University of Szeged (SZTE)

Institute of English & American Studies

Simkó Katalin Ilona

Collective - Distributive Ambiguity and Inquisitive Semantics

szakdolgozat / BA Thesis

Témavezető / Supervisor

Szécsényi Tibor

Szeged, 2013

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Abstract

Ambiguity is a natural part of language which appears in most human languages.

It occurs when a single word or phrase has more than one possible interpretation.

Its three main types are lexical, structural and semantic ambiguity depending on

the level at which the ambiguity is present. Ambiguous sentences cause

complications in semantic analysis, because of these alternative interpretations

that may influence the whole analysis differently. This paper deals with a type of

semantic ambiguity where the sentence may be interpreted for a group as a unit or

as its separate members, collective - distributive ambiguity. Multiple methods for

the representation of sentences with collective - distributive ambiguity are

discussed together with their positive features and disadvantages in a semantic

analysis. Inquisitive semantics is a relatively recent framework based on formal

logic and discourse representation theory. This framework is proposed for the

representation of these ambiguous sentences, as it combines the advantages of

previous models and also allows for different possibilities within one

representation. This feature is instrumental in this representation of collective -

distributive ambiguity and makes this framework able to adequately model it for

semantic analysis. This paper gives a summary of the main types of ambiguity and

their cause, reviews some earlier models for collective - distributive ambiguity,

introduces the framework of inquisitive semantics and presents a possibility for

the representation of collective - distributive ambiguity in this model.

Absztrakt

A többértelműségek természetes részei a nyelvnek, amik a legtöbb emberi

nyelvben előfordulnak. Akkor beszélhetünk többértelműségről, ha egyetlen

szóhoz vagy mondathoz több értelmezési lehetőség is tartozik. A

többértelműségek három fő típusa a lexikai, a strukturális és a szemantikai

többértelműség, attól függően, hogy a többértelműség ezek közül melyik szinten

jelenik meg. A többértelmű kifejezések komplikálják a szemantikai elemzést, a

több alternatív értelmezési lehetőség miatt, amelyek más-más hatással lehetnek a

teljes elemzésre. Ez a dolgozat főként a szemantikai többértelműségek egyik

fajtájával foglalkozik, amikor a mondat a benne szereplő individuumokra

egyenként vagy együtt, csoportként is érthető, ezek a kollektív - disztributív

többértelműségek. Több különböző reprezentációs lehetőség is bemutatásra kerül

a pozitívumaikkal és a szemantikai elemzés szempontjából hátrányos

tulajdonságaikkal együtt. Az inkvizitív szemantika egy viszonylag új rendszer,

ami főleg a formális logikára és a diskurzus reprezentáció elméletre alapszik. Ez a

rendszer alkalmas lehet ezen többértelmű mondatok reprezentációjára, mivel

kombinálja az előzményei előnyös tulajdonságait, valamint lehetővé teszi több

lehetőség egy reprezentáción belüli ábrázolását. Ez fontos a kollektív - disztributív

többértelműségek szemantikai elemzéshez szükséges modellezéséhez. Ez a

dolgozat összefoglalót ad a fő többértelműség típusokról és azok okairól, áttekint

néhány korábbi modellt a kollektív - disztributív többértelműségek kezelésére,

bevezeti az inkvizitív szemantika alapjait és egy lehetséges reprezentációt ad a

kollektív - disztributív többértelműségek elemzésére ebben a modellben.

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Contents

1 Introduction ................................................................................................. 1

2 Ambiguity .................................................................................................... 3

2.1 Lexical ambiguity .............................................................................. 3

2.2 Structural ambiguity .......................................................................... 4

2.3 Semantic ambiguity ........................................................................... 5

3 Collective - distributive ambiguity.............................................................. 7

3.1 Types of predicates ........................................................................... 7

3.2 Collective - distributive predicates ................................................... 9

3.3 Collective - distributive predicates in classical logic ...................... 11

4 Inquisitive Semantics ................................................................................ 14

4.1 Classical logic ................................................................................. 14

4.2 Dynamic logic ................................................................................. 16

4.3 Inquisitiveness ................................................................................. 17

4.4 Formal definition ............................................................................. 19

5 Collective - distributive ambiguity in inquisitive semantics..................... 22

5.1 Analysis ........................................................................................... 24

6 Conclusion ................................................................................................. 28

Bibliography .............................................................................................. 29

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1 Introduction

Semantics is the field of linguistics attempting to study and capture the meaning behind the

language humans use. This task can be accomplished in various ways. One of the main trends

of semantics is to formally represent meaning. This is usually done with the use of a pre-

established logical language. As logical languages are unambiguous and have clear-cut rules

and categories they can help in describing the meaning of natural language in a clear way as

well.

In the last few decades, a large number of attempts were made to create frameworks of

formal semantics that can adequately describe the meaning in human languages. This trend

only became stronger with computers becoming more and more widespread. People attempted

to formally model human language to such a degree that even computers could analyse and

produce it.

