## Scales, quantity & degree [2ex] lecture 1: quantifiers

Scales, quantity & degree

**Lecture 1: Quantifiers**
Rick Nouwen (Utrecht)
Scales, quantity and degree
S = hX, >i or S = hX, =i
2 Modified numerals
What is a quantifier?
Natural language determiners
Determiner phrases
Several parts that move
Predicate logical syncategorema
Generalised quantifiers
What is a quantifier?
articles: a, the, . .

determiners: every, most, . .

number words: one, two, three, . .

comparatives: fewer than five hundred, more than just a few, . .

superlatives: at most five, at least twelve, . .

PPs: between sixty and seventy, up to two hundred, . .

adjectives: (very) many, (too) few, . .

modifications: almost every, exactly five, . .

coordinations: most but not all, two or three, . .

monkey(s) is/are asleep in the zoo
‘I see nobody on the road,' said Alice.

‘I only wish I had such eyes,' the King remarked in a fretful tone.

‘To be able to see Nobody! And at that distance, too!
Lewis Carrol, Throught the looking glass, and what Alice found there
— so at once he went and took Aspirin
According to the story, there was a man with a headache, whosaw the advertisement:
nothing acts faster than AspirinTM
— so at once he went and took nothing.

(after Wilfrid Hodges, Logic, 1977)
According to the story, there was a man with a headache, whosaw the advertisement:
nothing acts faster than AspirinTM
— so at once he went and took nothing.

(after Wilfrid Hodges, Logic, 1977)
— so at once he went and took Aspirin
Quantifiers are obviously not referring terms
quantifiers ⇠ quantitiesOption 1: quantifiers stand proxy for quantitiesOption 2: quantifiers express relations between quantitiesgeneralised quantifier theory
In favour of a much richer theory of quantifier meaning
The psychology of words versus numbers
Bryant & Norman 1980; Beyth & Marom 1982; Wallsten et.al. 1986; Erev & Cohen 1990; Renooij & Witteman 1999
- expert knowledge communication
- communication guidelines (e.g. medical professions)
does/should an expert use words or numbers?how do words/phrases correspond with explicit cardinalityor frequencythis type of research presupposes that natural languagequantifiers express quantities
The psychology of words versus numbers
Bryant & Norman 1980; Beyth & Marom 1982; Wallsten et.al. 1986; Erev & Cohen 1990; Renooij & Witteman 1999
Hearers prefer numbersSpeakers prefer words
The psychology of words versus numbers
Bryant & Norman 1980; Beyth & Marom 1982; Wallsten et.al. 1986; Erev & Cohen 1990; Renooij & Witteman 1999
—————————————
many,quite a few, few, a few, veryfew, a lot, not many,several
The psychology of words versus numbers
Bryant & Norman 1980; Beyth & Marom 1982; Wallsten et.al. 1986; Erev & Cohen 1990; Renooij & Witteman 1999
Huge between-subject variationConsiderable overlap between words
Against quantifiers as words for amounts
Linda Moxey, Anthony Sanford (Glasgow)
Too many quantifiers (in a single free productionexperiment Moxey (1986) observed 182 different quantityexpressions)Moxey & Sanford 1993
each subject can assign just one number to one quantifier
on one occasion only (450 subjects)
no context effects, no comparison among quantifiers
results: impossible to distinguish between a few, only a few,
not many, few and very few
Support comes from studies on intensifiers such as very(Wright et.al. 1995)
Words and numbers
Quantifiers do not go proxy for numbersIn linguistic semantics the relation between quantifiers andquantities is a bit more complex, though
Nobody he, he, tii
A dilemma for compositionality
A dilemma for compositionality
Nobody he, he, tii
All mortals die.

Some men are mortal.

Some men die.

**Medieval studies on quantification**

All men are mortal

subject predicate
Question: What does the subject express?
The modern solution
Montague, Barwise & Cooper, Keenan & Stavi
Quantifiers as sets of sets
Let P be the set of people in the domain.

JeveryoneK = X.P ✓ X
JsomeoneK = X.P X , ;
JnooneK = X.P X = ;
Determiners as relations between sets
, ti, hhe, ti, tii
Generalised Quantifier Theory
Barwise & Cooper, Keenan, van Benthem, Westerstahl
a collection of linguistic and mathematical insightsproperties of linguistic and mathematical quantifierslinguistic universals concerning such propertiesmain focus is on generalisations
to a much lesser extent: processing aspects of quantifiersto a much lesser extent: linguistic properties of particularquantifiers
Example: the definiteness effect
Milsark 1977, Barwise & Cooper 1981, Keenan 1987
Weak quantifiersa. There are at least three gnomes in the garden.

b. There are some biscuits left in the fridge.

c. There are no aliens on mars.

d. There are fifty-two typos in the manuscript.

