TWICE VERSUS TWO TIMES IN PHRASES OF COMPARISON
By
Adam Michael Gobeski

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF ARTS
Linguistics
2011

ABSTRACT

TWICE VERSUS TWO TIMES IN PHRASES OF COMPARISON
By
Adam Michael Gobeski

The interaction of factor phrases—things like half, twice, and three times—with degree
constructions such as comparatives (i.e., comparisons of the form –er or more…than) and
equatives (comparisons of the form as…as) is an area that has seldom been touched upon. While
much has been written about comparatives in general, and some has been written about adverbial
versions of these factor phrases (in sentences such as John went to the store twice today), degreerelated factor phrases have been largely neglected. In particular, the distinction between factor
phrases like twice, which combine only with equatives, and phrases like two times, which
combine with both equatives and comparatives, has been overlooked. I will thus be examining
these factor phrases as they relate to degree constructions. I will argue that for many factor
phrases, like twice and half, Bierwisch (1989)’s prediction that they interact only with equatives
holds true, while in cases of the form n times this prediction breaks down. I will also argue that
factor phrases like twice involve a maximality operator, while factor phrases like two times
simply involve an existential operator.

ACKNOWLEDGEMENTS

This thesis would not have been completed without the help of a great many people. I
would like to thank them for their support, acknowledging that I could not have gotten where I
am without them.
I’d like to start by thanking my advisor, Dr. Marcin Morzycki, who got me interested in
semantics in the first place and who consistently believed in my abilities even when I sometimes
wavered. I’d also like thank the other members of my thesis committee, Dr. Alan Munn and Dr.
Alan Beretta; their comments and insights were always incredibly valuable.
In addition, I’d like to thank my fellow graduate students in Linguistics for giving me
support and lending an ear to my ideas, no matter how off the wall. In particular, I’d like to
thank anyone who listened to this thesis take shape over many meetings of the MSU Semantics
Group, and to anyone who listened to the early version of this at the ISOM 2008 workshop at the
Ohio State University. Your feedback was always appreciated.
Of course, thanks go to my parents, who got me to this point through their constant
unwavering support. In particular, credit goes to my father for once saying to me, “You’re
interested in all these languages. Why don’t you take a linguistics course?”
And finally, thanks to Brianne. Words can’t express.

iii

TABLE OF CONTENTS

1. INTRODUCTION

1

2. THE PROBLEM

2

3. PREVIOUS RESEARCH
Gradable Adjectives
Degrees
Syntactic Structures
Factor Phrases
Formal Semantics of Degree Phrases

4
4
5
11
15
20

4. TWICE VERSUS TWO TIMES IN PHRASES OF COMPARISON
Basic Assumptions
The Compositions of Twice and Two Times

21
21
25

5. CONCLUSION

33

6. REFERENCES

34

iv

Twice Versus Two Times in Phrases of Comparison

1 Introduction
The interaction of factor phrases—things like half, twice, and three times—with degree
constructions such as comparatives (i.e., comparisons of the form –er or more…than) and
equatives (comparisons of the form as…as) is an area that has seldom been touched upon. While
much has been written about comparatives in general, and some has been written about adverbial
versions of these factor phrases (in sentences such as John went to the store twice today), degreerelated factor phrases—those which interact with adjectives like tall and nimble—have been
largely neglected. In particular, the distinction between factor phrases like twice, which combine
only with equatives, and phrases like two times, which combine with both equatives and
comparatives, has been overlooked.
This thesis will therefore take a detailed look at these factor phrases as they relate to
phrases of comparison. I will argue that for many factor phrases, like twice and half, Bierwisch
(1989)’s prediction that they interact only with equatives holds true, while in cases of the form n
times—which, though a single basic form, is far more common than other factor phrases—this
prediction breaks down. I will also argue that FPs like twice involve a max operator, while FPs
like two times simply involve an existential operator.
Section 2 will outline the basic problem. Section 3 will discuss earlier writings about
factor phrases, adjectival comparison, and degrees in general. Section 4 will provide a syntactic
structure and denotations for the pertinent factor phrases, as well as showing how they work
compositionally in sentences. This section will also discuss a couple puzzles that are still

1

unresolved. Section 5 will conclude the thesis, while providing a possible direction for further
research.

2 The Problem
Look at the following sentences.

(1)

John is as tall as Mary.

(2)

John is taller than Mary.

(3)

John is the tallest.

