Subatomic quantification
Marcin Wągiel
OJ591 Topics in linguistics
1 / 91
Introduction
Ontological intuition dating back to Pre-Socratics
cf. Varzi (2016)
▶ entities are often made up of smaller entities (parts)
related to each other in a particular manner
Cognitive fact
Elkind et al. (1964), Kimchi (1993), Boisvert et al. (1999)
▶ humans conceive entities as being made up of smaller
entities related to each other in a particular manner
Figure 1: Part-whole perception (Elkind et al. 1964) 2 / 91
Introduction
Vital question
▶ to what extent is this fact relevant for natural language
semantics?
Claims
▶ natural language semantics is sensitive to subatomic
part-whole structures
▶ subatomic quantification (quantification over parts) is
subject to identical restrictions as quantification over
wholes
▶ some quantificational operations including counting
presuppose particular topological relations
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Outline
▶ Standard assumptions in lattice-theoretic approaches
▶ The three claims
1) Topological relations in natural language
2) General counting principles
3) Subatomic quantification
▶ Evidence
▶ cross-linguistic behavior of partitives
▶ Italian irregular plurals
▶ Polish half words
▶ multipliers such as English double
▶ Analysis
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Lattice-theoretic approaches to pluralities
Standard assumptions
▶ standard mereology
Link (1983) and many others
▶ only ⊑ and ⊔ ⇒ entities equivalent to sums of their parts
▶ opposing views
▶ mereotopology (Grimm 2012)
▶ probabilistic Type Theory with Records (Sutton &
Filip 2017)
▶ sorted domains ⇒ ⊑m × ⊑i, ⊑e × ⊑p
e.g., Link (1983), Bach (1986)
▶ opposing views
▶ situated part structure (Moltmann 1997, 1998)
▶ Iceberg semantics (Landman 2016)
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Lattice-theoretic approaches to pluralities
▶ no relationship between ⊑ and intuitive part-of relations
▶ “it should be this way”
e.g., Pianesi (2002), Champollion (2010)
▶ opposing views
▶ situated part structure (Moltmann 1997, 1998)
▶ Iceberg semantics (Landman 2016)
▶ atomicity: atoms ⇒ objects without proper parts
▶ opposing views
▶ natural units (Krifka 1989)
▶ Iceberg semantics (Landman 2016)
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Lattice-theoretic approaches to pluralities
Mereology
▶ study of parthood ⇒ parts and wholes
Leśniewski (1916), Leonard & Goodman (1940); Link (1983)
▶ set theory: set membership ∈ vs. subset relation ⊆ ⇒
{a} ̸= a
▶ mereology ⇒ no sets as abstract objects
▶ one primitive parthood relation ⊑
(1) Reflexivity
∀x[x ⊑ x]
(2) Transitivity
∀x∀y∀z[(x ⊑ y ∧ y ⊑ z) → x ⊑ z]
(3) Antisymmetry
∀x∀y[(x ⊑ y ∧ y ⊑ x) → x = y]
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Lattice-theoretic approaches to pluralities
Semi-lattice
▶ partial order
▶ parthood ⊑ and sum formation ⊔
a ⊔ b ⊔ c
a ⊔ ca ⊔ b b ⊔ c
a b c
Figure 2: Semi-lattice
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Lattice-theoretic approaches to pluralities
Atomicity
▶ proper parthood ⊏ ⇒ not reflexive
▶ atom ⊏ mereological concept
▶ atom ⇒ entity which has no proper parts
▶ atomic vs. atomless mereologies
(4) Proper part
x ⊏ y
def
= x ⊑ y ∧ ¬(y ⊑ x)
(5) Atom
∀x[atom(x) ↔ ¬∃y[y ⊏ x]]
(6) Atomicity
∀x∃y[y ⊑ x ∧ ¬∃z[z ⊏ y]]
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Mereotopological structures in natural language
Mereotopology
▶ mereotopology
Kuratowski (1922), Casati & Varzi (1999), Grimm (2012)
▶ mereology augmented with topological relations
▶ no atomicity understood as having no proper parts
▶ individual ⇒ a maximally strongly self-connected sum of
overlapping entities making up a whole
▶ semantics of number
▶ singular individuals ⇒ mereotopology, topological
relations between parts
▶ plural individuals ⇒ mereology, no topological
commitments
▶ further applications possible
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Mereotopological structures in natural language
▶ NL expressions sensitive to topological notions
▶ count nouns, aggregates, collective number
Grimm (2012)
▶ swarm nouns
Henderson (2017)
▶ Slavic derived aggregate nouns
Grimm & Dočekal (to appear)
▶ verbs of separation such as dismember, dismantle
▶ expressions involving quantification over parts
▶ part words
Wągiel (2018)
▶ multipliers
Wągiel (to appear)
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Mereotopology
Mereology + topological notions
Casati & Varzi (1999), Grimm (2012)
