that contain the clausal subjects. This in turn requires that the clausal subjects be

involved in the underlying relation. However, if the relation is restricted by the com-

parison class, then involvement in the underlying relation requires membership in the

comparison class. This last fact yields the presuppositions. A well-formed interpre-

tation of comparative and equative constructions requires the main and subordinate

clause subjects to be members of their respective comparison classes. Otherwise,

62

there would not be any means of assigning universal degrees to the main and subor-

dinate clauses.

The third effect of comparison classes can also be explained by a restriction of

the adjectival relation. Recall that when comparison classes differ from the main

to the subordinate clause, the result is an indirect comparison. For example, Esme

is as tall for a woman as Seymour is (tall) for a man can be true even when Sey-

mour is actually taller than Esme. Such indirect comparisons typically occur when

two adjectives have completely separate gradable properties (completely separate

rankings). This separation seems to hold for sentences like the ones above as long

as comparison classes are treated as restrictors. For example, under the restrictive

interpretation, the interpretation of the main clause adjectival relation (tall for a

woman) is completely disjoint from the intepretation of the subordinate clause rela-

tion (tall for a man). One is a relation that holds exclusively between women, the

other a relation that holds exclusively between men. With the relations being dis-

joint it is hardly surprising that the primary scale constructed from these relations is

completely different. The differences in the primary scale make a direct comparison

impossible.15

In summary, by hypothesizing that comparison classes restrict an underlying

adjectival relation, one can account for the effects that such classes have on the

standard of comparison, the introduction of presuppositions, and the coercion of

indirect comparisons.

63

2.2.6 Summary

The brief outline of my theory presented above is detailed enough to demon-

strate how I plan on accounting for the generalization mentioned in section 2.1. By

accounting for all these generalizations, this theory has a significant advantage over

other theories of comparison.

64

Notes

6Equatives and absolutives also demonstrate similar behavior with regard to tran-sitivity. Consider the following sentences.

(66) a. Esme is as planket as Seymour.

b. Seymour is as planket as Anne.

(67) Esme is as planket as Anne.

(68) a. Esme is not (quite) planket for a baby.

b. Seymour is (quite) planket for a baby.

(69) Seymour is more planket than Esme is.

The equatives in (66 & 67) behave identically to the comparative discussed above.The absolutives are slightly different since they do not allow for a direct comparisonbetween individuals. Still the effects of transitivity are present as long as the standardof comparison is taken to be the middle term in the transitive entailment. Forexample, the sentences in (68a & 68b) compare the subject with a contextuallydetermined standard. The sentence in (68a) claims that Esme is less planket thanthe standard while (68b) claims that Seymour is more planket than the standard.This standard can serve as the middle term in the transitive entailment, hence whyboth sentences together entail the truth of (69), even though the meaning of planketremains a mystery.

7 It should be noted that not all gradable adjectives demonstrate this kind ofambiguity. For instance, the adjective wet does not seem to relativize its meaning tocomparison classes. However, the main empirical point is that a significant numberof gradable adjectives do.

8Notice that the comparison class restricts the interpretation of smart as well.This point (discussed in McCawley, 1979) is not part of the present generalization.

9Seuren (1973) and Cresswell (1976) are two early examples where a hidden vari-able is interpreted in the subordinate clause.

10Bartsch & Vennemann (1972) do not actually hypothesize an ambiguity, ratherthey suggest that the main interpretation of the comparative morpheme involvesdirect comparison with indirect comparison being a secondary interpretation whendirect comparison is impossible. Even though there is no ambiguity in their theory,

65

the distinction between direct and indirect comparison remains arbitrary. This is notto say that the two interpretations are not related. In fact, properties of transitivityand asymmetry are found in both types of interpretation. However, the differencesdo not appear to be systematically related.

11Bartsch & Vennemann (1972) claim that although the more morpheme is gener-ally ambiguous between a direct and indirect interpretation, some people never getthe indirect comparison with the more morpheme.

12This makes the interpretation of gradable adjectives fundamentally different fromnon-gradable adjectives.

13Consider the following context, I thought Brad was very tall at six feet and twoinches. But he is actually only six feet tall. Still, six feet is pretty tall.

14 In this respect, my theory is exactly like Cresswell’s (1976), von Stechow’s(1984b), and Kennedy’s (1999).

15There are further detail in chapter 4 that help to explain other instances ofcoerced indirect comparisons such as, Seymour is taller for a man than he is widefor a man. Since the explanation for this sentence (although the same in principle)is a little more complex, I will forego the details in this outline.

66

CHAPTER 3A Brief Summary of Comparison in Linguistic Theory

This chapter strives to build a theoretical context for the theory of universal

degrees outlined at the end of chapter 2 and developed in detail in chapter 4. This

context serves two purposes. First, it creates a theoretical foundation for a theory

of universal degrees by outlining a series of alternative proposals that have had

considerable influence in the semantic community. Second, the theoretical context

provides a limit on the empirical issues discussed in this paper. The focus of this

thesis is on the model theoretic commitments needed to provide a coherent theory of

direct and indirect comparison. Issues related to movement, Negative Polarity Items

and differential expressions are peripheral to this focus.

In the two sections below, I first discuss other theories of comparison relevant

to a discussion about degrees, degree relations, comparison classes, orderings, and

adjectival incommensurability. These theories tackle problems similar to those dis-

cussed in chapter 2. The second section outlines some of the empirical topics that

I consider to be outside the scope of the present thesis. For each topic, a brief

justification is given for its omission.

3.1 Theories of Comparison

The semantic and syntactic literature on comparatives is extensive and varied.

Some of the topics covered include the syntax and semantics of ellipsis and covert

67

arguments (Gawron, 1995), cross-linguistic correlations between syntactic charac-

teristics and the grammaticalization of comparison (Stassen, 1985), the interaction

between comparison and modal operators (von Stechow, 1984a), the behavior of

quantifiers within subordinate clauses (Schwarzschild & Wilkinson, 2002; Bierwisch,

1987; Larson, 1988), contrasts between phrasal and sentential comparative construc-

tions (Hoeksema, 1983, 1984; Heim, 1985) and evidence for covert syntactic move-

ment (Heim, 2000; Hackl, 2000). In this section, I do not discuss the entire range of

topics related to comparative constructions, rather I limit my focus to the semantics

of the locus of comparison. I review proposals that provide at least a partial ac-

count of Wheeler’s Generalization and also a concrete hypothesis about the nature

of the hidden variable (Wheeler, 1972; Seuren, 1973, 1978; Cresswell, 1976; Bartsch

& Vennemann, 1972; Klein 1980, 1982, 1991; Kennedy, 1999; Bierwisch, 1987; von

Stechow, 1984a).16 As shown below, problems arise with these theories when consid-

ering the contrast between direct and indirect comparisons and the semantic effects

of comparison classes. Specifically, many of the theories do not provide an account

of indirect comparison (Wheeler, 1972; Seuren 1973, 1978, 1984; Cresswell, 1976;

Bierwisch, 1987; von Stechow, 1984b). For those that do provide an account, often

no explanation is given about the morphological similarities between direct and indi-

rect comparisons (Bartsch & Vennemann, 1972; Kennedy, 1999). Furthermore, there

is no theory that provides an adequate explanation of why comparison classes force

indirect comparisons.

68

The following discussion is divided into two sections, each based on a different

strategy for interpreting the locus of comparison. The first section discusses theo-

ries that relate comparisons to conjoined propositions with existential binding over

an adjectival variable. I have grouped these kinds of theories under the heading

conjunctive comparisons. The second section reviews theories not associated with

conjunctive comparisons. Generally such theories associate the locus of comparison

directly with a linear order. I have grouped these kinds of theories under the heading

non-conjunctive comparisons.

3.1.1 Conjunctive Comparisons

Many semantic theories of comparison interpret comparative constructions as a

conjunction of two propositions, one associated with the main clause stating that the

subject holds an adjectival property to a certain extent17 and the other associated

with the subordinate clause stating that the subject in the subordinate clause does

not hold the adjectival property to the same extent. Such theories also share another

property: both of the propositions contain a variable that is bound by an existential

quantifier. Thus for conjunctive theories, the truth conditions for Seymour is taller

than Esme is follows the general schema below.

(70) ∃α([[Seymour is α tall]] &¬[[Esme is α tall]])

The main distinction among such theories concerns what the variable α quantifies

over.

In the following three subsections, I review how these conjunctive theories of

comparison interact with the generalizations mentioned in chapter 2. I partition my

discussion with respect to how the theories treat the variable α. I begin with theories

69

that associate α with comparison classes, before discussing theories with extents and

delineations. With each type of theory, I first provide a bit of background on how

it is able to account for the meaning of comparative sentences before discussing the

empirical problems that arise in the context of the generalizations reviewed in the

previous chapter.

Comparison Classes.

Some theories of comparison base their analysis of comparative constructions

on comparison classes. Such a theory is proposed by Wheeler (1972). In this section

I review Wheeler’s account of comparative constructions. I first outline how the

semantics of comparison classes can be manipulated to provide a semantics for com-

parative constructions. I then discuss some of the advantages of Wheeler’s theory

before addressing some of the empirical problems.

To begin, understanding the details of Wheeler’s theory requires a brief review of

how comparison classes affect the interpretation of absolutive constructions. Recall

that comparison classes serve to relativize the meaning of gradable adjectives when

such adjectives appear in absolutive constructions. For example, the truth conditions

associated with large change in the sentences below as the comparison class changes.

(The for -clause restricts the comparison class.)

(71) a. Fido is large for a mammal.

b. Fido is large for a dog.

c. Fido is large for a maltese.

Fido can be large for a maltese while still being quite small for a dog or a mammal.

70

Wheeler (1972) maintains that the locus of comparison between the main clause

and the subordinate clause of a comparative relies on these types of comparison

classes. According to his analysis, understanding that Seymour is taller than Esme

requires understanding that there is a comparison class such that Seymour is tall

relative to this comparison class while Esme is not.18

For the purpose of exposition, let’s formalize this meaning in the following way.

Let’s suppose that TALL is a relation between individuals and comparison classes

such that [[TALL(a, C)]] means that a is tall considering the comparison class C.

With such a relation, the truth conditions for Seymour is taller than Esme is can be

characterized with the formula below, where C is a variable ranging over comparison

classes and s and e are Seymour and Esme respectively.19

(72) ∃C(TALL(s, C) &¬(TALL(e, C)))

This formula basically restates the conditions mentioned above. It is true if and

only if there is a comparison class such that Seymour is tall for this comparison

class but Esme is not. There are some interesting advantages to Wheeler’s rather

clever account of comparatives. For example, it does not involve any hypothesized

mechanisms or entities other than those already needed in the analysis of absolu-

tives. As I discussed earlier, it is impossible to determine what counts as tall or

beautiful without knowing the contextually determined comparison class. According

to Wheeler’s theory, understanding comparatives simply involves quantification over

such comparison classes.

Also, Wheeler provides an interesting account of his own generalization, one that

influences other conjunctive theories of comparison. Recall that Wheeler was the first

71

to observe that comparatives with nonsense words in the adjective position (such as

planket) still demonstrate the properties of transitivity and asymmetry despite the

fact that the nonsense words have no known interpretation. However, Wheeler’s

semantic representations of comparative constructions do not licence such transitive

and asymmetric entailments. Consider the following three sentences paired with

formulae that specify the truth conditions proposed by Wheeler.

(73) a. Seymour is taller than Esme is.

b. ∃C(TALL(s, C)&¬(TALL(e, C)))

(74) a. Esme is taller than Anne is.

b. ∃C(TALL(e, C)&¬(TALL(a, C)))

(75) a. Seymour is taller than Anne is.

b. ∃C(TALL(s, C)&¬(TALL(a, C)))

(76) a. Anne is taller than Seymour is.

b. ∃C(TALL(a, C)&¬(TALL(s, C)))

The truth of the formulae in (73b) and (74b) do not entail the truth of (75b), nor

does the truth of (75b) entail the falsity of (76b).

To account for his own generalization, Wheeler added a meta-linguistic20 axiom

similar to the one below. (Note that I have modified his axiom to suit the interpre-

tation given above. Also note that GA is the set of Gradable Adjectives and D is

the domain of individuals.)

72

(77) ∀ADJ : ADJ ∈ GA, ∀x, y ∈ D,

IF [∃C(ADJ(x, C)&¬ADJ(y, C))]

THEN [¬∃C ′(ADJ(y, C ′)&¬ADJ(x, C ′))]

The formula above claims that for all gradable adjectives (ADJ) and any individ-

uals a and b, if there is a comparison class C such that ADJ(a, C) is true and

(ADJ(b, C)) is false, then there is no comparison class C ′ such that ADJ(b, C ′) is

true and ADJ(a, C ′) is false.

Transitivity and asymmetry follow as a consequence of this axiom. For example,

the interpretation in (75b) entails that there is no comparison class C such that Anne

is tall and Seymour is not tall relative to C. This entailment is a direct contradiction

of (76b). Hence asymmetry follows as a consequence of the axiom.

A similar reasoning can be given for transitivity. Reconsider the truth conditions

given for the sentences in (73) through (75) above. By adopting Wheeler’s axiom,

the truth of (73b) now entails that there is no comparison class such that Esme is

tall and Seymour is not tall. The truth of (74b) entails that there is a comparison

class (call it C1 ) where Esme is tall and Anne is not tall. However, if (73b) is

true, then Seymour must also be tall relative to C1 . Thus, Seymour is tall relative

to C1 whereas Anne is not tall relative to C1 . Hence, (75b) must be true. The

truth of (73b) and (74b) entail the truth of (75b): transitivity holds. Furthermore,

this entailment is maintained no matter which adjective is substituted for TALL.

The entailment follows from the nature of comparison classes rather than from the

semantics of the adjective tall.

73

Despite the advantages of Wheeler’s theory in terms of accounting for transi-

tivity, asymmetry and deriving the interpretation of comparatives from the inter-

pretation of absolutives, there are some problems with his general approach. For

example, Wheeler’s theory is incompatible with the presence of multiple comparison

classes within one comparative construction. Furthermore, Wheeler cannot account

for comparisons that contain different adjectives in the main and subordinate clause.

Let me discuss each of these problems in turn.

In chapter 2, evidence was presented where two comparison classes appeared in

one comparative construction: one in each of its clauses. For example, consider the

sentence below repeated from the section on comparison classes.

(78) Esme is taller for a woman than Seymour is for a man.

In the sentence above, the comparison class in the main clause is completely different

from the one in the subordinate clause.

For Wheeler, the interpretation of comparatives crucially relies on maintaining

the same comparison class in both clauses. According to his interpretation, both

clauses must contain the same variable that is existential bound by the same quanti-

fier and this variable serves as the comparison-class argument. The presence of two

comparison classes is inconsistent with this type of analysis.

Perhaps a more troubling problem for Wheeler’s theory involves the differences

between direct and indirect comparisons.21 The challenge set out in the previous

chapter was to provide a uniform interpretation of the comparative morpheme that

derives both kinds of comparisons yet maintains their different effects. Wheeler does

74

not provide an adequate account of either direct or indirect comparison. Let me

demonstrate this with two examples.

For the first example, consider a context with hundreds of people where the five

ugliest people are also the five most intelligent people. Let me label these five, a, b, c,

d, and e, where a is Annick and e is Einstein. Let’s suppose that with regard to these

five the order in terms of both beauty and intelligence is the same. Both adjectives

order a over b over c over d over e. However the orderings are different for the other

people in the domain. The ordering associated with beauty orders everyone else in

the domain above a, b, c, d and e. The ordering associated with intelligence orders

everyone else in the domain below a, b, c, d and e. It is clear from this context that

a (along with b, c, d and e) is not beautiful, whereas e (along with b, c, d and a) is

very intelligent. With this context, consider the following sentence.

(79) Annick is more beautiful than Einstein is intelligent.

Einstein is at the top of the order in terms of intelligence (when the order includes

everyone, not just a, b, c, d, and e) whereas Annick is at the bottom in terms of

beauty (when the order includes everyone, not just a, b, c, d, and e). Like other

instances of indirect comparison, such a situation makes the sentence in (79) false.

However, an interpretation of this sentence according to Wheeler’s theory could

be true. For example, consider the expression below.

(80) ∃C(BEAUTIFUL(a, C) &¬(INTELLIGENT (e, C)))

A comparison class that satisfies this expression exists despite the fact that Annick

is among the least beautiful. One such comparison class is the set that contains a,

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b, c, d, and e and no others. (Out of this small group, Annick is the most beautiful

and Einstein is the least intelligent.) In summary, Wheeler’s theory makes the wrong

prediction with respect to indirect comparisons.

Similar problems occur with direct comparisons. Consider the context where

the widest man in the world is of average height. Let’s call him Jared. Furthermore

let’s assume that Jared is still taller than he is wide (as are most human beings).

Given these facts, the sentence below is clearly true.

(81) Jared is taller than he is wide.

Yet it is impossible for a Wheeler-style truth conditions for this sentence (given

below) to be true.

(82) ∃C(TALL(j, C) &¬(WIDE(j, C)))

Since Jared is the widest person, there is no comparison class where Jared is not

wide. A direct comparison is impossible.

In summary, although quantifying over comparison classes accounts for the truth

conditions of sentences with the same adjective in the main and subordinate clauses,

it encounters some problems when the adjectives differ. A theory like Wheeler’s that

quantifies over comparison classes cannot describe the truth conditions for direct and

indirect comparison.

Extents.

Similar to Wheeler’s approach to comparison classes, Seuren (1973, 1978, 1984)

also maintains a conjunctive interpretation of comparatives, however instead of quan-

tifying over comparison classes Seuren’s existential quantifier quantifies over extents.

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In this section, I first describe a possible definition of an extent. I then explore how

extents are used by Seuren to describe the truth conditions for comparative con-

structions. As shown below, extents can be used to provide an adequate account of

direct comparison, however, difficulties arise when considering indirect comparison

and the effects of comparison classes

To begin, Seuren (1973, 1978) did not formally define what an extent was. He

relied on intuitive notions understood in the context of paraphrases of absolutive and

comparative constructions. For example, a suitable paraphrase of Seymour is taller

than Esme is is the sentence The extent of Seymour’s height exceeds the extent of

Esme’s height. In the context of Generative Semantics, paraphrases were suitable

descriptions of the semantic content since paraphrases could serve as the underlying

structure of the comparative. However, in the present context extents need to be

defined more precisely.

Although Seuren does eventually give a more formal definition of extents (see

Seuren, 1984), for simplicity I will adopt a definition given by von Stechow (1984b),

Bierwisch (1987) and Kennedy (1999), all of whom relate extents to sets of degrees

or degree intervals. For example, if Seymour were five feet tall then the extent of his

tallness according to these theories would be the interval between zero and five feet

inclusive ([0′, 5′]) or equivalently the set of degrees (in feet and inches) between zero

and five (0, ..., 6′′..., 1′..., 1′6′′..., 2′, ...2′6′′..., 5′). I adopt this interpretation with the

assumption that it will not affect (either positively or negatively) the review put

forth in this section.

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Independent of the definition of extents, Seuren’s (1973) characterization of the

meaning of comparatives is similar to the interpretation of Seymour is taller than

Morag is given below.

(83) ∃e[(s is tall to e) &¬(m is tall to e)]

This formula can be paraphrased as follows: there exists an extent such that Sey-

mour is tall to that extent but Morag is not tall to the same extent. Of course

this characterization remains rather vague if is tall to is left unspecified. Let me

make this term a little more precise by taking advantage of the set representation of

extents. With such a representation, the is tall to relation can be defined through

a superset relation. If one assumes that there is a function (let’s call it map) that

takes individuals and measurement scales and maps them to extents in the scale that

represent the measurement of the individual (in other words, a measure function)

then ‘x is tall to e’ can be represented by the formula ‘map(x, TALL) ⊇ e’. This

formula states that the extent of x’s height is a superset of the extent e. With this

representation of the phrase x is tall to e, the truth conditions for Seymour is taller

than Morag is can be rewritten as follows.

(84) ∃e[(map(s, TALL) ⊇ e) &¬(map(m, TALL) ⊇ e)]

This formula is false only when the extent of Morag’s height contains all the degrees

in the extent of Seymour’s height.

One of the more interesting aspects of this proposal is that it is able to account

for direct comparisons in sentences like Seymour is taller than he is wide in a way

that Wheeler’s theory could not. With comparison classes, there are no means of

separating the interpretation of the variable from the meaning of the adjective. The

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interpretive effects of comparison classes are always relativized to the underlying

order provided by the adjective. In contrast, in a theory such as Seuren’s the ad-

jective simply describes on what measurement scale an object is measured without

necessarily putting any constraints on the ordering of individuals or on the number

of measurements in the scale. Nothing prevents two adjectives from being associ-

ated with the same measurement scale even when they are associated with different

dimensions of measurement (e.g., vertically oriented versus horizontally oriented).

Such a possibility can explain why tall and wide permit a direct comparison.

Each adjective could be associated with different orientations (one with a horizontal

linear orientation, the other with a vertical orientation), and yet be associated with

the same measurement scale (for example, the scale of inches and feet). In effect, the

semantics of the mapping function allows scales to be characterized independently

of the adjective. For example, the extent of Seymour’s height might be the set of

degrees between zero and five feet (map(s, TALL) = 0, ...6′′, ...1′, ...1′6′′, ...2′, ...5′)

while the extent of his width might be the set of degrees between zero and three feet

(map(s, WIDE) = 0, ...6′′, ...1′, ...3′). Both extents are in terms of feet and inches

despite the different orientation of the measurement.

Given this kind of scalar independence, let’s consider Seuren’s interpretation

of the sentence Seymour is taller than he is wide. The truth conditions for such a

sentence are equivalent to the formula below.

(85) ∃e[(map(s, TALL) ⊇ e) &¬(map(s, WIDE) ⊇ e)]

Given the assignment of Seymour’s width and height above, this formula is equivalent

to (86).

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(86) ∃e[0, ...1′, ...2′, ..., 5′ ⊇ e) &¬(0, ...1′, ...2′, ...3′ ⊇ e)

Such a formula is true since the extent consisting of the degrees between zero and

four feet is a subset of the extent of Seymour’s height but is a superset of his width.

Note that the truth of the sentence can be derived even though Seymour is quite

wide at three feet and yet quite short at five feet.

Putting the advantages of Seuren’s theory in terms of direct comparison aside,

there are some problems with this type of theory. Below I review two of these

problems: namely the lack of an interpretation of indirect comparison and unsub-

stantiated predictions with regard to the independence of measurement scales.22

I begin with problems concerning indirect comparisons. Seuren’s proposal in

its present form cannot account for such comparisons. To demonstrate this, let me

present an example sentence and then consider how Seuren’s theory might represent

its truth conditions. Let’s take a variant of the canonical examples of an indirect

comparison given in the introduction, such as Marag is more intelligent than Sydney

Crosby is talented. Remaining consistent with Seuren’s theory of extents, we could

represent the truth conditions of this sentence with the following formula.

(87) ∃e[(map(m, INTELLIGENT ) ⊇ e) &

¬(map(s, TALENTED) ⊇ e)]

However this formula does not adequately account for intuitions regarding this type

of sentence. Specifically, the formula given above is either trivially true or not well-

formed, depending on how one interprets the superset relation. Yet the sentence

itself is neither trivially true nor ill-formed. Let me explain why the truth conditions

for the formula are what they are. Extents of intelligence and talent are not on the

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same measurement scale, so for any extent that is a subset of the extent of Morag’s

intelligence it is impossible for it to be a subset of the extent of Sidney Crosby’s

talent. Thus, either the second half of the conjunct is trivially true when considering

extents of intelligence or is ill-defined by stipulation (extents from two measurement

scales cannot be compared).23

It is unclear how Seuren would be able to achieve an adequate interpretation

of indirect comparison. A crucial aspect of his theory is that he relies on extents

to compare individuals according to the selected measurement scale. Because of

this reliance, there is no obvious means of relativizing the representation of extents

to facilitate cross-adjectival comparisons. Yet, such a relativization is essential for

indirect comparison.

Although the lack of an account for indirect comparisons is undesirable, even

more damaging criticisms of Seuren’s proposal stem from the independence of the

measurement scale that proved so valuable in allowing direct comparisons. With the

scales of measurements being independent from the measure function, limits on what

can be measured do not put limits on the scale where the measurement is represented.

As a result, a theory such as Seuren’s does not provide an account of all of the

semantic effects of comparison classes. For example, recall that comparison classes

can force indirect comparisons, making direct comparisons inaccessible. Consider

the sentences below, some of which are repeated from section 2.1.2 on comparison

classes.

(88) a. Esme is taller for a woman than Seymour is for a man.

b. Esme is a taller woman than Seymour is a tall man.

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(89) a. Seymour is taller for a man than he is wide for a man.

b. Seymour is a taller man than he is a wide man.

The sentences in (88) and (89) do not allow direct comparisons. For example, the

sentences in (88) can be true even if Esme’s height is less than Seymour’s. A situation

that would be impossible with direct comparisons. Similarly, the sentences in (89)

can be false even when Seymour’s height is greater than his width. In fact, it must

be false if Seymour is short for a man but nonetheless quite wide. Seuren’s analysis

however leaves open the possibility of a perfectly coherent direct interpretation for

such sentences. For example, consider the sentence in (89a). Given the independence

of the measurement scale from the measure function, it is unlikely that the extent

of Seymour’s height would differ from the extent of his height for a man. Similarly,

it seems just as unlikely that the extent of Seymour’s width would differ from the

extent of his width for a man. Since Seymour is five feet tall and three feet wide the

truth conditions for the sentence in (89a) should be equivalent to following formula.

(90) ∃e[(0, ...1′, ...2′, ..., 5′ ⊇ e) &¬(0, ...1′, ...2′, ...3′ ⊇ e)]

Unfortunately for Seuren, this formula is no different from the one above that was

claimed to be equivalent to the interpretation of Seymour is taller than he is wide.

Yet the sentences clearly have different interpretations, one direct and the other

indirect.

In summary, Seuren (1973, 1978, 1984) provides an empirically adequate analysis

of direct comparisons but ultimately he does not develop an interesting explanation

of indirect comparisons. Furthermore, his analysis of direct comparison requires that

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the scale of degrees be independent from the adjective. This hypothesis is problematic

when considering the effect of comparison classes in forcing indirect comparisons.

Delineations.

In contrast to Wheeler (1972) and Seuren (1973), Klein (1980) proposes that the

existential quantifier involved in the interpretation of comparatives ranges over delin-

eators instead of comparison classes or extents. The precise nature of a delineator is

a little difficult to describe especially since the definition changes throughout Klein’s

work. Below, I briefly review the nature of delineators before discussing how Klein’s

theory fares with respect to the generalizations discussed in the previous chapter.

In his earlier work (Klein, 1980), delineators were adjectival modifiers that re-

duced the size of the comparison class to strengthen or weaken the application of

the adjective. To clarify exactly what this means, one must first understand how

Klein interpreted gradable adjectives. Klein viewed such adjectives as functions that

divided a contextually determined comparison class into three groups, those in the

high-end of the ordering associated with the adjective (the positive extension), those

in the low-end (the negative extension), and those that lie somewhere in between (the

middle extension). For the purpose of exposition, I will ignore the third category

since it is not a necessary component in the interpretation of comparatives.24

To better understand Klein’s proposal, consider the following characterization

of his original thesis. Klein (1980) hypothesized that the default interpretation of

a gradable adjective was a function that takes a comparison class and maps it to

the positive extension with respect to the adjective’s underlying order. I represent

such a function as follows: ADJ (C). Yet, this is not the only possible interpretation

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of an adjective in Klein’s system. An alternate interpretation exists that maps the

comparison class to the negative extension. I represent this function as follows,

ADJneg(C). Klein assumes that how the adjective divides the comparison class

depends on who or what constitutes the class. In other words, the dividing line of

who is on the positive or negative side of tall or beautiful changes from context to

context.

In Klein’s 1980 theory, delineators simply change how the adjective divides the

comparison class by modifying the comparison class through successive applications

of the adjective. A prototypical example of a delineator for Klein is the modifier

very. To gain a better understanding of delineators in general, I review Klein’s

interpretation of very in hope that it clarifies his original thesis.

Following an observation first made by Wheeler (1972), Klein noted that phrases

like very tall and very beautiful often can be paraphrased by the phrases tall for

someone who is tall or beautiful among those who are beautiful. An adequate inter-

pretation of very could take advantage of the successive application of the adjectives

in the paraphrases. This is exactly what Klein’s theory does. In his theory, very is a

function that modifies the adjective so that the new comparison class is the positive

extension of the original comparison class. For example, the interpretation of very

is equivalent to λADJ λC (ADJ(ADJ(C))) which makes the interpretation of very

tall equivalent to λC (TALL(TALL(C))). Given a comparison class C, this function

would yield the set of people who are tall among those who are tall in C.

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Klein relies on quantifying over delineators like very in his interpretation of

comparative constructions. For example, he represents the truth conditions of a

sentence like Seymour is taller than Esme is as follows.

(91) ∃D(s ∈ (D(TALL))(C)) &¬(e ∈ (D(TALL))(C))

The variable D ranges over delineators. Even with only one delineator defined, one

can see the potential of this formula to adequately account for the truth conditions

of comparatives. For example, the formula predicts that if Seymour is very tall and

Esme is not very tall then Seymour is taller than Esme. Such predictions seem

consistent with intuitions about comparison.

Although Klein (1980) originally defined all potential delineators through mod-

ification of comparison classes just like the interpretation of very, this kind of defi-

nition proved to be inconsistent.25 As an alternative characterization, Klein (1982,

1991) proposed a set of constraints on how any delineator could partition a set. He

then defined the set of delineators as the set of all functions that satisfied these

constraints.26 According to these constraints, a delineator is any function that mod-

ifies an adjective to provide an alternative partition of a comparison class consistent

with the adjective’s underlying ordering (the partition is an alternative in that it

differs from the adjectives default partition). For example, in modifying an adjective

like tall, all possible delineators move the dividing line of who counts as being tall

to include more or fewer people in the positive extension of the adjective. Whoever

ends up being in the positive extension of the modified adjective will always be taller

than everybody in the negative extension for all such delineators. Furthermore, any

delineator that includes more individuals in the positive extension for tall will also

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do so for all other gradable adjectives to which the delineator can apply. Likewise,

for any delineator that includes fewer individuals in the positive extension. As Klein

observed, all overt adjectival modifiers seem to satisfy these general conditions (very,

quite, somewhat, extremely, six feet, two feet, etc.).

I believe Klein’s analysis represents an advance with respect to direct and indi-

rect comparisons. Even when limited to overt delineators (any adjectival modifier),

one can begin to see the potential of accounting for direct and indirect comparison

with a single interpretation of the comparative morpheme.

For example, delineators such as very, extremely, quite, somewhat, nearly, and

almost are all relativized to the ordering of individuals provided by the gradable

adjective. Very yields an extension consisting of the higher end of the ordering

whether the order is in terms of beauty or intelligence. Similarly, nearly yields an

extension that includes the higher end of the negative extension of the adjective

independent of whether the adjective is tall or happy.

Interestingly, adjectives like intelligent and beautiful only accept overt modifiers

that relativize to the ordering provided by the gradable adjective. This fact explains

why sentences with these adjectives result in indirect comparisons. Consider the

sentence below with Klein’s truth conditions appearing immediately after.

(92) a. Esme is more talented than Morag is intelligent.

b. ∃D(e ∈ (D(TALENTED))(C)) &

¬(m ∈ (D(INTELLIGENT ))(C))

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If the sentence in (92) were true then according to Klein’s semantics there must be a

delineator D such that Esme is D talented and Morag is not D intelligent. Further-

more these delineators must relativize their meanings to the adjectival ordering since

talented and intelligent can only be modified by such delineators. As a result, Klein’s

interpretation compares Esme and Morag relative to their positions with respect to

the orderings in terms of talent and intelligence.

More details need to be given about delineators before the correct entailment

relations can be fully established,27 however it is clear that indirect comparisons

seems possible if certain adjectives are restricted to delineators that always relativize

their meaning to the adjectival ordering.

In contrast, Klein accounts for direct comparisons by expanding the set of delin-

eators to include those that do not relativize their meaning. Measure phrases such

as three feet and thirty inches determine a positive extension based on an external

constant. What counts as being three feet long or thirty inches wide does not change

as the comparison class changes. If an object is three feet wide in the context of

one comparison class, then it is three feet wide in all possible contexts no matter

what the comparison class might or might not contain. This is clearly different from

adjectival modifiers like very.

For Klein, a direct comparison is possible if the adjectives in the main and sub-

ordinate clauses can both be modified by the same kind of measure phrase. Consider

the sentence below with Klein’s truth conditions appearing immediately after.

(93) a. Seymour is taller than he is wide.

b. ∃D(s ∈ (D(TALL))(C)) &¬(s ∈ (D(WIDE))(C))

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This sentence is true if there is a delineator D such that Seymour is D tall but

not D wide. Unlike indirect comparisons, such a sentence is not restricted to only

considering modifiers like very and quite. For instance, the measure phrase five feet

can modify both wide and tall. This allows Klein’s interpretation of the sentence to

be true even if there is no relativized delineator D such that Seymour is D tall but

not D wide. For example, Seymour could be very wide at four feet but not very tall

at five feet. Such measurements would make (94) true.

(94) Seymour is five feet tall and he is not five feet wide.

The fact that the divide between direct and indirect comparison also patterns with

a divide between relativized and non-relativized modifiers is a powerful connection

that I believe is the key to providing a uniform interpretation of indirect and direct

comparison.

Although Klein exploits this parallel with some success, there are still some

problems with his theory. I discuss two of these problems below: one involving a

lack of ambiguity and another involving the effect of comparison classes in inducing

indirect readings.

To address the first, consider the sentences (95) and (96).

(95) Seymour is very wide but he is not very tall.

(96) Seymour is wider than he is tall.

Given the situation where Seymour is five feet tall and four feet wide, the sentence

in (95) is true. In contrast the sentence in (96) is not so easy to evaluate. It can be

true although such a reading is not preferred. As discussed above, normally direct

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comparisons are favoured over indirect when the adjectives in the main and subordi-

nate clauses are commensurable. Without contextual priming, most speakers would

consider the sentence in (96) to be false. Yet, according to Klein’s interpretation,

a false interpretation is not a possibility. The fact that a delineator like very can

make the conjunct in (95) true entails that the sentence in (96) must be true. Klein’s

uniform interpretation of comparatives does not permit ambiguous truth conditions

for a single sentence.

I believe that this problem arises because of the strong connection that Klein

makes between direct and indirect comparisons. Although it is empirically beneficial

to link the two types of comparison to a single interpretation, there still must be

some means of deriving an ambiguity. Klein’s theory in its present form is unable to

account for such an ambiguity.

The second problem for Klein’s theory pertains to the interaction between com-

parison classes and indirect comparison. As discussed above, comparison classes can

induce an indirect comparison when overtly appearing in both the main and subor-

dinate clauses. Yet, in Klein’s theory the direct comparisons should not be affected

by comparison classes since they are based on delineators that are not affected by

comparison classes. For example, consider the sentence below.

(97) Seymour is a five foot tall man but he is not a five foot wide man.

This sentence is true given the circumstance where Seymour is five feet tall and four

feet wide. In contrast, the following sentence is not true.

(98) Seymour is a taller man than he is a wide man.

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Nothing in Klein’s theory explains this contrast. The interpretation of the sentence

in (98) should be true if there exists a delineator D such that Seymour is a D tall

man but not a D wide man. As (97) demonstrates, such a delineator exists, namely

the delineator five foot. Klein’s theory suggests that sentences like (98) should permit

direct comparisons.

In summary, Klein’s theory of delineators is the first that attempts to unify direct

and indirect comparison. However, he is unable to explain why certain ambiguities

exist and why certain constructions do not allow direct comparisons.

3.1.2 Non-conjunctive comparisons

Unlike conjunctive comparisons, non-conjunctive comparisons relate measure-

ments in the main clause to measurements in the subordinate clause directly. This

direct relation is represented either as a strict linear order or a non-strict linear order.

Interestingly, a linear order is by definition transitive and asymmetric, hence there is

no need to invoke meta-principles to account for Wheeler’s generalization. Although

this marks a significant difference from the theories reviewed in the previous sec-

tion, many of the problems associated with conjunctive comparisons still persist for

non-conjunctive comparisons.

Below I highlight the potential benefits of using a linear order directly in the

interpretation of comparatives while also discussing some of the problems. I begin

with theories that implement degrees in their analysis before considering approaches

involving extents.

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Degrees.

Many theories relate the comparative morpheme to a strict ordering of degrees

symbolized by the greater-than symbol ‘>’. In such theories, the main and subordi-

nate clauses determine the degrees that are compared through this relation. Although

this analysis is common in the literature on comparison, many theories do not discuss

how differences in interpretation can be induced by different adjectives (see, Bartsch

& Vennemann, 1972; Hellan, 1981; Heim, 1985, 2000; Hackl 2000). Such theories are

too vague to adequately account for the generalizations discussed above and hence

are not relevant for the present purposes. Cresswell (1976) is the first theory with

degrees to provide a meaning for the comparative morpheme that is derived from

the adjectival phrase. Since his theory is the only one that contains sufficient detail

to address the issues discussed in the last chapter, it is his theory that I will take as

the prototypical example of a degree account.

Like the other theories with degrees in their ontology, Cresswell (1976) asso-

ciates the main and subordinate clause with degrees of measurement.28 In a sen-

tence such as Seymour is taller than Esme is, the main clause is associated with the

measurement of Seymour’s height while the subordinate clause is associated with a

measurement of Esme’s height. The adjective provides the connection between the

individuals and their respective measurements.

In contrast with the other theories, Cresswell’s measurements are not just a

value in a measurement scale. They are values plus the strict linear ordering that

places them in a scale. Thus, a measurement of Seymour’s height might be something

like the following ordered pair, 〈ds , > x 〉, where ds is the value of his height and ‘> x ’

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is a linear order on such values. (The subscript x distinguishes different types of

linear orders.) Similarly the measurement of Esme’s height would be something like

〈de , > x 〉.

