3 Polar O

pposition an

d the On

tology of Com

parison

Th

e objective of this ch

apter is to demon

strate that degrees sh

ould be form

alized as

intervals on

a scale rather th

an as poin

ts, as traditionally assu

med. U

sing cross-polar

anom

aly as the em

pirical basis for my claim

s, I will argu

e that gradable adjectives den

ote

fun

ctions from

objects to intervals on

a scale, or extents, an

d assum

e an on

tology wh

ich

distingu

ishes betw

een tw

o sorts of extents: positive exten

ts and n

egative extents (as in

Seu

ren 19

78, von

Stech

ow 19

84

b, and Löbn

er 199

0). I ch

aracterize the differen

ce

between

positive and n

egative adjectives as a sortal one: positive adjectives den

ote fun

ctions

from objects to positive exten

ts, and n

egative adjectives denote fu

nction

s from objects to

negative exten

ts. After settin

g this an

alysis into th

e seman

tic framew

ork developed in

chap

ter 2, I show

that cross-p

olar anom

aly can be exp

lained in

the sam

e way as

incom

men

surability: a con

sequen

ce of the sortal ch

aracterization of adjectival polarity is th

at

the ran

ges of positive and n

egative adjectives are disjoint; as a resu

lt, the com

parison relation

in exam

ples of cross-polar anom

aly is un

defined. Section

3.2 contin

ues w

ith an

examin

ation

of comparison

of deviation con

struction

s, wh

ich at first glan

ce appear to be coun

terexamples

to the proposals m

ade in section

3.1, but u

pon closer exam

ination

turn

out to fit in

natu

rally

with

the an

alysis of degree constru

ctions developed in

chapter 2. F

inally, section

3.3

demon

strates that th

e algebra of extents, in

conju

nction

with

the sem

antic an

alysis of

gradable adjectives as measu

re phrases, h

as the addition

al positive result of providin

g the

basis for an in

sightfu

l explanation

of the m

onoton

icity properties of gradable adjectives.

3.1 Com

parison an

d Polar O

pposition

3.1.1 The A

lgebra of Degrees

Th

e seman

tic analysis of gradable adjectives an

d degree constru

ctions developed in

chapter 2

characterized th

e interpretation

s of these expression

s in term

s of abstract degrees, or points

on a scale. T

he n

otion of a degree as a poin

t on a scale goes back at least to C

resswell (19

76),

187

wh

o suggests th

at “[w]h

en w

e make com

parisons w

e have in

min

d points on

a scale” (p.

266

). Cressw

ell builds an

algebra of degrees directly from th

e domain

of the adjective; in

chapter 2 (see section

2.1.3), I adopted a modified version

of this approach

, in w

hich

scales

and degrees are defin

ed abstractly in term

s of a linearly ordered set of poin

ts and a

dimen

sion. Specifically, I defin

ed a scale Sδ as a den

se, linearly ordered set of poin

ts along a

dimen

sion δ, an

d a degree as an elem

ent of th

e scale.

With

in th

is sort of ontology, gradable adjectives are tradition

ally analyzed as tw

o place

relations betw

een degrees an

d individu

als. In ch

apter 2 how

ever, I argued for an

alternative

approach to th

e seman

tics of gradable adjectives in w

hich

they are an

alyzed as measu

re

fun

ctions–

fun

ctions from

objects to degrees. Alth

ough

this approach

, as well as th

e

seman

tics of degree constru

ctions bu

ilt on top of it, differed from

the tradition

al analysis in

a nu

mber of im

portant respects, it n

evertheless m

aintain

ed the tradition

al assum

ptions

about th

e basic structu

re of scales and degrees. In

the follow

ing section

s, I will take a closer

look at these assu

mption

s, focusin

g on th

e representation

of adjectival polarity with

in a

degree algebra. In section

3.1.3, I will tu

rn to an

examin

ation of cross-polar an

omaly, w

here I

will dem

onstrate th

at an algebra of degrees th

at characterizes th

ese objects as points on

a

scale incorrectly predicts th

at comparatives con

structed ou

t of anton

ymou

s pairs of

adjectives shou

ld be perfectly interpretable.

3.1.2 Degrees an

d Polar O

pposition

As observed in

chapter 1 (see section

1.1.4), a characteristic of m

any (th

ough

not all) gradable

adjectives is that th

ey come in

pairs: tall - short, safe - dangerous, sharp - dull, bright - dim, etc.

In a basic sen

se, the m

eanin

gs of the m

embers of su

ch an

tonym

ous pairs are fu

ndam

entally

the sam

e: they apply to th

e same objects, an

d they provide th

e same kin

d of inform

ation

about th

e degree to wh

ich an

object possess some gradable property. T

he an

tonym

ous

adjectives tall and short, for exam

ple, both provide in

formation

about an

object’s heigh

t. At

the sam

e time, positive an

d negative adjectives represen

t different–

and in

some sen

se

complem

entary–

perspectives on th

e degree to wh

ich an

object possesses a gradable

188

property. Intu

itively, tall measu

res the h

eight an

object has; short m

easures th

e heigh

t an

object does not h

ave.

Th

e empirical m

anifestation

of this distin

ction in

perspective can be observed in

differences in

the in

terpretations of absolu

te and com

parative constru

ctions w

ith positive

and n

egative adjectives. Th

e basic analysis of th

e seman

tics of positive adjectives can be

illustrated by con

sidering in

terpretation of (1). T

he logical form

of (1) is (2), wh

ere slong

denotes a stan

dard of longn

ess relative to some com

parison property appropriate for T

he

Brothers K

aramazov (e.g., th

e property of being a R

ussian

novel). 1

(1)T

he Brothers K

aramazov is lon

g.

(2)abs(long(B

K))(slong )

Accordin

g to the tru

th con

ditions for th

e absolute con

struction

proposed in ch

apter 2, (1) is

true ju

st in case th

e degree to wh

ich T

he Brothers K

aramazov is lon

g is at least as great as the

standard valu

e; i.e., if and on

ly if the relation

illustrated in

(3) holds, w

here d

BK

is the degree

of The B

rothers K.’s len

gth.

(3)len

gth

: 0 -

--

--

--

--

- slong

--

--

--

--

--

dB

K -

--

--

--

--

- ∞

Absolu

te constru

ctions w

ith n

egative adjectives may be an

alyzed in basically th

e same

way, w

ith on

e importan

t modification

: the relation

between

the referen

ce value an

d the

standard valu

e mu

st be reversed. Th

is point is illu

strated by the an

alysis of (4).

(4)

The D

ream of a R

idiculous Man is sh

ort.

1See th

e discussion

of the derivation

of implicit stan

dards in ch

apter 2, section 2.3.2; I u

se

the variable slong rath

er than

the com

plex expression stn

d(long)(p

(The B

rothers Karam

azov)) for

perspicuity.

189

(5)abs(short(D

ream))(sshort )

Th

e logical representation

of (4) is (5), w

here ssh

ort denotes th

e standard of sh

ortness

appropriate for The D

ream of a R

idiculou

s Man

, wh

ich m

ay or may n

ot be the sam

e as the

standard of lon

gness in

a given con

text. Th

e crucial differen

ce between

(4) and (1) is th

at in

the form

er, the partial orderin

g relation associated w

ith th

e absolute con

struction

mu

st be

reversed. Con

sider, for example, a con

text in w

hich

the stan

dard of longn

ess for Ru

ssian

novels is 6

00

pages, and th

e standard of sh

ortness is 40

0 pages. 2 W

hat sh

ould be th

e case is

that an

y novel over 6

00

pages in len

gth cou

nts as lon

g, wh

ile any n

ovel under 40

0 pages in

length

coun

ts as short (see G

awron

199

5 for discussion

of this issu

e). Assu

min

g, then

, that

the orderin

g relation in

troduced by abs is reversed in

(5), (4) is true ju

st in case th

e relation

between

the referen

ce and stan

dard values illu

strated in (6

) holds.

(6)

leng

th: 0

-

--

--

dD

-

--

--

--

sshort -

--

--

--

--

--

--

--

--

--

- ∞

Com

paratives with

positive and n

egative adjectives show

exactly the sam

e difference

in orderin

g relations as absolu

te constru

ctions. C

onsider, for exam

ple, (7), wh

ich h

as the

logical representation

in (8).

(7)T

he Brothers K

aramazov is lon

ger than

The Idiot.

(8)

mo

re(long(BK

))(long(Idiot))

2In th

is context, a n

ovel wh

ose length

is between

400

and 6

00

pages is neith

er long n

or

short; in

Klein

’s (1980

) termin

ology, it falls in th

e “extension

gap” on th

e scale of length

. Th

is

example poin

ts out th

e fact that an

tonym

ous adjectives are, in

effect, contraries: althou

gh an

object

may be n

either lon

g nor sh

ort, no object is both

long an

d short. T

his relation

is enforced by a

general con

straint on

the relation

between

positive and n

egative standards; see B

ierwisch

1989

for

discussion.

190

(8) is tru

e just in

case the degree to w

hich

The B

rothers Karam

azov is long exceeds th

e

degree to wh

ich T

he Idiot is long. In

the con

text illustrated by (9

), wh

ere dI den

otes the

projection of T

he Idiot on a scale of len

gth an

d dB

K in

dicates The B

rothers Karam

azov’s

degree of length

, (7) is true.

(9)

leng

th: 0

-

--

--

--

--

- d

I -

--

--

--

--

- d

BK

--

--

--

--

--

∞

Negative com

paratives can be an

alyzed in th

e same w

ay as their positive cou

nterparts,

with

the exception

that th

e ordering relation

introdu

ced by the degree m

orphem

e mu

st be

reversed, as in n

egative absolute con

struction

s. Th

is is illustrated by (10

).

(10)

The Idiot is sh

orter than

The B

rothers Karam

azov.

(11)m

ore(short(Idiot))(short(B

K))

(10) is tru

e just in

case the degree to w

hich

The Idiot is sh

ort is ordered below th

e degree to

wh

ich T

he Brothers K

aramazov is sh

ort, as show

n in

(12).

(12)len

gth

: 0 -

--

--

--

--

- d

I -

--

--

--

--

- d

BK

--

--

--

--

--

∞

Pu

t anoth

er way, (10

) is true ju

st in case th

e degree to wh

ich T

he Idiot is short exceeds th

e

degree to wh

ich T

he Brothers K

aramazov is sh

ort, assum

ing th

at the defin

ition of “exceeds”

for degrees of shortn

ess is the in

verse of the defin

ition of “exceeds” for degrees of

longn

ess.

Th

e relational differen

ces between

positive and n

egative adjectives illustrated by

these exam

ples suggest a m

eans of form

ally representin

g adjectival polarity with

in th

e

algebra of degrees outlin

ed in section

3.1.1. As n

oted above, the m

embers of a positive an

d

negative pair of gradable adjectives n

ot only apply to th

e same objects (i.e., sh

are the sam

e

191

domain

), they also provide th

e same basic in

formation

about th

e objects to wh

ich th

ey apply:

they m

easure th

e degree to wh

ich an

object possesses some gradable property. A

noth

er

way of statin

g this observation

is that positive an

d negative pairs of adjectives m

easure

objects according to th

e same dim

ension

. Th

e crucial differen

ce between

positive and

negative adjectives, illu

strated by the exam

ples discussed above, is th

at the m

easurem

ents

they retu

rn perm

it complem

entary orderin

gs on th

eir domain

s. Th

is was m

ost clearly

show

n by th

e comparatives in

(7) and (10

): wh

en T

he Brothers K

aramazov “exceeds” T

he

Idiot with

respect to longn

ess, The Idiot “exceeds” T

he Brothers K

aramazov w

ith respect to

shortn

ess.