▶ connectedness c ⇒ primitive relation
▶ implied by overlap
(7) Reflexivity
∀x[c(x, x)]
(8) Symmetry
∀x∀y[c(x, y) ↔ c(y, x)]
(9) Parthood → connectedness
∀x∀y[x ⊑ y → ∀z[c(x, z) → c(z, y)]]
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Mereotopology
Mereology + topological notions
Casati & Varzi (1999), Grimm (2012)
▶ connectedness c ⇒ not transitive
▶ a and b ⇒ connected
▶ b and c ⇒ connected
▶ a and c ⇒ not connected
a b c
Figure 3: Connectedness and transitivity
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Mereotopology
Mereology + topological notions
Casati & Varzi (1999), Grimm (2012)
▶ internal part ⇒ entity included in a whole
▶ internal overlap ⇒ part of an entity included
▶ tangential overlap ⇒ ‘touching’ entities
(10) Internal part
ip(x, y)
def
= x ⊑ y ∧ ∀z[c(z, x) → o(z, y)]
(11) Internal overlap
io(x, y)
def
= ∃z[ip(z, x) ∧ ip(z, y)]
(12) Tangential overlap
to(x, y)
def
= o(x, y) ∧ ¬io(x, y)
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Mereotopology
a b
Figure 4: Internal part
a b
Figure 5: Internal overlap
a b
Figure 6: Tangential overlap
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Mereotopology
Mereology + topological notions
Casati & Varzi (1999), Grimm (2012)
▶ interior, exterior, closure, boundary
(13) Interior
ix
def
= ⊕X where X = {y : IP(y, x) = True}
(14) Exterior
ex
def
= i(−x)
(15) Closure
cx
def
= −(ex)
(16) Boundary
bx
def
= −(ix ⊕ ex)
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Mereotopology
a
Figure 7: Interior
a
Figure 8: Exterior
a
Figure 9: Closure
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Mereotopology
Self-connected entity
(17) sc(x)
def
= ∀yz[∀w(o(w, x) ↔ (o(w, y) ∨ o(w, z))) →
c(y, z)]
▶ any two parts that form the whole are connected to each
other
Strongly self-connected entity
(18) ssc(x)
def
= sc(x) ∧ sc(ix)
▶ entity’s interior is self-connected ⇒ excludes touching
objects
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Mereotopology
Maximally strongly self-connected relative to a property
(19) mssc(P)(x)
def
=
P(x) ∧ ssc(x) ∧ ∀y[P(y) ∧ ssc(y) ∧ o(y, x) ↔ y ⊑ x]
Strongly self-connected
▶ every part of the entity is connected to (overlaps) the
whole
Maximality
▶ anything else which has that property, is strongly
self-connected, and overlaps is part of it
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Mereotopology
Capturing objects
▶ integrated wholes ⇒ parthood and connectedness
▶ arbitrary sums ⇒ only parthood
a b c d
Figure 10: Wholes vs. sums
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Mereotopology
a ⊔ b ⊔ c
a ⊔ ca ⊔ b b ⊔ c
a b c
connectedness
parthood
Figure 11: Parthood and connectedness (based on Grimm 2012, p.
136)
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General counting principles
▶ mapping entities to numbers ⇒ 1-to-1 correspondence
▶ non-overlap ⇒ disjoint entities (Landman 2011, 2016)
▶ maximality ⇒ mereological exhaustivity
▶ integrity ⇒ individuated and integrated whole
Figure 12: Counting
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General counting principles
▶ illegal counting
▶ assigning a number to less than a whole entity
▶ summing up complementary parts
▶ overlapping entities
Figure 13: Illegal counting
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General counting principles
▶ independent evidence
Shipley & Shepperson (1990), Dehaene (1997)
▶ children between 3 and 4 years
▶ count only discrete integrated objects
Figure 14: Relevance of integrity in counting (Dehaene 1997, p.
60; adapted from Shipley and Shepperson 1990)
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General counting principles
▶ counting and measuring ⇒ independent operations
▶ distinct syntax and semantics (Rothstein 2017)
▶ counting indicates integrity
▶ measuring does not
▶ monotonic systems of measurement track part-whole
relations (Schwarzschild 2002) ⇒ not topological
relations
▶ numeral phrases ⇒ counting / measure ambiguity
▶ counting ⇒ measuring shift
▶ possible but restricted
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General counting rules
(20) Context: John is cooking with his child. They put
three whole apples on a table. John says:
a. There are three apples on the table…
b. Let’s count them together: one, two, three.
(21) Context: John is cooking with his child. They sliced
three apples and put the slices into a bowl. John says:
a. There are three apples in the bowl…
b. #Let’s count them together: one, two, three.
