event pluralities - day 2 - distributive...

Event pluralities - Day 2Distributive configurations

Patricia Cabredo Hofherr1 Lucia M. Tovena2

1UMR 7023CNRS & Université Paris 8

2Université Sorbonne Paris Cité – Paris VII

ESSLLI 2015 Barcelona

Outline

1 Introducing distributivityBasic notions and empirical issuesRefine the terminology

2 A typology of distributive configurationsFeatures and combinationsType 1 Distrib (whole–unit)Marking distributivity

3 Type 2 Cumul (Whole–whole)Instances and propertiesIdentity issuesAbout sendos

4 Type 3 Ratio (Unit–unit)Introducing ratiosTypes of ratios

Cabredo Hofherr/Tovena (CNRS/Paris 7) Event pluralities ESSLLI 2015 2 / 87

Introducing distributivity

Outline






Introducing distributivity Basic notions and empirical issues

Readings

Consider the girls in (1). The sentence has two readings.

(1) The girls carried a box

COLLECTIVE READING:– the nominal in subject position is interpreted referentially– one situation where the girls as a group carried a box.DISTRIBUTIVE READING:– the nominal in subject position is interpreted referentially, but it ispartitioned– possible multiplication effect on the nominal in object position– one situation where each individual girl carried a box (the sameor not).



Readings

Consider sentence (1) and the additional dimension of events.– The collective reading a single carrying event– The distributive reading many events of carrying.CUMULATIVE READING: cf. (2)

(2) The girls carried the boxes

– the nominals in subject and object positions are both plural andare interpreted referentially and scopelessly– the sentence is not specific about the number of subevents andof participants in each of them, but implies some subdistributionresulting in a plurality of events



Readings

The trademark of the cumulative reading is that there is nomultiplication of the nominals, but only their partition.Under the cumulative reading of (3), there are– all in all 3 girls– all in all 2 biscuits

(3) Three girls ate two biscuits

The difference wrt the collective reading, is that the sentencedescribes more than one event.



Example

Consider the readings of (3) from the point of view of eventsThree girls ate two biscuitsa. 3 events, each one of one girl eating two biscuits distb. 2 events, each one of three girls sharing and eating one biscuit distc. 1 event of three girls sharing and eating two biscuits colld. n events, each one of p girls eating m biscuits, n≥2, p≤3 and m<2.p and m do not have to have the same value in all the events cum

scoping issues (the difference btw cases a and b)



Example

Simple examples of predicative distributivity

(4) a. The men are beardedb. The girls slept

(5) The green players scored

(6) Every child ran to the door

The predicates in (4) and (6) necessarily distribute over theindividuals, not so for (5). There may be a single event in (6).Predicative distributivity will not be studied further.



Distributive configuration

A distributive reading can be forced by adding extra material.Sentence (7) is not ambiguous.

(7) Three girls ate two biscuits each

For a distributive reading to hold, there has to be a pairing of keyand share (Choe, 1987), although these expressions do not haveto be explicit. (not in all languages)sorting KEY is what forces the plural reading. Its function is similarto that of a wide scope taking expression in this respect. It definesthe domain over which distribution takes place.distributed SHARE designates the entity/entities that aredistributed, thus resembling a narrow scope taking expression.



Varieties of distributive configurations

quantified sorting key, e.g. English

(8) The girls ate two biscuits each

marked distributed share, e.g. Korean (Choe, 1987)

(9) emma-kamum+NOM

ai- tul-ekechild+PLU+to

phwungsen-hana-ssik-ulballoon+one+ssik+ACC

sacwu-esstaboughtMommy bought each child a balloon

ratios

(10) James Bond eats two olives per martini.



Identification of the components

Is the role of sorting key and distributed share reflected in theirmorphological form, semantic function, or syntactic positions?E.g. the sorting key c-command the distributed share.E.g. the sorting key is the domain that gets exhaustified in thedistribution. Distributed shares might get reused.



Identification of the components

Distributive readings involve event pluralities. Is event pluralitydirectly expressed, or is it a consequence of a partitioned pluralparticipant?Are these readings definable in terms of pairs of individuals, pairsindividual(s)–event, other?Are events part of the share?Are events always part of the distribution, be it as share or as key?