(1) through (3) all involve types of adjectival comparison, specifically that of height. The
construction of (1) is an equative. (2) is a comparative construction, and (3) is the superlative.
(3) is given here simply to complete the paradigm; I will not deal with this form any further.
The meanings of (1) and (2) are very similar. (2) is a true statement if and only if the
height of John exceeds the height of Mary. (1) is true if John’s height is greater than or equal to
Mary’s height. While that is not immediately intuitive, a sentence like (4),

(4)

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

shows this to be true. Thus, the only difference semantically between (1) and (2) is whether the
relation is either greater than or greater than or equal to.
However, despite this similarity, there are different patterns. For instance, while
comparatives can take bare measure phrases like three feet, equatives cannot.

2

(5)

John is three feet taller than Mary.

(6)

*John is three feet as tall as Mary.

(7)

*John is as three feet tall as Mary.

Some phrases, however, can combine with both comparatives and equatives. These
include factor phrases like two times.

(8)

John is two times taller than Mary.

(9)

John is two times as tall as Mary.

Yet not all factor phrases behave this way.

(10)

John is twice as tall as Mary.

(11)

*John is twice taller than Mary.

(12)

The red shirt is twice as expensive as the blue shirt.

(13)

*The red shirt is twice more expensive as the blue shirt.

On the surface, twice and two times appear to be identical in meaning—corresponding roughly to
the meaning y=2x, where y is the measure of the first thing (so in the case of (8), John’s height)
and x, the measure of the second (Mary’s height). So what would account for this difference?
Why is (8) acceptable and (11) unacceptable?

3

Before I answer that question, let’s take a step back and look at what’s already been
written about comparative adjective phrases, degrees, and factor phrases.

3 Previous Research

3.1 Gradable Adjectives
Looking back at (1) and (2),

(1)

John is as tall as Mary.

(2)

John is taller than Mary.

we note that they involve the gradable adjective tall and the equative and comparative degree
constructions, respectively. As Kamp (1975), Klein (1980), Kennedy (1997, 2001) and others
point out, gradable adjectives refer to properties. These properties are defined as scales which
are themselves defined by the ordering of the objects on that scale. So in (1) and (2), the height
of John and the height of Mary are both on the scale of tall-ness. By ordering objects such as
these on a given scale, we can then make comparisons among those objects. Thus it can be said
that, since John’s height is greater than Mary’s height on the scale of tall-ness, (2) is an
interpretable and true sentence. As for a sentence like (14),

(14)

John is tall.

4

an independent standard must be introduced into the scale of tall-ness, and if it is determined that
John’s height exceeds this standard, then (14) is true. What that standard is varies from situation
to situation, and while (14) may be true if John is four feet tall and he is being compared to the
set of people who are four feet tall or less, it will not be true if John is being compared to the set
of professional basketball players.
Another thing to note about gradable adjectives is that they can be divided into
subcategories determined by whether or not the scale is open or closed at either end. Tall-ness,
for instance, is closed on the bottom end, where heights are zero, and open on the top end—after
all, there is theoretically no limit as to how tall something can be. An adjective like dry, on the
other hand, is closed on both ends; once something is dry, it cannot become any drier. This is
something that can be observed more clearly with adverbs like completely or partly.

(15)

The towel is completely/partly dry.

(16)

*John is completely/partly tall.

In (15), completely and partly combine grammatically with the telic adjective dry. But in (16),
with atelic tall, the sentence is ungrammatical.

3.2 Degrees
Now the classical view of these types of adjectives (for the version Kennedy 1997 calls
the “Vague Predicate Analysis”, in which adjectives denote functions from objects to truth
values, see McConnell-Ginet 1973, Kamp 1975, Klein 1980, van Benthem 1983, and others; for
the version he calls the “Scalar Analysis”, in which adjectives denote functions from objects to

5

degrees, see Bartsch & Vennemann 1973, Seuren 1973, Cresswell 1976, Hellan 1981, von
Stechow 1984, Heim 1985, and others) is that their scales have domains that are partially ordered
in some way, usually through a series of points; thus in (1), the degree of John’s height is a point
on the scale that is above the degree of Mary’s height. If John is six feet tall, then the degree of
his height on the scale is a point at six feet. The advantage of thinking about degrees as points is
that they are easier to conceptualize and thus easier to manipulate; so in a sentence like (17),

(17)

John is three inches taller than Mary.

all that is necessary to find the degree of John’s height to take the degree of Mary’s height and
add three inches to it—each degree is a single point, not a range of them that must be further
manipulated.
The disadvantage, though, as Kennedy (1997) points out, is that it becomes very difficult
to explain the effects of negative adjectives, such as the following:

(18)

*Bob is five feet short.

If degrees are simply points on a scale, then it is not immediately clear why (18) is bad. After
all, whether an adjective is positive or negative should have no bearing on the specific points of a
scale. Additional machinery must be added to determine the ordering of degrees based on the
polarity of the adjective, such that the domain of a negative adjective is reversed from that of its
positive counterpart. So if we added another sentence, similar in meaning to (2),

6

(2)

John is taller than Mary.