Understanding and analysing natural languages is not an easy task. Even everyday

communication is full of misunderstandings, which means that linguistic analysis can be no

less difficult. A huge number of factors contribute to this, one of which is ambiguity.

Ambiguity is the phenomenon in language, when more than one meaning is associated

with a single sentence. In everyday communication, a lot of times we do not even perceive

these. In other cases it can cause serious misunderstandings, if the speaker intends one

possible meaning of the sentence, but the listener interprets it as something different. Again,

the participants may not notice the ambiguity of the sentence, unless the meaning of the

sentence is explicitly questioned.

Ambiguity may be the result of a number of factors, for example a certain word of the

sentence could be ambiguous, causing the whole unit to have more than one meaning or the

connections between the elements of a sentence may not be clear cut. The first is a type of

lexical ambiguity, while the second is a structural one, taking place at the level of syntax.

In this paper, I mostly discuss a different type of ambiguity, which belongs to semantic

ambiguities. The collective - distributive ambiguity occurs, when the predicate of the sentence

can refer to one of its plural arguments together as a unit or separately as individuals. For

example, the sentence The girls have three toys. may be interpreted so that the girls have a

total of three toys or each girl having three toys.

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Semantics tries to interpret and give representations for the sentences of natural

languages. This task is made even more difficult by such attributes of language. In the case of

sentences with collective - distributive ambiguity both the semantic interpretation and

representation require a lot of attention.

This paper will mostly focus on presenting possibilities for modelling collective -

distributive ambiguity. Finally, I propose the framework of inquisitive semantics to manage

the different readings of these ambiguous statements separately, but within one representation.

In Section 2, linguistic ambiguity is discussed together with its types and the problems

these ambiguous sentences may cause in a semantic analysis. Then, possible approaches to

collective - distributive ambiguity are presented in Section 3, reviewing its causes and a way

to describe this phenomenon in classical logic. Section 4 introduces the model of inquisitive

semantics, its origins of classical and dynamic logic, its innovations and possible uses. In

Section 5, I propose a way to model collective - distributive ambiguity in the framework of

inquisitive semantics utilizing the possibilities and propositions that are defined by the

framework. Finally, Section 6 provides a conclusion to the paper.

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2 Ambiguity

One of the many difficulties in the perception of human language lies in the ambiguous way

we use it. This can have an effect on the everyday conversations between people and also

impedes precise semantic modelling of human language. Ambiguous statements may lead to

misunderstandings and managing multiple meanings requires a very sophisticated semantic

framework.

“Ambiguity or ambiguous are the terms used to describe a word, phrase, or sentence

with multiple meanings” (Fromkin, Rodman, Hyams 2003:574). These elements can be

interpreted in more than one way, usually due to lexical or structural properties. There are at

least two alternative logical formulas that describe the same ambiguous sentence,

complicating their analysis. I analysed examples from main types of ambiguity and provided

syntactic trees and logical formulas to show their multiple interpretations.

Ambiguity can occur in a number of different ways. Some of the most frequent are the

lexical, structural and semantic ambiguities, which are discussed below. In this paper, I

mostly deal with a certain type of semantic ambiguity within a number of different semantic

frameworks.

2.1 Lexical ambiguity

(1) Time flies like an arrow.

In case of lexical ambiguity a word within the sentence has multiple meanings, like in the

well-known sentence in (1). To simplify this problem, I will only deal with two of the

possible interpretations of this sentence, a. ‘Time is moving very fast, like an arrow.’ and b.

‘A special type of flies, ‘time flies’ appreciate an arrow.’ The word time is ambiguous

between a noun (a.) and an adjective (b.); flies can be a third person singular verb (a.) or a

plural noun (b.) and like is a preposition in a. and a verb in b.

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2.2 Structural ambiguity

Another type of ambiguity is structural ambiguity. In these sentences we can find the reason

for ambiguity in syntax. Certain elements may connect to others in various alternative ways,

causing the whole sentence to have multiple interpretations. The sentence in (2) is ambiguous,

because the prepositional phrase with the book in (2a) refers to the action of hitting (‘It was

with the book that Sara hit the boy.’) while in (2b) it refers to the boy (‘It was the boy with the

book that Sara hit.’). The prepositional phrase could be the modifier of the verb as well as the

object of the sentence.

(2) Sara hit the boy with the book.

a. hit (Sara; the boy; with the book) b. hit (Sara; the boy with the book)

In (3), frightening may be interpreted as an adjective modifying people, (3a), or as the

main verb, (3b). These differences are also apparent if we look at the alternative syntactic

trees illustrating the possible interpretations.

(3) They are frightening people.

a. frightening (they; people) b. frightening people (they)

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2.3 Semantic ambiguity

Ambiguity can also be semantic. The sentence in (4) has two alternative interpretations, it is

semantically ambiguous. In (4a), all boys is interpreted as having a smaller scope than a girl;

then the sentence means, that there is one individual girl in the world that all the boys like. On

the other hand, in (4b) all boys is interpreted as having a bigger scope than a girl, then the

sentence has its conventional meaning where the liked girl may even be a different one for

every boy. This kind of ambiguity does not cause differences in the syntactic trees of the

alternative interpretations, but it can be seen in the differences of the corresponding logical

representations.