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Subatomic quantification
▶ natural language semantics is sensitive to the fact that
objects consist of parts
▶ linguistic expressions involving subatomic quantification
▶ whole adjectives (cf. Morzycki 2002)
▶ partitives such as part and half
▶ multipliers such as double (Wągiel to appear)
▶ enhanced mereological structure
▶ interaction between ⊑ associated with singularities and
pluralities
▶ interaction with additional topological relations ⇒
different mereotopological structures
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Subatomic quantification
▶ one universal mechanism allowing for counting
▶ applicable on different mereotopological levels
▶ interaction with specific properties of particular types of
entities
▶ quantification over wholes/parts ⇒ identical restrictions
▶ principles of non-overlap, maximality, and integrity
▶ structured parthood ⇒ counting of cognitively salient
parts
▶ parts ⇒ not necessarily topological commitments
▶ countability ⇒ only integrated entities
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Subatomic quantification
▶ counting of parts
▶ counted parts ⇒ maximal integrated entities
▶ counted parts cannot overlap
Figure 15: Counting of parts
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Subatomic quantification
▶ illegal counting of parts
▶ counting discontinuous parts of an object
▶ overlapping parts
Figure 16: Illegal counting of parts
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Partitive expressions
Argument for a unified mereology
▶ analogy between partitives involving singulars and plurals
Moltmann (1997, 1998)
▶ suggests unified part-whole structures
(22) a. Teil
part
des
of-the
Apfels
applegen
‘part of the apple’
b. Teil
part
der
of-the
Äpfel
apples
‘some of the apples’
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Partitive expressions
▶ in English the analogy does not hold
Schwarzschild (1996)
(23) a. part of the apple
b. #part of the apples
▶ systematic ⇒ attested in many languages
Germanic, Romance, Slavic, Celtic, Finno-Ugric, Semitic, Basque
(24) a. parte
part
del
of-the
muro
wall
‘part of the wall’
b. parte
part
dei
of-the
muri
walls
‘some of the walls’
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Partitive expressions
▶ Dutch
(25) a. deel
part
van
of
de
the
appel
apple
‘part of the apple’
b. deel
part
van
of
de
the
appels
apples
‘some of the apples’
▶ Russian
(26) a. čast’
part
jabloka
applegen
‘part of the apple’
b. čast’
part
jablok
applesgen
‘some of the apples’ 33 / 91
Partitive expressions
▶ Portuguese
(27) a. parte
part
da
the
maçã
apple
‘part of the apple’
b. parte
part
das
the
maçãs
apples
‘some of the apples’
▶ Irish
(28) a. cuid
part
den
from-the
úll
apple
‘part of the apple’
b. cuid
part
de
from
na
the
húlla
apples
‘some of the apples’ 34 / 91
Partitive expressions
▶ Hungarian
(29) a. az
the
alma
apple
egy
a
része
partposs
‘part of the apple’
b. az
the
almák
apples
egy
a
része
partposs
‘some of the apples’
▶ Hebrew
(30) a. xelek
part
me-ha-baxur
from-the-boy
‘part of the boy’
b. xelek
part
me-ha-baxur-im
from-the-boy-s
‘some of the boys’ 35 / 91
Partitive expressions
▶ Basque
(31) a. sagarraren
applegen
zati
part
bat
a
‘part of the apple’
b. sagarren
applesgenpart
zati
a
bat
‘some of the apples’
36 / 91
Partitive expressions
▶ proportional quantifiers and fractions ⇒ similar analogy
▶ systematic
▶ cross-linguistically widespread
(32) a. most of the apple
b. most of the apples
(33) a. half of the apple
b. half of the apples
(34) a. two thirds of the apple
b. two thirds of the apples
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Partitive expressions
▶ partitives involving number-neutral expressions
▶ object mass nouns
▶ pluralia tantum
▶ ambiguity between a singular and plural reading
▶ systematic ⇒ attested in many languages
(35) a. část
part
obuvi
footweargen
‘part of the footwear/some of the footwear’
b. část
part
nůžek
scissorsgen
‘part of the scissors/some of the scissors’
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Partitive expressions
▶ languages with general number such as Japanese
Sauerland & Yatsushiro (2004), Watanabe (2013)
▶ number-neutral nominal
▶ ambiguity between a singular and plural reading
(36) a. Ringo-no
apple-gen
ichibu-ga
part-nom
kusatteiru.
is.rotten
‘Part of the apple is rotten/Some of the apples
are rotten.’
b. Ringo-no
apple-gen
hotondo-ga
most-nom
kusatteiru.
is.rotten
‘Most of the apple(s) is/are rotten.’
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Partitive expressions
Counterargument for a unified mereology
Schwarzschild (1996)
▶ uncountability of part words in plural partitives
▶ only part-of-a-singularity reading
▶ systematic and cross-linguistically widespread
(37) a. tre
three
parti
parts
del
of-the
muro
wall
‘three parts of the wall’
b. #tre
three
parti
parts
dei
of-the
muri
walls
(i) * if counting walls
(ii) ✓ if counting parts of walls
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Partitive expressions
▶ animate nouns ⇒ stronger effects
(38) a. Parte
part
dei
of-the
ragazzi
boys
erano
were
in
in
Texas.
Texas
‘Some of the boys were in Texas.’
b. #Tre
three
parti
parts
dei
of-the
ragazzi
boys
erano
were
in
in
Texas.
Texas
(39) a. Część
part
chłopców
boysgen
śpi.
sleeps
‘Some of the boys sleep.’
b. #Trzy
three
części
parts
chłopców
boysgen
śpią.
sleep
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Partitive expressions
▶ exhaustive quantifiers and numeric contradictions
(40) a. #Trzy
three
połowy
halves
muru
wallgen
są
are
czerwone.
red
b. Trzy
three
połowy
halves
murów
wallsgen
są
are
czerwone.
red
‘Three halves of the walls are red.’