Th

e analysis of gradable adjectives as m

easure fu

nction

s sup

ports a form

al

representation

of adjectival polarity wh

ich captu

res the in

tuition

s about polar adjectives

outlin

ed here. T

he claim

that gradable adjectives den

ote measu

re fun

ctions is a claim

about

their sem

antic type: th

ey denote fu

nction

s from objects to degrees. P

ut an

other w

ay, the

range of a gradable adjective is a scale. S

ince scales are defin

ed as a linearly ordered set of

degrees along som

e dimen

sion (w

hich

corresponds to a gradable property su

ch as length or

density), w

e can captu

re the in

tuition

that positive an

d negative pairs of adjectives m

easure

objects according to th

e same dim

ension

by assum

ing th

at they defin

e identical m

appings

from th

e objects in th

eir domain

s to points on

a shared scale. T

o make th

is idea more

precise, I will assu

me th

e general prin

ciple in (13) (cf. R

ullm

ann

199

5).

(13)F

or any scale S

δ along dim

ension

δ, for any an

tonym

ous pair of gradable adjectives

ϕpos an

d ϕneg w

ith dim

ension

al parameter δ, an

d for any object x in

the dom

ain of ϕ

pos

and ϕ

neg : ϕpos (x) = ϕ

neg (x).

Th

e crucial differen

ce between

ϕp

os and ϕ

neg –

wh

at I referred to as a difference in

perspective–is that th

ey impose opposite orderin

gs on th

eir domain

s. Th

is difference can

be

represented in

terms of an

ordering distin

ction betw

een th

e ranges of ϕ

pos an

d ϕneg .

192

Specifically, I w

ill assum

e that th

e range of th

e positive is the set of degrees on

the scale

un

der the basic orderin

g provided by the scale (“<“), w

hile th

e range of th

e negative th

e set

of degrees on th

e scale un

der the reverse orderin

g (“>”). On

this view

, a negative adjective is

the du

al of its positive coun

terpart (cf. Cressw

ell 1976

, Gaw

ron 19

95, S

ánch

ez-Valen

cia

199

5, Ru

llman

n 19

95).

Th

e claim u

nderlyin

g this represen

tation of adjectival polarity is th

at positive and

negative pairs of adjectives m

ap the objects in

their dom

ains to degrees on

the sam

e basic

scale, but th

eir ranges are differen

tiated by virtue of bein

g associated with

opposite ordering

relations. S

ince th

ere is a one-to-on

e mappin

g between

the ran

ge of ϕpos an

d the ran

ge of

ϕneg (th

e identity fu

nction

), how

ever, the tw

o sets are isomorph

ic. In oth

er words, for an

y

scale S, th

e set of positive degrees on S

and th

e set of negative degrees on

S con

tain th

e

same objects. T

his h

as a very importan

t empirical resu

lt: it explains th

e validity of

statemen

ts like (14), a m

inim

al requirem

ent of an

y theory of th

e seman

tics of polar

adjectives and com

paratives.

(14)

The B

rothers Karam

azov is longer th

an T

he Idiot if and on

ly if The Idiot is sh

orter than

The B

rothers Karam

azov.

If positive and n

egative adjectives define iden

tical mappin

gs from objects in

their dom

ains to

degrees on a scale, bu

t differ the orderin

g they in

duce, th

en assu

min

g that th

e comparative

(and absolu

te) morph

emes “in

herit” th

e ordering associated w

ith th

e adjective that h

eads

the degree con

struction

, it follows th

at the coordin

ated propositions in

(14) (i.e., (7) and

(10)), are tru

e in th

e same situ

ations: w

hen

ever the degree of T

he Brothers K

’s length

exceeds the degree of T

he Idiot’s length

. 3

3If a negative adjective is th

e dual of its positive cou

nterpart, th

en th

e validity of (14) follows

from th

e Du

ality Prin

ciple: if a statemen

t ϕ is tru

e in all orders of type T

(partial order, total order,

etc.), then

its dual is also tru

e in all orders of type T

(see Landm

an 19

91:88).

193

193

3.1.3 Cross-polar A

nom

aly

Sen

tences su

ch as (15)-(18), w

hich

exemplify th

e phen

omen

on th

at I referred to in ch

apter 1

as “cross-polar anom

aly”, present an

interestin

g puzzle for th

e algebra of degrees discussed

in th

e previous tw

o sections, in

wh

ich degrees are represen

ted as points on

a scale. 4

(15)#

The B

rothers Karam

azov is longer th

an T

he Idiot is short.

(16)

#T

he Idiot is shorter th

an T

he Brothers K

aramazov is lon

g.

(17)#M

ars is closer than

Plu

to is distant.

(18)

#A brow

n dw

arf is dimm

er than

a blue gian

t is bright.

Th

ese examples in

dicate that com

paratives constru

cted out of a positive an

d negative pair of

adjectives are seman

tically anom

alous. C

rucially, (15)-(18

) contrast w

ith exam

ples of

comparative su

bdeletion in

volving adjectives of th

e same polarity, su

ch as (19

)-(20). 5

4Th

e anom

aly of senten

ces like (15)-(18) was first observed by H

ale (1970

), wh

o notes th

e

contrast betw

een (i) an

d (ii).

(i)T

he table is lon

ger than

it is wide.

(ii)#T

he table is lon

ger than

it is short.

See also Bierw

isch 19

89 for a discu

ssion of sim

ilar facts in G

erman

.

5Recall from

the discu

ssion in

chapter 1 th

at examples of cross-polar an

omaly also con

trast

with

superficially sim

ilar senten

ces like (i-ii), wh

ich I described as exam

ples of “comparison

of

deviation”.

(i)T

he Brothers K

aramazov is as lon

g as The D

ream of a R

idiculous Man is sh

ort.

(ii)A

brown

dwarf is m

ore dim th

an a blu

e giant is brigh

t.

I will retu

rn to a detailed discu

ssion of com

parison of deviation

in section

3.2.

194

194

(19)

Carm

en’s C

adillac is wider th

an M

ike’s Fiat is lon

g.

(20)

Fortu

nately, th

e ficus w

as shorter th

an th

e ceiling w

as low, so w

e were able to get it

into th

e room.

Th

e puzzle of cross-polar an

omaly is th

at the very sam

e assum

ptions w

hich

provided an

intu

itive characterization

of adjectival polarity, as well as an

explanation

of the validity of (14),

make th

e wron

g predictions in

the case of cross-polar an

omaly.

Exam

ples

of cross-p

olar an

omaly

are stru

cturally

instan

ces of

comp

arative

subdeletion

, therefore, assu

min

g the an

alysis of subdeletion

structu

res presented in

section

2.4.1.2 of chapter 2, th

e logical representation

of (15) is (21).

(21)m

ore(long(T

he Brothers K

))(max(λ

d[abs(short(The Idiot))(d)]))

Th

e standard valu

e in (21) is th

e degree introdu

ced by the com

parative clause: th

e maxim

al

degree d such

that T

he Idiot is at least as short as d. A

ccording to th

e analysis of adjectival

polarity outlin

ed in section

3.1.2, the adjectives long and short defin

e the sam

e mappin

g from

the objects in

their dom

ains to poin

ts on a scale of len

gth. It follow

s that degrees of longness

and shortness are th

e same objects, th

erefore the com

parative clause in

(21) shou

ld pick out

not on

ly the m

aximal degree in

the ran

ge of short that represen

ts The Idiot’s len

gth, bu

t also

the m

aximal degree in

the ran

ge of long th

at represents T

he Idiot’s length

, since th

ese two

degrees are in fact th

e same objects. T

hat is, even

thou

gh th

e orderings on

the ran

ges of

long and short are distin

ct, since degrees of lon

gness an

d degrees of shortn

ess are the sam

e

objects, the equ

ivalence in

(22) holds.

(22)m

ax(λd[abs(short(T

he Idiot))(d)]) = max(λ

d[abs(long(The Idiot))(d)])

195

195

If this is th

e case, how

ever, then

(15) shou

ld not on

ly be interpretable, it sh

ould be logically

equivalen

t to (7) and (10

). 6 Th

is is clearly the w

rong resu

lt.

Wh

at is importan

t to observe is that th

is result is n

ot dependen

t on th

e choice of th

e

specific adjectives long and short, it h

olds for any positive-n

egative pair. Sin

ce positive and

negative degrees den

ote the sam

e objects, any equ

ivalence relation

of the form

in (22)

holds, w

ith th

e result th

at the orderin

g relations sch

ematized in

(23) and (24) sh

ould be

defined, w

here d

eg is an

ordering relation

on degrees an

d dpos an

d dn

eg are positive and

negative degrees on

the sam

e scale.

(23)d

eg(d

pos )(dn

eg )

(24)

deg

(dneg )(d

pos )

Sim

ply put, th

ere is no property of an

algebra of degrees wh

ich explain

s the an

omaly of

comparatives con

structed ou

t of positive and n

egative pairs of adjectives. Th

is leads to the

followin

g conclu

sion: eith

er the assu

mption

that positive an

d negative degrees are th

e same

objects is incorrect, or th

e assum

ption th

at positive and n

egative adjectives share th

e same

scale is false. Sin

ce the latter (or som

e equivalen

t assum

ption) is requ

ired to explain th

e

validity of statemen

ts like (14) (cf. Ru

llman

n 19

95), it m

ust be th

e case that th

e former

needs to be reth

ough

t. 7

6Th

e same argu

men

t can be m

ade in th

e case of examples su

ch as (16

), in w

hich

the m

atrix

adjective is negative an

d the adjective in

the com

parative clause is positive.

7It shou

ld be observed that th

e problem of cross-polar an

omaly for a degree algebra is n

ot a

consequ

ence of th

e seman

tic analysis of com

paratives and gradable adjectives th

at I adopted in

chapter 2. T

he sam

e problems arise in

an approach

wh

ich assu

mes a relation

al seman

tics for

gradable adjectives and an

alyzes comparatives in

terms of restricted qu

antification

over degrees,

such

as the on

e discussed in

chapter 1. C

onsider, for exam

ple, the an

alysis of (i) with

in th

is type of

approach.

(i)#C

armen

is taller than

Mike is sh

ort.

196

196

3.1.4 On

the Un

availability of an In

comm

ensurability E

xplanation

Before I presen

t an altern

ative characterization

of degrees, I shou

ld address wh

at at first

glance appears to be a possible explan

ation of cross-polar an

omaly w

ithin

the algebra of

degrees outlin

ed here. T

his explan

ation rejects th

e claim th

at positive and n

egative pairs of

adjectives share th

e same scale, an

d explains th

e anom

aly of examples like (15)-(18) in

terms

of incom

men

surability. O

n th

is view, (15)-(18) are an

omalou

s for the sam

e reason th

at (25)-

(26) are: th

e two adjectives in

the su

bdeletion con

struction

are associated with

different

scales.

(25)#M

ike is taller than

Carm

en is clever.

(26)

#T

he Idiot is more tragic th

an m

y copy of The B

rothers Karam

azov is heavy.

Th

e explanation

of the an

omaly of exam

ples like (25) and (26

) developed in ch

apter 2

builds on

the sem

antic an

alysis of subdeletion

outlin

ed above. Accordin

g to these proposals,

the logical represen

tation of an

example like (25) is (27).

(ii)∃d[d >

ιd’.short(Mike,d’)][tall(C

armen,d)]

Accordin

g to (ii), (i) is true ju

st in case th

ere is a degree wh

ich exceeds th

e maxim

um

degree of

Mike’s sh

ortness, an

d Carm

en is th

at tall. Th

is mean

s that (i) sh

ould be tru

e in th

e situation

represented in

(iii), since d

M den

otes both th

e degree to wh

ich M

ike is short an

d the degree to

which M

ike is tall.

(iii)h

eigh

t: 0 -

--

--

--

--

- d

M -

--

--

--

--

- d

C -

--

--

--

--

- ∞

See K

enn

edy 199

7b for more detailed discu

ssion of th

e problem of cross-polar an

omaly for w

ithin

the con

text of a relational an

alysis of gradable adjectives.

197

197

(27)m

ore(tall(M

ike))(max(λ

d[abs(clever(Carm

en))(d))]))

Th

e standard valu

e introdu

ced by the com

parative clause in

(27) is the m

aximal degree d

such

that C

armen

is at least as clever as d, therefore, in

order to evaluate th

e truth

of (27),

an orderin

g mu

st be established betw

een th

is degree and th

e degree to wh

ich M

ike is tall.