Introducing distributivity Refine the terminology

Distributivity

Languages go to considerable lengths to express distributivity.The distributive dependency may be established between wholesums or individual units. Whole sums are the domains from whichthe arguments of the distributive relation gets their values,individual units provide such values.We define a term for each component.These terms are used to refer to the linguistic expressions and totheir semantic values.



key

The KEY is the domain over which distribution takes placeIt is semantically pluralIt has a countable domainThe expression is used referentially

(11) a. Les/ Ces filles ont mangé deux biscuits chacune (Fr.)the/these girls ate two biscuits

b. Cinq filles ont mangé deux biscuits chacune (Fr.)some/five girls ate two biscuits



k-unit

The K-UNIT is the singular/plural unit of the key that enters aninstance of the distributive relationAll the k-units have the same cardinality/size.By default, the single entities in the key domain are the units forthe distribution, e.g. (11).Default can be overridden by explicit information, e.g. (12).

(12) The girls sang a song two by two



sh-unit

The SH-UNIT is the singular/plural entity that get paired with k-unitsA sh-unit is not referentialAll the sh-units have the same cardinality.Numerals in the DP indicate the cardinality/size of each sh-unit

(13) a. Three girls ate a biscuit (each)b. Three girls ate many/several biscuits eachc. Three girls ate a little pasta (each)

The cardinality of the domain from which the sh-unit is taken,minimally is function of the cardinality of the key’s domain (moduloissues of reusability).



sum-share

The SUM-SHARE is the plural expression that introduces thedomain out of which the sh-unit are taken

(14) Los estudiantes han ganado sendos premios (Es.)the students won a prize each



The distributive relation

A distributive relation can be viewed as a dependency involvingentities from the domains of its arguments, e.g. having pairs ofk-unit–sh-unit in its domain.The cardinality of k-units and sh-units need not match, but eachset is homogeneousThe relation may specify the granularity of the distribution itself, orinstructions may come from cardinality information on thearguments, or both.




In first order logic, a distributive reading can be represented by auniversal quantifier taking scope over an existential (∀x∃y).This is the core of the proposal defended by Choe (1987).It is also the semantics of the distributive operator assumed inmany analyses.The universal quantifier may be replaced by a form ofmaximalisation.




The collection of pairs k-unit–sh-unit is the denotation of thedistributive relationThe components belong to the same clause.The paired units correspond to participants in a single event.Talking about ‘participants’ means that distributivity is mediated byevents and yields event pluralities.




In first order logic, the denotation of the distributive relation can beidentified with a subset of the denotation of the main predicate.This is a simplification, but think of the formula‘eat(girln, two-biscuit)’.A big issue is how to identify the distributive relation, whether itcoincides with a predicative relation expressed by the sentence.With the reification of events, they may be a domain of entitiesused as key or as sum-share.VPs can be sh-units



Interim summary

The key has a plural domain. If lexically expressed, it is referential.It is the independent value, and the sum-share is the dependentone.The distributive relation usually is extensional.K-unit and sh-unit are expressions used non-referentially, iflexically expressed.The cardinality of a sh-unit is often overtly expressed.The distributive relation holds of ‘anonymous’ pairs of units.


A typology of distributive configurations

Outline






A typology of distributive configurations Features and combinations

Four features for domains and units

We consider only sentences that are unambiguously distributive.Distributivity can be seen as a manifestation of the application of aprocedure for building pairs under a number of specificconstraints.Something in the sentence signals the application of theprocedure.




Typically, key, k-unit, sum-share and sh-unit are not all overtlyprovided in one distributive construction.Since it is a matter of building pairs, these four elements have apotential crucial role in the relation.We introduce a first set of four features that allow us to identifythese elements.Four features mark whether the respective components of therelation are provided.The first two concern the wholes, the second two concern theunits.Features keyF concerns the key.Features shareF concerns the sum-share.Features sh-unitF concerns the sh-unit.Features k-unitF concerns the k-unit.




They help us to capture the internal coherence of thephenomenon. They are used to identify relevant components ofthe meaning and/or behaviour of a collection of distributiveconfigurations and expressions that are understood as forming asystem.They help us to capture and classify points of contact with otherphenomena.



