Binominal each and collectivity by Mets Visser

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Binominal each and collectivity

by Mets Visser

Master thesis

Supervised by:

Jakub Dotlaˇ

cil - first supervisor

Asad Sayeed - second supervisor

Studentnumber: s1914464

Emailadress: metsvisser1992@gmail.com Coursecode: LTR999M30

Research Master Linguistics Rijksuniversiteit Groningen

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List of Figures

4.1 Syntactic tree for sentence (128): John watched every movie . . . 22 4.2 Syntactic tree for collective predicate sentence (140): John and Mary built a raft 24 4.3 Syntactic tree for distributive predicate sentence (141): John and Mary built a

raft . . . 25 4.4 Syntactic tree for fully distributive sentence with binominal each (143): John

and Mary watched two movies each. . . 26 4.5 Syntactic tree for fully distributive sentence with binominal each using individual

specific subevents (143): John and Mary watched two movies each. . . 33 4.6 Syntactic tree for collectivity retaining sentence with binominal each using

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Chapter 1

Introduction

In this master thesis I will analyse the binominal each construction. In this construction, the word ”each” (to be called ”binominal each” from now on) is added after one noun phrase, causing another noun phrase to distribute over it. An example sentence is (1). Its counterpart without binominal each is (2). This sentence is ambiguous between a collective and a distributive reading. Adding binominal each forces the distributive reading.

(1) The children bought three books each. (2) The children bought three books.

There already exists some literature on how to analyse this construction. The main goal of this thesis is to come up with an account for the problem that Blaheta pointed out in his master thesis (Blaheta, 2003). Usually, when binominal each is added to a VP-modifying prepositional phrase, it makes the VP distribute over a noun phrase, often the subject. An example is (3):

(3) The children watched a movie in two cinemas each.

There is no longer any collectivity going on, as soon as binominal each is introduced in the sentence. There is multiple movie-watching events.

But in some sentences involving collective predicates, there is still collectivity. For exam-ple, in (4), the modifier distributes over the noun phrase, but does not cause the whole VP to distribute. This is the phenomenon that I will analyse in this thesis. I will refer to sentences that display this phenomenon as ”collectivity-retaining”. The phenomenon will be called ”col-lectivity retention”.

(4) John and Mary lifted a piano with two jacks each.

1.1 Structure of this thesis

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Chapter 2

Binominal each

2.1 Basic case

Binominal each, and its counterparts in other languages, is a lexical item that languages can add after one noun phrase, causing another noun phrase to distribute over it. Sentences without binominal each are ambiguous between a distributive and a collective reading. In (5), there is a reading where they each watch a movie separately. In the other reading, they watch one movie together as a group.

(5) John and Mary watched one movie. (6) John and Mary watched one movie each.

When binominal each is added, as in (6), the second reading is no longer available. There is only the reading where they watch a movie individually. It is still ambiguous whether this movie is the same one for both of the individuals. This depends on the predicate that is used. Watching a movie can be done repeatedly using the same movie. If we use other predicates, however, things can be different. Building a raft, for example, can be done repeatedly as well, but in every case a new raft is built.

(7) John and Mary built one raft. (8) John and Mary built one raft each.

(7) is still ambiguous between a distributive and a collective reading. In the distributive reading, there are two rafts built: one by John and one by Mary. In (8), however, there is multiple events of building a raft. Unlike (6), there is no ambiguity whether there is one or multiple rafts built. The predicate building a raf t requires different rafts, as a raft can only be built once.

The word is called binominal, because it operates on two noun phrases. A distinction needs to be made between binominal (or adnominal) each, and adverbial (or floating) each. In the binominal version (9), the position of ”each” is immediately to the right of the noun phrase that is distributed. In the adverbial version, it occurs at the start of the verb phrase (10):

(9) John and Mary watched one movie each. (10) John and Mary each watched one movie.

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however, I will use the terminology that Blaheta uses in his master thesis (Blaheta, 2003). The Dist P hrase is the noun phrase that occurs immediately before ”each”. This is the noun phrase that is distributed over. The other noun phrase is called the Range P hrase . This is the noun phrase that distributes over the Dist Phrase. In the sentences above, the Range Phrase is J ohn and M ary. The Dist Phrase in (6) is one movie. The Dist Phrase in (8) is one raf t.

(11) [John and Mary]Rangewatched [one movie each]Dist.

Binominal each operates on two noun phrases: the Dist and the Range. It makes the Range Phrase distribute over the Dist Phrase. These noun phrases are related through a relation. In the sentences above, this is the transitive verb watch or the verb build, which is also transitive. They take two noun phrase arguments.

So, the anatomy of the binominal each construction consists of three parts: the Dist phrase, the Range phrase and the relation that connects the two to each other. Each of these elements in the construction is subject to some constraints. In the next section I discuss which constraints apply to each of these elements.

2.1.1 Structure of this chapter

In sections 2.2 and 2.3 I will discuss which constraints apply on the Dist and Range Phrases, respectively. Section 2.4 discusses different types of possible relations between the Dist and Range Phrase. So far we only saw transitive verbs. Ditransitives, verb modifiers and noun modifiers will also be discussed. In section 2.5, I will introduce distributivity and collectivity. Binominal each is a distributive operator and usually causes sentences to acquire a distributive reading. In section 2.5.5 I discuss why this is not always true and I introduce the main topic of this thesis. Section 2.6 briefly sums up the contents of this chapter.

2.2 Dist Phrase constraints

The Dist phrase needs to be a noun phrase. The binominal each construction causes the Range Phrase to distribute over the Dist Phrase. Therefore, the Dist Phrase needs to be indefinite, and it needs to have some kind of cardinality, which means that there needs to be some information about the number of items in the set that the Dist Phrase refers to (Blaheta, 2003) (Zimmermann, 2002). Compare these sentences:

(12) John and Mary watched [one movie each]. (13) John and Mary watched [a movie each]. (14) *John and Mary watched [the movie each].

Indefiniteness alone does not work. If the cardinality is not specified, the sentence is not grammatical.

(15) *John and Mary watched [movies each]. (16) *John and Mary watched [some movies each].

The cardinality constraint does not mean that a fixed number is needed. A range, for example, is also fine:

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(18) John and Mary watched [at least one movie each].

(19) John and Mary watched [between two and four movies each].

2.3 Range Phrase constraints

The Range phrase needs to be a noun phrase. The binominal each construction causes the Range Phrase to distribute over the Dist Phrase. The Range phrase therefore needs to be plural. If the Range Phrase is not plural, there is nothing to distribute. (Blaheta, 2003) (Zimmermann, 2002).

(20) [John and Mary] watched one movie each (21) [The students] watched one movie each (22) *[John] watched one movie each

(23) *[The student] watched one movie each

Definite plurals, like the ones in (20) and (21), always work. Indefinite plurals are also fine: (24) [Some students] watched one movie each.

(25) [Two students] watched one movie each.

Noun phrases containing quantifiers are sometimes fine, as long as they are not distributive themselves. The use of two distributive operators, where only one is necessary, is redundant and ungrammatical:

(26) ?[All students] watched one movie each. (27) *[Most students] watched one movie each. (28) *[Every student] watched one movie each. (29) *[Each student] watched one movie each.

2.4 Possible relations

The relation element of the binominal each construction links the Dist and Range phrases to each other. This is often a verb, but it does not need to be. Compare these sentences:

(30) John and Mary [watched] one movie each. (31) The students [bought] two books each. (32) Gloves [with] four fingers each

(33) Groups [of] fifteen people each

(34) John and Mary [followed] Bill [with] eight nominations each (35) John [cut] the tomatoes [into] three slices each

In the basic case, the Range is the subject, the Dist is the object, and the relation is a transitive verb:

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(37) [The students]Range [bought]Rel[two books each]Dist .

