EGG Summer School 2012
Introduction to Quantification (Class 5)
Quantification in the Event Domain
Heather Burnett (Université de Montréal/ÉNS, Paris)
[email protected]
https://sites.google.com/site/heathersusanburnett/home/teaching/
egg2012-quantification
1
Review
An introduction to the analysis of DPs as generalized quantifers:
• GQ theory. What is it and what is it good for?
• Some DPs that, with certain assumptions about the syntax-semantics interface,
present a puzzle for their analysis as GQs.
Today: Quantification outside the DP domain.
• A particular focus on the verbal quantification.
Questions:
1. Are there non-DP quantifiers? (yes. . . )
2. How can we develop an appropriate semantic analysis of degree and frequency
adverbs.
(1)
a.
b.
c.
d.
John
John
John
John
always pets the cat.
never pets the cat.
pet the cat three times.
often pets the cat.
3. What properties to non-DP quantifiers share with DP quantifiers?
Plan:
1. Introduction to event semantics.
2. Analyzing adverbs as GQs.
• Parallels between the DP and the VP domain.
3. The phenomenon of polyadic quantification: quantification over binary (and higher)
relations in natural language.
4. Polyadic event quantification: Quantifiers over events and individuals at the
same time.
• A study of the Quantification at a distance construction in Standard French.
(2)
Jean a beaucoup lu de livres.
Jean has a lot
read of books
≈ ‘John did a lot of book-reading and this event involved a lot of books’.
1
EGG Summer School 2012
2
Introduction to Quantification (Class 5)
Basic Event Semantics
Simple transitive sentences:
(3)
a.
b.
c.
John likes Mary.
Brutus stabbed Caesar.
Jones buttered the toast.
Simple analysis of the compositional semantics of (3):
(4)
a.
b.
c.
(5)
a.
b.
JlikesK = {hx, yi : x likes y}
JstabbedK = {hx, yi : x stabbed y}
JbutteredK = {hx, yi : x buttered y}
JJohnK = John/j
JBrutusK = Brutus/b
JBrutus stabbed CaesarK = 1 iff b ∈ {x : x stabbed c}
JBrutusK = b
Jstabbed Caesar K = {x : x stabbed c}
But what about:
JstabbedK = {hx, yi : x stabbed y} JCaesarK = c
(6)
Brutus stabbed Caesar quickly in the forum with a knife.
Or (Davidson, 1967):
(7)
Johnes buttered the toast slowly in the bathroom with a knife.
• slowly, in the bathroom, with a knife are VP adjuncts (semantically, modifiers).
(8)
a. Jones buttered the toast slowly in the bathroom.
b. Jones buttered the toast slowly.
c. Jones buttered the toast.
d. *Jones buttered.
e. *buttered the toast.
• But what do they modify?
– With the semantics that we have given above, verbs denote relations between
their arguments.
– These VP modifiers do not seem to be modifying the individual arguments of
the predicate. . .
Question: How to give a compositional semantics for expressions like slowly, in the
bathroom etc.?
• And then later: never, always, often. . .
A related question: What does the pronoun in (9) refer to?
2
EGG Summer School 2012
Introduction to Quantification (Class 5)
• It seems to be talking about the buttering action. . . but there is no such object in
the analysis of verbs as denoting relations between individuals!
(9)
Jones did it slowly, in the bathroom, with a knife.
Davidson (anticipated by Reichenbach (1947))’s answer:
• A (relatively) straightforward account of the linguistic data concerning modifiers
and pronouns such as in (9) is possible if we suppose that we can refer to actions
or events.
2.1
Compositional Event Semantics
Main proposal of event semantics:
The domain into which we interpret natural language sentences is sorted into
(at least) two kinds of objects: individuals (Jones, Caesar etc.) and events
(e1 , e2 . . . )
Definition 2.1 Model. An event semantics model M is a triple hDi , De , J·Ki, where
Di is a non-empty set to individuals, De is a non-empty set of events, and J·K is an
interpretation function.