(41) a. Obie
both
połowy
halves
muru
wallgen
są
are
czerwone.
red
‘Both halves of the wall are red.’
b. #Obie
both
połowy
halves
murów
wallsgen
są
are
czerwone.
red
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Partitive expressions
Summary of the attested patterns
▶ interpretative asymmetry in counting environments
▶ plural partitives ⇒ part-of-a-plurality reading
▶ count partitives ⇒ only part-of-a-singularity reading
singulars plurals
bare count bare count
subatomic quantification ✓ ✓ * ✓
quantification over wholes * * ✓ *
Table 1: Properties of partitive words
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Partitive expressions
Implications
Schwarzschild (1996)
▶ Italian and English do not disagree with respect to their
ontologies
▶ singularities and pluralities ⇒ two distinct mereological
structures
▶ part modeled as an existential ‘pieces’ quantifier reverse
of each ⇒ selects for plurality-denoting complements
44 / 91
Partitive expressions
Objection
▶ cardinals do not count pluralities ⇒ they count
singularities
▶ domain of quantification ⇒ set of atoms
e.g., Kratzer (1989), Chierchia (1998), Landman (2000)
▶ part words actually pattern with regular nominals
(42) a. three parts of the walls
(i) #three pluralities of parts of walls
(ii) plurality of three parts of walls
b. three walls
(i) #three pluralities of walls
(ii) plurality of three walls
45 / 91
Partitive expressions
Zeugma test
cf. Zwicky & Sadock (1975), Lasersohn (1995)
▶ indeterminacy (non-specificity) ⇒ no zeugma effect
▶ ambiguous expressions ⇒ zeugma effect
▶ part ⇒ not ambiguous with respect to ⊑m and ⊑i
(43) Ein
a
Teil
part
des
thegen
Apfels
applegen
und
and
der
thegen
Birnen
pearsgen
sind
are
verfault.
rotten
‘Part of the apple and some of the pears got spoiled.’
(44) Ein
a
Teil
part
der
thegen
Birnen
pearsgen
und
and
des
thegen
Apfels
applegen
sind
are
verfault.
rotten
‘Some of the pears and part of the apple got spoiled.’
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Italian irregular plurals
Inflectional class
▶ morphological and semantic idiosyncrasy
Acquaviva (2008)
▶ gender shift in the plural
(45) a. il
themasc.sg
tuo
yourmasc.sg
dito
fingersg
‘your finger’
b. le
thefem.pl
tue
yourfem.pl
dita
fingerpl
‘your fingers’
47 / 91
Italian irregular plurals
▶ nouns with both regular and irregular counterparts
(46) a. muro ∼
wallmasc.sg
muri ∼
wallmasc.pl
mura
wallfem.pl
‘wall ∼ walls ∼ walls (in a complex)’
b. osso ∼
bonemasc.sg
ossi ∼
bonemasc.pl
ossa
bonefem.pl
‘bone ∼ bones ∼ bones (in a skeleton)’
▶ irregular forms ⇒ collectivizers (Ojeda 1995) or inherently
encoding cohesion of referents (Acquaviva 2008)
▶ arguably a notion of connectedness of parts is involved
48 / 91
Italian irregular plurals
Observation
▶ partitives with irregular plurals ⇒ compatible with
cardinals
▶ quantification over parts of singularities or pluralities
(47) tre
three
parti
parts
delle
of-the
mura
wallcoll
‘three parts of the complex formed by the walls’
(i) ✓ if counting parts of walls
(ii) ✓ if counting individual walls
(iii) ✓ if counting continuous pluralities of walls
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Italian irregular plurals
(48) tre
three
parti
parts
delle
of-the
ossa
bonecoll
‘three parts of the skeleton formed by the bones’
(i) ✓ if counting bones
(ii) ✓ if counting parts of bones
(iii) ✓ if counting continuous pluralities of bones,
femur + knee, ulna + radius, and skull + neck
50 / 91
Italian irregular plurals
Italian partitives
▶ interaction between partitivity and number
▶ quantification over wholes
▶ subatomic quantification
▶ countability
singulars regular pl irregular pl
bare count bare count bare count
subatomic quantification ✓ ✓ * ✓ ✓ ✓
quantification over wholes * * ✓ * ✓ ✓
Table 2: Properties of Italian parte ‘part’
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Italian irregular plurals
Explanation
▶ interaction between partitives and number ⇒
(un)countability
▶ regular plurals ⇒ no topological relations between parts
▶ parts of a plurality do not form an integrated entity
▶ multiple overlapping parts of a plurality
▶ violation of the general counting rules ⇒ uncountability
▶ irregular plurals ⇒ connected parts
▶ parts of a plurality form a cohesive whole
▶ counting is possible as long as it operates on integrated
objects
52 / 91
Italian irregular plurals