Th

e adjectives tall and clever m

ap their argu

men

ts onto differen

t scales, how

ever, so the

ordering relation

introdu

ced by the com

parative is un

defined for th

ese two degrees,

resultin

g in sem

antic an

omaly.

Alth

ough

I will u

ltimately con

struct an

explanation

of cross-polar anom

aly in th

ese

terms, th

is type of explanation

does not appear to be available w

ithin

the tradition

al degree

algebra. Th

e explanation

of the in

comm

ensu

rability of examples like (25) an

d (26)

presupposes th

at the scales associated w

ith th

e adjectives clever and tall are distin

ct (i.e., they

are associated with

different dim

ension

al parameters; see th

e discussion

in ch

apter 1,

section 1.1.3). T

his is cru

cial to the explan

ation, becau

se wh

en th

e adjectives in a su

bdeletion

structu

re map th

eir argum

ents to poin

ts on th

e same scale, com

parison is possible, as

show

n by th

e examples discu

ssed above.

In order for an

incom

men

surability explan

ation of cross-polar an

omaly to go th

rough

,

then

, it mu

st be demon

strated that for an

y for anton

ymou

s pair of adjectives, there is n

o

mappin

g between

objects in th

e range of th

e positive adjective and objects in

the ran

ge of

the n

egative. If such

a mappin

g can be defin

ed, then

examples like (15)-(18) sh

ould be

amen

able to the sam

e sort of analysis as (19

) and (20

). Th

e problem is th

at it mu

st be the

case that a m

apping betw

een e.g. degrees of talln

ess and degrees of sh

ortness can

be

defined, oth

erwise th

e validity of (14) wou

ld remain

un

explained (see R

ullm

ann

199

5 for

additional discu

ssion of th

is point). T

he in

comm

ensu

rability explanation

fails because an

y

algebra of degrees wh

ich explain

s the validity of con

struction

s like (14)–wh

ich is a m

inim

al

requirem

ent of descriptive adequ

acy–shou

ld also permit a m

apping betw

een positive an

d

negative degrees, w

ith th

e consequ

ence th

at examples of cross-polar an

omaly sh

ould be fu

lly

198

198

interpretable.

Alth

ough

it fails to explain cross-polar an

omaly, th

e proposal that positive an

d negative

adjectives are incom

men

surable con

tains an

interestin

g hypoth

esis: that degrees of talln

ess

and degrees of sh

ortness are differen

t sorts of objects, with

the con

sequen

ce that th

e range

of a positive adjective is disjoint from

the ran

ge of its negative cou

nterpart. If th

is were th

e

case, then

cross-polar anom

aly could be explain

ed in th

e same term

s as incom

men

surability:

it wou

ld be impossible to evalu

ate the relation

introdu

ced by the com

parative morph

eme.

With

out an

ad hoc stipu

lation to th

is effect, how

ever, there is n

o way to bu

ild the n

ecessary

sortal difference betw

een p

ositive and n

egative degrees into a degree algebra: if

correspondin

g positive and n

egative degrees on th

e same scale den

ote the sam

e objects,

then

they are sortally in

distingu

ishable from

one an

other. In

the follow

ing section

, I will

develop an altern

ative algebra of degrees in w

hich

positive and n

egative degrees are analyzed

as distinct objects, an

d I will sh

ow th

at this distin

ction n

ot only provides an

insigh

tful

representation

of adjectival polarity and an

accoun

t of the validity of statem

ents like (14), it

also provides the basis for an explan

ation of cross-polar an

omaly.

3.1.5 The A

lgebra of Exten

ts

In order to develop a solu

tion to th

e problem of cross-polar an

omaly, I w

ill adopt an on

tology

of comparison

in w

hich

the projection

of an object on

a scale is represented n

ot as a discrete

point, bu

t rather as an

interval, wh

ich I w

ill refer to as an extent (cf. S

euren

1978, 19

84, von

Stech

ow 19

84

b, Bierw

isch 19

89

, Löbner 19

90

). In term

s of the an

alysis of gradable

adjectives developed here, th

e basic claim rem

ains th

e same: every gradable adjective

denotes a fu

nction

from objects to scalar valu

es, and scales are differen

tiated throu

gh

association w

ith a dim

ension

. Th

e difference is in

the stru

cture of th

e scalar values–w

hat I

have referred to u

p to now

as “degrees”. Wh

ereas the algebra of degrees ou

tlined in

section

3.1.2 represents scalar valu

es as a discrete points, th

e extent algebra th

at I will form

alize

199

199

below represen

ts degrees as intervals. A

s I will dem

onstrate, th

is modification

provides the

basis for an an

alysis of adjectival polarity wh

ich both

predicts the validity of statem

ents like

(14) above and supports an

explanation

of cross-polar anom

aly.

In term

s of basic structu

re, the n

otion of a scale can

be formalized as above.

Specifically, let a scale Sδ be a den

se, linearly ordered set of poin

ts along a dim

ension

δ which

may h

ave a min

imal elem

ent bu

t has n

o maxim

al elemen

t. 8 Since a scale is defin

ed as a sets

of points, an

extent on a scale can

be defined as a n

onem

pty, convex su

bset of the scale, i.e. a

subset of S

δ with

the follow

ing property: ∀

p1 ,p

2 ∈E∀

p3 ∈

Sδ [p

1 <p

3 <p

2 → p

3 ∈E

] (cf. Landm

an

199

1:110; th

is is simply th

e definition

of an in

terval for a linearly ordered set of poin

ts).

Fin

ally, I will defin

e a proper extent on a scale S

δ as a non

empty, con

vex proper subset of S

δ .

Th

e distinction

between

extents an

d proper extents n

ot crucial, rath

er it is made prim

arily to

simplify th

e definition

s of positive and n

egative extents presen

ted below, an

d to avoid the

question

of wh

at sort of extent th

e scale itself is.

Th

ese definition

s provide the basis for a sem

antic an

alysis of gradable adjectives as

fun

ctions from

individu

als to extents th

at has essen

tially the sam

e properties as the degree-

based analysis assu

med u

p to now

. In order to bu

ild a foun

dation for an

alternative th

eory of

adjectival polarity, I will m

ake a furth

er distinction

between

two sorts of exten

ts–positive

extents and negative extents–w

hich

are defined in

(28) and (29

).

(28)

A positive extent on

a scale is a proper extent w

hich

ranges from

the low

er end of th

e

scale to some positive poin

t.

(29)

A negative extent on

a scale is a proper extent w

hich

ranges from

some positive poin

t

to the u

pper end of th

e scale.

8Alth

ough

it seems appropriate to assu

me th

at the scales associated w

ith dim

ension

al

adjectives like long an

d wide h

ave a min

imal elem

ent, it is n

ot obvious th

at the sam

e holds of

adjectives like beautifu

l and creative. S

ince th

is point does n

ot affect the basic claim

s to be made

below, I h

ave left the qu

estion of w

heth

er scales have a m

inim

al elemen

t open.

200

200

To m

ake thin

gs more precise, assu

me th

at for any object a w

hich

can be ordered accordin

g

to some dim

ension

δ, there is a fu

nction

d from a to a u

niqu

e point on

the scale S

δ . 9 Let

the positive exten

t of a on S

δ (posδ (a)) be as defin

ed in (30

), and let th

e negative exten

t of a

on S

δ (neg

δ (a)) be as defined in

(31).

(30)

posδ (a) = {p ∈

Sδ | p ≤ d(a)}

(31)n

egδ (a) = {p ∈

Sδ | d(a) ≤ p}.

Accordin

g to the defin

itions in

(30) an

d (31), the positive an

d negative projection

s of an

object on a scale are (join

) complem

entary. T

his is illu

strated by (32), wh

ich sh

ows th

e

positive and n

egative extents of an

object a on th

e scale Sδ .

(32)S

δ : 0 -

--

--

--

posδ (a)

-

--

--

--

•-

--

--

--

neg

δ (a)-

--

--

--

> ∞

Given

these backgrou

nd assu

mption

s, we can

now

constru

ct a formal an

alysis of

adjectival polarity in term

s of an algebra of exten

ts. In section

3.1.2, I noted th

e followin

g

distinction

between

anton

ymou

s pairs of adjectives: althou

gh th

e mem

bers of a positive and

negative pair of adjectives m

easure objects accordin

g to the sam

e dimen

sion, th

ey represent

complem

entary perspectives on

the projection

of an object on

to scale. For exam

ple, the

senten

ces Carm

en is tall an

d Mike is short both

provide inform

ation abou

t the h

eight of

Carm

en an

d Mike, respectively, bu

t the in

formation

is qualitatively differen

t in each

case:

the positive adjective tall con

veys inform

ation abou

t the h

eight an

object has, w

hile th

e

negative adjective short con

veys inform

ation abou

t the h

eight an

object does not h

ave (cf. von

Stech

ow 19

84b:196

).

9Th

is point correspon

ds to a degree in th

e traditional algebra.

201

201

Th

e distinction

between

positive and n

egative extents provided by th

e algebra of

extents–

specifically,

the

comp

lemen

tarity illu

strated

in

(32)–p

rovides

a m

eans

of

representin

g adjectival polarity in a w

ay that captu

res these in

tuition

s. Specifically, I w

ill

claim th

at positive adjectives denote fu

nction

s from objects to positive exten

ts, and n

egative

adjectives denote fu

nction

s from objects to n

egative extents. P

ut an

other w

ay, I am claim

ing

that adjectival polarity is a sortal distin

ction betw

een positive an

d negative adjectives. A

ll

gradable adjectives denote fu

nction

s from objects to exten

ts, but th

e set of such

fun

ctions

can be sorted accordin

g to their ran

ge: positive adjectives denote gradable properties w

hose

range is th

e set of positive extents on

a scale S; n

egative adjectives denote gradable

properties wh

ose range is th

e set of negative exten

ts on S

. A con

sequen

ce of this approach

is that alth

ough

anton

ymou

s pairs of positive and n

egative adjectives define m

appings from

objects in th

eir domain

s to the sam

e scale, the positive projection

of an object on

a scale is

distinct from

the n

egative projection of th

at object on th

e same scale.

Th

is distinction

represents th

e fun

damen

tal difference betw

een th

is approach to

adjectival polarity and th

e one provided by th

e degree algebra outlin

ed in section

3.1.2: in th

e

former, bu

t not th

e latter, the ran

ges of a positive and n

egative pair of adjectives are disjoint.

As observed in

section 3.1.2, becau

se degrees are defined as poin

ts on a scale, th

ere is no w

ay

to differentiate positive an

d negative degrees. If gradable adjectives den

ote fun

ctions from

objects to points on

a scale, then

for any object a an

d any pair of positive an

d negative

adjectives ϕpos an

d ϕneg , ϕ

pos (a) and ϕ

neg (a) denote th

e same object. In

contrast, exten

ts have

additional stru

cture, n

amely th

e intervals exten

ding to th

e relevant en

d of the scale. A

s a

result, positive an

d negative exten

ts are distinct objects, an

d for any object a an

d any pair of

positive and n

egative adjectives ϕpos an

d ϕneg , ϕ

pos (a) ≠ ϕneg (a). P

ut an

other w

ay, positive and

negative exten

ts are different sorts of objects, distin

guish

ed in term

s of perspective. A

consequ

ence of th

is distinction

is that, for an

y scale S, th

e set of positive extents on

S an

d

the set of n

egative extents on

S are disjoin

t subsets of th

e total set of extents on

S. T

his

aspect of the exten

t algebra provides the fou

ndation

for the explan

ation of cross-polar

202

202

anom

aly that I w

ill present in

section 3.1.7.