• Transitive verbs denote ternary predicates: relations between individuals and events.
(10)
a.
b.
JstabK = {he, x, yi : e is a stabbing event, and x is the stabber and y is the
stabbee}.
JbutterK = {he, x, yi : e is a buttering event, and x is the butterer and y is
the butteree}.
Proposal re: syntax-semantics interface:
• In bare assertions, the event argument is saturated by an existential closure
operation.
• Suppose that this operation occurs at the vP level.
vP
∃
vP
DP
v0
Brutus v
VP
V
DP
stab Caesar
3
EGG Summer School 2012
Introduction to Quantification (Class 5)
JBrutus stabbed CaesarK = 1 iff There is some e : ST AB(e, b, c)
∃
JBrutus stabbed CaesarK = {e : ST AB(e, b, c)}
Jstabbed CaesarK = {he, xi : ST AB(e, x, c)}
JBrutusK = b
JstabK = {he, x, yi : ST AB(e, x, y)} JCaesarK = c
• Modifiers like quickly and in the forum can be properties of events.
(11)
JBrutus stabbed Caesar in the forumK = 1 iff There is some e1 : ST AB(e, b, c)
and e1 happened in the forum.
JBrutus stabbed Caesar in the forumK = 1 iff There is some e2 : ST AB(e2 , b, c) and e2 happened in the forum
JBrutus stabbed CaesarK = {e2 : ST AB(e2 , b, c)}
Jin the forumK = {e1 : e1 happened in the forum}
More about PP modifiers on Monday!
• Pronouns can refer to events.
(12)
3
JitKg = e845
Adverbial Quantifiers
Now what about:
(13)
a.
b.
c.
d.
John
John
John
John
always pets the cat.
never pets the cat.
pet the cat three times.
often pets the cat.
Proposal (à la de Swart (1991)):
VP Adverbs of quantity and degree denote generalized quantifiers (functions
from properties to truth values) over events!
From a technical point of view:
• Recall that the domain is sorted: we have the domain of events (De ) and the domain
of individuals (Di ).
• We encode in the meaning of the quantifier what its domain is.
(14)
JeveryoneK = the function EV ERY ON E :
a. For all P ⊆ Di , EVERYONE(P) = True iff Di ⊆ P .
b. λP i (Di ⊆ P )
Now we can propose a semantic analysis of always as an event quantifier that is parallel
to everyone (an individual quantifier).
4
EGG Summer School 2012
(15)
Introduction to Quantification (Class 5)
JalwaysK = the function ALW AY S :
a. For all P ⊆ De , (P) = True iff De ⊆ P .
b. λP e (De ⊆ P )
Compositional semantics:
• Instead of being existentially closed, the event argument is bound by the adverbial
quantifier at the vP level.
• Note that the structure that is interpreted is the base-generated structure (before
raising of the subject etc.).
(16)
Base-generated VP shell of John always pets the cat.
AdvP
Adv
vP
always
DP
v0
John v
VP
V
DP
pet the cat
(17)
Semantic interpretation of John always pets the cat.
JJohn always pets the catK = 1 iff De ⊆ {e : P ET (e, j, c)}
JalwaysK = λP e (De ⊆ P )
JJohn pets the catK = {e : P ET (e, j, c)}
JJohnK = j
Jpets the catK = {he, xi : P ET (e, x, c)}
JpetK = {he, x, yi : P ET (e, x, y)} Jthe catK = c
What other kinds of VP-Qs are there?
• Already we can see parallels in the lexicalization patterns across the DP and VP
domains:
(18)
(19)
(20)
SOME
a. JsomeoneK = λP i (Di ∩ P 6= Ø)
b. JsometimesK = λP e (De ∩ P 6= Ø)
NEVER/NO ONE
a. Jno oneK = λP i (Di ∩ P = Ø)
b. JneverK = λP e (De ∩ P = Ø)
FOUR X
5
EGG Summer School 2012
a.
b.