Conclusions
▶ part words can operate both at the atomic and subatomic
level of a part-whole structure
▶ partitives employ a general parthood relation
▶ countability results from the interaction between the
meaning of a part word and the meaning of a singular or
plural NP
▶ only integrated parts (proper or improper) of integrated
wholes can be assigned a number when counting
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Polish half words
Three distinct expressions
▶ morphologically derived from one another
(49) a. pół
root
‘half1’
b. poł-ow-a
root-derivational.suffix-inflectional.marker
‘half2’
c. poł-ów-k-a
root-derivational.suffix1-deriv.suffix2-infl.marker
‘half3’
54 / 91
Polish half words
▶ pół ⇒ incompatible with cumulative predicates
(50) a. pół
half1
jabłka
applegen
‘half of the apple’
b. pół
half1
stosu
pilegen
(jabłek)
(applesgen)
‘half of the pile (of apples)’
c. #pół
half1
jabłek
applesgen
d. #pół
half1
soku
juicegen
55 / 91
Polish half words
▶ połowa ⇒ no distributional restrictions
(51) a. połowa
half2
jabłka
applegen
‘half of the apple’
b. połowa
half2
stosu
pilegen
(jabłek)
(applesgen)
‘half of the pile (of apples)’
c. połowa
half2
jabłek
applesgen
‘half of the apples’
d. połowa
half2
soku
juicegen
‘half of the juice’
56 / 91
Polish half words
▶ połówka ⇒ compatible only with regular concrete
singulars
(52) a. połówka
half3
jabłka
applegen
‘half of the apple’
b. #połówka
half3
stosu
pilegen
(jabłek)
(applesgen)
c. #połówka
half3
jabłek
applesgen
d. #połówka
half3
soku
juicegen
57 / 91
Polish half words
Distribution of Polish half expressions
▶ three distinct categories
▶ collectives ⇒ put aside
▶ sensitivity to topological notions
singulars collectives plurals mass nouns
połowa ✓ ✓ ✓ ✓
pół ✓ ✓ * *
połówka ✓ * * *
Table 3: Distribution of Polish half -words
58 / 91
Polish half words
Observation
▶ available extensions of partitives differ ⇒ topological
sensitivity
(53) a. pół
half1
jabłka
applegen
‘half of the apple’ ✓ cont.part / ✓ discont.part
b. połowa
half2
jabłka
applegen
‘half of the apple’ ✓ cont.part / ✓ discont.part
c. połówka
half3
jabłka
applegen
‘half of the apple’ ✓ cont.part / # discont.part
59 / 91
Polish half words
Figure 17: Continuous half Figure 18: Discontinuous half
continuous part discontinous part
połowa ✓ ✓
pół ✓ ✓
połówka ✓ *
Table 4: Denotations of Polish half -words
60 / 91
Polish quarter words
More evidence
▶ Polish quarter-words ⇒ topological sensitivity
singulars collectives plurals mass nouns
jedna czwarta ✓ ✓ ✓ ✓
ćwierć ✓ ✓ * *
ćwiartka ✓ * * *
Table 5: Distribution of Polish quarter-words
continuous part discontinous part
jedna czwarta ✓ ✓
ćwierć ✓ ✓
ćwiartka ✓ *
Table 6: Denotations of Polish quarter-words 61 / 91
Cross-linguistic perspective
Diagnostics to detect topology-sensitive partitive expressions
▶ the flag test
▶ continuous vs. discontinuous parts
▶ easily distinguishable properties
Figure 19: Flag AB Figure 20: Flag ABA
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Cross-linguistic perspective
Different structures ⇒ similar semantic effect
cross-linguistically
▶ English
(54) a. Half the flag is red.
(i) AB
(ii) ABA
b. A half of the flag is red.
(i) AB
(ii) #ABA
63 / 91
Cross-linguistic perspective
▶ German
(55) a. Die
the
Hälfte
half
von
of
der
the
Fahne
flag
ist
is
rot.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
b. Die
the
eine
a/one
Hälfte
half
der
of-the
Fahne
flag
ist
is
rot.
red
‘The half of the flag is red.’
(i) AB
(ii) #ABA
64 / 91
Cross-linguistic perspective
▶ Dutch
(56) a. De
the
helft
half
van
of
de
the
vlag
flag
is
is
rood.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
b. De
the
halve
half
vlag
flag
is
is
rood.
red
‘The half of the flag is red.’
(i) AB
(ii) #ABA
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Cross-linguistic perspective
▶ Portuguese
(57) a. Metade
half
da
the
bandeira
flag
é
is
vermelha.
red
‘Half the flag is red.’
(i) AB
(ii) ABA
b. Meia
half
bandeira
flag
é
is
vermelha
red
‘A half of the flag is red.’
(i) AB
(ii) #ABA
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Cross-linguistic perspective
▶ Mandarin
(58) a. guó
national
qí
flag
de
DE
yí-bàn
one-half
shì
COP
hóng
red
de.
DE
‘Half the national flag is red.’
(i) AB
(ii) ABA
b. bàn-miàn
half-CL
guó
national
qí
flag
shì
COP
hóng
red
de.
DE
‘A half of the national flag is red.’