The binominal each construction requires two noun phrases that are linked to each other by means of a relation. Sentences with an intransitive verb involving binominal each therefore do not exist. There are verbs, however, that allow more than two noun phrases. Extra noun phrases or prepositional phrases can be added. These sentences display some flexibility when binominal each is introduced.

2.4.1 Ditransitives

Ditransitive verbs like ”give” take three noun phrase arguments. One of these has to be the Range phrase, another one has to be the Dist phrase. The binominal each construction does not allow all possibilities, though.

Sentence (38) is ambiguous. Both the subject ”John and Mary”, as in (39), as well as ”the children”, as in (40), can be the Range Phrase. This ambiguity arises as soon as there are multiple plural noun phrases in the sentence. Sentence (41) and (42) show that the Range can be both the subject or the indirect object.

(38) John and Mary gave the children one muffin each.

(39) [John and Mary]Range gave the children [one muffin each]Dist.

(40) John and Mary gave [the children]Range [one muffin each]Dist.

(41) John gave [the children]Range [one muffin each]Dist.

(42) [John and Mary]Range gave the child [one muffin each]Dist.

In these sentences, the Dist was the direct object. The indirect object, however, can also take the role of Dist:

(43) [John and Mary]Range gave [two children each]Dist a muffin.

There is a peculiar constraint here. The indirect object can only take the Dist role if the subject is the Range. Sentences where the indirect object is the Dist can not have the direct object as the Range. This sentence would have a meaning where there are two groups of two children and each group of children gets a muffin. Native speakers reject this possible meaning. They only accept the floating each version, where each child receives two muffins.

(44) John gave [two children each]Dist [two muffins]Range.

When we replace the indirect objects by their corresponding to-PPs, things are similar: (45) [John and Mary]Range gave [one muffin each]Dist to the children .

(46) John gave [one muffin each]Dist [to the children]Range.

(47) [John and Mary]Range gave [one muffin each]Dist to the child .

(48) [*floating only*] John gave [two muffins]Range [to two children each]Dist .

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2.4.2 Modifiers

The binominal each construction does not just occur in obligatory arguments of the verb. It can also occur in adjuncts modifying nouns or verbs. Often, these adjuncts are prepositional phrases. A distinction has to be made between noun modifiers and verb modifiers. Noun modifiers modify the Range phrase, while verb modifiers modify the event denoted by the verb.

2.4.3 Noun modifier

Some modifiers do not modify the event denoted by the verb, but modify the Range phrase. Noun modification is usually expressed by a preposition. The preposition usually signals a possessive/attributive relation, providing extra information about the individuals in the Range (Blaheta, 2003) (Zimmermann, 2002). Some examples:

(49) [Gloves]Range [with]Rel [four fingers each]Dist

(50) [Groups]Range [of]Rel [fifteen people each]Dist

(51) [Lists]Range [with]Rel [three names each]Dist

2.4.4 Verb modifier

The binominal each construction does not just occur in obligatory arguments of the verb. It can also occur in optional adjuncts, often in the form of prepositional phrases. These adjuncts take a particular role with the respect to the verb they modify. The binominal each construction can occur in a whole range of different roles:

(52) John hung [the curtains]Range[over two windows each]Dist. Role=location/result

(53) [The students]Range hit the teacher [three times each]Dist. Role=time

In the basic case, the Range phrase is the subject of the sentence, but other grammatical functions are also possible. In other sentences involving verb modifying prepositional phrases, the Range phrase can be the direct object:

(54) I cut [the tomatoes]Range into [four slices each]Dist.

(55) We bought [chicks]Range for [four cents each]Dist.

(56) John battles [his enemies]Range with [four armies each]Dist.

In all these cases involving binominal each in verb modifiers, the event that is modified is distributed across all individuals in the range. For every individual in the range, there is a separate event. The modifier then modifies each of these events. It does not matter whether the Range is the subject or the object.

In the section about ditransitives I claimed that the Range Phrase can take only the subject and the indirect object of a ditransitive sentence. It does not matter whether the sentence contains a dative or a to-PP. It is peculiar that the object position is not available for the Range phrase in ditransitive sentences.

(57) I cut [the tomatoes]Range into [four slices each]Dist.

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In (57), the meaning is that each of the tomatoes is cut into four slices. But in (58), there is no reading where each tomato is given to four children. This reading is grammatical, though. But native speakers reject it for pragmatic reasons. They prefer the floating-each reading where each tomato is given to four children. Thematic roles seem to have some influence on the availability of particular readings. This is not because of the syntax of binominal each, but because of pragmatic reasons.

2.5 Collectivity and distributivity

For some sentences, the relationship between the verb and its arguments can be ambiguous. (59) John and Mary built a raft.

Sentence (59) has two readings. In the first reading, John and Mary built one raft together, both taking part in the building. This reading is called the collective reading. In the other reading, John and Mary each built a raft, so there are two rafts involved. This reading is called the distributive reading. The Verb Phrase distributes over the individual in the subject.

There has been a lot of discussion about how these different readings should be analysed. Link (Link, 1983) introduces his * operator to create predicates that take groups/sums as arguments, and not just atoms. He claims that groups/sums are arranged in a mereological way, in a join semi-lattice. Atoms can form a group/sum by applying the sum operator: ⊕. The two readings of the (59) would be:

(60) [[build a raf t]](j ⊕ b)

(61) [[build a raf t]](j) ∧ [[build a raf t]](b)

Whether distributivity is caused by noun phrases or by the verb phrase is a point of dis-cussion. Some claim noun phrases are ambiguous between a single plural entity and a set of individuals. Others claim that the verb phrase causes the ambiguity. For a discussion, see (Landman, 2000) . Some verbs only take individuals as arguments. These are called distributive predicates. Others only take groups/sums arguments. Those are called collective predicates. A third group of predicates can take either individuals or groups/sums. These are called mixed predicates. I will discuss these three different classes shortly.

2.5.1 Distributive Predicates

The use of the word distributivity generally indicates the application of a predicate to the members or subsets of a set (Champollion, 2014). This concerns the entailment patterns of a predicate. Each of the different classes have different entailment patterns. This entailment pattern applies to distributive predicates (Nouwen, 2015)

(62) A predicate VP is distributive if and only if X and Y VP entails X VP and Y VP Purely distributive predicates take only atomic arguments. Examples are: smile , bewounded. However, these predicates can still take plural arguments. In that case, some theories assume some kind of distributive operator that performs the entailment pattern in (62) (Nouwen, 2015). A sentence like (63) would then get the logical representation of (64).

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2.5.2 Collective Predicates

Collectivity generally involves the notion of a predicate that applies to a plural entity as a whole, as opposed to applying to the individuals that form this entity (Champollion, 2014). Collective predicates lack the entailment pattern in (62). Collective predicates apply to plural arguments as a whole. It often does not even make sense to apply it to individuals, and may result in ungrammatical sentences:

(65) John and Mary met (66) * John met

meet is a collective predicate and does not apply to single individuals. Individuals cannot meet. Therefore, the extension of meet consists of only plural entities: sums or groups. They are combined using the sum ⊕ operator. (65) would have this representation:

(67) [[meet]](j ⊕ m)

2.5.3 Mixed Predicates

There is a third class of predicates that can take both singular and plural arguments. These predicates are called mixed predicates. Both individuals and sums/groups of those individuals can be in the extension of a mixed predicate.

(68) John lifted a chair

(69) John and Mary lifted a chair (70) [[lif t a chair]](j)

(71) [[lif t a chair]](j ⊕ m)

2.5.4 Collective predicates and ”each”

There is an interaction between these mixed predicates and the binominal each construction. Compare the sentences below. In the first sentence, the predicate is ambiguous. It can be interpreted both distributively and collectively. In the distributive reading, John is lifting one chair and Mary another. In the collective reading, they lift one chair together, as a group. The sentence is ambiguous between these two readings. In the second sentence, however, the ambiguity is resolved. The collective reading is no longer available. Binominal each in mixed predicate sentences forces the distributive reading.

