Various strategies of multiplication:
Differentials in equatives and comparatives
Mojmír Dočekal & Marcin Wągiel
SinFonIJA 10, Dubrovnik, 20-24.10.2017
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Introduction
Starting point
lexicons of many natural languages distinguish between
two types of adverbs of quantification
(1) a. twice/doubly (English)
b. deux fois/doublement (French)
c. dvaždy/vdvojne (Russian)
d. kétszer/kétszeresen (Hungarian)
puzzling contrasts
cross-linguistic semantic investigation
2 / 52
Introduction
Aim
an analysis of such expressions in two typologically distinct
languages
(2) a. dvakrát/dvojnásobně (Czech)
b. hai-lần/gấp-đôi (Vietnamese)
Focus
factor phrases in comparatives and equatives
Terminology
twice-type adverb ⇒ event numeral (EN)
doubly-type adverb ⇒ degree numeral (DN)
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Introduction
Puzzle
contrasts: acceptability of EN/DN in EQ
Claim
twice-type adverb ⇒ degree multiplier
doubly-type adverb ⇒ degree predicate
two distinct strategies of degree multiplication
implicit/explicit comparison: construction specific rather
than language parametrized
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Data
Variation: factor phrases and COMP/EQ (cf. Gobeski 2011)
English
(3) a. John is two times taller than Mary.
b. John is two times as tall as Mary.
c. *John is twice taller than Mary.
d. John is twice as tall as Mary.
e. *John is doubly taller than Mary.
f. *John is doubly as tall as Mary.
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Data
Macedonian
(4) a. Jon
Jon
e
is
dva
two
pati
times
po
more
visok
tall
od
from
Mari.
Mari
‘Jon is two times taller than Mari.’
b. *Jon
Jon
e
is
dva
two
pati
times
visok
tall
kolku
as
Mari.
Mari
c. Jon
Jon
e
is
duplo
doubly
po
more
visok
tall
od
from
Mari.
Mari
‘Jon is two times taller than Mari.’
d. *Jon
Jon
e
is
duplo
doubly
visok
tall
kolku
as
Mari.
Mari
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Data
German
(5) a. Hans
Hans
ist
is
zweimal
twice
größer
taller
als
than
Maria.
Maria
‘Hans is two times taller than Maria.’
b. Hans
Hans
ist
is
zweimal
twice
so
so
groß
tall
wie
how
Maria.
Maria
‘Hans is twice as tall as Maria.’
c. *Hans
Hans
ist
is
doppelt
doubly
größer
taller
als
than
Maria.
Maria
d. Hans
Hans
ist
is
doppelt
doubly
so
so
groß
tall
wie
how
Maria.
Maria
‘Hans is twice as tall as Maria.’
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Data
Polish
(6) a. Jan
Jan
jest
is
dwa
two
razy
times
wyższy
taller
niż
than
Maria.
Maria
‘Jan is two times taller than Maria.’
b. Jan
Jan
jest
is
dwa
two
razy
times
tak
so
wysoki
tall
jak
how
Maria.
Maria
‘Jan is twice as tall as Maria.’
c. *Jan
Jan
jest
is
podwójnie
doubly
wyższy
taller
niż
than
Maria.
Maria
d. *Jan
Jan
jest
is
podwójnie
doubly
tak
so
wysoki
tall
jak
how
Maria.
Maria
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Data
Czech
(7) a. Petr
Petr
je
is
dvakrát
twice
vyšší
taller
než
than
Marie.
Marie
‘Petr is two times taller than Marie.’
b. Petr
Petr
je
is
dvakrát
twice
tak
so
vysoký
tall
jako
how
Marie.
Marie
‘Petr is twice as tall as Marie.’
c. Petr
Petr
je
is
dvojnásobně
doubly
vyšší
taller
než
than
Marie.
Marie
‘Petr is two times taller than Marie.’
d. *Petr
Petr
je
is
dvojnásobně
doubly
tak
so
vysoký
tall
jako
how
Marie.
Marie
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Data
Vietnamese
(8) a. Petr
Petr
cao
tall
hơn
than
Marie
Marie
hai-lần.
twice
‘Petr is two times taller than Marie.’
b. Petr
Petr
cao
tall
Marie
Marie
hai-lần.
twice
‘Petr is twice as tall as Marie.’
c. Petr
Petr
cao
tall
hơn
than
gấp-đôi
doubly
Marie.
Marie
‘Petr is two times taller than Marie.’
d. Petr
Petr
cao
tall
gắp-đôi
doubly
Marie.