For Cresswell, the locus of comparison is a relation between measurements. He

represents this relation by writing more in all-caps (e.g., MORE). This relation

is defined only when the linear order associated with the measurement in the main

clause matches the linear order associated with the measurement in the subordinate

clause. When it is defined, it compares measurements by taking the strict linear order

from the measurement associated with the main clause and using this linear order

to compare the degree from the main clause with the degree from the subordinate

clause. Thus truth conditions for MORE(〈ds , > x 〉, 〈dm , > y〉) can be specified as

follows.

(99) MORE(〈ds , > x 〉, 〈dm , > y〉) is defined if and only if x = y.

When defined, it is true if and only if ds > x dm .

There are several interesting aspects to this proposal. First, Cresswell does

not need to invoke a meta-principle to account for Wheeler’s generalization. The

relation MORE compares degrees through the strict linear order associated with its

first argument. A strict linear order is always transitive and asymmetric, hence the

relation MORE itself is transitive and asymmetric. Transitivity and asymmetry are

inherited from the underlying strict linear ordering associated with the degrees.

Another interesting aspect of Cresswell’s analysis is that there are no means

of deriving a direct comparison for polar opposite adjectives or other types of in-

commensurable adjectives. In fact, in his theory polar opposite adjectives and other

92

types of incommensurability receive the same kind of treatment. Let me briefly

outline how this is the case.

For Cresswell, direct comparisons are only licensed when the measurements as-

sociated with the main and subordinate clauses share the same underlying strict

linear order. With this in mind, consider the sentences below with Cresswell’s truth

conditions appearing immediately after.

(100) a. Esme is more intelligent than Marilyn Monroe is talented.

b. MORE(〈de , > INTELL〉, 〈dm , > TALENT 〉)

(101) a. Esme is taller than Seymour is short.

b. MORE(〈de , > TALL〉, 〈ds , < TALL〉)

Both of the sentences above only receive an indirect interpretation. In both cases,

the degree associated with the main clause has a different strict linear order than the

degree associated with the subordinate clause. In the first sentence the two strict

linear orders do not even share the same domain (Cresswell assumes that degrees of

intelligence are different from degrees of talent). In the second, the two strict linear

orders share the same domain but order this domain differently (in fact the ordering

is reversed). For both interpretations, Cresswell’s MORE relation is not applicable.

Despite these advantages, some problems arise with Cresswell’s theory. For

example, Cresswell does not provide an interpretation for indirect comparison. Also,

like Seuren (1973), the independent nature of a scale of degrees make it difficult for

the theory to account for the effects of comparison classes on incommensurability.

Below I discuss both of these problems in more detail.

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I begin with the problem of indirect comparison. Within the main text at least,

Cresswell (1976) seems to maintain that the only grammatical interpretation of com-

paratives involves direct readings. He mentions the possibility of an indirect reading

only in a footnote, and even there he simply comments that such interpretations

must involve treating normally incompatible measurements as if they belonged to

the same measurement scale. In other words, he suggests that such readings are

metaphorical extensions of direct readings.

Others who worked on the degree analysis of comparatives did not treat the

indirect comparison so derivatively. Bartsch & Vennemann (1972), for example,

maintained that indirect comparisons constitute separate grammatical readings and

provided an analysis of such readings. Since Cresswell did not provide a concrete

proposal, I will review Bartsch & Vennemann’s analysis of indirect comparisons as

representative of the degree approach.

Bartsch & Vennemann (1972) hypothesized an ambiguity in the interpretation

of the comparative morpheme, one interpretation being direct and the other indirect.

Indirect interpretations have two properties different from those of direct interpreta-

tions. First, it has a function ‘∗’ that maps measurements to another measurement

on a secondary scale. Second, it only considers measurements that are above the

average value for the measurement scale associated with the adjective. As discussed

earlier, Bartsch & Vennemann believed that indirect comparisons had different im-

plications from direct comparisons. In particular, they believed that sentences like

Esme is more beautiful than Marie Curie is intelligent presuppose that Esme is beau-

tiful and Marie Curie is intelligent. Although I will present their analysis below, I

94

should remind the reader that this interpretation is based on an empirically incor-

rect generalization. The presuppositions discussed by Bartsch & Vennemann are not

connected to indirect comparisons. Putting this problem aside, let me discuss some

of the other aspects of their proposal.

In adapting Bartsch & Vennemann’s account to Cresswell’s proposal, the truth

conditions for the sentence Esme is more beautiful than Marie Curie is intelligent

should be equivalent to the following formula.

(102) MORE(∗(〈(de −NBEAUT ), > BEAUT 〉), ∗(〈(dc −N INTELL), > INTELL〉)),

where de is the degree of Esme’s beauty and dc is the degree of Marie Curie’s

intelligence.

The N ’s in this formula represent the contextually determined norm for each

adjective (under their proposal the norm is the average degree relative to the com-

parison class). The values associated with the main and subordinate clause are the

measurements normally associated with the direct interpretation less their respective

norms, where subtraction operates on degrees much like it does on numbers. The

comparison is done with respect to these relativized values.

The most important aspect of this proposal rests on the meaning of the func-

tion ‘∗’. Bartsch & Venneman (1972) do not give much detail about this function,

although they do make their intentions clear. The ‘∗’ function relativizes a measure-

ment to a numerical scale by factoring out the unit of measurement. This seems

easy enough to do with scales of weight or height. If the degree consists of pounds,

then the ‘∗’ value is that degree divided by 1 pound. The result is a numerical

value independent of the unit of measurement. For example, ∗(〈6po, > po〉) = 〈6, >〉,

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where ‘>’ is a strict ordering of the natural numbers. Similarly, if the degree con-

sists of inches, then the ‘∗’ value is that degree divided by 1 inch. For example,

∗(〈6′′, > inches〉) = 〈6, >〉.

Problems arise however when one considers the ‘∗’ function with respect to

beauty and intelligence. These rankings do not have common numerical values that

can be factored out. Furthermore, it is with these type of rankings that indirect

comparisons are most prevalent. Without proposing a detailed analysis of such ad-

jectives, the ‘∗’ function is about as descriptively adequate as Cresswell’s (1976)

footnote.

The second problem for the degree analysis is that it does not provide an ade-

quate account of why comparison classes induce indirect comparisons. Like Seuren

(1973), Kennedy (1999) and Bierwisch (1987), Cresswell (1976) accounts for direct

comparisons by allowing the measurement scales to exist separately from the adjec-

tive. Thus two adjectives can relate individuals to values on the same measurement

scale. This is precisely how Cresswell interprets tall and wide. Both adjectives relate

individuals to measurements (in inches, feet, meters etc.) on the same measurement

scale albeit in different ways (the measurement of Seymour’s height has a differ-

ent value than the measurement of his width). The shared scale explains why the

sentence Seymour is taller than he is wide can receive a direct interpretation: the

measurements of height involve the same strict linear ordering as measurements of

width. However this kind of explanation brings with it certain problems. Consider

the sentences below which do not have a direct interpretation.

(103) Seymour is taller for a man than he is wide for a man.

96

(104) Seymour is a taller man than he is a wide man.

The independence of the measurement scale makes the wrong kind of prediction for

these sentences. There is no reason why the value of Seymour’s height as a tall man

should be different from the value of his height as a person. The same holds for width.

Yet the sentences above cannot be evaluated in terms of a direct comparison.29

In summary, despite some interesting advantages of the degree approach to

comparison, it still suffers from the same kind of problems as conjunctive comparison.

The theory does not adequately account for indirect comparisons and it does not

sufficiently explain the effects of comparison classes on comparative constructions.

Extents.

Similar to Seuren’s (1973) theory, the proposals of von Stechow (1984b), Bier-

wisch (1987), and Kennedy (1999) all employ extents to account for comparative

constructions. The main difference being that instead of comparing extents through

conjunctive propositions, these authors compare the extents directly either through

a strict superset relation (Kennedy, 1999) or a set difference operator (von Stechow,

1984b; Bierwisch, 1987). In what follows, I first review the nature of extents and

measuring functions. I then demonstrate how such theories are homomorphic30 to

the degree analysis proposed by Cresswell (1976). As a result, comparing extents

through the subset relation shares the same benefits and problems as comparing

degrees through a linear order.

Since it is one of the clearer and more thorough presentations, I adopt Kennedy’s

(1999) analysis of extents as representative of the class as a whole. This is essentially

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an arbitrary choice although I do find it convenient to forgo the details of the dif-

ferentials discussed in von Stechow (1984b) and Bierwisch (1987). Still, this section

could be written taking any of the three authors as representative.31

Kennedy (1999) argues that gradable adjectives are measure functions in that

they map individuals to measurements. This idea was originally advanced in Bartsch

& Vennemann (1972). The main difference between Kennedy (1999) and Bartsch

& Vennemann (1972) is that Kennedy represented measurements as extents rather

than degrees. For Kennedy, an extent consists of a set of degrees from a single scale.

There are two types of extents in his theory (as well as von Stechow’s, 1984b), those

consisting of all the degrees between the zero element of the scale and a measurement,

and those consisting of all the degrees between the highest degree in the scale (or

infinity if there is no highest degree) and a measurement.32 The extents in (a) below

are examples of the first kind whereas the extents in (b) are examples of the second.

The scale in these examples consists of inches and feet.

(105) a. i. 0, ...6′′, ...1′..., 1′6′′..., 2′

ii. 0, ...6′′, ...1′..., 1′6′′..., 2′..., 2′6′′..., 3′..., 3′6′′

iii. 0, ...6′′, ...1′..., 1′6′′..., 2′..., 3′..., 4′..., 5′

b. i. 2′, ...2′6′′, ...3′..., 3′6′′..., 4′...,∞

ii. 3′6′′, ...4′, ...4′6′..., 5..., 5′6′′...,∞

iii. 5′, ...6′, ...7′..., 8′..., 9′..., 10′..., 11′...,∞

To understand why two kinds of extents are needed, I first need to explain

how Kennedy (1999) interpreted comparison constructions in general. Simplifying

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greatly, Kennedy basically associated the main clause and the subordinate clause

with extents. The extent associated with the main clause is obtained by applying

the main clause adjective to the clausal subject. The extent associated with the

subordinate clause is obtained by applying the subordinate adjective (which might

be identical to the main clause adjective) to the subject of the subordinate clause.33

These extents are compared with a modified superset relation. Unlike the regular

superset relation it has three possible results rather than two. Given any two sets,

the superset relation between them is either true, false or undefined. It is true if

and only if the first extent contains all the degrees in the second extent but not vice

versa (X ⊃ Y = 1 iff [X ∩ Y = Y ]&[X 6= Y ]). The relation is false if and only if

the second extent contains all the degrees in the first (X ⊃ Y = 0 iff X ∩ Y = X).

Otherwise, the relation is undefined.

Given these details, the truth conditions for Seymour is taller than Esme is ends

up being equivalent to the truth conditions for the formula below.

(106) TALL(s) ⊃ TALL(e)

As it so happens, TALL for Kennedy (1999) maps individuals to extents from zero

to the measurement of the individual’s height. For example, if Seymour were six feet

tall and Esme five feet, then the formula above would be extensionally equivalent to

the one below.

(107) 0..., 1′..., 2′..., 5′..., 6′ ⊃ 0..., 1′..., 2′..., 5′

This formula is true since the first extent contains all the degrees that the second

extent contains but not vice versa. In contrast, the sentence Esme is taller than

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Seymour is would be false. This sentence has truth conditions equivalent to the

formulae below.

(108) a. TALL(s) ⊃ TALL(e)

b. 0..., 1′..., 2′..., 5′ ⊃ 0..., 1′..., 2′, ..., 5′..., 6′

Given the heights of Seymour and Esme mentioned above, the second formula ends

up being extensionally equivalent to the first. As demonstrated, the second extent

contains all the degrees that the first contains, hence the falsity of both formulae.

As mentioned earlier, Kennedy’s (1999) system (like von Stechow’s, 1984b) con-

tains two kinds of extents. We have already seen an example involving one kind,

namely extents that contain the degrees from zero to a specified limit. I have not yet

given any examples involving the other kind of extents, the ones that contain all the

degrees from some measurement to the topmost element (or infinity). These extents

constitute the range of so-call negative gradable adjectives.

As discussed in Kennedy (1999), Bierwisch (1987) and Seuren (1978), gradable

adjectives can often be grouped in pairs that seem to relate to the same measurement

scale. For example, tall and short constitute one such pair. They both seem to

involve the scale of inches and feet. Within each pair, one adjective can often be

classified as positive, whereas the other can be classified as negative. A positive form

can be identified by the fact that it allows measure phrases such as John is four feet

tall. A negative form cannot take such phrases (*John is four feet short). In our

example pair, the adjective tall is positive whereas short is negative.

Short unlike tall maps individuals to extents that contain all the degrees be-

tween the height of the individual and the highest degree in the scale (or infinity).

100

Thus, while TALL(s) yields the extent 0..., 1′..., 2′..., 5′..., 6′, SHORT (s) yields the

extent 6′..., 7′..., 8′..., 9′...,∞. Such a representation allows the superset relation to

correctly interpret sentences such as Esme is shorter than Seymour is. According to

the analysis outlined above, this sentence should have truth conditions equivalent to

the following formula.

(109) SHORT (e) ⊃ SHORT (s)

Given Kennedy’s interpretation of negative adjectives, this formula is equivalent to

the one below.

(110) 5′..., 6′..., 7′..., 8′...,∞ ⊃ 6′..., 7′..., 8′...,∞

This formula is true since the first extent contains all the degrees in the second

(and not vice versa). In contrast, the sentence Seymour is shorter than Esme is is

false under Kennedy’s analysis. This sentence has truth conditions equivalent to the

following formula.

(111) a. SHORT (s) ⊃ SHORT (e)

b. 6′..., 7′..., 8′...,∞ ⊃ 5′..., 6′..., 7′..., 8...,∞

The second formula is a further specification of the first. Note that in this formula,

the second extent contains all the degrees that the first contains, hence its falsity.

One benefit of Kennedy’s method of comparison is that comparative construc-

tions that contain polar opposite adjectives in the main and subordinate clause are

ill-formed under the standard interpretation. For example, the sentence Seymour

is taller than Esme is short is not interpretable under the current analysis. Such a

sentence would be equivalent to the following formula.

101

(112) TALL(s) ⊃ SHORT (e)

By applying TALL and SHORT to s and e respectively, we obtain the formula in

(113).

(113) 0..., 1′..., 2′..., 6′ ⊃ 5′..., 6′..., 7′..., 8...,∞

Notice that the first extent does not contain all the degrees in the second. Further-

more, the second does not contain all the degrees in the first. Hence, by Kennedy’s

definition the comparison is not defined.

Like Bartsch & Vennemann (1972), Kennedy provides an alternate interpreta-

tion of such sentences that compares Seymour’s deviation from the norm of tallness

to Esme’s deviation from the norm of shortness. The details of this interpretation

are basically identical to Bartsch & Vennemann’s so I will not review them here.

There are several benefits and problems with Kennedy’s analysis that seem

to overlap with Cresswell’s (1976) proposal involving degrees. This should not be

surprising. Like Cresswell’s measurements, Kennedy’s extents not only encode a

precise measurement (the upper bound for positive extents, the lower bound for

negative extents) but they also partially encode other elements in the scale. Positive

extents include everything in the scale below the measurement, negative extents

include every thing in the scale above the measurement. Hence, the subset relation

not only compares measurements but also partially compares whether the scales

to which the measurements belong are compatible. Essentially the quirky truth

conditions for Kennedy’s subset relation check to see if two extents are formed from

the same scale and match in polarity. Cresswell’s MORE relation did the same

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thing when comparing degrees. In fact, one can prove that the two systems are

equivalent.34 I provide a proof to such an effect in the Appendix.

As a result of this equivalency, the criticisms advanced against Cresswell’s sys-

tem hold equally well for Kennedy’s. Due to the independence of the scales from

the measure functions, Kennedy cannot explain why tall and wide cannot compare

measurements directly when constrained by a comparison class or a nominal, as in

Seymour is taller for a boy than he is wide or Seymour is a taller boy than he is a

wide boy. Like Cresswell’s theory, Kennedy’s theory is not compatible with extents

changing due to the introduction of comparison classes. Also, due to the vague spec-

ification of the ‘∗’ operator borrowed from Bartsch & Vennemann (1972), indirect

comparison is not given a precise interpretation especially when sentences involve

non-measurable attributes such as beauty and talent.

3.1.3 Summary of Problems

There are two main problems that almost all theories of comparison encounter.

First, in order to account for direct comparisons, most theories separate the scale

of measurement from the interpretation of the adjective. In other words, the scale

exists independently of the adjective. However, such a separation renders it difficult

to explain the effect of comparison classes forcing indirect comparisons. Comparison

classes seem to redefine the scale of measurement by modifying the adjective.

Second, no theory is able to provide an adequate account of indirect comparison.

Most theories do not even attempt to provide an account (Cresswell, 1976; Seuren,

1973; Wheeler, 1972; von Stechow, 1984) and those that do (Klein, 1980, 1982;

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Bartsch & Vennemann, 1972; Kennedy, 1999) either have unsubstantiated predictions

or are too vaguely specified to make any predictions.

With these problems clearly defined, the goal of a semantic theory of comparison

becomes evident. An adequate theory must be able to provide an empirically accurate

account of indirect comparison that is able to explain the effect of comparison classes

in comparative constructions. It is with this goal in mind that I present my theory

in chapter 4.

3.2 Apologia: An Explanation of Omissions

Some of the major issues discussed in the literature on comparatives have not

been discussed in the review above and will not be discussed in the rest of this thesis.

Some of the issues are omitted since a theory of universal degrees does not offer any

further insight into the phenomena. For example, any of the theoretical accounts of

the behaviour of quantifiers in subordinate clauses is consistent with the basic ontol-

ogy for degrees developed in the next chapter.35 Also, universal degrees have nothing

new to contribute to the basic explanations of the interaction between subordinate

clauses and Negative Polarity Items (NPIs) such as ever and anyone.36 37 How-

ever, two of these omitted issues require some justification since they have played

a significant role in the development of previous semantic theories of comparison,

both in terms of the comparative relation and the ontology of degrees. For exam-

ple, much debate within the literature has centered around whether the comparative

morpheme should be interpreted as a generalized quantifier or not (see Cresswell,

1976; von Stechow, 1984; Kennedy, 1999; Heim, 2000; Hackl, 2000). Similarly, many

authors believe that differential expressions represent a major data point that must

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be accounted for by any adequate theory of comparison (see, Hellan, 1981; von Ste-

chow, 1984; Klein, 1991). In the two sections that follow, I explain why I do not

address either of these issues.

3.2.1 Generalized Quantification Over Sets of Degrees

One of the key issues in the literature on comparison is whether the comparative

morpheme should be treated as a generalized quantifier operating on sets of degrees.

Such a thesis has been supported by Cresswell (1976), Moltmann (1992), Hackl

(2000), and Heim (2000) and refuted by von Stechow (1984a) and Kennedy (1999).

As correctly characterized by Kennedy (1999), the key empirical issues involved in

this debate concern whether the interpretation of the comparative morpheme leads

to scopal ambiguities or not. If comparison involved generalized quantification, then

by analogy to nominal quantifiers and quantifier raising, scopal ambiguities would

be expected. Yet, if comparison involved no such quantification then there should

be no such ambiguities.38

Much of the evidence points to a lack of ambiguity. As Kennedy (1999) notes,

the comparative morpheme demonstrates no clear-cut ambiguity with respect to

negation, nominal quantifiers (see also Heim, 2000), nor most intensional operators

(such as think, want, and believe). Also, as noted by von Stechow (1984a), most of

the ambiguities substantiated within the comparative literature (see Russell, 1905;

Cresswell, 1976; Hellan, 1981; Hoeksema, 1984; Larson, 1988) can be accounted for

through the interpretation of the subordinate clause rather than the comparative

morpheme itself.

105

However there are some exceptions. There are a few intensional operators that

demonstrate evidence of an ambiguity. As discussed in Heim (2000) and Hackl

(2000),39 verbs such as require and allow40 trigger an ambiguity when their comple-

ments contain comparative clauses.41

It seems to me that the empirical evidence cannot decide between these two

options. By adopting a quantificational perspective on comparison (as Heim, 2000,

does), one is able to explain the ambiguity that exists with a couple of lexical items,

but it remains a mystery why other sentential contexts do not demonstrate any

ambiguity. By adopting a non-quantificational perspective (as Kennedy, 1999, does)

one can explain the lack of ambiguity in most syntactic contexts, but one cannot

explain why an ambiguity appears with a few choice lexical items.

Choosing arbitrarily, I will adopt a position where the comparative morpheme

is not interpreted as a generalized quantifier. By doing so, I will be able to provide

an interpretation (much like Kennedy, 1999, does) based on the surface position

of the comparative morpheme. No covert syntactic movement will be required. I

believe that this will simplify the presentation somewhat. However, it should be

noted that the issues and generalizations addressed in the next two chapters are

not dependent on whether the comparative morpheme is a generalized quantifier

or not. Both kinds of interpretation involve a partial ordering of degrees, and it

is the nature of the partial order and the ontology of the degrees that is relevant

when addressing questions about comparison classes and contrasts between direct

and indirect comparison.

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3.2.2 Differentials

Many theories of comparison take differential expressions as an essential data

point for building an adequate account of comparative constructions (see, Hellan,

1981; von Stechow, 1984a, and Schwarzschild & Wilkinson, 2002). Differential ex-

pressions modify the comparative morpheme specifying a difference between two

measurements. As examples of such expressions, the phrases two inches, three min-

utes and much appear before the comparative morpheme in the sentences below.

(114) a. Seymour is two inches taller than Sam is.

b. This meeting was three minutes longer than the last one was.

c. Morag is much more beautiful than Susi.

Theories such as Hellan’s (1981), von Stechow’s (1984b), and Schwarzschild & Wilkin-

son’s (2002) suggest that such expressions favour a treatment of the locus of compar-

ison as a relation that compares differences, even when the comparative morpheme

appears without modification. Unlike these works (and like Klein, 1980, 1982), I do

not consider differential expressions to be that important of a factor in motivating a

semantics for the locus of comparison. There are three reasons to treat differential

expressions as a minor data point. First, a large majority of adjectives do not per-

mit differential expressions other than much. Second, differential expressions cannot

appear in all the same syntactic environments as more basic forms of comparison.

Thus, they truly seem to require a separate account. Third, theories of differential

comparison are dependent on more basic concepts involving partial orders. In de-

veloping a theory that does not compare differences, I still remain consistent with

a theory that provides a more detailed account. By avoiding the issues involved in

107

differential expressions, I can focus on more central issues for a general theory of

comparison, such as the nature of grammaticalized partial orders. In what follows, I

briefly outline each of these reasons in more detail.

There seems to be a limited number of adjectives that allow differential ex-

pressions as modifiers (that is, differential expressions other than much). All such

adjectives are associated with non-linguistic measurements, such as time (minutes),

distance (feet), area (square feet), volume (cubic feet), temperature (degrees Cel-

sius), colour (shades) and loudness (decibels). The list below is nearly exhaustive

(modulo the colour terms).

(115) a. tall, short, wide, narrow, high, low, deep, shallow, long, large, big, small,

hot, warm, cold, loud, quiet, red, yellow, blue, dark, and light

b. two feet taller, two feet shorter, two feet wider, one inch narrower, one

foot higher, two inches lower, six feet deeper, one inch shallower, two feet

longer, three cubic feet larger, one square foot smaller, three degrees hot-

ter, two degrees warmer, two degree colder, ten decibels louder, 5 decibels

quieter, three shades greener, two shades darker etc.

In contrast, adjectives that do not allow such expressions cannot be listed in such a

limited space. Not only do they include all lexical adjectives not mentioned above

(happy, sad, handsome, ugly, intelligent, stupid, funny, serious, clear, opaque, hard,

soft, lazy, social, common, etc.), they also include adjectives formed by the mor-

phemes -y (cloudy, misty, angry, watery, muddy, grainy, dirty, filthy, etc.), -ly (manly,

womanly, etc.), -ed (worried, frustrated, disturbed, disillusioned, interested, etc.), -ing

(interesting, fascinating, indulging, intriguing, disturbing, etc.), -ive (creative, active,

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etc.), -al (musical, fundamental, instrumental, etc.), -ic (artistic, fantastic, etc.),

and -ful (beautiful, sorrowful, hopeful, etc.). Given this distribution of differential

expressions, I believe it is reasonable to treat non-differential comparatives from a

more general perspective, reserving differential comparisons as special cases related

to external measurement devices.

Not only are differential expressions limited to a certain class of adjectives, they

are also limited to a certain class of syntactic constructions. Differential expressions

(other than much) are only permitted in sentences when the comparative involves a

predicative use of the adjective. Attributive constructions do not allow such expres-

sions. For example, the sentences below all contain attributive uses of the gradable

adjective. However, only the sentences without the differential modifiers are well-

formed.

(116) a. Seymour is a taller man than Harold.

b. ?? Seymour is a one inch taller man than Harold.

c. * Seymour is an exactly one inch taller man than Harold.

(117) a. This is a warmer room than the one we were in yesterday.

b. ?? This is a ten degree warmer room than the one we were in yesterday.

c. * This is an exactly ten degree warmer room than the one we were in

yesterday.

Such a contrast demonstrates two facts. First, there is systematic difference between

basic comparatives and differential comparatives. This contrast suggests that basic

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comparatives should not be treated as involving hidden differential expressions. Sec-

ond, differential expressions only appear in a subset of syntactic environments. This

suggests once again that they should be treated as a special case separate from more

basic forms of comparison.

Finally, putting aside issues about the marginal status of differentials, I would

like to draw attention to the fact that most differential expressions can be integrated

into a relation that does not measure differences. As a consequence differentials need

not influence the semantics of the locus of comparison. For instance, let’s assume that

basic comparisons involve a partial ordering relation () that relates a measurement

associated with the main clause (call it dmain) with a measurement associated with

the subordinate clause (call it dsub). Hence the truth conditions of most comparatives

can be characterized by the following expression.

(118) dmain dsub

As noted by Hellan (1981) and von Stechow (1984), if a scale is linearly ordered then

an addition operator (+) can be defined through the linear ordering of the scale.

This can be done with the conditions outlined below.

(119) Conditions on + relative to . ∀d1 , d2 ,

1. d1 + d2 = d2 + d1

2. d1 + d2 d1

3. if d1 d2 then ∃d(d1 = d2 + d)

4. if d1 d2 then ∀d(d1 + d d2 + d)

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So conceived, differential expressions can simply be viewed as modifiers on the degrees

associated with the subordinate clause. The non-differential representation of the

locus of comparison can be kept constant. For example, the truth conditions of

Seymour is two inches taller than Esme is can be represented as follows.

(120) dmain d2 ′′ + dsub

Differentials can be built on top of a more basic theory of comparison that only

involves comparison through a partial order.

In summary, differential expressions are marginalized both lexically and syntac-

tically. Furthermore, an account of such expressions can be built on top of a more

basic semantics of comparison that does not involve addition or the measurement of

differences. For these reasons, I will adopt a position which assumes that the locus of

comparison involves comparing two degrees through a basic partial order. Semantic

issues involving addition and differential expressions will be treated as special cases

that build on top of these more basic concepts.

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Notes

16Theories that do not address these generalizations or simply adopt the theo-retical machinery of earlier works will not be considered in detail (Larson, 1988;Schwarzschild & Wilkinson, 2002; Hellan, 1981; Hackl, 2000; Heim, 1985, 2000;Gawron, 1995; etc.)

17The term extent is being used here in its non-technical sense.

18Note that the truth or falsity of every comparative in Wheeler’s analysis can bedecided based on the comparison class consisting only of the two individuals beingcompared. For example, ‘a is taller than b’ is true if and only if a is tall relative tothe comparison class a, b and b is not tall relative to the same comparison class.

19Wheeler actually preferred a non-conjunctive system of comparison withoutquantification over comparison classes. In this theory, the subordinate clauses wereinterpreted as comparison classes. However, a conjunctive version of his theory ismentioned by him as an equivalent alternative. Furthermore, the comparison-classanalysis of comparatives is probably best understood through the conjunctive anal-ysis, especially given its parallelisms to other theories of comparison.

20Technically he classified this principle as meta-linguistic knowledge rather thanas part of the grammatical system. However, including it in the grammar makeslittle difference to his general account.

21As discussed by Cresswell (1976), comparisons with two different adjectives inthe main and subordinate clauses provides the central motivation for adopting adegree analysis or some type of equivalent.

22There are several other minor problems with Seuren’s theory that I do notaddress. As an example, unlike Wheeler (1972) Seuren does not provide a meta-principle on extents that would guarantee transitive and asymmetric entailmentsindependent of the interpretation of the adjective. However, since this flaw can beeasily remedied by simply adapting Wheeler’s meta-principle for comparison classesto extents, I will not discuss this problem here.

23Seuren (1984) prefers the latter tactic. In adapting his discussion to the presentpresentation, Seuren basically stipulates that for A ⊇ B to be false, B must be asuperset of A.

112

24The partition of a comparison class into three is a property of gradable adjec-tives that is often over-emphasized within Klein’s work (1980, 1982). His theory ofdelineations is consistent with a two-way partition of a comparison class. Further-more not all gradable adjectives support a tripartite division. As mentioned in theintroduction, adjectives like full, empty, wet and dry seem to support a bipartitedivision.

25Successive applications allows for delineators such as the following:D1 = λADJ λC (ADJ(ADJneg(C))).This delineator is almost the same as very except that the first application ofthe adjective selects for the negative extension rather than the positive. The re-sult of applying this delineator to an adjective like tall would be equivalent toλC(TALL(TALLneg(C))). As opposed to very tall, which when applied to a com-parison class yielded the taller people among those who are tall, this function yieldsthe set of taller people among those who are short. Interestingly, the two sets arecompletely distinct. This is problematic for Klein’s original representation of com-paratives. Recall that Klein represented the truth conditions of the sentence Seymouris taller than Esme is as ∃D(s ∈ (D(TALL))(C)) &¬(e ∈ (D(TALL))(C)). Dueto D1, this formula ends up being true when Seymour is tall for a short person yetEsme is tall for a tall person. D1 applied to TALL will only contain the people whoare tall among the short people. It will not contain anyone who is not short.

26 It is interesting to note that Klein’s main constraint called the Consistency Postu-late is basically identical to Wheeler’s meta-principle constraining comparison classes:The only difference is that instead of quantifying over comparison classes, this postu-late quantifies over delineators. For example, compare the definition of the constraintbelow to Wheeler’s meta-principle specified earlier in this section.

(121) CONSISTENCY POSTULATE:∀x, y, ADJ, CIF ∃D((x ∈ (D(ADJ))(C))&¬(y ∈ (D(ADJ))(C)))THEN ¬∃D′((y ∈ (D′(ADJ))(C))&¬(x ∈ (D′(ADJ))(C)))

27The entailment relations can be derived in the following way. First, a looseordering can be given to the overt delineators like very, somewhat, and nearly bycomparing how much of the positive part of the extension they include. The adjectivenearly usually yields an extension that is larger than somewhat (when applied tothe same adjective) which yields an extension larger than very etc. An order canassigned through these extensions. For any two delineators D and D′, If when applied

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to any adjective (ADJ) and context (C) D yields an extension that is a subset of[(D′(ADJ))(C)], then D is greater than D′. By this ordering, very is greater thansomewhat which is greater than nearly. It seems reasonable to conclude that ifsomeone is very beautiful then they are more than somewhat beautiful since very isgreater than somewhat. Similarly if someone is somewhat beautiful then they aremore than nearly beautiful. With this ordering in mind, consider the interpretationin (92). This interpretation states that there is a delineator D such that Esme is Dbeautiful and Morag is not D intelligent. Suppose there is a D2 such that Morag isnot D2 intelligent. By the consistency postulate, D must be greater than D2 . HenceEsme must be more than D2 beautiful.

28To be more accurate, he interprets them as degree predicates, however treatingCresswell as if he associates them directly with degrees will it make it easier toanalyze his theory.

29There is one possible explanation for the lack of a direct reading. In Bartsch& Vennemann (1972) the indirect interpretation crucially involves an average value(symbolized by N above) that is calculated in terms of a comparison class. Thedirect interpretation has no such average value. Thus perhaps the existence of overtcomparison classes is only compatible with the indirect interpretation.

I do not find this explanation to be very convincing. It rests on the assumptionthat an average value is needed for indirect comparisons but not direct comparisons.Yet this assumption is not empirically well motivated. Recall that Bartsch & Ven-neman (1972) (like Kennedy, 1999) use deviation from an average value to explainwhy the sentence Seymour is more tall than he is wide presupposes that Seymouris tall and wide. As I have already discussed, these presuppositions do not holdfor all indirect comparison. It is perfectly acceptable to inform someone of my ownlack of intelligence with the sentence Medusa is more beautiful than I am intelligent.The evidence for a distinction between direct and indirect comparison in terms ofcalculating average values seems rather weak.

Note also that this kind of explanation would suggest that the sentence Seymouris a taller man than Pat is actually involves an indirect comparison. Although thispresents no empirical difficulties as far as I can see, it does run contrary to generalassumptions. I do not press the issue here since I will be advancing a thesis wheredirect and indirect comparison have no grammatical distinction.

30 In fact, they are monomorphic since the homomorphism is injective.

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31von Stechow (1984b) is essentially equivalent to Kennedy (1999) except it doesnot provide as much detail about measure functions and the like. Also von Stechow’slocus of comparison is more on par with Hellan’s (1981) analysis involving differentials(an extent a is greater than b if there is a positive difference between a and b).Differentials will not be discussed in this thesis so adopting von Stechow’s analysiswould take us a little off topic. Bierwisch (1987), which is as detailed as Kennedy(1999), achieves the same basic results as Kennedy although by slightly differentmeans. Crucially he makes use of the properties of additions and subtraction in hisalgebra of extents. Although these differences might be significant in many empiricalrespects, such differences are not significant in terms of the criticisms advanced inthis section. Hence they will not be discussed here.

32Bierwisch (1987) only had one kind of extent, the one consisting of all the degreesbetween the zero element any (positive) degree d.

33Of course the details are much more complicated. The extent associated withthe subordinate clause is actually chosen from a set of extents that are abstractedout of the embedded sentence. The maximal extent is chosen from this set.

34There are several syntactic differences that result in some semantic ambiguitiesin Kennedy’s (1999) proposal that are not present in Cresswell (1976). Also, Cress-well (1976) discusses a greater variety of comparative constructions that Kennedy’sanalysis does not address. However, these differences are not of much concern forthe present discussion.

35The ontology of degrees is not even an issue for theories that employ semanticmechanisms that essentially broaden the scope of the embedded quantifier as inBierwisch (1987) and Larson (1988). Whether one employs universal degree or notmakes no difference. However, Schwarzschild & Wilkinson (2002) use an intervalbased semantics to account for the behaviour of quantifiers. If the universal scaleserves as the base scale for the intervals, then this account is compatible with thebasic ontology developed in chapter 4.

36Subordinate clauses licence such items. Consider the following two grammaticalsentences,

(122) Seymour is taller than I ever thought.

(123) Seymour is taller than anyone thought.

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37Seuren (1973, 1978) tries to explain NPIs in terms of a hidden sentential nega-tions in the subordinate clause. Hoeksema (1983) gives a more standard explanationwith regards to the comparative morpheme being interpreted as a downward entailingfunction (cf. Ladusaw, 1979). von Stechow (1984a) suggest that the interpretationof than itself is downward entailing.

38Although Kennedy hits upon the right empirical issues with regard to gener-alized quantification, I do not agree that such data is relevant to quantificationalrepresentations in general. Theories without generalized quantification do not neces-sarily predict that the comparative morpheme can move syntactically to take scopein different positions. Hence such theories do not necessarily predict any ambiguities.Furthermore, logical expressions representing the truth conditions of comparativesthrough existential quantification are generally equivalent to representations that donot use any quantifiers. If logical forms simply serve as short-hand for a model the-oretic interpretation (as argued by Montague, 1974), then the debate between usinglogical representations with or without quantification seems nonsensical.

39The data was originally mentioned in Gawron (1995).

40The verb to limit also triggers such ambiguities.

41Heim (2000) does not actually discuss the possibility that the ambiguities mightbe a result of the interpretation of the differential phrase exactly n rather than thecomparative morpheme. von Stechow (1984) proposes that such phrases have quan-tificational force while the comparative morpheme does not. Still, not all of Heim’sexamples involve differential phrases. As noted by Heim (2000), the ambiguitiespersist with the use of the comparative morpheme less.

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CHAPTER 4A Unified Account of Direct and Indirect Comparison

In this chapter, I present a unified theory of direct and indirect comparison. The

key to this unified account is a Universal Scale called ‘Ω’. This scale contains de-

grees (hereon universal degrees) that are isomorphic to the rational numbers between

0 and 1 (inclusive). I propose that comparative and equative morphemes compare

two individuals through these degrees, however these degrees are not simply mea-

surements. Rather, they represent the position an individual occupies on a more

primary scale such as a scale of beauty, intelligence, height or width. The higher an

individual is in the primary scale, the closer the universal degree is to the highest

degree in the universal scale. The lower an individual is in the primary scale, the

closer the universal degree is to the lowest degree in the universal scale. (Note unlike

scales in other theories, primary scales do not order measurements. In a sense, they

are more basic than measurement scales.)

One of the more interesting properties of comparing individuals through univer-

sal degrees is that comparisons can be made directly even when the primary scales

for two individuals are completely different. For example, there are two primary

scales in the sentence Esme is more intelligent than Sidney Crosby is talented : one

is associated with intelligence, the other with talent. According to the semantics

that I develop in this chapter, a truth value will be assigned to this sentence based

on whether Esme occupies a higher position in the scale of intelligence than Sidney

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Crosby occupies in the scale of talent. If Esme occupies a higher position, then the

universal degree that represents this position will be closer to the highest degree in

the scale (that is, closer than the degree assigned to Sidney Crosby). Hence the

universal degree assigned to Esme will be strictly greater than the one assigned to

Sidney Crosby.