(72) John and Mary lifted one chair (73) John and Mary lifted one chair each

2.5.5 Pianos and jacks

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(74) John and Mary lifted a piano with one jack each.

In this situation, the verb phrase is not distributed. Usually in binominal each sentences, the Range Phrase also distributes the Verb Phrase. But in this sentence there is only one event of lifting a piano. The distributive reading (where each of the individuals lifts a piano) is not available. It is pragmatically odd because one person can not lift a piano on their own. However, (74) is grammatical and perfectly fine from a pragmatic point of view. So the other reading must be available, the reading where John and Mary collectively lift the piano. The paradox here is that each of them uses one jack to carry this collective action out. So, binominal each in the verb modifier causes the modifier to distribute across the individuals in the range, but leaves the verb phrase itself collective. This is an interesting phenomenon that has not been accounted for before. From now on, this class of sentences with collectivity and distributivity mixed, will be called ”collectivity-retaining” and I will look into them in more detail in the next section.

2.6 Conclusion

The binominal each construction has three elements: • Range Phrase - which distributes

• Dist Phrase - which is distributed over by the Range Phrase • Relation - which links the Range and Dist Phrases to each other

The Dist and Range Phrases are constrained by some restrictions. The Dist Phrase has to be indefinite and requires some kind of cardinality. The Range Phrase needs to be plural. There is a wide range of possible relations that the binominal each construction appears in: transitive, ditransitive, verb modifier and noun modifier are covered here.

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Chapter 3

Collectivityretaining sentences

-Data analysis

In this chapter I will explore the range of different possibilities that collectivity retention allows for, analyse them and come up with an explanation. I will look into two different sources of explaining this phenomenon: the verb and the semantic role of the Dist Phrase. First, I will check whether collectivity retention occurs with all possible verb classes. Then I will check whether some semantic roles occur in collectivity-retaining sentences more often than in others.

Before doing that, I will first define the notion of ”collectivity retention”.

Definition 1. Sentences involving binominal each in which at least one argument/modifier of the verb is interpreted in a collective way, are called ”collectivity-retaining”.

Collectivity-retaining sentences also have a counterpart. Those sentences will be called ”fully distributive”:

Definition 2. Sentences involving binominal each in which every argument/modifier of the verb is interpreted in a distributive way, are called ”fully distributive”.

Sometimes I will mark sentences to show whether they are interpreted in a retaining way or in a fully distributive way. Sentences that are not marked have a collectivity-retaining interpretation. Sentences that are interpreted only in a fully distributive way are marked with [FD]. Apart from that, stars and question marks are used, as usual, to mark ungrammatical and questionably grammatical sentences.

3.0.1 Structure of this chapter

In section 3.1 I discuss Aktionsart verb classes. I first introduce the four verb classes used in this classification. Then, I check in which classes collectivity-retaining sentences occur often and in which they do not. In section 3.2 I discuss the semantic role of the Dist Phrase. I look for patterns concerning which semantic roles the Dist Phrase takes. In 3.3 I draw conclusions from the discussed data.

3.1 Aktionsart

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aspect of a verb tells something about the way in which it is structured in time. Lexical aspect should not be confused with grammatical aspect, which for example the progressive form is about. Lexical aspect is an inherent property of a verb, whereas grammatical aspect can be changed mostly independent of the verb.

3.1.1 Vendlerian classes

Vendler (Vendler, 1957) separated verbs into four different classes: states, activities, accom-plishments and achievements. I will use the definitions from Rothstein (Rothstein, 2008). Verb phrases are classified according to two features: telicity and dynamicity. The telicity of a verb phrase depends on whether it has an internal culmination point. If it has, the verb phrase is telic. Otherwise it is atelic. The dynamicity of a verb phrase depends on whether the verb phrase has internal stages. A verb phrase e0 has stages in case some stage e can be identified that develops into e0.

Furthermore, three more properties are used to talk about verb phrases with respect to this classification: cumulativity, quantization and homogeneity. In contrast to telicity and dynamicity, however, they do not define the classes. An event denoted by a verb phrase is cumulative if the sum of two of those events is also an instance of that event. An event e is quantized if there exists no subevent e0 of e such that e0 is also an instance of e. An event is homogeneous if all subevents e0 of e are also instances of e.

States

States are atelic, cumulative and non-dynamic. Every instant of a state is equal to any other instance. They are homogeneous down to those instants. Some examples:

(75) The sky is blue (76) I know the answer Activities

Activities are also atelic, but in contrast to states they have stages. They are therefore dynamic. They are also homogeneous, but down to minimal intervals instead of instants, like states are.

(77) John dances (78) Mary builds houses Accomplishments

Accomplishments share the dynamicity with activities. Stages can also be identified. The difference from activities is that with accomplishment, these stages in the end develop into a culmination point. They are therefore telic. They are therefore not cumulative and not homogeneous, but they are quantized.

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Achievements

Achievements are also telic, like accomplishments. However, unlike accomplishments, they do not have stages. They occur in instants or very small intervals. They are heterogeneous and non-cumulative.

(81) John arrived at the station (82) The bomb exploded

3.1.2 Collectivity retention

When it comes to sentences involving collectivity retention with binominal each, it seems that some Aktionsart classes are more common than others. Accomplishments and achievements are more common than activities and states:

(83) John and Mary lifted a piano with two jacks each. (accomplishment, instrument) (84) The soldiers weighed the coins on two scales each. (accomplishment, instrument) (85) The students founded a company with two investments each. (achievement, instrument) (86) The students elected a president with three votes each. (achievement, instrument)

It seems that both accomplishments and achievements are both able to occur in collectvity retaining binominal each sentences. Collectivity-retaining sentences occur far more often in sentences of which the verb is an accomplishment or achievement. Collectivity retention with activities and states are not that common.

Accomplishments

It seems that accomplishments are not problematic in collectivity-retaining sentences with bi-nominal each:

(87) John and Mary lifted a piano with two jacks each. (88) The soldiers weighed the coins on two scales each.

There are several tests to check whether verbs are accomplishments. Accomplishments are able to occur in the present continuous, they can be modified by ”in x time” phrases but not by ”for x time”. When ”almost” is added, it can apply to both the starting point of the event, or the endpoint. Accomplishments also take aspectual verbs like ”resume”.

(89) John and Mary were lifting a piano with two jacks each.

(90) John and Mary lifted a piano with two jacks each in/*for an hour. (91) John and Mary almost lifted a piano with two jacks each. (start/end) (92) John and Mary resumed lifting a piano with two jacks each.

(93) The soldiers were weighing the coins on two scales each.

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Achievements

Achievements are also common in collectivity-retaining sentences with binominal each:

(97) The students founded a company with two investments each. (achievement, instrument) (98) The students elected a president with three votes each. (achievement, instrument)

Achievements behave differently in the same tests compared to accomplishments. Achieve-ments do not occur in the progressive. When modified by ”almost”, there is no longer ambiguity whether the startingpoint or the endpoint has been reached. Achievements occur in instants or very small intervals and there is no such thing as a distinction between startingpoint and endpoint. Finally, achievements cannot take an aspectual verb like ”resume”.

(99) *The students were founding a company with two investments each.

(100) The students founded a company with two investments each in/*for an hour. (101) The students almost founded a company with two investments each. (end) (102) *The students resumed founding a company with two investments each. (103) *The students were electing a president with three votes each.

(104) The students elected a president with three votes each in/*for an hour. (105) The students almost elected a president with three votes each. (end) (106) *The students resumed electing a president with three votes each. Activities

The accomplishment sentences can be turned into activities by using bare plurals instead of indefinites.