(19)

Mary is shorter than John.

then the ordering of the domains (adapted from Kennedy 1997) would be as in (20):

(20)

Dtall = {a, b, Mary, c, John}
Dshort = {John, c, Mary, b, a}

where a, b, and c are other degrees on the scale. This is workable, but then it becomes unclear
why the cross-polar anomaly (i.e., a sentence which compares a positive adjective with a
negative adjective) in (21), with the following logical representation, is bad.

(21)

*John is taller than Bob is short.
∃d[(d(tall))(John) ∧ ¬(d(short))(Mary)]

As Kennedy (1997) states, there is a function d that would cause (21) to be both interpretable and
true; the function would partition the domains of the adjectives as in (22).

(22)

posd(tall) = {Mary, c, John}
negd(tall) = {a, b}
posd(short) = {b, a}
negd(short) = {John, c, Mary}
7

Thus in (22), the positive domain of the function d for tall would consist of the elements Mary, c,
and John, while the negative domain would consist of a and b. Since the ordering of the
elements for short is simply the reverse of tall, the positive domain for short, for example, is b
and a—the same elements as the negative domain of tall, simply reversed. But if adjectives
simply distinguish between positive and negative types based on the ordering of the degree
objects within the domain, then (21) becomes acceptable—an outcome that is not desired.
However, Kennedy (1997, 2001), following from Seuren (1978, 1984) and von Stechow
(1984), offers a different interpretation. In this version, degrees are not points on a scale but are
instead intervals. Thus, instead of degrees representing points, they represent sets of points.
Positive and negative degrees differ, therefore, by which subset of the scale they represent. So
for a positive degree, for any point contained within the degree, every point below that point is
also contained within the degree.

(23)

POS(S) = {d ⊆ S | ∃p1 ∈ d∀p2 ∈ S[p2 ≤ p1 → p2 ∈ d]}

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

8

Negative degrees are very similar, but whereas positive degrees go to the bottom of the
scale, negative degrees go to the top. In formal terms,

(24)

NEG(S) = {d ⊆ S | ∃p1 ∈ d∀p2 ∈ S[p1 ≤ p2 → p2 ∈ d]}

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

(25)

Furthermore, for any given object x on a scale S, the maximal element of the positive degree is
equal to the minimal element of the negative degree.

(26)

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

So with a sentence like (2), Mary’s height is less than John’s height, making Mary’s height a
proper subset of John’s height. In formal terms with a positive adjective,

9

(27)

tall(Mary) ⊂ tall(John)

and with a negative adjective,

(28)

short(John) ⊂ short(Mary).

Thinking of degrees as intervals now allows us to explain the unacceptability of (21).
John’s height is a positive degree, whereas Bob’s height is a negative degree. Since neither is a
proper subset of the other, the sentence becomes uninterpretable.

(29)

short(Bob) ⊂ tall(John)

(30)

One additional point is to make explicit the max operator referenced in (26). Following
von Stechow (1984) and Rullmann (1995), among others, I assume that there is a maximality
operator which selects the maximal point of a degree interval, and strictly speaking it is this
maximal point which comparatives select. In other words, if John is six feet tall, then he is also

10

five feet tall, four feet tall, three feet tall, etc., but the important part of John’s height is the
maximal point at six feet.

3.3 Syntactic Structures
The “classical view” (as Bhatt & Pancheva 2004 call it) of degree clause structures like
comparatives comes from sources like Selkirk (1970), Bowers (1975), Bresnan (1973),
Jackendoff (1977), Hellan (1981), Heim (2000), and others. In this syntax, the degree clause and
the degree morpheme form a constituent. So in the case of (2),

(2)

John is taller than Mary.

the degree clause than Mary combines with the morpheme –er, which then combines further up
the tree with the gradable adjective. The fact that degree morphemes require specific
complementary words in the degree clause (than in the case of comparatives, as in the case of
equatives) is evidence for this constituency.

11

(31)

1

An argument against this, as Bhatt & Pancheva (2004) point out, is that in general, the degree
morpheme and the degree clause cannot appear together—a fact against constituency.

(32)

*John is [more than Mary] tall.

There is also the matter of the type clash in (31). DegP is of type <<d,t>, t>, while A is of type
<d, <e,t>>. In order to resolve this, DegP needs to Quantifier Raise (QR), leaving behind a
trace of type d (this also assumes a type of semantic binding, like the version described in Heim
& Kratzer 1998).
Kennedy (1997), following from others like Abney (1987), Larson (1988), Corver
(1990), and Grimshaw (2005), offers an alternate interpretation. In his version, the AP and the
degree morpheme form a constituent which then combines with the degree clause.