(4) All boys like a girl.

Another semantic ambiguity type is the collective - distributive ambiguity. In these

cases, a plural argument of the predicate can be read as distributive (understood separately for

each individual) or as collective (together, as a group). The sentence in (5) is ambiguous

between the collective and the distributive readings. While the collective reading suggests that

the boys carried a piano together, the distributive interpretation would mean that every boy

carried a separate piano. This kind of ambiguity is not representable at the level of syntax, and

formally modelling it in semantics is also difficult.

(5) The boys carry a piano upstairs.

Ambiguity is not only problematic in English. Lexical, structural and semantic

ambiguities are present in a number of languages. While the occurrence of certain types of

ambiguous statements is dependent on the specific language, ambiguity itself is universal. All

languages have words with multiple meanings or similar sounding ones that may cause

ambiguity in oral communication. Structurally and semantically ambiguous sentences also

a. ∃y∀x[(boy(x)˄girl(y)) → like(x;y)]

‘There is one girl that every single boy likes.’

b. ∀x∃y[(boy(x)˄girl(y)) → like(x;y)]

‘All boys have a girl that they like.’

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emerge in most languages. These create obstacles in the semantic analysis of all these

languages.

These few examples are enough to show that there are various types of ambiguity, not

all of which can be represented by traditionally linguistic methods, such as syntactic trees.

The logical formulas can represent these alternative interpretations in a sentence, by

‘translating’ the meaning of the sentence with formal precision.

The main semantic problem in dealing with ambiguous structures is that every time an

ambiguous element comes up, all the possible interpretations have to be analysed and the

analysis has to continue on two or more threads from then on. In case the sentence has more

than two possible interpretations, we have to deal with even more lines of analysis. These

multiple threads get divided even further every time another ambiguous element comes up.

This complicates and prolongs the analysis exponentially. As our language use is full of these

ambiguities, these problematic sentences will keep coming up throughout our analysis,

making it less and less manageable. All of these threads of analysis have to be dealt with,

unless one of the interpretations gets contradicted later on, in which case that one thread and

the branches originating from it can be omitted.

A number of approaches were taken to give an adequate semantic analysis for

ambiguous statements; for example (Reyle, 1993) analysed ambiguous sentences using

Discourse Representation Theory and underspecification. He deals with the ambiguity by not

making the interpretation of either reading explicit in his model while the sentence is

ambiguous. He specifies it later, once it is disambiguated by the context.

Another commonly used method of representing ambiguous meaning is by adding the

alternative interpretations together using disjunctions. This has the advantage of precisely

including all possible meanings in the analysis, but there is also a big disadvantage. A lot of

times one of the alternatives for interpretation is included in another. For example, in the case

of I wore my red socks and boots. the possibilities of the first interpretation, where both the

socks and the boots are red are already part of the second, where only the socks are

necessarily red. This means that by only using disjunctions, we lose one of the readings of

these ambiguous sentences in the semantic representation.

The next section of this paper deals with collective - distributive ambiguities in more

detail, taking a look at alternative ways of representing it in semantics and classical logic.

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3 Collective - distributive ambiguity

The collective - distributive ambiguity is a type of semantic ambiguity where a plural

argument of a predicate can be understood as separate individuals distinctly (distributive

interpretation) or as these individuals together (collective interpretation). This type of

ambiguity arises with a variety of predicates and different types of arguments.

3.1 Types of predicates

In 1984, Godehard Link published a paper on the collective and distributive interpretations of

nouns and developed a concept of groups. As a response, Fred Landman (1989) develops an

approach to these plural noun phrases that can represent this collective - distributive problem

as well and furthers the notion of groups. “The distinction between the distributive and the

collective reading of a sentence […] is mainly a lexical property of the verb involved.”

(Landman, 1989:559) Predicates are categorized as lexically singular, lexically plural or

mixed.

a. Lexically singular predicates can only denote sets of singular individuals, they are

inherently distributive. For example the verb see is lexically singular; it can only be

interpreted separately for each individual in the group. In The boys saw the car. each boy saw

the car independent of the others. In case of distributive predicates, if the predicate is true for

the sum of the things in its argument, then it is also true for them individually.

(6) Chris and David are actors. ⇒ Chris is an actor. and David is an actor.

actor(c+d) ⇒ actor(c) ˄ actor(d)

b. Lexically plural predicates take sets of plural individuals, they are inherently

collective. The verb meet can only be understood collectively. The predicate cannot be true

for the individuals separately, only in a group reading.

(7) Rick and Shane met. ⇏ Rick met. and Shane met.

met(r+s) ⇏ met(r) ˄ met(s)

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c. The third type of predicates, mixed predicates take sets of singular or plural

individuals, therefore they can be understood collectively or distributively, which often makes

them ambiguous when used with plural noun phrases. For example, the sentence The boys

carry a piano upstairs. is ambiguous between its collective and distributive readings. If

understood collectively, the boys carry a piano together; while, if understood distributively

the boys each carry a piano separately.