Combinations

The four features allows us to characterise three cases:Type 1 Distrib: overt key and sh-unit (i.e. a whole and a unit)This first case characterises often discussed distributiveconfigurations.Type 2 Cumul: key and sum-share (i.e. the two wholes)The second case resembles cumulativity insofar as the key andthe sum-share are overtly given.Type 3 Ratio: k-unit and sh-unit (i.e. the two units)The third case is the expression of a ratio, where k-unit andsh-unit are what is given

Apparently, no whole–unit case where the whole is the sum-share andthe unit is the k-unit.


A typology of distributive configurations Type 1 Distrib (whole–unit)

Type 1 Distrib (whole–unit)

Type 1 Distrib is the case most commonly discussed in the literatureand will be discussed cursorily. There is a variety of linguisticexpressions. The distributive reading is enforced because:

a universal quantifier occurs separated from the key, so-calledfloated and binominal cases.an expression is reduplicated, e.g. a numeral (Day 5).



So called ‘floated’ quantifiers

In floated cases, at surface level the quantifier occurs after theconstituent expressing the key (and the auxiliary) (15)The key is the subjectThe quantifiers are tout or chacun in French.The verb can be intransitive and unaccusative (15b)

(15) a. Les enfants ont tous pris une glace vanille (Fr.)the children all had a vanilla icecream

b. Les arbres sont tous tombés (Fr.)the trees all fell down

c. Two girls have each carried three bags

Analyses differ. One possibility is to assume that the quantifierrealises a distributive operator, and the VP is the sh-unit.



So called ‘binominal’ quantifiers

In binominal cases, the quantifier is postposed to both key andsh-unitThe quantifier is chacun in French.The verb is transitive or ditransitive

(16) a. Les filles ont pris un biscuit chacune (Fr.)the girls took a biscuit each

b. Two girls have carried three bags each

Note that chacune agrees in gender with filles in (16a).Analyses differ. The quantifier may be seen to form a constituentwith the sh-unit.



Reduplication

The distributive reading is enforced because an expression isreduplicated, e.g. a numeral in Georgian (17) (Gil, 1988)

(17) a. Ormatwo-erg

k’acmaman-ERG

samithree-ABS

c̆antasuitcase-ABS

c’aiGo.carried-3sg

Two men carried three suitcasesb. Orma

two-ergk’acmaman-ERG

sam-samithree-dist-ABS

c̆antasuitcase-ABS

c’aiGo.carried-3sg

i. Two men carried three suitcases each.ii. Two men carried suitcases three at a time.iii. Two men carried sets of three suitcases

The reduplicated numeral sets the size of the sh-unit.


A typology of distributive configurations Marking distributivity

A feature for marking

The distributive only reading of the sentence can be overtlyexpressed by marking a component.This information can be stored as a separate feature in aclassification of distributive configurations, call it markF.At its origin, this is a typological distinction between languages(Gil, 2005).The two cases most commonly discussed in the literature are keymarked, cf. French and English (8) and (16), and sh-unit marked,cf. Korean (9) and Georgian (17).



A feature for marking

Marking may encode information on the granularity of the relationThe type 1 Distrib exhibits both subcases.The type 2 Cumul is sum-share marked.The type 3 Ratio is k-unit marked.

We do not make cross-linguistic claims about Types 2 and 3.



Marking

Type 1 Distrib Key-marked cases:floated and binominal cases

I The key is marked by a quantifier(-like) marker that acts as thedistributive operator.

I The quantifier is separated from the key and might form aconstituent with the sh-unit.

I We consider them key-marked cases because the marker agreeswith the key in gender, if agreement is overt.



Marking

Type 1 Distrib Share-marked cases:there is a preposition-like marker, e.g. nga in Albanian (18) from(Rushiti, 2015)

(18) Dytwo

fëmijëtchildren

morëntook

ngaDIST

trithree

torte.cakes

two children took three cakes each



Marking

there is a suffix-like marker, e.g. ssik in Korean (19) from (Choe,1987)

(19) a. Noray-han-kok-ulsong-one-CL-ACC

sonim-motwu-kaguest-all-NOM

pwul-ess-tasing-pst-DEC

the guests all sang a (single) songb. Noray-han-kok-ssik-ul

song-one-CL-ssik-ACCsonim-motwu-kaguest-all-NOM

pwul-ess-tasing-pst-DEC

the guests each sang a song



Interim summary

Three types of distributive configurations:Type 1 Distrib: overt key and sh-unit (i.e. a whole and a unit)Type 2 Cumul: key and sum-share (i.e. the two wholes)Type 3 Ratio: k-unit and sh-unit (i.e. the two units)