Introduction to Quantification (Class 5)
Jfour thingsK = λP i (| Di ∩ P |= 4)
Jfour timesK = λP e (| De ∩ P |= 4)
• Other pairs: many/often; few/seldom etc.
Note: DP-Qs and VP-Qs are not completely parallel:
• To form a DP-GQ, we simply combine a determiner with any kind of NP.
(21)
a.
b.
c.
Five cats were in the yard.
Five dogs were in the yard.
Five times were sufficient
– To form a VP-GQ, we need a particular ‘event-describing’ NP, like times.
– The syntactic construction of GQs over events is very restricted.
(22)
3.1
John pet the cat five times.
a. *John pet the cat five pettings/strokes. . .
Summary
• The introduction of events as an ontological category was first motivated by the
observation of some similarities that these objects seem to have with the members
of the domain of individuals.
Individuals can be modified by predicates
(23)
The black cat on the mat
Davidson (1967): Events can be modified by predicates.
(24)
Jones buttered the toast slowly, deliberately, in the bathroom, with a
knife, at midnight
Individuals can be referred to with pronouns.
(25)
He likes her
Davidson (1967): Events can be referred to with pronouns.
(26)
Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight
• Other similarities (count/mass ≡ telic/non-telic). . .
Furthermore: We can obtain a simple and elegant analysis of the semantics of VP
adverbs of quantity if we analyze them as denoting generalized quantifiers over events.
• This analysis reflects a series of parallel lexicalization patterns between quantifiers
over individuals and quantifiers over events.
(Sets) of individuals can be arguments of generalized quantifiers
(27)
Five poets are vegetarians
6
EGG Summer School 2012
(28)
Introduction to Quantification (Class 5)
5 POETS({x : V EGET ARIAN (x)})
de Swart (1991): Sets of events can be arguments of adverbial quantifiers.
(29)
Brutus stabbed Caesar five times
(30)
1 iff | {e : ST ABBIN G(e)&AGEN T (e, b)&T HEM E(e, c)} |= 5
(1 iff | {e : ST ABBIN G(e, b, c)} |= 5)
For the rest of the class: I present yet another similarity between elements that
constitute the individual and event domain.
• Natural language has unreducible binary quantification not only in the individual
domain (van Benthem (1989); Keenan (1992)), but also in the event domain.
• This fact is exemplified by the Quantification at a Distance construction in Standard
European French.
The QAD Construction
• In French, individual quantification can be realized by use of an adnominal quantifier
(ex. beaucoup ‘a lot’) that selects a DP headed by the determiner de ‘of’.
(31)
J’ai lu beaucoup de livres
I have read a lot
of books
‘I read a lot of books’
Canonical Quantification
• The quantifier may also be placed in an adverbial position to form a sentence that
seems synonymous with (31).
(32)
J’ai beaucoup lu de livres.
I have a lot
read of books
‘I read a lot of books’
Quantification at a Distance
Since it was first noticed by Kayne (1975), the QAD construction has been frequently
studied for the standard dialect of European French (Kayne (1975); Obenauer (1983);
Rizzi (1990), Doetjes (1995) inter alia).
Proposal:
• QAD sentences in Standard French involve properly binary quantification by the
quantificational adverb beaucoup over <event, object> pairs denoted by the verb
phrase.
4
Binary Generalized Quantifiers
Binary Quantifier: A quantifier that takes a binary relation as an argument, not a
property.
(33)
λR(Q(R)) = 1 iff . . .
Actually: Sentences like No student likes every book can be analyzed as involving a type
< 2 > generalized quantifier, one that expresses a property of a binary relation, in this
case LIKE.
7
EGG Summer School 2012
Introduction to Quantification (Class 5)
• There is no need to propose that No student likes every book involves a binary
quantifier
• Such a quantifier is straightforwardly reducible to the iteration of two unary quantifiers: first every book is applied to LIKE and then no student is applied to the
result (see class 3).