The following is true in case of the distributive reading, but false if interpreted

collectively:

(8) Lewis and Simon carried a piano upstairs. ⇒ Lewis carried a piano upstairs. ˄ Simon

carries a piano upstairs.

carried a piano upstairs(l+s) ⇒ carried a piano upstairs(l) ˄ carried a piano

upstairs(s)

Certain adverbials and determiners can trigger distributive or collective readings of a

previously ambiguous sentence. The boys carried a piano upstairs together. can only be

interpreted collectively, while The boys each carried a piano upstairs. can only be

distributive.

Landman’s proposition to solve this problem is by introducing groups, which are

collections of individuals that “do not in general inherit the properties that their individual

members have” (Landman, 1989:572). Landman suggests that for example committees are

distinct from the sums of their singular members, which makes the following conclusion seem

awkward:

(9) The committee consists of Paul and Chris. ˄ Paul and Chris are judges. ⇒ The committee

are judges.

Landman extends this group understanding to plurals as well, distinguishing them from

sums which actually refer to the individuals within. In his framework, “distributive readings

refer to sums, while collective readings do not refer to sums but to groups” (Landman,

1989:593).

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In summary, Landman’s classification of predicates is not in terms of distributivity or

collectivity but in terms of what kind of entities a predicate can take. Some verbs can only

take individuals, others only take groups, and the rest may take groups or individuals. This is

where ambiguity arises.

Landman also argues for multiple levels of groups instead of just one as Link suggests.

As conjunction is recursive, not only sums can be conjoined to form groups, but groups too.

Then these groups of groups can form further groups and so on. He takes his example from

the writings of Karl Marx (Landman, 1989:579), the Farmers and the City-proletarians being

a type-1 group that form the Working Classes, a type-2 group. The State is a made up of this

group of groups, the Working Classes and the Exploiting Classes, making it a type-3 group.

Marx then introduces the States of Central Europe, made up of these type-3 groups, therefore

it must be type-4, a group of groups of groups of groups. This process iterates, but it is highly

dependent on context; in another context, the state could simply refer to a group of

individuals, making it a type-1 group. This is the problem of granularity. A possible solution

to this problem is proposed by Carl Pollard by using hyperintensions to achieve finer

granuality (Pollard, 2007).

3.2 Collective - distributive predicates

Collective - distributive ambiguity may occur in a number of different ways. In most cases the

predicate has to be Landman’s mixed type predicate and the argument a plural noun phrase

for this ambiguity to arise, but it is not specified, which argument of the predicate has to have

these attributes and different kinds of noun phrases can be in these positions. Sometimes even

collective predicates can gain a distributive reading due to higher level groups as their

argument.

a. Perhaps the most common case when sentences that are ambiguous between collective

and distributive readings happen, is when a mixed type predicate takes a plural noun with a

definite determiner as its subject.

(10) The boys carry a piano upstairs.

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b. The subject of a mixed type predicate can also be a plural, quantified noun. The

sentence in (11) is ambiguous. In the collective reading, the three boys together bought four

chairs in total, while in the distributive reading the three boys bought four chairs each, making

the total number of chairs bought 12.

(11) Three boys bought four chairs.

c. Coordinated names in the subject position can also cause this type of ambiguity. (12)

has two different interpretations. It can mean that Joe and Zoey are married to each other in a

collective reading or they can be married to different people in a distributive reading.

(12) Joe and Zoey are married.

d. The plural noun is not always the subject of the sentence; it can be a different

argument of the predicate as well. In (13), the collective - distributive ambiguity cannot be

caused by the subject. In its collective interpretation two apples were bought in total while

distributively each kid received two apples.

(13) I bought two apples for my two kids.

e. Coordinated, type-2 groups can cause further ambiguities, even if the predicate is not

mixed type. In case of (14), the fact that the collective group (the boys and the girls) is made

up of groups itself means that the sentence can be interpreted distributively (‘The boys met to

discuss the problem.’ and ‘The girls met to discuss the problem.’) as well as collectively (‘The

boys met with the girls to discuss the problem.’) as the type of the predicate would suggest.

(14) The boys and the girls met to discuss the problem.

The sentence in (15) has three alternative interpretations. Probably the most intuitive,

distributive interpretation, is that ‘the members of the Jones family hate the members of the

Smith family and the other way around’. The second, also distributive interpretation is, that

‘the members of the Jones family hate each other’ and ‘the members of the Smith family hate

each other’ as well. The collective reading of this sentence is that the ‘Joneses and the Smiths

all hate everyone within both families’. The two different distributive meanings are caused by

the properties of ‘each other’ and the coordination of groups.

(15) The Joneses and the Smiths hate each other.