(i) AB
(ii) #ABA
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Multipliers
Neglected class of numerical expressions
▶ cross-linguistically widespread category
▶ attested also in non-IE languages
(59) a. double
b. doppelt German
c. doppio Italian
d. dvojnoj Russian
e. dvigubas Lithuanian
f. dupla Hungarian
g. shuāng Mandarin
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Multipliers
Expressions dedicated to counting parts
▶ entailment ⇒ complex inner structure
(60) a. The Pschent is a double crown.
b. ⊨The Pschent consists of two parts.
Figure 21: Pschent Figure 22: Deshret Figure 23: Hedjet
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Multipliers
More examples
▶ some frequent collocates in COCA
(61) a. double bracket
b. double sink
c. double tomb
d. double canoe
e. double flute
f. double chin
g. double layer
h. double glazing
i. double rainbow
j. double star
k. double hamburger
l. double shotgun
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Multipliers
Non-trivial quantificational behavior
▶ quantification over parts rather than wholes
▶ adjectival properties
▶ modified NPs ⇒ always countable (Universal Packager)
(62) a. three crowns
b. three double crowns
(63) a. #three coffees
b. three double coffees
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Multipliers
Relationship between multipliers and cardinals
▶ Slavic and Baltic multipliers ⇒ derived from numeral
roots
▶ multiplicative affix ⇒ classifier
(64) a. dv-a
numeral.root-infl.marker
‘two’ Russian
b. dv-oj-n-oj
numeral.root-stem-mult.suffix-infl.marker
‘double’ Russian
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Multipliers
▶ Slavic and Baltic multipliers
(65) a. dwa Polish
b. podwójny
(66) a. dva Czech
b. dvojitý
(67) a. dva BCS
b. dvostruki
(68) a. du Lithuanian
b. dvigubas
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Multipliers
Quantification over cognitively salient parts
▶ self-sufficient parts ⇒ property comparable to the whole
▶ essential parts
Possible extensions
▶ mass nouns ⇒ quantification over parts of portions
▶ event nominals ⇒ parts of events
▶ role nouns ⇒ parts of roles
Zobel (2017)
(69) a. double vodka
b. double murder
c. double agent
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Data summary
Cross-linguistic distribution of partitives
▶ singulars and plurals ⇒ unified part-whole structures
▶ differences ⇒ topological notions
Italian irregular plurals
▶ countability ⇒ sensitive to integrity
▶ both at the subatomic and superatomic level
Polish half words
▶ topological sensitivity
▶ expressed formally
Multipliers
▶ numerical expressions devised to count parts
▶ identical constraints on counting
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Analysis
Count nouns
▶ mssc entities ⇒ integrated wholes ⇒ no atomicity
(70) Count noun
apple = λx[mssc(apple)(x)]
Pluralization
▶ presupposition ⇒ mssc predicates
▶ algebraic closure (Link 1983)
▶ no topological constraints
(71) PL = λP . Pmssc[∗
P]
(72) apples = PL ( apple ) =
λx
[
∗
(
λy[mssc(apple)(y)]
)
(x)
]
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Analysis
Cardinals
▶ complex expressions ⇒ derived from numeral roots
▶ predicate modifiers
Ionin & Matushansky (2006), Chierchia (2010)
▶ classifier semantics ⇒ shift from names of numbers
Rothstein (2013), Sudo (2016)
▶ classifier CL# ⇒ measure function #(P)
▶ require mssc predicates ⇒ counts integrated wholes
(73) Measure function #(P)
∀P∀x[#(P)(x) = 1 iff mssc(P)(x)]
(74) Cardinal numeral
two = CL# (
√
tw ) =
λP. Pmssc λx[*P(x) ∧ #(P)(x) = 2]
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Analysis
Multipliers
▶ complex expressions ⇒ derived from numeral roots
▶ names of numbers ⇒ predicate modifiers
▶ classifier CL⊞ ⇒ measure function ⊞(P)
▶ count essential parts of mssc entities
(75) Measure function ⊞(P)
∀P∀x[⊞(P)(x) = 1 iff
mssc(P)(x)∧∃y[y ⊑ x∧essential(P)(y)∧#(y) = 1]]
(76) Polish multiplier
podwójny = CL⊞ (
√
dw ) =
λP. Pmssc λx[P(x) ∧ ⊞(P)(x) = 2]
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Analysis
Partitives
▶ partitive constraint ⇒ entity-denoting embedded DP
▶ part words
▶ partitivity ⇒ proper parthood (Barker 1998)
▶ half words
▶ vague ⇒ correspond to ≈ 50%
▶ contextually conditioned measure function µ similar to
more (Bale & Barner 2009)