(107) [John and Mary]Rangeare carrying pianos around with two jacks each.

(108) John and Mary are carrying [pianos]Range around with two jacks each.

(109) [The soldiers]Range are weighing coins on two scales each.

(110) The soldiers are weighing [coins]Range on two scales each.

When bare plurals are used instead of indefinites, it seems that this plural might take over the role of Range Phrase. The first case is ambiguous between the two readings (107) and (108): Both ”John and Mary” and ”pianos” can serve as Range Phrase. In (107), the collectivity is retained. In (108), there is also a reading available in which they only individually carry pianos. In the second case, ”coins” can also be the Range phrase and this is in fact much more pragmatically plausible. Activities do not necessarily have to occur with direct objects. We can remove them:

(111) John and Mary are carrying with two jacks each. (112) The soldiers are weighing on two scales each.

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States

States are not easy to modify, even with regular modifiers that make the verb distribute. I found only one possible example of state modification that involves collectivity retention: (113) John and Mary liked each other for three reasons each.

States do not occur in the progressive. They also do not take aspectual verbs like ”resume”. Modification with ”for x time” is allowed. Modification with ”in x time” is also allowed. In that case it will modify the starting point. Addition of ”almost” is also allowed. In that case it also modifies the starting point.

(114) *John and Mary are liking each other for three reasons each.

(115) John and Mary liked each other for three reasons each for/*in three months. (116) *John and Mary almost liked each other for three reasons each.

(117) *John and Mary resumed liking each other for three reasons each.

The tests do not fully reveal the pattern of states. In-modification does not seem to work at all. Also, adding ”almost” results in an ungrammatical sentence. I am not sure whether this is actually a state.

3.1.3 Conclusion

From the data so far, we can conclude that accomplishments and achievements are common in collectivity-retaining sentences. Achievements are less common. Sentences with achievements that are collectivity-retaining do exist, but they can be ambiguous. The meaning is not obvious. States are even less common.

3.2 Semantic role of the Dist Phrase

It seems that not all kinds of modifiers seem to be able to modify the verb in collectivity-retaining sentences. It turns out that the semantic role that the modifier takes is important. Possible roles

In sentences involving accomplishments, the modifier can take instrument and origin roles: (118) John and Mary lifted a piano [with two jacks each]instrument

(119) John and Mary lifted a piano [from one side each]origin

Achievements are less likely to occur in collectivity-retaining sentences. Compared to achievements, they lack dynamicity. They are momentary events instead of interval events. Because of this, achievements in general take modifiers less easily. They cannot be modified by origin modifiers, but instruments are no problem.

(120) The students founded a company [with two investments each]instrument

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Impossible roles

Time and location modifiers cannot appear in collectivity-retaining sentences: (122) [FD] John and Mary lifted a piano [for two hours each]time

(123) *John and Mary lifted a piano [in two rooms each]location

(124) [FD] The students elected a president [for three years each]time

(125) [FD] The students founded a company [in two countries each]location

When binominal each occurs in verb modifiers that are of type location or time, the col-lectivity retention reading is no longer available. Instead, they get a fully distributive reading. In (122), there are two piano liftings. First John lifts a piano for two hours, then Mary lifts the same piano for two hours. (123) is ungrammatical, as a lifting event cannot happen at two locations at the same time. (124) is fully distributive as well. This sentence has the reading that each student elected a different president for 3 years. (125) is also fully distributive. The sentence means that that for every student, there is a company found in two countries.

This difference between roles can be explained in terms of the events the verbs denote. Time and location modifiers with binominal each cause the event, or Verb Phrase, to distribute. One piano lifting event cannot happen at multiple times or multiple locations. If multiple times or locations are mentioned in a sentence, there are multiple events involved: one for each time/location.

With instruments or origin roles, the situation is different. There can be one lifting event in which multiple instruments (one jack for every lifter) or multiple origins (every lifter lifts from another side) are involved.

3.3 Restrictions

From the examples above it is clear that collectivity-retaining sentences are quite restricted. There are two constraints that such sentences need to meet:

• The verb needs to take at least three arguments: a Dist Phrase, a Range Phrase and any number of other Noun or Prepositional Phrases

• At least one argument/modifier other than the Range or Dist Phrase should be interpreted in a collective way.

Besides these strict requirements, there are some general tendencies that can be noticed regarding collectivity-retaining sentences:

• Verbs of type accomplishment and achievement are more common than activities and states

• Location and time roles cause the Verb Phrase to distribute. Therefore, they cannot occur in collectivity-retaining sentences, but cause the sentence to have a fully distributive reading instead.

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Chapter 4

Semantic analysis

In this chapter I provide a semantic analysis of binominal each that is able to account for collectivity-retaining sentences. To do that, I will use Champollion’s framework (Champollion, 2015) called Quantificational Event Semantics (QES) that combines type-theoretic representa-tions and events in an elegant and mathematically clean way.

4.1 Quantificational Event Semantics

Quantificational Event Semantics makes use of events in the Neo-Davidsonian way. Verbs denote event predicates. They take event variables as arguments. In Neo-Davidsonian methods, opposed to Davidsonian methods, not just modifiers but also the arguments of the verb are treated as functions from events to entities. These functions are called semantic roles. These functions are conjoined to the event predicate. This is the Neo-Davidsonian representation for a simple sentence:

(126) John buttered the toast with the knife

(127) ∃e[butter(e) ∧ agent(e) = j ∧ theme(e) = t ∧ instrument(e) = k]

Quantificational Event Semantics is different compared to standard approaches. Standard approaches take the event variable to have widest scope in the sentence. Champollion argues that the event should always have lowest scope. To make this happen, he introduces a different closure operator. (Champollion, 2015) Also, in QES, noun phrases first combine with their semantic role. They then combine with the verb. In standard approaches this is done the other way around.

4.1.1 Types

Champollion’s system is a typetheoretic system. It uses three basic types: e for entities, t for truth values, and v for events. Noun phrases are translated into generalized quantifiers of type het, ti. Verb denotations are also type raised, like generalized quantifiers. Instead of type v they are of type hvt, ti.

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Category Type Example lexical entry Interpretation

DPs het, ti [[two movies]] λP.∃X[X ⊆ movies ∧ |X| = 2 ∧ P (X)] Verbs hvt, ti [[watch]] λf.∃e[watch(e) ∧ f (e)]

Roles hett, hvtt, vttii [[theme]] λQλV λf.Q(λx.V (λe.[f (e) ∧ th(e) = x])) Closure hv, ti closure operator λe.true

Table 4.1: Categories, types and interpretations for some example lexical entries

expression is of type hvtt, vtti. Arguments and modifiers keep combining with this verb phrase denotation. When all arguments and modifiers have combined with the verb phrase, the closure operator applies. The resulting sentence is then of type t.

Table (4.1) contains categories, types and interpretations for some example lexical entries. The next section shows how the semantic composition is done using an example sentence.

4.1.2 Example sentence with quantifiers

I will show the syntactic tree and the derivation of an example sentence (128) involving a quantifier noun phrase to show that the existential quantifier binding the event gets indeed the lowest scope. Below are the lexical entries that are used.

The derivation along with the syntactic tree (4.1) of example sentence (128) are written below. The derivation makes it clear that the existential quantifier binding the event has the lowest scope.

(128) John watched every movie

These are the lexical entries that are used in the compositional process: (129) [[john]] = λP.P (j)

(130) [[watch]] = λf.∃e[watch(e) ∧ f (e)]

(131) [[every movie]] = λP.∀x[movie(x) ∧ P (x)]

(132) [[theme]] = λQλV λf.Q(λx.V (λe.[f (e) ∧ th(e) = x])) (133) [[agent]] = λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x])) (134) closure operator = λe.true

4.1.3 Pluralities

In the QES framework, pluralities are treated as sets. It uses the type α that ranges over type e expressions for singular entities and type he, ti for pluralities (sets). The expression ”four girls”, for example would get the following denotation:

(135) λP.∃X[X ⊆ girls ∧ |X| = 4 ∧ P (X)] X is of type α and P is of type hα, ti.