As should be evident, indirect comparisons are easily accommodated into such a

semantics. The interpretation of such sentences is almost equivalent to paraphrases

that accurately reflect speaker intuitions. A sentence such as Medusa is more beautiful

than I am intelligent is true if Medusa occupies a higher position in the scale of beauty

than I occupy in the scale of intelligence. Otherwise it is false.

The interpretation of direct comparisons such as Seymour is taller than he is

wide are slightly more complicated. As I discuss in more detail in chapter 6, the

rankings associated with adjectives like tall and wide have more structure than those

associated with adjectives like beautiful and intelligent. Tall and wide are connected

to the same (non-linguistic) measurement system (a scale of inches and feet). This

measurement system affects the composition of the primary scales associated with

tall and wide in two ways. First it adds measurements such as 2 feet to both of

the primary scales. The degrees associated with these measurements will be ordered

in the primary scales in the same way that they are ordered in the non-linguistic

measurement system: the degree associated with 3 feet will be greater than the one

associated with 2 feet which will be greater than the one associated with 1 foot, so

on and so forth. Second, the measurement system ensures that the measurements

in the primary scales define the cardinality of the scalar domain. In other words,

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every individual’s width and height is equivalent to some measurement and this

measurement determines the individual’s position in the primary scale. As a result,

the scales for tall and wide will be structured in the exact same way, even though

the measurements of certain individuals will be quite different.

Considering the influence of the measurement systems on the primary scales,

sentences such as Seymour is taller than he is wide can receive the same kind of

analysis as indirect comparisons. The truth of such sentences can be determined

by comparing the universal degree that represents Seymour’s position relative to

the scale of heights to the universal degree that represents his position relative to

the scale of widths. The only difference from indirect comparisons is that the po-

sition of Seymour in the scale of heights is determined by the measurement of his

height. Similarly, the position of Seymour in the scale of widths is determined by the

measurement of his width. If the measurement of Seymour’s height is greater than

the measurement of his width, then the universal degree associated with Seymour’s

height will be greater than the one associated with his width.

Additional support for this unified account comes from overt restriction of com-

parison classes. According to my theory, direct comparison is an artifact of mea-

surement systems. The influence of measurements on the primary scales structure

these scales in identical ways. Accordingly, if the primary scales did not contain

measurements then direct comparisons should be impossible. The scales of widths

and heights should no longer be structured in the same way nor should the position

of an individual within the scale be dependent on his measurement. Prepositional

phrases such as for a boy restrict the comparison class of a gradable adjective. By

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restricting the comparison class one also restricts the primary scale. Thus tall for

a boy is associated with a primary scale that contains degrees only related to boys.

There are no degrees associated with measurements. Sentences such as Seymour is

taller for a boy than he is wide for a boy are evaluated with primary scales that are

not influenced by measurements. Interestingly, such sentences do not permit direct

comparisons. The sentence is false if Seymour is quite wide at 4 feet but only of av-

erage height at 5 feet: this despite the fact the measurement of his height is greater

than the measurement of his width.

This chapter provides the foundations for a unified account of direct and indirect

interpretations. The outline is as follows. In the first section, I discuss how primary

scales can be created from gradable adjectives. I suggest that gradable adjectives are

associated with relations between individuals. For example, the adjective beautiful

will be associated with the relation x has as much beauty as y. Employing Cresswell’s

(1976) methodology, I demonstrate how a linear order of equivalence classes can be

formed from such a relation. In the second section, I define the Universal Scale Ω.

This scale will consist of degrees that are isomorphic to the set of rational numbers

between 0 and 1. Also in this section, I develop a function that can map a degree in

a primary scale to a universal degree in Ω. This mapping preserves the underlying

order established by the primary scale. In other words, for any two degrees a and

b in the primary scale, a is greater than b according to the primary scale if and

only if the universal degree associated with a is greater than the one associated with

b in Ω. In the third section, I present my interpretation of the comparative and

equative morphemes. These interpretations compare two individuals by comparing

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the positions the individuals occupy in their respective primary scales. As I discuss

in chapter 5 and chapter 6, this interpretation accounts for both direct and indirect

comparisons.

4.1 The Primary Scale

The hypothesis that gradable adjectives are associated with binary relations has

often been discussed in the literature on comparison (see Klein, 1991, and Cresswell,

1976), however I believe I am the first to propose that such adjectives actually

denote this type of relation. In this section, I outline the details of this proposal. I

suggest that the interpretation of every gradable adjective consists of a domain D of

individuals and a Graph G, a subset of D×D. The scales that are more relevant to the

interpretation of comparative sentences can be formed from these relations. Contrary

to other theories (Bartsch & Vennemann 1972, Cresswell 1976, and Kennedy 1999),

the interpretation of gradable adjectives does not (directly) relate individuals to

degrees or extents. In what follows, I demonstrate how interpreting adjectives as

relations defines the right kind of primary scale that can be used to account for

direct and indirect comparisons.

I begin by reviewing Cresswell’s proposal for developing an ontology of degrees

for abstract scales like beauty and manliness. As I discuss, Cresswell creates such

degrees from a base relation. For example, a scale of beauty can be defined through

a relation that encodes who has as much beauty as someone else. This can be

done in the following way: first individuals are collapsed into equivalence classes.

Those who are equally as beautiful will occupy the same equivalence class. Second,

the equivalence classes are ordered linearly based on the underlying relation: one

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equivalence class is ordered above another if and only if the members of the first

have as much beauty as the members of the second. This linear order of equivalence

classes represents the primary scale for beauty, where the equivalence classes are

degrees and the ordering of equivalence classes is the linear ordering that establishes

the scalar relation. As I discuss at the end of this section, although this definition of

a primary scale may seem circular or a little redundant, it is neither.

4.1.1 Cresswell’s Ontology

Cresswell (1976) proposed a very influential theory concerning the ontology of

degrees. He suggested that degrees could be created through an underlying relation

between individuals. In this theory, degrees are associated with equivalence classes

consisting of the individuals in the relation. Furthermore, the linear ordering of the

degrees is associated with an ordering of the equivalence classes that is a congruence

relation with respect to the underlying relation.

Before discussing the details of this rather complex ontology, it is relevant to

review Cresswell’s motivations. Cresswell did not think that this ontology was im-

portant for measurable dimensions like height or length. These dimensions could

plausibly be associated with measurements such as feet and inches that are ordered

linearly in much the same way they are ordered on a ruler or measuring tape. Cress-

well was much more concerned with dimensions that could not be so easily associated

with measurements, such as beauty and manliness. Consider the following quote.

...Must we postulate the kalon as a degree of beauty or the andron as adegree of manliness? Degrees of beauty may be all right for the purposesof illustration but may seem objectionable in hard-core metaphysics...(Cresswell, 1976)

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Cresswell’s goal was to keep the semantics of degrees that proved so useful in analyz-

ing comparative constructions with adjectives such as tall and yet rid oneself of any

commitments to the existence of abstract measurements such as kalons or androns.

As linguists, one must be reminded of the philosophical concerns involving unin-

tuitive objects. Allowing for degrees of beauty to be objects in the ontology suggests

that such degrees exist in the same way as individuals, or perhaps even more trou-

blesome, in the same way as inches and feet. Yet there is a clear intuitive difference

between such objects. One can point to individuals and even inches on a ruler, but

not to degrees of beauty.

Linguistically speaking, such hypotheses do not seem objectionable if they are

suitably supported by data. Prior to any empirical support, a kalon is no less im-

plausible than electrons or verb phrases, except that the existence of the latter two

are established components of successful scientific theories. However, this is a moot

point. Cresswell’s concerns led to a very influential hypothesis whose applications

go well beyond the rather minor concern of quelling objections of philosophers inter-

ested in hard-core metaphysics. As I discuss in this chapter, his method of creating

scales from relations can be applied generally and need not be limited to abstract

adjectives.

Creating Scales from an Underlying Relation.

As observed by Cresswell (1976), if one has the conceptual abilities to determine

who has more of a certain quality than another, then one can develop a scale based

on this distinction. For example, most people are able to determine whether one

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individual has as much beauty as another. Sometimes the distinction is more diffi-

cult to make (given two people of near equal beauty) and sometimes the criterion

for beauty shifts in different contexts (yielding different results), but still in most

contexts a decision can be made. From this conceptual ability one can define the

following relation, where D is a contextually limited domain of individuals:

〈D, 〈x, y〉 : x, y ∈ D & x has as much beauty as y〉

This relation has some interesting properties. First, it is transitive. For any z, w,

and v, if 〈z, w〉 and 〈w, v〉 are in the graph of the relation, then so is 〈z, v〉. This

follows from the transitive properties of the concept has as much beauty as. Second,

it is reflexive. Given any individual, z, it follows almost tautologically that z has

as much beauty as z has. Thus for all z ∈ D, 〈z, z〉 is in the graph of the relation.

Third it is connected. Given any two individuals, z and w, one can compare their

beauty. Hence either 〈z, w〉 or 〈w, z〉 is in the graph of the relation.42

Since the relation is transitive, reflexive and connected, it fits the criteria of

being a connected quasi order. In what follows, I review how one can create scales

from an underlying connected quasi order. Although this analysis in linguistics was

first proposed by Cresswell (1976), such a derivation is well-known in mathematics

where the resulting structure is often called a quotient algebra (for discussion and

examples see Bell & Slomson, 1969 and Bell & Machover, 1977). Although this term

has been used in linguistics (see Klein, 1991), I prefer the term quotient structure.

(Technically, an algebra has operations as well as an ordering relation. These simple

quotient structures only have an ordering relation.)43

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There are two basic steps in developing a scale based on a quasi order. First,

one must define equivalence classes over the individuals in the quasi order. All the

individuals within a single equivalence class must be similar to each other in terms of

their behavior in the relation. Second, one must create a relation between equivalence

classes that is congruent to the original quasi order of individuals. Below I discuss

the details of each step.

From Relations to Scales: Some Examples.

To make the formation of scales from quasi orders more accessible, perhaps it will

be useful to give some examples of underlying connected quasi orders and the type

of scales I plan on associating with them. In what follows, I provide two examples.

In these examples, I keep the number of elements in the domain of the quasi orders

rather small. This allows the quasi orders to be graphically represented.

Below, I use a diagram to represent quasi orders. This saves space and allows for

a clear presentation of the informational content contained within the quasi order.

Let me state a few conventions that I adopt. First I do not represent reflexivity

graphically. Rather this property will be assumed for all elements in the diagram.

Also, if 〈a, b〉 is a member of the quasi order but 〈b, a〉 is not, then a will be placed

above b in the diagram. Furthermore, a line will connect a to b either directly or

through a path of other elements. In contrast, if both 〈a, b〉 and 〈b, a〉 are members

of the quasi order then a will be placed to the right or left of b. Both elements will

appear on the same level.44

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With these conventions in place, let’s consider a quasi order ζ with the domain

Dζ = a, b, c, d, e, f, g, h. A relation between these elements can be represented by

the diagram below.

e

qqqqqqqMMMMMMM

b

qqqqqqqMMMMMMM

YYYYYYYYYYYYYYYYYYY d

eeeeeeeeeeeeeeeeeee

qqqqqqqMMMMMMM

c

MMMMMMM

YYYYYYYYYYYYYYYYYYY a

qqqqqqqMMMMMMM f

qqqqqqq

eeeeeeeeeeeeeeeeeee

g h

To become familiar with this representation of the quasi order, let me list some of the

ordered pairs that are members of the quasi order according to the diagram. First,

since e is above a and there is a path from e to a, this means that 〈e, a〉 is a member

of the quasi order while 〈a, e〉 is not. Similar reasoning establishes 〈b, a〉 and 〈d, h〉

as members and 〈a, b〉 and 〈h, d〉 as non-members. Second, since c appears on the

same level as a and f , this means that 〈a, c〉, 〈c, a〉, 〈c, f〉, 〈f, c〉, 〈a, f〉, and 〈f, a〉

are all members of the quasi-order.

As I discuss in detail in the two sections following this section, this kind of quasi

order can be associated with a linear order of equivalence classes, otherwise known as

a quotient structure. This association can be described as follows: first equivalence

classes can be formed by collapsing together in a set all the individuals that are

on the same level. In doing this, the equivalence classes contain all the members

for which ζ holds symmetrically. Note that elements that have no others to their

left or right will be the sole member of their equivalence class. Second, a relation

that orders the equivalence classes can be defined through the original graphical

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representation of the quasi order. For example, one can order one equivalence class,

call it A, over another, call it B, if and only if a member of A is above a member of B

in the graphical representation of the quasi order. Using this method of formation on

the quasi order depicted above, one obtains the following linear order of equivalence

classes.

e

b, d

a, c, f

g, h

This quotient structure can be used as a primary scale where the equivalence classes

are the degrees in the scale. The top-most degree is the set e and the bottom-most

degree is the set g, h.

To illustrate this process again, consider the quasi order ξ with the domain

Dξ = a, b, c, d, e, f, g, h. The relation between elements in this domain can be

represented by the following diagram.

a

MMMMMMM b

qqqqqqq

c

hhhhhhhhhhhhh

VVVVVVVVVVVVV

d

MMMMMMM

YYYYYYYYYYYYYYYYYYY e

qqqqqqqMMMMMMM f

qqqqqqq

eeeeeeeeeeeeeeeeeee

g h

Keeping the same conventions introduced with the previous example, this diagram

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implies that 〈a, b〉 and 〈b, a〉 are members of ξ. Also, 〈a, c〉 and 〈a, e〉 are members

although 〈c, a〉 and 〈e, a〉 are not.

As with the example above, a scale can be created from this quasi order by

collapsing the elements of ξ into equivalence classes and then ordering the equivalence

classes based on the diagram representing ξ. Thus every element on the same line

belongs to the same equivalence class. These equivalence classes are then ordered

with respect to the original quasi order. For this example, the resulting quotient

structure will have the following form.

a, b

c

d, e, f

g, h

Once again, this quotient structure can be used as a scale where a, b is the topmost

element and g, h is the bottommost. In the next two sections, I outline the formal

details of a (non-graphical) derivation of scales: first discussing the formation of

equivalence classes from the underlying quasi order and then addressing how to

create a linear order of such equivalence classes.

Equivalence Classes.

The first step in forming a scale is to define equivalence classes that serve as the

degrees in the scale. To do this, I develop an equivalency relation between members

in the domain of the original relation. I then group into sets all the individuals that

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are equal to each other according to this relation. These sets define the equivalence

classes.

As presented in Cresswell (1976), one can define an equivalency relation based

on how individuals are related to others in the domain. Two individuals a and b are

equivalent to each other if and only if every individual to which a is related, b is also

related and vice versa. This is stated more precisely below, where the equivalency

relation is symbolized by ‘∼’. (Like Klein, 1991, I will use the symbol ‘ζ’ to represent

the underlying relation, in this case the quasi order.)

a ∼ b iff ∀x(ζ(a, x) ↔ ζ(b, x) and ζ(x, a) ↔ ζ(x, b))

A more intuitive definition of this equivalency relation can be restated in terms of

substitution. Two individuals are equivalent in terms of the relation ζ if and only if

they can substitute for one another without changing the truth values of statements

involving ζ. Below I give a more precise definition, where Φ is any sentence containing

the relation ζ in combination with any names or logical constants (∧, ∨, ¬, etc.).

Also ‘Φ(a/b)’ means substitute a for b in the sentence Φ.

a ∼ b iff ∀Φ(Φ is equivalent to Φ(a/b) and Φ(b/a))

Essentially, this definition states that two individuals are equivalent to each other if

and only if they are identical under substitution.45

With this equivalency relation, equivalence classes can be formed by grouping

all the individuals that are equivalent to each other into the same set. For any quasi

order ζ, we can define the set of equivalence classes in the following way.

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The set of equivalence classes Eζ for ζ:X ⊆ Dζ : ∀x, y ∈ X(x ∼ y) & ¬∃Y ((X ⊂ Y ⊆ Dζ) &

∀z, w ∈ Y (z ∼ w))

This set contains all subsets of the domain of ζ such that every individual in the

subset is equivalent to every other individual in the subset, and there is no superset

that has the same property. In other words, every subset is the maximal subset for

which the equivalency between individuals holds.

Another way of defining the set of equivalence classes is by defining a function

from the individuals in the domain of the relation ζ onto an equivalence class. Since

this function will be useful in describing the linear order required for scales, let

me redefine the set of equivalence class with this function. Consider the following

definition.

Let eζ be a function from Dζ to POW(Dζ) such that∀x ∈ ζ (eζ(x) = y : y ∈ Dζ & x ∼ y).

With this function the set of equivalence classes can be defined as follows.

The set of equivalence classes Eζ for ζ:X ⊆ Dζ : ∃x ∈ Dζ (X = eζ(x))

The function eζ defines the set maximally so no other additional properties are needed

to individuate the equivalence class.

A Linear Order.

The next step in creating a scale is to introduce a linear ordering on the set

of equivalence classes. A linear order has all the properties of the greater-than-

or-equal relation with regard to numbers. It is connected, transitive, reflexive and

anti-symmetric. So, to linearly order the equivalence classes one needs to develop

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a relation that has these four properties. This can be done by basing the relation

on the underlying connected quasi order between individuals. For example, for any

quasi order ζ that defines a set of equivalence classes Eζ , one can define a linear

order ζ in the following way:

Defining a Linear Order on Eζ :∀X, Y ∈ Eζ (X ζ Y ) iff ∃x, y[(x ∈ X) & (y ∈ Y ) & ζ(x, y)]

This linear order can also be defined using the function eζ as below.

Defining a Linear Order on Eζ :∀x, y ∈ Dζ ((eζ(x) ζ eζ(y)) iff ζ(x, y))

As a result of either definition, an equivalence class X is greater-than or equal to an

equivalence class Y if and only if the members of x bear the relation ζ to the members

of Y . To give a more relevant example, if ζ were the relation has as much beauty as,

then the equivalence class X would be greater-than or equal to the equivalence class

Y if and only if the members of X have as much beauty as the members of Y .

As demonstrated below, ζ is connected, transitive, reflexive and antisymmetric

as long as ζ is a connected quasi order.

ζ is connected: Choose any two equivalence classes that are membersof Eζ , where ζ/∼ = 〈Eζ , ζ〉. Call these equivalence classes A and B.Choose an arbitrary member of A, call it a, and an arbitrary member ofB, call it b. Since a and b are members of the domain of ζ and ζ is total,it follows that either 〈a, b〉 or 〈b, a〉 is a member of ζ. By definition of ζ

it follows that either A ζ B or B ζ A. Hence, any two equivalenceclass are ordered with respect to ζ .

ζ is transitive: Choose any three member of Eζ , call them A, B andC. Suppose that A ζ B and B ζ C. It follows from the definitionof ζ that there is some a ∈ A, some b ∈ B, and some c ∈ C suchthat ζ(a, b) and ζ(b, c). Since ζ is transitive, it follows that ζ(a, c). Thus,

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A ζ C.

ζ is reflexive: Choose any member of Dζ and call it a. Since ζis reflexive it follows that ζ(a, a). By definition of ζ , it follows theeζ(a) ζ eζ(a). Since a was chosen arbitrarily, one can conclude thatfor all x ∈ Dζ , eζ(x) ζ eζ(x). But this mean that for all X in Eζ ,X ζ X.

ζ is antisymmetric: Choose any two member of Eζ and call them Aand B. Suppose that A ζ B and B ζ A. This means that thereis some member of A, call it a, and some member of B, call it b, suchthat ζ(a, b) and ζ(b, a). Thus by definition of ∼ it follows that a ∼ b.However, this means that a and b belong to the same equivalence class.Hence A = B.

Note that in this proof, three of the four properties (connectivity, transitivity, and

reflexivity) are inherited from the underlying quasi. The definition of the linear order

simply preserves these properties when ordering the equivalence classes. In contrast,

the equivalence classes themselves yield the emergence of the fourth property: anti-

symmetry.

Since ζ is connected, transitive, reflexive and antisymmetric, it has all the

properties of a linear order.

4.1.2 Questions of Circularity & Redundancy

Before discussing other issues concerning Cresswell’s construction of scales from

quasi orders, I would like to address a potential confusion. On the surface, it seems as

if Cresswell’s derivation might be open to an accusation of circularity or redundancy.

Scales of beauty used to analyze sentences such as Esme is as beautiful as Morag is

are based on a relation that encodes whether Esme has as much beauty as Morag.

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I do not think that Cresswell’s analysis is circular nor redundant. Below I briefly

outline why.

Creating Scales from Relations is not Circular.

In constructing scales for beauty and manliness, it seems as if Cresswell is defin-

ing a semantics for comparatives and equatives using a relation specified in terms of

comparative and equative sentences. However, as Cresswell states in detail, there is a

difference between the concept of comparison and the semantics of how comparative

and equative sentences are given truth values.

The underlying relation does not require an analysis of comparatives to define

the relation. All it assumes is that given two individuals, speakers can tell if one has

as much of a certain property as the other. Stated otherwise, the underlying relation

requires that speakers have the conceptual ability to compare individuals in terms of

beauty, height, width, intelligence etc., without necessarily using language to make

this comparison.

Note that this much has to be assumed under any analysis of comparative and

equative constructions, whether a theory derives degrees or considers them to be

privative. To consistently map individuals to degrees that represent their height

with respect to others, one needs to conceptually recognize the relation between

individuals in terms of height. Those who are taller are mapped to higher degrees

than those who are shorter. No theory of comparison tackles how speakers are able

to maintain this consistency. Yet given this ability to recognize differences between

individuals, one can provide a fairly convincing analysis of how comparative and

equative constructions are assigned a truth value.

133

Let me clarify this point by discussing conceptual abilities outside the limited

realm of language. I assume that even those without language (monkeys, cats, dogs)

are able to compare two objects or individuals in terms of a certain property (to tell

which food bowl has more, or which potential mates are more suitable/beautiful).

Clearly, such individuals have the conceptual ability to compare without the linguistic

ability. All that is needed to build the underlying relation is this conceptual ability.

It is an unfortunate burden of presentation that it is difficult to express this concept

without using some type of comparative or equative construction.

Creating Scales from Relations is not Redundant.

Having dispensed with the idea that Cresswell’s derivation of scales is circular,

there still remains the potential that it is redundant. If one has a relation that can

distinguish who has as much beauty as another, why would one need to convert such

a relation into degrees to provide an analysis for the two sentences below?

(124) Esme is as intelligent as Morag.

(125) Esme is more intelligent than Morag.

Clearly such sentences can be provided with truth conditions defined solely through

the underlying relation. For example, suppose that ‘ι’ is the underlying relation that

encodes who has as much intelligence as another. The formulae below accurately

describe the truth conditions of the sentences above.

(126) ι(e,m), where e is Esme and m is Morag. (Truth conditions for (124))

(127) ι(e,m) & ¬ι(m, e), where e is Esme and m is Morag. (Truth conditions for

(125))

134

The formula in (126) is true if and only if Esme has as much intelligence as Morag.

The formula in (127) is true if and only if Esme has as much intelligence as Morag but

Morag does not have as much intelligence as Esme. With this possible representation

of the truth conditions, the question becomes why should one construct a scale in

order to provide an analysis of comparatives and equatives when such a scale does

not seem to be necessary?

This critique would be warranted except that it is factually mistaken. Although

the non-degree approach works for the sentences above, it becomes problematic when

there are two different adjectives in the main clause and the subordinate clause, as

with the sentences below.

(128) Seymour is as tall as he is wide.

(129) Esme is as beautiful as she is intelligent.

Such sentences are difficult to interpret since it is unclear how the two different

quasi-orders are related to each other through the comparison. One option would

be to form two sets from the main clause and the subordinate clause by abstracting

out one of the argument positions and then comparing the two sets. Certainly such

an option works when the main and subordinate clause contain the same adjective.

For example, an alternative representation of the truth conditions for Esme is as

intelligent as Morag would be the formula below.

(130) x : ι(e, x) ⊇ y : ι(m, y)

This formula is true if and only if the set of people that Esme is as intelligent as is

a superset of the set that Morag is as intelligent as.

135

Such a representation would produce the following kind of representation for

the sentences in (128) and (129), where ‘τ ’, ‘ω’, ‘β’, and ‘ι’ are the underlying quasi-

orders for tall, wide, beautiful, and intelligent respectively.

(131) x : τ(s, x) ⊇ y : ω(s, y)

(132) x : β(e, x) ⊇ y : ι(e, y)

The formula in (131) is true if and only if the set of people Seymour is as tall as is

equal to or a superset of the set that he as wide as. The formula in (132) is true if

and only if the set of people Esme is as beautiful as is equal to or a superset of the

set that she is as intelligent as.

Although such a representation is possible, it gives the wrong truth conditions

for such sentences. For example, suppose that Seymour is shorter and wider than

Alex, yet Seymour’s height is still greater than his width. Given this situation, the

representation in (131) could never be true. The set of people that Seymour is as

tall as would not contain Alex. The set of people that Seymour is as wide as (or

wider than) would contain Alex. Hence the first set could never be a superset of the

second. Yet the sentence in (128) is true under such conditions.

Similar results follow for the representation in (132). Consider the situation

where Esme is the second most beautiful person and the second most intelligent.

However the one person who is more beautiful than her is also one of the most

unintelligent. Let’s call this person Morag. Given this situation, the set of people

that Esme is as beautiful as (or more beautiful than) can never be a superset of the

set of people that she is as intelligent as (or more intelligent than). The first set will

136

not contain Morag but the second set will. The formula in (132) is false, yet the

sentence in (129) is true under such conditions.

Without degrees and scales (or some equivalent structure such as delineations

cf. Klein, 1980, 1982, 1991), it is not obvious how one would deal with compar-

isons involving two different adjectives. In contrast degrees and scales allow some

interesting avenues of exploration. As Cresswell (1976) demonstrated with regard

to direct comparisons, degrees and scales can help provide a semantics for sentences

with two different adjectives that is identical to an analysis of sentences with only

one adjective. The goal of this chapter is to do the same for indirect comparisons.

To summarize, building scales from quasi orders is not redundant since such

quasi orders do not contain a sufficient amount of structure to provide an adequate

analysis of all types of comparison. The addition of degrees and scales provides the

necessarily structure to give a unified account of comparison.

4.2 The Universal Scale

In this section, I develop a function that is able to map degrees onto rational

numbers that encode the position of the degree in its scale. For now, I only define

the function for degrees that are in the domain of a finite linear order.46 It is unclear

whether a similar function could be created for infinite cardinalities.

In what follows, I first define a Universal Scale that is isomorphic to the rational

numbers between 0 and 1.47 I then develop a function that takes linear orders as

arguments and yields functions from elements in the original linear order to elements

in the Universal Scale. This function is a homomorphism from the original scale onto

137

the Universal Scale. In other words, the relation between elements in the original

scale is the same as the relation between elements in the Universal Scale.

I begin by defining the Universal Scale. To limit confusion about the claims

I am making in this section, let me distance myself slightly from the term rational

number. Instead, I will define a linear order called Ω that is isomorphic to the linear

order on the rational numbers between zero and one, inclusive. This linear order will

constitute what I call the Universal Scale.

Ω will be defined as follows:

The Universal Scale Ω:

Let Ω be the pair 〈Dω,〉, where Dω is defined as a set of degrees in a one-to-one relation with the set of rational numbers between 0 and 1 inclusive.

Each d in Dω will be labeled with the rational number with which it isin a one-to-one correspondence. For example, d 1

3, d 3

20, and d 25

116will be

the degrees in one-to-one correspondence with 13, 3

20, and 25

116respectively.

The linear order orders elements in Dω in the same way that ≥ ordersthe rational numbers. Thus for all dx and dy in Dω, dx dy if and onlyif x ≥ y.

With this definition of the Universal Scale, I can now define a function that maps ev-

ery element in a scale onto an element in the Universal Scale that encodes its relative

position. Since the mapping changes as different kinds of scales are considered, I rel-

ativize this function to the scale under consideration. To do this, I develop a function

called H (Gothic ‘H’) that takes scales as arguments and yields homomorphisms.

I begin by specifying the domain of this function.

138

The domain of H:

Let the set of all possible scales with a finite domain be denoted byΣ. Each member of Σ is an ordered pair whose first coordinate is thedomain of the scale and whose second coordinate is the linear order onthat domain.

Next I will define the co-domain of the function.

The co-domain of H:

Let H be the set of functions such that for each member h of H there isa finite linear order, call it γ, such that h is a homomorphism from γ toΩ. In other words, for all x and y that are members of γ, x ≥ γ y if andonly if h(x) h(y).

With the domain and co-domain so specified, I can state the function H as follows:

The Universal Homomorphism H:

Let H be a function from Σ to H such that for each member γ of Σ, H(γ)has the following properties (For simplicity, let me represent this functionas Hγ):

1. For all x and y in Dγ, x ≥ γ y if and only if Hγ(x) Hγ(y).2. For all x in Dγ, Hγ(x) = d z

y,

where z = |y : y ∈ Dγ & x ≥ γ y| and y = |Dγ|

According to this function, each element d in a linear scale γ is mapped to the degree

in the Universal Scale that is in one-to-one correspondence with the rational number

with the following two properties: first, its numerator is equal to 1 plus the number

of elements less than d in the original linear order γ. Second, its denominator is

equal to the cardinality of the domain of γ.

139

Some examples of how this function works might help clarify these properties.

Consider the linear order δ = 〈Dδ,≥ δ〉, where Dδ is the set a, b, c, d, e, f, g. Sup-

pose that the ordering of these elements by ≥ δ is reflected in the following Hasse

diagram.

a

b

c

d

e

f

g

The result of applying the function H to this linear order is the function Hδ. This

function yields the following results when applied to the elements of δ:

Hδ(a) = d 77,

Hδ(b) = d 67,

Hδ(c) = d 57,

Hδ(d) = d 47,

Hδ(e) = d 37,

Hδ(f) = d 27,

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Hδ(g) = d 17.

Note the parallelism between the original scale and the Universal Degrees in the

range of the function Hδ. Just as the following relations are true with regard to δ,

a ≥ δ b,

c ≥ δ g,

b ≥ δ f ,

...so too are the counterparts with regard to Ω,

d 77 d 6

7,

d 57 d 1

7,

d 67 d 2

7.

The function Hδ preserves the ordering established by ≥ δ.

As another example, consider the linear order β = 〈Dβ,≥ β〉, where Dβ is the

set d1 , d2 , d3 , ...., d23 , d24. Thus Dβ has 24 elements. Suppose that ≥ β orders the

elements of Dβ in the following way: dn ≥ β dm if and only if n ≥ m.

The result of applying the function H to β is the function Hβ. This function

yields the following values when applied to the elements of β.

Hβ(d24 ) = d 2424

,

Hβ(d23 ) = d 2324

,

Hβ(d22 ) = d 2224

,

...

141

Hβ(d3 ) = d 324

,

Hβ(d2 ) = d 224

,

Hβ(d1 ) = d 124

.

As with the previous example, there is a parallelism between the elements in domain

of β and elements in the range of Hβ. Just as the following relations are true in β,

d22 ≥ β d9

d10 ≥ β d3

d14 ≥ β d1

...so too are the counterparts in Ω.

d 2224 d 9

24

d 1024 d 3

24

d 1424 d 1

24

The function Hβ preserves the original ordering established by β. As demonstrated

in the sections below, the function H will play an important role in the interpretation

of comparative and equative morphemes.

4.3 The Interpretation of More and As

Below, I outline my interpretation of the comparative and equative morphemes

as well as their subordinate clauses. These interpretations are based on the assump-

tion that all gradable adjectives are interpreted as relations between individuals. The

comparative and equative morphemes play two roles with respect to this relation.

First, they convert the underlying relation into a universal scale allowing comparisons

142

to be calculated with respect to this scale. Second, they transform the relation along

with the subordinate clause into a set of individuals. (By transform I mean that

the interpretation of the morphemes are functions that take the underlying relation

and the interpretation of the subordinate clause as arguments and yield a set.) The

set interpretation is the canonical interpretation for adjectives. In other words, the

comparative and equative morphemes basically convert gradable adjectives into an

interpretation more in-line with non-gradable adjectives.

4.3.1 The Interpretation of MORE

The comparative morpheme comes in two forms: either it appears as the word

more or the suffix -er. On the surface, these two forms seem to convey the same

meaning. Furthermore, their complementary distribution suggests that they are

allomorphs of the same morpheme. I label this underlying morpheme MORE and I

interpret this morpheme as follows:

Interpretation of the Comparative Morpheme:

Where d is a variable that ranges over elements in the Universal ScaleΩ, ζ is a variable that ranges over quasi-orders, and x is a variable thatranges over individuals.[[MORE]] = λd λζ λx(Hζ/∼(eζ(x)) d)

There are three arguments (lambda abstracted variables) in this interpretation of the

comparative morpheme: one ranging over universal degrees, another ranging over

quasi orders and a third ranging over individuals. To gain a better understanding of

this interpretation, I discuss the role of each argument in the following three section.

143

The Gradable Adjective Argument.

As stated previously, I interpret all gradable adjectives as quasi orders. In terms

of the linear order of lambda abstracted variables in the formula above, a quasi order

serves as the second argument to the comparative morpheme (represented by the

variable ζ).

There are two aspects of the interpretation that are influenced by the quasi

order, one involves the formation of the primary scale and the other the formation of

a function from equivalence classes to universal degrees. I discuss each aspect below.

One of the more important properties of the interpretation of MORE is that it

takes a quasi order and converts it into a quotient structure. In the interpretation

given above, the conversion of the quasi order into its quotient structure is symbolized

by the subscript on the variable ‘ζ’ (the subscript ‘/ ∼’). ‘ζ/∼’ is the linear order of

equivalence classes generated from ‘ζ’.

The quotient structure aids in the creation of a well-defined function from equiv-

alence classes to universal degrees. The combination of the quotient structure with

the Universal Homomorphism (‘H’) yields such a function (‘Hζ/∼ ’). This function is

applied to eζ(x) and is partly responsible for the formation of the set that will even-

tually represent the adjectival interpretation. Recall from the previous sections that

‘eζ ’ is a function from individuals to the equivalence classes in ‘ζ/∼’ that contains

the individual. Thus, ‘eζ(x)’ is the equivalence class that contains x in ‘ζ/∼’. To

avoid too many subscripts in the representation, I will usually write ‘x’ to symbolize

‘eζ(x)’, where ‘ζ’ is the quasi subscripted in ‘H’.

144

Note that the Universal Homomorphism when applied to x provides the first

argument to the strict linear order ‘’ (the strict linear order of universal degrees).

‘’ represents the central relation of comparison. It is derived from the (non-strict)

linear order in the standard way: x y if and only if x y & y x.

Set Formation.

The variable x in the interpretation above serves an integral role in the creation

of the set (or function) that is eventually formed after the comparative morpheme

combines with the subordinate clause and gradable adjective. This set contains all

the individuals whose universal degree (calculated through ‘Hζ/∼ ’) is strictly greater

than the degree picked-out by the subordinate clause.

The resulting set combines with the rest of the comparative sentence in the

same way that a non-gradable adjective combines with nouns and predicates in non-

comparative contexts.48

The Degree Argument.

The degree variable d represents the contribution of the subordinate clause in

the interpretation of comparatives. As should be obvious from the nature of the vari-

able, I treat subordinate clauses as degrees (Note, I present the actual semantics for

the subordinate clause in section 4.3.3). A rather unintuitive aspect of this presen-

tation is that the subordinate clause serves as the first argument to the comparative

morpheme. Since this might be controversial, let me provide a few justifications.

Below I argue that subordinate clauses demonstrate some properties of syntactic

displacement. I then suggest that the lowest (i.e. most embedded) surface position

of the subordinate clause is the original argument position.

145

To begin, there is some evidence that subordinate clauses in comparatives are

displaced or moved from their underlying argument position. For example, such

clauses can appear in a variety of different surface positions as demonstrated in

(133).

(133) a. Seymour is a taller man than Jon is.

b. Seymour was a stronger man yesterday than Jon was.

c. Seymour was required to be stronger last year than Jon.

In (133a) the subordinate clause appears after the noun that the adjective modifies.

In (133b) the subordinate clause is adjoined after the adverbial modifier yesterday. In

(133c) it appear after the adverbial modifier last year. It can appear in this position

even when the adverbial modifier semantically applies to the higher VP. For example,

(133c) has an interpretation that is similar to (134) below, even under the reading

where there are no requirements put on Jon (see Gawron, 1995, for a discussion of

such readings).

(134) Last year, Seymour was required to be stronger than Jon.

Movement would help to explain the variety of syntactic positions for subordinate

clauses while also maintaining a uniform functional representation of MORE.

Interestingly, this variation in position seems to be subject to island constraints.

For example, when the adjective is embedded in a known island for movement,

subordinate clauses do not have the same kind of distributional flexibility. Consider

the sentences in (135).

146

(135) a. Last Year, Esme was required to be more than four years old and stronger

than Jon.

b. ?? Esme was required to be more than four years old and stronger last

year than Jon.

c. Yesterday, Seymour fulfilled the requirement to be stronger than Jon.

d. ?? Seymour fulfilled the requirement to be stronger yesterday than Jon.

The sentences in (135a) and (135c) are perfectly acceptable. In contrast, the sen-

tences in (135b) and (135d) are not acceptable, at least not as paraphrases of the

sentences in (135a) and (135c). In both of the unacceptable sentences, the subordi-

nate clause is adjoined above an adverbial phrase while the comparative morpheme

is embedded in a known island for movement: the coordinate structure in (135b) and

the complex NP in (135d). If the subordinate clause were moved, then this type of

distribution would fall out naturally from general constraints on movement.49

If one accepts the thesis that subordinate clauses are displaced from their under-

lying argument position (as I do), then the most logical location for this argument

position is as an internal argument to the comparative morpheme. Such a position

can be overtly occupied by subordinate clauses when the complement of than is a

measure phrase such as four feet or a little as in (136). In fact, sometimes this

position is obligatory for such phrases (see (136d) below).

