Von Stechow: Degree Semantics
13/09/2008
TOPICS IN DEGREE SEMANTICS: 4 LECTURES
HANDOUT 1: DEGREES
ARNIM VON STECHOW, TÜBINGEN
1.
Plot of the Lectures ......................................................................................................... 1
2.
Reading........................................................................................................................... 2
3.
What are degrees? ........................................................................................................... 3
4.
Comparative as relation between degrees ........................................................................ 4
5.
Degrees as measures and differentials ............................................................................. 5
5.1. Weight ..................................................................................................................... 5
5.2. Spatial degrees ......................................................................................................... 7
5.3. Negative adjectives and degrees ............................................................................... 8
5.4. Anomalies ................................................................................................................ 9
5.5. Interval Semantics: extents ..................................................................................... 10
6.
Notes on the literature ................................................................................................... 11
7.
References .................................................................................................................... 12
1.
PLOT OF THE LECTURES
Every semanticists agrees that the notion of degree is used when it comes to the semantics
of comparative constructions as the following ones:
(1)
The table is longer than the drawer is wide.
(2)
I thought your yacht is longer than it is.
A colleague, who recently gave a course on degrees in another summer school, told me that
he found papers on degree semantics confusing. In particular, there is much talk about
intervals of degrees, which are occasionally even called degrees.
I think the terminology used in the literature including my own papers is indeed
sometimes confusing. So the first lecture will be devoted entirely to the ontology of
degrees. I will say that degrees are equivalence classes of individuals. And degree scales
must therefore be sequences of such classes. These classes have to be constructed. We will
apply the notions to comparative construction and define such things as the addition or the
difference of degrees. The problem of anomalies (incommensurability) is introduced.
The second lecture is about the syntax and the LF of degree constructions. It is based
on Bresnan’s syntax. The semantic language will be a typed language, an intensional version
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of Church’s (1934) type-language, the same tool that is used in E. Stabler’s lectures.
I will motivate the assumption that adjectives should be analysed as relations between
individuals and degrees, a modified semantics of Cresswell’s. I the first lecture we started
with Kennedy’s semantics that assumed adjectives to be function from individuals to degrees.
This caused complications for the syntax of embedded degree clauses. The semantics of the
comparative reached will that advocated in Heim’s papers, viz. that the comparative expresses
the proper subset relation.
The third and the forth lecture addresses the problem of quantifiers in than-clauses.
We encounter the following distribution: NPIs are licensed in than-clauses, strong quantifiers
like every girl give raise to unattested readings when interpreted in the than-clause, they give
correct readings when they take matrix scope. Negative quantifiers and downward entailing
quantifiers in than-clause are not understandable; they would make sense if they took matrix
scope. Non-standard quantifiers like most girls/the fewest girls produce anomalies in the
than-clause. The “positive” variant most girls is correctly interpreted with matrix scope, the
negative variant cannot be interpreted at all. The same behaviour is observed with frequency
adverbs and modals.
Two attempts to derive the data are discussed: the theory of Schwarzschild &
Wilkinson and that of Heim. Both theories have a method to lift degree adjectives to the
quantifier level. The first theory is too rigid because it gives all quantifiers in than-clauses
matrix scope, which cannot be correct for NPIs, “negative” quantifiers, most modals and
frequency adverbs. The refinement by Heim still has a problem with negation.
I wanted to discuss a recent attempt to overcome the problem by S. Beck, but there
wasn’t enough time. I don’t see that this attempt solves all the problems mentioned. So the
problem of quantifiers in than-clauses remains unsolved and is a topic for further research.
2.
READING
Basics in semantics: (Heim and Kratzer, 1998)
Construction of degrees:
(Cresswell, 1976: sect. 4)
(download)
(Klein, 1980: sect. 4)
(download)
Interval semantics and anomalies:
(von Stechow, 1984a: sect. ) (download)
(Kennedy, 2001)
(download)
(more explicitly in chap. 4 and 5 of (Kennedy, 1997) (download)
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(Heim, 2000)
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(download)
The papers marked as “download” are available on the server.