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S t ∀x[movie(x) →

∃e[watch(e) ∧ ag(e) = j ∧ th(e) = x]]

VP hvt, ti λf.∀x[movie(x) →

∃e[watch(e) ∧ f (e) ∧ ag(e) = j ∧ th(e) = x]]

VP hvt, ti λf.∀x[movie(x) → ∃e[watch(e) ∧ f (e) ∧ th(e) = x]]

DP hvtt, vtti λV λf.∀x[movie(x) → V (λe.[f (e) ∧ th(e) = x])]

[th] hett, hvtt, vttii λQλV λf. Q(λx.V (λe.[f (e) ∧th(e) = x])) every movie het, ti λP.∀x[movie(x) → P (x)] watch hvt, ti λf.∃e[watch(e) ∧ f (e)] DP hvtt, vtti

λV λf V (λe[f (e) ∧ ag(e) = j]) [ag] hett, hvtt, vttii λQλV λf. Q(λx.V (λe.[f (e) ∧ag(e) = x])) John het, ti λP.P (j) closure hv, ti λe.true

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(136) λP.P ({j, m})

To shift it to its distributive interpretation, we use the DistShift operator which is defined as:

(137) [[DistShif t]] = λQλP.Q(λX.∀y ∈ X → P (y))

To get the distributive interpretation of the noun phrase, we apply the Distshift operator to the denotation in (136). The result is (138).

(138) λP.∀y[y ∈ {j, m} → P (y)]

Using this ambiguity in the Noun Phrase, we can assign two interpretations for sentences involving mixed predicates, like (139). Taking the collective interpretation and the distributive interpretation of the Noun Phrases, respectively, we arrive at (140) and (141).

(139) John and Mary built a raft

(140) ∃x[raf t(x) ∧ ∃e[build(e) ∧ ag(e) = {j, m} ∧ th(e) = x]]

(141) ∀y[y ∈ {j, m} → ∃x[raf t(x) ∧ ∃e[build(e) ∧ ag(e) = y ∧ th(e) = x]]]

The syntactic tree and semantic composition for (140) is shown in (4.2). For its distributive counterpart (141), the tree and composition are shown in (4.3).

4.2 Binominal each in fully distributive sentences

In the QES framework, all noun phrases that are arguments or modifiers of the verb phrase, apply one by one to the verb denotation, returning a new verb phrase denotation with the added modifier/argument. Binominal each needs to relate two of such arguments to each other: the Dist and Range phrase. What binominal each needs to do is distribute the Range Phrase over the Dist Phrase. But making the Range Phrase distribute makes the process slightly more complicated. (142) is the lexical entry for binominal each I came up with. It contains four lambda operators. For each lambda operator, the type is written below.

(142) [[each]] = λDisthvtt,vttiλRelhvt,tiλRangeRolehett,hvtt,vttiiλRangehet,ti

.RangeRole(DistShif t(Range))(Dist(Rel))

I will explain how binominal each works in the QES framework using (143) as an example sentence. This sentence is fully distributive. Collectivity-retaining sentences will be accounted for in the next section. (144) is the meaning representation we are looking for:

(143) John and Mary watched two movies each.

(144) ∀y[y ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ ag(e) = y ∧ th(e) = X]]]

4.2.1 Derivation

These are the lexical entries that are used in the compositional process. The syntactic tree shows how the lexical entries are combined.

(145) [[J ohn and M ary]] = λP.P ({j, m})

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S t ∃x[raf t(x)∧

∃e[build(e) ∧ ag(e) = {j, m} ∧ th(e) = x]]

VP hvt, ti λf.∃x[raf t(x)∧

∃e[build(e) ∧ f (e) ∧ ag(e) = {j, m} ∧ th(e) = x]]

VP hvt, ti λf.∃x[raf t(x)∧ ∃e[build(e) ∧ f (e) ∧ th(e) = x]]

DP hvtt, vtti λV λf.∃x[raf t(x)∧ V (λe.[f (e) ∧ th(e) = x])]

[th] hett, hvtt, vttii λQλV λf. Q(λx.V (λe.[f (e) ∧th(e) = x])) a raft het, ti λP.∃x[raf t(x) ∧ P (x)] build hvt, ti λf.∃e[build(e) ∧ f (e)] DP hvtt, vtti

λV λf V (λe[f (e) ∧ ag(e) = {j, m}]) [ag] hett, hvtt, vttii

λQλV λf. Q(λx.V (λe.[f (e)

∧ag(e) = x])) John and Mary

het, ti λP.P ({j, m}) closure

hv, ti λe.true

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S t ∀y[y ∈ {j, m} →

∃x[raf t(x) ∧ ∃e[build(e) ∧ th(e) = x ∧ ag(e) = y]]]

VP hvt, ti λf.∀y[y ∈ {j, m} →

∃x[raf t(x) ∧ ∃e[build(e) ∧ th(e) = x ∧ f (e) ∧ ag(e) = y]]]

VP hvt, ti λf.∃x[raf t(x)∧ ∃e[build(e) ∧ f (e) ∧ th(e) = x]]

DP hvtt, vtti λV λf.∃x[raf t(x)∧ V (λe.[f (e) ∧ th(e) = x])]

[th] hett, hvtt, vttii λQλV λf. Q(λx.V (λe.[f (e) ∧th(e) = x])) a raft het, ti λP.∃x[raf t(x) ∧ P (x)] build hvt, ti λf.∃e[build(e) ∧ f (e)] DP hvtt, vtti λV λf ∀y[y ∈ {j, m} → V (λe.f (e) ∧ ag(e) = y])]

[ag] hett, hvtt, vttii

λQλV λf. Q(λx.V (λe.[f (e)

∧ag(e) = x])) John and Mary

het, ti

λP.∀y[y ∈ {j, m} → P (y)] closure

hv, ti λe.true

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S (166) (164) VP (156) (154) [[each]] λDistλRelλRangeRoleλRange .RangeRole(DistShif t(Range))(Dist(Rel)) DistPhrase (153) [[theme]] [[two movies]] [[watch]] Rel [[agent]] RangeRole [[john and mary]]

Range [[closure]]

Figure 4.4: Syntactic tree for fully distributive sentence with binominal each (143): John and Mary watched two movies each.

(147) [[watch]] = λf.∃e[watch(e) ∧ f (e)]

(148) [[two movies]] = λP.∃X[X ⊆ movies ∧ |X| = 2 ∧ P (X)] (149) [[theme]] = λQλV λf.Q(λx.V (λe.[f (e) ∧ th(e) = x])) (150) [[agent]] = λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x])) (151) [[closure]] = λe.true

Before we use the denotation of binominal each, the Dist Phrase first combines with its semantic role. We combine two movies with its semantic role theme.

(152) [[theme]]([[two movies]]) = λQλV λf.Q(λx.V (λe.[f (e)∧th(e) = x]))(λP.∃X[X ⊆ movies∧ |X| = 2 ∧ P (X)])

(153) [[theme]]([[two movies]]) = λV λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ V (λe.[f (e) ∧ th(e) = X])] each takes the resulting phrase (the Dist Phrase) as its first argument and combines with it. (154) [[two movies each]] = λRelλRangeRoleλRange.RangeRole(DistShif t(Range))

(λV λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ V (λe.[f (e) ∧ th(e) = X])](Rel))

The second argument is the relation Rel. Inside the denotation of each ,the Dist Phrase applies to this argument.