(136) a. Esme is more than four feet tall.

b. Esme is taller than four feet.

c. Seymour is more than a little beautiful.

147

d. * Seymour is more beautiful than a little.

In (136a) and (136c) the subordinate clause combines with the comparative mor-

pheme before the comparative combines with the gradable adjective. This is the

lowest overt position of the subordinate clause. For this reason, I treat this position

as the underlying argument position.50

In summary, I have argued for two assumptions in this section. First, I used

evidence from the distribution of subordinate clauses to support my claim that such

clauses are displaced from their underlying argument position. Second, I suggested

that the lowest and most embedded overt position of such clauses represents their

actual argument position.51

4.3.2 The Interpretation of As

The interpretation of the equative morpheme as is quite similar to the interpre-

tation of the comparative but with two differences. First the order of the arguments

is different. Second the locus of comparison is translated as a partial order rather

than a strict order. Below I first present the interpretation of the equative before

discussing these two differences.

Unlike the comparative morpheme, the equative morpheme has the same phono-

logical form in any syntactic context: namely as. Unfortunately the head of the sub-

ordinate clause also has this phonological form. To avoid confusing the two terms, I

subscript the equative morpheme with EQ. I provide the following interpretation of

this morpheme.

148

Interpretation of the Equative Morpheme:

Where ζ is a variable that ranges over quasi orders, d is a variable thatranges over universal degrees, and x is a variable that ranges over indi-viduals in the domain:[[asEQ ]] = λζ λd λx(Hζ/∼(x) d)

Most aspects of this interpretation have already been discussed in the section on the

interpretation of the comparative. Only the two differences mentioned above warrant

any further discussion. Below I first address how the equative morpheme interacts

with its subordinate clause before discussing how it represents the equality relation

between degrees.

The first difference between equative and comparative morphemes concerns the

arrangement of the arguments. Unlike the comparative morpheme, the equative first

applies to the gradable adjective before applying to the subordinate clause. The rea-

son for this is simple. Unlike the comparative, there is no evidence that subordinate

clauses ever serve as the first argument to the equative morpheme. Consider the

sentences in (137).

(137) a. Seymour is as tall as four feet.

b. * Seymour is as as four feet tall.

The first sentence is acceptable when the subordinate clause appears after the grad-

able adjective. In contrast the second is completely unacceptable.52

Another difference between the equative and the comparative is the fact that

the equative uses a partial order instead of a strict order. This relation is chosen over

149

the strict order to capture facts about equality. For example, in most circumstances

the sentence in (138) would imply that Esme and Seymour are equally intelligent.

(138) Seymour is as intelligent as Esme.

With a strict order between degrees, this type of interpretation would not be possi-

ble. If the degree of Seymour’s intelligence were equivalent to the degree of Esme’s

intelligence, then neither could be strictly order over the other (if dx = dy then

dx dy and dy dx ). In contrast, a partial order between degree permits equality

(if dx = dy then dx dy and dy dx ). However, such facts about equative as-

sertions brings up the question of whether equality might be more appropriate for

the equative than the greater-than-or-equal relation. Empirically this does not seem

to be warranted. Implications of equality are pragmatic in nature (see Altas, 1984,

for a thorough discussion of such issues). For example, such implications are easily

defeated with additional co-text as in (139).

(139) Seymour is as intelligent as Patrick. In fact, he is even more intelligent.

For this reason, the weaker interpretation seems more appropriate for the equative

morpheme.

4.3.3 Interpreting Subordinate Clauses

The interpretation of subordinate clauses is a very complex issue. In this section

I do not address subordinate clauses in general but only address the type of clauses

that are relevant for indirect comparison. I address other subordinate clauses in

chapter 7. The difference between indirect comparisons and other types of compar-

ison is that indirect comparisons generally involve sentences that contain different

adjectives in the main and subordinate clauses, as with the sentences below.

150

(140) Jon is more excited than the universe is large.

(141) Morag is as beautiful as Sydney Crosby is talented.

(142) Tony is more anxious than his sister is carefree.

As demonstrated with these sentences, the subordinate clauses basically have a full

clausal structure: a subject and a predicate. Bresnan called such sentences instances

of Comparative Subdeletion (CS). I will adopt Bresnan’s terminology in the discus-

sion below. In what follows, I discuss the clausal structure of these subordinate

clauses and how they are generally treated within the literature on comparison. I

basically propose the same kind of interpretation as the rest of the literature with

minor alterations to adapt such proposals to my interpretation of gradable adjectives

as quasi orders.

Sentential Complements with a Hidden Variable.

Within the semantic literature (see Bresnan, 1973, 1975; Cresswell, 1976; Kennedy,

1999 among others), instances of Comparative Subdeletion are treated as if than

and as have sentential complements, although such complements are considered to

be slightly different from non-subordinate sentences. Most theories hypothesize that

the degree modifier position in subordinate clauses is occupied by a hidden variable.

Below, I outline why this thesis is so pervasive.

The main reason to hypothesize that than and as have a sentential complement

stems from the fact that such complements appear to be identical to other sentential

structures. For example, consider the three sentences below.

(143) Donald is taller than the table is long.

151

(144) Donald is as tall as the table is long.

(145) The table is long.

The first two sentences are CS constructions while the third is an absolutive construc-

tion. As is apparent from the examples, the absolutive construction is practically

identical to the complement of both as and than.

However there is an important difference between the absolutive construction

and the other two sentences, one that leads to a greater understanding of the un-

derlying structure. The absolutive construction permits degree modifiers while the

CS constructions do not. For example, note the contrast between the two sets of

sentences below.

(146) a. The table is six feet long.

b. The table is very long.

(147) a. i. * Donald is taller than the table is six feet long.

ii. * Donald is as tall as the table is six feet long.

b. i. * Donald is taller than the table is very long.

ii. * Donald is as tall as the table is very long.

The absolutive constructions in (146) allow modifiers such as six feet and very. In

contrast, the CS constructions in (147) are not acceptable with such modifiers. As

discussed by Bresnan (1975), if one hypothesizes in CS constructions that a hidden

argument occupies the syntactic position normally occupied by the degree modifier,

then these distributional facts would be expected.

152

In summary, subordinate clauses in CS constructions seem to have all the overt

properties of being sentential except that such such clauses do not permit overt

degree modifiers. To explain these properties, I follow Bresnan in maintaining that

such clause are indeed sentential but that the degree modifier position is occupied

by a phonologically null syntactic object.

Subordinate Clauses as Universal Degrees.

Accepting that subordinate clauses contain a sentential structure with a hidden

variable, let me present a possible interpretation of the subordinate structures that

is consistent with this fact. The basic ideas presented in this section are adopted

from Cresswell (1976), von Stechow (1984b), and Kennedy (1999). All three of these

authors treat the hidden variable as a degree variable. All three also abstract the

degree variable to form a set of degrees. Also, all three use the elements of this set

to determine the truth value of the sentence through the locus of comparison. In

terms of this last step, I follow Kennedy (1999) in assuming that a unique degree is

selected from the set and this degree serves as the argument to the comparative.

There are two assumptions that I make to achieve this interpretation. The first

concerns how the gradable adjective interacts with the hidden degree variable. The

second explains how the hidden variable is abstracted to form a set. I discuss each

of these assumptions below.

Since gradable adjectives do not operate on degrees directly within my theory, I

must assume that there is a hidden function that converts the gradable adjective in

the subordinate clause so that it can interact with the hidden variable. I label this

hidden function ABS for absolute. (This function will prove useful when providing an

153

interpretation for absolutive constructions.) This hidden function combines with the

gradable adjective before the hidden degree variable is adjoined. Thus the phrase the

table is long in the subordinate clause than the table is long will have the following

structure.

TP

qqqqqqqVVVVVVVVVVVVV

DP

qqqqqqqMMMMMMM TP

MMMMMMM

qqqqqqq

the table is AP

MMMMMMM

qqqqqqq

d ??

qqqqqqqMMMMMMM

ABS long

The most important aspect of this structure is the degree variable appearing within

the adjectival phrase. Taking into consideration the presence of this variable, I

provide ABS with the following interpretation.

The Interpretation of ABS:

Where ζ is a variable that ranges over quasi orders, d is a variable thatranges over elements in the universal scale, and x is a variable that rangesover individuals,

[[ABS]] = λζ λd λx(Hζ/∼(x) d)

This interpretation is identical to the interpretation of the equative morpheme. In

essence, ABS behaves like a hidden equative. The main difference between the

equative and ABS arises in how it interacts with other elements in the subordinate

154

clause. Only the ABS function combines with a hidden variable that is abstracted

to form a set. Let me describe this process in more detail.

As noted by Chomsky (1977) and McConnell-Ginet (1973), there is a syntactic

parallelism between the abstraction of a degree in a subordinate clause of a compar-

ative and the abstraction of a variable over individuals in the restrictive clauses of

nominals. I maintain this parallelism in the semantic domain. Just as that I saw is

interpreted as a set in the man that I saw so too is the table is long in the phrase

than the table is long.

I use a hidden operator to represent this set formation in an effort to make

this parallelism clear. I suggest that this operator originates in the degree modifier

position and moves to the top of the subordinate clause. The movement or chain

(much like the movement or chain in the restrictive clauses of the nominals) leads

to the island violations demonstrated in chapter 2. This operator moves to the top

of the subordinate clause leaving a degree variable in its place. The operator then

abstracts this variable to form a set. I use the symbol ‘operator’ to explicitly

represent this abstraction in the syntax and semantics.

The heads of the subordinate clauses (as and than) will be interpreted as func-

tions that operate on the set formed by ‘operator’. Their role will be to select the

largest degree in the set as determined by the linear order in the universal scale

Ω. The largest element in a partial order is often called the supremum. The function

that chooses the largest degree is sometimes represented as sup. The heads of the

subordinate clause will simply be identified with this function: [[as/than]] = sup.

155

With the subordinate heads, the hidden function ABS, and the nature of the

hidden variable so specified, one can provide the following interpretation for a sub-

ordinate clause like than Esme is beautiful. For the purpose of the example, I ignore

the interpretation of tense or intensional elements. I also assume that β is the un-

derlying quasi order associated with beauty. Furthermore, for the sake of a complete

interpretation I will arbitrarily assign the universal degree d 410

to the equivalence

class containing Esme.

sup (d : d 410 d) = d 4

10

llllllllllRRRRRRRRRR

[[than]]= sup

d : d 410 d

llllllllll

::::

::::

::::

::

operatord

λx[Hβ/∼(x) d](Esme)= (Hβ/∼(eβ(esme)) d)

= (d 410 d)

llllllllllRRRRRRRRRR

[[Esme]]= Esme

λx[Hβ/∼(x) d]

llllllllllRRRRRRRRRR

is λx[Hβ/∼(x) d]

llllllllllRRRRRRRRRR

d λdλx[Hβ/∼(x) d]

llllllllll

::::

::::

::::

::

[[ABS]]= λζλx[Hζ/∼(x) d]

[[ beautiful ]] = β

156

The interpretation of the subordinate clause ends up being a degree: more specifi-

cally, the degree in the universal scale that represents Esme’s position on the primary

scale. In general, when the subordinate clause contains a simple absolutive construc-

tion, the interpretation of such a clause will simply be equal to the universal degree

that represents the position of the clausal subject in the primary scale.

4.4 Summary

In this chapter, I explained how a scale of equivalence classes (a quotient struc-

ture) could be created from a quasi order. Furthermore, I described how equivalence

classes in this scale could be mapped to universal degrees that encode their position. I

then provided an interpretation of the comparative and equative morphemes that ex-

ploited this mapping procedure along with the quotient structure. As I demonstrate

in chapter 5 and 6, this interpretation yields both direct and indirect comparisons.

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Notes

42See Hellan (1981) for a discussion of issues concerning connectivity.

43Krantz, Suppes, & Tversky (1971) discuss such structures with respect to mea-surement theory. Landman (1991) discusses some application of such structures withrespect to ordering equivalence classes of events across possible worlds. However hecalled such structures equivalence structures.

44 It is this convention that makes these diagrams different from traditional HasseDiagrams. Generally, elements on the same level of a Hasse diagram are not recip-rocally related.

45 Interestingly, Cresswell’s definition can be used to define an equivalency relationno matter what the properties of the underlying relation are. In fact, although Iam using the definition to build an equivalency relation on quasi orders, Cresswell(1976) actually used the definition on asymmetric and transitive relations. Thisdemonstrates the flexibility and utility of such a definition. Other methods of defin-ing the equivalency relation are more dependent on the properties of quasi ordersand hence will only work for such relations. For example, Klein (1991) defines hisequivalency relation as follows:

a ∼ b iff ζ(a, b) & ζ(b, a)

Although these two equivalency relations are defined differently, the end result isidentical when limiting oneself to quasi orders. For example, suppose that there aretwo elements a and b in the domain of ζ such that a ∼ b according to Cresswell’sdefinition. It follows that ∀x(ζ(a, x) ↔ ζ(b, x) and ζ(x, a) ↔ ζ(x, b)). Since ζ isreflexive, it follows that 〈a, a〉 and 〈b, b〉 are members of ζ. As a consequence ofCresswell’s definition, it follows that 〈a, b〉 and 〈b, a〉 are members of ζ. Yet thisimplies that a ∼ b according to Klein’s definition. Similarly, suppose that thereare two elements a and b in the domain of ζ such that a ∼ b according to Klein’sdefinition. It follows that ζ(a, b) & ζ(b, a). Choose any element c in ζ. Suppose that〈a, c〉 is a member of ζ. Since ζ is transitive and 〈b, a〉 is a member of ζ, it followsthat 〈b, c〉 is a member of ζ. Suppose that 〈c, a〉 is a member of ζ. By transitivity andthe fact that 〈a, b〉 is a member of ζ, it follows that 〈c, b〉 is a member of ζ. Similarreasoning demonstrates that if 〈b, c〉 is in ζ then 〈a, c〉 must also be in ζ and if 〈c, b〉is in ζ then 〈c, a〉 must also be in ζ. Since c was chosen arbitrarily it follows that for

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all x ζ(a, x) ↔ ζ(b, x) and ζ(x, a) ↔ ζ(x, b). Hence, a ∼ b according to Cresswell’sdefinition.

46 In other words, linear orders that have a finite domain.

47 I make this distinction, although I acknowledge that being isomorphic to therational number system might be the same as being the rational number system. SeeMakkai (1999) for a discussion.

48Alternatively, one could view this as a characteristic function that formed theset. Either way, the resulting interpretation after the comparative morpheme hasbeen combined with the gradable adjective and the subordinate clause is a standardpredicate interpretation that is common to all adjectives.

49There is one counterexample to a movement analysis. If a movement analysiswere true, then it would not be able to explain why degree arguments are able tomove out of the NPs that the adjectives modify. This phenomena is demonstratedby (133a). Fortunately, this is not only a problem for displacement of subordinateclauses, it is also a problem for abstraction of degree variables from within thoseclauses. Recall from chapter 2 that abstraction of the degree variable in a subordi-nate clause was impossible if the variable was contained within an island. However,abstraction is permitted when the variable is embedded in an NP that it modifies.For example, the subordinate clause in (148) is well-formed whereas the clause in(149) is not. Note, it is sentences such as these that led Bresnan to hypothesizea fundamental difference between Comparative Deletion and Comparative Subdele-tion. Comparative Subdeletion violated the Complex NP constraint.

(148) Mary is a taller woman [than Jon is a d tall man].

(149) * Mary is a taller woman [than Jon is a very intelligent man and a tall man].

One possible conclusion to draw from these facts is that movement out of the adjec-tival phrase of an NP does not result in any kind of syntactic violations. Of course,one still needs to explain the ungrammaticality of *How tall is Jon a man while alsomaintaining the grammaticality of How tall a man is Jon. See Matushansky (2002)for a discussion.

50 It should be noted that nothing crucial to the interpretation of the locus ofcomparison relies on this assumption of movement. Although I believe that this

159

assumption is correct, if in the end it proves to be misguided, my basic analysis ofdirect and indirect comparison can still be maintained.

51A workable alternative to the displacement of subordinate clauses is to have thecomparative phrase move covertly and then have late adjunction of the subordinateclause. See Bhatt & Pancheva (2004) for a defence of this alternative.

52Note, evidence that the subordinate clause moves from its argument position isless convincing for equative constructions. Consider the sentences in (150).

(150) a. Esme is as intelligent as Einstein is.

b. ? Esme was as intelligent yesterday as Einstein.

c. * Esme is an as intelligent woman as Einstein.

d. ? Esme was required to be as intelligent last year as Einstein.

Unlike the comparative examples, the last three sentences in (150) either soundparticularly awkward or are completely unacceptable. I have no explanation of whythe equative would behave so differently, however the evidence does support the orderof the arguments in the interpretation above. In almost all syntactic circumstancesthe subordinate clause appears immediately after the gradable adjective. However,there is one exception to this generalization. The subordinate clause can appearafter the nominal in the following kind of construction: Seymour is as intelligent achild as Esme is. This kind of construction also appear with Wh-questions: Howintelligent a child is Seymour?.

160

CHAPTER 5An Account of Indirect Comparison

The interpretation of the comparative and equative morphemes given in chapter

4 provide a useful and robust semantics for indirect comparison. The basic account is

simple. The interpretation of the comparative and equative morphemes compare the

positions two individuals occupy on their respective primary scales. The Universal

Scale is able to encode positions and order them linearly. Thus, positions can be

compared in much the same way that two rational numbers can be compared.

In this chapter, I explore the predictions of this theory with respect to sentences

and situations that yield clear and systematic truth value judgments. As I hope to

show, a theory with universal degrees accurately accounts for speaker intuitions.

5.1 Controlling Comparison Classes

In building a degree account of indirect comparison, I follow the general practice

of first considering sentences that yield clear judgments with respect to a given

situation. I then discuss examples where judgments are not so well defined.53 Such

a strategy entails ignoring the most prototypical examples of indirect comparison as

in (151a) and (151b).

(151) a. Esme is more beautiful than Einstein was intelligent.

b. Unfortunately, I’m as talented as Medusa is beautiful.

161

These examples have open-ended and large comparison classes. In (151a) Esme’s

beauty is evaluated in terms of the beauty of all contextually relevant men, women

and children. Such a comparison class varies among speakers and changes from

context to context. Similarly, Einstein’s intelligence is evaluated in terms of the

intelligence of people in general (or perhaps in certain contexts physicists in general).

Once again such a comparison class might be different for different speakers and might

also depend on the life experience of the speaker or audience. The indeterminate

nature of the comparison classes in these sentences make them less than ideal for

building a semantic theory of comparison.

To correct for the variability of comparison classes, I propose using sentences

that overtly restrict the comparison class in both the main and subordinate clause.

For example, consider the sentences below.

(152) a. Esme is more beautiful for a committee member than Seymour is intelli-

gent for a committee member.

b. Esme is more beautiful for a committee member than Seymour is intelli-

gent.

c. i. Esme is more beautiful for a woman than Seymour is intelligent for

a man.

ii. Sidney Crosby is more talented for a hockey player than Medusa is

ugly for a Gorgon.

d. i. Esme is a more beautiful committee member than Seymour is an

intelligent committee member.

ii. Esme is a more beautiful woman than Seymour is an intelligent man.

162

iii. Sidney Crosby is a more talented hockey player than Medusa is a

ugly Gorgon.

In (152a) to (152c) a for -clause restricts the comparison class in both the main and

subordinate clause. Often the same for -clause restricts both clauses, whether it is

specified twice as in (152a) or only once as in (152b). (Note, (152a) and (152b) are

basically synonymous.) However, sometimes two different for -clauses can appear in

a comparative where one restricts the main clause and the other the subordinate

clause. This is demonstrated with the two sentences in (152c). In each sentence,

a for -clause appears in both the main and subordinate clause. As demonstrated

in (152d), Not only can comparison classes be restricted by for -clauses, but they

can also be restricted by the nominals that the adjectives modify. As mentioned in

chapter 2, even though such nominals are not identical to the comparison classes,

they still influence the value of such classes. The comparison class is usually a subset

of the denotation of the nominal. Considering this fact, it is rather unsurprising that

the sentences in (152d) paraphrase those in (152a) and (152c).

In assessing the viability of a theory with universal degrees, I use sentences

that either contain a for -clause or a modified nominal. Such sentences yield less

variable truth value judgments. By using constructions that restrict the composition

of comparison classes, I hope to demonstrate a deeper systematicity with regard to

the judgments about indirect comparison.

5.2 Interpreting Comparison Classes

As is apparent from the previous section, the semantics of comparison classes

plays a central role in my theory of indirect comparison. In this section I specify

163

how I treat such classes and the for -clauses that modify them. Like Klein (1980), I

will assume that there is a contextually determined variable C that interacts with

the adjective to influence the domain of comparison.54 Below I outline how such a

variable can be integrated into a theory with universal degrees and how for -clauses

can influence the value of such a variable.

To begin, recall that the underlying interpretation of all gradable adjectives is

a relation. Each relation R can be represented as an ordered pair consisting of a

domain and a graph (a set of ordered pairs): R = 〈DR, GR〉. Relations serve as the

basis for the primary scale (the quotient structure). I propose that the comparison

class restricts such underlying relations. As a result of this restriction, the domain of

the relation will only consist of elements from the comparison class while the graph

will only consist of ordered pairs where each member of the pair is also an element

in the comparison class.

To formalize this idea more precisely, I use the typical restriction symbol ‘’.

This symbol will represent a function from sets to a function on relations. I will

define it as follows.

Definition of :

Let be a function from sets to a function on relations. For any setA, ( A) maps any relation R (R = 〈DR, GR〉) to (R A) (R A =〈(DR ∩ A), (GR ∩ (A× A))〉).

With this definition, the result of restricting a relation by a comparison class is a

relation whose domain is a subset of the comparison class and whose graph only

relates members of the comparison class.

164

One can account for the effect of comparison classes by always interpreting

gradable adjectives as if they have combined with such a restrictor function. This

is exactly what I propose. I use the comparison class variables C and C ′ as the

restricting sets. I then assume that each gradable adjective is restricted as soon as it

enters into the derivation. Thus, for any gradable adjective Z with the interpretation

ζ = 〈Dζ , Gζ〉, Z immediately combines with a restricting function as represented

below.

ζ C = 〈(Dζ ∩ C), (Gζ ∩ (C × C))〉

qqqqqqqMMMMMMM

[[Z]] = ζ C

The primary scale is created from this restricted relation. As a consequence the

comparison class indirectly determines the elements in the scale (the equivalence

classes) and how those elements relate to one another (the linear order).

Having introduced the idea of a contextually determined variable, it remains to

be shown how an optional for -clause influences the value assigned to this variable. In

the remainder of this section I outline one possibility that is consistent with known

facts.

To keep the representation of comparatives simple, I suggest that for -clauses do

not affect the semantic values of comparative sentences directly. Rather, I propose

that they introduce a presupposition that each member of the comparison class has

a certain property and that each member of the contextual domain that has that

property is a member of the comparison class. To implement the semantic vacuity

of such phrases, I simply interpret them as identity functions (functions that map

165

every element in its domain onto itself). As a result, for -clauses do not have any

affect on the interpretation of a sentence other than the presuppositional content

that influences the value of the comparison class.

Before specifying this possible interpretation more formally, I would like to high-

light that this is only one possibility. The content of the variable could also be fixed

semantically by interpreting the for -clause as a proposition that is adjoined to the

comparative through conjunction. This proposition could simply assert that The

content of C has the property P where the value of P depends on the nominal that

serves as the argument to the preposition. I do not see how determining the value

semantically or through presupposition can be distinguished empirically. Normally

presuppositions project in certain constructions such as when the presupposing sen-

tence appears as part of a question or embedded under negation or in the antecedent

of a conditional. The presupposition in this case states the value of a hidden variable.

It is difficult to test whether this information projects or not since it has no obvious

consequences outside of the comparative sentence. I choose to adopt the presuppo-

sitional theory for practical reasons. The absence of any true semantic content for

the prepositional phrases allows for a much clearer presentation of the data.

The details of the presuppositional theory are as follows. First there must be

a separate for preposition that specifically relates to comparison classes. Let’s call

this preposition forcc, where the cc are the initials for comparison class. Forcc takes

two arguments: a comparison class variable C and a set. The variable has no overt

correlate. As for the set, its value is fixed by the NP complement.

166

Note, I assign the NP a set interpretation despite the fact that the NP contains

an indefinite determiner. This is not necessarily problematic in and of itself. Of-

ten indefinite NPs in predicate positions are interpreted as sets (see, Partee, 1987,

Winter, 2001, and Landman, 2004, for a discussion). Interestingly, the NPs in for -

clauses have much in common with NPs in predicate positions. For example, they

both have number marking that agrees with other nominals in the sentence. Consider

the sentences in (153) and (154).

(153) a. Seymour and Jon are boys.

b. ?? Seymour and Jon are a boy.

c. ?? Seymour is boys.

d. Seymour is a boy.

(154) a. Seymour and Jon are tall for boys.

b. ?? Seymour and Jon are tall for a boy.

c. ?? Seymour is tall for boys.

d. Seymour is tall for a boy.

In (153), the sentences are odd if the indefinite NPs do not agree with the subject

of the predicate in number. Similarly, in (154) the absolutive constructions are odd

if the indefinite NPs do not agree in number with the clausal subject. Considering

this parallel,55 it seems to be a reasonable hypothesis that the indefinite NPs in

for -clauses are interpreted as sets.

Given that forcc takes a variable and a set as an argument, the semantics for

the prepositional phrase can be stated as follows.

167

Interpretation of forcc

For all predicates P , comparison classes C and sets A:[[forcc(C)(A)]](P ) = P ,and is defined if and only if (C = A).56

With this interpretation of the for -clause, an AP such as tall forcc a boy has the

following interpretation:

[[tall forcc a boy]]=[[tall( C) forcc a boy]] (Spelling out the hidden variable in the AP)=([[forcc]](C) ([[a boy]])) ([[tall( C)]])= [[tall( C)]]

Presupposition: C = [[a boy]],where [[a boy]] is the set of boys in the model.

In terms of non-presuppositional interpretation, tall forcc a boy is equivalent to the

AP tall. However, the presupposition restricts the value of the comparison class to

the set of boys and this comparison class affects the value assigned to tall. As a result,

the value assigned to tall forcc a boy can be quite different from the value assigned

to tall. For the rest of this chapter, I assume this interpretation of for -clauses.

5.3 Examples of Indirect Comparison

To demonstrate how the semantics given above can account for indirect compar-

ison, I present two situations where the truth or falsity of cross-scalar comparisons

seems rather clear. I then demonstrate how the interpretation of MORE and asEQ

correctly accounts for these judgments. For the sake of simplicity, I restrict the do-

main of the situations to rather small numbers. Such a restriction simplifies graphical

representations and sharpens linguistic judgments. Also in the first situation, I limit

168

the domain so that no two people have a gradable property to the same extent. As a

result, the underlying quasi orders are isomorphic to the primary scales. I feel that

this makes the situation a little more accessible. In the second situation, I allow for

individuals to have a gradable property to an equal extent. As a result, the quasi

orders are quite different from the quotient structures. Although this situation is a

little more complex, judgments are still quite clear.

I begin by constructing the simpler situation. To create such a context I will

invent a fictitious committee consisting of ten members. In our model, each member

will be associated with a letter from a through j. All the sentences I use involve

statements that compare certain members of the committee: namely Betty, Heather,

Ida and Evelin. The letter b will represent the interpretation of Betty, the letter h

the interpretation of Heather, the letter i the interpretation Ida and the letter e the

interpretation of Evelin. An important characteristic of this fictitious committee is

that each member differs from the others in terms of beauty and intelligence. In fact,

the order from the most beautiful to the least is as follows:

a → b → c → d → e → f → g → h → i → j.

The individual a has more beauty than b who has more beauty than c and so on and

so forth. In contrast, the order from the most intelligent to the least is as follows:

i → f → j → g → h → a → d → b → e → c.

The individual i has more intelligence than f who has more intelligence than j and

so on and so forth.

There are two quasi orders that are relevant in this context. The first is the

quasi order ‘β’ whose domain consists of people and whose graph consists of all the

169

ordered pairs 〈x, y〉 such that x has as much beauty as y. The second quasi order is

‘ι’ whose domain also consists of people and whose graph consists of all the ordered

pairs 〈x, y〉 such that x has as much intelligence as y. In all of the sentences presented

in this section, the comparison class C will be overtly limited to the set of committee

members.

The context, as it is currently constructed, determines the following graphical

representation of the quasi order (β C) as well as the following representation of

the quotient structure (β C)/∼. Furthermore, the homomorphism H(βC )/∼ yields

the following degrees in Ω.

170

β C (β C)/∼ Ω

a a d 1010

b b d 910

c c d 810

d d d 710

e e d 610

f f d 510

g g d 410

h h d 310

i i d 210

j j d 110

The degrees on the far right represent the position of the committee members in a

scale of beauty (for committee members).

Similar to the effects on ‘β’, such a context also determines the following graph-

ical representation of the quasi-order (ι C) as well as the quotient structure

(ι C)/∼. Also, the homomorphism H(βC )/∼ yields the following degrees in Ω.

171

ι C (ι C)/∼ Ω

i i d 1010

f f d 910

j j d 810

g g d 710

h h d 610

a a d 510

d d d 410

b b d 310

e e d 210

c c d 110

The degree on the far right represent the position of the committee members in the

scale of intelligence.

With these scales in mind, consider the four sentences listed below.

(155) a. Betty is more beautiful for a committee member than Heather is intelli-

gent.

b. Betty is more intelligent for a committee member than Evelin is beautiful.

172

c. (Unfortunately) Ida is as beautiful for a committee member as Evelin is

intelligent.

d. Heather is as beautiful for a committee member as she is intelligent.

Given the situation specified above, most speakers consider the sentences in (155a)

and (155c) to be true. Since Betty is the second most beautiful among the committee

members whereas Heather is only the fifth most intelligent, it follows that Betty is

more beautiful for a committee member than Heather is intelligent. Also, since Ida

is the second least beautiful and Evelin is the second least intelligent, it follows that

Ida is as beautiful for a committee member as Evelin is intelligent.

In contrast to these two sentences, most speakers consider (155b) and (155d) to

be false. Betty is the third least intelligent committee member whereas Evelin is in

the fifth most beautiful, hence Betty is not more intelligent for a committee member

than Heather is beautiful. Also, Heather is at the lower end of the scale in terms of

beauty but in the middle in terms of intelligence. These facts suggest that she is not

as beautiful (for a committee member) as she is intelligent.

These empirical results are consistent with predictions of the theory presented

in chapter 4. To demonstrate this, let’s consider how these sentences would be

interpreted in this context. Note that to make the derivation slightly more readable

I leave out the compositional integration of the prepositional phrase for a committee

member. Recall that this phrase simply passes up the value of its sister (via the

identity function) and presupposes that the variable C is equal to the set of committee

members. I will assume throughout the derivation that C = a, b, c, d, e, f, g, h, i, j.

173

Also, I ignore the interpretation of tense or any other modal elements for the time

being.

Below I present the derivation of the truth value for (155a) in several steps.

First I derive the value of the subordinate clause than Heather is intelligent. Then

I derive the value for the entire sentence Betty is more beautiful for a committee

member than Heather is intelligent.

The subordinate clause contains many elements that are not overtly represented:

the degree variable ‘d’, the degree operator ‘operatord ’, the function ABS and the

comparison class C. When all the hidden elements are spelled out, the subordinate

clause ends up having the following syntactic structure:

(156) Subordinate Clause

qqqqqqqMMMMMMM

than S

qqqqqqqMMMMMMM

operatord S

qqqqqqqMMMMMMM

Heather TP

qqqqqqqMMMMMMM

is AP

qqqqqqqMMMMMMM

d ??

qqqqqqqMMMMMMM

ABS GA

qqqqqqqMMMMMMM

intelligent C

I provide the interpretation of this subordinate clause in segments: first interpreting

the AP [d [ABS [intelligent C]]].

174

[[d ABS intelligent C]]= ([[ABS]]([[intelligent]] C))(d)= ((λζ λd λx(Hζ/∼(x) d)(ι C))(d)= (λd λx(H(ιC )/∼(x) d))(d)= λx(H(ιC )/∼(x) d)

This phrase is interpreted as a set (or the characteristic function that defines the

set). This is a standard interpretation for adjectival phrases.

Having interpreted the AP predicate, I now provide an interpretation for the

entire phrase [than [ operatord [Heather is [d ABS intelligent C]]]].

[[ than operatord Heather is d ABS intelligent C]]= ([[than]])[(operatord)(([[d ABS intelligent]])([[Heather]]))]= ([[than]])[(operatord)((λx(H(ιC )/∼(x) d))(h))]

= ([[than]])[(operatord)((H(ιC )/∼(h) d))]= ([[than]])[(operatord)(d 6

10 d)]

= ([[than]])(d : (d 610 d))

= sup(d : (d 610 d))

= d 610

The interpretation of this subordinate clause is equivalent to the universal degree

that represents Heather’s position in the scale of intelligence restricted to committee

members. Since she is the fifth most intelligent out of ten people, her position is

represented by the degree ‘d 610

’.

With this interpretation of the subordinate clause, the following meaning can

now be assigned to the entire sentence Betty is more beautiful for a committee member

than Heather is intelligent. Recall that the subordinate clause in this sentence serves

as the first argument to the comparative morpheme. Thus, this sentence will be

interpreted with the following structure.

175

(157) S

qqqqqqqMMMMMMM

Betty TP

qqqqqqqMMMMMMM

is AP

YYYYYYYYYYYYYYYYYYY

PP

qqqqqqqMMMMMMM

AP

qqqqqqqMMMMMMM for a committee member

DegP

qqqqqqq

2222

2222

2222

A

qqqqqqqMMMMMMM

more beautiful C

SC

qqqqqqqMMMMMMM

than Heatheris intelligent

Also, recall that C is equal to the set a, b, c, d, e, f, g, h, i, j. Given this structure,

the interpretation of this sentence can be calculated as follows.

[[ Betty is more beautiful for a committee member than Heather is intelligent]]= (([[MORE]]([[than Heather is intelligent]]))([[beautiful]] C))([[Betty]])= (([[MORE]](d 6

10))(β C))(b)

= ((λd (λζ λx(Hζ/∼(x) d))(d 610

))(β C))(b)

= (λζ λx(Hζ/∼(x) d 610

)(β C))(b)

= (λx(H(βC )/∼(x) d 610

))(b)

= (H(βC )/∼(b) d 610

)

= (d 910 d 6

10)

= 1

176

As is apparent from the second last line of the derivation, the truth conditions of the

sentence are derived from a comparison of two universal degrees: one that represents

Betty’s position on a scale of beauty (for a committee member) and another that

represents Heather’s position on a scale of intelligence (for a committee member).

Since Betty’s relative position is higher, the sentence is true.

The derivation of (155b), Betty is more intelligent for a committee member than

Evelin is beautiful, is almost identical to the derivation of (155a). The subordinate

clause is interpreted as a universal degree, one that represents Evelin’s position in

the scale of beauty, and the truth or falsity of the entire sentence depends on the

comparison of two universal degrees. The derivation is given below.

Interpretation of the AP in the subordinate clause:

[[d ABS beautiful C]]= ([[ABS]]([[beautiful]] C))(d)= ((λζ λd λx(Hζ/∼(x) d)(β C))(d)= (λd λx(H(βC )/∼(x) d))(d)= λx(H(βC )/∼(x) d)

Interpretation of the subordinate clause:

[[ than operatord Evelin is d ABS beautiful C]]= ([[than]])[(operatord)(([[dABS beautiful]])([[Evelin]]))]= ([[than]])[(operatord)((λx(H(βC )/∼(x) d))(e))]= ([[than]])[(operatord)((H(βC )/∼(e) d))]= ([[than]])[(operatord)(d 6

10 d)]

= ([[than]])(d : (d 610 d))

= sup(d : (d 610 d))

= d 610

177

Interpretation of the entire sentence:

[[ Betty is more intelligent for a committee member than Evelin is beautiful]]= (([[more]]([[than Evelin is beautiful]]))([[intelligent]] C))([[Betty]])= (([[more]](d 6

10))(ι C))(b)

= ((λd (λζ λx(Hζ/∼(x) d))(d 610

))(ι C))(b)

= (λζ λx(Hζ/∼(x) d 610

)(ι C))(b)

= (λx(H(ιC )/∼(x) d 610

))(b)

= (H(ιC )/∼(b) d 610

)

= (d 310 d 6

10)

= 0

The truth conditions of the sentence reduce to a comparison of two universal degrees.

The first represents Betty’s position with regard to intelligence (for a committee

member) and the second represents Evelin’s position with regard to beauty (for a

committee member). Since Evelin is lower on the scale of intelligence than Betty is

on the scale of beauty, the sentence is false.

Note that the truth conditions of both of the comparative sentences reduce to

a comparison of two universal degrees that represent positions in different scales.

In fact, since the subordinate clause only contains a simple absolutive sentence, the

truth conditions for both sentences can be simply represented by comparing the

results of the functions that assign universal degrees to individuals. For example,

the truth values of both sentences is equivalent to the values assigned to formulae

(158a-ii) and (158b-ii).

(158) a. i. Betty is more beautiful for a committee member than Heather is

intelligent.

ii. (H(βC )/∼(b) H(ιC )/∼(h))

178

b. i. Betty is more intelligent for a committee member than Evelin is beau-

tiful.

ii. (H(ιC )/∼(b) H(βC )/∼(e))

In the rest of this chapter, I will sometimes take advantage of this equivalency to

avoid recalculating the compositionally derived interpretation.

Similar to the comparative constructions, the equative sentences in (155c) and

(155d) have an interpretation that also depends on a comparison of two universal

degrees. The first represents the position of the subject in the main clause with

respect to beauty whereas the second represents the position of the subject in the

subordinate clause with respect to intelligence. However there are two differences

between these sentences and the comparatives: first equatives have a slightly different

syntactic structure than the comparatives. Although the structure of the subordinate

clauses are identical, the way this clause combines with the rest of the sentence differs

slightly. In equative constructions, the subordinate clause merges after the gradable

adjective. This yields the following syntactic structure for Ida is as beautiful for a

committee member as Evelin is intelligent.