Strong quantifiersa. *There is every student in the classroom.

b. *There are most biscuits on this plate.

c. *There are less than half the gnomes in the garden.

d. *There are not all aliens on mars.

The weak/strong distinction can be made formally explicit interms of formal properties of quantifiers
Strong versus weak

**Symmetry: **Q(X)(Y) $ Q(Y)(X)

Strong quantifiers are not symmetrical:If every student is a spy, then every spy is a studentIf most students are spies, then most spies are students
Weak quantifiers are symmetrical:If some students are spies, then some spies are studentsIf no students are spies, then no spies are students
Existential-there sentences only admit symmetrical quantifiers
Generalised Quantifier Theory
is/are asleep in the zoo
Assumption of homogeneity within the class of GQs
Isomorphism invariance
For any U and U0,
if ⇡ : U ! U0 is a bijection, then
QU(X, Y) ! QU0(⇡(X))(⇡(Y))
Isomorphism invariance
{a, c, d} {a} < 10 $
Q(⇡(X))(⇡(Y)) $
{b, e, a} {b} < 10 $
Logicality is the notion that purely logical operators are notabout particular entities but are topic neutralvan Benthem: Quantifiers are expressions that satisfyisomorphism invarianceMost of the mathematical work on GQs concerns suchlogical quantifiers
JJohnK = A.A(j)JEvery . . but JohnK = A. B.(B [ {j}) ✓ A
With some extra assumptions: logical quantifiers are thosethat rely solely on cardinality
Logicality: the tree of numbers
For a pair A and B, let pos(A, B) be A B and neg(A, B) be A B
JeveryK(A)(B) $ neg(A, B) = 0JnoK(A)(B) $ pos(A, B) = 0JmostK(A)(B) pos(A, B) > neg(A, B)Jmore than 2K(A)(B) $ pos(A, B) > 2
hneg(A, B), pos(A, B)i
Generalised Quantifier Theory
Logicality represents extreme view within GQTThere exists a subclass of logical quantifiersThese only express a relation between their arguments and a
Limited applicability to natural languageThere is more to quantifiers than cardinalityEven the purported logical ones
Imagine a conference of lawyers and policemen where normally 60lawyers and 40 policemen attend. Also, on average, only 10attendants are women. This year, there are only 20 lawyers, but astaggering 80 policemen. Strikingly, all the lawyers happen to bewomen and all the policemen are men.

Many is non-extensional
Logicality assumes that at the heart of quantifier semanticsis cardinality comparisonThe arguments of a quantifier are taken for grantedJmanyK = A. B. A B > m
Many lawyer attended the meeting this year.

Many women attended the meeting this year.

Many is non-extensional
Logicality assumes that at the heart of quantifier semanticsis cardinality comparisonThe arguments of a quantifier are taken for grantedJmanyK = A. B. A B > m
Many lawyer attended the meeting this year.

Many women attended the meeting this year.

Imagine a conference of lawyers and policemen where normally 60lawyers and 40 policemen attend. Also, on average, only 10attendants are women. This year, there are only 20 lawyers, but astaggering 80 policemen. Strikingly, all the lawyers happen to bewomen and all the policemen are men.

Many is non-extensional
JmanyK = Ahs,he,tii. Bhs,he,tii.

8w 2 CONB : A(w⇤) B(w⇤) > A(w) B(w)
Shalom Lappin, 2000
Newstead & Coventry 2000
There are many marbles in the bowl
Newstead & Coventry 2000
There are many marbles in the bowl
"The studies all used a task in which participants rated the appropriatenessof quantifiers describing the number of balls in a bowl. The size of the ballswas found to have an effect: Identical numbers of balls were given differentratings depending on ball size." Newstead & Coventry 2000
Newstead & Coventry 2000
There are many marbles in the bowl
"The results are interpreted as indicating that quantifiers carry little specificmeaning in themselves but instead derive their meaning from the context inwhich they occur." Newstead & Coventry 2000
The vagueness of many
The conclusion of Newstead & Coventry becomes trivial ifwe do not think of many as a quantifierbut if we instead think of it as a relative adjectivei.e. what N&C uncovered is simply standardcontext-dependence and vaguenessthe effects are clearly limited to a subset of the set of GQs
There are more than seven marbles in the bowl
Many as a relative adjective
John is a tall basketball player
John is a tall toddler
There are many ants in my back garden
There are many mole hills in my back garden
A predicate is vagueIf it has borderline cases (and if there are cases that areborderline borderline, etc.)If it can be part of a sorites paradox
Vagueness in relative adjective
Key semantic contrast:
the positive form: John is tall
the comparative form: John is taller than Bill
The Similarity ConstraintWhen x and y differ to only a very small degree in the property that a vaguepredicate g is used to express, we are unable or unwilling to judge the propo-sition that x is g true and that y is g false.