Th

e ontological assu

mption

s that I h

ave outlin

ed here h

ave their roots in

the w

ork of

Seu

ren (19

78) (see also Seu

ren 19

84), wh

o analyzes gradable adjectives as relation

s between

individu

als and exten

ts and m

akes a distinction

between

positive and n

egative extents alon

g

the lin

es of the on

e proposed in (28) an

d (29); th

is approach is also adopted in

von S

techow

1984b an

d Löbner 19

90

. 10 Th

e importan

t contribu

tion th

at the w

ork reported here m

akes

to this lin

e of research is th

at it introdu

ces a strong em

pirical argum

ent–th

e phen

omen

on

of cross-polar anom

aly–for formalizin

g scalar values in

terms of in

tervals and, in

particular,

for usin

g the sortal distin

ction betw

een positive an

d negative exten

ts as the basis for a th

eory

of adjectival polarity (see also the discu

ssion of th

e distribution

of measu

re phrases in

section 3.1.8). B

efore I go throu

gh th

e explanation

of cross-polar anom

aly, how

ever, I will

show

how

the algebra of exten

ts provides an explan

ation for th

e different relation

al

characteristics of positive an

d negative adjectives observed in

section 3.1.2, an

d how

it

accoun

ts for the validity of statem

ents like (14).

3.1.6 E

xtents an

d Polar O

pposition

Th

e formal represen

tation of adjectival polarity proposed in

the previou

s section h

as an

importan

t consequ

ence: it derives th

e relational differen

ce between

positive and n

egative

pairs of gradable adjectives, elimin

ating th

e need to assu

me th

at the com

parative and

absolute degree m

orphem

es “inh

erit” their orderin

g relations from

the adjectives th

ey

combin

e with

. Instead, w

e can adopt a u

niform

set of truth

condition

s for degree

morph

emes as in

(33)-(36), w

here th

e ordering relation

s on exten

ts {>,<,≥} are defined in

a

standard B

oolean fash

ion, as in

(37)-(39).

10Bierw

isch (19

89

) also formalizes scalar valu

es as intervals, rath

er than

points, bu

t

Bierw

isch’s system

does not m

ake the distin

ction betw

een positive an

d negative exten

ts advocated

here.

203

203

(33)||abs(eR )(eS )|| = 1 iff eR ≥ eS

(34)

||mo

re(eR )(eS )|| = 1 iff eR > eS

(35)||less(eR )(eS )|| = 1 iff eR < eS

(36)

||as(eR )(eS )|| = 1 iff eR ≥ eS

(37)[e1 > e2 ] iff [e1 ∩

e2 = e2 & e1 ≠ e2 ]

(38)

[e1 < e2 ] iff [e1 ∩ e2 = e1 &

e1 ≠ e2 ]

(39)

[e1 ≥ e2 ] iff [e1 ∩ e2 = e2 ]

Con

sider, for example, th

e analysis of positive an

d negative absolu

te constru

ctions,

such

as (40) an

d (41), in th

e context illu

strated by (42), wh

ere long(BK

) denotes th

e positive

extent of T

he Brothers K

aramazov on

the scale of len

gth, short(B

K) den

otes its negative

extent, an

d slong an

d sshort denote stan

dards of longn

ess and sh

ortness (for e.g. R

ussian

novels) respectively. 11

(40

)T

he Brothers K

aramazov is lon

g.

(41)

The B

rothers Karam

azov is short.

(42)

leng

th:

0 -

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

> ∞

0 -

--

--

--

- long(B

K) -

--

--

--

--

--

•-

--

--

short(BK

)-

--

> ∞

0 -

--

--

--

slong -

--

--

--

•

•-

--

--

--

--

--

--

sshort -

--

--

--

--

->

∞

Th

e logical representation

s of (40) an

d (41) are (43) and (44) respectively.

11As above, I assu

me th

at slong an

d sshort are derived as a fun

ction of th

e mean

ing of th

e

adjective and a con

textually determ

ined com

parison property, as described in

chapter 2, section

2.3.

204

204

(43)

abs(long(BK

))(slong )

(44

)abs(short(B

K))(sshort )

Accordin

g to the tru

th con

ditions for th

e absolute in

(33), (40) is tru

e just in

case the exten

t

to wh

ich T

he Brothers K

aramazov is lon

g is at least as great as the stan

dard value. In

the

context in

(42), (40) is tru

e, because long(B

K) ≥ slong h

olds.

Th

e negative absolu

te in (41) can

be analyzed in

exactly the sam

e way: it is tru

e just

in case th

e extent to w

hich

The B

rothers Karam

azov is short is at least as great as th

e

standard of sh

ortness. (43) is false in

the con

text illustrated by (42), becau

se short(BK

) ≥

sshort does not h

old. Wh

at is importan

t to observe is that th

e same orderin

g relation is u

sed

to calculate th

e truth

of both th

e positive and n

egative absolute con

struction

s. Th

is

contrasts w

ith th

e analysis of th

is senten

ce in th

e algebra of degrees discussed in

section

3.1.2, wh

ich requ

ired the orderin

g relation associated w

ith th

e absolute to be reversed for

negative adjectives.

Th

e analysis of positive an

d negative com

paratives is similar. C

onsider, for exam

ple,

(45) an

d (46

) in th

e context represen

ted by (47), w

here lon

g(BK

) and short(B

K) are as

defined above, an

d long(Idiot) and short(Idiot) represen

t the positive an

d negative exten

ts of

The Idiot’s len

gth, respectively.

(45)

The B

rothers Karam

azov is longer th

an T

he Idiot.

(46

)T

he Idiot is shorter th

an T

he Brothers K

aramazov.

(47)

leng

th:0

-

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

> ∞

0 -

--

--

--

- long(B

K) -

--

--

--

--

--

•-

--

--

short(BK

)-

--

> ∞

0 -

--

--

--

- long(Idiot)

-

--

--

-•-

--

--

--

short(Idiot)-

--

--

> ∞

Th

e logical representation

of (45) is (48).

205

205

(48

)m

ore(long(B

K))(long(Idiot))

Accordin

g to the tru

th con

ditions for th

e comparative in

(34), (4

5) is true ju

st in case

long(B

K) > lon

g(Idiot), i.e., just in

case the positive projection

of The B

rothers Karam

azov on

the scale of len

gth exceeds th

e positive projection of T

he Idiot on th

e scale of length

. Th

ese

condition

s are met in

the con

text represented by (47), so (45) is tru

e. Again

, there is n

o

need to assu

me a ch

ange in

the orderin

g relation associated w

ith th

e comparative: th

e

structu

re of extents is su

ch th

at the sam

e relation can

be used to ch

aracterize the tru

th

condition

s of both positive an

d negative com

paratives.

Th

e negative com

parative (46) is treated in

the sam

e way. T

he logical form

of (46) is

(49

).

(49

)m

ore(short(Idiot))(short(B

K))

(46) is tru

e just in

case short(Idiot) > short(BK

) holds, i.e., ju

st in case th

e negative projection

of The Idiot on

the scale of len

gth exceeds th

e negative projection

of The B

rothers Karam

azov

on th

e scale of length

. Th

is relation also h

olds in (47), so (46

) is true.

Th

is discussion

of (45) and (46

) illustrate an

other im

portant aspect of th

e analysis,

nam

ely that it explain

s the validity of con

struction

s like (50).

(50)

The Idiot is sh

orter than

The B

rothers Karam

azov iff The B

rothers Karam

azov is longer

than

The Idiot.

(50) can

be paraphrased in

the follow

ing w

ay: “the exten

t to wh

ich T

he Idiot is short exceeds

the exten

t to wh

ich T

he Brothers K

aramazov is sh

ort if and on

ly if the exten

t to wh

ich T

he

Brothers K

aramazov is lon

g exceeds the exten

t to wh

ich T

he Idiot is long”. M

ore generally,

206

206

statemen

ts like (50) can

be viewed as su

bstitution

instan

ces of (51), wh

ere ϕpos an

d ϕneg are

anton

ymou

s gradable adjectives.

(51)ϕ

pos (a) > ϕpos (b) iff ϕ

neg (b) > ϕn

eg (a)

Th

e validity of (51) follows from

the claim

that th

e positive and n

egative projections of an

object on a scale are join

complem

entary. In

section 3.1.4, I claim

ed that th

e positive and

negative projection

s of an object on

to a scale along dim

ension

δ, posδ (a) an

d neg

δ (a),

correspond to (30

) and (31), repeated below

, wh

ere d is a fun

ction from

an object to a poin

t

on th

e scale.

(30)

posδ (a) = {p ∈

Sδ | p ≤ d(a)}

(31)n

egδ (a) = {p ∈

Sδ | d(a) ≤ p}

From

these defin

itions, w

e can defin

e the com

plemen

ts of positive and n

egative extents as

follows:

(52)–

neg

δ (x) = posδ (x) - {d(x)}

(53)–

posδ (x) = n

egδ (x) - {d(x)}

Retu

rnin

g to (51), since th

e result of applyin

g ϕpos an

d ϕneg to an

object is a positive or

negative exten

ts on th

e scale shared by ϕ

pos and ϕ

neg , we can

rewrite (51) as th

e more gen

eral

statemen

t in (54), an

d use th

e equivalen

ces in (52) an

d (53) to prove its validity.

(54)

posδ (a) > po

sδ (b) iff neg

δ (b) > neg

δ (a)

If posδ (a) > po

sδ (b), then

posδ (a) - {d(a)} > po

sδ (b) - {d(b)}, since d(a) an

d d(b) are the

207

207

maxim

al elemen

ts of pos(a) an

d pos(b), respectively. It follow

s that –

neg

δ (a) > –neg

δ (b),

by substitu

tion, an

d finally th

at neg

δ (b) > neg

δ (a), by contraposition

. Th

e other direction

of

the bicon

ditional can

be proved in exactly th

e same w

ay.

3.1.7 Cross-P

olar An

omaly R

evisited

We are n

ow in

a position to explain

cross-polar anom

aly. Recall th

at in section

3.1.3 I

investigated th

e possibility of an explan

ation of th

is anom

aly with

in a degree algebra in

terms

of incom

men

surability. T

he basic idea beh

ind th

is type of explanation

is that degrees of e.g.

tallness and shortness, like degrees of e.g. tallness an

d cleverness, shou

ld be analyzed as distin

ct

objects in differen

t ordered sets. If this w

ere the case, th

en th

e relation in

troduced by th

e

comparative m

orphem

e wou

ld be un

defined, an

d cross-polar anom

aly could be explain

ed in

the sam

e way as in

comm

ensu

rability. I observed that th

is explanation

is un

available if

degrees are defined as poin

ts on a scale, becau

se the algebra of degrees does n

ot make a

distinction

between

positive and n

egative degrees: for any scale, th

e set of positive degrees

and th

e set of negative degrees are isom

orphic. A

s a result, an

y expression w

hich

is a

substitu

tion in

stance of (23) or (24) (repeated below

), wh

ich sch

ematically represen

t the

interpretation

of examples of cross-polar an

omaly, sh

ould be perfectly in

terpretable.

(23)d

eg(d

pos )(dn

eg )

(24)

deg

(dneg )(d

pos )

In con

trast, the algebra of exten

ts makes a stru

ctural distin

ction betw

een positive an

d

negative exten

ts: positive and n

egative extents are differen

t sorts of objects, wh

ich

represent com

plemen

tary perspectives on th

e projection of an

object on a scale. T

his

distinction

was u

sed to characterize adjectival polarity as a sortal distin

ction: positive

adjectives denote fu

nction

s from objects to positive exten

ts; negative adjectives den

ote

208

208

fun

ctions from

objects to negative exten

ts. Th

e crucial con

sequen

ce of these assu

mption

s

is that th

e range of a positive adjective is disjoin

t from th

e range of its n

egative coun

terpart.

If the sets of positive an

d negative exten

ts are disjoint, th

en an

y expression th

at is a

substitu

tion in

stance of (55) or (56

), wh

ere deg

is an orderin

g on exten

ts and epos an

d epos are

positive and n

egative extents, w

ill fail to return

a truth

value, sin

ce the orderin

g relation

introdu

ced by deg

will be u

ndefin

ed.

(55)d

eg(epos )(en

eg )

(56)

deg

(epos )(eneg )

Th

e end resu

lt is that w

ithin

the algebra of exten

ts, cross-polar anom

aly can be explain

ed in

the sam

e terms as in

comm

ensu

rability: the degree relation

introdu

ced by the com

parative

morph

eme is u

ndefin

ed for the com

pared extents.

For illu

stration, con

sider an exam

ple like (57), wh

ich h

as the logical represen

tation in

(58).