3.
WHAT ARE DEGREES?
We read that degrees are numbers or they are intervals of numbers. This is confusing. We
want to say what degrees are. If A is a degree relation, A-degrees are equivalence classes of
objects that are not distinguishable with respect to A-ness. For some A-degrees we have
measures, for others we don’t, though we could perhaps. Intervals are sequences of degrees
and must therefore distinguished from degrees at least terminologically.
The classical reconstruction to the notion of degrees goes back to at least Carnap and
ultimately to Frege/Russell. We rely on (Cresswell, 1976). Let us start with the construction
of degrees of beauty. We assume that we are empirically given a relation >beau “prettier than”.
The relation is antis-symmetric and transitive. F(>beau) is the field of the relation.
x =beau y “x is exactly as beautiful as y”
(3)
(x,y F(>beau)) [x =beau y iff (z F(>beau)) [x >beau z iff y >beau z ] & [z >beau x iff
z >beau y ]
•
=beau is an equivalence relation.
The degrees of beauty are the equivalence classes generated by =beau.
(4)
Degrees of beauty
(x F(>beau)) [x]beau = {y | y =beau x}
Degbeau = {[x]beau | x F(>beau)}
In terms of types degrees are predicates, i.e., the have type et. But we introduce a new type
for degrees, viz. d.
[Marilyn]beau is the degree of Marilyn’s beauty, and “Marilyn is beautiful to degree
[Marilyn]beau” says something entirely trivial, namely that Marilyn is as beautiful as she is.
Precisely this triviality shows that the notion of degree is innocuous. Degrees are simply sets
of objects, no numbers, intervals and the like.
(5)
Comparative relation A:
a relation between individuals, anti-symmetric and transitive
( 6)
Degrees of sort A: The equivalence classes generated by A
( 7)
Type of degrees: d
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Domain of d: Dd: Union of DegA for any comparative relation A
4.
COMPARATIVE AS RELATION BETWEEN DEGREES
Next we order degrees by carrying over the relation >beau to degrees.
(d,d’ Degbeau) d >[beau] d’:= (x d)(y d’) x >beau y
(9)
>[beau] is a higher order relation, i.e. a relation between degrees (sets).
Suppose next we want to express the statement
(10)
Liz is prettier than Marilyn
via a a comparison of degrees. Than this could be done as follows:
(11)
The degree to which Liz is pretty > the degree to which Marilyn is pretty
The most straightforward way to formalize this is to say that the adjective pretty maps its
argument into the degree of its beauty.
(12)
Adjectives
(preliminary)
beautiful/pretty : x.[x]beau
x.[x]beau is often referred to as a measure function and written as x.BEAUTY(x).
Many semanticists assume that degree adjectives express measure function (e.g. (Kennedy,
1997)).
The comparative would then be the relation >[A] between two degrees:
(13)
Comparative
type d(dt)
erA : d DegA.d’ DegA.d’ > d
The analysis of sentence (10):
(14)
erbeau [(than) pretty(Marilyn)] pretty(Liz)
BEAUTY(Liz) >[beau] BEAUTY(Marilyn)
No notion of measurement is involved. To call a function such as pretty a measure function
as frequently is, strictly speaking, a misnomer if we adopt this construction. (14) says that
Liz belongs to the people we judge to be more beautiful than Marilyn. No notion of distance
is involved.
Degrees and relations between them depend on the comparative relation A that
generates them. This dependency should be made clear by appropriate subscripts. In the
literature this is usually omitted.
•
What about positive adjectives?
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Marilyn is pretty
5.
DEGREES AS MEASURES AND DIFFERENTIALS
We start with weight. This avoids the problem of (a)nomalies arising with spatial degrees.
5.1.
Weight
Weights are degrees with measures like kg and g. We construct these.
(16)
The goose weighs more than the duck.
[more (than) [the duck weighs]] [the goose weighs]
(17)
The goose weighs 5 kilo.
(18)
The duck weighs 3 kilo.