Marie
‘Petr is twice as tall as Marie.’
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Data
Attested patterns
Czech
EN DN
COMP
EQ *
Vietnamese
EN DN
COMP
EQ
German
EN DN
COMP *
EQ
Macedonian
EN DN
COMP
EQ * *
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Data
Attested patterns
Polish
EN DN
COMP *
EQ *
English
EN1 EN2 DN
COMP * *
EQ *
possibly more patterns to be observed
Generalization:
event numerals: always compatible with COMP/EQ
degree numerals: more variation
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Puzzle
EN/DN asymmetry in COMP/EQ
Czech
EN DN
COMP
EQ *
Vietnamese
EN DN
COMP
EQ
Proposed solution:
two distinct strategies of degree multiplication
interaction with the explicit/implicit mode of comparison
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More data
Quantification over events
Czech
event numerals ⇒
degree numerals ⇒ *
(9) a. Petr
Petr
napsal
wrote
mamince
letter
dopis
for-mother
dvakrát.
twice
‘Petr wrote the letter to his mother twice.’
b. *Petr
Petr
napsal
wrote
mamince
letter
dopis
for-mother
dvojnásobně.
doubly
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More data
Quantification over events
Vietnamese
event numerals ⇒
degree numerals ⇒ *
(10) a. Petr
Petr
đã
already
viết
write
thư
letter
cho
for
mẹ
mother
hai-lần
twice
‘Petr wrote the letter to his mother twice.’
b. *Petr
Petr
đã
already
viết
write
thư
letter
gấp-đôi
doubly
cho
for
mẹ.
mother
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More data
Typal compatibility
Czech stacked numerals
(11) a. Petrovi
for-Petr
se
refl
to
this
třikrát
thrice
dvojnásobně
doubly
vyplatilo.
payed-off
‘For Petr it payed off doubly three times.’
b. *Petrovi
for-Petr
se
refl
to
this
dvojnásobně
doubly
třikrát
thrice
vyplatilo.
paid-off
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More data
Predicate position
Czech adjectival degree numerals
(12) …škoda
damage
dosahuje
reaches
asi
approximately
50
50
tisíc
thousand
korun.
crowns
Hodnota
value
uchráněného
saved
majetku
property
je
is
dvojnásobná.
double
‘…the damages reach approximately 50 000 CZK. The
value of saved property is twice as high.’ (CNC)
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More data
Generalization
event numerals
quantification over events and degree environments
our focus: degree environments
future research: unified analysis
degree numerals
only degree environments
predicate position
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Framework
General assumptions
ontology: degrees (type d) ordered into scales
scale: D, >, DIM
D: a set of degrees
>: an ordering relation on D
DIM: a dimension of measurement, e.g., height
interval-based approach to degrees (Kennedy 2001,
Schwarzschild & Wilkinson 2002)
measure functions associate entities with scales (Solt 2014)
semantics of gradable adjectives
(13) tall = λdλx.µheight(x) ≥ d
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Framework
Syntactic structure of comparatives
small DegP view (Bresnan 1973, Heim 2000)
-er + than-clause ⇒ constituent at LF
DegP ⇒ argument of the gradable predicate
(14) AP
DegP
Deg
-er
PP
[than…]
A
tall
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Framework
Clausal comparatives
elided clause ⇒ the maximal interval corresponding to a
standard of comparison (Pancheva 2006)
(15) a. Petr is taller than Marie.
b. LF: [IP [IP Petr is d1-tall] [ DegP -er1 [PP than
[CP Marie is d-tall]]]]
than-clause: a free relative interpreted as a definite
description of degrees ⇒ type d (Heim 2000)
-er: degree quantifier, type d, t , d, t , t
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Framework
Remedy for the type mismatch
non-trivial semantics for than (cf. von Stechow 1984,
Rullmann 1995)
than as a partitive preposition in the domain of degrees
(Pancheva 2006)
(16) than = λd λd.d is part of d d, d, t
than: d → a set of degrees which d is member of
example
standard of comparison ⇒ 170 cm
than-clause ⇒ a set of degrees in the interval 0–170 cm
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Framework
(17) DegP
d, t , t
Deg
d, t , d, t , t
-er
PP
d, t
P
d, d, t
than
CP
d
Marie is d1-tall
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Proposal
Comparative
standard view: A > B (von Stechow 1984, Heim 2000,
Schwarzschild 2008)
our proposal
factor COMP: A ≥ B instead of A > B
similar approach: percentage differential COMP (Gobeski
& Morzycki 2017)
(18) -er× = λD λD.max(D) ≥ max(D ) d, t , d, t , t
pragmatic strengthening: ≥ =
independent motivation: differentials
source (neo-Gricean view)
competition between numerals in factor phrases
⇒ scalar implicature (Horn 1972)
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Proposal
Equative
elided clause ⇒ the maximal interval corresponding to a
standard of comparison
(19) a. Petr is as tall as Marie.