1

These types derive from Heim (2000).
12

(33)

This structure also has the advantage that no type shifts need to occur in order for proper
interpretation to happen. There is the worry, though, that in this structure, AP is now of type
<e,d>, which is not a workable general type for any adjective not contained within a degree
phrase. There is also the question of why sometimes the degree morpheme and the degree clause
take scope together without the AP. Heim (2000) (by way of Bhatt & Pancheva 2004) gives the
2

following example :

(34)

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

2

The abbreviations, taken from Heim (2000): p = the paper; longw(p,d) = x is long to
degree d in world w; Acc(w) = the set of worlds accessible from w. The world argument is
suppressed when it is the utterance world.
13

a.

required > -er: required [[exactly 5 pages –er than that]1 [the paper be t1long]
∀w ∈ Acc: max{d: longw(p,d)} = 15 pages

b.

-er > required: [exactly 5 pages –er than that]1 [required [the paper be t1long]]
max{d: ∀w ∈ Acc: longw(p,d)} = 15 pages

In (34)a, the paper is exactly fifteen pages long in every acceptable world. In (34)b, the paper is
exactly fifteen pages long in those worlds where it is the shortest (that is, the paper must be at
least fifteen pages long). More importantly, (34)b gives a reading where the degree morpheme
and the degree clause take scope without the AP.
Bhatt & Pancheva (2004) suggest that this contradictory evidence can be resolved by
what they call “late merger of the degree clause.” In this version, the degree morpheme begins
as a sister to the gradable predicate before moving later in the derivation to become a sister of the
degree clause, leaving a copy behind. It is this copy that is pronounced, while the degree clause
merges countercyclically after the degree morpheme has moved covertly to its scope position.
The degree clause then merges as an argument to the QR-ed degree morpheme. Thus the degree
morpheme is interpreted in its scope position but pronounced in its base position.

14

(35)

3

I will not delve into the implications of this theory any further, as it should not directly affect the
goal of this thesis; it is merely presented for sake of completion.

3.4 Factor Phrases
The most extensive analysis of factor phrases (words like twice and seven times) comes
from Bierwisch (1989), who observes a number of characteristics about them. He notes that for
a sentence like (36),

(36)

Hans ist doppelt so groß wie Eva.
Hans is twice as tall as Eva.

3

This tree is taken from Bhatt & Pancheva (2004).
15

4

Hellan (1981) and von Stechow (1984) assume the following representation:

(37)

∃x1 ∃x2 ∃x3 [[HANS [TALL x1]] ∧ [EVA [TALL x2]] ∧ [x3 = 2] ∧ [x1 =
x2×x3]

In other words, there are three degrees x1, x2, and x3, such that the degree to which Hans is tall is
equivalent to the degree of Eva’s tall-ness multiplied by a factor of two. While this does not take
into account an example like (4),

(4)

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

in that (37) assumes that Hans’s height is exactly twice Eva’s without the possibility of it being
greater, this can be easily fixed.

(38)

∃x1 ∃x2 ∃x3 [[HANS [TALL x1]] ∧ [EVA [TALL x2]] ∧ [x3 = 2] ∧ [x1 ≥
x2 × x3]

Although the formal representations have changed somewhat since Bierwisch (1989), the basic
principle shown in (38) is still usable. Bierwisch further states that this seems to imply that

4

Hellan (1981)’s work comes to me via von Stechow (1984) and Bierwisch (1989).
16

equative phrases are based on multiplication and differential comparatives on addition, as
evidenced by (5) and (6), where (6) seems to rule out the possibility of addition or subtraction in
equatives via measure phrases, while (5) is based on addition and subtraction.

(5)

John is three feet taller than Mary.

(6)

*John is three feet as tall as Mary.

(39)

Formal representation of (5) via Bierwisch (1989):
∃x1 ∃x2 ∃x3 [[JOHN [TALL x1]] ∧ [MARY [TALL x2]] ∧ [x3 = 3 ft] ∧ [x1 = x2
+ x3]

This of course excludes the fact that comparatives can combine with factor phrases of the form n
times without difficulty, as in (8).

(8)

John is two times taller than Mary.

This may be a special case, however; I will go into further detail in section 4.
Bierwisch (1989) also points out that in general, equatives of negative adjectives cannot
take factor phrases. This however only applies to dimensional adjectives like tall and long;
evaluative adjectives like good and pretty are unaffected.

(40)

Dieses Brett ist dreimal so long wie der Tisch.
This board is three times as long as the table.

(41)

?Dieses Brett ist dreimal so kurz wie der Tisch.

