Three sources of vagueness in degree constructions

Galit W. Sassoon, Ben Gurion University of the Negev

Since Russell (1905), semanticists often characterize gradable predicates as mapping entities

to real numbers r (Kennedy 1999). The mapping is additive wrt a dimension (Klein 1991).

For example, the degree function of long, f

long

, is 'additive wrt length'. It represents ratios between

quantities of length in entities – the fact that the length of the concatenation (placing end to end)

of any two entities d

1

and d

2

(symbolized as d

1

length

d

2

) equals the sum of lengths of the two

separate entities (f

long

(d

1

length

d

2

) = f

long

(d

1

) + f

long

(d

2

)). This analysis provides straightforward

semantic accounts of numerical degree predicates (NDPs; like 2 meters tall) ratio predicates (like

twice as happy as Sam), and difference predicates (2 meters shorter).

Yet, many predicates don't license NDPs (#two meters short; #two degrees warm /beautiful

/happy), rendering the numerical analysis unintuitive. Moreover, there is much indeterminacy

concerning the (presumed) mapping of entities to numbers. Given the real interval [0,1], why

would one have a degree 0.25 rather than say 0.242 in

happy? (Kamp and Partee 1995); which set

of real numbers forms the degrees of happy? Moltmann (2006) concludes that only the few

predicates that license NDPs map entities to numbers. Conversely, I propose that any gradable

predicate (including

happy) maps entities to numbers, but no mapping (including that of tall!) is

fully specified, resulting in a limited distribution of NDPs, ratio-modifiers, and unit names. Let

me explain these claims in more details.

Let the set W

c

consist of worlds that given the knowledge in some actual context c (the

common knowledge of some community of speakers out of the blue) may still be the actual world

(Stalnaker 1975). We cannot count directly quantities of the 'stuff' denoted by mass nouns

(

height, heat, happiness). These quantities have no known values (like 1,2,3,..) Thus, objects d

with a non-zero quantity of height (say, the meter) should be mapped to different numerals in

different worlds (

w

1

,w

2

W

c

: f

tall,w1

(d) f

tall,w2

(d)). Still, meter rulers tell us the ratios between

entities' heights, and in any w, f

tall,w

represents these ratios (in every wW

c

, entities with n times

d's height are mapped to the numeral nf

tall,w

(d)). All tall's functions in W

c

, then, yield the same

ratios between entities' degrees (these ratios are known numbers). Let's call all objects, whose

height equals that of the meter, 'meter unit objects'. I propose that an entity d falls under NDPs

like 2 meters tall iff the ratio between d's degree in tall and the meter unit-objects' degree in tall,

r

m,w

, is 2 (wW

c

: f

tall,w

(d)=2r

m,w

). So it is not the case that Dan is 2 meters tall iff f

tall

maps

Dan to 2. The value to which f

tall

maps Dan is unknown (n: wW

c

, f

tall,w

([[Dan]]

w

)=n). We

feel that we have knowledge about entities' degrees in tall only because the following two

preconditions hold:

(i) The ratios between entities' degrees are known numbers (d

1

,d

2

, n: wW,

f

tall,w

(d

1

)=nf

tall,w

(d

2

)), and

(ii) There is an agreed-upon set of unit-objects s.t. any d is associated with a known

number representing the ratio between d's degree and the unit-objects' degree in tall.

Violations of (ii) : Lack of agreed-upon unit-objects

Consider happy or heavy (understood as feels heavy). Even if one speaker treats certain internal

states as unit-objects, no other speaker has access to these states. So no object d can be s.t. it

would be agreed-upon by all the community that d is a unit-object. My proposal predicts that the

lack of conventional unit-objects will prevent the possibility of determining numbers for entities.

This proposal is superior to non-numerical theories (cf. Moltmann 2006) because it accounts

for the compatibility of happy with ratio and difference modifiers. For example, the felicity of

Dan is twice as happy as Sam shows that the ratios between happiness degrees can be treated as

meaningful (it is true iff wW

c

: f

happy,w

([[Dan]]

w

)=2×f

happy,w

([[Sam]]

w

)). Generally, we don't

need to know entities' degrees, only the ordering or ratios between their potential degrees.