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3.3 Collective - distributive predicates in classical logic

Being a type of semantic ambiguity, the collective - distributive ambiguity is not present in

syntactic representations, and it is also hard to formally model in semantics, partially due to

the problem of granuality. This means that in most frameworks “not enough meaning

distinctions are available” (Pollard, 2007:1). Also, as with other types of ambiguity, the

alternative readings are hard to represent adequately, without overcomplicating the model

while also keeping the possible understandings of the sentence in our analysis.

Based on Márta Maleczki’s modelling of Hungarian noun phrases (Maleczki, 1995), the

following is a possibility for the interpretation and representation of English in classical logic.

In this view, the individuals are placed in a structured lattice, each representing an individual

or a group of individuals. Different types of noun phrases refer to different points in this

lattice. In this model, we imagine a set of possible worlds with two individuals, Anna and

Becky. The representations show their relation to the predicate ‘sang a song’.

(16) One girl sang a song. (exactly one)

The top item in this, ({a,b}), represents the two individuals singing together as a group;

({a}) represents only Anna singing a song; similarly, ({b}) represent only Becky singing a

song; while ∅ represents a situation, where neither Anna nor Becky sang a song. The sentence

in (16) is interpreted as true in the cases marked in this illustration; either if Anna sang a song

or if Becky sang a song, as it gives no information about either of them, just the number of

individuals singing.

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(17) Anna sang a song.

The sentence in (17) is interpreted as true either if she sang a song alone or if she sang a

song together with Becky. The sentence does not give any information about Becky, nor does

it say, whether Anna was singing individually or as part of a group. In both cases the

sentence, Anna sang a song. is true.

(18) Only Anna sang a song.

There is only one possibility that makes this sentence true. Anna has to be the one and

only individual in the world who sang a song, no one else can be singing, so the sentence

gains a distributive interpretation.

(19) A girl sang a song.

The only case that makes the sentence in (19) false is if no one sang a song. It makes no

restriction to either one of the individuals and even if they sang together, the sentence is true.

The indefinite article does not designate any of the individuals in particular; neither does it

restrict the number of individuals. It only excludes the option where no one sings.

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(20) Anna did not sing a song.

The sentence in (20) provides no information related to Becky, so the sentence is true in

case Becky alone sang a song and also if no one sang a song. The only restriction is that Anna

cannot be singing.

(21) The girls sang a song.

a. sang a song ({a}) ˄ sang a song ({b}) b. sang a song ({a,b})

The ambiguous nature of the sentence in (21) causes it to have two distinct

interpretations. In the distributive reading, Anna and Becky both sing a song each; this

reading is true only if both of the cases marked in (21a) are true at the same time. Interpreted

collectively (21b), Anna and Becky have to have sung together to make the sentence true.

Although the combination of the representations in (21a) and (21b) are similar to the

one in (19), they are not the same. As the black outline around {a} and {b} represents in

(21a), the distributive reading of this sentence is only true if both Anna and Becky sing. While

the sentence in (19) is true if only Anna sings and Becky does not, and also the other way

around, (21) is not true in these cases. In (21), {a} and {b} are not options on their, only

combined.

This structured lattice approach makes the representation of noun phrases more

transparent and makes use of the group understanding in the collective interpretation of

sentences while taking us one step closer to an actual transcription into the logical formulas.

Unfortunately, this framework is still not capable of managing the alternative interpretations

of the ambiguous sentence in a single representation.

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4 Inquisitive Semantics

Inquisitive semantics is a relatively new approach of dynamic semantics strongly based on

predicate and first-order logic systems. This new model of describing language in a formal

way was introduced in 2009, by Jeroen Groenendijk and Floris Roelofsen. This approach

allows for a better understanding of how we can process distinct possibilities within a

statement and the dynamic changes that occur in human conversations.

In the simplest understanding, the meaning of a sentence is described as its informative

content. In this traditional sense, the meaning of a sentence is known, if in every single

possible world we can decide whether the statement is true or false. The meaning of complex

statements can be determined taking into consideration the meaning of the simple statements

within them and their connections. This truth conditional view of meaning is the most popular

starting point for current semantic models.

4.1 Classical logic

This formal approach to semantics was proposed by Richard Montague, who in his paper

published in 1970 theorized, that English and all other natural languages can be treated as

formal languages. Two essential principles of this theory are the truth-conditional aspects of

meaning and the Principle of Compositionality: "The meaning of a whole is a function of the

meanings of its parts and their mode of syntactic combination." (Portner and Partee, 2002:1).

The meaning of a statement can be described by the truth values assigned to them in every

possible world which depend on the truth-conditions of the statement. The truth-conditions of

a statement express the state of affairs needed in a possible world to make the statement true.

Compound statements can be formed by using logical operators – like negation ( ),

conjunction (˄) or disjunction (˅) – to combine atomic statements, like It is sunny or It is

raining – denoted by p and q – or other compound statements. While conjunction and

disjunction combine two atomic statements to form a compound statement, negation needs

only a single argument.