17

This board is three times as short as the table.
(42)

Der Film ist dreimal so gut wie das Buch.
The film is three times as good as the book.

(43)

Der Film is dreimal so schlecht wie das Buch.
The film is three times as bad as the book.

The acceptability of (43) versus the unacceptability of (41) may be due to the fact that
multiplying degrees of quality is imprecise and thus does not require the precision that
multiplying degrees of dimension does; thus negative adjectives are more likely to lead to an
unacceptability judgment in those cases. This is presumably related to sentences (18) and the
reason why it too is unacceptable.

(18)

*Bob is five feet short.

An alternate possibility is that evaluative adjectives are essentially closed scales; that is, while it
is theoretically possible to imagine something of infinite length, there is a point at which, say,
perfect beauty is reached (with the definition of perfection varying from individual to individual)
and that thus there is no such thing as infinite beauty, hyperbolic sentences aside.
5

As far as the syntax of factor phrases, Bierwisch (1989) assumes the following structure :

5

I have adapted Bierwisch’s original structure to more closely mirror the structure in (31),
but all positions within the tree remain the same. The types, however, are mine.
18

(44)

The factor phrase (FP) is a modifier of the DegP, not an argument, and thus attaches as in (44).
One final point regarding Bierwisch (1989): two of the words he lists as factor phrases
are half and double. Double does not behave like other factor phrases, requiring special
language to be accommodated.

(45)

*John is double as tall as Mary.

(46)

*John is double taller than Mary.

(47)

John is double the height of Mary.

Based on this patterning, we can tentatively place double in a separate category from other factor
phrases, one that we will not deal with any further here. Half, however, behaves differently
depending on the environment. In some cases it can pattern along the same lines as twice.

19

(48)

John is half as tall as Mary.

(49)

*John is half taller than Mary.

Kennedy & McNally (2005) note, however, that it can occur on its own in a manner similar to
words like mostly.

(50)

The glass is half full.

(51)

Her eyes were half closed.

This may simply be a case of two different instances of the word half. Whether they are distinct
or whether the version in (48) and (49) is related to the version in (50) and (51) is beyond the
scope of this thesis.
Beyond Bierwisch (1989), factor phrases rarely come up in the literature. Kennedy
(1997, 2001) does not mention them. Doetjes (1997, 2007) deals only with n times phrases in
relation to adverbial times (e.g., John went to the store three times today), while Rett (2008) only
refers them in a footnote (which I will return to in section 4).

3.5 Formal Semantics of Degree Phrases
Although there are a number of different approaches as to how to build up degree phrases
compositionally, there are certain areas that are common to all of them. It is agreed, for instance,
that comparatives introduce a greater-than relation between two objects, while equatives
introduce a greater-than-or-equal-to relation. Taking Kennedy & McNally (2005)’s denotations
as an example,

20

(52)

[[er/more than dc]] = λG<d,et>λx. ∃d[d > dc ∧ G(d)(x)]

(53)

[[as as dc]] = λG<d,et>λx. ∃d[d ≥ dc ∧ G(d)(x)]

where G(d)(x) is the gradable adjective and dc is the degree introduced by the degree clause, we
see that, for instance, (52) simply introduces a degree which is greater than the degree dc and
states that, using tall as the gradable adjective, the property of tall-ness of an individual x is
equal to d (i.e., tall(x) = d). One thing not included in (52) and (53) is the max operator from
von Stechow (1984), but in order for their denotations to have any meaning, the max operator
must be assumed for the degrees being compared. There are some variations in denotations
(Meier 2003, for example, makes the maximality operator explicit, i.e., max(d1) > max(d2)), but
this approach is basically the same as the others.

4 Twice versus Two Times in Phrases of Comparison

4.1 Basic Assumptions
Let’s look at the data again.

(8)

John is two times taller than Mary.

(9)

John is two times as tall as Mary.

(10)

John is twice as tall as Mary.

(11)

*John is twice taller than Mary.

21

There is an asymmetry here that needs to be explained. Why should two times be acceptable
with either comparatives or adjectives while twice only works with equatives?
The first thing to observe is that (8) and (9) actually mean the same thing. After all, if
John is six feet tall and Mary is three feet tall, (8) is still true. This bears a closer look. In the
case of (1) and (2),

(1)

John is as tall as Mary.

(2)

John is taller than Mary.

they do not mean the same thing. Extrapolating from this, one would expect (8) to be true only if
John’s height were greater than six feet. This is, on the face of it, odd. And it is not just English
that exhibits this quirk: in Macedonian, for example, the meaning of (8) and (9) can only be
expressed with the comparative form (Monica Markovski, p.c.).