Type 1 Distrib can be:key markedsh-unit marked


Type 2 Cumul (Whole–whole)

Outline






Type 2 Cumul (Whole–whole) Instances and properties

Instances of Type 2 Cumul (Whole–whole)

Spanish sendos

(20) Los estudiantes han escrito sendos libros (Es.)the students wrote a book each

English respective, French respectif

(21) The students carried their respective suitcases

Key and sum-share are provided, these are the expressionsdenoting the wholes.K-unit and sh-unit are not overtly provided. They arehomogeneous.



Distributive reading

Sentence (22) is about a plurality of prize winning events. Nocollective reading of the nominals.

(22) Los estudiantes han ganado sendos premios (=(14))the students won a prize each

Being distributive, sendos cannot be the complement of a verbthat takes collective arguments

(23) a. Trajeron sendos librosthey took their respective books

b. *Reunieron sendos librosthey gathered together their respective books



Spanish sendos

The key must be plural, cf. (24) vs. (22) and (25).

(24) Leí (ambos /*sendos) librosI read both books

(25) a. Juan y María han ganado sendos premios (coord)Juan and María won a prize each

b. Dos amigos han ganado sendos premios (nb. plu)two friends both won a prize



Sum-share marker

Sendos behaves like a determiner. It occurs before a noun thatmust be bare and plural, and overtly agrees with it in gender.

(26) a. Los niños montaban en sendas bicicletasthe children rode on their bicycles

b. * Los niños montaban en sendos bicicletasthe children rode on their bicycles

It does not have pronominal use.It is a sum-share marker.



Sum-share marker

Sendos covertly sets to one the granularity of the relation, i.e. thesize of sh-unit AND k-unit.( 6= sh-unit markers)Numerals are allowed on the key, and define the cardinality of thewhole key.

(27) Los tres hombres se comieron sendas manzanasthe three men ate an apple each

Numerals are not allowed on the sum-share, this is the dependentdomain (6= sh-unit)Sh-units cannot be ‘reused’. In this, sendos behaves like othersh-unit markers, cf. the marker -gáa of Tlingit (Cable, 2014).


Type 2 Cumul (Whole–whole) Identity issues

Binding conditions

In sh-unit marked cases of Type 1 Distrib, the syntactic relationbetween the key and a distributive numeral is known to be local.The relation between the key and sendos, a sum-share marker,can be likened to anaphora.



Binding conditions

Bosque (1992) has argued that sendos behaves like an anaphora,subject to principle A, i.e. to the same principle as reflexives andreciprocals. It must be bound by a plural c-commandingantecedent.A noticeable difference is that the antecedents of sendos licensequantitative information, rather than identity information.What Bosque calls anaphora could be termed a case ofcovariation, like with dependent indefinites. (Day 5)



Binding conditions (Bosque 1992)

The key/antecedent (mostly) outscopes sendosIt can be an empty category controlled from outside the clause byovert material

(28) a. Los abogados permitieron a los clientes [PRO consultar consendos asesores fiscales]the lawers allowed the clients to consult their respective financialcounsellors

b. Los abogados promitieron a los clientes [PRO consultar consendos asesores fiscales]the lawyers promised to the clients to consult their respectivefinancial counsellors



Binding conditions

The arguments of a distributive relation are participants in the sameevent. This restricts binding options.

clausemate binder

(29) Los tres chicosi dijeron que sus dos amigosj habíanescrito sendas∗i/j novelasthe three boys said their friends had written a novel each

c-commanding antecedent

(30) *Los amigos de tus padresi conducían sendos∗i Mercedesthe friends of your parents all drove Mercedes



Binding conditions

no split antecedent

(31) a. *Juani habló con Maríaj sobre sendosi+j proyectos para elfuturoJuan discussed with María their projects for the future

b. Juani y Maríaj hablaron sobre sendosi+j proyectos para elfuturoJuan and María discussed their projects for the future



A feature for identity

A distributive relation holds of ‘anonymous’ pairs of units. It doesnot require access to the identity of the individuals.The use of information on their identity exceeds thecharacterisation of distributivity and must be stored separately.Adding a feature of this type allows us to capture a crucialdifference between sendos and respective, and more generallybetween pure distributivity and forms of reciprocity and the like.