(19)
The goose weighs 2 kilo more than the duck
Assume for objects a, b the operation of fusion (put a and b on the scales and consider them
as one object) a+b. For different objects a1,…an we know:
a1 <weigh a1+a2 < a1+ a2 + a3 <weigh …. <weigh a1 + a2 +…+an
(20)
Degrees of weight
[x]weigh = {y | x =weigh y} = WEIGHT(x)
(21)
The verb weighs as measure function:
weighsed : x.WEIGHT(x)
(22)
a. moreA
d DA.d’ DA. d’ >[A] d
like –er (preliminary)
b. lessA : d DA.d’ DA. d >[A] d’
(23)
The goose weighs more/less than the duck.
[the goosee weighsed] [dt mored(dt)/less (than) [d the ducke weighsed ]]
WEIGHT(g) >[weigh] WEIGHT(d)/ WEIGHT(d) >[weigh] WEIGHT(g)
Measures of weight
We assume an operation +weigh of addition for the objects in F(>weigh). x +weigh y is the object
we get when we put x and y together on the scales. We know from experience that x +weigh y
>weigh x (same for y).
We extend +weigh to degrees of weight:
(24)
Let d, d’ be in Dweigh. d +[weigh] d’ = {u F(>weigh) | (u1 d)(u2 d’) u1 +weigh u2 =
u}
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We can define multiples of weights: For any d Dweigh: 2d = d +[weigh] d. Generally for n >
1: nd = (n-1)d +[weigh] d. The weight unit is the kilo in Paris, call it kg.
(25)
The standard sequence based on [kg]weigh Kranz 1971
1[kg]weigh, 2[kg]weigh, 3[kg]weigh…
It follows that the sequence is ordered by <[weigh]: 1[kg]weigh < WEIGH 2[kg]weigh,…
(26)
Units of measurement
kg (or g, or mg etc.) is the unit of weight. A comparative relation A may have a unit u
of measurement, but for most comparative relations no unit is established.
measures are degrees based on units of measurement.
Fractions of units: 0.5[kg]weigh, 0.21[kg]weigh.
We assume the existence of n[kg]weigh and have a dense scale.
(27)
The goose weighs 5 kilo
WEIGHT(g) = 5[kg]weigh
Intuitively the expression 5 kilo should be the name of the degree 5[kg]weigh. The actual
syntax requires it to be a degree predicate.
(28)
5 kilodt as a degree predicate
(preliminary)
d Dweigh. d = 5[kg]weigh
(29)
The goose weighs 5 kilo.
5 kilo [weighs [the goose]]
WEIGHT(g) = 5[kg]weigh
Differentials are degrees that indicate the distance between degrees.
(30)
The goose weighs 2 kilo more than the duck.
WEIGHT(g) > WEIGHT(d) & DIFFweigh(WEIGHT(g), WEIGHT(d)) = 2 [kg]
( 31)
Addition and Difference of Degrees
Let d1, d2 be in DA.
a. d1 +[A] d2 = {u F(>A) | (u1 d1)(u2 d2) u =A u1 +A u2}
b. DIFFA(d1, d2) = that A-degree d3 s.t. d1 + d = d2 or d2 + d = d1.
DIFFweigh (5 [kg]weigh , 3 [kg]weigh) = 2 [kg]weigh = DIFF (3 [kg]weigh, 5 [kg]weigh)
(32)
Differential PPs
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by 2 kilo(ddt)(ddt) = Rddt.d Dweigh.d’ Dweigh.R(d)(d’) & DIFF(d,d’) = 2[kg]weigh
(33)
The duck weighs 2 kilo less than the goose.
[[[by 2 kilo] less] [(than) the goose weighs]][the duck weighs]
WEIGHT(g) >[weigh] WEIGHT(d) & DIFF(WEIGHT(g), WEIGHT(d)) = 2 [kg]weigh
( 34)
Scales
Let A be a comparative relation. The A-scale A is the sequence of the A-degrees
ordered by <A.
( 35)
Degrees based on units serve two purposes:
(a) they localise objects relatively to other object, they are comparative concepts.