▶ different measures for different NPs ⇒ number, volume
(77) PART = λyλx[x ⊏ y]
(78) HALF = λyλx[x ⊏ y ∧ µ(x) ≈ µ(y) × 0.5]
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Analysis
Partitioning
▶ partitioning function π ⇒ non-overlap
▶ relative atomicity ⇒ irrelevant
▶ multiple possible partitions
(79) Partitioning function π
for any P and any x and y in π(P)
¬∃z[z ⊑ x ∧ z ⊑ y]
Individuation
▶ individuation of parts ⇒ non-overlap + integrity
▶ individuating element IND ⇒ π + mssc
(80) Individuating element
IND = λPλx[mssc
(
π(P)
)
(x)]
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Analysis
Partitive words
▶ bare partitivity ⇒ topological neutrality
▶ interaction ⇒ topological sensitivity, individuation
(81) German topology-neutral part word Teil
Teil = λyλx[x ⊏ y]
(82) Polish topology-neutral half word połowa
połowa = λyλx[x ⊏ y ∧ µ(x) ≈ µ(y) × 0.5]
(83) Polish topology-sensitive half word pół
pół = λy . ymssc λx[x ⊏ y ∧ µ(x) ≈ µ(y) × 0.5]
(84) Polish individuating suffix -k-k-
= IND = λPλx[mssc
(
π(P)
)
(x)]
(85) Polish individuating half word połówka
połówka = -k- ( pół )
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Analysis
Polish topology-neutral proportional partitive
(86) połowa
half
jabłka
applegen
(87) ⟨e, t⟩
⟨e, ⟨e, t⟩⟩√
połowa
‘half’
e
⟨⟨e, t⟩, e⟩
DEF
⟨e, t⟩
jabłko
‘apple’
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Analysis
Polish topology-sensitive proportional partitive
(88) połówka
half-k
jabłka
applegen
(89) ⟨e, t⟩
⟨⟨e, t⟩, ⟨e, t⟩⟩
-k-
IND
⟨e, t⟩
⟨e, ⟨e, t⟩⟩√
pół
‘half’
e
⟨⟨e, t⟩, e⟩
DEF
⟨e, t⟩
jabłko
‘apple’
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Analysis
German count explicit partitive
(90) zwei
two
Teile
parts
des
thegen
Apfels
applegen
(91) ⟨e, t⟩
⟨⟨e, t⟩, ⟨e, t⟩⟩
n√
zw
2
⟨n, ⟨⟨e, t⟩, ⟨e, t⟩⟩⟩
CL#
⟨e, t⟩
⟨⟨e, t⟩, ⟨e, t⟩⟩
IND
⟨e, t⟩
⟨e, ⟨e, t⟩⟩
Teil
‘part’
e
⟨⟨e, t⟩, e⟩
DEF
⟨e, t⟩
Apfel
‘apple’
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Analysis
Polish multiplier phrase modified by the cardinal
(92) trzy
three
podwójne
double
hamburgery
hamburgers
(93)
⟨e, t⟩
⟨⟨e, t⟩, ⟨e, t⟩⟩
n√
trz
3
⟨n, ⟨⟨e, t⟩, ⟨e, t⟩⟩⟩
CL#
⟨⟨e, t⟩, ⟨e, t⟩⟩
n√
dw
2
⟨n, ⟨⟨e, t⟩, ⟨e, t⟩⟩⟩
CL⊞
⟨e, t⟩
hamburger
‘hamburger’
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Conclusion
Claims
▶ NL semantics ⇒ sensitive to subatomic part-whole
structures
▶ quantification over parts and wholes ⇒ identical
restrictions
▶ counting ⇒ presupposes particular topological relations
Countability
▶ only integrated parts of integrated wholes ⇒ number
▶ improper ⇒ quantification over wholes
▶ proper ⇒ subatomic quantification
86 / 91
Conclusion
Novel evidence
▶ cross-linguistic distribution of partitives
▶ Italian irregular plurals
▶ Polish half words
▶ multipliers
Consequences
▶ mereotopological approach
▶ generalized system of quantification
▶ classifier semantics for numeral expressions
87 / 91
Conclusion
Further investigation
▶ more expressions sensitive to subatomic parthood
▶ adjectives such as whole, entire, complete
▶ adverbs such as wholly, partially
▶ verbs of separation such as dismember, dismantle
▶ cross-linguistic investigation
▶ English: part of ∼ a part of, half of ∼ half a(n)
▶ German: halb ∼ Hälfte
▶ French: part ∼ partie
▶ structured parthood
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References
Acquaviva, P. (2008). Lexical Plurals: A Morphosemantic Approach. Oxford University Press, Oxford.
Bach, E. (1986). The algebra of events. Linguistics and Philosophy, 9(1):5–16.
Bale, A. C. and Barner, D. (2009). The interpretation of functional heads: Using comparatives to explore the
mass/count distinction. Journal of Semantics, 26(3):217–252.
Barker, C. (1998). Partitives, double genitives and anti-uniqueness. Natural Language & Linguistic Theory,
16(4):679–717.
Boisvert, M., Standing, L., and Moller, L. (1999). Successful part-whole perception in young children using
multiple-choice tests. The Journal of Genetic Psychology, 160(2):167–180.
Casati, R. and Varzi, A. C. (1999). Parts and Places: The Structures of Spatial Representation. MIT Press,
Cambridge, MA.
Champollion, L. (2010). Parts of a Whole: Distributivity as a Bridge between Aspect and Measurement. PhD
thesis, University of Pennsylvania.
Chierchia, G. (2010). Mass nouns, vagueness and semantic variation. Synthese, 174(1):99–149.
Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. Oxford University Press, New York,
NY.
Elkind, D., Koegler, R. R., and Go, E. (1964). Studies in perceptual development: Ii. part-whole perception. Child
Development, pages 81–90.
Grimm, S. (2012). Number and Individuation. PhD thesis, Stanford University, California.
Grimm, S. and Dočekal, M. (to appear). Counting aggregates, groups and kinds: Countability from the perspective
of a morphologically complex language. In Counting and Measuring Across Languages. Cambridge University
Press, Cambridge.
Henderson, R. (2017). Swarms: Spatiotemporal grouping across domains. Natural Language & Linguistic Theory,
35(1):161–203.
Ionin, T. and Matushansky, O. (2006). The composition of complex cardinals. Journal of Semantics,
23(4):315–360.
89 / 91
References
Kimchi, R. (1993). Basic-level categorization and part-whole perception in children. Bulletin of the Psychonomic
Society, 31(1):23–26.
Krifka, M. (1989). Nominal reference, temporal constitution and quantification in event semantics. In Bartsch, R.,
van Benthem, J., and von Emde Boas, P., editors, Semantics and Contextual Expression, pages 75–115. Foris
Publications, Dordrecht.
Landman, F. (2011). Count nouns – mass nouns, neat nouns – mess nouns. Baltic International Yearbook of
Cognition, Logic and Communication, 6(1):12.
Landman, F. (2016). Iceberg semantics for count nouns and mass nouns: Classifiers, measures and portions. Baltic
International Yearbook of Cognition, Logic and Communication, 11(1):6.
Lasersohn, P. (1995). Plurality, Conjunction and Events. Kluwer Academic Publishers, Boston.
Link, G. (1983). The logical analysis of plural and mass nouns: A lattice–theoretical approach. In Bäuerle, R.,
Schwarze, C., and von Stechow, A., editors, Meaning, Use, and Interpretation of Language, pages 302–323.
Mouton de Gruyter, Berlin.
Moltmann, F. (1997). Parts and Wholes in Semantics. Oxford University Press, New York, NY.
Moltmann, F. (1998). Part structures, integrity, and the mass-count distinction. Synthese, 116(1):75–111.
Morzycki, M. (2002). Wholes and their covers. In Jackson, B., editor, Proceedings of Semantics and Linguistic
Theory 12, pages 184–203. CLC Publications, Ithaca, NY.
Ojeda, A. E. (1995). The semantics of the Italian double plural. Journal of Semantics, 12(3):213–237.
Pianesi, F. (2002). Friederike Moltmann, ‘Parts and Wholes in Semantics’. Linguistics and Philosophy,
25(1):97–120.
Rothstein, S. (2013). A Fregean semantics for number words. In Aloni, M., Franke, M., and Roelofsen, F., editors,
Proceedings of the 19th Amsterdam Colloquium, pages 179–186.
Rothstein, S. (2017). Semantics for Counting and Measuring. Cambridge University Press, Cambridge.
Sauerland, U. and Yatsushiro, K. (2004). A silent noun in partitives. In Moulton, K. and Wolf, M., editors,
Proceedings of the North East Linguistic Society 34, pages 505–516. GLSA, Amherst, MA.
90 / 91
References
Schwarzschild, R. (1996). Pluralities. Kluwer Academic Publishers, Dordrecht.
Schwarzschild, R. (2002). The grammar of measurement. In Jackson, B., editor, Proceedings of Semantics and
Linguistic Theory 12, pages 225–245. CLC Publications, Ithaca, NY.
Shipley, E. F. and Shepperson, B. (1990). Countable entities: Developmental changes. Cognition, 34(2):109–136.
Sudo, Y. (2016). The semantic role of classifiers in Japanese. Baltic International Yearbook of Cognition, Logic
and Communication, 11(1):1–15.
Sutton, P. R. and Filip, H. (2017). A probabilistic, mereological account of the mass/count distinction. In Hansen,
H. H., Murray, S. E., Sadrzadeh, M., and Zeevat, H., editors, Logic, Language, and Computation. 11th
International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2015, Tbilisi, Georgia,
September 21-26, 2015, pages 146–170. Springer, Berlin.
Varzi, A. C. (2016). Mereology. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics
Research Lab, Stanford University.
Wągiel, M. (to appear-a). Entities, events, and their parts: The semantics of multipliers in Slavic. In Radeva-Bork,
T. and Kosta, P., editors, Current Developments in Slavic Linguistics: Twenty Years After. Peter Lang,
Frankfurt am Main.
Watanabe, A. (2013). Count syntax and the partitivity. In 19th ICL Papers, pages 1–18. Département de
Linguistique de l’Université de Genève, Geneva.
Zobel, S. (2017). The sensitivity of natural language to the distinction between class nouns and role nouns. In
Proceedings of Semantics and Linguistic Theory 27, pages 438–458. CLC Publications, Ithaca, NY.
Zwicky, A., Sadock, J., and Kimball, J. (1975). Ambiguity tests and how to fail them. In Kimball, J. P., editor,
Syntax and Semantics, pages 1–36. Academic Press, New York, NY.
91 / 91