Cross-linguistic compositional classifier semantics for numerical expressions Marcin Wągiel ReSSP, La Garrotxa, Oct 5–7 2018 1 / 59 Introduction What’s this all about? numerals simplex forms multiple uses semantically complex expressions numerical expressions understudied yet revealing morphological complexity derived complex meanings unified semantics core meaning ⇒ reference to numbers shifting operators classifier semantics 2 / 59 Introduction Various uses of numerals Bultinck (2003), Geurts (2006), Rothstein (2013, 2017) no unique meaning? polysemous? (1) a. Five is the fourth Fibonacci number. arithmetical b. Five ducks entered the lobby. quantifying c. These are five buckets. predicative d. the five girls adjectival e. five pounds of buckwheat measure f. Chanel number five label 3 / 59 Introduction Derivationally complex numerical expressions in Polish (2) a. dwóch two studentów students ‘two students’ 2 entities b. dwoje twogndr studentów students ‘two students (male and female)’ 2 entities gender inference c. dwójka twocoll studentów students ‘group of two students’ 2 entities collectivity 4 / 59 Introduction Derivationally complex numerical expressions in Polish (3) a. dwóch two studentów students ‘two students’ 2 entities b. podwójna double korona crown ‘double crown’ *2 entities 2 parts c. dwukrotny two-time prezydent president ‘two-time president’ *2 entities 2 events 5 / 59 Introduction Mainstream research Barwise & Cooper (1981), Scha (1981), Landman (2004) English data numerous prolific insights influential theories focus: derivationally simplex expressions English: no rich morphology derivationally complex expressions ⇒ neglected 6 / 59 Introduction Basic cardinal numerals gender morphology non-trivial semantic effects semantic compositionality grammatical gender ⇒ classifier system Complex numerical expressions morphological complexity non-trivial quantificational effects semantic compositionality different affixes ⇒ distinct classifiers 7 / 59 Cardinals Recent research gender as a grammaticalized classifier system Serbo-Croatian neuter nouns Arsenijević (2016) Arabic numerals Fassi Fehri (2016, 2018) Focus virile/non-virile alternation in Polish cardinal numerals (4) a. Pięć fivenv dziewczyn girlsnv spało. slept b. Pięciu fivev chłopców boysv spało. slept 8 / 59 Cardinals Novel data referential use of cardinals cardinals in contexts calling for numerical arguments cross-linguistic correspondences Claims Polish cardinals ⇒ reference depends on gender gender on cardinals ⇒ grammaticalized classifier semantic asymmetry non-virile cardinals ⇒ names of number concepts virile cardinals ⇒ classifier semantics 9 / 59 Cardinals Markedness virile forms ⇒ morphologically and semantically marked (5) a. dw-a root-nv.marker b. dw-a-j root-marker-v.marker numeral roots ⇒ often homophonous to non-virile forms (6) a. √ pięć b. pięć-∅ root-nv.marker c. pięci-u root-v.marker 10 / 59 Cardinals Semantic functions of English cardinal numerals Rothstein (2013, 2017) nominal modifiers (7) The four cats lay on the sofa. predicates (8) My reasons for saying this are four. names of number concepts (9) Six is bigger than two. 11 / 59 Cardinals Polish cardinals virile/non-virile ⇒ not reducible to syntactic agreement both forms can modify NPs and occur in predicate position (10) a. Tych these pięć fivenv dziewczyn girlsnv przyszło. came b. Tych these pięciu fivev chłopców boysv przyszło. came (11) a. Tych these dziewczyn girlsnv było was pięć. fivenv b. Tych these chłopców boysv było was pięciu. fivev 12 / 59 Cardinals