b. LF: [IP [IP Petr is d1-tall] [DegP as… as1 [CP Marie
is d-tall]]]
no preposition/complementizer
English: no standard marker than
Slavic: wh-element, e.g., Czech jako (‘how’)
Vietnamese: no marking
25 / 52
Proposal
Equative
COMP: degree quantifier, type d, t , d, t , t
our proposal
EQ: operates on d ⇒ DegP takes the CP directly
similar approach: percentage EQ (Gobeski & Morzycki
2017)
(20) as… as = λdλD.max(D) = d d, d, t , t
COMP vs. EQ: not just > vs. ≥ / = (Rett 2013)
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Proposal
(21) DegP
d, t , t
Deg
d, d, t , t
as… as
CP
d
Marie is d1-tall
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Proposal
Strategies of multiplication
multiplicative
basic
operation from a degree to a degree multiplied by n
predicative
characteristic function of degrees equal to a contextually
salient degree dc multiplied by n
dc = the MAX value of a standard on a proper scale
Predicate Modification (Heim & Kratzer 1998)
applies to degree predicates
28 / 52
Proposal
Semantics for EN/DN
event numeral ⇒ degree multiplier
(22) a. dvakrát/hai-lần = λd.2 × d d, d
b. EN = λnλd.n × d n, d, d
degree numeral ⇒ degree predicate
(23) a. dvojnásobně/gấp-đôi = λd.d = 2 × dc d, t
b. DN = λnλd.d = n × dc n, d, t
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Czech factor comparatives
(24) DegP
d, t , t
Deg
d, t , d, t , t
-ší
‘-er’
PP
d, t
P
d, d, t
než
‘than’
CP
d
EN
d, d
dvakrát
‘twice’
CP
d
Marie is d1-tall
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Czech factor comparatives
Composition
CP: the maximal interval to which Marie is tall
EN: the interval is multiplied by 2
než: CP → set of degrees which are part of that interval
-ší : max operation picks the maximal interval to which
Marie is tall multiplied by 2
Strengthening
competition: dvakrát and higher EN
scalar implicature: ≥ =
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Czech factor comparatives
(25)
DegP
d, t , t
Deg
d, t , d, t , t
-ší
‘-er’
PP
d, t
DN
d, t
dvojnásobně
‘doubly’
PP
d, t
P
d, d, t
než
‘than’
CP
d
Marie is d1-tall
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Czech factor comparatives
Composition
CP: the maximal interval to which Marie is tall
než: CP → set of degrees which are part of that interval
DN + PP: Predicate Modification
each member of the set multiplied by 2
set of degrees that are 2 × bigger than Marie’s height
-ší : max operation picks the maximal interval to which
Marie is tall multiplied by 2
Strengthening
competition: dvojnásobně and higher DN
scalar implicature: ≥ =
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Czech factor comparatives
Interpretation
(26) a. (7-a)/(7-c) = max(λd.µheight(Petr) ≥ d) ≥
max(λd .d = 2 × µheight(Marie))
b. max(λd.µheight(Petr) ≥ d) =
max(λd .d = 2 × µheight(Marie))
true iff the degree to which Petr is tall = the degree to
which Marie is tall multiplied by 2
exemplary scenario
max(µheight(Marie)) = 90
max(µheight(Petr)) = 180
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Czech factor equatives
(27) DegP
d, t , t
Deg
d, d, t , t
tak… jako
‘as…as’
CP
d
EN
d, d
dvakrát
‘twice’
CP
d
Marie is d1-tall
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Czech factor equatives
Composition
CP: the maximal interval to which Marie is tall
EN: the interval is multiplied by 2
no partitive preposition to shift the modified CP
no additional max operation required
tak…jako: equates the maximal interval to which Marie is
tall multiplied by 2 to another degree
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Czech factor equatives
Interpretation
(28) (7-b) = max(λd.µheight(Petr) ≥ d) =
max(λd .d = 2 × µheight(Marie))
true iff the degree to which Petr is tall = the degree to
which Marie is tall multiplied by 2
exemplary scenario
max(µheight(Marie)) = 90
max(µheight(Petr)) = 180
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Czech factor equatives
(29) DegP
d, t , t
Deg
d, d, t , t
tak… jako
‘as…as’
DN
d, t
dvojnásobně
‘doubly’
CP
d
Marie is d1-tall
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Czech factor equatives
Why are degree numerals * in EQ?