179

(159) S

qqqqqqqMMMMMMM

Ida TP

qqqqqqqMMMMMMM

is AP

YYYYYYYYYYYYYYYYYYY

SC

qqqqqqqMMMMMMM

??

qqqqqqqVVVVVVVVVVVVV as Evelin

is intelligent

??

qqqqqqqMMMMMMM PP

qqqqqqqMMMMMMM

as A

qqqqqqqMMMMMMM for a committee

member

beautiful C

The syntactic structure for Heather is as beautiful for a committee member as she

is intelligent is identical to the one represented in (159) modulo the change in the

lexical items.

The second difference between the comparatives and the equatives is that in

equatives the two universal degrees are compared by a greater-than-or-equal relation

rather than a strictly greater-than relation. This allows for the sentence to be true

when the two degrees are equal. In fact due to pragmatic considerations, the equative

normally claims that the two values are equal.

Taking note of these differences, consider the derivation of the interpretations for

(155c) and (155d) given below. As with the comparatives, I divide the calculations

into three separate sections.

180

Interpretation of (155c):

Ida is as beautiful for a committee member as Evelin is intelligent

Interpretation of the AP in the subordinate clause:

[[d ABS intelligent C]]= ([[ABS]]([[intelligent]] C))(d)= ((λζ λd λx(Hζ/∼(x) d)(ι C))(d)= (λd λx(H(ιC )/∼(x) d))(d)= λx(H(ιC )/∼(x) d)

Interpretation of the subordinate clause:

[[ as operatord Evelin is d ABS intelligent C]]= ([[as]])[(operatord)(([[d ABS intelligent]])([[Evelin]]))]= ([[than]])[(operatord)((λx(H(ιC )/∼(x) d))(e))]= ([[as]])[(operatord)((H(ιC )/∼(e) d))]= ([[as]])[(operatord)(d 2

10 d)]

= ([[as]])(d : (d 210 d))

= sup(d : (d 210 d))

= d 210

Interpretation of the entire sentence:

[[ Ida is as beautiful for a committee member as Evelin is intelligent]]= (([[asEQ ]]([[beautiful]] C))([[as Evelin is intelligent]]))([[Ida]])= ((λζ (λd λx(Hζ/∼(x) d))(β C))([[as Evelin is intelligent]]))(i)= (λdλx(H(βC )/∼(x) d)d 2

10)(i)

= λx(H(βC )/∼(x) d 210

)(i)

= (H(βC )/∼ (i) d 210

)

= (d 210 d 2

10)

= 1

181

Interpretation of (155d):

Heather is as beautiful for a committee member as she is intelligent

Interpretation of the AP in the subordinate clause:

[[d ABS intelligent C]]= ([[ABS]]([[intelligent]] C))(d)= ((λζ λd λx(Hζ/∼(x) d)(ι C))(d)= (λd λx(H(ιC )/∼(x) d))(d)= λx(H(ιC )/∼(x) d)

Interpretation of the subordinate clause:

[[ as operatord she is d ABS intelligent C]]= ([[as]])[(operatord)(([[d ABS intelligent]])([[she]]))]= ([[as]])[(operatord)((λx(H(ιC )/∼(x) d))(h))]

= ([[as]])[(operatord)((H(ιC )/∼(h) d))]= ([[as]])[(operatord)(d 6

10 d)]

= ([[as]])(d : (d 610 d))

= sup(d : (d 610 d))

= d 610

Interpretation of the entire sentence:

[[ Heather is as beautiful for a committee member as she is intelligent]]= (([[asEQ ]]([[beautiful]] C))([[as she is intelligent]]))([[Heather]])= ((λζ (λd λx(Hζ/∼(x) d))(β C))([[as she is intelligent]]))(h)= (λdλx(H(βC )/∼(x) d)d 6

10)(h)

= λx(H(βC )/∼(x) d 610

)(h)

= (H(βC )/∼(h) d 610

)

= (d 310 d 6

10)

= 0

182

As demonstrated above, the sentence in (155c) is true since the position of Ida relative

to the scale of beauty is the same as the position of Evelin relative to the scale of

intelligence. The universal degrees are able to represent these positions. In contrast,

with the sentence in (155d) the position of Heather relative to the scale of beauty is

lower than her position relative to the scale of intelligence. Hence why the sentence

is false.

Note that as with the comparatives, the truth conditions of both of the equative

sentences reduce to a comparison of two universal degrees. Like the comparatives,

these truth conditions are equivalent to those of the formulae in (160a-ii) and (160b-

ii).

(160) a. i. (Unfortunately) Ida is as beautiful for a committee member as Evelin

is intelligent.

ii. (H(βC )/∼ (i) H(ιC )/∼(e))

b. i. Heather is as beautiful for a committee member as she is intelligent.

ii. (H(βC )/∼(h) H(ιC )/∼(h))

As with the comparative, I will take advantage of this equivalency to sometimes

avoid repetition of the derivation.

In summary, a theory with universal degrees accurately accounts for the truth

conditions of comparative and equative sentences when considering simple situations

where the domain of people is limited and equality is not an issue. However, a

question still remains about whether a more complex situation would yield different

results. As I demonstrate below, adding more complexity to the situation does not

183

significantly change either the truth value judgments of the speakers nor the truth

value assignments of the semantic theory.

Let’s consider one example. Let me expand the committee mention above so

that it contains five additional members on top of the original ten. Let’s denotes

these individuals as a′, b′, c′, d′, and e′. Suppose that a′ and b′ are equally as beautiful

as Betty (denoted by b). Suppose that c′ is equally as beautiful as c, d′ is equally

as beautiful as d, and e′ is equally as beautiful as e (Evelin). Keeping the relations

between the original ten members the same, the following diagram represents the

expanded quasi order associated with beauty.

184

β C

a

hhhhhhhhhhhhh

VVVVVVVVVVVVV

b

MMMMMMM

YYYYYYYYYYYYYYYYYYY b′

qqqqqqqMMMMMMM a′

qqqqqqq

eeeeeeeeeeeeeeeeeee

c

VVVVVVVVVVVVV c′

hhhhhhhhhhhhh

d

VVVVVVVVVVVVV d′

hhhhhhhhhhhhh

e

MMMMMMM e′

qqqqqqq

f

g

h

i

j

Since, b, b′ and a′ all have as much beauty as each other, they occupy the same

level in the diagram. Similar reasoning explains why c and c′ are on the same level:

likewise for d and d′ and e and e′.

The quotient structure collapses the individuals that are equally as beautiful into

equivalence classes. These equivalence classes can then be associated with a degree

in the universal domain that represents their position in the quotient structure. The

scale of equivalence classes and the associated degrees in Ω are given below.

185

(β C)/∼ Ω

a d 1010

b, b′, a′ d 910

c, c′ d 810

d, d′ d 710

e, e′ d 610

f d 510

g d 410

h d 310

i d 210

j d 110

Although the expanded quasi order changes the composition of the equivalence

classes, it does not change how the individuals are mapped to universal degrees.

Betty’s beauty and Evelin’s beauty (for committee members) are still represented by

the universal degrees d 910

and d 610

respectively.

Similar affect can be demonstrated with the quasi order associated with intelli-

gence. Suppose out of the five extra members, a′ is equally as intelligent as Heather

(h), while b′ and c′ are equally as intelligent as Betty (b). Also suppose that d′ is

186

equally as intelligent as f and e′ is as equally intelligent as e (Evelin). Keeping the re-

lations between the original ten members the same, the following diagram represents

with expanded quasi order associated with intelligence.

ι C

i

qqqqqqqMMMMMMM

f

MMMMMMM d′

qqqqqqq

j

g

qqqqqqqMMMMMMM

h

MMMMMMM a′

qqqqqqq

a

d

hhhhhhhhhhhhh

VVVVVVVVVVVVV

b

MMMMMMM

YYYYYYYYYYYYYYYYYYY b′

qqqqqqqMMMMMMM c′

qqqqqqq

eeeeeeeeeeeeeeeeeee

e

MMMMMMM e′

qqqqqqq

c

Since b, b′ and c′ have as much intelligence as each other, they occupy the same level

in the diagram. Likewise for f and d′, h and a′, and e and e′.

The quotient structure of this quasi order has the same amount of equivalence

classes as the reduced quasi order. The only difference is in the composition of the

equivalence classes. Below I give the quasi order and the associated degrees in Ω.

187

(ι C/∼ Ω

i d 1010

f, d′ d 910

j d 810

g d 710

h, a′ d 610

a d 510

d d 410

b, b′, c′ d 310

e, e′ d 210

c d 110

The addition of the five members does not effect the value of the universal degrees

that are assigned to the original ten members of the committee as long as the addi-

tional five are equivalent to one of the original ten. The quotient structure absorbs

the new members of the committee into the equivalence classes. The assignment of

the universal degree is based on the number of equivalence classes in the domain

rather than the number of individuals in the quasi order. As a result, the values of

the four sentences interpreted above remain the same in the expanded model as they

were in the reduced model.

188

Interestingly, speaker judgments do not seem to change with the addition of

individuals equivalent to others already in the domain. However there is one qualifi-

cation: the size of the equivalence classes must remain significantly smaller than the

number of equivalence classes in the quotient structure. As I discuss in section 5.4,

when the addition of individuals to the domain results in a significant expansion of

an equivalence class, there are some effects on speaker intuitions.

To summarize the examples presented in this section, the interpretation of the

comparative and equative morphemes basically reduced the truth conditions of the

comparative and equative sentences to a comparison of two universal degrees. These

degrees represent the positions of the main clause subject and the subordinate clause

subject in their respective primary scales. The comparative sentences assert that the

universal degree associated with the main clause is strictly greater than the one

associated with the subordinate clause. In contrast, the equative sentences assert

that the universal degree associated with the main clause is greater than or equal

to the one associated with the subordinate clause. As demonstrated in this section,

these truth conditions accurately account for speaker intuitions.

5.4 Potential Problems with Indirect Comparison

The present theory of comparison yields results that are consistent with speaker

intuitions while maintaining a simple and clear account of indirect comparisons.

Despite these advantages there are two potential problems. The first involves the

level of precision inherent in the linear order associated with ‘Ω’. Speaker intuitions

are not always clear with indirect comparisons even when the number of equivalence

classes is relatively low. I will call this problem the problem of vagueness. The second

189

potential problem involves the effects of large equivalence classes on relatively small

domains. The judgments of many speakers are influenced by the size of certain

equivalence classes and the present theory predicts that there should not be any

influence at all.

Below I discuss each of these problems and some potential solutions. To answer

the first problem, I suggest a minor alterations to the present theory that will intro-

duce a certain amount of vagueness. The source of the theoretical difficulties rests

with the absolute and precise linear order of the set of universal degrees. However, it

is quite trivial to weaken the precision of this order. In contrast to the first problem,

a solution to the second involves quite a major change to the present theory, one that

would see the removal of the primary dimension. If the universal degrees are based

directly on the quasi orders rather than the quotient structures, then the potential

size of an equivalence class will affect the universal degree assignment.

5.4.1 The Problem of Vagueness.

The problem of vagueness is best exemplified when the comparison classes for

the main and subordinate clauses differ. Below I construct a situation that allows for

such a difference. This situation involves two committees with different sized mem-

berships: one with nine members and the other with five members. As I demonstrate,

there are many sentences that are not problematic for such a situation. However,

the semantic interpretation of the equative and comparative morpheme is only con-

sistent with speaker judgments when there is a significant difference between two

universal degrees. As the difference becomes smaller, theoretical predictions are no

longer supported by speaker intuitions.

190

I begin by outlining the relevant situation. It is fairly easy to adjust the sit-

uations and sentences presented above so that the primary scale associated with

the main and subordinate clauses differ. Instead of having one committee and only

one for -clause, a second committee can be added with a second for -clause. In the

situation specified below, there are two committees instead of only one. I will call

the first Committee A and the second Committee B. Committee A will consist of

nine members denoted by the letters a through i. Committee B will consist of only

six member denoted by the letters a′ through e′. In the sentences presented below,

members of Committee A will be compared in terms of their beauty. The order of

individuals in terms this quality is as follows:

a → b → c → d → e → f → g → h → i

The individual a has more beauty than b who has more beauty than c and so on and

so forth.

In contrast, members of Committee B will be compared in terms of their talent.

The order of individuals is specified as follows:

a′ → b′ → c′ → d′ → e′

The individual a′ has more talent than b′ who has more talent than c′ and so on and

so forth.

These orders give rise to the quasi-orders, quotient structures, and associated

universal degrees depicted in the diagrams below.

191

Beauty and Committee A

β C (β C)/∼ Ω

a a d 99

b b d 89

c c d 79

d d d 69

e e d 59

f f d 49

g g d 39

h h d 29

i j d 19

192

Talent and Committee B

τ C ′ (τ C ′)/∼ Ω

a′ a′ d 55

b′ b′ d 45

c′ c′ d 35

d′ d′ d 25

e′ e′ d 15

Note that in the diagrams above, β is the quasi order that contains all the ordered

pairs 〈x, y〉 such that x has as much beauty as y. Also, τ is the quasi order that con-

tains all the ordered pairs 〈x, y〉 such that x has as much talent as y. Furthermore, C

is equal to the set of all Committee A members (the set a, b, c, d, e, f, g, h, i), while

C ′ is equal to to the set of all Committee B members (the set a, b, c, d, e, f, g, h, i).

With these quasi-orders, quotient structures and universal degrees so specified, con-

sider the following sentences.

(161) a. Brenda is more talented for a member of committee B than Evelin is

beautiful for a member of committee A.

b. Betty is more beautiful for a member of committee A than Cindy is

talented for a member of committee B.

It is not hard to find a context where judgments of sentences like these become quite

natural. For example, imagine that there are two contests, one a beauty contest

193

and the other a talent contest. The members of Committee A participate in the first

while the members of Committee B participate in the second. The sentences in (161)

might arise in discussions about who did better in their respective contests.

In such a context, most speaker find both the sentences in (161a) and (161b) to

be true when [[Brenda]] = b′, [[Betty]] = b, [[Cindy]] = c′, and [[Evelin]] = e. Since

Brenda is the second most talented member of Committee B while Evelin is only the

fifth most beautiful on Committee A, it follows that Brenda is more talented than

Evelin is beautiful. Similarly, since Betty is the second most beautiful on Committee

A while Cindy is in the middle of the pack in terms of talent, it follows that Betty

is more beautiful than Cindy is talented.

These judgments are predicted by the current theory. To demonstrate this,

consider the formulae below.

(162) a. Truth Conditions for Brenda is more talented for a member of Committee

B than Evelin is beautiful for a member of Committee A:

(H(τC ′)/∼(b′) H(βC )/∼(e))

= (d 45 d 5

9)

= 1, since 45

= 3645

> 2545

= 59

b. Truth Conditions for Betty is more beautiful for a member of Committee

A than Cindy is talented for a member of Committee B :

(H(βC )/∼(b) H(τC ′)/∼(c′))

= (d 89 d 3

5)

= 1, since 89

= 4045

> 2745

= 35.

194

As discussed in the last section, the formulae in (162a) and (162b) are equivalent

to the truth conditions that are assigned to (161a) and (161b). With these truth

conditions, the sentence in (161a) is true since the universal degree that represents

Brenda’s position relative to the level of talent for members Committee B is greater

than the universal degree that represents Evelin’s position relative to the level of

beauty for the members of committee A. Similarly (161b) is true since the universal

degree that represents Betty’s position in the scale of beauty is greater than the

universal degree that represents Cindy’s position in the scale of talent. In summary

the current theory is able to explain intuitions about cross-scalar comparisons even

when the two scalar domains differ significantly in size and in composition.

Unfortunately, the current theory cannot explain all the intuitions concerning

these types of sentences. Predictions of the theory go awry when the universal degrees

are relatively close in value. For example, consider the following sentence.

(163) a. Evelin is as beautiful (for a member of committee A) as Cindy is talented

(for a member of committee B).

b. Among their respective committees, Evelin is as beautiful as Cindy is

talented.

The number of Committee A members that are more beautiful than Evelin is equal

to the number that are less beautiful: there are four above and four below. Evelin

is in the middle of this scale. Similarly for Cindy, the number of Committee B

members that are more talented than her is equal to the number that are less talented:

there are two above and two below. She also is in the middle of the scale. Many

speakers feel that such a situation renders the sentence in (163a) true. Also, many

195

do not find the sentence true or at least to not have strong intuitions concerning

such sentences. These judgments are problematic for the current theory. According

to the interpretation given for the equative morpheme above, (163a) should be false.

To see why, consider the following formula.

(164) Truth Conditions for Evelin is as beautiful (for a member of committee A) as

Cindy is talented (for a member of committee B):

(H(βC )/∼(e) H(τC ′)/∼(c′)

= (d 59 d 3

5)

= 0, since 59

= 2545

2745

= 35.

As mentioned in the previous section, this formula is equivalent to the truth condi-

tions assigned to equative sentence in (163a). Notice that due to the different values

for the denominators in mapping the scales to universal degrees, the degree assigned

to the middle of the scale in terms of beauty ends up being less than the degree

assigned to the middle of the scale in terms of talent. In fact, as the scalar domains

get smaller the degree assigned to the middle of the scale gets larger. Yet, speakers

often (although not always) treat the middle as if it is assigned to the same degree.

In summary, the present account seem to make the wrong prediction with regard

to these types of comparison. The problem, I believe, lies with the degree of precision

in the fractional domain.

5.4.2 Introducing Vagueness.

Luckily, the interpretation of the comparative and equative morpheme can be

easily weakened to account for speaker intuitions. Below I outline one possibility for

doing this.

196

Instead of basing the interpretation of comparatives and equatives on the linear

order ‘’, such an interpretation can be based on a much weaker ordering relation.

The linear order ‘’ considers two values to be equal by identity. However, another

ordering relation can be developed based on near identity, where two values are

equivalent if the are close to each other in the linear order. To create such a relation,

I will first need to expand the domain of Ω so that it is isomorphic to the domain of

rational numbers (instead of just the rational numbers between zero and one). This

will allow me to include the operations of addition and subtraction over the domain.

Thus Ω will be redefined as follows:

Definition of the Universal Scale Ω:

Let Ω be the ordered quadruplet 〈Dω,, +ω,−ω〉, where Dω is defined asa set of degrees in a one-to-one relation with the set of rational numbers.

Each d in Dω will be labeled with the rational number with which it isin a one-to-one correspondence. For example, d 1

3, d 3

20, and d 25

116will be

the degrees in one-to-one correspondence with 13, 3

20, and 25

116respectively.

The linear order orders elements in Dω in the same way that ≥ ordersthe rational numbers. Thus for all dx and dy in Dω, dx dy if and onlyif x ≥ y.

The addition operator +ω applies to Dω in the same way that addition ap-plies to rational numbers. Thus for all dx and dy in Dω, dx +ω dy = dx+y .

The subtraction operator −ω applies to Dω in the same way that sub-traction applies to rational numbers. Thus for all dx and dy in Dω,dx − ω dy = dx−y .

197

With addition and subtraction defined, the ordering relation can be weakened in the

following way:

Definition of the Vague Ordering ±dc :57

For all dx and dy in Dω, dx ±dc dy if and only if dx (dy + dc)or dx (dy − dc), where dc is a contextually determined variable thatranges over values in Dω.

Definition of the Vague Strict Ordering ±dc :

For all dx and dy in Dω, dx ±dc dy if and only if dx ±dc dy and it isnot the case that dy ±dc dx .

Note that the Vague Ordering is not a partial order since it is no longer anti-

symmetric. The Vague Strict Order is not a strict linear order since it is no longer a

total order. The lack of these properties are a necessary part of the imprecise nature

of the new ordering relation.

These two new ordering relations can be used to redefine the interpretation of

MORE and asEQ. The interpretation can remain essentially the same except that

‘’ will be replaced by ‘ ±dc ’.

Interpretation of the Comparative Morpheme and Equative Mor-pheme:

Where d is a variable that ranges over universal degrees, ζ is a variablethat ranges over quasi-orders, and x is a variable that ranges over indi-vidual:

[[MORE]] = λd λζ λx(Hζ/∼(x) ±dc d)

[[asEQ ]] = λζ λd λx(Hζ/∼(x) ±dc d)

198

By allowing two values to be equivalent if they are near each other in the linear

order, we can account for the equative examples given above: Evelin is as beautiful

for a member of Committee A as Cindy is talented for a member of Committee B. As

long as dc is greater than or equal to d 245

then the universal degree that represents

Evelin’s beauty (d 2545

) will be equivalent to the degree that represents Cindy’s talent

(d 2745

).

Furthermore, by making the value of dc able to shift from context to context,

the level of precision can be adjusted. This might account for speaker variability

concerning judgments that involve comparison of positions that are relatively close

to being equal in their respective scales.

Throughout the rest of this thesis, I will leave this interpretation as a possible

alternative to the first interpretation given above. However for the examples in the

rest of this thesis, vagueness does not generally play a role in the interpretation. In

other words, with most of the examples an interpretation with the Vague Ordering

is equivalent to the interpretation without the Vague Ordering. For simplicity, I will

generally use the original (and precise) interpretation.

5.4.3 The Problem of Equivalence Classes.

Another problem for the present approach involves the size of equivalence classes.

In most circumstances, the number of individuals that are equivalent or even near

equivalent to each other in terms of a certain property is much less than the number

of individuals in the domain. As we have already demonstrated above, absorbing in-

dividuals into equivalence classes can often correctly account for speaker intuitions.

However, sometimes the size of the equivalence classes are so large that they effect

199

speaker judgments. Below I present one such example and explain why such judg-

ments constitute a problem for the interpretation of MORE and asEQ given above.

I begin by setting up the problematic situation. Suppose there is a committee

with seven members denoted by the first seven letters of the alphabet a, b, c, d,

e, f , and g. Suppose that within this committee, a is the most beautiful, b is the

second most beautiful, while the rest have the same amount of beauty and are below

a and b. This situation creates the following quasi order for beauty (restricted to the

committee):

β C

a

b

ccccccccccccccccccccccccc

hhhhhhhhhhhhh

VVVVVVVVVVVVV

[[[[[[[[[[[[[[[[[[[[[[[[[

c d e f g

This diagram is bottom heavy, in the sense that there is a significant amount of

individuals on the bottom level.

Continuing with the situation, suppose that within this committee, a is the

least intelligent, while b is slightly more intelligent than a but less intelligent than

the rest of the committee. Also, the members of the rest of the committee are

equally as intelligent as each other. This situation creates the following quasi order

for intelligence (restricted to the committee):

200

ι C

c

[[[[[[[[[[[[[[[[[[[[[[[[[ d

VVVVVVVVVVVVV e f

hhhhhhhhhhhhhg

ccccccccccccccccccccccccc

b

a

In contrast to the other diagram, this one is top heavy.

Given the derivation of the primary scale specified above and the definition

of the Universal Homomorphism, the following graphical representations depict the

primary scales (quotient structure) and the associated universal degrees for beauty

and intelligence (restricted to the committee):

(β C)/∼ Ω

a d 33

b d 23

c, d, e, f, g d 13

(ι C)/∼ Ω

c, d, e, f, g d 33

b d 23

a d 13

An interesting consequence of these scales and this situation it that the universal

201

degree associated with the individual b in terms of her beauty will be identical to

her universal degree in terms of her intelligence: H(βC )/∼(b) = H(ιC )/∼(b) = d 23.

As a consequence, the present theory predicts that the following sentence should be

true, where Betty denotes b.

(165) Betty is as intelligent for a committee member as she is beautiful.

The interpretation of the sentence in (165) is equivalent to the following formula:

(166) H(βC )/∼(b) H(ιC )/∼(b)

This formula represents a comparison of Betty’s position on the scale of beauty to

her position on the scale of intelligence. The derivation of the scale of beauty and

intelligence collapses the individuals c, d, e, f , and g into one equivalence class.

Betty’s position in both scales ends up being the same. She belongs to the second

equivalence class from the top. Hence why the sentence is interpreted as being true.

Generally, speakers do not consider the sentence in (165) to be true. In fact,

most speaker consider Betty to be more beautiful for a committee member than she

is intelligent. I suspect that the basis for the judgment stems from the fact that there

are more people that Betty exceeds in beauty than there are people that she exceeds

in intelligence. Whatever the explanation is, the significant effect of the equivalence

classes on judgments is not expected in the present framework.

5.4.4 Abandoning the Primary Scale.

In this section I briefly outline a possible solution to the problem of equivalence

classes. This solution involves giving up the primary scale and mapping individ-

uals to a universal degree based on the underlying quasi order. Ultimately, I do

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not adopt this solution. Equivalence classes and the primary scale prove to be very

valuable for providing an account of direct comparison and thus should not be aban-

doned. Nonetheless, it will be beneficial to see how this problem could potentially

be overcome.

As I mentioned earlier, large equivalence classes create rather odd situations.

Generally, the majority of people can be distinguished from each other in terms of

a certain quality. However, when such classes are large they seem to effect speaker

judgments. One potential means of accounting for these judgments involves basing

the universal degree directly on the quasi order.

As the theory is currently specified, individuals are mapped to degrees with

respect to the positions their equivalence classes occupy in the quotient structure. For

any individual x and quasi order ζ, the universal degree assigned to x is isomorphic

to a rational number that has a denominator equal to the cardinality of the domain of

ζ’s quotient structure and a numerator equal to one plus the number of equivalence

classes x’s equivalence class dominates in ζ’s quotient structure.

Although this kind of mapping yields interesting results, basing the universal

degree on the quotient structure is not necessary to provide an adequate semantics of

indirect comparison. Instead, the degree could be based directly on the underlying

quasi-order. This can be done in the following way.

For any individual x and quasi order ζ, the universal degree assigned to x is

isomorphic to a rational number that has a denominator equal to the cardinality

of the domain of ζ and a numerator equal to one plus the number of individuals

y that have the following property: 〈x, y〉 is a member of ζ and 〈y, x〉 is not a

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member of ζ. Such an assignment leads to the following redefinition of the Universal

Homomorphism.

Redefinition of the Universal Homomorphism ‘H’:

Let H be a function from quasi orders to functions from individuals toUniversal degrees with the following properties.

For all quasi orders ζ and all w ∈ Dζ , Hζ(w) = d xy

where y is equal to

the cardinality of the domain of ζ (y = |Dζ |) and x is equal to one plusthe cardinality of z : 〈w, z〉 ∈ ζ & ¬(〈z, w〉 ∈ ζ) (x = 1 + |z : 〈w, z〉 ∈ζ & ¬(〈z, w〉 ∈ ζ)|).

This redefinition changes the universal degree assigned to Betty in terms of beauty

and intelligence. The cardinalities of the domains of the quasi orders ‘β C’ and

‘ι C’ are both equal to seven. The set z : 〈b, z〉 ∈ (β C) & ¬(〈z, b〉 ∈ (β C))

is equal to c, d, e, f, g. The cardinality of this set is five. Thus the universal degree

that represents Betty’s position relative to beauty for a committee member is d 67.

In contrast, the set z : 〈b, z〉 ∈ (ι C) & ¬(〈z, b〉 ∈ (ι C)) is equal to a.

The cardinality of this set is one. Thus the universal degree that represents Betty’s

position relative to intelligence is d 27.

As a result of these new degree assignments the sentence in (165) (Betty is

as intelligent for a committee member as she is beautiful) is now predicted to be

equivalent to the following formula.

(167) H(βC )(b) H(ιC )(b)

This formula is false since d 27

is not greater than or equal to d 67.

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In summary, speaker judgments can be explained within the framework of the

present theory if the primary scales are ignored when mapping individuals to uni-

versal degrees. Although this might seem desirable given the example above, I feel

that I should add a note of caution. Not all situations or adjectives demonstrate

this kind of effect with equivalence classes. As I discussed in section 5.3, in most

situations the number of individuals in each equivalence class does not vary signifi-

cantly: rather individuals are distributed more evenly. In such situations, collapsing

individuals into equivalence classes correctly predicts that the number of individuals

in each equivalence class does not effect speaker judgments. Also, certain gradable

properties more amenable to equality are sometimes unaffected by the content of

an equivalence class. These include adjectives like tall and wide where individuals

are equal if they share the same measurement. Consider the following situation.

Suppose that every member of a committee has one of three heights: 5′10′′, 5′11′′,

or 6′. Also, suppose that every member of the same committee has one of three

widths: 2′3′′, 2′5′′, or 2′7′′. If a member of this committee is five feet and eleven

inches tall and two feet and five inches wide, then one could claim that he is as tall

for a committee member as he is wide. He is associated with the middle value with

respect to both of the gradable properties. Note that this claim can be true even

if three people occupy the low end of heights and seven the high end, while seven

people occupy the low end of widths and three the high end. The important fact

is that the person can be associated with the middle value in both scales. Thus,

sometimes in certain situations the number of individuals can be ignored in favour of

the equivalence classes. For this reason, I am reticent about giving up on equivalence

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classes completely. Also, as I discuss in detail in the section on direct comparison,

equivalence classes are integral to providing a unified account of direct and indirect

comparison.

5.5 Summary of Indirect Comparison

The semantics given for the comparative and equative morphemes allow for

individuals to be compared through universal degrees that represented their positions

in a primary scale. The primary scales are created from the underlying quasi-orders

associated with the gradable adjectives in the main and subordinate clauses. The

interpretation given for equative and comparative sentences has truth conditions that

are based on a comparison of the two universal degrees associated with the main and

subordinate clauses. For comparatives, the sentences are true if and only if the

degree associated with the main clause is strictly greater than the one associated

with the subordinate clause. In contrast, the interpretation of equatives permits the

two degrees to be equal.

For the most part, comparison of these universal degrees accurately reflects

speaker intuitions. I believe that the same kind of analysis given for the sentences

above can be extended to more typical examples of cross-scalar comparisons such as

those in (168).

(168) a. Unfortunately I’m as intelligent as Medusa is beautiful.

b. Sidney Crosby is more talented than Einstein was intelligent.

The only difference between these sentences and the ones presented in the previous

sections is that these sentences do not have any overt restriction on the comparison

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classes. As a result, the comparison class for such sentences are possibly (even proba-

bly) quite large and hard to determine. In making claims about my intelligence, I am

generally limiting the scale to only include human beings. In making claims about

Sidney Crosby’s talent (a famous Canadian hockey player) I am generally limiting

the scale to other hockey players. These types of scales might include hundreds or

thousands of relevant members instead of the five or ten in the example sentences

discussed above. Such comparison classes would also be highly variable on speaker

experience. If someone only met individuals that were highly skilled at hockey, they

might not consider Sidney Crosby to be all that talented.

The size and indeterminacy of the comparison classes in these sentences make

it hard to provide a complete analysis. However, in principle, the interpretation of

the comparative and equative morphemes specified above should yield the same kind

of analysis for these sentences as it did for the more restricted examples. The truth

conditions of such sentences depend on a comparison between two universal degrees:

one that represents the position of the main clause subject in its primary scale and

another that represents the position of the subordinate clause subject in its primary

scale. The only difference is in the size of the primary scale and hence the size of the

denominators in the fractions.

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Notes

53This strategy of basing a theory on clear examples before considering more con-troversial sentences is discussed in Chomsky (1957).

54 I accept Klein’s analysis with little argument and implicitly reject Ludlow’s(1989). Ludlow argues that comparison classes are determined grammatically byprepositional phrases and modified nouns. I actually do not think there is a strongargument for either position. Klein (1980) draws a nice parallel between the con-textual determination of comparison classes and the determination of the domainof a quantifier. The latter is often treated as a contextually determined variableand hence the parallelism favours a pragmatic analysis of comparison classes. How-ever, as Ludlow discusses, overt prepositional phrases and modified nouns seem tostrictly determine comparison classes regardless of context. Ludlow suggests thatsuch systematic grammatical interactions are an unexpected property of pragmati-cally determined variables. I believe that this debates rests on a much deeper andmore interesting question: namely what does it mean to be a contextually deter-mined variable? What are the properties that distinguish it for more grammaticalprocesses? Until this question is addressed I find the debate to be moot. However,this is outside of the scope of this thesis. The present analysis of comparatives is notdependent on a pragmatic account of comparison classes.

55Note, the parallel is that the oddness of the (b) and (c) examples contrast withthe well-formed sentences in (a) and (d). However, there is a difference. The (b) and(c) sentences in (153) are much less acceptable than their counterparts in (154).

56Actually, a more accurate definedness condition would be to make C the largestpossible subset of A. Recall that when modified nominals and for -clauses appearin the same sentence, then the comparison class is not identical to the nominalcomplement of the for -clause. This is due to the restriction that the value of thecomparison class must be a subset of the modified nominal. I use this definition tosimplify the representation.

57These definitions are inspired by other theories who introduce this kind of im-precision. For example, see Schwarzschild & Wilkinson (2002).

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CHAPTER 6An Account of Direct Comparison

The most notable property of adjectives that permit a direct comparison is their

association with measurement systems. For example, heights, lengths and widths all

can be measured in terms of inches, feet, centimeters, and meters. Similarly, earli-

ness, lateness, age and length of time can be measured in minutes, hours, days and

weeks. Every author who provides an account of direct comparison takes advan-

tage of this property, either by using the common measurement system to justify a

scale independent of the adjectives (see Cresswell, 1976; Kennedy, 1999; and Bartsch

& Vennemann, 1972) or by using the existence of such a measurement system to

explain the semantics of degree modifiers such as three feet, two inches, and four

hours (see Klein, 1982). Like these theories, I base my explanation of direct com-

parison on the existence of measurements; however there are some key differences.

First, unlike Cresswell (1976), Kennedy (1999), and Bartsch & Vennemann (1972),

I do not hypothesize that these measurements constitute degrees or name degrees.

Rather measurements will have the same ontological status as individuals. Second,

unlike Klein (1982), I do not analyze direct comparisons by manipulating the effect

of degree modifiers. Rather, I suggest that measurement systems affect the compo-

sition of the primary scales which in turn affect the assignment of universal degrees.

It is this influence on the assignments of the universal degrees that explains direct

comparisons.

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In this chapter, I describe the influence of measurements on the primary scale

and universal degrees. This influence stems from allowing measurements to partici-

pate in the underlying quasi orders in the same way that other individuals participate

in these relations. As I discuss in section 6.4, quasi orders that contain measurements

produce quotient structures that are isomorphic to the measurement system. Such a

result has interesting consequences when two gradable adjectives are associated with

the same measurement system. For example, tall and wide both permit a modifi-

cation by phrases referring to feet or inches. If both of the quasi orders associated

with tall and wide contain measurements of inches and feet, then both of the quo-

tient structures associated with tall and wide will be isomorphic to a measurement

system of inches and feet and hence isomorphic to each other as well. The two quasi

orders will still differ in terms of how they order individuals (one individual might

be taller than another but the opposite might hold in terms of width) and how they

relate individuals to measurements (most individuals have different measurements

for their width and height), but they will be the same in that each individual will be

equivalent to one and only one measurement. It is this aspect of the quasi order that

establishes the isomorphism to the measurement system. One of the consequences of

this isomorphism is that the assignment of equivalence classes to universal degrees

will be systematically related to the measurement that is contained within the equiv-

alence class. For example, an equivalence class containing the measurement 3 inches

will be mapped to the same universal degree whether the equivalence class belongs to

the quotient structure associated with tall or the one associated with wide. Hence, a

comparison of universal degrees is equivalent to a comparison of measurements. The

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effect of the measurements on the quasi orders and quotient structures derives direct

comparisons without changing the semantic interpretation of the comparative and

equative morphemes.

A prediction of this theory is that direct comparisons depend on measurement

systems. This prediction is confirmed by two sources of evidence. First, the in-

troduction of a hypothetical measurement system induces a direct comparison for

sentences that under other circumstances only permit indirect comparisons. Second,

prepositional phrases that restrict measurements from the underlying quasi order

force indirect comparisons.

6.1 Measurements in Language

Before addressing the issue of direct comparisons, it might be useful to discuss

terms like three feet, three years, three minutes, and three degrees. Such terms are

generally called measure phrases in the literature and I will follow this tradition

here. In what follows, I draw attention to two different uses of measure phrases. As

I discuss, their interpretation is different when they appear in subject position as

opposed to when they are used as degree or differential modifiers.

Distributionally measure phrases fall into two categories. They appear in the

same position as degree modifiers like very, somewhat, and a little and they also

appear in subject positions like other nominals such as Seymour, Esme and the boy.

Consider the sentences in (169) compared to those in (170).

(169) a. Mary is [very/somewhat/a little/seven feet ] tall.

b. Mary is [somewhat/a little/seven feet ] taller than Esme.

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c. Despite what you say, I believe that [Esme/Seymour/the boy/six feet ] is

tall.

d. Mary is somewhat taller than [Esme/Seymour/the boy/seven feet ].

(170) a. * Mary is [Esme/Seymour/the boy ] tall.

b. * Mary is [Seymour/the boy ] taller than Esme.

c. * Mary is taller than [very/somewhat/a little].

d. * Despite what you say, I believe that [very/somewhat/a little] is tall.

As demonstrated by these sentences, measure phrases pattern with degree modifiers

and noun phrases (as shown in (169)) even though noun phrases and degree mod-

ifiers are otherwise in complementary distribution (as shown in (170)). Such facts

point to two different roles for measure phrases. Not only do such phrases have an

interpretation that is similar to degree modifiers such as somewhat, they also have an

interpretation similar to noun phrases. For direct comparisons, the role of measure

phrases as noun phrases is particularly important. Let me briefly address some of

the syntactic characteristics of this role in more detail.

An interesting fact about the nominal behavior is that measure phrases are sin-

gular even when the nouns that refer to them have plural morphology. Furthermore

measure phrases cannot contain determiners. Consider the sentences in (171).

(171) a. Seven feet is tall.

b. ?? Seven feet are tall.

c. Those seven feet are wide.

d. * Those seven feet is wide.

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In these sentences, only (171a) is a statement about measurements. When there is

plural agreement as in (171b) and (171c), the phrase seven feet refers to actual feet.