Asymmetry in numerical contexts virile forms cannot name numbers (12) a. liczba number pięć fivenv b. #liczba number pięciu fivev virile forms cannot appear in a counting list (13) a. jeden, onenv dwa, twonv trzy, threenv cztery, fournv pięć. . . fivenv b. #jeden, onev dwaj, twov trzej, threev czterej, fourv pięciu. . . fivev 13 / 59 Cardinals virile forms cannot appear in mathematical statements (14) a. Dwa two razy times pięć fivenv równa equals się refl dziesięć. ten b. *Dwa two razy times pięciu fivev równa equals się refl dziesięć. ten similar other marked forms (15) a. Dwa two razy times dwa twonv równa equals się refl cztery. four b. *Dwa two razy times dwie twofem równa equals się refl cztery. four 14 / 59 Cardinals Non-virile forms as names of numbers ⇒ distinctive properties resist adjectival modification cf. Babby (1987), Miechowicz-Mathiasen (2011) (16) a. dobrei good pięći five butelek bottles b. *liczba number dobrei good pięći five incompatible with the universal quantifier cf. Corbett (1978), Gvozdanović (1999), Miechowicz-Mathiasen (2011) (17) a. wszystkie all pięć five butelek bottles b. *liczba number wszystkie all pięć five 15 / 59 Interim summary Polish cardinal numerals virile/non-virile alternation non-virile forms unmarked felicitous as modifiers and predicates felicitous as names of number concepts when used as names of numbers ⇒ distinctive properties virile forms marked felicitous as modifiers and predicates infelicitous as names of number concepts incompatible with numerical contexts 16 / 59 Cross-linguistic perspective Bulgarian virile cardinals Cinque & Krapova (2007), Pancheva (2018) special forms for numerals 2–10 ending in -ma or -ima infelicitous in contexts calling for numerical arguments (18) a. Tezi these pet twonv ženi womennv dojdoxa. came b. Tezi these petima fivev mâže menv dojdoxa. came (19) a. Deset ten deleno divided na on pet fivenv e is dva. two ‘Ten divided by two is five.’ b. #Deset ten deleno divided na on petima fivev e is dva. two 17 / 59 Cross-linguistic perspective Slovak masculine animate cardinals special numeral forms in -aja, -ia or -i infelicitous in contexts calling for numerical arguments (20) a. Týchto these päť fivenma žien womennma prišlo. came b. Tito these piati fivema muži menma prišli. came (21) a. Dvakrát two.times päť fivenma je is desať. ten ‘Two times five is ten.’ b. #Dvakrát two.times piati fivema je is desať. ten 18 / 59 Cross-linguistic perspective Specialized numerals across languages Hurford (1998, 2001) attributive numerals ⇒ modification counting numerals ⇒ reference to number concepts language number attributive counting German 2 zwei zwo Maltese 2 żewg tnejn Chinese 2 liˇang èr Hungarian 2 két kettö Basque 2 bi biga 19 / 59 Cross-linguistic perspective Interaction between gender and reference in Arabic specialized numeral forms only feminine cardinals are felicitous in numeric contexts Fassi Fehri (2017) (22) a. taalat-at-un three-fem-nom t-usawii fem-equals 2+1 2+1 ‘Three equals 2+1.’ Arabic b. *taalat-un three-nom t-usawii fem-equals 2+1 2+1 20 / 59 Cross-linguistic perspective Classifiers numerals and classifiers are always adjacent Greenberg (1972) language ordering Vietnamese [num-cl]-n Thai n-[num-cl] Ibidio [cl-num]-n Bodo n-[cl-num] classifiers are often suffixes on numerals Aikhenvald (2000) (23) pajluhua-na one-cl yahui dog ‘one dog’ Yucuna 21 / 59 Cross-linguistic perspective Classifier constructions in Japanese bare numerals cannot be used as modifiers or predicates Sudo (2016) (24) a. *ichi-no one-gen hana flower b. ichi-rin-no one-cl-gen hana flower ‘one flower’ Japanese (25) a. *kyoo-no today-gen okyakusan-wa guest-top juu-ni-da. ten-two-cop b. kyoo-no today-gen okyakusan-wa guest-top juu-ni-nin-da. ten-two-cl-cop ‘The number of guests today is twelve.’ 