(57)#

The B

rothers Karam

azov is longer th

an T

he Dream

of a Ridiculous M

an is short.

(58)

mo

re(long(The B

rothers K))(m

ax(λe[abs(short(T

he Dream

of a Ridiculous M

an))(e)]))

Th

e denotation

of the com

parative clause in

(58) is th

e maxim

al extent e su

ch th

at The

Dream

of a Ridiculous M

an is at least as short as e, w

hich

is a negative exten

t. Th

e reference

value, h

owever, is derived by applyin

g the adjective long to th

e denotation

of The B

rothers K,

return

ing th

e positive extent of T

he Brothers K

’s length

. Sin

ce positive and n

egative extents

come from

disjoint sets, th

e relation in

troduced by m

ore is u

ndefin

ed for its argum

ents,

and th

e structu

re is anom

alous.

Th

e same explan

ation applies to sen

tences in

wh

ich th

e polar adjectives are reversed,

such

as (59).

209

209

(59)

#T

he Dream

of a Ridiculous M

an is shorter th

an T

he Brothers K

aramazov is lon

g.

(60

)m

ore(short(T

he Dream

of a Ridiculous M

an))(max(λ

e[abs(long(The B

rothers K))(e)]))

In th

is example, th

e standard valu

e is a positive extent–th

e maxim

al extent e su

ch th

at The

Brothers K

aramazov is at least as lon

g as e–and th

e reference valu

e is a negative exten

t–the

negative projection

of The D

ream of a R

idiculou

s Man

on a scale of len

gth. A

gain, th

e

argum

ents su

pplied to the com

parative morph

eme com

e from disjoin

t sets, so the

comparison

relation is u

ndefin

ed, and th

e structu

re is anom

alous.

It shou

ld be observed that th

e analysis of cross-polar an

omaly th

at I have ou

tlined

here m

akes a more gen

eral prediction: if su

bdeletion stru

ctures are in

terpreted by directly

supplyin

g the den

otation of th

e comparative clau

se as the stan

dard value, th

en an

y instan

ce

of comparative su

bdeletion in

wh

ich th

e adjective in th

e main

clause is of differen

t polarity

from th

e adjective in th

e than-clause sh

ould be an

omalou

s, not ju

st examples in

volving

anton

ymou

s pairs such

as (15)-(18). Th

e followin

g min

imal pairs verify th

is prediction.

(61)

Un

fortun

ately, the ficu

s turn

ed out to be taller th

an th

e ceiling w

as high

.

(62)

#Un

fortun

ately, the ficu

s turn

ed out to be taller th

an th

e ceiling w

as low.

(63)

Luckily, th

e ficus tu

rned ou

t to be shorter th

an th

e doorway w

as low.

(64

)#Lu

ckily, the ficu

s turn

ed out to be sh

orter than

the doorw

ay was h

igh.

Th

e logical representation

of the com

parative in (6

2) is (65).

(65)

mo

re(tall(the ficus)(m

ax(λe[abs(low

(the ceiling))(e)]))

(62) is an

omalou

s for the sam

e reason as th

e examples discu

ssed above: the degrees

introdu

ced by the argu

men

ts of mo

re are not m

embers of th

e same ordered set, th

erefore

210210

the com

parative relation is u

ndefin

ed. Cross-polar an

omaly is th

us a m

ore general

phen

omen

on th

an th

e original facts su

ggest: it does not requ

ire that th

e two adjectives in

the com

parative are mem

bers of a specific positive-negative pair, on

ly that th

ey are of

different polarity. In

deed, the an

alysis proposed here predicts th

at this sh

ould be th

e case.

To su

mm

arize, the explan

ation of cross-polar an

omaly ou

tlined h

ere is available

because th

e algebra of extents perm

its a sortal distinction

between

positive and n

egative

extents to be m

ade at a very basic, structu

ral level. More im

portantly, th

e fact that th

is

anom

aly is observed in th

e first place provides support for th

e hypoth

esis that “degrees”

shou

ld be characterized as in

tervals on a scale, rath

er than

as points on

a scale. If adjectival

polarity is represented as a sortal distin

ction betw

een positive an

d negative adjectives, an

d if

positive and n

egative extents are distin

ct objects, then

we predict th

at comparatives

constru

cted out of polar opposites sh

ould be an

omalou

s. Th

at is, any form

ula w

hich

is a

substitu

tion in

stance of (55) an

d (56) sh

ould fail to retu

rn a tru

th valu

e, because th

e relation

introdu

ced by deg

will be u

ndefin

ed. Ultim

ately, the cru

cial compon

ent of th

e analysis is

the distin

ction betw

een positive an

d negative exten

ts provided by the exten

t algebra, as

similar resu

lts can be derived u

sing altern

ative approaches to th

e seman

tics of comparatives.

For exam

ple, in K

enn

edy 199

7b I assum

e a relational sem

antics for gradable adjectives an

d a

quan

tificational an

alysis of comparatives, an

d I present an

accoun

t of cross-polar which is very

similar to th

e one proposed h

ere. Th

e end resu

lt, then

, is that th

e phen

omen

on of cross-

polar anom

aly provides a compellin

g empirical argu

men

t for formalizin

g “degrees” as

intervals on

a scale–i.e., as extents–rath

er than

as points on

a scale, and for adoptin

g the

sortal characterization

of adjectival polarity described in th

is section (an

d advocated in Seu

ren

1978, 19

84, von S

techow

1984b, an

d Löbner 19

90

).

3.1.8 The D

istribution of M

easure Phrases

As observed by von

Stech

ow (19

84b), the algebra of exten

ts also provides an explan

ation for

211211

the con

trast between

examples like (6

6) an

d (67).

(66

)M

y Cadillac is 8 feet lon

g.

(67)

#My F

iat is 5 feet short.

If we assu

me th

at measu

re phrases den

ote boun

ded extents (see th

e discussion

of

comparison

of deviation in

section 3.2), an

d that dim

ension

al adjectives such

as long, tall, and

wide are associated w

ith scales w

ith a m

inim

al elemen

t, then

this con

trast can be explain

ed

in th

e same w

ay as cross-polar anom

aly: the orderin

g relation in

troduced by th

e absolute

morph

eme is u

ndefin

ed in (6

7) but n

ot in (6

6).

Th

e explanation

run

s as follows. If th

e scale of length

has a m

inim

al elemen

t, then

according to th

e definition

s of scales and positive an

d negative exten

ts in section

3.1.5,

positive extents on

the scale of len

gth are bou

nded bu

t negative exten

ts on th

e scale of

length

are not (becau

se scales have n

o maxim

al elemen

t). It follows th

at the objects in

the

range of long are th

e sort of extents th

at can be referred to by a m

easure ph

rase, but objects

in th

e range of short are n

ot. It follows th

at the logical represen

tation of (6

6), sh

own

in (6

8),

is interpretable, becau

se the partial orderin

g between

the positive exten

t of my C

adillac’s

longn

ess and th

e extent den

oted by 8 feet is defined.

(68

)abs(long(m

y Cadillac))(8 feet)

In con

trast, since th

e negative exten

t of my F

iat’s shortn

ess and th

e boun

ded extent

denoted by 4 feet are objects in

disjoint sets, th

e ordering in

troduced by th

e absolute in

the

logical representation

of (67), sh

own

below in

(69

), is not defin

ed.

(69

)abs(short(m

y Fiat))(5 feet)

212212

Becau

se the orderin

g relation in

(69

) is un

defined, (6

7) is correctly predicted to be

anom

alous.

If this an

alysis is correct, then

it suggests an

explanation

for the fact th

at wh

ile (67) is

anom

alous, (70

) is not.

(70)

My fiat is sh

orter than

8 feet.

Stru

cturally, (70

) is a phrasal com

parative. Given

the an

alysis of phrasal com

paratives

presented in

chapter 2, th

e logical representation

of (70) sh

ould be (71).

(71)m

ore(short(m

y Fiat))(short(5 feet))

Th

e crucial differen

ce between

(71) and (6

9) is th

at the stan

dard value in

the latter is

provided directly by the m

easure ph

rase, wh

ile the stan

dard value in

the form

er is derived by

applying th

e adjective short to the m

easure ph

rase. If it can be sh

own

that th

e result of th

is

operation is th

e negative exten

t that ran

ges from th

e point on

the scale correspon

ding to 5

feet to the u

pper end of th

e scale, then

we w

ill have an

explanation

for this con

trast. I will

leave this as a poin

t for futu

re investigation

.

3.1.9 P

ositives that Look Like Negatives

Ch

ris Barker (person

al comm

un

ication) observes th

at senten

ces like (72)-(74) appear to be

coun

terexamples to th

e claim th

at comparatives form

ed out of positive an

d negative pairs of

adjectives are anom

alous.

(72)Y

our C

is sharper th

an you

r D is flat.

(73)M

y watch

is faster than

your w

atch is slow

.

213213

(74)

She w

as earlier than

I was late.

If the adjectives in

(72)-(74) are actual positive-n

egative pairs, then

the an

alysis outlin

ed in

the previou

s sections in

correctly predicts that th

ese senten

ces shou

ld be anom

alous. T

here

are at least three reason

s to believe that th

e adjectives in (72)-(74) are n

ot positive-negative

pairs, how

ever, but rath

er pairs of positive adjectives. If this is tru

e, then

they are

comparable n

ot to examples of cross-polar an

omaly, bu

t rather to typical exam

ples of

comparative su

bdeletion (su

ch as (19

) and (20

) above), and so are n

ot coun

terexamples to

the claim

s made in

this section

.

T

he first argu

men

t involves in

terpretation. (72)-(74) are n

onan

omalou

s only if th

e

adjectives are interpreted in

such

a way as to m

easure th

e divergence from

some com

mon

point of referen

ce–a “pure ton

e” in (72), th

e “actual tim

e” in (73), an

d “on tim

e” in (74).

Th

is is particularly clear in

the case of (73), w

hich

can on

ly mean

that th

e extent to w

hich

my

watch

is faster than

the actu

al time is greater th

an th

e extent to w

hich

your w

atch is slow

er

than

the actu

al time. (73) can

not be u

sed to make th

e claim th

at my w

atch is m

ore

temporally advan

ced than

yours, w

hich

wou

ld be true in

a situation

in w

hich

both of ou

r

watch

es were slow

. An

interpretation

of (73) along th

ese lines is an

omalou

s in th

e same w

ay

that (75) is:

(75)#M

y car is faster than

your car is slow

.

Th

e second argu

men

t comes from

the distribu

tion of overt m

easure ph

rases. A

characteristic of n

egative adjectives is that th

ey cann

ot be modified by overt m

easure ph

rases

(see section 3.1.8):

(76)

#Mr. R

eich is 5 feet sh

ort.

(77)#M

aureen

was drivin

g 14 mph

slow.

214214

Both

of the adjectives in

each of (72)-(74) perm

it measu

re phrases, h

owever, w

hen

they are

interpreted in

the w

ay described above, so that th

ey measu

re divergence from

a reference

point. (78)-(80

) illustrate th

is point.

(78)

You

r C is 30 H

z flat/sharp.

(79)

My w

atch is 10 m

inutes fast/slow.

(80

)Sh

e was an hour early/late.

Th

e third argu

men

t makes u

se of the observation

that statem

ents w

hose logical

forms are su

bstitution

instan

ces of (51), repeated below, are valid for an

y positive-negative

pair of adjectives.

(51)ϕ

pos (a) > ϕpos (b) iff ϕ

neg (b) > ϕn

eg (a)

If flat-sharp, fast-slow, an

d early-late, are actual positive-n

egative pairs, then

(81)-(83) shou

ld be

valid.

(81)

You

r A is sh

arper than

your D

iff your D

is flatter than

your A

.

(82)

My w

atch is faster th

an you

rs iff yours is slow

er than

min

e.

(83)

She w

as earlier than

I was iff I w

as later than

she w

as.