The same result occurs when a determiner is added. In fact, to refer to measure-

ments, the nominal must have singular agreement and must never be modified by a

determiner. In this respect, measure phrases in the subject position are exactly like

names. The only noticeable difference is that measure phrases refer to measurements

rather than to human beings or institutions. These syntactic facts are relevant to

the arguments presented in section 6.3.2 where I suggest that measurements have

the same ontological status as individuals.58

In summary, measure phrases have two uses: they can be used as nouns or

as degree modifiers. When used as nouns, such phrases demonstrate the syntactic

characteristics of proper names.

6.2 What are Measurement Systems?

To provide a complete analysis of direct comparisons, it is important to discuss

the nature of the measurement systems that are involved in such comparisons. I

should qualify that by measurement systems I am literally referring to the invented

scales that are shared by a society and that provide objective measurements. For

example, the Celsius and Fahrenheit scales provide measurements of temperature.

Scales of feet and inches or meters and centimeters provide measurements of length or

distance. Squared feet or meters provide measurements of area, cubic feet or meters

measurements of space. Minutes, hours, days, and years all provide measurements

of time.

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Interestingly only two kinds of measurement systems are involved in direct com-

parison: measurement systems associated with length or distance and systems as-

sociated with time. An example of the former has been presented numerous times

already, but another example appears in (172a). An example of the latter appears

in (172b).

(172) a. The door is taller than it is wide.

b. Anne was as early for the meeting as Seymour was late.59

In (172a) the measurement of the doors length (perhaps in terms of inches) is being

compared directly with a measurement of its width. In (172b), a measurement of

Anne’s earliness (perhaps in terms of minutes) is being compared to a measurement

of Seymour’s lateness.

There are two aspects of measurements of distance and time that I believe to be

important for establishing a direct comparison. First, different types of adjectives

can be connected to the same measurement system. Second, the measurement sys-

tems have a beginning point (otherwise known as a zero element). Also, although

the measurement systems themselves are often dense, people use the measurement

systems as if they were discrete. In other words, for practical purposes people often

round up measurements to the closest inch, foot, day or year, and treat measure-

ments as if they were a multiple of the contextually determined base (whether inches,

feet, minutes or days). In what follows, I discuss both of these properties in more

detail.

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6.2.1 Two Adjectives, One Measurement System

Direct comparisons (like indirect comparisons) occur with sentences that have

two different adjectives in the main and subordinate clauses. The difference between

direct and indirect comparisons is that the two adjectives in direct comparisons are

associated with the same measurement system. Evidence for this association can be

gathered from the type of degree modifiers the adjectives permit. In this section, I

review some of this evidence.

As discussed in Klein (1982), adjectives like wide, high, tall, long and deep can

all be modified by measure phrases such as 60 centimeters, three feet, 10 inches. Such

degree modifiers are not permitted for any other adjective. For example, in contrast

with the sentences in (174), the sentences in (173) are well-formed.

(173) a. The door is three feet wide.

b. The board is three feet long.

c. Seymour is three feet tall.

d. The ceiling is three feet high.

e. The grave is three feet deep.

(174) a. * Mary is three feet beautiful.

b. * Seymour is three feet late.

c. * Esme is three feet intelligent.

The fact that the adjectives in (173) are grammatical with this type of degree modifier

supports the claim that all these adjectives share a connection to a measurement

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system that other adjectives do not: a system of inches, centimeters, feet and other

units involving linear space.

The same kind of relationship is observed with the adjectives long (as in time),

old, late, and early. Each of these adjectives can be modified by measure phrases such

as three years, sixteen months, thirty days and ten minutes. No other adjectives can

be modified by such phrases. For example, all the sentences in (175) are well-formed,

while those in (176) are not.

(175) a. That newborn is only fifteen minutes old.

b. Seymour was fifteen minutes late for the meeting.

c. Seymour was fifteen minutes early for the meeting.

d. The meeting was fifteen minutes long.

(176) a. * That newborn is only fifteen minutes beautiful.

b. * Seymour is fifteen minutes intelligent.

c. * Esme is fifteen minutes tall.

Once again, this relationship supports the claim that all the adjectives in (175) are

associated with a measurement system that the adjectives in (176) cannot access.

In summary, the adjectives involved in direct comparisons are similar in that

they are all related to the same kind of measurement system. As I discuss below,

this property is essential for maintaining a unified account of indirect and direct

comparisons.

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6.2.2 Measurements of Time and Linear Space: A Well-Ordered System

Another quality that I believe to be relevant for direct comparisons concerns how

speakers treat measurement systems of time and linear space. Such measurement

systems are often treated as if they were well ordered: that is, as if they had a minimal

measurement that was ordered below all others and as if every measurement in the

system had a unique successor.60 In this section, I discuss this property in more

detail.

The idea that measurement systems involving time and space have a starting

point is rather uncontroversial. Any measurement system that applies to time or

linear distance usually has a smallest measurement (whether it be zero, one second,

one inch, etc.). Hopefully anyone who has knowledge of such scales would concede

at least this fact.61 A more controversial property involves the claim that people

(at least sometimes) conceive of measurements as being limited in fine-grainedness.

Below I try to justify this claim. But first, let me outline by example exactly what

I mean by “limitations in fine-grainedness.”

Measurements are often used as if they were limited in terms of how small the

basic unit can be. For example, heights are normally given in terms of inches and

feet, but not in terms of quarter inches. Reports of peoples’ heights are rounded off

(up or down) to the closest inch. Similarly, distances between cities are normally

given only in miles. As with heights, reports of distances between two cities are often

rounded off (up or down) to the closest mile. Units of measurement below the mile

are not relevant. The same kind of limit in the unit of measurement can be observed

with time. Reports of lateness (or earliness) are commonly rounded off to the nearest

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minute. Similarly reports of age are often given in terms of years, not months or

days.62 Depending on the context, people ignore lesser units of measurement even if

those units are part of the same scale.

Not only are reports rounded off to a basic unit, but judgments are affected by

this rounding off. For example, if Seymour and Brad report their heights as being

six feet and two inches, most people would accept the following two statements in

(177) as true.

(177) a. Seymour is as tall as Brad and Brad is as tall as Seymour.

b. Seymour and Brad are equally as tall.

People would accept these statements even if they knew that Seymour and Brad’s

heights differed by an eighth of an inch.

Similarly, if Seymour and Brad both report their age as 30 years, most people

would accept the following two statements in (178) as being true.

(178) a. Seymour is as old as Brad and Brad is as old as Seymour.

b. Seymour and Brad are the same age.

They would accept these statements even if they knew that Seymour and Brad were

not born on the same day.

Examples like these are common. Two mountains can be talked about as having

the same height despite differing by a few inches. The distances between two sets of

cities can be talked about as being equal despite the fact that they might differ by

several feet. Two individuals can be equally as late despite arriving a few seconds

apart.

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These kinds of examples demonstrate what I mean by the claim that people

treat measurement systems as if they are limited in their fine-grainedness. The fact

that people round off measurements to a certain unit and then also (at least some-

times) treat comparisons as if they are only relevant according to these (rounded

off) measurements is an interesting characteristic of the way we use the concept of

measurement. People treat measurement systems as if there is a base unit (contex-

tually or conventionally defined) and each measurement is equivalent to a (natural

number) multiple of this base. To be clear, I am not claiming that measurement

systems of time and distance need to be objectively limited in the precision of their

measurements, rather I am only claiming that in practice people often conceive or

use measurements systems as if they were limited in their precision.

One way to account for this evidence is to hypothesize that people think of (or

are able to think of) these measurement systems as well ordered systems with a base

unit: a well ordered system is a system that is isomorphic to a subset of the natural

numbers.63 For example, in terms of height the base unit is an inch which also

serves as the smallest measurement. All other measurements are (natural number)

multiples of this inch. Hence after an inch, the next measurement is two inches, then

three inches, and so on and so forth. For any measurement, the next measurement is

defined by an increase in the multiple: for a measurement of n inches, the successor

could be defined as n + 1 inches.

A similar example can be given for time. The base measurement may be a

minute. If so, then the next two measurements are two minutes followed by three

minutes, and so on and so forth. In fact, in general if the base unit is b then the next

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two measurements are two b followed by three b, etc. The terms two b and three b

simply name the next measurement.

Further support for this concept of measurement systems can be demonstrated

by the creation of novel bases. As discussed by Bierwisch (1987), almost anything

that can be predicated of tall, wide or long can become a measurement of height,

width and length. For example, consider the sentences in (179).

(179) a. Matchsticks are not very long.

b. Apples are not very tall.

In (179) I give two arbitrary examples of nouns that can be used in the subject

position for the predicates to be long, to be tall and to be wide. The objects named by

these nouns can also be used as a base for a new measurement system. For example,

in taking the length of a prototypical matchstick one can define a measurement

system in terms of matchsticks. The base measurement will be the height of one

matchstick. Other measurements will be multiples of this height. Like the intuitive

treatment of inches or minutes discussed above, this measurement system is well

ordered: it has a base unit and for any measurement there is a unique successor (the

next measurement in the scale). The first measurement is named by one matchstick.

The next by two matchsticks, followed by three matchsticks and so on and so forth.

In the same way, one can define a measurement system based on the height of a

prototypical apple.64 With these novel measurement systems, the sentences in (180)

involving more conventional measurement systems can be rephrased as in (181).

(180) a. An apple is three inches tall.

b. Jon is six inches taller than Seymour is.

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c. The new tiles are six inches narrower than the old tiles.

(181) a. An apple is two matchsticks tall.

b. Jon is four matchsticks taller than Seymour is.

c. Jon is one apple taller than Seymour is.

d. The tiles are six matchsticks (two apples) narrower than the old tiles.

In using the novel measurement systems there is no change in truth conditions of

the sentences. Once one knows how to use one matchstick to measure height, one

also knows the meaning of four matchsticks, five matchsticks, etc. Furthermore, one

knows that an object that is four matchsticks long is shorter than an object that

is five matchsticks long. Note that unlike more conventional measurement systems,

there is no question about in-between measurements. The fine-grainedness of the

measurement system is dependent on the size of the base.

In summary, the way people treat established and novel measurement systems

of time and linear space supports the idea that such systems are considered to be

well-ordered and limited in their fine-grainedness. Furthermore, this limit in fine-

grainedness often determines how people judge comparisons.

6.3 Two Assumptions about Measurements

There are two assumptions about measurements that facilitate an extension of

the interpretations of the comparative and equative morphemes to direct compar-

isons. First, for any context the domain of measurements must be finite (although

arbitrarily so). Second, measurements must participate in the underlying quasi or-

ders as if they were individuals. In other words, they must be able to have as much

height as certain individuals or not have as much height. In this section I explain why

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these assumptions are required while also attempting to provide some independent

justifications.

6.3.1 Domain of Measurements is Finite

As I demonstrate in section 6.4, measurement systems influence the composition

of certain quotient structures by placing each measurement in a unique equivalence

class. Also, as explained above, the function that maps equivalence classes to uni-

versal degrees crucially involves calculating the cardinality of the domain of the

quotient structure. This cardinality becomes the denominator of the fraction that is

isomorphic to the assigned universal degree.

These two facts create a potential problem. If the domain of measurements is

infinite, then the domain of the quotient structure would be infinite.65 This entails

that the denominator will also be infinite. However, it is not clear that a rational

number with an infinite denominator is definable. To eliminate this problem I will

assume that the domain of measurements is always finite within any given context.

This will insure that the universal homomorphism is defined for any primary scale.

This assumption is arbitrary. Obviously the actual domain of measurements

is infinite and there is no non-linguistic reason to limit this domain. However, in

any context, the infinite nature of a measurement system is never needed to justify

a comparison nor is it needed to calculate the truth conditions of non-technical

statements. Thus, there is no theoretical disadvantage to adopting this assumption.

In contrast, by adopting this assumption, a unified account of direct and indirect

comparison becomes possible.

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6.3.2 Measurements as Individuals

As noted by Cresswell (1976) there seems to be something different about the

way we talk about measurements of height, width, length and age as opposed to

beauty, talent and intelligence. In what follows, I discuss these differences and suggest

that they support the treatment of such measurements as having the same ontological

status as individuals, events, states, and other hypothesized entities in the semantic

model.66

Scales related to width, height, length and age are quite different from scales re-

lated to beauty, intelligence and talent. For example, measurements of width, height,

length and age can be denoted by nouns such as inches, feet, hours and days. There

are no corresponding nouns for beauty, intelligence or talent. Furthermore, mea-

surements of width, height, length and age can be predicated by adjectival phrases.

Below I address each of these differences in more detail.

To begin, there is a certain class of nouns that can be used to refer to mea-

surements. Furthermore these measurements generally belong to scales involved in

direct comparison. For example, nouns such as inches, feet, and meters can refer to

measurements of width and height. Perhaps not surprisingly, adjectives such as tall

and wide permit direct comparisons. Similarly nouns such as hours, days, and years

can refer to measurements of age and length (of time). As with tall and wide, the

adjectives early and late also permit direct comparisons.67

Like Klein (1982), I do not believe it is a coincidence that adjectives involved in

direct comparison are associated with nouns that can refer to measurements. Unlike

Klein (1982), who exploits the ability of such nouns to form measure phrases,68 I

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believe the explanation is much more fundamental. Nouns are generally interpreted

as denoting individuals in the domain of discourse (the domain of the model). If

measurements of height (such as inches) and age (such as days) belong to the domain

then it would be expected that nouns would be able to denote such measurements. If

within the semantic model, measurements have the same ontological status as doors,

birthday parties, tables, boys, girls, men and women, then nouns would be expected

to denote these measurements in the same way the noun table denotes the set of

tables or the noun boys denotes the set of boys.

In contrast, the lack of nominal correlates for measurements of beauty, intelli-

gence and talent could be explained by the fact that there are no measurements for

these gradable properties.69 Thus in hypothesizing that measurements of height and

age have the same ontological status as individual men or women while measure-

ments of beauty and intelligence do not, one can explain why nouns are able to refer

to the former but not the latter.

Further support for the close connection between individuals and measurements

comes from predicative facts. Like noun phrases that refer to individuals, noun

phrases that refer to measurements of height, width, length and age can appear as

subjects of adjectival predicates such as is tall, is short, and is long. Some examples

are given in (182).

(182) a. ...I thought Seymour was six feet and five inches tall, but it turns out

that he is only six feet and four inches. Still, six feet and four inches is

quite tall.

224

b. ...I know that Jon is more than three feet tall. In fact, he is five feet.

But, five feet is still quite short.

c. ...Instead of seeing Anne in 10 months, I’ll see her in three. Still, three

months is quite long.

d. ...So I guess Brad was only 30 minutes late instead of 45. Still 30 minutes

is quite late.

In each of these conversational snippets, a measure phrase serves as a clausal subject.

These examples suggest that the individuals denoted by measure phrases participate

in the predicates in much the same way as the individuals denoted by names such as

Seymour and Jon. In other words, the predicates is quite tall, is quite short, is quite

long and is quite late apply to measurements as if they were individuals.70

In summary, the fact that measurements of width, height, length and age are

associated with certain kinds of nouns and the fact that measure phrases can be

subjects of adjectival phrases support the hypothesis that measurements associated

with these scales have the same status as individuals within the semantic theory.

6.4 Explaining Direct Comparisons

The assumptions presented above about measurement systems yield direct com-

parisons when paired with a semantics involving universal degrees. In what follows,

I demonstrate how measurement systems can have such an effect. I first set up a

situation that involves a comparison of height and width. I then contrast how the

measurement system adds to the complexity of quasi orders that are relevant to

such a situation. As I hope to show, the semantic analysis for the comparative and

equative morphemes yield the right truth conditions for a variety of sentences where

225

direct comparison is involved (as long as a well ordered measurement system figures

into the composition of the primary scale).

To begin, let me specify a situation involving seven individuals: six will be

represented by the letters a through f . The seventh will be called Seymour and will

be represented by the letter s. In this situation, Seymour is quite short at five feet

and two inches but quite wide at three feet. The other six individuals are all taller

than Seymour: a is the tallest at six feet and three inches, b the second tallest at

six feet and two inches, and c the third tallest at six feet, while the individuals d, e

and f are all five feet and ten inches tall. Given these individuals and these height

specifications, the quasi order that encodes the has as much height as relation would

have the following graphical representation if it were limited to individuals. Note, I

will use τ to represent this quasi order.

τ

a

b

c

qqqqqqqMMMMMMM

d

MMMMMMM e f

qqqqqqq

s

This diagram shows a quasi ordering of seven elements. However, given the assump-

tions outlined above about measurement systems, the quasi order should be much

more complex. Measurements should be represented in the quasi order just like the

226

individuals. Thus, measurements such as five feet should have as much height as

individuals that are five feet and under. Similarly, individuals that are taller than

or as tall as five feet should have as much height as the measurement five feet. Also,

in accordance with how people usually treat measurements of height (of people), the

measurement system under consideration should normally be limited to inches.

With these additions, the number of elements in the domain of the quasi order

increases significantly. At a minimum, all the measurements from one inch to six

foot and three inches should be included in the domain (75 extra individuals). This

increase in the number of elements changes the composition of the quasi order. As can

be seen in the diagram below, the number of levels in the graphical representation

of the quasi order increases from 5 to a number that extents beyond the confines

of the page. (Note that in the diagram below, dotted lines indicate a gap in the

representation of the quasi order.)

227

τ

6′4′′

qqqqqqqMMMMMMM

6′3′′

VVVVVVVVVVVVV a

hhhhhhhhhhhhh

6′2′′

MMMMMMM b

qqqqqqq

6′1′′

qqqqqqqMMMMMMM

6′

MMMMMMM c

qqqqqqq

5′11′′

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

5′10′′

VVVVVVVVVVVVV d

MMMMMMM e

qqqqqqqf

hhhhhhhhhhhhh

5′9′′

5′8′′

5′7′′

5′3′′

qqqqqqqMMMMMMM

5′2′′

MMMMMMM s

qqqqqqq

5′1′′

In contrast to the diagram without measurements, there is an increase in the number

228

of levels and also an increase in the degree of separation between certain individuals.

For example, 5 levels separate s from d, e, and f whereas in the diagram without

measurements these individuals were only separated by one level. Also, in the dia-

gram above each individual is on the same level as one and only one measurement

(but not vice versa).

This kind of effect with measurement systems can also occur with the quasi order

associated with wide. For example, suppose that in the current situation, Seymour

is the widest at three feet, f is the second widest at two feet and five inches, followed

by b at two feet and two inches. The remaining individuals are all equally as wide

at two feet and one inch. Limiting the quasi order ‘ω’ to people, the relation has as

much width as would maintain the following order.

ω

s

f

b

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

a c d e

This diagram shows a quasi ordering of seven elements.

With measurements participating in the quasi order, the graphical representa-

tion changes significantly. Consider the diagram below. (Once again, I only partially

represent the diagram since there is not enough space for a full representation.)

229

ω

3′1′′

qqqqqqqMMMMMMM

3′

MMMMMMM s

qqqqqqq

2′11′′

2′10′′

2′6′′

MMMMMMM

qqqqqqq

2′5′′

MMMMMMM f

qqqqqqq

2′4′′

2′3′′

MMMMMMM

qqqqqqq

2′2′′

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY b

MMMMMMM

qqqqqqq

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee

2′1′′

VVVVVVVVVVVVV a

MMMMMMM c d

qqqqqqqe

hhhhhhhhhhhhh

2′

As with τ , the number of elements in the quasi order increases significantly. Fur-

thermore, every individual is on the same level as a measurement.

These more complex quasi orders are useful for providing an account of direct

comparison. However, as mentioned earlier in this chapter they are useful insofar as

230

a limit is put on the number of measurements in the domain of discourse. This limit

can be set by arbitrarily choosing an upper bound for the measurement. For present

purposes, I will set the upper bound at six feet and eight inches (eighty inches),

although there is nothing important about this choice. As long as the upper bound

is taller than the tallest person then a direct comparison will be possible.71 (Recall

that sentences that prefer a direct comparison also have an indirect interpretation.

Failure to create the right conditions for a direct comparison simply results in this

indirect comparison. Thus, in explaining how to get direct comparisons, I need to

only outline how such an interpretation could be possible rather than outlining why

it is necessary.) With this arbitrary upper bound, the quasi orders associated with

height and width will only involve measurements that are equal to or below six feet

and eight inches.

One of the consequences of having this upper bound is that it defines the number

of equivalence classes in the resulting quotient structure. Recall that, given the way

people usually treat measurements, every individual will be equivalent in height and

width to some measurement in inches. Since equivalence classes contain all the

individuals that are equal in height (for the quotient structure based on height)

or width (for the quotient structure based on width), then each equivalence class

will contain at least one measurement. Also, since no two measurements have the

same height or width, it follow that each equivalence class will only contain one

measurement. In the diagrams below, I give a partial representation of the quasi

orders associated with height and width (τ & ω) and a partial representation of the

resulting quotient structures (τ /∼ & ω/∼).

231

τ τ /∼

6′8′′ 6′8′′

6′4′′

qqqqqqqMMMMMMM 6′4′′

6′3′′

VVVVVVVVVVVVV a

hhhhhhhhhhhhh6′3′′, a

6′2′′

MMMMMMM b

qqqqqqq6′2′′, b

6′1′′

qqqqqqqMMMMMMM 6′1′′

6′

MMMMMMM c

qqqqqqq6′, c

5′11′′

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV 5′11′′

5′10′′

VVVVVVVVVVVVV c

MMMMMMM e

qqqqqqqf

hhhhhhhhhhhhh5′10′′, c, e, f

5′9′′ 5′9′′

5′3′′

qqqqqqqMMMMMMM 5′3′′

5′2′′

MMMMMMM s

qqqqqqq5′2′′, s

5′1′′ 5′1′′

3′ 3′

0′1′′ 0′1′′

232

ω ω/∼

6′8′′ 6′8′′

5′2′′ 5′2′′

3′1′′

qqqqqqqMMMMMMM 3′1′′

3′

MMMMMMM s

qqqqqqq3′, s

2′11′′ 2′11′′

2′6′′

MMMMMMM

qqqqqqq2′6′′

2′5′′

MMMMMMM f

qqqqqqq2′5′′, f

2′4′′ 2′4′′

2′3′′

MMMMMMM

qqqqqqq2′3′′

2′2′′

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY b

MMMMMMM

qqqqqqq

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee 2′2′′, b

2′1′′

VVVVVVVVVVVVV a

MMMMMMM c d

qqqqqqqe

hhhhhhhhhhhhh2′1′′, a, c, d, e

2′ 2′

0′2′′ 0′2′′

0′1′′ 0′1′′

As with previous examples, individuals that are graphically represented as being on

the same level in the quasi order are grouped into the same equivalence class in

233

the resulting quotient structure. In this partial representation, every measurement

forms its own equivalence class and every individual is a member of an equivalence

class that contains one measurement. As before, dotted lines indicate gaps in the

representation.

This one to one correspondence to the measurement system has some interesting

consequences with respect to the assignment of universal degrees. Recall that each

equivalence class Z in a quotient structure 〈E,≥〉 is mapped to a universal degree

d xy

where x is equal to one plus the number of equivalence classes Z dominates

(|Y : Z ≥ Y |) and where y is equal to the number of equivalence classes in the

domain of the quotient structure (|E|). Consider the quotient structures τ /∼ and

ω/∼. Both quotient structures have equivalence classes that contain at least one and

only one measurement. As a result, the cardinality of the domain of each quotient

structure is defined by the number of measurements in the domain. With six foot

eight (or eighty inches) set as the upper bound for the measurement system, the

cardinality of both domains is 80. Furthermore, for both of the quotient structures

the order of equivalence classes is isomorphic to the order of measurements in the

measurement system. For any two measurements, x and y, if x is greater than y in

the measurement system then the equivalence class that contains x will be above the

equivalence class that contains y in the quotient structure. This holds for τ /∼ and

ω/∼. As a consequence, for any equivalence class Z that contains the measurement

x, the number of equivalence classes Z dominates will be equal to the number of

measurements below x in the measurement system. Considering all these facts, the

end result is that for any measurement x, the equivalence class that contains x in the

234

quotient structure τ /∼ will be mapped to the same universal degree as the equivalence

class that contains x in the quotient structure ω/∼. In other words, the two quotient

structures are isomorphic to each other despite the fact that the content of the

equivalence classes (in terms of people, not measurements) differs quite significantly.

This assignment pattern with respect to measurements and universal degrees

affects the truth conditions for comparative and equative sentences. According to the

interpretation given in chapter 4, the truth conditions for comparatives and equatives

are equivalent to a comparison of two universal degrees. Comparative sentences are

true if and only if the universal degree associated with the main clause is greater than

the one associated with the subordinate clause. Equative sentences are true if and

only if the universal degree associated with the main clause is greater than or equal to

the one associated with the subordinate clause. Furthermore, the universal degrees

are represent the relative positions of the equivalence classes containing the clausal

subjects in their respective quotient structures. The quotient structures are created

from the quasi orders associated with the adjectives in the main and subordinate

clauses. If the adjectives in the main and subordinate clauses are tall and wide

and if these adjectives are affected by the measurement system in the appropriate

way, then the universal degrees for both the main and subordinate clause will be

isomorphic to the position of the measurement contained in the equivalence class.

As a consequence, if the measurement in the equivalence class containing the subject

of the main clause is greater than the measurement in the equivalence class containing

the subject of the subordinate clause, then the universal degree associated with the

main clause will be greater than the one associated with the subordinate clause. A

235

comparison in terms of universal degrees is equivalent to a comparison in terms of

measurements. This is exactly what is wanted to account for direct comparisons.

To understand this parallelism in more detail let’s consider some examples. The

sentences in (183) allow for direct comparisons.

(183) a. Seymour is taller than he is wide.

b. Seymour is wider than he is tall.

c. Seymour is as wide as he is tall.

In the current situation, where Seymour is five feet and two inches tall but three

feet wide, the sentence in (183a) is true where as (183b) and (183c) are false. At

least this is the case for the more salient reading of these sentences.72 Given the

interpretation for the comparative and equative morpheme specified in section 4.3,

the truth conditions of these three sentences will be based upon a comparison of

the universal degrees that represent the positions of Seymour’s equivalence classes

in the quotient structures associated with heights and widths. The Universal Homo-

morphism yields a function that maps Seymour’s equivalence classes to the universal

degrees that represents these positions. Below, I show some of the more relevant

equivalence classes in the quotient structures of heights and widths on the left hand

side while also giving the relevant universal degrees on the right-hand side. Recall

that for this context, the number of measurements has an arbitrary upper bound,

namely 6’8” (or 80 inches).

236

τ /∼ Ω

6′8′′ d 8080

6′4′′ d 7680

6′3′′, a d 7580

6′2′′, b d 7480

6′1′′ d 7380

6′, c d 7280

5′11′′ d 7180

5′10′′, c, e, f d 7080

5′9′′ d 6980

5′3′′ d 6380

5′2′′, s d 6280

3′ d 3680

2′6′′ d 3080

0′1′′ d 180

237

ω/∼ Ω

6′8′′ d 8080

5′10′′ d 7080

5′2′′ d 6280

3′1′′ d 3780

3′, s d 3680

2′11′′ d 3580

2′6′′ d 3080

2′5′′, f d 2980

2′4′′ d 2880

2′3′′, g d 2780

2′2′′, b d 2680

2′1′′, a, c, d, e d 2580

2′ d 2480

0′1′′ d 180

238

As a result of the upper bound, each equivalence class in both of the quotient struc-

tures is mapped to a universal degree that is isomorphic to a fraction of the form x80

,

where x is a natural number. Due to the effect of the measurements on the quasi

order (and hence the quotient structure) each equivalence class will contain only one

measurement, let’s call this measurement m. Furthermore, each equivalence class

will be placed above n other equivalence classes, where n is the number of measure-

ments that m is greater than. Thus, the value of x will always equal n+1, no matter

which quotient structure the equivalence class is in. Stated otherwise, for all X and

Y such that X is a member of the quotient structure τ /∼ and Y is a member of

the quotient structure ω/∼, if a measurement m is a member of both X and Y then

Hτ/∼(X) will be identical to Hω/∼(Y ). X and Y will be mapped to the same univer-

sal degree. For example, if X contained the measurement 3′ and Y also contain the

same measurement, then both equivalence classes would be mapped to d 3680

.

This mapping has consequences for the interpretation of (183a). If the compar-

ison class is broad enough (i.e., if it is larger than Dτ and Dω, then (τ C) and

(ω C) will simply be equivalent to τ and ω. The comparison class will have no effect

on the quasi order or the quotient structure. (Note that this possibility is left open

since there is no prepositional phrase that overtly restricts the comparison class.) As

a result, the following truth conditions represent the interpretation of (183a).

(184) Truth conditions for Seymour is taller than he is wide:

H(τC )/∼(s) H(ωC )/∼(s)

= Hτ/∼(s) Hω/∼(s) (since (τ C) = τ and (ω C) = ω)

239

= d 6280 d 36

80

= 1

The truth of the sentence is based on a comparison of two universal degrees. Since

Seymour is five feet and two inches tall, the measurement 5′2′′ is a member of his

equivalence class for height. Thus, the universal degree that represents Seymour’s

position in the quotient structure is d 6280

. Furthermore, since Seymour is three feet

wide, the measurement 3′ is a member of the equivalence class for width. As a

consequence, the universal degree that represents Seymour’s position in the quotient

structure is d 3680

. The sentence is true since d 6280

is greater than d 3680

. This hold despite

the fact that Seymour is quite wide and yet not tall.

Like (183a) the truth conditions for (183b) and (183c) will be based upon a com-

parison of two universal degrees (one that represents Seymour’s height and another

that represents his width). The truth conditions for (183b) and (183c) are given in

(185a) and (185b).

(185) a. Truth conditions for Seymour is wider than he is tall :

H(ωC )/∼(s) H(τC )/∼(s)

= Hω/∼(s) Hτ/∼(s) (since (τ C) = τ and (ω C) = ω)

= d 3680 d 62

80

= 0

b. Truth conditions for Seymour is as wide as he is tall :

H(ωC )/∼(s) H(τC )/∼(s)

= Hω/∼(s) Hτ/∼(s) (since (τ C) = τ and (ω C) = ω)

240

= d 3680 d 62

80

= 0

The sentences in (183b) and (183c) are false since the universal degree that represents

Seymour’s width (d 3680

) is less than the one that represents his height (d 6280

).

In summary, measurement systems affect quasi orders in such a way that a com-

parison of universal degrees becomes equivalent to a comparison of measurements.

This is what leads to the characteristics that define direct comparisons. Interestingly,

the semantics for the comparative and equative morpheme (that were developed to

account for indirect comparison) do not change. Direct comparisons can simply be

derived from the nature of the quasi orders. Furthermore this kind of effect should

not be limited to adjectives such as tall and wide. Other quasi orders should be

similarly affected by measurements, including ones associated with measurements of

time. Minutes and/or hours should be able to structure quotient structures associ-

ated with earliness and lateness in the same way that inches structure the quotient

structures associated with width and height. Indeed, such a structure would explain

why the sentence in (186) would be true if Mary were five minutes late and Esme

five minutes early for their respective doctors’ appointments.

(186) Mary was as late as Esme was early.

Like the interpretation of sentences with tall and wide, the universal degrees as-

signed to the equivalence classes will be isomorphic to the measurements contained

in them, hence a comparison of universal degrees will be equivalent to a comparison

of measurements: with the sentence above most likely this would be a comparison

of minutes.

241

I should highlight before concluding this section that a direct comparison de-

pends on three contingent properties being met. First, the comparison class variable

must not exclude measurements from the underlying quasi orders. As I discuss in

section 6.5.2, it is possible for such an exclusion to occur and when it does a direct

comparison is not available. Second, both equivalence classes must have the same

upper bound. Note that this is not a necessary fact about the semantic system. In

principle, two quasi orders need not share the same upper bound, although I should

qualify that it is extremely probable for the same upper bound to be specified in

both quasi orders since both of their domains are based on the same model in the

same context. Third, every individual must be equivalent to a member of a discrete

set of measurements. Once again, it is possible for people to treat an individual as if

he is not equivalent to any measurement, although, as noted earlier, people normally

do not treat measurements in this way. Without these three contingent properties

a direct comparison would be impossible. However, the contingency of these prop-

erties is not empirically problematic since indirect comparisons, although strained,

are possible for these types of sentences. Hence, all that is needed to explain direct

comparisons is the possibility of a comparison that is equivalent to one based on

measurements. This kind of interpretation should not be forced by the semantic

system.

6.4.1 Short and Low: A Problem for Klein

Before presenting further support for this theory of direct comparison, I would

like to demonstrate how the current proposal accounts for direct comparisons in-

volving negative gradable adjectives such as low and short. The only other proposal

242

that provides a universal semantics for direct and indirect comparison, namely Klein

(1980, 1982), has difficulty dealing with these kinds of comparison. In what follows,

I first outline Klein’s difficulties before demonstrating how the current theory can

account for these types of examples.

Klein (1980, 1982) bases his account of direct comparison on the existence of

measure phrases as degree modifiers. Certain adjectival pairs permit direct com-

parisons because they can be modified by the same set of measure phrases. Other

adjectival pairs do not permit direct comparisons because they cannot be modified

by the same set of measure phrases. For example, a comparative such as Seymour is

taller than he is wide has a direct interpretation since both tall and wide can be mod-

ified by measure phrases such as four feet and twenty inches. (Recall that according

to Klein’s theory this sentence is true if and only if there is a degree modifier D such

that Seymour is D tall but he is not D wide. With measure phrases serving as the

degree modifiers, such truth conditions reduce to a comparison of measurement.)

Unfortunately for Klein, negative gradable adjectives are not easy to incorporate

into this kind of explanation of direct comparison. As discussed by Seuren (1978)

and Kennedy (1999, 2005) among others, negative counterparts to positive gradable

adjectives such as tall and wide do not permit measure phases as modifiers. Consider

the sentences in (187).73

(187) a. i. Seymour is five feet tall.

ii. ?? Seymour is five feet short.

b. i. Seymour is three feet wide.

ii. ?? Seymour is three feet narrow.

243

c. i. The ceiling is eight feet high.

ii. ?? The ceiling is eight feet low.

d. i. The grave is six feet deep.

ii. ?? The grave is six feet shallow.

In each of the sentence pairs, one member is not well-formed: namely the sentence

with the negative gradable adjective. This fact is problematic for Klein. Since for him

direct comparisons depend on shared measure phrases and since negative gradable

adjectives do not permit modification by measure phrases, his theory would have

difficulty accounting for direct comparisons that involve negative gradable adjectives.

For example, sentences such as Esme is shorter than the ceiling is low have a direct

interpretation. Such a sentence is true in a situation where the ceiling is 5′6′′ high and

Esme is 5′5′′ tall even though these measurements mean that Esme is not short and

yet the ceiling is low. However, this type of a sentence cannot have truth conditions

based on sentences such as Esme is five feet short and the ceiling is not five feet low.

Such sentences are not acceptable.

Unlike Klein’s account, the explanation for direct comparison given in this chap-

ter does not rely on degree modification. As a consequence, the lack of modification

for negative gradable adjectives does not present a problem. However, it still remains

to be shown how the current proposal can account for these facts.

Only one additional assumption is needed to explain this kind of comparison.

The adjectives short and low must be systematically related to their positive coun-

terparts tall and high in that the quasi orders associated with short and low must

be the inverse of the quasi orders associated with tall and high. In other words, a

244

requirement for being a negative gradable adjective is that in all models the interpre-

tation of the negative is the inverse of its positive. This requirement can be stated

more formally as the Polarity Requirement.

Polarity Requirement:

If ζ and ξ are polar opposites then in all models the following conditionholds.

For all 〈x, y〉, 〈x, y〉 is a member of ζ if and only if 〈y, x〉 is a member ofξ.

This requirement shapes the content of the quasi orders associated with short and

low. Note, the inverse relationship between positive and negative gradable adjectives

does have some empirical support. Given any two adjectives in polar opposition (let’s

call them A and B) and any two individuals (let’s call them x and y), if follows that

x is A-er than y if and only if y is B-er than x. In fact, statements of this nature

seem to be tautological. See the examples in (188).

(188) a. Seymour is taller than Esme if and only if Esme is shorter than Seymour.

b. This ceiling is higher than this ladder if and only if the ladder is lower

than the ceiling.

Each of these sentences is true in any given situation.

With the Polarity Requirement, one can characterize the quasi orders associated

with short and low by appealing to the quasi orders associated with tall and high.

Let’s label these quasi orders as σ (short), λ (low), τ (tall), and γ (high). In the

diagrams below, I give a possible (partial) representation of the diagrams for τ and

γ on the right. On the left, I give the related diagrams for σ and λ. As with the

245

height examples in the pervious section, I decided to represent the fine-grainedness

of the measurement system in terms of inches, however I changed the maximal mea-

surement to 10′ (120 inches) instead of 6′8′′ (80 inches). This upper bound seems

more appropriate for ceilings.

τ σ

10′ 0′1′′

5′6′′

qqqqqqqMMMMMMM 5′2′′

MMMMMMM

qqqqqqq

5′5′′

MMMMMMM e

qqqqqqq5′3′′

MMMMMMM s

qqqqqqq

5′4′′

qqqqqqqMMMMMMM 5′4′′

qqqqqqqMMMMMMM

5′3′′

MMMMMMM s

qqqqqqq5′5′′

MMMMMMM e

qqqqqqq

5′2′′ 5′6′′

0′1′′ 10′

246

γ λ

10′

VVVVVVVVVVVVV l

MMMMMMM k

qqqqqqqh

hhhhhhhhhhhhh0′1′′

9′11′′ 5′5′′

MMMMMMM

qqqqqqq

5′7′′

MMMMMMM

qqqqqqq5′6′′

MMMMMMM c

qqqqqqq

5′6′′

MMMMMMM c

qqqqqqq5′7′′

5′5′′ 9′11′′

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

0′1′′ 10′ l k h

These quasi orders determine the following quotient structures.