Diachronically, sendos derives from the distributive numeralsingulos (one at the time) of late Latin.Sendos distributes by setting up pairs in an independent way. Thepairs individual-basket are not required to be in the denotation ofanother relation in (32).

(32) Nos fueron enviadas sendas cestas de navidadwe were sent a Christmas basket each




For the sake of this discussion, identification is deemed possible if wedispose of at least two independent properties that single out anindividual in the restriction, or we use a rigid designator.

(33) a. A girl that you met yesterday, called this morningb. Louise called this morning




Respective requires identification. The pairs are validated on thebasis of further information external to the distributive relation, e.g.the pairs are in the denotation of another relation, be itpossession, taste preferences, etc., cf. the required possessive in(34) and (35).

(34) On nous a envoyé nos paniers de Noël respectifs (Fr.)each of us was sent her/his own Christmas basket

(35) The boys took their respective bags

The possessive is not required in all languages, cf. (36).

(36) Bisogna mettere i pupazzetti nelle rispettive caselle (It.)the toy animals must be put in their respective holes




Information on the identity of the individuals is not used, but thisdoes not mean that the context is opaque. sendos can often befaithfully translated by respective in languages without this form.When the context makes it clear that the identity of the sh-units ispragmatically irrelevant, translating sendos by respective is nolonger felicitous, cf. (37).

(37) Los hombres se comieron sendas manzanas the men ate an apple each #the men ate their respective apples


Type 2 Cumul (Whole–whole) About sendos

Summarising

The distributive configuration enforced by sendos can be characterisedwith three conditions.

The first condition is on the size of the sh-unit, as stated in (38).

(38) Condition on granularitysendos sets to one the size of the sh-unit that enters thedistribution.

This condition results in covertly requiring homogeneouscardinality for all the implicit sh-units.



Summarising

The second condition bears on the dependency betweenarguments regarding information about the cardinality of thewholes, cf. (39).

(39) Condition on pluralitysendos requires a plural antecedent, the key. Thecardinality of the referent of the sendos phrase dependson that of the key.

The cardinality of the phrase does not match the cardinalityexpressed by the distributive numeral. The fact that thespecification of the cardinality of the sum-share must be recoveredfrom the key, provides a reason why anaphoric links are relevant.



Summarising

Conditions (38) and (39) have fairly uncontroversial content, cf.(Bosque, 1992; Aranovich, 2003)The anaphoric relation concerns quantitative information, notreferential one. (6= reflexives and reciprocals)



Summarising

Recall that sendos covertly sets to one the size of sh-unit ANDk-unit.Let’s suppose sendos is a distributive operator. Both itsarguments are used up, atomic unit by atomic unit.

1 There is numerical information that goes from the antecedent downthe spine of a syntactic representation, and concerns the cardinalityof the sum-share.

2 There seems also to be numerical information that goes up fromthe sendos phrase to the antecedent and concerns the cardinalityof the k-unit.

This bidirectional flow of quantitative information is not what isusually assumed for anaphoric relations.



Summarising

The need for quantitative information going up can be captured viaa third constraint, on quantitative symmetry of the arguments ofthe relation, cf. (40).

(40) Condition on quantitative symmetrysendos sets up a distributive dependency where k-unit andsh-unit match in size.

Alternatively, it could be assumed that the size of k-units is theatom by default. (Day 4)


Type 3 Ratio (Unit–unit)

Outline






Type 3 Ratio (Unit–unit) Introducing ratios

Instances of Type 3 Ratio (Unit–unit)

Italian constructions with prepositions per and a, and the Englishcounterparts

(41) a. James Bond ha mangiato due olive per martini (It.)b. James Bond ate two olives per martini

(42) a. Hanno dato tre fette a testa (It.)b. they gave out three slices apiece

Both k-unit and sh-unit are overtly provided.Key and sum-share are not provided, and not necessarilycontextually assumed.