• Binary quantifiers that are equivalent to iterated application of unary quantifiers
are known as Fregean (or reducible) in the literature.
van Benthem (1989); Keenan (1992): Some binary quantifiers found in natural language are unreducible, i.e. not equivalent to the composition of two unary quantifiers.
• Keenan (1992) argues that the different-different construction (34) must be analyzed in terms of a binary quantifier, which he then proves to be unreducible.
(34)
Different students answered different questions on the exam
Because of examples such as (34), English is said to go beyond the Frege boundary.
• We will see that properly binary quantifiers appear in the verbal domain as well!
5
Quantification at a Distance
Obenauer (1983): QAD sentences can be used in only a subset of the contexts in which
Canonical Quantification sentences are used.
• QAD sentences show a Multiplicity of Events requirement.
(35)
Multiplicity of Events Requirement:
QAD sentences are only true in contexts involving multiple events
• QAD involves event quantification.
• I present one test for the ME requirement, for others, see Obenauer (1983) and
Burnett (2012).
Test 1: Point Adverbials
• To test for adverbial quantification: Insert a prepositional phrase forcing a
single event context.
– Sentences with canonical quantification are compatible in single-event contexts.
(36)
Dans cette cassette, il a trouvé beaucoup de pièces d’or
In
this casette, he has found a lot
of pieces of gold
‘In this casette, he found a lot of gold pieces’
(37)
En soulevant le couvercle, il a trouvé beaucoup de pièces d’or
In lifting
the lid,
he has found a lot
of pieces of gold
‘Lifting the lid, he found a lot of gold pieces’
(Obenauer (1983: 78, his (42))
8
EGG Summer School 2012
Introduction to Quantification (Class 5)
• QAD sentences with PPs forcing a single-event reading are ungrammatical.
(38)
*Dans cette cassette, il a beaucoup trouvé de pièces d’or
In
this casette, he has a lot
found of pieces of gold
(39)
*En soulevant le couvercle, il a beaucoup trouvé de pièces d’or
In lifting
the lid,
he has a lot
found of pieces of gold
(Obenauer (1983: 78, his (43))
• QAD sentences with PPs suggesting a context where there are many events are fine.
(40)
Dans cette caverne, il a beaucoup trouvé de pièces d’or
In
this caverne, he has a lot
found of pieces of gold
‘In this caverne, he found a lot of gold pieces’
(41)
En cherchant partout,
il a beaucoup trouvé de pièces d’or
In searching everywhere, he has a lot
found of pieces of gold
‘Searching everywhere, he found a lot of gold pieces’
(Obenauer (1983: 78, his (45))
Summary:
• Quantification in QAD sentences actually involves quantification over an event variable: they are only true in contexts involving many events.
Is QAD simply adverbial quantification (like with always)?
(42)
J J’ai beaucoup lu de livresK = BCP e (∃x(Reading (e, I, x) & Book(x)))
• beaucoup is in an adverbial position.
• Obenauer’s observation: All the quantifiers that participate in QAD independently exist as adverbs.
5.1
The Multiplicity of Objects Requirement
Pure event quantification is too simple!
• QAD sentences are subject to an additional requirement:
(43)
Multiplicity of Objects Requirement:
QAD sentences are only true in contexts involving multiple objects
QAD also involves quantification over the direct object
• (44) cannot be uttered in a context in which I called only my own mother many
times.
(44)
J’ai beaucoup appelé de mères
I have a lot
called of mothers
QAD sentences with multiple events involving one (or even few) object(s) are judged
false.
9
EGG Summer School 2012
(45)
Introduction to Quantification (Class 5)
J’ai lu mon livre préféré plusieurs fois la semaine passée
I have read my book favourite many
times the week
last
‘I read my favourite book many times last week’
(46)#J’ai beaucoup lu de livres
I have a lot
read of books
Conclusion:
Quantification in QAD with beaucoup in Standard French seems to be, at the
same time, over the event variable, and over the object variable.