(b) they measure the distance of an object from the beginning of the scale 0. Any
object that weighs 2 kilos has a distance of 2 kilos from 0.
5.2.
Spatial degrees
Spatial degrees are the most pervasive due to the different dimensions of space. Spatial degree
adjectives: tall/short, wide/narrow, long/short, high/low, long/short…
The conventional units of measurement are m (meter), cm (centimetre) and so on.
The construction of degrees is as before. The interesting fact is that comparison can abstract
from the dimension i.e. the direction of measurement.
(36)
The table is longer than the drawer is wide (by 20 cm)
(37)
The snake is longer than the bear is thick (by 80cm).
•
Note incidentally: The examples show that we need degrees in the semantics. We
cannot express the truth-conditions of these by analysing comparative adjectives like
longer simply as a relation between individuals.
(38)
Positive spatial adjectives
tall/heigh:
x.HEIGTH(x) = x.{u F(>tall) | u =tall x}
wide/broad: x.WIDTH(x) = x.{u F(>wide) | u =wide x}
long: = x.LENGTH(x) = x.{u F(>long) | u =long x}
The unit of measurement is m, the meter in Paris. This generates the degrees
[m]tall = {u F(>tall) | u =tall m}
[m]wide = {u F(>wide) | u =wide m}
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and so on. We assume primitive relations +A of concatenation of the objects in these
degrees: we juxtapose them in vertical, horizontal etc. direction and form multiples of
spatial degrees. We can define addition of degrees and multiples thereof. The construction
is as before. Thus we have: n[m]tall etc.
(39)
Alla is 10cm taller than Sveta
[[[by 10 cm] er] [(than) Sveta tall]] [Alla tall]
HEIGHT(a) >[heigh] HEIGHT(s) & DIFF[heigh](HEIGHT(a), HEIGHT(s)) = 10[cm]tall
Quantities of spatial degrees: Occasionally we compare spatial degrees of a different
direction of measurement. We abstract from the direction.
( 40)
The table is longer than the drawer is wide.
HEIGHT(t) >? WIDTH(d)
( 41)
undefined!
Quantities of degrees.
Let d be a spatial degree based on the relation >A and the unit of measurement u, i.e.
d = n[u]A. Then QU(d) = n[u].
(42)
The table is longer than the drawer is wide.
[[[by 10 cm] er] QU[the drawer wide]] [QU[the table long]]
QU(HEIGHT(t)) > QU(WIDTH(d)) & DIFF(QU(HEIGHT(t)), QU(WIDTH(d))) =
10[cm]
The relation > holds of two quantities of degrees m[u] and n[u] iff m > n. QU is an
operation of accommodation applied at LF.
5.3.
(43)
Negative adjectives and degrees
Negative spatial adjectives
short/low:
x.SHORTNESS(x) = x.{u F(>short) | u =shortx}
etc.
Pairs like tall/short are called antonyms.
(44)
The shortness scale short: The degrees in Dtall ordered by decreasing length, i.e. for
any two neighbours d, d’ in the sequence: d >[tall] d’, i.e. d <[short] d’ iff d >[tall] d’.
short has a maximum, but no minimum.
(45)
short : …….1[m] <[short] <…<[short]
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F(>short) = F(>tall) and SHORTNESS(x) = HEIGHT(x) for any x F(>shor), but there is no unit
of shortness nor is quantity of shortness, because that would be the distance from the
minimum. But there is difference of shortness, and these are degrees of tallness.
(46)
•
(47)
DIFFtall(SH(x), SH(x)) = DIFF(HEIGHT(x),HEIGHT(y))
Differences of degrees are always positive degrees.
Sveta is shorter than Alla.
[er [All short]] [Sveta short]
SH(s) >[short] SH(a)
iff HEIGHT(s) >[short] HEIGHT(a)
iff HEIGHT(s) <[tall] HEIGHT(a)
(48)
Sveta is 10 cm shorter than Alla.