Negation is understood as the equivalent of ‘no’ or ‘not’ in English, so intuitively,

negation turns the truth value of a statement to the opposite: if p is true, then p is false and if

p is false, then p is true.

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The logical equivalent of ‘and’ is conjunction, an operation with two arguments, usually

expressed with the operator ˄. A compound statement of conjunction with two atomic

statements as arguments – p˄q (It is sunny and it is raining) – can only be true if both

arguments – both p (it is sunny) and q (it is raining) – are true; in every other case, the

compound statement is false.

Disjunction is the equivalent of English ‘or’. Its usual notation is ˅, and it is used to

connect two statements, like conjunction. A compound statement of disjunction - p˅q (It is

sunny or it is raining) – is false only if both arguments are false – the possible world, where

the value assigned to both p and q is false (0) – and true in every other case.

(22)

a. ¬p b. p˄q c. p˅q

(22) uses Groenendijk and Roelofsen’s visualization of the truth values, indices and

states. The dark grey circles mark possible worlds, or indices; the numbers inside them

represent the truth values of the statements – p and q -, the first number corresponding to the

truth value of p in that world and the second to that of q. The lighter grey, highlighted areas

show in which of the possible worlds the given compound formula is true.

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4.2 Dynamic logic

While this simple model of classical logic describes the meaning of isolated sentences rather

successfully, it fails to capture the dynamic changes of meaning in human communication.

Dynamic semantic models overcome this difficulty by focusing on the potential of the

sentence to change the common ground. According to this view, the conversational

participants’ common knowledge – common ground – is changed in some way by each

utterance. The effect of a sentence on the common ground is understood as its meaning, thus

making this model dynamic, as it changes throughout the conversation.

In their 1991 paper, Dynamic Predicate Logic, Jeroen Groenendijk and Martin Stokhof

suggest a framework that deals with meaning in discourse. Examining the effects of new

information on the common ground of the speech participants, they conclude that the meaning

of a statement can be determined by looking at the differences between the common ground

before and after the statement was uttered. DPL is a successful attempt to provide solutions to

problems dealt with in Discourse Representation Theory (Kamp, 1981), but do so in a more

conventional and compositional manner. Probably the best known of these problems is the

inadequacy of standard first-order predicate logic to describe the anaphoric relations of the

“donkey-sentences” properly, which is solved in Groenendijk and Stokhof’s work.

In this dynamic view, the meaning of a statement is its potential to change the common

ground. After the statement, the new common ground is the intersection of the previous

common ground and the proposition of the statement.

(23) a.

+ q →

b.

+ q →

17

In (23) the change caused by the statement q is shown in the representation of two

different, possible common grounds. It is clear in both cases that the new common ground is

made up of the intersection of the previous common ground and the proposition of q. In (23a),

the original common ground consists of three indices: both p and q are true, p is false and q is

true, and both p and q are false. The common ground is changed by the new statement, q, in a

way, that only two indices remain, the ones where q was originally true: both p and q are true

and p is false, q is true. The common ground in (23b) consists of two indices: p is false, q is

true and both p and q are false. The statement q changes this by only leaving one index: p is

false, q is true.

4.3 Inquisitiveness

Inquisitive semantics adds yet another layer to this model, by describing “the interactive use

of language in exchanging information” (Groenendijk and Roelofsen, 2009:2). According to

their theory, one participant cannot change the common ground alone; they can merely

suggest changes that the others react to. This way, propositions are the proposals of the

participants to change the common ground in one specific way or they may invite the others

to choose from different alternatives.

In inquisitive semantics, propositions are made up of one or more possibilities which are

sets of possible worlds – or, using inquisitive semantics’ terminology, sets of indices. These

possibilities are the different alternatives for changing the common ground. A proposition is

inquisitive, if it invites the other participants of the conversation to choose between different

alternatives, so the proposition consists of at least two possibilities. This way the meaning of a

sentence gets an interactive aspect, the common ground can only be changed through the

cooperation of the conversation participants.

Inquisitiveness elaborates this model by distinguishing the possibilities that can make

the statement true. For example in case of a simple disjunction, like p˅q, it does not treat all

three indices where it is true (0,1; 1,0 and 1,1) as one unit, but separates the two possibilities

that can make the statement true. One possibility includes all the indices where p is true and

the other includes all the indices where q is true.

18

(24) p˅q in inquisitive semantics

Statements such as It is sunny or it is raining. consist of two possibilities. There are two

distinct alternatives that make the statement true: one possibility includes all the indices,

where It is sunny is true (the value assigned to p is 1; indices 1,1 and 1,0) and the other

possibility includes all the indices where It is raining is true (the value assigned to q is 1;

indices 1,1 and 0,1). The two possibilities together include all the indices that traditionally

describe a disjunctive statement and exclude the 0,0 index as well.

This semantic model works by the same principles in more complicated models as well;

ones that involve more indices and more complicated statements. Apart from being capable of

representing statements of predicate logic, inquisitive semantics is also usable in first-order

logic (Ciardelli, 2010).