• A lot of events
• A lot of objects
5.2
Analysis of the semantics of adverbial beaucoup
Following Peters and Westerstahl (2006), I assume that what differentiates degree quantifiers like beaucoup from other quantifiers like trois fois ‘three times’ is that degree
quantifiers are extremely context dependent:
• They require a contextual ‘standard’ parameter for the truth of sentences containing
them to be evaluated.
(47)
Let s1 ∈ N.
For all P ∈ P(De ∪ Di ) BCPs11 (P ) = 1 iff | P |> s1
(48)
JBrutus a beaucoup poignardé César K = 1 iff BCPs11 ({e : Stabbing(e, B, C)}) = 1
Proposal (Burnett (2012)):
The adverbial quantifier BCP 1 is extended to deal with binary relations in the following
way:
(49)
Let s, t ∈ N such that 0 < s, t <| De ∪ Di |,
SF
For all R ∈ P(De × Di ), BCPs,t
(R) = 1 iff BCPs1 (Dom(R)) = 1 &
1
BCPt (Ran(R)) = 1
• BCP SF takes a set of <event, object > pairs and yields true just in case the
cardinality of the set of first co-ordinates is a lot, and the cardinality of the set of
second co-ordinates is also a lot.
(50)
JJ’ai beaucoup lu de livresK = 1 iff | {e : Reading (e, I, x) & Book(x) |> se &
| {x : Reading (e, I, x) & Book(x) |> ti
Arguments for this Analysis
• Captures the multiplicity of events requirement and the multiplicity of objects requirement: they are straightforwardly built into the meaning of the quantifier.
Main argument for a binary quantification approach to QAD:
(51)
Theorem Burnett (2012):
BCP SF is unreducible to any iteration of unary quantifiers.
10
EGG Summer School 2012
Introduction to Quantification (Class 5)
• The quantifier that we need to analyze the semantics of QAD in Standard French
is not Fregean.
• A binary approach to the analysis of QAD sentences in Standard French is necessary
to get the truth conditions of QAD sentences right.
5.3
Summary
• The quantification involved in QAD is binary quantification over the event argument
and the direct object.
• Such an analysis is necessary to account for the semantics of the construction since
the binary extension of beaucoup is not reducible to the composition of unary quantifiers.
Consequences: These results indicate:
• The structure of the event domain is sufficiently similar to that of the individual
domain to permit properly binary quantification over <event, object> pairs.
• Natural language goes ‘beyond the Frege boundary’ in both the domain of events
and the domain of individuals.
References
Burnett, H. (2012). The role of microvariation in the study of semantic universals. Journal
of Semantics, 29:1–38.
Davidson, D. (1967). The logical form of action sentences. In Rescher, N., editor, The
logic of decision and action, pages 81–95. Pittsburgh University press, Pittsburgh.
de Swart, H. (1991). Adverbs of Quantification: A Generalized Quantifier Approach. PhD
thesis, University of Groningen.
Doetjes, J. (1995). Quantification at a distance and iteration. In Beckman, J., editor,
Proceedings of NELS 25, pages 111–126. University of Massachusetts, Amherst: GLSA.
Kayne, R. (1975). French Syntax. Cambridge Mass: MIT Press.
Keenan, E. (1992). Beyond the frege boundary. Linguistics and Philosophy, 15:199–221.
Obenauer, H.-G. (1983). Une quantification canonique: la quantification à distance.
Langue française, 58:66–88.
Parsons, T. (1990). Events in the Semantics of English. Cambridge: MIT Press.
Peters, S. and Westerstahl, D. (2006). Quantifiers in logic and language. Oxford University Press, Oxford.
Reichenbach, H. (1947). Elements of symbolic logic. MacMillan, London.
Rizzi, L. (1990). Relativized Minimality. Cambridge: MIT Press.
van Benthem, J. (1989). Polyadic quantifiers. Linguistics and Philsophy, 12:437–465.
11