[[by 10 cm er] [Alla short]] [Sveta short]
SH(s) >[short] SH(a) & DIFF[tall]( SH(s), SH(a)) = 10[cm]tall
iff HEIGHT(s) >[short] HEIGHT(a) & DIFF[tall]( HEIGHT (s), HEIGHT (a)) =
10[cm]tall
iff HEIGHT(s) <[tall] HEIGHT(a) & DIFF[tall]( HEIGHT (s), HEIGHT (a)) =
10[cm]tall
5.4.
( 49)
Anomalies
*Sveta is 1.60m short
[Sveta]short = 1.60 [m]short ?
n [m]short should be n-times more short than [m]short. In order to define these multiples we
would need an operation +short such that for any x, y F(>short) x +short y >short x (resp. y).
There is no natural way of defining that.
•
There is no unit for shortness. Nor are there multiples of degrees of shortness. The
same holds for other negative adjectives.
(50)
a. Alla is taller than Sveta is (tall).
b. *Alla is taller than Sveta is short.
c. Sveta is shorter than Alla is (short)
d. Alla is shorter than Sveta is tall.
The unacceptability of (50b) is explained: we cannot compare degrees on the length and the
shortness scale (Cresswell).
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er? [(than) Sveta short] [Alla tall]
HEIGHT(a) >[short] SH(s)
?
HEIGHT(a) >[tall] SH(s)
?
SH(s) = HEIGHT(s), but we don’t know what which scale the comparative should choose.
The two statements mean the opposite.
•
(52)
The semantics predict that (50d) is unacceptable!
a. The rope is longer than the gap is wide.
b. *The rope is longer than the gap is narrow.
c. The rope is shorter than the gap is narrow.
5.5.
Interval Semantics: extents
(von Stechow, 1984a) and (Kennedy, 2001) say that adjectives assign individuals intervals.
They follow Seuren and call the intervals extents. Positive adjective assign individuals
positive extents, their antonyms assign them negative extents. Positive extents are initial
segments on a degree scale, negative extents are final segments of degree scale.
(53)
Adjective as functions from individuals to extents: type e(dt)
tall : x.(0,HEIGHT(x))
short : x.(HEIGHT(x), )
Von Stechow and Kennedy are not very explicit what the points of these intervals are. von
Stechow denotes positive A-extents as <A 0, n> and negative A-extents as <A n, >. Kennedy
uses a similar notation. A more explicit notation would be:
(54)
tall : x.d Dtall.HEIGHT(x) [tall] d.
short : x.d Dtall.HEIGHT(x) [tall] d.
We say that the points of these intervals are degrees, i.e., equivalence classes of individuals.
The comparative can be defined as strict subset relation in interval semantics:
(55)
Comparative in interval semantics
(Heim, 2000)
er : Ddt.D’dt.D D’
(56)
Sveta is shorter than Alla.
er [(than) Alla short] [Sveta short]
[d Dtall.HEIGHT(Alla) tall] d] [d Dtall.HEIGHT(Sveta) tall] d]
|------------------------------------Alla+++++++++++++++++++++> 10
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|------------------Sveta+++++++++++++++++++++++++++++++> Measure phrases must be of the quantifier type in interval semantics:
(57)
Measure phrases in interval semantics
1.70 m(dt)t : Ddt: D an initial segment of tall. D(1.70 [m]tall)
•
(58)
The type lifting only has the purpose to make sure that D is a positive extent.
Sveta is 1.60 m tall.
HEIGHT(Sveta) [tall] 1.60[m]tall
The semantic accounts for decreasing monotonicity, i.e. the sentence entails:
(59)
Sveta is d tall, for any d < 1.60m
6.
NOTES ON THE LITERATURE
The basic idea for the construction of degrees goes back to Frege’s constructions of numbers.
To construct the number 3 we start with a particular set, say x , say {Ede, Ora, Orin}, and
define 3 as all the sets that are indistinguishable from x with respect to the relation >much, i.e.
the have the same number of elements. The operation + of concatenating non-overlaping sets
is .