Theories differ both in whether they derive (2) and in how: some derive (2) as afunction of language use (e.g. Soames 1999); some as a by-product of epistemicuncertainty (e.g. Williamson 1994); and some as a feature of the meaning of vaguepredicates (e.g. Raffman 1994, 1996; Fara 2000).

My goal in this paper is to take a close look at two ways of expressing
comparison, which differ in both their morphosyntactic properties and seman-tic/pragmatic properties, with the goal of showing how they can help us assesstheories of vagueness andexplanations of the Similarity Constraint on the one hand,and semantic analyses of the positive and the comparative forms (and the relationbetween them) Compar
on the other.

will suggest Vagueness
that the Similarity Constraint
(and so featuresKennedy
agueness more generally) is due to a semantic property of vague
predicates, and that this property is a feature of the positive form but not the com-parative form. This can be easily accommodated if both forms are derived from amore abstract source, but it is difficult (though perhaps not impossible) to explainif the comparative is derived from the positive.

implicit us is big,
Uranus is bigger than Neptune.

#Uranus is big, compared to
Consider the asymmetric size relation between the planets Uranus and Venus, asdetermined by
(To .make differences
in diameter easily perceptible, I will represent the sizes of the planets as concentriccircles in the figures to follow.)
any of the explicit comparison constructions in (9).

Figure 1: Uranus (51,118 km) vs. Venus (12,100 km)
Figure 2: Uranus (51,118 km) vs. Neptune (49,500)
A speaker might describe this relation by uttering one of the following sentences:
Uranus is bigger than Neptune.

Neptune is smaller than Uranus.

Uranus is the bigger one/of the two.

Neptune is the smaller one/of the two.

In contrast, the implicit comparison constructions in (10) are infelicitous: they donot support crisp judgments.4
a. #Uranus is big compared to Neptune.

b. #Neptune is small compared to Uranus.

c. #Uranus is the big one.

d. #Neptune is the small one.

At first glance, the infelicity of these sentences as descriptions of the sce-
nario in Figure 2 appears to follow straightforwardly, given that the kinds of judg-ments involved in evaluating them are exactly the kind of judgments that the Sim-larity Constraint makes reference to. If this constraint applies to any context ofevaluation of a vague predicate, the similarity in size between Uranus and Neptunemeans that either both planets must be in the positive extension of the predicateor both must be in the negative extension. If the semantic characterization givenabove for compared to sentences is correct, then (10a-b) are infelicitous becausethey violate the constraint that in every context of evaluation, both the positive andnegative extension of the predicate should be non-empty. Similarly, (10c-d) violatethe presuppositions of the definite, since it must be the case (according to (2)) either
4van Rooij (this volume) claims that implicit comparisons are false in crisp judgment contexts.

My own judgment about the truth or falsity of the examples in (10) in the context of Figure 2 is not so
clear, in contrast to my judgment of their (un)acceptability. Given that there is a natural pragmatic
explanation for the facts, as described in the text, I prefer to characterize the examples in (10) as
infelicitous rather than false.

Vagueness with Many
There are many mole hills in my garden
There are more mole hills in my garden than in yours
Many as a relative adjective
cf. Solt 2006, 2007
Distribution of very: modifies adjectives
I am very (*much) tall
I like you very *(much)
I am very *(much) into heavy metal music
I found very many (*much) mistakes
His good qualities are many
The flaws in the proposal were many and serious
Vagueness and Informativity
To use words specifically you also have to avoid vague terms like"many," "few," and "difficult." In their place you should use precisewords like "four" or "illegal." What does many mean? It meansdifferent things to different people. But four (whether it is four or fourmillion) is a measurable amount. If you wanted to refer to how manywidgets your company had you might be tempted to reply "a lot" if youfound that your company had an entire warehouse full of them.

However, your boss might respond "that's not even a year's worth." [.]The words "many," "large," and "important" mean something differentto each of us. As a writer you should constantly strive to selectsimple, straightforward words that mean the same thing to mostpeople.

(University of Florida Precize Writing Guide)
Vagueness and informativity
Be specific with numbers and avoid vague terms like many, alot, and most.