Violations of (i) : Lack of knowledge about ratios between degrees

While we may feel acknowledged of the ratios between, say, our degrees of happiness in

separate occasions, we can hardly ever feel acknowledged of the ratios between degrees of

entities in predicates like short. This is illustrated by the fact that ratio modifiers are less

acceptable with short than with tall or with long (as in Dan is twice as tall as Sam vs. #Dan is

twice as short as Sam, and as Google search-results show). In accordance, the present analysis

predicts that, in the lack of knowledge concerning ratios between degrees, numerical degree

predicates will not be licensed (as in

*two meters short).

Still, numerical degree predicates

are fine in the comparative (as in two meters shorter). In

actual contexts, we can positively say that Dan's degree in short is n meters bigger than Sam's iff

Sam's degree in tall is n meters bigger than Dan's. Elsewhere (Salt 18), I show that any function

that linearly reverses and linearly transforms the degrees of f

tall

can predict these facts. I.e., I

propose that for any wW

c

there is a constant Tran

short,w

, s.t. f

short,w

assigns any d the degree

(Tran

short,w

–f

tall,w

(d)) (so Dan is taller iff Sam is shorter); the transformation value, Tran

short

, is

unknown (n: wW

c

, Tran

tall,w

=n). Therefore, if in c tall maps some d to 2 meters

(wW

c

, f

tall,w

(d)=2r

m,w

), short maps d to Tran

short

–2 meters (f

short,w

(d)= Tran

short,w

– 2r

m,w

). So

in the lack of knowledge about Tran

short

(it varies across W

c

), we can't say which entities are 2

meters short in c (d: wW

c

, f

short,w

(d)=2r

m,w

). However, in computing degree-differences,

the transformation values cancel one another: wW

c

, d

2

is 2 meters taller than d

1

(f

tall,w

maps d

2

to some n and d

1

to n–2r

m,w

) iff wW

c

, d

1

is 2 meters shorter (f

short,w

maps d

2

to Tran

short,w

–

n and d

1

to Tran

short,w

–(n–2r

m,w

); the degree difference is still 2r

m,w

.) Thus, we can felicitously

say that entity-pairs fall, or don't fall, under 'two meters shorter'.

Last, but not least, my proposal is superior to other accounts of the licensing of numerical

degree predicates (cf. von Stechow 1984; Kennedy 1999), because it can capture facts pertaining

to positive predicates, like

warm. Positive predicates may have transformation values, too, which

(among other things) render, e.g., #

2 degrees warm, but not 2 degrees warmer, infelicitous.

Cross linguistic variations with respect to the licensing of numerical degree predicates is

expected, since languages may vary as to whether predicates like heavy or warm measure

external or internal states (or both), and whether the measure is transformed or not.

A third (but different) source of vagueness

I proposed that despite the fact that, e.g., f

tall,w

differs across worlds in W

c

, we have knowledge

about the ratios and ordering between entities' degrees in predicates like tall (so there is no

denotation-gap in predicates like two meters tall or taller). Similarly, in previous vagueness-

based gradability theories (Kamp 1975; Fine 1975), the denotation of taller does not vary across

valuations in a vagueness-model. Yet, sometimes we do not know the truth value of statements

like Dan is (two inches) taller than Sam. I submit that this vagueness is due to a different source.

I propose that individuals are distinguished by their property values (the values that the degree

functions assign to them). For instance, if the referent of Dan in w

1

is 1.87 meters tall, and the

referent of Dan in w

2

is 1.86 meters tall, I say (following Lewis 1986) that the name Dan refers to

two different individuals in these two worlds. However, if in w

1

and w

2

the referent of Dan is

1.87 meters tall, and identical in all the other property values, even if 1.87 counts as ‘tall’ in w

1

but not in w

2

, I still say (unlike Lewis 1986) that the name Dan denotes the same individual in

these two worlds (it is only our interpretation of the word tall that has changed). I do take

individuals to be real entities, identified with their ‘real’ properties. So it is invariably determined

for each two individuals in D what their heights are. However, when we use proper names, we do

not know exactly which individuals in D they refer to (since we do not know all of their property

values). When we do not know the heights of these individuals, we may easily not know how

their heights compare. If Dan's height is not accessible to me (its referent is 1.87m tall in w

1

,

1.86m tall in w

2

, etc.), I may not know whether Dan is taller than Sam it true or not.