Th

ese statemen

ts are not valid, h

owever, on

the relevan

t interpretation

. Alth

ough

(83), for

example, is valid if late an

d early describe the relative tem

poral ordering of tw

o individu

als

(with

respect to some even

t), it is not valid if th

e adjectives describe deviation from

some

reference poin

t indicatin

g “on tim

e”. On

the latter in

terpretation, th

e first conju

nct w

ould

be true in

a context in

wh

ich w

e were both

early, but sh

e was earlier th

an I w

as, wh

ile the

215215

second con

jun

ct wou

ld be false, because n

either of u

s was late. T

his latter in

terpretation is

the on

e the adjectives m

ust h

ave in (74), h

owever: if th

ey are interpreted so th

at (83) valid,

(74) is anom

alous.

Th

e data discussed h

ere support th

e conclu

sion th

at both m

embers of th

e adjective

pairs in (72)-(74) are positive: adjectives like sharp an

d flat (as well as fast-slow

and early-late,

on th

e relevant in

terpretations), w

hich

measu

re divergence from

some referen

ce point (for

example, th

e point at w

hich

a tone is n

either sh

arp nor flat), n

ot only project th

eir

argum

ents on

to the sam

e scale, they defin

e the sam

e sorts of projections on

to a scale. 12” If

the adjectives in

(72)-(74) are sortally the sam

e, as the facts in

dicate, then

these sen

tences

do not represen

t coun

terexamples to th

e analysis developed in

this section

.

3.1.10 Sum

mary

Th

e extent approach

to the sem

antics of gradable adjectives con

sists of two prin

cipal claims.

First, gradable adjectives den

ote relations betw

een in

dividuals an

d extents. A

n exten

t is

defined as an

interval on

a scale, and a stru

ctural distin

ction is m

ade between

two sorts of

extents: positive exten

ts and n

egative extents. S

econd, positive an

d negative adjectives are

distingu

ished by th

e sort of their exten

t argum

ents: positive adjectives den

ote relations

between

individu

als and positive exten

ts; negative adjectives den

ote relations betw

een

individu

als and n

egative extents.

Th

e discussion

so far has sh

own

this an

alysis to have several in

teresting resu

lts. It

12A n

um

ber of other in

teresting qu

estions m

ust be left for fu

ture w

ork. Tw

o in particu

lar

shou

ld be addressed. First, are th

e characteristics of th

e pairs in (72)-(74) tru

e of all adjectives wh

ich

do not m

ake reference to a stan

dard in th

e absolute form

? For exam

ple, the tru

th valu

e of that note

is sharp is not depen

dent on

context of u

tterance in

the w

ay that th

e truth

value of that note is loud

is:

the form

er is true as lon

g as the projection

of that note on a scale of “pitch

” is non

-nu

ll. Second, h

ow

shou

ld the am

biguity of an

adjective like slow, w

hich

has both

a “true” n

egative interpretation

(as in

(75)) and an

interpretation

in term

s of divergence from

a reference poin

t (as in (73)), be accou

nted

for?

216216

explains th

e validity of statemen

ts like (14), satisfyin

g the m

inim

al requirem

ent of

descriptive adequacy, an

d in addition

, it provides a un

iform accou

nt of th

e seman

tics of

positive and n

egative adjectives in both

the absolu

te and com

parative forms. M

ost

importan

tly, it provides a principled explan

ation of cross-polar an

omaly w

ithin

a general

analysis of in

comm

ensu

rability.

3.2 Com

parison of deviation

3.2.1 The Sem

antic C

haracteristics of Com

parison of D

eviation

As n

oted in ch

apter 1, any an

alysis of cross-polar anom

aly mu

st also accoun

t for the existen

ce

of comparatives in

volving positive an

d negative pairs of adjectives w

hich

are superficially

similar to exam

ples of cross-polar anom

aly but are n

ot anom

alous. (84)-(87) exem

plify

senten

ces of this type, w

hich

I referred to as “comparison

of deviation” con

struction

s.

(84

)W

illiam is as tall as R

obert is short.

(85)

Fran

cis is as reticent as H

ilary is long-w

inded.

(86

)San

Fran

cisco Bay is m

ore shallow

than

Mon

terey Bay is deep.

(87)

Th

e Ten

derloin is m

ore dirty than

Pacific H

eights is clean

.

Com

parison of deviation

and cross-polar an

omaly are lin

ked in th

e followin

g way: an

explanation

of cross-polar anom

aly mu

st not h

ave the con

sequen

ce that com

parison of

deviation sen

tences are predicted to be u

ngram

matical; sim

ilarly, it shou

ld not be th

e case

that an

accoun

t of comparison

of deviation perm

its non

-anom

alous in

terpretations of cross-

polar anom

aly. 13

In ch

apter 1 (section 1.1.4.3), I n

oted three im

portant ch

aracteristics distingu

ish th

e

13Recall from

the discu

ssion in

chapter 1, section

1.23, that th

is was a problem

for the vagu

e

predicate analysis.

217217

former from

the latter. T

he first differen

ce involves basic in

terpretation. U

nlike stan

dard

comparative con

struction

s, wh

ich com

pare the projection

s of two objects on

to a scale,

comparison

of deviation con

struction

s compare th

e extents to w

hich

two objects differ from

a relevant stan

dard value. T

his is m

ost clearly illustrated by th

e equative con

struction

(84).

(84) can on

ly mean

that th

e extent to w

hich

William

exceeds some stan

dard of tallness is

(relatively) the sam

e as the exten

t to wh

ich R

obert exceeds some stan

dard of shortn

ess; (84)

cann

ot mean

that W

illiam an

d Robert are equ

al in h

eight. T

his in

terpretation sh

ould be

contrasted w

ith th

at of a more typical exam

ple of equative su

bdeletion, su

ch as (88), in

wh

ich it is asserted th

at the h

eight an

d width

of the doorw

ay are the sam

e.

(88

)T

he doorway is as tall as it is w

ide.

Th

e comparatives in

(86) an

d (87) have sim

ilar interpretation

s. (86), for exam

ple, mean

s

that th

e extent to w

hich

the T

enderloin

exceeds a standard of dirtin

ess is greater than

the

extent to w

hich

Pacific H

eights exceeds a stan

dard of cleanlin

ess.

Th

at these sen

tences h

ave only in

terpretations of th

is sort is made clear by th

e

second differen

ce between

them

and stan

dard comparative con

struction

s. Wh

ereas

standard com

paratives do not en

tail that th

e property predicated of the com

pared objects is

true in

the absolu

te sense, com

parison of deviation

constru

ctions do carry th

is entailm

ent.

For exam

ple, (84

) entails th

at William

is tall and th

at Robert is sh

ort, but (8

8) en

tails

neith

er that th

e doorway is tall n

or that it is w

ide: this sen

tence cou

ld be truth

fully u

sed to

describe a one foot by on

e foot openin

g. Sim

ilarly, wh

ile (86) en

tails that S

an F

rancisco B

ay

is shallow

and M

onterey B

ay is deep, (89) does n

ot, thou

gh th

is inform

ation is con

veyed as a

cancelable im

plicature, as show

n by (9

0).

(89

)San

Fran

cisco Bay is sh

allower th

an M

onterey B

ay.

(90

)San

Fran

cisco Bay is sh

allower th

an M

onterey B

ay, thou

gh th

ey’re both fairly deep.

218218

In con

trast, denyin

g that S

an F

rancisco B

ay is shallow

after an u

tterance of (8

6) is

contradictory:

(91)

San

Fran

cisco Bay is m

ore shallow

than

Mon

terey Bay is deep. #S

an F

rancisco B

ay

isn’t sh

allow, th

ough

.

Fin

ally, comparison

of deviation in

terpretations do n

ot license m

orphological

incorporation

of the adjective an

d the com

parative morph

eme. (86

) and (87) con

trast with

(92) an

d (93), respectively, w

hich

are instan

ces of cross-polar anom

aly.

(92)

#San F

rancisco B

ay is shallow

er than

Mon

terey Bay is deep.

(93)

#Th

e Ten

derloin is dirtier th

an P

acific Heigh

ts is clean.

On

the su

rface, comparison

of deviation con

struction

s appear to be problematic for

the an

alysis of cross-polar anom

aly developed in section

3.1, wh

ich predicted th

at any

comparative con

structed ou

t of a positive and n

egative pair of adjectives shou

ld trigger

incom

men

surability. O

n closer in

spection, h

owever, com

parison of deviation

constru

ctions

like (84)-(87) actually provide in

teresting su

pport for the an

alysis of cross-polar anom

aly.

Con

sider, for example, th

e interpretation

of (84). Wh

at is crucial to observe is th

at this

senten

ce does not sim

ply permit th

e type of interpretation

discussed above, in

the com

pared

objects are the exten

ts to wh

ich th

e compared in

dividuals exceed appropriate stan

dards of

tallness an

d shortn

ess, (84) has only th

is type of interpretation

. In particu

lar, (84) cannot

mean

that th

e compared in

dividuals are (at least) equ

al in h

eight. T

his latter readin

g, wh

ich

wou

ld correspond to th

e standard in

terpretation of th

e equative con

struction

, is correctly

ruled ou

t by the an

alysis of cross-polar anom

aly developed above.

To see w

hy, con

sider the logical represen

tation of (84) on

this in

terpretation:

219219

(94

)as(tall(W

illiam))(m

ax(λe[abs(short(R

obert))(e)]))

Assu

min

g the tru

th con

ditions for th

e equative given

in section

3.1.6, th

e interpretation

of

(84) represented by (9

4) is ruled ou

t as an in

stance of cross-polar an

omaly: th

e argum

ents

of as are not elem

ents of th

e same ordered set, so th

e partial ordering relation

introdu

ced

by the degree m

orphem

e is un

defined. A

similar case can

be made for (85). T

he pu

zzle

that rem

ains to be solved is h

ow th

e comparison

of deviation in

terpretation arises in

the first

place. I will address th

is question

in th

e next section

.

3.2.2 Differen

tial Exten

ts and C

omparison

of Deviation

As n

oted above, the basic in

terpretation of com

parison of deviation

constru

ctions do n

ot

involve com

parison betw

een “absolu

te” extents (w

here an

absolute exten

t is the projection

of some object on

to a scale), but rath

er comparison

between

the exten

ts to wh

ich tw

o

objects each exceed an

appropriate standard. C

onsider, for exam

ple, (84), repeated below, in

a context in

wh

ich th

e positive projection of W

illiam on

a scale of heigh

t is tall(w) an

d the

negative projection

of Robert on

a scale of heigh

t is short(r), as show

n in

(95).

(84

)W

illiam is as tall as R

obert is short.

(95)

Heig

ht:

0 -

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

> ∞

tall(w

):0

-

--

--

--

--

stall -

--

--

--

-•-

- e1 -

-•

short(r): •

--

e2 -

-•-

--

--

--

--

- sshort

--

--

--

--

->

∞

If the stan

dards of tallness an

d shortn

ess in th

is context are stall an

d sshort , respectively, then

tall(w) = e

1 •stall , and short(r) = e2 •sshort , w

here “•” is a con

catenation

operation on

extents,

220220

defined in

terms of set differen

ce as in (9

6). 14

(96

)ed •e = ιe’[e’ −

e = ed ]

If absolute exten

ts can be partition

ed as in (9

5), then

e1 denotes th

e difference betw

een

tall(w) an

d stall , and e2 den

otes the differen

ce between

short(r) and sshort . T

hat is, e1 an

d e2

represent th

e extents to w

hich

Bradley an

d Reich

exceed standards of talln

ess and

shortn

ess, respectively.

I wou

ld like to suggest th

at comparison

of deviation in

volves comparison

of such

“differential exten

ts”. Specifically, th

e comparison

of deviation in

terpretation of (8

4)

involves a relation

between

e1 and e2 in

(95). B

uildin

g on th

e standard in

terpretation of th

e

equative, th

e comparison

of deviation in

terpretation of (84) can

be characterized as in

(97):

(97)

||(84)|| = 1 iff e1 ≥e2 , w

here e1 •stall = tall(w

) and e2 •sshort = short(r).

Note th

at in order to evalu

ate the orderin

g relation in

(97), it m

ust be th

e case that th

ere is

a mappin

g between

e1 and e

2 and tw

o extents th

at share an

endpoin

t, e.g. e’1 and e’2 ,

respectively, in(9

8).