(Associated Press Stylebook and Libel Manual)
All three can give informative answers:
(i): the number of guests was 34
(ii): the number of guests was satisfactory
(iii): the number of guests was sufficient
How to be precise
Message: vagueness (and imprecision) leads to uninformativity
(i) There were exactly 34 people at my party last week.

(ii) There were many people at my party last week.

(iii) There were more than 10 people at my party last week.

All three sentences are about how many people attended myparty. But only (i) gives a precise answer.

How to be precise
Message: vagueness (and imprecision) leads to uninformativity
(i) There were exactly 34 people at my party last week.

(ii) There were many people at my party last week.

(iii) There were more than 10 people at my party last week.

All three sentences are about how many people attended myparty. But only (i) gives a precise answer.

All three can give informative answers:
(i): the number of guests was 34
(ii): the number of guests was satisfactory
(iii): the number of guests was sufficient
Moxey and Sanford's perspective approach
Moxey & Sanford 2000
Logicality has it that quantifiers express cardinalityrelationsMoxey and Sanford: Quantifiers express a perspective ona quantity
Rather than providing a quantitythey describe it from a certain perspective
Moxey and Sanford's perspective approach
Sanford et al. 2002
In the train disaster, a few people were seriously injured, whichis a #good/bad thing.

In the train disaster, few people were seriously injured, which isa good/#bad thing.

Moxey and Sanford's perspective approach
Sanford et al. 2002
Thankfully, not quite all passengers died in the crash.

#Thankfully, almost all passengers died in the crash.

Thankfully, few passengers died in the crash.

#Thankfully, a few passengers died in the crash.

Moxey and Sanford 1993, Nouwen 2003 — complement anaphora
Nearly all of the fans went to the match.

They cheered their team on at every opportunity.

Not quite all of the fans went to the match.

They watched it at home on TV instead.

Few students got this question right. For example, Bill didn't /?Bill did.

A few students got this question right. For example, Bill did /???Bill didn't
Perspective effects and GQT
A set of sets Q is MON" iff Q(X) X ✓ X0 ) Q(X0)
A set of sets Q is MON# iff Q(X) X0 ✓ X ) Q(X0)
a few passengers is MON"
few passengers is MON#
Perspective and monotonicity
Most students went to the party.

They had a lot of fun.

Most students went to the party.

#They were too busy.

Very few of the students went to the party.

They (still) had a lot of fun.

Very few of the students went to the party.

They were too busy.

You have to answer almost all questions correctly if you want to pass#You have to answer not quite all questions correctly if you want to
almost X entails but does not assert ¬Xnot quite X entails and asserts ¬X
Schwenter 2002, Horn 2002
The limits of monotonicity
Thankfully, not quite all politicians are corrupt#Thankfully, almost all politicians are corrupt
The limits of monotonicity
Thankfully, not quite all politicians are corrupt#Thankfully, almost all politicians are corrupt
You have to answer almost all questions correctly if you want to pass#You have to answer not quite all questions correctly if you want to
almost X entails but does not assert ¬Xnot quite X entails and asserts ¬X
Schwenter 2002, Horn 2002
The limits of monotonicity
SALE! Up to 60% reduction!
#SALE! At most 60% reduction!
There is more to quantifiers than cardinalityor cardinality comparison
vagueness and context-dependence
There is no one semantic template for quantity expressions
Next: some more complicationsBut: the tools of GQT are indirectly relevant
The syntactic force of hhe, ti, ti
Quantifiers (type hhe, ti, ti) may move and adjoin at a
higher node (of type t)leaving behind a trace of type e
Quantifier raising
8x[student(x) ! help(j, x)]
Q.8x[student(x) ! Q(x)]
y. x.help(x, y) z
Someone loves everyone
x. y.love(y, x) z
Someone loves everyone
x. y.love(y, x) z
Someone loves everyone
x. y.love(y, x) z
The versatility of QR
The versatility of QR
Operators that raise are of type hh↵, ti, tiThere is ↵-lambda abstraction over the landing site's sisterTraces are of a simplex type (↵)
Example: the comparative
What does the comparative express?
First attempt: JtallerK = x. y.y's height>x's heightJohn is taller than Bill is true iff John's height exceeds Bill's
John is taller than 6'The table is longer than the room is wide
Second attempt: JtallerK = d. y.y's height>d
d. y.y's height>d
How does a than-clause denote a degree? (It doesn't)
than WHi Bill is ti tall
*John is healthier than Mary wants to do fitness in order to beHow healthy does Mary want to do fitness in order to be?
Consequence: J-erK = Dhd,ti. y.y's height> max(D)
Comparative morphology as a generalised quantifier
J-erK = Dhd,ti. D0hd,ti.max(D0) > max(D)
John is td hd,he,tii
(This draft is 10 pages.)The paper is required to be exactly 5 pages longer than that.