type-driven incompatibility
DN: type d, t
CP: type d
no partitive preposition to shift the CP ⇒ no node d, t
Predicate Modification: unavailable
Function Application: DN ( CP ) ⇒ type t, # with Deg
inevitable type mismatch ⇒ ungrammaticality
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Vietnamese comparatives
(30) DegP
d, t , t
Deg
d, t , d, t , t
∅
PP
d, t
DN
d, t
gắp-đôi
‘doubly’
PP
d, t
P
d, d, t
hơn
‘than’
CP
d
EN
d, d
hai-lần
‘twice’
CP
d
Marie is d1-tall
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Explicit vs. implicit comparison
Modes of comparison (Kennedy 2007)
explicit: standard of comparison ⇒ degree
(31) Petr is taller than Marie.
(32) compd = λdλgλx.max(g)(x) ≥ d
implicit: standard of comparison ⇒ individual
(33) Compared to Marie, Petr is tall.
(34) compe = λyλgλx.max(g)(x) ≥ max(g)(y)
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Explicit vs. implicit comparison
Degree parameter in Vietnamese (Beck at al. 2009)
subcomparatives
(35) Xe-ô-tô
cl-car
to
big
hơn
than
đường
road
hẹp
narrow
này.
this
‘The car is bigger than the road is narrow.’
degree questions
(36) Anh-ấy
he
thông-minh
smart
thế-nào?
much-how
‘How smart is he?’
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Explicit vs. implicit comparison
Degree parameter in Vietnamese (Beck at al. 2009)
negative islands are ungrammatical
(37) *Petr
Petr
thông-minh
smart
hơn
than
không-ai-cả
nobody
ở
aux
trong
inside
lớp.
class
‘*Petr is smarter than nobody in his class.’
Explicit/implicit mode of comparison
construction specific rather than language parametrized
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Explicit vs. implicit comparison
Scenario 1
book A is 400 pages long, book B is 200 pages long
(38) a. Quyển
book
A
A
dầy
thick
quyển
book
B
B
hai-lần.
twice
‘Book A is twice as long as book B.’
b. Quyển
book
A
A
dầy
thick
gấp-đôi
dubly
quển
book
B.
B
‘Book A is twice as long as book B.’
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Explicit vs. implicit comparison
Scenario 2
book A is 400 pages long, book B is 250 pages long
(39) a. #Quyển
book
A
A
dầy
thick
quyển
book
B
B
hai-lần.
twice
‘Book A is twice as long as book B.’
b. Quyển
book
A
A
dầy
thick
gấp-đôi
dubly
quển
book
B.
B
‘Book A is twice as long as book B.’
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Explicit vs. implicit comparison
Vietnamese equatives
deficient marking
underspecification
ambiguity: explicit/implicit comparison
distinct linearizations ⇒ different structures
Vietnamese EN: explicit comparison
Vietnamese DN: implicit (covers explicit)
Implicit comparison
null ImplComp: type e, d, t , t
individual → degree serving as a standard
EQ ordering manipulates the context ⇒ the positive form
is true of both standard and correlate
vague value
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Vietnamese explicit equatives
(40) DegP
d, t , t
Deg
d, d, t , t
∅
CP
d
EN
d, d
hai-lần
‘twice’
CP
d
Marie is d1-tall
47 / 52
Vietnamese implicit equatives
(41)
ImplCompP
d, t , t
ImplComp
e, d, t , t
∅
DP
e
Marie
AP
d, t
DN
d, t
gấp-đôi
‘doubly’
AP
d, t
cao
‘tall’
48 / 52
Vietnamese explicit equative structures
Different equative structures
sufficient marking: equative verbs
explicit mode of comparison
(42) Petr
Petr
cao
tall
bằng/nhu
equal
Marie.
Marie
‘Petr is equal to Marie in height.’
(43) *Petr
Petr
cao
tall
bằng/nhu
equal
Marie
Marie
gấp-đôi
doubly
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Conclusion
Observations:
factor phrases target comparatives and equatives
factor phrases involve event numerals and degree numerals
EN/DN: cross-linguistic variation in COMP and EQ
Proposal:
2 strategies of multiplication
EN: multiplicative d, d
DN: predicative d, t
interaction with the implicit/explicit mode of comparison
more interactions to be explored
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Thanks!
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