(54)

Jon je po visok od Mari.
John is taller than Mary.

(55)

Jon je visok kolku Mari.
John is as tall as Mary.

(56)

Jon je dva puti po visok od Mari.
John is two times taller than/as tall as Mary.

(57)

*Jon je dva puti visok kolku Mari.
John is two times as tall as Mary.

22

So the question remains: why do (8) and (9) mean the same thing? Given that (1) and (2) do not,
the answer must lie within the factor phrase. First, I shall specify where the factor phrase
attaches to the tree.

Adapting Bierwisch (1989)’s syntactic structure with my types to the

Kennedy syntax gives us a tree like (58):

(58)

The factor phrase attaches at Deg’ in order to gain access to the degree within; this also has the
advantage of corroborating Rett (2008)’s independent assumption that factor phrases attach at
Deg’, in order to account for their ability to affect the semantics in terms of its evaluative
markedness. A sentence is evaluative if it refers to a degree that exceeds a contextually specified
standard. So (59) is marked for evaluativity, while (60) is not:

23

(59)

Paul is tall.

(60)

Paul is as tall as Chris.

(61)

Paul is as short as Chris.

(59) is evaluative because it implies a standard of tallness that Paul exceeds it. (60) is nonevaluative. It merely states that Paul’s height is greater than Chris’s; it makes no mention of
where either is in relation to a standard of tallness. (61), meanwhile, is evaluative because the
negative adjective implies that Paul is short, which therefore means he exceeds the standard for
shortness. Evaluativity can be tested by determining whether the sentence entails the
corresponding positive construction; that is, (59) entails that Paul is tall, while (60) does not: it
makes no reference to a contextually supplied standard. (61), however, does, as the use of the
negative adjective short refers to a standard of shortness and entails that Paul is short. This
evaluative property (EVAL) is located within Deg’: this allows it to affect the evaluativity of an
expression within a DegP, rather than a whole CP. For instance, the evaluativity of (61) is not
affected by the addition of a modifier such as at least:

(62)

Paul is at least as short as Chris.

Even though the semantics of the complete expression should be non-evaluative (i.e., (62) does
not mutually entail that Paul is as tall as Chris), (62) is still evaluative because it still refers to the
contextually-supplied standard of shortness, so while (62) does not entail that Paul is as tall as

24

Chris, it does entail that Paul is short. This illustrates why EVAL is located within Deg’.
However, with a sentence like (63),

(63)

Paul is twice as short as Chris, but neither one is short.

the factor modifier does affect the evaluativity. The factor phrase in (63) does not preserve the
evaluativity that would otherwise be present. This leads Rett to believe that the factor phrase is
affecting the semantics and thus must be base-generated in Deg’.
Placing FP as an adjunct to Deg’ means that it is of type <<d, <e,t>>, <d, <e,t>>>.
This gives it access to the merged gradable adjective and degree morpheme, allowing it to
modify the degree relation; any further up the tree and the factor phrase would have to become
needlessly complicated to enact the required modification, as I will show in section 4.2.

4.2 The Compositions of Twice and Two Times
Let’s assume for now that Bierwisch (1989)’s intuition regarding factor phrases is
correct, and that they generally only combine with equatives, in the way of words like twice and
6

half. A way to include equatives and exclude comparatives is thus necessary. The key
difference between comparatives and equatives, remember, is the presence of > versus ≥. Thus a
denotation for the equative degree morpheme as, which is of type <<e,d>, <d, <e,t>>>, would
7

look like (64).

6

It should be noted that I do not believe this assumption to be correct, as evidenced by
sentences like (8); however, it serves as a useful starting point.
7
I will use Kennedy’s version of the degree construction, as with the fewest moving parts
it will be easier to see what is occurring with the factor phrase.
25

(64)

[[as]] = λG<e,d>λdλx [G(x) ≥ d]

This denotation simply takes the gradable adjective G(x) (which is of type <e,d>, meaning that it
takes an individual and returns a degree, not a truth value) and states that the degree the adjective
returns is greater than or equal to another degree (which, using (10) as the sentence being
interpreted, will later be supplied by the degree clause as Mary). Combining as with the
adjective tall (which I assume simply is denoted as λx[tall(x)], i.e., “the height of x is some
certain degree”) gives us (65).

(65)

[[as tall]] = [[as]] ([[tall]])
= λdλx [tall(x) ≥ d]

Now we need a denotation for twice, one that loosely means something like “x is tall to a degree
greater than or equal to another degree d multiplied by a factor of 2”. The key thing, once again,
is to ensure that equatives are allowed and comparatives disallowed. With that in mind, I suggest
a denotation for twice like (66):

(66)

[[twice]] = λF<d,<e,t>>λdλx [max{dx: F(dx)(x)} ≥ 2d]

This denotation takes in Deg’, which is of type <d, <e,t>>, and introduces another degree dx, the
maximum of which is greater than or equal to the degree d times 2. This additional degree
allows us to compare our two degrees tall(x) and d.