247

τ /∼ σ/∼

10′ 0′1′′

5′6′′ 5′2′′

5′5′′, e 5′3′′, s

5′4′′ 5′4′′

5′3′′, s 5′5′′, e

5′2′′ 5′6′′

0′1′′ 10′

γ/∼ λ/∼

10′, l, k, h 0′1′′

9′11′′ 5′5′′

5′7′′ 5′6′′, c

5′6′′, c 5′7′′

5′5′′ 9′11′′

0′1′′ 10′, l, k, h

248

Note that the equivalence classes determined by polar opposite adjectives are com-

pletely equivalent. The domain for τ /∼ is the same as the domain for σ/∼. Similarly,

the domain for γ/∼ is the same as the domain for λ/∼. The only difference between

the resulting quotient structures is that the equivalence classes are ordered oppo-

sitely. An interesting aspect of this opposite ordering is that the negative adjectives

reverse the ordering of measurements. In the quotient structures associated with the

positive adjectives (τ /∼ and γ/∼) the equivalence class that contains the measure-

ment 5′6′′ is greater than the one that contains 5′5′′. As demonstrated in the last

section, a consequence of this ordering is that the universal degree associated with

the equivalence class containing 5′6′′ is greater than the universal degree associated

with the one containing 5′5′′. In σ/∼ and λ/∼ the opposite holds. The equivalence

class that contains 5′5′′ is greater than the one that contains 5′6′′. As a result, the

universal degree associated with the equivalence class containing 5′5′′ is greater than

the universal degree associated with the one containing 5′6′′.

Like the positive adjectives, there is an isomorphic relationship between the uni-

versal degrees associated with negative adjectival quotient structures and the mea-

surements contained within the equivalence classes. However, unlike the positives

the ordering of the universal degrees is opposite to the ordering of measurements.

This creates a different relationship between universal degrees and measurements.

Recall that for the positive adjectives, if every equivalence class contains a measure-

ment then the universal degree can be calculated with respect to this measurement:

specifically an equivalence class containing the measurement m will be mapped to

249

the universal degree d n+1u

, where u is equal to the number of measurements and n

is equal to the number of measurements below m. This relationship changes for the

negative adjectives. Since the measurements are arranged in the opposite order, for

any given equivalence class that contains the measurement m, the number of equiv-

alence classes that it dominates will be equal to the number of measurements that

are above m (within this limited domain). Thus the universal degree assigned to this

equivalence class will be d k+1u

where k is the number of measurements greater than

m.

Let’s consider some examples. With the upper bound arbitrarily set at 120

inches, an equivalence class X containing the measurement 5′5′′ (65 inches) will be

mapped to d 56120

if X is a member of the quotient structure associated with short or

low. In contrast, if X were a member of the quotient structure associated with tall

or high, then it would be mapped to d 65120

(= d 1324

).

Having spelled out the consequences of the Polarity Requirement, let’s reconsider

a sentence with two negative gradable adjectives.

(189) Esme is shorter than the ceiling is low.

With the semantics specified above, the truth conditions of this sentence are based

on a comparison of two universal degrees, one that represents the position of the

equivalence class containing Esme in the quotient structure σ/∼ and another that

represents the position of the equivalence class containing the ceiling in the quotient

structure λ/∼. If the first is greater than the second then the sentence is true,

otherwise it is false. As with the positive adjectives considered above, I am assuming

that σ C = σ and that λ C = λ.

250

Since Esme is five feet and five inches tall, the measurement 5′5′′ will be a

member of her equivalence class. As a consequence, the equivalence class containing

Esme will be mapped to d 56120

. Furthermore since the ceiling is five feet and six

inches high, the measurement 5′6′′ will be a member of its equivalence class. As

a consequence, the equivalence class will be mapped to d 55120

. The first universal

degree is greater than the second since the measurement of Esme’s height is less

than the measurement of the ceilings height. As with the positive adjectives, the

measurement system influences the composition of the quasi orders and quotient

structures such that a comparison of universal degrees is equivalent to a comparison

of measurements.

In summary, unlike Klein’s analysis the current proposal does not base direct

comparisons on degree modification by measure phrases. Rather, direct comparison

results from the effect of measurement systems on the composition of quasi orders and

quotient structures. This effect applies equally to negative and positive adjectives.

6.4.2 Cross Polar Anomaly

Before discussing further support for the theory presented in this chapter, I

would like to address how the current proposal interprets comparatives that contain

two polar opposite adjectives. Both Cresswell (1976) and Kennedy (1999) consider

it an empirical advantage of their theories that they provide the same explanation

of why direct comparisons are impossible for sentences with polar-opposite adjec-

tives (such as the sentence in (190a))) and incommensurable adjectives (such as the

sentence in 190b)).

(190) a. Seymour is taller than Mary is short.

251

b. Esme is more beautiful than Mary is talented.

Like these theories, the semantics developed in chapter 4 also treats the interpretation

of these two types of sentences in the same manner. According to the interpretation

of the comparative morpheme, the sentence in (190b) is true or false based on the

position of Esme’s equivalence class in the quotient structure associated with beauti-

ful and the position of Mary’s in the quotient structure associated with talent. Since

beautiful and talented do not share a measurement system, the comparison cannot

have a direct interpretation.

Unlike the sentence in (190b), (190a) contains two adjectives that are associated

with the same measurement system. Such a shared measurement system often results

in direct comparisons: an interpretation where the sentence is true if and only if

the measurement of Seymour’s tallness (his height in inches) is greater than the

measurement of Mary’s shortness (her height in inches). However, (190a) does not

permit such a comparison. Let me briefly outline how the theory presented above

explains the lack of a direct comparison for such constructions.

There are two possibilities to consider when interpreting the sentence in (190b):

one where the comparison class contains measurements and another where it does

not. With the later possibility, a direct comparison would be impossible since such

comparisons crucially rely on measurements participating in the underlying quasi

order. However, even if the measurements were included in both quasi orders, direct

comparisons would still not be permitted. As described in the previous section, the

measurements in the quasi orders associated with short and tall affect the assign-

ment of universal degrees in different ways. The quotient structure created from

252

the interpretation of tall is a direct reflection of the ordering of measurements in

the measurement system. If a measurement x is greater than a measurement y in

the measurement system, then the universal degree assigned to the equivalence class

containing x will be greater than the one assigned to the equivalence class containing

y. As a result the higher end of the scale will be associated with large measurements.

In contrast, the quotient structure associated with short is a reverse image of the

ordering of measurements. If a measurement x is greater than a measurement y in

the measurement system, then the universal degree assigned to the equivalence class

containing x will be less than the one assigned to the equivalence class containing y.

As a result, the higher end of the scale will be associated with small measurements.

Hence, a consequence of these differences is that the link between measurements

and the assignment of universal degrees is not the same for the two quotient struc-

tures. This lack of a shared link makes it impossible to derive a direct comparison.

As demonstrated above, direct comparisons are only possible when both quotient

structures maintain the same link between measurements and universal degrees.

In summary, whether measurements are part of the underlying quasi order or

not, sentences with polar opposite adjectives will never be able to have a direct in-

terpretation. Like Kennedy (1999) and Cresswell (1976) such sentences will grouped

together with instances of so-called “incommensurability”.

6.5 Further Support

Above, I demonstrated the plausibility of a universal interpretation for direct

and indirect comparison. By manipulating the effect of measurement systems on

quasi orders, a uniform interpretation of the comparative and equative morphemes

253

can be maintained. In this section I discuss additional empirical support for this

theory of direct comparison. There are two sources of evidence. The first involves

manipulating pragmatic and contextual factors to artificially create direct compar-

isons. If direct comparisons are simply a consequence of measurement systems, then

by inventing measurement systems one should be able to derive direct comparisons

for sentences that normally would never allow for such an interpretation. The sec-

ond source involves manipulating grammatical structure to force indirect compar-

isons. According to the theory described above, direct comparisons are only possible

when measurements are part of the underlying quasi order. Hence, such compar-

isons should be impossible when measurements are overtly excluded from the quasi

order. Prepositional phases such as for a boy specify a comparison class that does

not contain any measurements (only boys). As I discuss below, when such overt

specifications of the comparison class are used, direct comparisons are impossible.

6.5.1 Coercing Direct Comparisons

The semantic theory presented in chapter 4 accounts for direct comparisons by

appealing to the conceptual nature of measurement systems. In contexts where well-

ordered measurement systems are isomorphic to quotient structures, a comparison

of universal degrees is equivalent to a comparison of measurements. A consequence

of this proposal is that invented or conjectured measurement systems should be

able to derive direct comparisons from sentences that normally only allow indirect

comparisons.74 As I show in this section, this prediction is borne out. Below I first

present a prototypical example of indirect comparison. I then establish the possibility

254

of a measurement system that could be associated with both of the adjectives in ques-

tion. This measurement system is only hypothetical (and in reality very unlikely).

Still, when considering such a measurement system, speaker intuitions change. A

sentence that previously did not permit a direct comparison suddenly demonstrates

the implicational properties of such comparisons.

To begin, consider the sentence in (191).

(191) Esme is angrier than she is happy.

This sentence is true if Esme is (very / quite / somewhat) angry but not (very / quite

/ somewhat) happy. Furthermore, it is impossible for Esme to be somewhat happy

but not angry and still be angrier than she is happy. These entailments properties

are characteristic indirect comparisons.

In a normal context, the lack of a direct comparison is expected since there

is no measurement system that is shared by angry and happy. However, such a

measurement system does not seem to be completely impossible. Let me present one

possibility for a shared measurement system.

Although unlikely, it is possible that the intensity of anger and happiness could

be associated with blood flow to a certain region of the brain. The higher the level of

anger or happiness, the higher the level of blood flow to that region of the brain. By

measuring the level of blood flow in a particular region (level of cerebral blood flow

or CBF) one can obtain a measurement of anger or happiness. Measurements can

be calculated in terms of milliliters per minute per 100 grams of brain tissue. For

the sake of argument, let’s say measurements of CBF can be rounded up to whole

numbers such as 41, 35 and 26 milliliters per minute per 100 grams of brain tissue.

255

With such a measurement system in place, it is possible for blood flow in the

anger region of Esme’s brain to be 30 milliliters per minute per 100 grams while the

measurement in her happiness region might only be 25. Furthermore, it is possible

that most other people are quite a bit angrier than Esme, with a localized cerebral

blood flow above 40. Also it is possible that most other people are sadder than Esme

with a localized cerebral blood flow below 25. In other words, people in general are by

nature more angry than happy. Given this situation, Esme would be considered quite

happy but not all that angry, at least not when compared to other people. However,

given the measurements of Esme’s blood flow, the sentence in (191), repeated in

(192), is true.

(192) Esme is angrier than she is happy.

This sentence is true in a situation where indirect comparisons would not normally

be permitted.

Note that the implausibility that such a measurement system could ever be

established does not weaken the argument under consideration. The interesting con-

sequence of this example is that contexts where such a measurement system is con-

ceptually possible suddenly permit direct comparisons. In summary, there is a close

connection between non-linguistic measurement systems and direct comparisons.

6.5.2 Comparison Classes and Indirect Comparisons

The most convincing evidence for the current analysis of direct comparisons in-

volves sentences where such a comparison is impossible. As I discussed in chapters

2 and 3, the addition of prepositional phrases that determine comparison classes

often force indirect interpretations even when the sentences employ adjectives that

256

normally prefer direct comparisons. In this section I explain how the current theory

accounts for such a phenomenon without any modification. As I discuss below, the

key to this explanation is the inclusion of measurements as individuals. Comparison

classes restrict the domain of the underlying quasi order. If the comparison class does

not contain any measurements, then the quasi order will not contain any measure-

ments. As a result, the measurements will not be able to enter into the equivalence

classes in the quotient structure and hence, there will be no isomorphism between

the assignment of universal degrees and measurements.

Let me introduce some example sentences to describe this account in more detail.

Consider the sentences in (193) below.

(193) a. Seymour is taller than he is wide.

b. Jon is taller than his boat is long.

Both of the sentences in (193) prefer a direct comparison. For example, in evaluating

the sentence in (193a), Seymour is taller than he is wide as long as the measurement

of his height exceeds the measurement of his width. This holds even if Seymour is

wider and shorter than most other men. The same is true of the sentence in (193b).

If the measurement of Jon’s height is greater than the measurement of the boat’s

length then Jon is taller than the boat is long. This holds independently of the boat’s

relative length compared to other boats. Both of the sentences in (193) have truth

conditions that can be stated in terms of measurements. Interestingly, the addition

of prepositional phrases changes the truth conditions for these sentences. This is

demonstrated with the sentences in (194).

(194) a. Seymour is taller for a man than he is wide for a man.

257

b. Jon is taller for a man than his boat is long for a boat.

Unlike the sentence in (193a), the sentence in (194a) can be false even when the

measurement of Seymour’s height is greater than the measurement of his width. In

particular, if Seymour is quite short at four feet and quite wide at three feet then

he is taller than he is wide but he is not taller for a man than he is wide for a man.

Similarly, unlike the sentence (193b), the sentence in (194b) can be true even when

Jon’s height is not greater than the boat’s length. For example, Jon could be very

tall for a man at six feet and four inches but the boat could be not very long for a

boat at nine feet. Given this situation, Jon is taller for a man than the boat is long

for a boat.

A semantics with universal degrees predicts this kind of effect with for -clauses.

Recall that a for -clause basically sets the value of the comparison class for the main

and subordinate clause. As a consequence, the comparison class for the main clause

and the subordinate clause in (194a) will be equal to the set of all men in the context.

For the sentence in (194b), the main clause will have its comparison class limited

to the set of men and the subordinate will have its comparison class limited to the

set of boats. These comparison class values contrast with the sentences in (193)

where no restriction is overtly present. In other words, the comparison classes for

the sentences in (193) can contain degrees whereas the comparison classes in (194)

cannot contain degrees. Such a difference leads to different truth conditions.

The effect of for -clauses is probably best understood with an example. Consider

the following situation. Suppose that the individuals a through s represent the men

in the current context, whereas a′ through s′ represent the boats. The letter s will

258

represent Seymour, the letter j will represent Jon, and the letter b′ will represent

Jon’s boat. For the sentences in (193) the comparison class can be the entire domain.

Thus, restricting the underlying quasi order by the comparison class does not change

the composition of the quasi order. If τ were the quasi order associated with tall,

ω the quasi order associated with wide and ξ the quasi order associated with long,

then (τ C), (ω C) and (ξ C) would be equivalent to τ , ω and ξ respectively.

As a result, the quotient structures associated with tall, wide and long would be

isomorphic to the measurement system used to measure width, height and length.

Below I give a partial graphical representation of the quasi orders and their quotient

structures. Note, for convenience I only represent individuals in the quasi order that

are either measurements, boats or men. A complete representation would involve

many other individuals. (As with the previous diagrams, dotted lines represent gaps

in the representation.)

259

τ τ /∼

6′6′′

qqqqqqqMMMMMMM 6′6′′

6′5′′

MMMMMMM

VVVVVVVVVVVVV a

qqqqqqqMMMMMMM b

qqqqqqq

hhhhhhhhhhhhh6′5′′, a, b

6′4′′

MMMMMMM

VVVVVVVVVVVVV j

qqqqqqqMMMMMMM r

qqqqqqq

hhhhhhhhhhhhh6′4′′, j, r

6′3′′

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY d

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV e

MMMMMMM

qqqqqqq

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee 6′3′′, d, e

6′2′′

MMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY f

MMMMMMM

VVVVVVVVVVVVV g

qqqqqqqMMMMMMM h

qqqqqqq

hhhhhhhhhhhhhi

qqqqqqq

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee 6′2′′, f, g, h, i

6′1′′

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY q

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV c

MMMMMMM

qqqqqqq

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee 6′1′′, q, c

6′

MMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY k

MMMMMMM

VVVVVVVVVVVVV l

qqqqqqqMMMMMMM m

qqqqqqq

hhhhhhhhhhhhhn

qqqqqqq

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee 6′, k, l,m, n

5′11′′ o p 5′11′′, o, p

5′1′′

qqqqqqqMMMMMMM 5′1′′

5′ s 5′, s

260

ξ ξ/∼

20′1′′

qqqqqqqMMMMMMM 20′1′′

20′ a′ s′ 20′, a′, s′

18′ j′ r′ 18′, j′, r′

16′ d′ e′ 6′3′′, d′, e′

15′ f ′ g′ h′ i′ 15′, f ′, g′, h′, i′

12′ q′ c′ 12′, q′, c′

11′ 11′

9′2′′ 9′2′′

9′1′′

qqqqqqqMMMMMMM 9′1′′

9′

MMMMMMM p′ b′

qqqqqqq9′, b′, p′

8′11′′ 8′11′′

8′ l′ m′ n′ o′ 8′, l′, m′, n′, o′

261

ω ω/∼

6′6′′

qqqqqqqMMMMMMM 6′6′′

6′5′′

MMMMMMM

VVVVVVVVVVVVV b′

qqqqqqqMMMMMMM e′

qqqqqqq

hhhhhhhhhhhhh6′5′′, b′, e′

6′4′′ f ′ r′ 6′4′′, f ′, r′

3′1′′

qqqqqqqMMMMMMM 3′1′′

3′

MMMMMMM i s

qqqqqqq3′, i, s

2′9′′ 2′9′′

2′6′′

MMMMMMM

YYYYYYYYYYYYYYYYYYY

[[[[[[[[[[[[[[[[[[[[[[[[[ b

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY c

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV d

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhhMMMMMMM e

ccccccccccccccccccccccccc

eeeeeeeeeeeeeeeeeee

qqqqqqq2′6′′, b, c, d, e

2′5′′

MMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY

[[[[[[[[[[[[[[[[[[[[[[[[[ f

qqqqqqqMMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY g

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhh

qqqqqqqMMMMMMM h

ccccccccccccccccccccccccc

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhh

qqqqqqq2′5′′, f, g, h

2′4′′

MMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY j

MMMMMMM

VVVVVVVVVVVVV k

qqqqqqqMMMMMMM l

hhhhhhhhhhhhh

qqqqqqqm

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhh

qqqqqqq2′4′′, j, k, l, m

2′3′′

VVVVVVVVVVVVV n

qqqqqqqMMMMMMM o

hhhhhhhhhhhhh2′3′′, n, o

2′2′′

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY p

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhhMMMMMMM 2′2′′, p

2′1′′

VVVVVVVVVVVVV q

MMMMMMM r

qqqqqqqa

hhhhhhhhhhhhh2′1′′, q, r, a

2′ 2′

262

There are some important properties that should be highlighted in these quasi orders

and quotient structures. Notice that in the quotient structure associated with height,

Seymour (s) is in an equivalence class that contains the measurement 5′, whereas

in the quotient structure associated with width, he is in an equivalence class that

contains the measurement 3′. Since the quotient structures are isomorphic to the

measurement system, it follows that the Seymour’s equivalence class with respect

to height will be mapped to a greater universal degree than his equivalence class

with respect to width. This fact makes the sentence in (193a) true. In contrast,

the equivalence class that contains Jon (j) in the quotient structure associated with

height also contains the measurement 6′4′′. Furthermore, the equivalence class that

contains Jon’s boat (b′) in the quotient structure associated with length also contains

the measurement 9′. As a consequence, the universal degree assigned to Jon’s equiv-

alence class will be less than the universal degree assigned to his boat’s equivalence

class. This fact makes the sentence in (193b) false.

Now consider the same quasi orders when they are restricted by comparison

classes containing only men or boats.

263

τ C (C is the set of men) (τ C)/∼

a

VVVVVVVVVVVVV b

hhhhhhhhhhhhha, b

j

VVVVVVVVVVVVV r

hhhhhhhhhhhhhj, r

d

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY e

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhhMMMMMMM d, e

f

MMMMMMM

YYYYYYYYYYYYYYYYYYY g

VVVVVVVVVVVVV h

hhhhhhhhhhhhhi

qqqqqqq

eeeeeeeeeeeeeeeeeee f, g, h, i

q

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY c

MMMMMMM

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee q, c

k

MMMMMMM

YYYYYYYYYYYYYYYYYYY l

VVVVVVVVVVVVV m

hhhhhhhhhhhhhn

qqqqqqq

eeeeeeeeeeeeeeeeeee k, l, m, n

o

MMMMMMM p

qqqqqqqo, p

s s

264

ξ C ′ (C’ is the set of boats) (ξ C)/∼

a′

VVVVVVVVVVVVV s′

hhhhhhhhhhhhha′, s′

j′

VVVVVVVVVVVVV r′

hhhhhhhhhhhhhj′, r′

d′

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY e′

MMMMMMM

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee d′, e′

f ′

MMMMMMM

YYYYYYYYYYYYYYYYYYY g′

VVVVVVVVVVVVV h′

hhhhhhhhhhhhhi′

qqqqqqq

eeeeeeeeeeeeeeeeeee f ′, g′, h′, i′

q′

VVVVVVVVVVVVV c′

hhhhhhhhhhhhhq′, c′

b′

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY p′

MMMMMMM

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee b′, p′

l′ m′ n′ o′ l′, m′, n′, o′

265

ω C (C is the set of men) (ω C)/∼

i

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY s

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhhMMMMMMM i, s

b

MMMMMMM

VVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY c

MMMMMMM

VVVVVVVVVVVVV d

hhhhhhhhhhhhh

qqqqqqqe

eeeeeeeeeeeeeeeeeee

hhhhhhhhhhhhh

qqqqqqqb, c, d, e

f

qqqqqqqVVVVVVVVVVVVV

YYYYYYYYYYYYYYYYYYY g

hhhhhhhhhhhhh

qqqqqqqMMMMMMM

VVVVVVVVVVVVV h

MMMMMMM

hhhhhhhhhhhhh

eeeeeeeeeeeeeeeeeee f, g, h

j

MMMMMMM

YYYYYYYYYYYYYYYYYYY k

VVVVVVVVVVVVV l

hhhhhhhhhhhhhm

eeeeeeeeeeeeeeeeeee

qqqqqqqj, k, l,m

n

MMMMMMM o

qqqqqqqn, o

p

qqqqqqqMMMMMMM p

q r a q, r, a

Since measurements are neither men nor boats, the resulting quotient structures will

not contain equivalence classes that involve measurements. As a result, the quotient

structures no longer have an isomorphic relationship with the measurement system.

This changes how the equivalence classes are mapped to universal degrees. Below

I restate the three quotient structures to the left while also giving the associated

universal degree to the right.

266

(τ C)/∼ Ω

a, b d 88

j, r d 78

d, e d 68

f, g, h, i d 58

q, c d 48

k, l, m, n d 38

o, p d 28

s d 18

267

(ξ C ′)/∼ Ω

a′, s′ d 77

j′, r′ d 67

d′, e′ d 57

f ′, g′, h′, i′ d 47

q′, c′ d 37

b′, p′ d 27

l′, m′, n′, o′ d 17

268

(ω C)/∼ Ω

s, i d 77

b, c, d, e d 67

f, g, h d 57

j, k, l,m d 47

n, o d 37

p d 27

q, r, a d 17

These assignments of universal degrees contrast sharply with the unrestricted quasi

orders. With the restricted quasi orders, the universal degree assigned to the equiv-

alence class containing Seymour with respect to height is no longer greater than the

universal degree assigned to his equivalence class with respect to width. The first is

d 18

while the second is d 77. Furthermore, the universal degree assigned to the equiva-

lence class containing Jon’s boat with respect to length is no longer greater than the

one assigned to Jon’s equivalence class with respect to height. As a consequence of

these assignments, the sentence in (194a) is false whereas the sentence in (194b) is

true. Formulae representing the truth conditions for the sentences in (194) are given

below.

269

(195) a. Truth conditions for Seymour is taller for a man than he is wide for a

man:

H(τC )/∼(s) H(ωC )/∼(s)

= (d 18 d 7

7)

= 0 (where C is the set of men:

a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s)

b. Truth conditions for Jon is taller for a man than his boat is long for a

boat :

H(τC )/∼(j) H(ξC ′)/∼(b′)

= (d 78 d 2

7)

= 1 (where C ′ is the set of boats:

a′, b′, c′, d′, e′, f ′, g′, h′, i′, j′, k′, l′, m′, n′, o′, p′, q′, r′, s′)

As shown, (194a) is false despite the fact that the measurement of Seymour’s height

is greater than his width. Also (194b) is true despite the fact that the measurement

of Jon’s height is not greater than the measurement of his boat’s length.

This kind of result carries over to other constructions without prepositional

phrases. Recall that nominals to which the gradable adjectives serve as modifiers

often pragmatically restrict the comparison class. In general, the comparison class

must be a subset of the denotation of the nominal. With this assumption in mind,

consider the sentence in (196).

(196) Seymour is a taller man than he is a wide man.

270

Like (194a) and unlike (193a), the sentence in (196) is false in a situation where

Seymour is quite short at five feet but quite wide at three feet. This fact can be ex-

plained by assuming that the comparison classes for both the main and subordinate

clause only contain men. In this sentence, the quasi-orders and quotient structures

are the same as the ones employed in the derivation of (194a). The only difference

with such a sentence is that the adjective is used attributively rather than as a pred-

icate. As a result, the complex adjectival phrase containing the comparative, the

gradable adjective and the subordinate clause combines with the nominal through

intersection. However, this does not change the overall truth conditions of the sen-

tence. The derivation of the interpretation for (196) is given below. (Note, I will

assume that the indefinite in the predicate position is interpreted as a set.)

Interpretation of the AP in the subordinate clause:

[[ wide ]]= [[d ABS wide C]] (where C is the set of men:a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s)= ([[ABS]]([[wide]] C))(d)= ((λζ λd λx(Hζ/∼(x) d)(ω C))(d)= (λd λx(H(ωC )/∼(x) d))(d)= λx(H(ωC )/∼(x) d)

Interpretation of the subordinate clause:

[[than he is a wide man]]= [[ than operatord he is a [d ABS wide C] man ]]= ([[than]])[(operatord)(([[dABS wide]] ∩ [[man]])([[he]]))]= ([[than]])[(operatord)(((λx(H(ωC )/∼(x) d)) ∩ (λx MAN(x)))(s))]= ([[than]])[(operatord)((λx(H(ωC )/∼(x) d)& MAN(x))(s))]= ([[than]])[(operatord)((H(ωC )/∼(s) d)& MAN(s))]

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= ([[than]])[(operatord)((d 77 d)& MAN(s))]

= ([[than]])(d : (d 77 d)& MAN(s))

= sup(d : (d 77 d)& MAN(s))

= d 77

Interpretation of the entire sentence:

[[ Seymour is a taller man than he is a wide man]]= ((([[more]]([[than he is a wide man]]))([[tall]] C)) ∩ [[man]])([[Seymour]])(where C is equal to the set of men)= ((([[more]]([[than he is a wide man]]))([[tall]] C)) ∩ (λx MAN(x)))([[Seymour]])= ((([[more]](d 7

7))(τ C)) ∩ (λx MAN(x)))([[Seymour]])

= ((((λd (λζ λx(Hζ/∼(x) d)))(d 77))(τ C)) ∩ (λx MAN(x)))(s)

= ((((λd (λζ λx(Hζ/∼(x) d)))(d 77))(τ C)) ∩ (λx MAN(x)))(s)

= ((λζ (λx(Hζ/∼(x) d 77))(τ C)) ∩ (λx MAN(x)))(s)

= ((λx(H(τC )/∼(x) d 77)) ∩ (λx MAN(x)))(s)

= λx((H(τC )/∼(x) d 77) & MAN(x))(s)

= ((H(τC )/∼(s) d 77) & MAN(s))

= ((d 18 d 7

7) & MAN(s))

= 0(since (d 18 d 7

7) = 0)

The truth conditions for (196) are based on a comparison of two universal degrees.

Since the nominal restricts measurements from the comparison classes, the result is

an indirect comparison.

In summary, the theory presented in chapter 4 is able to explain why a restric-

tion in comparison classes would result in indirect comparisons. Direct comparisons

depend on measurements participating in quasi orders and quotient structures in

much the same way that other individuals participate in the quasi orders and quo-

tient structures. By restricting the comparison classes to men or boats, the resulting

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quasi orders will only contain men or boats. They will no longer contain measure-

ments. As a result, direct comparisons are no longer possible. This aspect of the

current theory is quite important. As mentioned in chapter 2, no other theory of

comparison adequately explains why both prepositional phrases and nominal modifi-

cation force indirect comparisons. Theories such as Bartsch & Vennemann’s (1972),

Seuren’s (1973, 1978), Cresswell’s (1976), von Stechow’s (1984a), and Kennedy’s

(1999) all treat adjectives like tall, wide, and long as if they directly relate individ-

uals to measurements. As their theories currently stand, there is no obvious way

for comparison classes to interfere with this relation. In a theory such as Klein’s

(1980, 1982) that is based on degree modifiers, no explanation is given of why degree

modifiers in attributive instances of the adjectival phrases should be different from

the degree modifiers in predicative instances of adjectival phrases. (Why should the

effect of the degree modifier in Seymour is five feet tall be any different from the

effect of the one in Seymour is a five foot tall man?). However, such a theory would

have to maintain a difference in order to account for the contrast between Seymour

is taller than he is wide and Seymour is a taller man than he is a wide man. With

the interpretation given in chapter 4, the differences between such sentences can be

explain by the effects of comparison classes on the underlying quasi order.

6.6 Concluding remarks on Direct Comparisons

In summary, a direct interpretation of comparative and equative expressions is

possible with the current semantic proposal as long as measurements are treated as

having the same ontological status as individuals and as long as the measurement

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system is conceived (perceived) as a well-ordering of a finite number of measure-

ments. As demonstrated above, by introducing measurements into the domain of

the quasi order, the resulting quotient structure is isomorphic to the ordering of

measurements in the measurement system. This holds whether the underlying quasi

order is associated with width or height. Since the quotient structure is isomorphic

to the measurement system, the universal degrees assigned to equivalence classes of

the quotient structure will also be isomorphic to the measurement system. Hence

a comparison of two universal degrees is equivalent to a comparison of two mea-

surements, even when the measurements are associated with two different kinds of

adjectives.

Further support for this theory comes from two sources. First, sentences that

normally do not permit a direct comparison are suddenly able to licence such a

comparison when a new measurement system is introduced into the context that

interacts with the gradable adjectives in the main and subordinate clause. Second,

overt restriction of comparison classes forces indirect comparisons. By excluding

measurements from the underlying quasi order, an isomorphism between the quotient

structure and the measurement system can no longer be maintained.75

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Notes

58The syntax and semantic implications of measure phrases will only be touchedupon in this section. For a more detail discussion of the various issues the reader isreferred to Schwarzschild (2002) and Nakanishi (2003).

59See Kennedy (1999) for an interesting discussion about the adjectives early andlate and the lack of cross-polar anomaly.

60That is to say, every measure has a successor other than the largest measure ifone exists.

61Note, this is not a fact about all scales. For example, measurements of tempera-ture do not necessarily have a starting point: despite the existence of an absolute zeropoint. In principle such a scale is isomorphic to the integers (positive and negative),so there is no smallest degree that is below all others.

62The notable exception is the age of babies and toddlers.

63The system is discrete and has a minimal element.

64 In fact, there is some precedence for this kind of measurement system. Thecartoon characters called Smurfs are reportedly three apples tall. Also, haut de troispommes is an idiom in French.

65This fact is a contingent property related to how I treat measurement systemsand how I construct quotient structures. In general this property does not hold. Forexample, the quotient structure based on mod n can convert an infinite set of naturalnumbers to a finite quotient structure (a quotient structure with n elements in thedomain).

66To be clear, by ontological status I mean the status entities have within the se-mantic theory rather than the actual world. I make no claims about the existence (ornon-existence) of measurements in the actual world nor do I make any claims aboutthe existence (or non-existence) of individuals. See, Bach (1986) for a discussionabout the difference between the ontology of a semantic model versus the ontologyof the real world.

67Note that although early and late intuitively seem like they are in polar opposi-tion to one another much like tall and short, linguistic evidence does not favour such

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an analysis. Both adjectives permit measure phrases which is not typical of polaropposite adjectives. See Kennedy (1999) for a more detailed discussion about whythese two adjectives are not polar opposites.

68Klein (1982) uses the nouns that quantify over measurements to build up measurephrases such as five feet. These measure phrases are used as delineators that partitiona set into those who are above the measurement and those who are below. Howindividuals are assigned a measurement depends on which adjective provides theoriginal ordering. Thus, direct comparisons can be assigned an interpretation byquantifying over these kinds of delineators. Seymour is taller than he is wide istrue if and only if there is some delineator x such that Seymour is x tall and he isnot x wide. For example, if Seymour were five feet tall and two feet wide then thedelineator five feet would make the direct comparison true.

69Measurements only arise through the calculation of equivalence classes and theassignment of universal degrees that occurs when evaluating an equative or com-parative sentences. Such measurements are derived from the interpretation of thecomparative and equative morphemes and do not exist independently as part of thedomain of discourse.

70Unfortunately, measurement noun-phrases do not act like other noun phrases inall syntactic contexts. For example, the sentences in (197a) and (197c) are typicalexamples of comparative constructions. In contrast the sentences in (197b) and(197d) are slightly deviant.

(197) a. John is taller than Seymour is.

b. ? John is taller than three feet is.

c. The meeting was longer than that movie was.

d. ? The meeting was longer than 4 hours is.

Despite this slight difference, the ability of measure phrases to appear as subjects toadjectival predicates suggests that they should be treated the same as other individ-uals that participate in the adjectival property.

71The setting of an arbitrary upper bound could be accomplished with the com-parison class variable or the contextually limited domain of discourse. In any givencontext the comparison class or the domain of discourse might only contain a subsetof possible measurements.

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72As I discussed in chapter 2, it is possible (although a little difficult) to have anindirect comparison with these sentences. However, I will temporarily ignore thispossibility for now.

73See Winter (2005) for a recent discussion of such contrasts. His discussion is muchbroader than the discussion in the comparative literature since he also discusses theinteraction between measure phases and prepositional phrases. For example, notethe difference in acceptability between six feet behind and *six feet near.

74To be fair, this might also be a prediction of Klein’s (1980, 1982) theory althoughthe details of his theory would have to spelled out in greater detail. Particularly, ifdegree modifiers can be novelly created by a new measurement system, then Kleinwould also predict the emergence of direct comparisons. In general, this predictionholds for any theory that bases direct comparison on the existence of measurementsystems.

75Another potential source of evidence concerns adjectives that are associated withdifferent measurement systems but the same kind of partition. Judgments are diffi-cult in this regard, however Bartsch & Vennemann (1972) claim that it is possible tocompare two objects or the same object through different measurement systems andthat these comparison pattern with prototypical direct comparison. One example inthis regard is I am taller in centimeters than I am in inches. Some consider sucha sentence true since the number of centimeters that measure my height is greaterthan the number of inches. In the current theory such a comparison is possible aslong as the two measurement systems have the same upper bound.

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CHAPTER 7Residual Issues

The semantic system described in chapters 4, 5, and 6 constitutes a research

program more than a complete theory. In this final chapter, I explore other aspects

of comparison that are not directly related to direct and indirect comparison. In

what follows, I first discuss the interpretation of comparative with only one overt

instance of a gradable adjective. I then present a possible interpretation for absolutive

constructions and degree modifiers.

7.1 Comparisons with the Same Adjective

I have not yet discussed how a theory of universal degrees accounts for com-

parative sentences that contain only one adjective, such as the sentences in (198)

below.

(198) a. Seymour is taller than Esme is.

b. Esme is more intelligent than Sally is.

c. Sally is more beautiful than Mary.

In this section, I propose that such sentences have truth conditions that are similar

to more specified comparatives such as Seymour is taller than Esme is wide. Not

only does such a hypothesis provide accurate truth conditions for such sentences,

it also explains why sentences such as those in (199) have an interpretation that is

similar to prototypical indirect comparisons.

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(199) a. Seymour is taller for a boy than Esme is for a girl.

b. Esme is more intelligent for a member of committee A than Sally is for

a member of committee B.

c. Sally is more beautiful for a housewife than Mary is for a model.

In what follows, I explain how the truth conditions assigned to comparative con-

structions presented in chapter 4 accounts for sentences with only one overt instance

of an adjective. I then discuss the effect of comparison classes on such sentences.

To begin, there are a variety of opinions and theories concerning the interpreta-

tion of subordinate clauses for sentences with only one overt gradable adjective such

those in (198). Kennedy (1999) claims that the interpretation of such subordinate

clauses is different from the interpretations of the same kind of clauses in other com-

parative constructions such as Seymour is taller than Esme is wide. Other linguists

such as Hoeksema (1983, 1984) and Hankamer (1973) maintain that the interpreta-

tion is different for the subordinate clauses in (198c) but not necessarily for the ones

in (198a) and (198b). Still others such as Rooth (1992) and Gawron (1995) maintain

that subordinate clauses have the same semantic interpretation across the board: a

full clausal interpretation.

Although opinions differ with respect to how the subordinate clauses should

be treated, opinions seem to be remarkably the same when considering the overall

interpretation of such sentences. All authors recognize a fundamental parallelism

between the sentences in (198), repeated in (200), and the pragmatically awkward

sentences in (201).

(200) a. Seymour is taller than Esme is.

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b. Esme is more intelligent than Sally is.

c. Sally is more beautiful than Mary.

(201) a. Seymour is taller than Esme is tall.

b. Esme is more intelligent than Sally is intelligent.

c. Sally is more beautiful than Mary is beautiful.

No matter how one treats subordinate clauses, the truth conditions for the sentences

in (200) should be identical to the truth conditions for the sentences in (201). In this

section, I exploit this parallelism in order to provide a brief outline of how a theory

with universal degrees can account for sentences with only one instance of a gradable

adjective. I use the predicted truth conditions for sentences such as those in (201) as

the truth conditions that should be assigned to sentences like those in (200). This

allow me to remain agnostic on how one should treat the structural properties of

different kinds of subordinate clauses, while still enabling me to extend the theory

presented in chapter 4 to a variety of different sentence constructions.