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(156) [[watch two movies each]] = λRangeRoleλRange.RangeRole(DistShif t(Range)) (λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]])

For incorporating the Range Phrase, we slightly deviate from the usual approach. Usually in the QES framework, the het, ti is first combined with the role denotation. The result, a DP phrase, is then incorporated in the composition of the VP. With this preliminary version of binominal each, this does not work, as the DistShift operator only takes het, ti type denotations as arguments. We therefore have to make an exception to the usual way of sentence composition. To incorporate the Range Phrase, we first combine the whole thing with the Role of the Range Phrase.

(157) λRangeRoleλRange.RangeRole(DistShif t(Range))

(λf ∃X[X ⊆ movies∧|X| = 2∧∃e[watch(e)∧f (e)∧th(e) = X]])(λQλV λf.Q(λx.V (λe.[f (e)∧ ag(e) = x])))

(158) λRange.λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x]))(DistShif t(Range)) (λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]]) Then, we plug in the het, ti of the Range.

(159) λRange.λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x]))(DistShif t(Range))

(λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]])(λP.P ({j, m})) (160) λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x]))(DistShif t(λP.P ({j, m})))

(λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]]) We also plug in the denotation of the DistShift-operator.

(161) λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x]))(λQλP.Q(λX.∀y ∈ X → P (y))(λP.P ({j, m}))) (λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]])

Now, the DistShift-operator applies to the het, ti denotation of the Range Phrase: (162) λQλV λf.Q(λx.V (λe.[f (e) ∧ ag(e) = x]))(λP.∀y[y ∈ {j, m} → P (y)])

(λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]])

Then, the role denotation of the Range Phrase applies to the distributive Range Phrase het, ti denotation:

(163) λV λf.∀y[y ∈ {j, m} → V (λe.[f (e) ∧ ag(e) = y])]

(λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ f (e) ∧ th(e) = X]]) The next step is applying the resulting phrase to the VP-denotation:

(164) λf.∀y[y ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ ag(e) = y ∧ f (e) ∧ th(e) = X]]]

The last step is applying the closure operator:

(165) λf.∀y[y ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e[watch(e) ∧ ag(e) = y ∧ f (e) ∧ th(e) = X]]](λe.true)

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4.3 The logical form of collectivity-retaining sentences

I will extend the current framework in such a way that it is able to account for collectivity-retaining sentences with binominal each. Before providing a lexical entry for each and showing an example derivation, we need to know what the logical form of these sentences look like. There are multiple possibilities.

4.3.1 Possible logical forms

The logical form of collectivity-retaining sentences is not as straightforward as the one of fully distributive sentences. In fully distributive sentences, the Range Phrase is only interpreted distributively. In collectivity-retaining sentences, the Range Phrase is interpreted in two ways at the same time. On the one hand, the Range Phrase is interpreted collectively as an argu-ment/modifier of the event. On the other hand it is interpreted distributively because binominal each distributes the Dist Phrase across all individuals in the Range.

This double behaviour has to be accounted for in the logical form. There are multiple ways to do this. (167) is the sentence of which we want to obtain the logical form in the end. As a starting point, we could take the logical form of a simple transitive sentence like (168). The result will be (169). It captures the fact that John and Marry participate in the lifting event in a collective way. This sentence can be derived with the current framework.

(167) John and Mary lifted a piano with two jacks each (168) John and Mary lifted a piano

(169) ∃x[piano(x) ∧ ∃e[lif t(e) ∧ th(e) = x ∧ ag(e) = {j, m}]]

Now we have to add the verb modifier that takes the instrument role. In (170), the modifier without binominal each is added. Its corresponding logical form is (171).

(170) John and Mary lifted a piano with two jacks

(171) ∃z[2jacks(z) ∧ ∃x[piano(x) ∧ ∃e[lif t(e) ∧ th(e) = x ∧ ag(e) = {j, m} ∧ instr(e) = z]]] (171) captures the fact that two jacks were used to do the lifting. This is not what we want but it sheds some light on the relation between the agent and the instrument role. In (171), this relation is very general. It only tells us that there are two jacks that are used by John and Mary together to lift the piano. Nothing is said about the way in which the jacks are distributed across the agents.

There are multiple ways in which this can be done. We have to somehow express the dual nature of the Range Phrase in the logical form. In the first two approaches, I mention both the collective and the distributive behaviour. In the third approach, I mention only the distributive behaviour and assume that the collective behaviour follows implicitly. In the fourth approach, I mention the collective behaviour explicitly and handle the distributive behaviour in the definition of the subevent.

4.3.2 Role of the Range Phrase specified

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two jacks per individual agent. There are two agents, so a total of four jacks is used to lift the piano.

(172) John and Mary lifted a piano with two jacks each

But incorporating this participant-orientation makes the framework more complex. Entries for roles would need to contain other roles. If we would allow that, we could get these logical forms for sentence (172):

(173) ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ ∀y[y ∈ ag(e) → ∃z[2jacks(z) ∧ sub(e0, e) ∧ instr(e0) = z]]]]

(174) ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ ∀y[y ∈ {j, m} → ∃z[2jacks(z) ∧ sub(e0, e) ∧ instr(e0) = z]]]]

Both are intuitive meaning representations of sentence (172). sub(e0, e) is a subevent func-tion that returns true if e0 is a subevent of e. In both representations, there are subevents for every agent of the lifting event. The instrument modifies this subevent.

Both representations capture the intuitive meaning. In the first one, it is specifically stated that the participant, that distributes over the instrument, is the agent of the lifting event. I will call this one the ”participant specific” representation. This option is discussed in this chapter. In the second one, nothing is said about the role of the participant that is distributed across. I will call this one the ”participant general” representation. This option is discussed in the next section, section 4.3.3. For both options, however, we need a new entry for roles.

In our current framework, roles combine with het, ti denotations and then combine with the verb phrase. An example of a lexical entry of such a role is (175).

(175) [[theme]] = λQλV λf.Q(λx.V (λe.[f (e) ∧ th(e) = x]))

However, this type of lexical entry for roles is not able to account for agent-orientation. It does not introduce subevents and modification of that either. So we need to have additional lexical entries for roles.

With role-specific entries for roles, we can model roles that are oriented towards a specific role. In (172), for example, the instruments are distributed across the agent. The instrument role is oriented towards the agent. We can get logical forms like (173).

The entry for an agent-oriented instrument role would be this:

(176) [[instr]] = λQλV λf.V (λe.[f (e) ∧ ∀y[y ∈ ag(e) → Q(λx.sub(e0, e) ∧ instr(e0) = x)]]) If we combine it with its Noun Phrase of type het, ti we get (178).

(177) [[two jacks]] = λP.∃X[X ⊆ jacks ∧ |X| = 2 ∧ P (X)]

(178) λV λf.V (λe.[f (e)∧∀y[y ∈ ag(e) → ∃X[X ⊆ jacks∧|X| = 2∧sub(e0, e)∧instr(e0) = X]]]) Problems

The problem with specific role oriented entries for roles is they do not generalise well. For example, the instrument role is often oriented towards the agent, but this does not necessarily have to be the case. It can also be directed towards the theme/patient:

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4.3.3 The Range Phrase itself specified

To avoid this problem, we could use representations that mention just the Range Phrase instead of the role that the Range Phrase takes. For (172) we would get this representation:

(181) ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ ∀y[y ∈ {j, m} → ∃z[2jacks(z) ∧ sub(e0, e) ∧ instr(e0) = z]]]]

To get this kind of representation, we need to somehow combine the Role entry with the Range Phrase. We could add an extra argument of type het, ti for the Range:

(182) [[instr]] = λRλQλV λf.V (λe.[f (e)∧R(λz.∀y[y ∈ z → Q(λx.sub(e0, e)∧instr(e0) = x))]]) If we combine this with the collective denotation of John and Mary, we will get this: (183) [[john and mary]] = λP.P ({j, m})

(184) λQλV λf.V (λe.[f (e) ∧ ∀y[y ∈ {j, m} → Q(λx.sub(e0, e) ∧ instr(e0) = x)]]) Problems

The participant-general role entries avoid the problem that the participant-specific roles have. Both agent-oriented and theme-oriented instrument roles have the same lexical entry. However, there is a more severe problem.