26

(67)

[[twice as tall]] = [[twice]] ([[as tall]])
= λdλx [max{dx: tall(x) ≥ dx} ≥ 2d]

Functionally, dx is equal to tall(x), as the maximal value of dx must be equal to the smallest value
of tall(x) that we’re interested in. So for all intents and purposes, tall(x) ≥ 2d, which is the
desired outcome. The rest of the denotation continues as normal.

(68)

8

[[John is twice as tall as Mary]] = [[twice as tall]] ([[than
Mary]])([[John]])
= λx [max{dx: tall(x) ≥ dx} ≥ 2(dm)([[John]])
= 1 iff max{dx: tall(John) ≥ dx} ≥ 2(dm)
= 1 iff tall(John) ≥ 2(dm)

Since max(dx) is equal to tall(John), the redundant information is removed, leaving simply
tall(John).
However, when a comparative attempts to combine with twice, it breaks down.

(69)

8

[[twice taller]] = [[twice]] ([[taller]])

dm represents the degree of Mary’s height and is the end result of [[as Mary]].
27

= λdλx [max{dx: tall(x) > dx} ≥ 2d]

The reason (69) breaks down is that twice requires the maximum degree that is smaller than
tall(x), and there is no such number—after all, is dx equal to tall(x) – 1? tall(x) – 0.1? tall(x) –
0.0000000000000000000000000000000000001? There is no way of determining a value for dx.
To put it in mathematical terms, dx approaches tall(x) from below but never reaches it. This then
explains why (10) is acceptable and (11) is not. Furthermore, this also corroborates Bierwisch
(1989)’s observation that factor phrases only appear with equatives, not comparatives. The
denotation for half is simply a matter of changing the factor.

(70)

[[half]] = λF<d,<e,t>>λdλx [max{dx: F(dx)(x)} ≥ 1/2 d]

This still leaves the matter of n times phrases unresolved. However, using twice as a
guide, we can modify (66) slightly to allow compatibility with both equatives and comparatives.

(71)

9

[[n times]] = λF<d,<e,t>>λdλx. ∃dx [F(dx)(x) ∧ dx ≥ nd]

(71) takes the Deg’ and states that it is larger than some degree dx, and that dx is greater than or
equal to a degree d multiplied by some factor n. So, interpreting (9) gives us the following:

9

I am not concerned in this thesis how n times is built up compositionally. Suffice to say
that the number that the factor in the phrase ends up being replaces n in (71).
28

(72)

[[two times as tall]] = [[two times]] ([[as tall]])
= λdλx. ∃dx [tall(x) ≥ dx ∧ dx ≥ 2d]
[[John is two times as tall as Mary]] = [[two times as tall]]
([[as Mary]])([[John]])
= 1 iff ∃dx [tall(John) ≥ dx ∧ dx ≥ 2dm]
= 1 iff tall(John) ≥ 2dm

Like (68), tall(John) is equal to dx, and thus the redundant information is removed, leaving the
final line of (72). This is precisely the desired result for (9).
Combining two times with the comparative, however, gives a subtly different result.

(73)

[[two times taller]] = [[two times]] ([[taller]])
= λdλx. ∃dx [tall(x) > dx ∧ dx ≥ 2d]

According to (73), the sentences in (8) and (9) do not technically mean the same thing. It is in
fact the case that tall(John) cannot be exactly equal to twice Mary’s height. tall(John) must be
slightly larger than 2dm. However, it is also true that (73) does not specify the size of the
interval between tall(x) and dx. Thus tall(x) only needs to be an atom taller than dx for this to
hold true. This means that tall(x) can round down to the value of dx—pragmatically, their values
are identical. It seems to be virtually impossible to construct an example where (73) fails

29

pragmatically. Even when dealing with incredibly small objects like atoms, it is still possible for
tall(x) and d(x) to be pragmatically equivalent. For example, if there is an atom of hydrogen
with an atomic weight of 1.00794 atomic mass units (u) and a molecule of hydrogen gas (which
consists of two atoms of hydrogen) with an atomic weight of 2.01588 u, we can still say (74),

(74)

This molecule of H2 is two times heavier than that atom of hydrogen.

because even at this level of smallness, there is no way of knowing if the molecule actually
weighs 2.01588…1 and we have simply rounded down. Thus pragmatically, a sentence like (8)
will look like (75).