Let me begin by considering how a theory with universal degrees would represent

the truth conditions for the sentence in (202).

(202) Seymour is more intelligent than Jon is.

In describing the truth conditions for such a sentence, I assume that the main and

subordinate clause have the same comparison class. This is not a necessary condition,

however it is the pragmatically favoured condition.76 Considering the parallelism

discussed above, the sentence in (202) should have truth conditions similar to (203)

below.

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(203) Seymour is more intelligent than Jon is intelligent.

In this sentence, the same gradable adjective appears in both the main and subordi-

nate clause. In terms of the theory outlined in chapter 4, this implies that the same

quasi order is involved in both clauses. This aspect of the interpretation is demon-

strated in the formula below, where ‘ι’ represents the have as much intelligence as

relation. (Note, ‘ι’ appears on both sides of the comparative relation.) This formula

represents the truth conditions for the sentence in (203).

(204) H(ιC )/∼(s) H(ιC )/∼(j)

In (204), the values of ‘H(ιC )/∼(s)’ and ‘H(ιC )/∼(j)’ are determined by the positions of

the equivalence classes containing Seymour and Jon in the quotient structure based

on the interpretation of intelligent (restricted by the comparison class variable).

Since both clauses share the same gradable adjective and comparison class, it follows

that both clauses also share the same quotient structure. As a result, ‘H(ιC )/∼(s)’

is strictly greater than ‘H(ιC )/∼(j)’ if and only if the equivalence class containing

Seymour dominates the equivalence class containing Jon in the quotient structure

based on ‘(ι C)/∼’. Furthermore, due to how equivalence classes are formed, the

equivalence class containing Seymour dominates the one containing Jon if and only

if Seymour has (at least) as much intelligence as Jon but Jon does not have as

much intelligence as Seymour. This fact accurately describes the truth conditions

that seem to be empirically required. Seymour is taller than Jon if and only if

Seymour has at least as much intelligence as Jon but Jon does not have as much

intelligence as Seymour. Sentences with one adjective simply become a special (and

almost trivial) instance of the application of the comparative morpheme. Unlike

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direct and indirect comparison, the main and subordinate clause have exactly the

same quotient structure. Due to this fact, a comparison of universal degrees ends up

being equivalent to a direct comparison of equivalence classes, which in turn is based

on a direct relation between individuals.

Unlike (202), the sentence in (205) is slightly more complex.

(205) Seymour is more intelligent for a member of committee A than Jon is for a

member of committee B.

In hypothesizing parallel truth conditions for such sentences, (205) should have an

interpretation that is identical to (206).

(206) Seymour is more intelligent for a member of committee A than Jon is intel-

ligent for a member of committee B.

The sentence in (206) is assigned a truth value based on a comparison of two universal

degrees. Like the sentence in (203), the main and subordinate clauses share the

same underlying quasi-order. However unlike (203), the universal degrees will not

be derived from equivalence classes belonging to the same quotient structure. The

two prepositional phrases put different restrictions on the comparison classes in the

main and subordinate clauses. Thus, the quasi order associated with intelligence is

be restricted to members of committee A for the main clause while in contrast it is

restricted to members of committee B for the subordinate clause. As a consequence,

the comparison involves two different scales: much like prototypical instances of

indirect comparison.

Like indirect comparisons, it should be possible for the sentence to be false even

if Seymour is in fact more intelligent than Jon. Indeed this seems to be the case.

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For example, suppose that committee A was composed of the seven most intelligent

people in the world and that committee B was composed of seven people with less

than average intelligence. Furthermore, suppose that Seymour is the least intelligent

of committee A while Jon is the fourth most intelligent on committee B. Under such

circumstances, the sentence in (205) is false, despite the fact that Seymour is more

intelligent than Jon.

The semantics outlined in chapter 4 correctly predicts such intuitions. According

to the interpretation of comparative morpheme, the sentence in (206) has truth

conditions equivalent to the following formula.

(207) H(ιC )/∼(s) H(ιC ′)/∼(j),

Where C is the set of members of committee A and C ′ is the set of members

of committee B.

In this formula, ‘H(ιC )/∼(s)’ represents the position of Seymour’s equivalence class in

a scale of intelligence relativized to committee A. For simplicity, let’s assume that in

the situation described above every individual constitutes his own equivalence class.

Since Seymour is the least intelligent of seven, ‘ ‘H(ιC )/∼(s)’ is equal to d 17. Further-

more, in the formula above ‘H(ιC ′)/∼(j)’ represents the position of Jon’s equivalence

class in a scale of intelligence relativized to committee B. Since Jon is the fourth

most intelligent, ‘H(ιC ′)/∼(j)’ is equal to d 47. Since, (d 1

7 d 4

7) is false, the sentence

in (205) should also be false.

In summary, comparatives with only one adjective end up being special cases

of comparatives with two adjectives. The only difference between the two is that

the main and subordinate clause share the same underlying quasi order. As a result

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of this shared quasi order, sometimes the truth conditions are trivial (when the

comparison classes for both clauses are the same) and sometimes they mirror the

behavior of indirect comparisons (when the comparison classes differ).

7.2 The Absolutive Construction

Having provided an interpretation for the comparative and equative construc-

tions it seems appropriate to also provide an interpretation for the absolutive con-

struction where no overt comparative or equative morphemes are present. Examples

of such sentences appear in (208).77

(208) a. Seymour is intelligent.

b. Esme is beautiful.

c. Jon is tall.

I have nothing special to say about this construction other than the fact that it can be

easily incorporated into the current framework. I essential adopt Kennedy’s (1999)

proposal that absolutive constructions involve a hidden comparative morpheme and

degree variable. Despite the usual problematic consequences of hypothesizing hidden

morphemes,78 such a hypothesis seems to be necessary if one wants to maintain a

degree analysis: since gradable adjectives are interpreted as quasi orders, such quasi

orders must be modified to be able to serve as (one place) predicates.

To begin, I assume that a sentence such as Esme is beautiful has the following

structure.

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TP

qqqqqqqVVVVVVVVVVVVV

DP

qqqqqqqMMMMMMM TP

MMMMMMM

qqqqqqq

Esme is AP

MMMMMMM

qqqqqqq

dstandard ??

qqqqqqqMMMMMMM

ABS

qqqqqqqMMMMMMM

beautiful C

Between the subject and gradable adjective there are two phonetically null elements:

a degree variable that represents the standard of comparison and an absolutive mor-

pheme that aids in the conversion of the quasi order into a one place predicate.

To interpret this structure, one requires an interpretation of the absolutive mor-

pheme and the degree variable. An interpretation of ABS was already introduced

when I discussed the interpretation of subordinate clauses. This interpretation is

repeated below.

The Interpretation of ABS:

Where ζ is a variable that ranges over quasi orders, d is a variable thatranges over elements in the universal scale, and x is a variable that rangesover individuals,

[[ABS]] = λζ λd λx(Hζ/∼(x) d)

Like the equative morpheme, the ABS morpheme takes a quasi order and a universal

degree as arguments and yields a set of individuals.

285

The interpretation of ‘dstandard ’ is a little more complex. Like Bartsch & Venne-

mann (1972), Bierwisch (1987), Kennedy (1999), I propose that this degree variable

has a value that is determined by contextual factors. However, unlike these authors

I do not believed that this value changes for different gradable adjectives. With uni-

versal degrees, the standard value for beautiful can be the same as the standard value

for intelligent or tall. For example, the standard might be contextually set at d 23.

This standard would partition a set of individuals into those whose universal degree

(relative to the interpretation of the adjective) is greater than or equal to d 23

and

those who universal degree is less than d 23. This partition can apply no matter which

gradable adjective appears in the absolutive construction. The absolutive predicate

will be true of the individuals that are contained within equivalence classes in the top

third of the quotient structure created from the gradable adjective. In contexts that

have a higher standard, the value could be set at d 34. In contexts that have a lower

standard, the value could be set at d 12. However, in any context the standard value

would be the same for all gradable adjectives, whether it relates to height, beauty,

talent or intelligence.

Given the outline of the absolutive sketched-out above, one can interpret Esme

is beautiful as follows. (Note, the denotation of Esme is represented by the letter e

while the denotation of beautiful is represented by the Greek letter ‘β’.)

[[Esme is beautiful]]=([[Esme]])(dstandard(ABS([[beautiful]] C)))=([[Esme]])(dstandard(λζ λd λx(Hζ/∼(x) d)(β C)))=(e)(dstandard(λd λx(H(βC )/∼(x) d)))=(e)(λx(H(βC )/∼(x) dstandard))

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=(H(βC )/∼(e) dstandard)

The truth or falsity of this sentence depends on the universal degree that is assigned

to Esme’s equivalence class in the quotient structure based on ‘β’. If the assigned

degree is above the standard then the sentence is true, if it is below the standard it

is false.79

7.3 Degree Modifiers as Universal Degrees

Another issue that has not yet been addressed concerns the interpretation of

degree modifiers such as very, somewhat and quite. Interestingly, such modifiers seem

to have the same effect on adjectival constructions no matter which gradable adjective

is being modified. In this section, I briefly discuss some possible ways to maintain

this consistency. I first address an interpretation that is similar to Wheeler’s (1972),

Bartsch & Vennemann’s (1972) and Klein’s (1980), before considering an alternative

that interprets such modifiers as universal degrees. Since the interpretation of degree

modifiers is a little bit of a side issue independent of the main thesis, the discussion

here will be brief and the theories will not be presented in detail. My goal is merely

to outline possible avenues of interpretation rather than present a full theory.

Wheeler (1972) was the first to observe that one can provide empirically ade-

quate interpretation of the modifier very by allowing the modifier to manipulate the

implicit comparison class in a sentence. He recognized that predicates such as very

tall and very beautiful are quite similar in meaning to unmodified adjectives with

restricted comparison classes. For example, very tall is similar in meaning to tall for

someone who is tall or tall among the tall. Also, very beautiful is similar in meaning

to beautiful for someone who is beautiful or beautiful among the beautiful. Wheeler

287

exploited these paraphrases by simply interpreting very as a function that restricts

the comparison class to those who are in the positive extension of the adjective it

modifies. One of the more interesting aspects of this kind of interpretation is that

it allows for successive application of the modifier. For instance, just as very tall

can be paraphrased by tall among the tall, so too can very very tall be paraphrased

by the predicate tall among the very tall. The ability of this type of interpretation

to account for cyclic application of degree modifiers has led many to adopt simi-

lar interpretations of such modifiers (for example, see Klein, 1980, 1982; Bartsch &

Vennemann, 1972; Bierwisch, 1987).

It is important to note that any theory that has comparison classes can take

advantage of this type of interpretation: including the theory discussed in chapter

4. However, the empirical support for cyclic application of degree modifiers is not

as strong as it first appears. Although very can combine recursively, it can only do

so with other instances of very. It cannot combine with other degree modifiers such

as quite, somewhat, or a little bit, nor can such degree modifiers combine cyclically

with each other. For example, consider the following sentences.

(209) a. ?? Jon is quite quite tall.

b. ?? Jon is somewhat somewhat tall.

c. ?? Jon is a little bit a little bit tall.

d. ?? Jon is somewhat quite tall.

e. ?? Jon is very somewhat tall.

f. ?? Jon is quite very tall.

g. ?? Jon is very a little bit tall.

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None of the sentences in (209) have a coherent interpretation.

Given the lack of cyclic application for most modifiers, I would like to suggest an

alternative to the view that restricts comparison classes. With universal degrees, it

is possible to maintain an interpretation of very, somewhat and quite that relativizes

to the gradable adjective while also assigning such modifiers a rigid value. One can

simply interpret modifiers as universal degrees and let the absolutive morpheme take

this degree as an argument in the same way that it takes ‘dstandard ’ as an argument.

Thus, just as someone is tall if the universal degree assigned to their equivalence

class is above the standard, one could be very tall if the universal degree assigned

to their equivalence class is above the interpretation of very. The exact value of the

degree modifiers would have to be determined empirically, however estimated values

that are reasonably close could be as follows: the interpretation of very could be

the universal degree d 89, the interpretation of quite could be slightly lower at d 5

6, the

interpretation of a little bit could be d 35

and the interpretation of somewhat could

be d 12.

There is some empirical motivation for such a theory of modification. A theory

which simply restricts comparison classes would have to provide a complex interpre-

tation for the following sentences.

(210) a. Jon is more than very tall.

b. Esme is more than somewhat beautiful.

c. Sidney Crosby is more than a little bit talented.

In contrast, if degree modifiers were interpreted as universal degrees then these con-

structions would have a very simple interpretation. The universal degree assigned

289

to the modifier could simply serve as the degree argument to the comparative mor-

pheme.

In summary, standard interpretations of degree modification can be incorporated

into the current theory of comparison. However, the existence of universal degrees

allows for an alternative theory which might be advantageous when interpreting the

affect of degree modifiers within comparative constructions.

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Notes

76Contextual factors can prime for a reading where a comparison is done with con-trasting comparison classes. However, clearly the default reading of such sentencesassumes that both clauses share the same comparison class.

77Note I will only address the most prevalent interpretation of gradable adjectivesin absolutive constructions: interpretations that seem to be influenced by contextu-ally determined comparison classes and standard values. I will not address adjectivessuch as empty, full, wet and open. Such adjectives require a separate treatment, atleast within the present framework. Adjectives such as these that do not demon-strate gradability in their absolutive sense. Furthermore, their interpretation doesnot exhibit the same degree of dependence on contextual factors, nor can their inter-pretation be influenced by overt comparison classes. See Kennedy (2005) for a moredetailed discussion.

78Hypothesizing hidden morphemes in one language is not in itself problematic.However, it is a strong cross-linguistic generalization that absolutive constructionsare never accompanied by overt modifiers or morphemes of any sort. Kennedy (2005)claims that there are some exceptions.

79Note that although I outline a degree analysis of absolutive constructions inthis section, this analysis is by no means necessary. One can easily treat absolutiveadjectives as unmodified quasi orders if so desire. Such an analysis would not involvea hidden absolutive morpheme. Instead, the quasi order could combine with a hiddenvariable that ranges over individuals in quasi order’s domain. The value assigned tothis variable could represent the standard value for the quasi order. One could labelthis variable ‘xstandard ’ instead of ‘dstandard ’. Such an interpretation would provide thefollowing structure for Esme is beautiful.

291

TP

qqqqqqqVVVVVVVVVVVVV

DP

qqqqqqqMMMMMMM TP

MMMMMMM

qqqqqqq

Esme is AP

MMMMMMM

qqqqqqq

xstandard

qqqqqqqMMMMMMM

beautiful C

If the variable combines with the two place relation in the standard way (argumentabsorbtion), the result would be a one place predicate that is true of all the individ-uals either equivalent to or ordered above ‘xstandard ’ in the quasi order. The value ofthe standard could be contextually determined in much the same way as the degreevariable.

Although there are some advantages of this possible interpretation in terms of thereduction of hidden morphemes, one loses the parallelism between adjective phrasesin subordinate clauses and those in absolutive clauses. For this reason I prefer thedegree analysis.

292

CHAPTER 8Conclusion

The theory presented in this thesis is empirically supported in many different

respects. In this chapter I review some of this empirical support. First, I discuss how

the theory provided a unified interpretation of the comparative and equative mor-

phemes while also maintaining a difference between direct and indirect comparisons.

Second, I discuss some of the other empirical issues that were resolved as a by-product

of this account of direct and indirect comparison. For example, the interpretation

of the comparative and equative morphemes provide a simple account of Wheeler’s

Generalization. Also, the underlying structural characteristics of subordinate clauses

explains the parallelism between such clauses and Wh-questions. Finally, the inter-

pretation assigned to comparison classes explains why such classes affect standard

values for comparison, induce presuppositions and force indirect comparisons.

8.1 Direct versus Indirect Comparisons

The main empirical hurdle set out in chapter 2 was to provide a unified inter-

pretation for the comparative and equative morpheme but yet still account for the

differences between a direct and indirect comparison. In chapter 4, I provided such

an interpretation. I proposed that the comparative and equative morphemes yield

truth conditions that are dependent on a comparison of two universal degrees: one

associated with the main clause and the other with the subordinate clause. The

values of the two universal degrees depend on the interaction between the clausal

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subjects and the primary scales associated with each clause. Let me outline this

dependency in more detail.

The main and subordinate clause both contain a gradable adjective that is inter-

preted as a quasi order. This quasi order is used to build a quotient structure where

the individuals in the domain of the quasi orders are collapsed into several different

equivalence classes and then these equivalence classes are ordered linearly in a way

that is congruent to the original quasi order. The quotient structure constitutes the

primary scale.

Given these primary scales, one can associate each clause with a universal degree

in the follow way. The main clause can be associated with the universal degree

that encodes the position of the equivalence class containing the main-clause subject

in the primary scale created from the gradable adjective in the main clause. The

subordinate clause can be associated with the universal degree that encodes the

position of the equivalence class containing the subordinate-clause subject in the

primary scale created from the gradable adjective in the subordinate clause. Thus,

by comparing universal degrees, one compares two positions in two primary scales.

With this degree assignment, one can account for the difference between the

interpretation of comparative and equative sentences by changing how the univer-

sal degrees are compared for each type of sentence. The comparative morpheme

compares two universal degrees by a strictly greater-than relation: a comparative

sentence is true if and only if the position represented by the universal degree asso-

ciated with the main clause is strictly greater than the position represented by the

universal degree associated with the subordinate clause. The equative morpheme

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compares two universal degrees by the greater-than-or-equal relation: an equative

sentence is true if and only if the position represented by the universal degree as-

sociated with the main clause is greater-than-or-equal to the position represented

by the universal degree associated with the subordinate clause. These truth condi-

tions for comparative and equative sentences are the same in any given context and

with any given gradable adjectives. There is no difference in the interpretation of

the morphemes when one is evaluating a direct comparison as opposed to an indi-

rect comparison. Rather the only difference between direct and indirect comparisons

arises in the composition of the primary scales.

With direct comparisons, the quasi orders associated with the main and sub-

ordinate clauses have the same set of measurements in their domain. These mea-

surements are ordered in identical ways (although the other individuals in the two

domains need not be). Given some conventions about how people normally treat

measurement systems, the measurements in the quasi orders will affect the compo-

sition of the quotient structures in two ways. First, due to how equivalence classes

are formed, each equivalence class will have one and only one measurement. Second,

due to the congruence of the quotient structures to the quasi orders, the ordering of

equivalence classes will be systematically related to the ordering of the measurements

contained in the equivalence classes. For positive gradable adjectives, an equivalence

class A will be ranked above an equivalence class B if and only if the measure-

ment in A is greater than the measurement in B. For negative gradable adjectives,

an equivalence class A will be ranked above an equivalence class B if and only if

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the measurement in A is less than the measurement in B. As a result of the sys-

tematic connection between an ordering of equivalence classes and an ordering of

measurements, the position of the equivalence classes in the quotient structures will

be identical to the position of the measurements in the measurement system. This

holds for any quotient structure based on a quasi order that contains measurements.

For example, the position of the equivalence class containing 5′ in the quotient struc-

ture associated with tall will be equivalent to the position of 5′ in the measurement

system of inches and feet. Similarly, the position of the equivalence class containing

5′ in the quotient structure associated with wide will be equivalent to the position

of 5′ in the measurement system of inches and feet. Thus, given certain conventions

on how people treat measurement systems, comparing the positions of two equiva-

lence classes in their respective quotient structures is equivalent to comparing the

two measurements in the equivalence classes. In other words, a comparison of two

universal degrees is equivalent to a comparison of measurements.

This connection between measurement systems and primary scales explains the

truth conditions that direct comparisons invoke. A sentence such as Seymour is taller

than he is wide is true as long as the measurement of Seymour’s height is greater

than the measurement of his width. It does not matter if Seymour might not be

very tall compared to other men or that he might be quite wide compared to other

men. Under a direct interpretation, his ranking in comparison to other individuals is

inconsequential to the truth conditions of the sentence. Measurements are the only

thing that matter.

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This all changes with indirect comparisons such as Seymour is taller for a man

than he is wide for a man. With such comparisons there is no shared measurement

system for the quasi orders in the main and subordinate clauses. (Note, due to the

restriction of the comparison classes, the quasi orders will only have men in their

domain: no measurements.) As a result, the equivalence between a comparison of

measurements and a comparison of the positions of equivalence classes disappears.

Rather, without measurements, the primary scales encode how certain individuals

are ranked above others in terms of the gradable property. The position of the

equivalence class containing Seymour in the quotient structure associated with tall

for a man reflects how high Seymour is in a ranking of men in terms of height. The

position of the equivalence class in the quotient structure associated with wide for a

man reflects how high Seymour is in a ranking of men in terms of width. Thus, if

Seymour is not tall for a man but he is wide for a man, then he cannot be taller for

a man than he is wide for a man. A comparison of universal degrees compares how

the clausal subjects are ranked with respect to other individuals in the comparison

class.

In summary, there is only one interpretation for the equative and comparative

morphemes. This interpretation involves a comparison of universal degrees. How-

ever, a comparison of universal degrees can be equivalent to a comparison of mea-

surements when the same measurements participate in both underlying quasi orders

(and when people treat these measurements in certain ways). When these conditions

are met, the result is a direct comparison. When they are not met, the result is an

indirect comparison. Note, an important aspect of this explanation is that a single

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sentence can have both a direct and indirect interpretation. The choice of inter-

pretation depends on whether the interpretation of the contextually set comparison

class variable contains measurements or not. Thus, ambiguous interpretations are

predicted as long as the context permits two different possible assignments for the

comparison class variable.

8.2 Other Empirical Generalizations

Not only does the theory developed in chapter 4 explain the difference between

direct and indirect comparison, it also accounts for the other empirical generalization

discussed in chapter 2. For example, it accounts for Wheelers Generalization, for the

parallelism between Wh-questions and subordinate clauses, and for the three effects

of comparison classes on sentences with gradable adjectives. In what follows, I briefly

review the account for each of these generalizations.

To begin, let me address Wheeler’s Generalization. Recall that according to

Wheeler’s Generalization comparative constructions maintain transitive and asym-

metric entailments even when the interpretation of the gradable adjective is unknown

or is not normally consistent with a gradable property. Like Bartsch & Vennemann

(1972), Cresswell (1976), von Stechow (1984b), and Kennedy (1999), the interpre-

tation assigned to comparative constructions in chapter 4 compares two degrees by

using a strict linear order. This order is transitive and asymmetric by definition.

Hence, transitive and asymmetric entailments are predicted to exist due to the na-

ture of the relation used to compare two universal degrees.

Unlike the account of Wheeler’s Generalization, an account of the parallelism be-

tween Wh-questions and subordinate clauses does not follow from the interpretation

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of the comparative and equative morphemes. Rather it follows from the interpre-

tation and syntactic structure assigned to subordinate clauses. Recall that just as

How long did you make the claim he was? is ungrammatical so too is Seymour is

taller than you made the claim he was. Gradable adjectives in subordinate clauses

cannot be contained within islands for movement. The syntactic structure and se-

mantic interpretation of subordinate clauses discussed in chapter 4 can explain this

fact. (Note, that the solution advanced in chapter 4 is basically identical to the

solution presented in Kennedy, 1999.) Like Wh-questions, subordinate clause have

an operator that moves from a position near the gradable adjective to a position just

below the complementizer. This movement (or the resulting chain of the movement)

is subject to island constraints in much the same way that Wh-movement is sub-

ject to such constraints. Hence, subordinate clauses that have gradable adjectives

embedded in an island for movement are as ungrammatical as parallel structures in

Wh-questions.

As a final empirical point in favour of the current theory, the interpretation of

comparison classes advanced in chapter 4 explains the three different effects such

classes have on sentences with gradable adjectives. It explains why they influence

the value of the standard of comparison in absolutive constructions (contrast big for

a mouse and big for an elephant). It explains why they induce presuppositions for

both absolutive and comparative constructions (Fido is big for a dog presupposes

that Fido is a dog, as does Fido is bigger for a dog than Seymour is for a boy). It

accounts for why they sometimes force indirect comparisons (eg., the sentence Esme

is taller for a woman than Seymour is for a man has a different interpretation than

299

Esme is taller than Seymour is). Recall in chapter 4, I proposed that the comparison

class variable should be interpreted as a set that restricts the interpretation of the

gradable adjective (i.e., it restricts the quasi order assigned to the gradable adjective).

The result of this restriction is a quasi order that only contains members of the

comparison class. The primary scale (quotient structure) based on this restricted

quasi order only contains equivalence classes composed of members of the comparison

class. In summary, by restricting the interpretation of the adjective, the comparison

class determines the composition of the primary scale.

This effect on the primary scale is what explains the three different influences of

these classes on absolutive and comparative constructions. For example, in absolutive

constructions such as Jon is tall or Esme is intelligent, the subject is considered to

be a member of the predicate if he or she is ranked above the standard value. The

standard value (under almost all theories including my own) is a degree associated

to a value just above the middle of the primary scale. This holds across adjectives.

Since comparison classes influence the composition of primary scales then they also

influence what counts as the middle for such scales. Hence they effect the assignment

of the standard value. For example, the middle of the scale is different for big for

an elephant and big for a mouse because the primary scales are different: one scale

only contains equivalence classes composed of elephants, the other equivalence classes

composed of mice.

As with the standard of comparison, the influence of comparison classes on

primary scales can also explain why such classes induce certain presuppositions.

According to the semantics outlined in chapter 4, in calculating the truth or falsity

300

of absolutive and comparative sentences, the clausal subjects must be mapped to

equivalence classes in the primary scale and then the equivalence classes to universal

degrees. Thus to even evaluate such sentences, the clausal subject is required be a

member of one of the equivalence classes in the primary scale. But for an individual

to be a member of an equivalence class requires membership in the comparison class.

Hence to even evaluate whether such sentences are true or false requires that the

clausal subjects are members of the comparison classes. This explains why sentences

such as Fido is big for a dog and Fido is bigger for a dog than Seymour is for a boy

presuppose that Fido is a dog. If Fido was not a dog then these sentences could not

be assigned a truth value.

Just as the composition of primary scales induce presuppositions, so too can

they induce indirect comparisons. Recall that when the positions of two individu-

als are compared relative to two separate scales that do not share a measurement

system the result is an indirect comparison. If the comparison classes in the main

and subordinate clauses are different and if the comparison classes do not contain

measurements, then the primary scales for the main and subordinate clauses must

also be different (and must not have equivalence classes containing measurements).

Since comparison classes determine the composition of the primary scales, two differ-

ent comparison classes implies two different primary scales. Such facts explain why

sentences such as Esme is taller for a woman than Seymour is for a man and Bob is

more intelligent for a linguist than Jack is for a physicist have interpretations and

implications that parallel prototypical examples of indirect comparison.

301

In summary, the three influences of comparison classes reduce to one underlying

influence: namely comparison classes restrict the interpretation of the quasi order

and hence determine the composition of the primary scale.

302

Notes

76Contextual factors can prime for a reading where a comparison is done with con-trasting comparison classes. However, clearly the default reading of such sentencesassumes that both clauses share the same comparison class.

77Note I will only address the most prevalent interpretation of gradable adjectivesin absolutive constructions: interpretations that seem to be influenced by contextu-ally determined comparison classes and standard values. I will not address adjectivessuch as empty, full, wet and open. Such adjectives require a separate treatment, atleast within the present framework. Adjectives such as these that do not demon-strate gradability in their absolutive sense. Furthermore, their interpretation doesnot exhibit the same degree of dependence on contextual factors, nor can their inter-pretation be influenced by overt comparison classes. See Kennedy (2005) for a moredetailed discussion.

78Hypothesizing hidden morphemes in one language is not in itself problematic.However, it is a strong cross-linguistic generalization that absolutive constructionsare never accompanied by overt modifiers or morphemes of any sort. Kennedy (2005)claims that there are some exceptions.

79Note that although I outline a degree analysis of absolutive constructions inthis section, this analysis is by no means necessary. One can easily treat absolutiveadjectives as unmodified quasi orders if so desire. Such an analysis would not involvea hidden absolutive morpheme. Instead, the quasi order could combine with a hiddenvariable that ranges over individuals in quasi order’s domain. The value assigned tothis variable could represent the standard value for the quasi order. One could labelthis variable ‘xstandard ’ instead of ‘dstandard ’. Such an interpretation would provide thefollowing structure for Esme is beautiful.

303

TP

qqqqqqqVVVVVVVVVVVVV

DP

qqqqqqqMMMMMMM TP

MMMMMMM

qqqqqqq

Esme is AP

MMMMMMM

qqqqqqq

xstandard

qqqqqqqMMMMMMM

beautiful C

If the variable combines with the two place relation in the standard way (argumentabsorbtion), the result would be a one place predicate that is true of all the individ-uals either equivalent to or ordered above ‘xstandard ’ in the quasi order. The value ofthe standard could be contextually determined in much the same way as the degreevariable.

Although there are some advantages of this possible interpretation in terms of thereduction of hidden morphemes, one loses the parallelism between adjective phrasesin subordinate clauses and those in absolutive clauses. For this reason I prefer thedegree analysis.

304

Appendix:Kennedy and Cresswell, A Proof of Equivalency

In what follows, I first discuss how each extent in Kennedy’s (1999) theory can

be mapped injectively to a degree in Cresswell’s (1976) theory. Once this mapping is

specified, Kennedy’s superset relation ends up being equivalent to Cresswell’s MORE

relation.

Kennedy’s proposal suggests that adjectives are functions from individuals to

extents. The extents consist of degrees from a scale. Hence, an underlying assump-

tion of his semantics is that there is an array of measurement scales from which

extents can be formed. I will denote these scales as D1 , D2 , D3 etc., where each Dn

consists of a domain and the canonical linear ordering of that domain. The canonical

linear ordering will be symbolized by ≥ Dn . To avoid potential complications with

Kennedy’s definition of the superset relation, I will assume that the degrees in a

scale are unique to that scale. In other words, degrees of beauty are different from

degrees of anger, degrees of time (minutes and seconds) or degrees of length (inches

and feet).80 Following Kennedy’s notation, I will use subscripts to denote the scale

with which a degree is associated. Thus dD1 is a degree from the scale D1 .

Given this notation, a mapping can be specified from each extent in Kennedy’s

(1999) system to a degree in Cresswell’s (1976) system. I will first specify this map-

ping for extents that contain all the degrees between zero and some measurement

305

before discussing extents from some specified measurement to infinity. Using the no-

tation from the previous paragraph, we can represent such extents with the following

kind of set, d ∈ Dn : dx ≥ Dn d, where dx is any degree in Dn . With this set

representation, one can specify a function h that maps each set of this form onto a

degree in Cresswell’s system. Recall that for Cresswell (1976) a degree is an ordered

pair consisting of a measurement and a linear ordering. I will define the function h,

so that it maps sets of degrees in the following way.

(211) ∀Dn ,∀dx ∈ Dn h(d ∈ Dn : dx ≥ Dn d) = 〈dx , > Dn 〉, where ∀x, y(x >

Dn y) iff (x ≥ Dn y)&¬(y ≥ Dn x).

Extents that contain all the degrees from some specified measurement to the

upper limit of the scale will be mapped differently. These extents can be represented

with the following set notation, d ∈ Dn : d ≥ Dn dx, where dx is any degree in

Dn . I will define h so that it maps sets of this form in the following way.

(212) ∀Dn ,∀dx ∈ Dn h(d ∈ Dn : d ≥ Dn dx) = 〈dx , < Dn 〉, where ∀x, y(x <

Dn y) iff (y ≥ Dn x)&¬(x ≥ Dn y).

With h so specified, I can state the following equivalency.

(213) ∀ez , ey (ez ⊃ ey) iff (MORE(h(ez ), h(ey)))

The proof of this equivalency is as follows. (Recall from the section on Cresswell,

1976, that MORE(〈d1 , R1 〉, 〈d2 , R2 〉) is defined when R1 = R2 , otherwise it is

undefined. When define, it is true if and only if d1 R1 d2 , otherwise it is false.)

(214) PROOF

a. LEFT TO RIGHT:

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i. Suppose (ez ⊃ ey) is undefined. Thus, either ez and ey containdegrees from different scale (call the scales Dz and Dy respectively)or one of the extents is of the form d ∈ Dn : dx ≥ Dn d and theother of the form d ∈ Dn : d ≥ Dn dx. Suppose the former isthe case. Then h(ez ) would equal 〈dz , R1Dz 〉 and h(ey) would equal〈dy , R2Dy〉, where R1 and R2 could be either > or <. However,R1Dz cannot equal R2Dy since by assumption Dz does not equal Dy .Hence (MORE(h(ez ), h(ey))) is undefined. Considering the othercase, suppose that one of the extents is of the form d ∈ Dn : dx ≥Dn d and the other of the form d ∈ Dn : d ≥ Dn dx. Even if thedegrees were from the same scale, they would not represent the sameordering, since h would map one extent to 〈x, > Dn 〉 and the otherto 〈w, < Dm 〉. The relation < Dm cannot equal > Dn even if m = n.Hence (MORE(h(ez ), h(ey))) is undefined.

ii. Suppose (ez ⊃ ey) is true. Then there are two cases. Either ez

and ey are both of the form d ∈ Dn : dx ≥ Dn d or they areboth of the form d ∈ Dn : d ≥ Dn dx. If the former, thenlet dz and dy be the upper limits of ez and ey respectively. Since(ez ⊃ ey) is true, dy must be a member of ez and dz must notbe a member of ey . Thus, dz ≥ Dn dy . Also it does not hold thatdy ≥ Dn dz . By the definition of h, this means that dz > Dn dy . Thisentails that MORE(〈dz , > Dn 〉, 〈dy , > Dn 〉) is true which means that(MORE(h(ez ), h(ey))) is true. Suppose that ez and ey are both ofthe form d ∈ Dn : d ≥ Dn dx. Let dz and dy be the lower limits ofez and ey respectively. Since (ez ⊃ ey) is true, dy must be a memberof ez and dz must not be a member of ey . Thus, dy ≥ Dn dz but¬(dz ≥ Dn dy). By the definition of h, this means that dz < Dn dy .This entails that MORE(〈dz , < Dn 〉, 〈dy , < Dn 〉) is true which meansthat (MORE(h(ez ), h(ey))) is true.

iii. Suppose(ez ⊃ ey) is false. Then there are two cases. Either ez

and ey are both of the form d ∈ Dn : dx ≥ Dn d or they areboth of the form d ∈ Dn : d ≥ Dn dx. If the former, thenlet dz and dy be the upper limits of ez and ey respectively. Since(ez ⊃ ey) is false, dz must be a member of ey . Thus, dy ≥ Dn dz .By the definition of h, this means that it cannot be the case thatdz > Dn dy . This entails that MORE(〈dz , > Dn 〉, 〈dy , > Dn 〉) is false,which means that (MORE(h(ez ), h(ey))) is false. Suppose that ez

and ey are both of the form d ∈ Dn : d ≥ Dn dx. Let dz and

307

dy be the lower limits of ez and ey respectively. Since (ez ⊃ ey)is false, dz must be a member of ey . Thus, dz ≥ Dn dy . By thedefinition of h, this means that it is false that dz < Dn dy . Thisentails that MORE(〈dz , < Dn 〉, 〈dy , < Dn 〉) is false, which means that(MORE(h(ez ), h(ey))) is also false.

b. RIGHT TO LEFT:

i. Suppose (MORE(h(ez ), h(ey))) is undefined. Then, either h(ez ) andh(ey) contain ordering relations from different scale, or they haveopposite orderings on the same scale. If the former, then ez and ey

contain degrees from different scales. Hence (ez ⊃ ey) is undefined.If the latter, then one of the extents (ez or ey) is of the form d ∈Dn : dx ≥ Dn d and the other of the form d ∈ Dn : d ≥ Dn dx.Hence (ez ⊃ ey) is undefined.

ii. Suppose (MORE(h(ez ), h(ey))) is true. Then there is a Dw and adz , dy ∈ Dw such that h(ez ) = 〈dz , RDw 〉 and h(ey) = 〈dy , RDw 〉and dz RDw dy , where R is either > or <. Suppose R is >. Thenez would be the set d ∈ Dw : dz ≥ Dw d and ey would be theset d ∈ Dw : dy ≥ Dw d. But since dz > dy by assumption,this means that all d in ey are also in ez but not vice versa. Hence(ez ⊃ ey) is true. Alternatively suppose that R is <. Then ez

would be the set d ∈ Dw : d ≥ Dw dz and ey would be theset d ∈ Dw : d ≥ Dw dy. But since dz < dy by assumption,this means that all d in ey are also in ez but not vice versa. Hence(ez ⊃ ey) is true.

iii. Suppose (MORE(h(ez ), h(ey))) is false. Then there is a Dw and adz , dy ∈ Dw such that h(ez ) = 〈dz , RDw 〉 and h(ey) = 〈dy , RDw 〉 andit is not the case that dz RDw dy , where R is either > or <. SupposeR is >. Then ez would be the set d ∈ Dw : dz ≥ Dw d and ey wouldbe the set d ∈ Dw : dy ≥ Dw d. But since it is not the case thatdz > dy , then it must be the case that dy ≥ dz . This means thatall the d’s in ez are also in ey . Hence (ez ⊃ ey) is false. Alternativelysuppose that R is <. Then ez would be the set d ∈ Dw : d ≥ Dw dzand ey would be the set d ∈ Dw : d ≥ Dw dy. But since it is notthe case that dz < dy , then then it must be the case that dy ≥ dz .This means that all the d’s in ez are also in ey . Hence (ez ⊃ ey) isfalse.

308

The function h ends up being a monomorphism81 with respect to the interpretation

of the comparative morpheme.

309

Notes

80Note that without this assumption, there remains a possibility that two adjec-tives associated with two different scales could be commensurable under Kennedy’sdefinition of the superset relation. This is an empirically undesirable result.

81A monomorphism is a homomorphism that is injective. Note that if Cresswell’s(1976) system contained all the same scales as Kennedy’s (1999), then the two sys-tems would be isomorphic. There is nothing the prohibits Kennedy and Cresswellfrom having the same scales but there is nothing that forces the requirement either.

310

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