14Som

e kind of con

catenation

operation on

extents is in

dependen

tly required to in

terpret

“differential” com

paratives like (i) and (ii) (see von

Stechow

1984a, H

ellan 19

81).

(i)W

illiam is 2 feet taller th

an R

obert.

(ii)San

Fran

cisco Bay is over 150

fathom

s shallow

er than

Mon

terey Bay.

Intu

itively, a senten

ce like (i) asserts that th

e extent to w

hich

William

is tall exceeds the exten

t to

wh

ich R

obert is tall by the am

oun

t denoted by 2 feet.

221221

(98

)h

eigh

t:0

-

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

--

> ∞

•-

- e’1

--

•

•-

- e’2

-

-•

Th

is could be accom

plished in

the follow

ing w

ay. In addition

to the set of positive an

d

negative exten

ts on a scale S

δ , we can

define a set of bou

nded exten

ts: the set of bou

nded

proper intervals on

the scale. 15 A

ssum

ing th

at an explicit th

eory of measu

remen

t can be

overlaid onto th

e theory of scales (see K

lein 19

91 for som

e promisin

g initial proposals in

this

direction), w

e can m

ake the n

atural assu

mp

tion th

at all boun

ded extents of equ

al

measu

remen

t are equivalen

t (i.e., we can

group bou

nded exten

ts into equ

ivalence classes

based on th

eir measu

remen

ts). If this is th

e case, then

e1 and e2 in

(95) are equ

ivalent to e’1

and e’2 in

(98) respectively, an

d it follows th

at if e’1 ≥e’2 , th

en e1 ≥

e2 .

15Th

is assum

ption en

tails that for an

y scale with

a min

imal elem

ent, th

e set of positive

extents is a su

bset of the set of bou

nded exten

ts. If the h

ypothesis th

at measu

re phrases den

ote

boun

ded extents is correct (see section

3.1.8), w

e make th

e followin

g prediction: on

ly positive

adjectives that m

ap their argu

men

ts onto scales w

ith a m

inim

al elemen

t will perm

it measu

re

phrases, sin

ce only in

these cases w

ill the correspon

ding positive exten

ts be boun

ded. Th

is

prediction appears to be born

e out by w

ell-know

n con

trasts like (i) and (ii), assu

min

g that th

e scale

associated with w

arm (a scale alon

g a dimen

sion of “tem

perature”) h

as no low

er boun

d.

(i)#It’s 50

degrees warm

today.

(ii)T

he kitch

en is 50

degrees warm

er than

the basem

ent.

Th

e importan

ce of (ii) is its acceptability indicates th

at a standard of m

easurem

ent is defin

ed for the

scale onto w

hich

warm

maps its argu

men

ts, thu

s the u

nacceptability of (i) can

not be du

e to an

absence of su

ch a stan

dard (as could be argu

ed for an adjective like beau

tiful, for exam

ple). Even

thou

gh th

ere may be an

actual low

er boun

d on th

e temperatu

re of objects in th

e un

iverse, it seems

reasonable to assu

me th

at the scale on

to wh

ich th

e adjective warm

maps its argu

men

t does not h

ave

a lower bou

nd, given

the fact th

at a senten

ce such

as Absolute zero is w

armer than nothing is perfectly

interpretable.

222222

In order for th

e general approach

to comparison

of deviation ou

tlined h

ere to go

throu

gh, h

owever, it m

ust be sh

own

that com

parison of differen

tial extents in

examples

involvin

g p

ositive an

d

negative

adjectives

(such

as

(84

)-(87))

does

not

result

in

incom

men

surability. T

hat is, it m

ust be sh

own

that differen

tial extents are sortally th

e

same, regardless of w

heth

er they are su

bintervals of positive exten

ts or subin

tervals of

negative exten

ts, so that th

e comparison

relation is defin

ed in exam

ples like (84)-(87). In

fact, this resu

lt follows from

the defin

ition of exten

t concaten

ation in

(96

). Tw

o initial

condition

s are importan

t. First, sin

ce we are on

ly interested in

cases in w

hich

ed •e is the

argum

ent of a gradable predicate, it m

ust be th

e case that e’ is eith

er positive or negative,

therefore e’ m

ust be a proper exten

t. Secon

d, e and e’ m

ust be exten

ts of the sam

e sort:

both positive or both

negative. T

his follow

s from th

e definition

of an exten

t as a convex

subset of a totally ordered set of poin

ts: if e.g. e and e’ w

ere of opposite polarity, then

the

value of e’ - e, w

hich

equals –(e’ ∩

e), wou

ld not be con

vex, and th

erefore not an

extent.

Assu

min

g these in

itial condition

s, two qu

estions m

ust be an

swered: w

hat is th

e sort

of ed wh

en e is positive, an

d wh

at is the sort of ed w

hen

e is negative? In

the case in

wh

ich e

is positive, ed mu

st be a boun

ded extent (i.e., a proper su

bset of S - e), oth

erwise e’ w

ould n

ot

satisfy the con

dition th

at it be a proper extent. W

hen

e is negative, ed again

mu

st be

boun

ded in order to en

sure th

at e’ is a proper extent. T

he con

clusion

is that a differen

tial

extent is bou

nded w

hen

concaten

ated with

either a positive or a n

egative extent, a fact w

hich

I interpret as in

dicating th

at differential exten

ts are (structu

rally) of the sam

e sort. 16 Th

is is

consisten

t with

the h

ypothesis m

ention

ed above that all differen

tial extents are elem

ents of

16Eviden

ce in su

pport of this con

clusion

is provided by examples like (i) an

d (ii), wh

ich sh

ow

that differen

tial extents can

measu

re differences betw

een com

pared positive extents an

d differences

between

compared n

egative extents (cf. th

e anom

aly of measu

re phrases w

ith n

egative absolute

adjectives, discussed in

section 3.1.8).

(i)T

he plan

e was 5 kilom

eters high

er than

the h

elicopter.

(ii)T

he h

elicopter was 5 kilom

eters lower th

an th

e plane.

223223

the set of bou

nded exten

ts on a scale. If all differen

tial extents are m

embers of th

e same

set of boun

ded extents, an

d assum

ing th

at mappin

gs of the sort defin

ed above can be

determin

ed, it shou

ld be possible to establish an

ordering relation

between

any tw

o

differential exten

ts. Th

erefore the com

parison relation

associated with

comparison

of

deviation con

struction

s shou

ld be defined.

To su

mm

arize the discu

ssion so far, I h

ave suggested th

at comparison

of deviation

involves a relation

between

differential exten

ts, rather th

an a relation

between

absolute

extents, as in

standard com

parative/equative in

terpretations. C

omparison

of deviation is

possible in con

texts in w

hich

standard com

parison triggers in

comm

ensu

rability because

differential exten

ts are sortally the sam

e, regardless of the sort of th

e extent w

ith w

hich

they are con

catenated. A

positive result of th

is analysis is th

at it explains w

hy com

parison of

deviation in

terpretations h

ave the en

tailmen

ts they do. F

ocusin

g on (84), if tall(b) = e1 •stall ,

then

tall(b) ≥ stall . Accordin

g to the sem

antics of th

e absolute adopted in

chapter 2, it follow

s

that th

e statemen

t William

is tall is true. M

ore generally, if th

e differential exten

ts wh

ich

are compared in

comparison

of deviation con

struction

s denote th

e difference betw

een th

e

absolute exten

t of an object an

d an exten

t wh

ich den

otes an appropriate stan

dard, then

the

truth

condition

s for the absolu

te are indepen

dently satisfied. 17

17A fin

al question

remain

s un

answ

ered: wh

at is the differen

ce in com

positional sem

antics

between

standard com

parative constru

ctions an

d comparison

of deviation con

struction

s? Alth

ough

this is a qu

estion th

at I will h

ave to leave for futu

re work, a con

sideration of degree m

odifiers such

as

very, fairly and qu

ite suggests an

initially prom

ising an

swer to th

is question

. Like comparison

of

deviation con

struction

s, degree modifiers also in

volve comparison

of extents w

hich

measu

re the

difference betw

een th

e absolute projection

of an object on

a scale and a stan

dard value. C

onsider, for

example, a sen

tence like (i).

(i)W

illiam is very tall.

The m

eanin

g of (i) can be accu

rately and in

tuitively paraphrased as in

(ii).

(ii)T

he exten

t to wh

ich W

illiam exceeds a stan

dard of tallness is large.

224224

Before con

cludin

g this section

, it shou

ld be observed that com

parison of deviation

is

not u

niqu

e to comparatives in

volving polar opposites; it represen

ts a possible interpretation

of subdeletion

structu

res in gen

eral, as show

n by (9

9).

(99

)T

he B

ay Bridge is as lon

g as the E

mpire State B

uildin

g is tall.

(99

) is ambigu

ous betw

een a stan

dard equative in

terpretation, in

wh

ich it is asserted th

at

the len

gth of th

e Bay B

ridge equals th

e heigh

t of the E

mpire S

tate Bu

ilding (w

hich

is false),

and a com

parison of deviation

interpretation

, in w

hich

the tw

o objects are asserted to deviate

to a (relatively) equal exten

t from th

e appropriate standards of lon

gness an

d tallness (w

hich

is true). T

his am

biguity is predicted by th

e analysis of com

parison of deviation

that I h

ave

outlin

ed here. N

othin

g about th

e analysis restricts it to exam

ples involvin

g positive and

negative pairs of adjectives. A

s a result, in

examples w

hich

do not in

volve adjectives of

opposite polarity, such

as (99

), it shou

ld not on

ly be possible to constru

ct a comparison

of

deviation in

terpretation, it sh

ould also be possible to con

struct an

interpretation

in term

s of

Th

at (i) has su

ch a m

eanin

g is supported by its en

tailmen

ts: (i) clearly entails th

at William

is tall (the

same can

be said of e.g., William

is fairly tall, William

is quite tall). Note as w

ell that th

e negation

of

(i) does not en

tail that W

illiam is not tall, on

ly that th

e extent to w

hich

he exceeds a stan

dard of

tallness is not large. T

hat is, th

e negation

of (i) is consisten

t with

a situation

in w

hich

William

is

short, or on

e in w

hich

he’s ju

st not very tall, as sh

own

by (iii)-(iv).

(iii)W

illiam is tall, bu

t he’s not very tall.

(iv)W

illiam is n

ot very tall; in fact, h

e’s quite sh

ort.

Given

th

e sim

ilarity in

m

eanin

g betw

een

degree m

odifiers an

d com

parison

of

deviation

constru

ctions, a plau

sible hypoth

esis is that com

parison of deviation

indicates th

at the com

parative

morph

emes m

ore, less, and as h

ave, in addition

to their stan

dard interpretation

s as comparative

degree morph

emes, in

terpretations as degree m

odifiers like very and fairly, an

d it is the latter

interpretation

that is in

volved in com

parison of deviation

constru

ctions.

225225

the stan

dard mean

ing of th

e equative m

orphem

e. 18

3.2.3 Summ

ary

Th

e analysis of com

parison of deviation

proposed here claim

s that com

parison of deviation

constru

ctions differ from

standard com

paratives in th

at they in

volve comparison

of

differential exten

ts, rather th

an com

parison of th

e absolute projection

s of two objects on

a

scale. Differen

tial extents m

easure th

e difference betw

een th

e absolute projection

of an

object and a stan

dard value, an

d as a result, th

e truth

condition

s for the absolu

te are satisfied

wh

enever th

e truth

condition

s for the com

parison of deviation

constru

ction are satisfied.

Cru

cially, since all differen

tial extents are sortally th

e same, an

d so elemen

ts of the sam

e

ordered set (the set of bou

nded exten

ts), the com

parison relation

introdu

ced by comparison

of deviation con

struction

s is defined, an

d the sen

tences do n

ot trigger cross-polar anom

aly.