(75)

[[John is two times taller than Mary]] = [[two times taller]]
([[than Mary]])([[John]])
= 1 iff ∃dx [tall(John) > dx ∧ dx ≥ 2dm]
= 1 iff tall(John) ≥ 2dm

Two times and twice will also work with closed-scale adjectives like dry.

(76)

Dara is twice as dry as Michael.

(77)

Dara is two times as dry as Michael.

(78)

[[twice as dry]] = λdλx [max{dx: dry(x) ≥ dx} ≥ 2d]

10

10

I am assuming that dry, similarly to tall, is an adjective that maps individuals to degrees
of dryness. In (78) and (79), dm represents the degree of Michael’s dryness.
30

[[Dara is twice as dry as Michael]] = [[twice as dry]]
([[as Michael]])([[Dara]])
= λx [max{dx: dry(x) ≥ dx} ≥ 2(dm)([[Dara]])
= 1 iff dry(Dara) ≥ 2(dm)

(79)

[[Dara is two times as dry as Michael]] = [[two times as dry]]
([[as Michael]])([[Dara]])
= 1 iff ∃dx [dry(Dara) ≥ dx ∧ dx ≥ 2dm]
= 1 iff dry(Dara) ≥ 2dm

This, once again, is exactly the desired result.
With closed-scale adjectives, twice and two times, just like ordinary measure phrases,
work in both directions. (78) and (79) show the composition of sentences involving dry, and
(82) and (83) show the composition with wet.

(80)

Michael is twice as wet as Dara.

(81)

Michael is two times as wet as Dara.

(82)

[[Michael is twice as wet as Dara]] = [[twice as wet]]
([[as Dara]])([[Michael]])
= λx [max{dx: wet(x) ≥ dx} ≥ 2(dd)([[Michael]])
= 1 iff wet(Michael) ≥ 2(dd)

31

(83)

[[Michael is two times as wet as Dara]] = [[two times as wet]]
([[as Dara]])([[Michael]])
= 1 iff ∃dx [wet(Michael) ≥ dx ∧ dx ≥ 2dd]
= 1 iff wet(Michael) ≥ 2dd

Moreover, with open scale adjectives like tall, their negative counterparts also work as
desired.

(84)

Mary is twice as short as John.

(85)

[[twice as short]] = [[twice]] ([[as short]])
= λdλx [max{dx: short(x) ≥ dx} ≥ 2d]

In (85), we note that the degrees on the scale of shortness are exactly what we want—for
instance, if we assume that Mary is three feet tall and John is six feet tall, then plugging the
numbers into (85) means that max{dx: 3 ≥ dx ≥ 12}. Since the scale of shortness is reversed
relative to the scale of tallness, this is correct, since three would be larger on the shortness scale
(i.e., further from the “starting point” of infinity) than twelve.
There is however a concern related to fractional factors. Although half has been
demonstrated in (70) to be fine, there is a puzzle with phrases like a third. While (86), the
equative form, follows the pattern, (87) and (88) do not.

(86)

Andrew is a third as tall as Mitch.

32

(87)

Andrew is a third taller than Mitch.

(88)

Andrew is a third [less tall/shorter] than Mitch.

The meaning of (87) and (88) loosely translate to (89) and (90), respectively.

(89)

tall(Andrew) = tall(Mitch) + 1/3(tall(Mitch))

(90)

tall(Andrew) = tall(Mitch) – 1/3(tall(Mitch))

(87) and (88) behave more like comparatives with measure phrases, looking more like (91) than
what we would expect from factor phrases.

(91)

Andrew is 12 inches taller than Mitch.
tall(Andrew) = tall(Mitch) + 12 in.

This puzzle has yet to be satisfactorily explained.

5 Conclusion
In this thesis I have argued that factor phrases like twice and two times do not in
fact mean the same thing but have subtly different denotations. I have also stated that factor
phrases attach to degree constructions as an adjunct to Deg’ in Kennedy (1997) and others’
framework. This leads to further support of this analysis of degree constructions. Moreover,
although Bierwisch (1989) has done some work on factor phrases, I have taken many of his
assumptions further, supported them where appropriate, and altered them where necessary to fit

33

the data. Given the data regarding both adjectives like tall and dry and the observations made by
Bierwisch (1989) and Rett (2008), my denotations of twice and two times appear to be correct.
Further work in this area may also include continuing this line of research with other
factor phrases such as double, which does not follow the pattern established by twice, two times,
etc., and potentially investigating the relations between these factor phrases and adverbial
versions found in eventive and temporal sentences. There is also the curious behavior of
fractional factor phrases, which do not follow the pattern of phrases like twice. However, my
analysis can hopefully be extended to account for these as well.

34

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35

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