19

4.4 Formal definition

To formally link together statements with logical indices, states and possibilities, Groenendijk

and Roelofsen use support (⊨). Their formal definition is the following:

1. σ⊨p iff ∀v∈σ : v(p) = 1

2. σ⊨¬ ϕ iff ∀τ⊆σ : τ⊭ ϕ

3. σ⊨ϕ˅ψ iff σ⊨ϕ or σ⊨ψ

4. σ⊨ϕ˄ψ iff σ⊨ϕ and σ⊨ψ

5. σ⊨ϕ→ψ iff ∀τ⊆σ : if τ⊨ϕ then τ⊨ψ

Where v is an index (a possible world), “a binary valuation for the atomic sentences in the

language” (Groenendijk and Roelofsen, 2009:6); σ and τ are states, “non-empty sets of

indices”; p, ϕ and ψ are statements, propositions expressed by a sentence. A set of all indices

is denoted by ω and a set of all states by S.

Sentences are made up of one or more possibilities, all of which combined formulate the

proposition expressed by the sentence. Formally:

A possibility for φ is a maximal state supporting φ, that is, a state that

supports φ and is not properly included in any other state supporting φ.

The proposition expressed by φ, denoted ⌊φ⌋, is the set of possibilities for

φ. (Groenendijk and Roelofsen, 2009:7)

1. A state σ supports an atomic sentence iff every index in σ makes p true

(25)

The informational state shown in (25) supports p, because in all the indices it includes

(1,1 and 1,0) p is true. It does not support q, because in one of the indices included (1,0)

q is false.

20

2. A state σ supports a negation ¬ϕ iff no substate of σ supports ϕ

The informational state in (22a) supports ¬p, because in none of the included indices

does p=1.

3. A state σ supports a disjunction iff it supports at least one of the disjuncts

(26) a. b. c. d.

e. f.

The statement p˅q is supported by the informational states in (26a), (26b), (26c), (26d)

and (26e) as (26a) and (26b) support p and q respectively; (26c) supports both p and q.

The states in (26d) and (26e) are the maximal states supporting p and q. (26f) does not

support p˅q, as it contains an index, (0,0), where neither p, nor q is true, making p˅q

false as well.

21

4. A state σ supports a conjunction iff it supports both conjuncts

(27) a. b.

(27a) supports p˄q as all of its indices support both p and q. (27b) does not support

p˄q, as in index 0,1 p is not true and in index 1,0 q is not true.

5. A state σ supports an implication ϕ→ψ iff every subset of σ that supports ϕ also

supports ψ

(28) a. b.

The informational state in (28a) supports p→q, as it consists of one index, where both p

and q are true, which also makes the implication true in this world, therefore this state

supports the implication p→q. The state in (28b) does not support p→q, because in one

of the indices, (1,0), p is true while q is false. This makes the implication false in that

index and so the state does not support p→q.

22

5 Collective - distributive ambiguity in inquisitive semantics

As shown in the previous sections of this paper, the semantic analysis of ambiguous sentences

is problematic for number of reasons. First of all, as it was shown by Pollard (2007) a higher

order logic system is needed to give an adequately fine-grained representation for all possible

readings of the sentence. But even with these, a more fundamental problem arises, namely,

how to formally model a sentence that has more than one possible meaning. How to manage

alternative meanings that belong to a sentence in semantic analysis without making the system

overly complicated? Groenendijk and Roelofsen’s model of inquisitive semantics offers a

possible solution to this problem.

During semantic analysis of larger pieces of text (for example conversations), the

sentences making it up are analysed linearly. In a dynamic view of semantics, these new

sentences change the common ground of the speakers in some way; the meaning of a sentence

is equal to its common ground changing potential. Each statement adds some extra

information or removes certain things from the common ground, which in this way changes

all the way through the communication. This makes the analysis fairy linear, each new

sentence is analysed, it changes the common ground and then we move on to the next

sentence until we reach the end of our analysis.

This process becomes problematic when we encounter an ambiguous element; a

sentence that has multiple meanings, alternative ways of changing the common ground. If we

want to give a thorough semantic analysis, we have to take into consideration all the possible

readings, investigate how each would change the common ground and move on to the next

sentence keeping all the alternative possibilities for the common ground in mind. From this

point on, we have to analyse all sentences in light of all the possibilities for our ambiguous

sentence.

Coming across another ambiguous statement further complicates things, as we have to

analyse all its possible readings in all the possibilities for the common ground, which were

caused by the first ambiguous statement. The number of alternatives grows exponentially;

even if all the ambiguous elements have just two alternative readings, after two of these

sentences we will have four, after three sentences eight, and after five ambiguous sentences

we will have 32 different alternatives for the common ground.

23

This is a serious problem, as it severely prolongs the analysis and makes it substantially

more complicated. As ambiguous sentences are fairly common in communication – even if

we do not notice them a lot of times – this problem does require a solution in order to make

semantic analysis more economical.