⇤[max( d.long(p, d))] = 15pp
the paper should be 15 pages longmax( .⇤long(p, d)) = 15pp
the minimum number of pages that is acceptable for the paper is 15
(This draft is 10 pages.)The paper is allowed to be exactly 5 pages longer than that.

[max( d.long(p, d))] = 15pp
it's okay if the paper turns out 15 pages longmax( . long(p, d)) = 15pp
the upper page limit is 15 pages
(This draft is 10 pages.)The paper is required to be less long than that.

the paper should be shorterthe minimum number of pages that is acceptable for the paper < 10
The second reading might be difficult to get, but it is available. Try:(For the Nigella Lawson version of this cake I used 6 sticks of butter.) The Delia Smithversion requires less butter than that.

Not all intensional verbs behave similarly. Heim's examples:
The paper should be less long than that.

#It's not required for it to be as long as that
The paper is supposed to be less long than that.

#It's not required for it to be as long as that
I want the paper to be less long than that.

#I don't require it to be as long as that
(My prediction: Bill will break the world record long jump(8m95cm). It turned out he only jumped 8m80cm.) I predictedBill to jump exactly 15cm further than that.

New interim summary
There is more to quantifiers than quantitySome quantity expressions simply are not quantifiersSome quantifiers are not quantity expressions
almost all / not quite all
There is no homogeneous class of quantifiers, unless we
focus on very narrow properties
syntactic mobility
Three supposedly equivalent determiners
more than twoat least three
A universal structure for quantified expressions
hhe, ti, hhe, ti, tii
hhhe, ti, hhe, ti, tii, hhe, ti, hhe, ti, tiii
hhe, ti, hhe, ti, tii
The structure of modified numerals
more than three students
The structure of modified numerals: Hackl 2001
(i) #More than one student is meeting
(ii) At least two students are meeting
(iii) #More than 9 people got married on Saturday
(iv) At least 10 people got married on Saturday
(v) #John separated more than one animal
(vi) John separated at least two animals
Hackl's proposal: [ more than [ two NP VP ] ]
The structure of modified numerals: Focus-sensitivity
Krifka 1999, Geurts & Nouwen 2007
At least three boys left. (Maybe four)
At least three boys left. (Maybe some girls too)
At least it isn't raining. (Maybe the sun will even shine)
This behaviour is unexpected if at least modifies three.

Modified numerals show signs of DegP movement
Bill needs to score fewer than 10 points to win.

Bill will win only if he doesn't score 10 or more points (available, but unlikely)The minimum number of points Bill needs to score to win is 9 or fewer.

Bill is allowed to eat fewer than 10 cookies.

It's okay if Bill eats 9 or fewer cookies. (available)Bill shouldn't eat more than 9 cookies. (available)
Hackl's comparative semantics
JmanyK = d. X. Y.9x[#x = d X(x) Y(x)]
3 pages { [ [ 3 many ] pages ] { Y.9x[page(x) Y(x) #x = 3]
John wrote more than three pages.

hhe, ti, hhe, ti, tii
Hackl's comparative semantics
[ allowed [ [fewer than 10 ] [ d [[d many] cookies] [ x [ Bill eatx ] ] ] ]]
J d Bill eat d many cookiesK =
d.9x[cookies(x) eat(b, x) #x = d]
Jfewer than 10K = D.max(D) < 10
Sentence: [max( d.9x[cookies(x) eat(b, x) #x = d]) < 10]
Hackl's comparative semantics
Sentence: max( d. 9x[cookies(x) eat(b, x) #x = d]) < 10
The semantics of modified numerals
Geurts & Nouwen 2007
According to a simple GQT analysisJat least threeK = Jmore than twoK. This turns out wrong for
several reasons.

Specificity(i) I will invite at most two people, namely Cody and Vic.

(ii) I will invite fewer than two people, namely Cody and Vic.

The semantics of modified numerals
Geurts & Nouwen 2007
According to a simple GQT analysisJat least threeK = Jmore than twoK. This turns out wrong for
several reasons.

InferenceThere are 10 marbles in the bag
) There are more than 4 marbles in the bag?? ) There are at least 4 marbles in the bag
Why are there so many different modified numerals?
more than 100 pages
fewer than 100 pages
as many as 100 pages
as few as 100 pages
100 pages or more
at least 100 pages
100 pages or fewer
at most 100 pages
maximally 100 pages
no fewer than 100 pages
minimally 100 pages
no more than 100 pages
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