Vagueness in degree constructions

Conference Paper (PDF Available) · January 2009with 20 Reads
Conference: Sinn und Bedeutung 13
Abstract
This paper presents a novel semantic analysis of unit names and gradable adjectives, inspired by measurement theory (Krantz et al 1971). Based on measurement theory's typology of measures, I claim that different predicates are associated with different types of measures whose special characteristics, together with features of the relations denoted by unit names, explain the puzzling limited distribution of measure phrases.
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.

Supplementary resources

  • Conference Paper
    Full-text available
    What should an adequate representation of individuals (elements of the domain of discourse) be like, within vagueness models with degrees? This paper explores the hypothesis that individuals are distinguished by their property values, i.e. the extents to which they satisfy gradable properties. First, as Lewis (1986) argues, cross-world identity is intuitively implausible between individuals differing in their property values, e.g., their height, weight, etc. ('intrinsic properties'). However, cross-world identity is intuitively plausible between individuals sharing the same property values (the same heights, weights, etc.) , even if they differ along extrinsic, relational properties, e.g., if they are considered 'tall' in one world, but not in another, due to variance in the cutoff point of tall across the two worlds. A representation of individuals by their property values captures the intuitively sharp distinction between these two cases. Second, this proposal captures the intuitive difference between cases we tend to call 'ignorance' and cases we tend to call 'vagueness' (for Williamson 1994, cases of 'accidental' versus cases of 'inherent' ignorance), which are usually modeled with the same formal means. While vagueness/inherent ignorance (about the truth value of statements like Dan is tall) arises due to partial information regarding cutoff points of vague predicates like tall, accidental ignorance (say, about the truth value of statements like Dan is two meters tall or Dan is taller than Sam) arises due to partial information regarding property values (e.g., the height) of referents of arguments like Dan ('discourse entities'). A representation of individuals by their property values, then, captures the intuitive distinctions between both phenomena.
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