We need the het, ti denotation of the Range Phrase twice in the compositional process. First we need it in the denotation for the participant-general instrument role and then we need it again for the agent role itself. Usually, in compositional semantics this is not allowed.

We could solve this issue by assuming the Range Phrase is incorporated into the Role denotation through binding, instead of the usual functional application. The process of binding is discussed in the section 4.3.5.

Even if we assume the Range Phrase is bound, there is still the problem of Role entry redundancy. The current approach requires additional lexical entries for roles. The usual entries for roles, like (175) do not work. We therefore try to resolve an issue that is caused by binominal each by adding lexical entries for roles. This seems redundant and not the right way to go. In section 4.3.5 I will discuss another method that also requires binding the Range Phrase, but doesn’t need additional lexical entries for roles.

4.3.4 Implicit Collectivity

In another possible representation, only the distributive behaviour is mentioned. The collective behaviour is assumed to be implicit:

(185) ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ∀y[y ∈ {j, m} → ∃X ⊆ jacks.|X| = 2 ∧ ∃e0[sub(e0, e) ∧ instr(e0) = X ∧ ag(e0) = y]]]]

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Problems

The problem with (185) is that it is too weak. It would be true, for example, if John, Bill and Mary lifted a piano together and John and Mary used two jacks each and Bill did not. The sentence should be false in that case, intuitively. The other methods mentioned so far did not suffer from this issue as both the collective and distributive behaviour were mentioned in the logical form explicitly. It seems that we somehow need to encode both the collective and the distributive behaviour explicitly in the logical form.

Another issue with the methods provided so far is the fact that they only work for collectivity-retaining sentences. We would like to have a treatment of each that is able to handle fully distributive and collectvity retaining sentences with only a single entry.

4.3.5 Individual specific subevents

In this section, I will provide such a treatment. This approach makes use of event summation: sums of events are also events. This allows us to create similar logical forms for both types of sentences.

(186) John and Mary watched two movies each

(187) John and Mary lifted a piano with two jacks each

(188) ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃er[sub(er, e) ∧ instr(er) = X]]]]]

(189) ∃e[watch(e) ∧ ag(e) = {j, m} ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e_r[sub(er, e) ∧ th(er) = X]]]]

Every sentence with binominal each is concerned with two kinds of events. There is one main event ( e in the entry) that captures everything that is not within scope of binominal each. Then, for every individual in the Range, there is another event, which is a subevent (er

in the formula) of the main event e. This means that for all arguments and modifiers in the sentence, it has to either modify the main event e or the subevent e0. In other words, it has to be either inside, or outside the scope of binominal each.

With binominal each, the Dist Phrase is the only phrase that is interpreted inside the scope of binominal each. The nature of the subevent er captures the fact that the subevents

are distributed across each individual in the Range Phrase. The presence of other phrases determines whether a sentence is fully distrubutive or collectivity-retaining:

Definition 3. Sentences involving binominal each in which all arguments/modifiers apply to the subevents are called ”fully distributive”

Definition 4. Sentences involving binominal each in which at least one argument/modifier applies to the main event are called ”collectivity-retaining”

The logical forms look a bit different than the ones we saw so far. In the first two approached, the range phrase was incorporated in both the main event and the subevents. In the third approach, we modeled the range phrase only in the subevent part of the logical form and assumed that the collective behaviour in the main event part is implicit. Now that we found out that this does not work, we came up with these logical forms in which the Range Phrase is only mentioned in the main event. The role of the range in the subevents is still needed though. This is done by the subevent er. Where e is the main event, there is a subevent er for each

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4.4 Implementation of individual specific subevents

4.4.1 Binding the Range Phrase

To avoid using the Range Phrase in the compositional process twice, I assume that the Range Phrase is incorporated in the entry for each by binding instead of functional application. This move has two other advantages. First of all, the lexical entry is more simple, as we no longer need lambdas for the Range Phrase and the role of the Range Phrase. Secondly, the compositional process can now be aligned with the syntactic tree, we do not need to use different trees for the syntax of the sentence and the compositional process.

Burzio (Burzio, 1986) provided some evidence indeed that binominal each indeed binds it antecedent. With binominal each, the antecedent is the Range Phrase. The argument that Burzio provides for his claim is that the italian version of binominal each ciascuno requires gender agreement with its antecedent. ciascuno is used when the Range Phrase refers to masculine entities, ciascuna is used when feminine entities are referred to.

4.4.2 Lexical entry for each

We need a new entry for each: (190). The Range Phrase R is incorporated through binding. We no longer need the lambdas to incorporate the Range Phrase. We have only one lambda left, for the Dist Phrase.

(190) [[eachR]] = λDistλV λf.V (λe[f (e) ∧ R(λz.∀r[r ∈ z → Dist(λf0∃er[issub(e, z, r, er) ∧

f0(er)])])(closure)])

each combines with a DP of type hvtt, vtti and returns a phrase that is also of type hvtt, vtti. After combining with the Dist Phrase, the result can combine first with the relation (”watch” or ”lift a piano”) and then with the Range Phrase (”John and Mary”), exactly like the syntactic tree dictates.

The function issub is defined as follows:

(191) [[issub]] = λeλzλrλer.r ∈ z ∧ ∀θ[θ(e) = z → θ(er) = r]

This definition splits up an event with a plural role into multiple subevents, where each of these subevents has its slot for that role filled with one of the individuals. So in the case of (186), there is one watching event that is done by John and Mary, which is the agent. Then, there is an individual specific subevent for each individual in the plural role: one subevent for John and one subevent for Mary. Each of these subevents has an agent (the respective individual) and a theme (two movies).

With collectivity retaining sentences, things are similar. The difference is that there is additional roles modifying the main event. See definition 3 and 4. In the case of sentence (187), there is one lifting event that involves a plural agent role: John and Mary. Additional, there is a theme: a piano. This theme role modifies the main event, because the piano is lifted by both of them together. Then there are subevents for each of the individuals in the Range: one for John and for Mary. Each of these subevents is modified by an instrument phrase: with two jacks.

4.4.3 Derivation: fully distributive sentence

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S (202) VP (200) VP (198) Dist Phrase (196) [[each]] (193) DP [[th]] [[two movies]] [[watched]] Rel DP Range [[ag]] [[john and mary]] [[closure]]

Figure 4.5: Syntactic tree for fully distributive sentence with binominal each using individual specific subevents (143): John and Mary watched two movies each.

(193) [[each]] = λDistλV λf.V (λe[f (e) ∧ ∀r[r ∈ {j, m} → Dist(λf0∃e_r[issub(e, {j, m} , r, er) ∧

f0(er)])(closure)]])

Then, we incorporate the Dist Phrase in the meaning. As this is the only operator that applies at the subevent level, we can also fill in the closure operator.

(194) [[two movies each]] = λDistλV λf.V (λe[f (e)∧∀r[r ∈ {j, m} → Dist(λf0∃er[issub(e, {j, m} , r, er)∧

f0(er)])(λe.true)]])(λV λf ∃X[X ⊆ movies ∧ |X| = 2 ∧ V (λe.[f (e) ∧ th(e) = X]]))

(195) [[two movies each]] = λV λf.V (λe[f (e)∧∀r[r ∈ {j, m} → λV λf ∃X[X ⊆ movies∧|X| = 2 ∧ V (λe.[f (e) ∧ th(e) = X])](λf0∃e_r[issub(e, {j, m} , r, er) ∧ f0(er)])(λe.true)]])

(196) [[two movies each]] = λV λf.V (λe[f (e) ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ th(er) = X]]])

We now have a regular DP of type hvtt, vtti and we can apply it to watch:

(197) [[watch two movies each]] = λV λf.V (λe[f (e)∧∀r[r ∈ {j, m} → ∃X[X ⊆ movies∧|X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ th(er) = X]]])(λf.∃e[watch(e) ∧ f (e)])

(198) [[watch two movies each]] = λf.∃e[f (e) ∧ watch(e) ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃e_r[issub(e, {j, m} , r, er) ∧ th(er) = X]]]]

Then, we apply the Range Phrase.