The account given here closely follows the exposition in (Klein, 1980). Klein however
doesn’t want to construct degrees from primitive relations >A. He thinks that the predicative
function of adjectives is basic and constructs >A. We don’t discuss his construction here. Once
>A is given, we can proceed as indicated here.
Cresswell’s construction of degrees proceeds in two steps. First he constructs
equivalence classes exactly as we have described it here. This degrees are equivalence classes
objects. In the second step he defined distances d between degrees. A distance d is a real
number. The second step of the construction is not made explicit and not clear to me.
There is a tradition in semantics that treat measures as numbers. (Krifka, 1989: section
8) is very explicit. A measure function m is a function from individuals (that are closed under
mereological fusion) into numbers. Degrees are classes of objects mapped to the same
number, i.e. a degree d has is a set xe.[m(x) = n] for some number n. Thus Krifka’s degrees
are equivalence classes as well. Call | d | the cardinality of d, i.e.
| d | = n. d = x[m(x) = n]
Krifka gives the following definitions (accommodated to our notation):
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d < d’ iff | d | < | d’ |
d + d’ = x.[m(x) = | d | + | d’ |]
n • d = x[m(x) = n • | d | ]
Adjectives are relations between individuals and degrees:
(60)
Krifka’s talld(et)
d.x.x is tall to degree d, i.e. x d
( 61)
Krifka’s measure phrase
10 cmd : x[cm(x) = 10]
•
(62)
cm is a special measure function.
Krifka’s comparative
er : Ad(et).d.Ddt.x.(d’)[D(d’) & A(d+d’)(x)]
•
Krifka follows (von Stechow, 1984b) in making the differential an argument of the
comparative.
•
Alla is taller than Sveta
•
Exd Alla er tall d Ld. Sveta d-tall
•
Place for the differential degree
( 63)
Alla is 10 cm taller than Sveta (is tall)
Alla [[[er tall] 10 cm] [(than) d. Sveta d-tall]]
(d’)[cm(Sveta) = |d’| & cm(Alla) = |d’| + 10 ]
•
The representation still misses the dimension of measurement, i.e. the vertical
direction against the gravitation vector.
For (Kennedy, 1997) degrees are points on a scale ordered along some dimension A. He
does not say what these points are. The dimension seems the same as our degree relations.
His terminology is compatible with the present view that degrees are equivalence classes
determined by A. (Kennedy, 1997) follows (Seuren, 1973) and (von Stechow, 1984a) in
calling degree intervals extents.
7.
REFERENCES
Cresswell, M. J. 1976. The Semantics of Degree. In Montague Grammar, ed. B. Partee, 261292. New York: Academic Press.
Heim, Irene, and Kratzer, Angelika. 1998. Semantics in Generative Grammar: Blackwell
Textbooks in Linguistics. Oxford/Malden, MA: Blackwell.
Heim, Irene. 2000. Degree Operators and Scope. Paper presented at SALT 10, Cornell
University.
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Kennedy, Chris. 1997. Projecting the Adjective: The Syntax and Semantics of Gradability and
Comparison, University of California: PhD dissertation, published 1999 by Garland
Press.
Kennedy, Christopher. 2001. Polar Opposition and the Ontology of 'Degrees'. Linguistics and
Philosophy 24:33-70.
Klein, Ewan. 1980. A semantics for positive and comparative adjectives. Linguistics and
Philosophy 4.1:1-45.
Krifka, Manfred. 1989. Nominalreferenz und Zeitkonstitution: Studien zur Theoretischen
Linguistik. München: Wilhelm Fink.
Seuren, Pieter A. M. 1973. The Comparative. In Generative Grammar in Europe, eds. F.
Kiefer and N. Ruwet, 528-564. Dordrecht: Reidel.
von Stechow, Arnim. 1984a. My Reaction to Cresswell's, Hellan's, Hoeksema's and Seuren's
Comments. Journal of Semantics 3:183-199.
von Stechow, Arnim. 1984b. Comparing Semantic Theories of Comparison. Journal of
Semantics 3:1-77.
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