Instances of Type 3 Ratio (Unit–unit)

The per construction has the form ‘sh-unit per k-unit’The apiece cases have the form ‘sh-unit a k-unit’ tooA third construction with the form ‘sh-unit (per) ogni k-unit’contains the universal quantifier ogni (every), cf. (43).

(43) a. Guadagna duecento euro ogni sei contratti (It.)b. S/he earns 200E every six contracts



Ratios

A ratio is the result of comparing two quantities multiplicatively.Sentences like (41) say that the quantity of martinis was half thatof olives, restricted to the context.Ratios are relationships between pairs of numbers. The numbersin (41) express the constant relation between the size of a k-unitand that of a sh-unit.The multiplicative comparison implies a plurality of events.



Distributive dependencies and ratios

Recall that in a distributive dependency:the key is the (quantity) set of individuals over which the sh-unitsare distributed. Its cardinality is contextually available.The key is also the quantity with respect to which another quantity(the sum-share) is directly proportional.The size of the sh-unit is the coefficient of proportionality.



Availability of the key

Distributive configurations of Type 1 Distrib, such as in (44), are casesof ratios where the key is contextually given.

(44) Les trois enfants ont tous mangé quatre biscuits (Fr.)the three children have all eaten four biscuitstotal nb of biscuits = total nb of children × 4

The cardinality of the sum-share is equal to the product of thecardinality of the key and the constant represented by the size of thesh-unit, assuming that sh-units or their members do not get reused.



Properties of ratios

A ratio is scale invariant.e.g. the quantity of olives was the double of that of martinis inBond’s drinking setup defined in (41), no matter how manymartinis he drank in total.A ratio expresses a distribution rule.It states a comparison between the progression of a sequence ofnumbers representing quantities that add up to make a key, andthe progression of a sequence of numbers representing quantitiesthat add up to make a sum-share.The linguistic expressions of the units are understoodintensionally.


Type 3 Ratio (Unit–unit) Types of ratios

Types of ratios

The domains of sh-unit and k-unit can be disjoint or shared. Each casepairs with a separate type of ratio.

When k-unit and sh-unit denote in disjoint domains, it is aPART-TO-PART RATIO (45).

(45) due parti di idrogeno per una di ossigeno (www)two parts hydrogen to one part oxygen



Types of ratios

When the sh-unit expression denotes in a subset of the k-unitdomain, it is a PART-TO-WHOLE RATIO (46).

(46) a. [...] nel test per determinare gli anticorpi, mi restaquindi un tasso di errore potenziale da uno a tre casiogni 2.000 (www)

b. I am therefore left with a potential error rate of one tothree per 2,000 for the antibody test

The quantitative information contributed by the k-unit plays therole of ‘whole’ locally, i.e. in a pair and not in the multiplicativecomparison.



The domains of the units

In either type of ratio, the restrictor of the k-unit NP contributesprimary information, and usually cannot simply be contextuallyretrieved.As for the sh-unit, information is partially retrievable in thepart-to-whole case, with an effect that somewhat resemblesconservativity in generalised quantifier theory.Percent expressions fall in the part-to-whole group, but the issueof their conservativity is more complicated.



Word order

The order of the two units does not correspond to the standardorder in distributive configurations in Italian and English.

I The sh-unit is provided as the first argument of the ratio expression.I The k-unit is the second argument, the ratio marker (if any) is

prefixed to it.

This order constraint does not hold across languages.




The distributive relation cannot be identified with the relation expressedby the main predicate. The tuples in the distributive relation are not(proper) part of the set of tuples in the denotation of the predicate

The sh-unit is integrated in the argument structure/event structureof the sentence.The k-unit, be it with per, a or ogni, occupies a non-argumentalposition, more precisely it is not a participant or a parameter of theevent. It does not get case.




The two units instantiate participants in a common abstract(super)event, in the part-to-part ratio.The distributive relation, and the events, are an abstraction, e.g. adrinking situation in (41), where a martini is sipped and the olivesare eaten.There is a plurality of (super)events, or possibly of ‘cases’ in Lewis(1975)’s sense.



Ratios in Italian

We are going to review some properties of the three instances ofType 3 Ratio constructions in Italian.