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S (213) VP (211) VP (209) Dist Phrase (207) [[each]] (204) DP [[instr]] [[with two jacks]]

Rel (208) DP [[th]] [[a piano]] [[lif t]] DP Range [[ag]] [[john and mary]] [[closure]]

Figure 4.6: Syntactic tree for collectivity retaining sentence with binominal each using individ-ual specific subevents (167): John and Mary lifted a piano with two jacks each.

(200) λf.∃e[watch(e) ∧ f (e) ∧ ag(e) = {j, m} ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ movie ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ th(er) = X]]]]

We then apply the closure operator.

(201) λf.∃e[watch(e) ∧ f (e) ∧ ag(e) = {j, m} ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ th(er) = X]]]](λe.true)

(202) ∃e[watch(e) ∧ ag(e) = {j, m} ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ movies ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ th(er) = X]]]]

4.4.4 Derivation: collectivity-retaining sentence

Also for collectivity-retaining sentences, we will first bind the Range Phrase:

(204) [[each]] = λDistλV λf.V (λe[f (e) ∧ ∀r[r ∈ {j, m} → Dist(λf0∃e_r[issub(e, {j, m} , r, er) ∧

f0(er)])](closure)])

Then, we incorporate the Dist Phrase in the meaning. As this is the only operator that applies at the subevent level, we can also fill in the closure operator.

(205) [[with two jacks each]] = λDistλV λf.V (λe[f (e)∧∀r[r ∈ {j, m} → Dist(λf0∃er[issub(e, {j, m} , r, er)∧

f0(er)])](λe.true)])(λV λf ∃X[X ⊆ jacks ∧ |X| = 2 ∧ V (λe.[f (e) ∧ instr(e) = X]]))

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(207) [[with two jacks each]] = λV λf.V (λe[f (e) ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ instr(er) = X]]])

We now have a regular DP of type hvtt, vtti and we can apply it to lif t a piano:

(208) [[lif t a piano with two jacks each]] = λV λf.V (λe[f (e) ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks∧|X| = 2∧∃er[issub(e, {j, m} , r, er)∧instr(er) = X]]])(λf.∃p[piano(p)∧∃e[lif t(e)∧

f (e) ∧ th(e) = p]])

(209) [[lif t a piano with two jacks each]] = λf ∃p[piano(p) ∧ ∃e[lif t(e) ∧ f (e) ∧ th(e) = p ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃e_r[issub(e, {j, m} , r, er) ∧ instr(er) = X]]]]]

Then, we apply the Range Phrase

(210) λV λf.V (λe[f (e) ∧ ag(e) = {j, m}])(λf ∃p[piano(p) ∧ ∃e[lif t(e) ∧ f (e) ∧ th(e) = p ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ instr(er) = X]]]]])

(211) λf ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ f (e) ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ instr(er) = X]]]]]

In the end, we apply the closure operator:

(212) λf ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ f (e) ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ instr(er) = X]]]]](λe.true)

(213) ∃p[piano(p) ∧ ∃e[lif t(e) ∧ th(e) = p ∧ ag(e) = {j, m} ∧ ∀r[r ∈ {j, m} → ∃X[X ⊆ jacks ∧ |X| = 2 ∧ ∃er[issub(e, {j, m} , r, er) ∧ instr(er) = X]]]]]

4.5 Noun modifiers

We now have a solid theory of binominal each when it occurs in an argument or modifier of the verb. This theory covers cases of transitives, ditransitives and VP modifying prepositional phrases. There is, however, one class that is not covered yet. Prepositional phrases that modify nouns can also occur with binominal each:

(214) John wears two gloves with four fingers each

Our current theory does not work because it only covers sentences in which the Dist and Range Phrase are related through an event variable. In noun modifying prepositional phrases, events are not available. Another type of relation is at play here. In (214) this relation is denoted by the preposition with.

It is not hard to extend our theory to account for noun modification as well. The lexical entry that we use for each can be simplified a lot, as all the complexity that is introduced by the machinery for handling events and thematic roles can be removed. In the end, we only want to relate two noun phrases (Dist and Range) to each other through a relation. This will be our lexical entry:

(215) [[each]] = λRelλDistλRange.Rel(Dist)(DistShif t(Range))

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4.5.1 Derivation

I will now show the derivation for an example noun phrase: [[two gloves with f our f ingers each]]. The other entries are the familiar ones:

(217) [[f our f ingers]] = λP.∃X[X ⊆ f ingers ∧ |X| = 4 ∧ P (X)] (218) [[two gloves]] = λP.∃Z[Z ⊆ gloves ∧ |Z| = 2 ∧ P (Z)]

First, we plug in the relation and the Dist Phrase, and apply the relation to the Dist Phrase. (219) [[with f our f ingers each]] = λRange[λQλRλP.R(λx.P (x)∧Q(λy.have(x, y)))](λP.∃X[X ⊆

f ingers ∧ |X| = 4 ∧ P (X)])(DistShif t(Range))

(220) [[with f our f ingers each]] = λRangeλRλP.R(λx.P (x) ∧ ∃X[X ⊆ f ingers ∧ |X| = 4 ∧ have(x, X)])(DistShif t(Range))

We apply the DistShift operator to the Range Phrase:

(221) [[DistShif t(two gloves)]] = λP.∃Z[Z ⊆ gloves ∧ |Z| = 2 ∧ ∀y[y ∈ Z → P (y)]] We then plug this in as the Range, and apply the Relation to it.

(222) [[two gloves with f our f ingers each]] = λRangeλRλP.R(λx.P (x) ∧ ∃X[X ⊆ f ingers ∧ |X| = 4 ∧ have(x, X)])(λP.∃Z[Z ⊆ gloves ∧ |Z| = 2 ∧ ∀y[y ∈ Z → P (y)]])

(223) [[two gloves with f our f ingers each]] = λP ∃Z[Z ⊆ gloves ∧ |Z| = 2 ∧ ∀y[y ∈ Z → P (y) ∧ ∃X[X ⊆ f ingers ∧ |X| = 4 ∧ have(y, X)]]]

4.5.2 Problems

There is a problem with logical forms like (223), though. The P (y) is in scope of the universal quantifier that distributes the Range Phrase. This means that the resulting Noun Phrase can only be interpreted in a distributive way. It cannot have the collective interpretation. This is of course not surprising, as the DistShift operator applied to the Range Phrase in the compositional process, allowing only the distributive reading.

We will run into problems with sentences like (224) and (225). (224) [Two groups of three people each] met each other

(225) [Two groups of three people each] separated

In these sentences, the Range Phrase two groups is interpreted distributively with respect to binominal each in the noun modifier, but collectively with respect to the verb. The current framework cannot account for these sentences, because the QES framework resolves distributiv-ity on the Noun Phrase, using the DistShift-operator. So Noun Phrases in the QES framework are either distributive or collective. Noun Phrases which are both are not possible in the cur-rent framework. This means that sentences like (226), which display a mixture of collective and distributive predicates, all applied to the same plural subject, cannot be accounted for:

(226) The students woke up, gathered, smiled, carried a piano upstairs and then separated. In sentence (224) and (225), the meaning of the Noun Phrase will be this:

Binominal each and collectivity by Mets Visser