3.3 The Logical P

olarity of Gradable A

djectives

3.3.1 The M

onoton

icity Properties of P

olar Adjectives

In ch

apter 1 (section 1.1.4.1), I observed th

at negative polarity item

licensin

g and en

tailmen

t

patterns in

dicate that positive adjectives gen

erate mon

otone decreasin

g contexts, w

hile

negative adjectives gen

erate mon

otone in

creasing con

texts (see Seu

ren 19

78, Ladu

saw

1979

, Linebarger 19

80, Sán

chez-V

alencia 19

96

). (100

)-(107) review

the cru

cial data.

18Note th

at a comp

arative version of (9

9) also p

ermits a com

parison

of deviation

interpretation

if incorporation

does not occu

r:

(i)T

he B

ay Bridge is m

ore long th

an th

e Em

pire State B

uildin

g is tall.

226226

(100

)It is difficu

lt/*easy for him

to admit th

at he h

as ever been w

rong.

(101)

It wou

ld be foolish/*clever of h

er to even bother to lift a finger to help.

(102)

It is strange/*typical th

at any of those papers w

ere accepted.

(103)

It’s lame/*cool th

at you even h

ave to talk to any of these people at all.

(104

)It’s dan

gerous to drive in

Rom

e. ⇒ It’s dan

gerous to drive fast in

Rom

e.

(105)

It’s safe to drive in D

es Moin

es. ⇐It’s safe to drive fast in

Des M

oines.

(106

)It’s stran

ge to see Fran

ces playing electric gu

itar. ⇒ It’s stran

ge to see Fran

ces

playing electric gu

itar poorly.

(107)

It’s comm

on to see F

rances playin

g electric guitar. ⇐

It’s comm

on to see F

rances

playing electric gu

itar poorly.

Mon

otonicity properties represen

t one of several factors w

hich

have tradition

ally been u

sed

to classify gradable adjectives according to th

eir “logical polarity” (in th

e sense of H

. Klein

199

6): adjectives w

hich

license n

egative polarity items an

d down

ward en

tailmen

ts in clau

sal

complem

ents, su

ch as difficu

lt and strange, are classified as n

egative, wh

ile adjectives wh

ich

do not licen

se negative polarity item

s but do perm

it upw

ard inferen

ces, such

as easy and

comm

on, are classified as positive (see Seu

ren 19

78 for early discussion

of this issu

e). Th

e

goal of this section

is to show

that th

e mon

otonicity properties of polar adjectives follow

directly from th

e theory of polarity developed in

section 3.1, in

wh

ich polar adjectives are

sorted according to th

eir range: positive adjectives den

ote fun

ctions from

individu

als to

positive extents; n

egative adjectives denote fu

nction

s from in

dividuals to n

egative extents.

3.3.2 Degrees an

d Mon

otonicity

Recall from

the discu

ssion in

section 3.1 th

at logical polarity (in th

e sense discu

ssed above) is

represented in

a degree algebra as a distinction

in th

e range of gradable adjectives. A

s

defined above, a scale S

δ is a linearly ordered set of degrees alon

g some dim

ension

. If we

227227

assum

e that a positive adjective preserves th

e order on S

δ , wh

ile a negative on

e reverses it,

then

for any scale S

δ and an

y pair of positive and n

egative adjectives associated with

Sδ , th

e

set of degrees associated with

the positive D

pos and th

e set of degrees associated with

the

negative D

neg stan

d in th

e dual relation

. Dpos an

d Dn

eg are differentiated by orderin

g, but

their m

embersh

ip is the sam

e. Moreover, sin

ce there is a on

e-to-one m

apping betw

een th

e

two sets (th

e identity fu

nction

), Dpos an

d Dn

eg are isomorph

ic. A positive resu

lt of this

analysis is th

at it predicts that statem

ents like (10

8) are valid.

(108

)T

he Dream

of a Ridicu

lous M

an is shorter th

an T

he Brothers K

aramazov if an

d only if

The B

rothers Karam

azov is longer th

an T

he Dream

of a Ridiculous M

an.

Degrees of lon

gness an

d shortn

ess are the sam

e objects, therefore, given

the differen

ce in

the orderin

g relations associated w

ith positive an

d negative adjectives, th

e two con

jun

cts in

(108) are tru

e in exactly th

e same situ

ations.

Th

is representation

of logical polarity has an

additional positive resu

lt: it entails th

at

negative adjectives are m

onoton

e decreasing. T

o see wh

y, consider an

arbitrary case of

ordering alon

g a dimen

sion, for exam

ple, safety. If b is safer than

a, then

the relation

in

(109

) holds.

(109

)a <

safety b

If the orderin

g relation associated w

ith a n

egative adjectives is the du

al of the relation

associated with

its positive coun

terpart, then

given th

e analysis of gradable adjectives as

fun

ctions th

at map in

dividuals to degrees, w

hen

ever (110) h

olds, (111) also holds.

(110)

safe(a) < safe(b)

(111)dangerou

s(b) < dangerous(a)

228228

Given

the defin

itions in

(112), it follows th

at positive adjectives denote m

onoton

e increasin

g

fun

ctions an

d negative adjectives den

ote mon

otone decreasin

g fun

ctions.

(112)a.

A fu

nction

f is monotone increasing iff: a < b →

f(a) < f(b)

b.A

fun

ction f is m

onotone decreasing iff: a < b → f(b) < f(a)

Alth

ough

this is a positive resu

lt, it mu

st be acknow

ledged that th

e mon

otonicity of

negative adjectives does n

ot follow from

an in

dependen

t aspect of the algebra of degrees.

Rath

er, it is a definition

al property of negative adjectives: th

at negative adjectives are scale

reversing is assu

med in

order to constru

ct a seman

tics wh

ich correctly accou

nts for th

e

interpretation

s of positive and n

egative adjectives in th

e absolute an

d comparative form

s.

Noth

ing abou

t the system

itself requires n

egatives to be mon

otone decreasin

g; rather, it is

the data w

hich

force this assu

mption

. An

alternative situ

ation w

ould be on

e in w

hich

indepen

dently m

otivated characteristics of th

e ontology h

ave the addition

al consequ

ence th

at

negative adjectives are scale reversin

g. Th

e discussion

of cross-polar anom

aly in section

3.1

demon

strated that th

e distinction

between

positive and n

egative extents provided by an

extent algebra is in

dependen

tly motivated, as it su

pports an explan

ation of cross-polar

anom

aly (someth

ing th

e algebra of degrees fails to do). In th

e followin

g section, I w

ill show

that th

is distinction

has th

e additional positive resu

lt of deriving th

e mon

otonicity properties

of gradable adjectives.

3.3.3 Exten

ts and M

onoton

icity

Th

e conclu

sion of section

3.1 was th

at scalar values (“degrees”) sh

ould be form

alized in

terms of in

tervals of a scale, as originally proposed by S

euren

(1978) (see also von

Stech

ow

1984b an

d Löbner 19

90

), and polar opposition

shou

ld be represented as a sortal distin

ction

229229

between

positive and n

egative adjectives: positive adjectives denote fu

nction

s wh

ose range is

the set of positive exten

ts Epos ; n

egative adjectives denote fu

nction

s wh

ose range is th

e set of

negative exten

ts En

eg . In addition

to supportin

g a principled explan

ation of cross-polar

anom

aly, this sortal ch

aracterization of adjectival polarity h

as the positive con

sequen

ce of

entailin

g that positive adjectives are m

onoton

e increasin

g fun

ctions an

d negative adjectives

are mon

otone decreasin

g fun

ctions.

Con

sider again th

e case of an orderin

g along a dim

ension

of safety, in w

hich

an object

a is ordered below an

object b, as in (113)

(113)a <

safety b

Wh

en th

e relation betw

een a an

d b show

n in

(113) holds, th

en th

e values of safe(a) an

d

safe(b), and dangerous(a) an

d dangerous(b) (i.e., the positive an

d negative exten

ts of a and b on

the scale of safety) are as sh

own

in (114).

(114)

safety:0

--

--

--

--

--

-a

--

--

--

--

-b

--

--

--

--

--

--

->

∞

safe(a):0

--

--

--

--

--

-•

safe(b):0

--

--

--

--

--

--

--

--

--

--

-•

dangerous(a): •

--

--

--

--

--

--

--

--

--

--

--

->

∞

dangerous(b):

•-

--

--

--

--

--

--

> ∞

Assu

min

g a standard B

oolean orderin

g on exten

ts as in (115), w

hen

ever (114) holds, th

e

relations in

(116) an

d (117) also obtain.

(115)[e < e’] iff [e ∩

e’ = e and e ≠ e’]

(116)

safe(a) < safe(b)

(117)dangerou

s(b) < dangerous(a)

230230

Th

e positive adjective safe preserves the orderin

g on a an

d b, but th

e negative adjective

dangerou

s reverses it. Th

erefore safe is mon

otone in

creasing, an

d dangerou

s is mon

otone

decreasing. T

his resu

lt does not follow

from som

e property of safe and dangerous; rath

er it is

a general con

sequen

ce of the h

ypothesis th

at logical polarity is represented as a sortal

distinction

between

gradable adjectives. Wh

at distingu

ishes th

is result from

the on

e

obtained in

the degree approach

is that it follow

s directly from th

e algebra of extents, n

ot

from prior assu

mption

s about im

plicit ordering relation

s associated with

the adjectives safe

and dangerous. 19

3.3.4 Summ

ary

Takin

g differences in

mon

otonicity properties of gradable adjectives as a startin

g point, th

is

section con

sidered two altern

ative representation

s of logical polarity. Th

e degree analysis

makes th

e assum

ption th

at negative adjectives are scale reversin

g at a basic level; rather th

an

explainin

g wh

y negative adjectives are m

onoton

e decreasing, it defin

es them

as such

. Th

e

extent approach

, on th

e other h

and, derives th

e mon

otonicity properties of polar adjectives

from th

e sortal distinction

between

positive and n

egative adjectives provided by the algebra

of extents. T

his approach

is indepen

dently m

otivated because, u

nlike th

e degree algebra, it

supports an

explanation

of cross-polar anom

aly.

It is worth

notin

g that th

e proposals I have m

ade here do n

ot predict wh

ether an

adjective is going to be positive or n

egative–nor are th

ey inten

ded to. Th

e goal is rather to

present a form

al representation

of adjectival polarity wh

ich accou

nts for th

e logical

properties of gradable adjectives in as robu

st and explan

atorily adequate a w

ay as possible.

19If the n

omin

al determin

ers many an

d few are an

alyzed as positive and n

egative gradable

predicates, respectively (Klein

1980

, see also Sapir 19

44), then

it shou

ld be possible to explain th

eir

mon

otonicity properties in

the sam

e way.

231231

Th

e facts I have con

sidered in th

is paper show

that a form

al system in

wh

ich polarity is

represented as a sortal distin

ction betw

een gradable adjectives ach

ieves this goal.

3.4 Con

clusion

Th

e pu

zzle wh

ich form

ed the startin

g poin

t for this ch

apter w

as the an

omaly of

comparatives form

ed of positive and n

egative pairs of adjectives. Observin

g that th

is

anom

aly is un

explained if degrees are an

alyzed as points on

a scale, I claimed th

at degrees

shou

ld instead be form

alized as intervals on

a scale, or extents, an

d I made a stru

ctural

distinction

between

two sorts of exten

ts: positive extents, w

hich

range from

the low

er end

of the scale to som

e positive point, an

d negative exten

ts, wh

ich ran

ge from som

e point to

the u

pper end of th

e scale. Adjectival polarity w

as then

characterized as a sortal distin

ction

between

positive and n

egative adjectives: positive adjectives denote fu

nction

s from objects to

positive extents; n

egative adjectives denote fu

nction

s from objects to n

egative extents. T

his

approach n

ot only su

pported a principled an

alysis of cross-polar anom

aly, it also explained th

e

different relation

al characteristics of positive an

d negative adjectives, an

d provided the basis

for an explan

ation of th

e interpretation

of comparison

of deviation con

struction

s. Fin

ally,

the algebra of exten

ts and th

e sortal characterization

of adjectival polarity has th

e additional

positive result of en

tailing th

at positive adjectives denote m

onoton

e increasin

g fun

ctions an

d

negative adjectives den

ote mon

otone decreasin

g fun

ctions.

232232