Subatomic quantification Marcin Wągiel OJ591 Topics in linguistics 1 / 84 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 / 84 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 3 / 84 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 4 / 84 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) 5 / 84 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) 6 / 84 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] 7 / 84 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 8 / 84 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]] 9 / 84 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 10 / 84 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) 11 / 84 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)]] 12 / 84 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 13 / 84 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) 14 / 84 Mereotopology a b Figure 4: Internal part a b Figure 5: Internal overlap a b Figure 6: Tangential overlap 15 / 84 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) 16 / 84 Mereotopology a Figure 7: Interior a Figure 8: Exterior a Figure 9: Closure 17 / 84 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 18 / 84 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 19 / 84 Mereotopology Capturing objects ▶ integrated wholes ⇒ parthood and connectedness ▶ arbitrary sums ⇒ only parthood a b c d Figure 10: Wholes vs. sums 20 / 84 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) 21 / 84 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 22 / 84 General counting principles ▶ illegal counting ▶ assigning a number to less than a whole entity ▶ summing up complementary parts ▶ overlapping entities Figure 13: Illegal counting 23 / 84 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) 24 / 84 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 25 / 84 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. 26 / 84 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 27 / 84 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 28 / 84 Subatomic quantification ▶ counting of parts ▶ counted parts ⇒ maximal integrated entities ▶ counted parts cannot overlap Figure 15: Counting of parts 29 / 84 Subatomic quantification ▶ illegal counting of parts ▶ counting discontinuous parts of an object ▶ overlapping parts Figure 16: Illegal counting of parts 30 / 84 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’ 31 / 84 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’ 32 / 84 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 / 84 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 / 84 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 / 84 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 / 84 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 37 / 84 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’ 38 / 84 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.’ 39 / 84 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 40 / 84 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 41 / 84 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 42 / 84 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 43 / 84 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 / 84 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 / 84 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.’ 46 / 84 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 / 84 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 / 84 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 49 / 84 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 / 84 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’ 51 / 84 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 / 84 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 53 / 84 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 / 84 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 / 84 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 / 84 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 / 84 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 / 84 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 / 84 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 / 84 Multipliers Neglected class of numerical expressions ▶ cross-linguistically widespread category ▶ attested also in non-IE languages (54) a. double b. doppelt German c. doppio Italian d. dvojnoj Russian e. dvigubas Lithuanian f. dupla Hungarian g. shuāng Mandarin 61 / 84 Multipliers Expressions dedicated to counting parts ▶ entailment ⇒ complex inner structure (55) a. The Pschent is a double crown. b. ⊨The Pschent consists of two parts. Figure 19: Pschent Figure 20: Deshret Figure 21: Hedjet 62 / 84 Multipliers More examples ▶ some frequent collocates in COCA (56) 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 63 / 84 Multipliers Non-trivial quantificational behavior ▶ quantification over parts rather than wholes ▶ adjectival properties ▶ modified NPs ⇒ always countable (Universal Packager) (57) a. three crowns b. three double crowns (58) a. #three coffees b. three double coffees 64 / 84 Multipliers Relationship between multipliers and cardinals ▶ Slavic and Baltic multipliers ⇒ derived from numeral roots ▶ multiplicative affix ⇒ classifier (59) a. dv-a numeral.root-infl.marker ‘two’ Russian b. dv-oj-n-oj numeral.root-stem-mult.suffix-infl.marker ‘double’ Russian 65 / 84 Multipliers ▶ Slavic and Baltic multipliers (60) a. dwa Polish b. podwójny (61) a. dva Czech b. dvojitý (62) a. dva BCS b. dvostruki (63) a. du Lithuanian b. dvigubas 66 / 84 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) (64) a. double vodka b. double murder c. double agent 67 / 84 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 68 / 84 Analysis Count nouns ▶ mssc entities ⇒ integrated wholes ⇒ no atomicity (65) Count noun apple = λx[mssc(apple)(x)] Pluralization ▶ presupposition ⇒ mssc predicates ▶ algebraic closure (Link 1983) ▶ no topological constraints (66) PL = λP . Pmssc[∗ P] (67) apples = PL ( apple ) = λx [ ∗ ( λy[mssc(apple)(y)] ) (x) ] 69 / 84 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 (68) Measure function #(P) ∀P∀x[#(P)(x) = 1 iff mssc(P)(x)] (69) Cardinal numeral two = CL# ( √ tw ) = λP. Pmssc λx[*P(x) ∧ #(P)(x) = 2] 70 / 84 Analysis Multipliers ▶ complex expressions ⇒ derived from numeral roots ▶ names of numbers ⇒ predicate modifiers ▶ classifier CL⊞ ⇒ measure function ⊞(P) ▶ count essential parts of mssc entities (70) Measure function ⊞(P) ∀P∀x[⊞(P)(x) = 1 iff mssc(P)(x)∧∃y[y ⊑ x∧essential(P)(y)∧#(y) = 1]] (71) Polish multiplier podwójny = CL⊞ ( √ dw ) = λP. Pmssc λx[P(x) ∧ ⊞(P)(x) = 2] 71 / 84 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 (72) PART = λyλx[x ⊏ y] (73) HALF = λyλx[x ⊏ y ∧ µ(x) ≈ µ(y) × 0.5] 72 / 84 Analysis Partitioning ▶ partitioning function π ⇒ non-overlap ▶ relative atomicity ⇒ irrelevant ▶ multiple possible partitions (74) 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 (75) Individuating element IND = λPλx[mssc ( π(P) ) (x)] 73 / 84 Analysis Partitive words ▶ bare partitivity ⇒ topological neutrality ▶ interaction ⇒ topological sensitivity, individuation (76) German topology-neutral part word Teil Teil = λyλx[x ⊏ y] (77) Polish topology-neutral half word połowa połowa = λyλx[x ⊏ y ∧ µ(x) ≈ µ(y) × 0.5] (78) Polish topology-sensitive half word pół pół = λy . ymssc λx[x ⊏ y ∧ µ(x) ≈ µ(y) × 0.5] (79) Polish individuating suffix -k-k- = IND = λPλx[mssc ( π(P) ) (x)] (80) Polish individuating half word połówka połówka = -k- ( pół ) 74 / 84 Analysis Polish topology-neutral proportional partitive (81) połowa half jabłka applegen (82) ⟨e, t⟩ ⟨e, ⟨e, t⟩⟩√ połowa ‘half’ e ⟨⟨e, t⟩, e⟩ DEF ⟨e, t⟩ jabłko ‘apple’ 75 / 84 Analysis Polish topology-sensitive proportional partitive (83) połówka half-k jabłka applegen (84) ⟨e, t⟩ ⟨⟨e, t⟩, ⟨e, t⟩⟩ -k- IND ⟨e, t⟩ ⟨e, ⟨e, t⟩⟩√ pół ‘half’ e ⟨⟨e, t⟩, e⟩ DEF ⟨e, t⟩ jabłko ‘apple’ 76 / 84 Analysis German count explicit partitive (85) zwei two Teile parts des thegen Apfels applegen (86) ⟨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’ 77 / 84 Analysis Polish multiplier phrase modified by the cardinal (87) trzy three podwójne double hamburgery hamburgers (88) ⟨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’ 78 / 84 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 79 / 84 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 80 / 84 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 81 / 84 References Acquaviva, P. 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