The per configuration

The per configuration expresses a part-to-part ratio, the two unitshave distinct domains, cf. (41).per is a preposition that forms a constituent with the k-unit.The preposition per has other uses.Special conditions primarily concern the k-unit.




A k-unit is a bare singular count noun (41). It contributes– a property that restricts the domain and– a default quantity equal to oneThe noun is count.The sh-unit cannot be a bare singular noun, the quantity must beprovided overtly.




A k-unit cannot be a numeral determiner + noun (47)

(47) # Sono quattro scarpe per due bambiniit is four shoes per two children

A k-unit can be a numeral only if the sh-unit is just a numeral (48)

(48) Ne ho distribuiti tre per dueI handed them out three per two




The key is not required and the per constituent does not have anantecedent.It is possible to have multiple k-units, each introduced by per.

(49) Fanno 30 euro per notte per personait is 30E per night per person



The a configuration

Similarities with the per configurationThe a configuration expresses a part-to-part ratio.a is a preposition that forms a constituent with the k-unit.The units have disjoint domains.The preposition a has other uses, even more than preposition per.



The a configuration

Differences with respect to the per configurationThe key may be discourse givenTypically, k-units are temporal/spatial units, or part names.

(50) a. Ci va diverse volte al giornos/he goes there several times a day

b. Costa settemila euro al metro quadroit costs seven thousand euros per square metre

c. Costa tre euro al pezzoit costs three euros apiece

The k-unit is a definite DP, when the head noun serves as astandard unit of measure, or it is a numeral.Exception: head nouns persona and testa are used bare.



The ogni configuration

The ogni configuration expresses a part-to-part ratio or apart-to-whole ratio.ogni (every) is a determiner that forms a constituent with thek-unit.The units have disjoint or shared domains.The k-unit usually contains a numeralThe key is usually discourse given



The ogni configuration

ogni can be preceded by per, cf. (51).

(51) [...] il dividendo è di 0,104 euro per ogni azione (www)the dividend is 0.104 euros per share

ogni is needed if the size of the k-unit is overtly specified.per does not cooccur with ogni in a proportion, i.e. an equationthat says that two ratios are equivalent

(52) [...] in pratica è come se ci fossero 35 fallimenti al giorno,tre ogni due ore (www)in practice it is as if there were 35 bankruptcies a day,three every two hours.



Considerations about the key

In a distributive dependency, the key is the set that getsexhaustified. In a ratio, no set gets exhaustified.A new criterion for the key:The progression that leads to increasing sums of units is indexedon one of the two series, the value from that series is the k-unitand the whole series is the key.The key is the independent value that plays a leading role in theinformation conveyed by the sentence. Example (41) is about theway James Bond drank martinis, not about the way he ate olives.The progression on the key series matches the increase in thenumber of superevents.



Considerations about the key

The constraint of exhaustivity on the key can be turned into aconstraint on using the k-unit as information for identifying cells ofa potential partition on the plurality of superevents.The key is the domain always associated with the partition onsuperevents.This new definition of key extends to distributive dependenciesand ratio expressions.In the case of a ratio, the partition is not extensional



Summing up ratios

K-unit and sh-unit are the only required components.The k-unit is overtly provided. It is not a participant or parameterof the event described by the sentence.The distributive relation cannot coincide with a predicative relationexpressed by the sentence.



Summing up ratios

In Italian, there appear to be two main types of ratio expressions:– the per and a configurations express a distributive rule. They areinstances of part-to-part ratios.– the ogni configuration expresses an inference from an overallevaluation. It can instantiate part-to-whole and part-to-part ratios.The ogni configuration can be used with a contextually identifiablekey.



General summary

Distributive configurations describe pluralities of events.The key is the domain distributed over. The denotation of thedistributive relation is crucially built on it.The atoms in the key domain work as k-units, unless overtlymodified by expressions such as two by two.



General summary

Examples of different distributive configurations have promptedthe need of a classification of distributive dependencies.We can use features to pin down points of variation amongdistributive dependencies taken in a broad sense.



General summary

The (non-)use of information on individuals distinguishesdistributive sendos in Spanish from other expressions such asreciprocals.Distributive dependencies are traditionally analysed as holding onextensional domains. Ratios are intensional.In Italian, there appear to be two main types of ratio expressions,the per/a configuration and the ogni configuration, neither of whichis extensional



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