22 / 59 Cross-linguistic perspective Classifier constructions in Japanese classifier constructions do not fit unambiguously numerical contexts (26) a. juu ten waru divide.by ni-wa two-top go-da. five-cop ‘Ten divided by two is five.’ b. *juu-ko ten-cl waru divide.by ni-ko-wa two-cl-top go-ko-da. five-cl-cop 23 / 59 Cross-linguistic perspective Partly classifier languages Bale & Coon (2014) intriguing data from Mi’gmaq and Chol some cardinals require classifiers others do not different cardinals ⇒ distinct semantics (27) a. na’n five (*te’s)-ijig cl-agr ji’nm-ug man-pl ‘five men’ Mi’gmaq b. asugom six *(te’s)-ijig cl-agr ji’nm-ug man-pl ‘six men’ 24 / 59 Background Standard approach Borer (2005), Chierchia (1998, 2010), Rothstein (2010), Li (2011), Scontras (2013) mass-like semantics of nouns in classifier languages classifiers compensate semantic deficits of nouns Alternative view Krifka (1995), Bale & Coon (2014), Sudo (2016) different semantics of numerals in classifier languages classifiers compensate semantic deficits of numerals 25 / 59 Background Measure functions Krifka (1989, 1990, 1995) operations that map entities to numbers liter maps an entity to a number of liters of that entity (28) a. MeasureP = λx[n(x) ∧ µ(x) = n] b. two liters of water = λx[water(x) ∧ liter(x) = 2] natural unit/object unit operation #(P) maps a plurality to a number of individuals #(P) is compatible only with quantized predicates (29) a. NumeralP = λx[*n(x) ∧ #(n)(x) = n] b. two cats = λx[*cat(x) ∧ #(cat)(x) = 2] 26 / 59 Background Cardinal numerals as predicates Rothstein (2012, 2013, 2017) names of numbers abstract objects of a primitive type n Scha (1981) (30) a. Numeral = n b. two = 2 cardinal predicates the type of adjectives e, t Landman (2003) (31) a. Numeral = λx[|x| = n] b. two = λx[|x| = 2] 27 / 59 Background shifting mechanism relating n and e, t Fregean property theory Chierchia (1985) properties are basic ∩ and ∪ ⇒ shifts between properties and numbers (32) a. Numeral e,t = λx[|x| = n] b. Numeral n = ∩ λx[|x| = n] c. ∪∩ Numeral e,t = Numeral e,t (33) a. two e,t = λx[|x| = 2] b. two n = ∩ two e,t = 2 28 / 59 Proposal Morpho-syntactic assumptions roots are category-free Halle & Marantz (1993) numeral head assigns gender adding more structure is possible Semantic assumptions names of numbers are basic overt/covert element cl (‘classifier’) shift: n ⇒ e, t , e, t measure function #(P) atomicity requirement ⇒ presupposition at(P) classifier construction ⇒ modifier or predicate 29 / 59 Proposal Composition of Polish cardinals numeral roots ⇒ type n names of numbers often homophonous to non-virile forms never homophonous to virile forms cardinal suffixes operations on numbers ⇒ type n, n inflectional morphology virile forms ⇒ marked markedness ⇒ specific meaning non-virile marker ⇒ [nv] gender value cl can be applied later in the derivation virile marker ⇒ [v] gender value + cl 30 / 59 Proposal Semantics of numeral roots singular terms (34) a. √ Numeral = n b. √ pięć = 5 Semantics of cl classifier element ⇒ shift measure function atomicity presupposition pluralization (35) cl = λnλPλx.at(P)[*P(x) ∧ #(P)(x) = n] 31 / 59 Proposal Cardinal suffixes operators of type n, n attach to numeral roots yield a value enlarged via addition or multiplication compatible only with natural numbers (36) a. pięt-naście five-teen ‘fifteen’ b. *pół-naście half-teen (37) a. -naście = λn.integer(n)[n + 10] b. -dziesiąt = λn.integer(n)[n × 10] 32 / 59 Proposal Composition of number-denoting pięć (‘five’) (38) numeralPn 5 numeral [nv] -∅ √ pięć-n 5 33 / 59 Proposal Composition of classifier pięć (‘five’) (39) numeralP e,t , e,t λPλx.at(P)[*P(x) ∧ #(P)(x) = 5] cl n, e,t , e,t λnλPλx.at(P)[*P(x) ∧ #(P)(x) = n] numeralPn 5 numeral [nv] -∅ √ pięć-n 5 34 / 59 Proposal Composition of number-denoting piętnaście (‘fifteen’) (40) numeralPn 15 numeral [nv] -e n 15 -nast- n,n λn.integer(n)[n + 10] √ pięć-n 5 35 / 59 Proposal Composition of classifier piętnaście (‘fifteen’) (41) numeralP e,t , e,t λPλx.at(P)[*P(x) ∧ #(P)(x) = 15] cl n, e,t , e,t λnλPλx.at(P)[*P(x) ∧ #(P)(x) = n] numeralPn 15 numeral [nv] -e n 15 -nast- n,n λn.integer(n)[n + 10]36 / 59 Proposal Composition of virile pięciu (‘five’) (42) numeralP e,t , e,t λPλx.at(P)[*P(x) ∧ #(P)(x) = 5] numeral [v] cl n, e,t , e,t λnλPλx.at(P)[*P(x) ∧ #(P)(x) = n] -u √ pięć-n 5 37 / 59 Proposal Composition of virile piętnastu (‘fifteen’) (43) numeralP e,t , e,t λPλx.at(P)[*P(x) ∧ #(P)(x) = 15] numeral [v] cl n, e,t , e,t λnλPλx.at(P)[*P(x) ∧ #(P)(x) = n] -u n 15 -nast- n,n λn.integer(n)[n + 10] √ pięć-n 5 38 / 59 Extensions Unified semantics one mechanism to derive distinct meanings non-classifier languages classifier languages interaction between classifier semantics and gender cardinals complex numerical expressions different morphemes ⇒ distinct classifiers 39 / 59 Numerical expressions Slavic derivationally complex numerical expressions Dočekal (2012, 2013), Wągiel (2014, 2015, to appear) denumeral group nouns (44) dw-ój-k-a two-suffix studentów students ‘group of two students’ Polish taxonomic numerals (45) dv-oj-í two-suffix pivo beer ‘two kinds of beer’ Czech 40 / 59 Numerical expressions Slavic derivationally complex numerical expressions Dočekal (2012, 2013), Wągiel (2014, 2015, to appear) numerals specialized for pluralia tantum (46) dv-oj-e two-suffix makaze scissors ‘two pairs of scissors’ BCS mixed gender numerals (47) dw-oj-e two-suffix studentów students ‘two students (one male and one female)’ Polish 41 / 59 Numerical expressions Slavic derivationally complex numerical expressions Dočekal (2012, 2013), Wągiel (2014, 2015, to appear) multipliers (48) dv-oj-n-oj two-suffix gamburger hamburger ‘double hamburger’ Russian frequency numerals (49) dw-u-krotn-y two-suffix prezydent president ‘two-time president’ Polish 42 / 59 Numerical expressions Denumeral group nouns obligatory collective inferences (50) a. Dwóch two studentów students napisało wrote esej. essay ‘Two students wrote an essay.’ (i) collective: total = 1 essay (ii) distributive: total = 2 essays b. Dwójka twocoll studentów students napisała wrote esej. essay ‘A group of two students wrote an essay.’ (i) collective: total = 1 essay (ii) *distributive: total = 2 essays 43 / 59 Numerical expressions Denumeral group nouns incompatible with distributive predicates (51) a. Dwóch two studentów students jest is wysokich. tall ‘Two students are tall.’ b. Dwóch two studentów students spłodziło fathered czworo four dzieci. children ‘Two students fathered four children.’ (52) a. #Dwójka twocoll studentów students jest is wysoka. tall b. #Dwójka twocoll studentów students spłodziła fathered czworo four dzieci. children 44 / 59 Numerical expressions Frequency numerals hidden ‘becoming’ event ⇒ acquiring a property (53) a. Obama Obama to this dwukrotny two-time prezydent. president ‘Obama is a two-time president.’ b. Obama Obama został became prezydentem president dwa two razy. times ‘Obama became a president twice.’ Figure 1: Obama’s presidency 45 / 59 Numerical expressions Frequency numerals only stage-level properties can be repetitively acquired and lost the again test (54) a. Obama Obama ponownie again został became prezydentem. president ‘Obama became a president again.’ b. #Obama Obama ponownie again został became mężczyzną. man ‘Obama became a man again.’ 46 / 59 Multipliers Expressions dedicated to counting parts entailment ⇒ complex inner structure (55) a. Pszent to podwójna korona. Pschent this double crown ‘The Pschent is a double crown.’ b. Pszent składa się z dwóch części. Pschent consists from two parts ‘The Pschent consists of two parts.’ Figure 2: Pschent Figure 3: Deshret Figure 4: Hedjet 47 / 59 Cross-linguistic perspective Complex numerical expressions in English no rich morphology but still many different expressions overlooked (56) a. two b. twosome c. twofold d. two-time e. two times f. twice g. double h. doubly 48 / 59 Cross-linguistic perspective Multipliers cross-linguistically widespread category attested also in non-IE languages (57) a. double b. doppelt German c. doppio Italian d. dvigubas Lithuanian e. dupla Hungarian f. shu¯ang Mandarin 49 / 59 Cross-linguistic perspective Multipliers Romance, Germanic, Hungarian ⇒ Latin borrowings derivationally complex in many languages language number cardinal multiplier Russian 2 dva dvojnoj Lithuanian 2 du dvigubas Finnish 2 kaksi kaksinkertainen 50 / 59 Proposal Composition of complex numerical expressions Composition of complex numeral expressions the same compositional mechanism numeral root ⇒ object of type n different suffixes ⇒ different classifiers collective affix cl selecting for groups group-formation ⇒ collective inference measure function ⇒ counts groups taxonomic affix cl selecting for subkinds kind shift ⇒ taxonomic meaning measure function ⇒ counts subkinds 51 / 59 Proposal Composition of complex numerical expressions multiplicative affix cl selecting for parts access to the subatomic part-whole structure measure function ⇒ counts salient comparable parts frequency affix cl selecting for ‘becoming’ events association of ‘becoming’ events with an individual measure function ⇒ counts events of acquiring a property 52 / 59 Conclusion Observations Polish cardinal numerals gender plays a role in reference to numbers non-virile ⇒ can name numbers marked forms ⇒ cannot name numbers classifier constructions can only be used as predicates and modifiers unlike bare numerals cannot name numbers cross-linguistic correspondences different forms for attributive and counting numerals Arabic feminine numerals Japanese classifier constructions 53 / 59 Conclusion Proposal Polish cardinal numerals non-virile ⇒ names of numbers or cardinal predicates marked forms ⇒ only cardinal predicates claim virile gender ⇒ grammaticalized classifier ingredients numeral roots: names of number concepts born at type n cl: shift n ⇒ e, t + measure function virile/non-virile alternation virile numeral head introduces cl non-virile numeral head does not (cl can be added later) 54 / 59 Conclusion Extensions an further research cross-linguistic applications gender on cardinals classifier constructions complex numerical expressions complex morphology ⇒ complex semantics different affixes ⇒ distinct classifiers (morpho)syntax-semantics interface gender value and cl ⇒ one node or more? relationship between gender and quantification cf. 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