One of the most basic properties of inquisitive semantics can be useful in representing

this problem of alternatives in ambiguity. In inquisitive semantics, statements are represented

not only by their informative content, but by layers of possibilities that represent the

alternatives that can make them true. For example, a simple disjunction like p˅q is true on the

one hand if p is true and alternatively if q is true. These are represented in this framework as

alternative possibilities.

(29)

These possibilities in inquisitive semantics make this framework fitting for the analysis

of ambiguous sentences. The alternative readings of the ambiguous sentence are separable in

this representation, but they are modelled within one unit of it. This way, the representation of

an ambiguous sentence is no more complex than the representation of a disjunction. It keeps

the analysis linear without “losing” the alternative readings.

The possibilities in inquisitive semantics may be good for the analysis of other types of

ambiguity as well. The aim of this paper is to use the framework for modelling the collective -

distributive ambiguity in English.

Sentences with collective - distributive ambiguity usually have two possible

interpretations. The collective reading means that the predicate is true for the argument as a

group, while in the case of the distributive reading, the predicate is true for the individuals in

the argument separately. These two readings describe two separate possibilities.

In the following section, the ambiguous sentence from Section 3.3 is analysed using

inquisitive semantics. This incorporates the results from the previous analysis using classical

logic and the structured lattice of individuals and adds an inquisitive layer to it.

24

5.1 Analysis

(30) The girls sang a song.

a. sang a song ({a}) ˄ sang a song ({b}) b. sang a song ({a,b})

As discussed before, the sentence in (30) has two readings, a distributive one, (30a), in which

Anna and Becky each sang a song on their own, and a collective one, (30b), which treats them

together as a group, this reading is true only if they sang a song together. Without any context,

this sentence is completely ambiguous between the two readings, we have no indication of

which interpretation is intended and in most cases only one or the other reading is actually

true.

The use of these structured lattices makes for a good semantic representation, but to

analyse these sentences in inquisitive semantics, this model needs to be converted into a set of

possible worlds. Staying with the two individuals and the predicate, this can be achieved by

establishing the three statements whose assigned truth-values influence the truth-value

assigned to the alternative interpretations of the sentence. These statements of first order logic

are converted into the atomic statements p, q and r to facilitate their representation in

inquisitive. This way it is sufficient to use predicate logic in inquisitive semantics, instead of

the more complex system of first order inquisitive semantics.

25

(31) a. Anna and Becky sang a song together.

sang a song ({a,b})=p

b. Anna sang a song alone.

sang a song ({a})=q

c. Becky sang a song alone.

sang a song ({b})=r

(31a) is closely related to the collective reading of the sentence, while the atomic

sentences in (31b) and (31c) help in determining the truth-values assigned to the distributive

reading in the possible indices. These three atomic statements in (31) are enough to establish a

set of indices in which the different readings of the sentence can be distinguished.

(32)

(32) illustrates the possible combinations of the truth-values of three atomic statements’

in Groenendijk and Roelofsen’s style. In all the indices, the first number represents the truth-

value of (31a), the collective reading of the sentence; the second is that of (31b), Anne

distributively and the third is the truth-value assigned to (31c), Becky singing distributively.

As previously, the circles represent possible worlds, indices. In these indices, the

statements with a 1 value assigned to them are true and the statements with a 0 value assigned

to them are false. (32) is a representation of all the possible indices based on these three

atomic statements. The possibilities belonging to the alternative readings of the sentence will

consist of the different combinations of these indices.

26

(33)

The inquisitive representation of the two alternative readings of The girls sang a song.

is shown in (33). The two readings are represented as two different possibilities in the

inquisitive framework. These possibilities are basically the alternative ways the sentence can

be true. Just as a simple disjunction, like p˅q has two possibilities in it: it can be true in case

p is true and alternatively if q is true, these ambiguous sentences have possibilities related to

their alternative readings.

The sentence in (30) has a distributive reading (30a), which is represented in (33) by the

possibility containing two indices, 1,1,1 and 0,1,1. The distributive reading requires both q

and r to be true, and is not influenced by the truth-value of p. The sentence is true

distributively, if Anna and Becky both sang a song alone and it does not matter whether they

also sang a song together or not.

The collective reading (30b) is represented by the possibility of four indices, 1,1,1;

1,1,0; 1,0,1 and 1,0,0 in (33). In all these indices, the truth-value of p is true, the truth-value of

q and r is irrelevant. The collective reading of the sentence is true if the girls only sang a song

together, if Anna or Becky sang a song alone as well as together and if they both sang a song

alone and one together.

Both of these are possibilities in the terms of inquisitive semantics, they are maximal

states supporting a statement. The first one supports (30a), the distributive reading of the

sentence, while the second supports (30b), the collective reading. They are maximal, as they

are not properly included in any other state. These two possibilities combined make up the

proposition expressed by (30), The girls sang a song.

27

The ‘classical’ meaning of a sentence, its informative content remains intact in form of

its proposition, the sum of all possible meanings. The framework of inquisitive semantics is

capable of dealing with statements in predicate and first-order logic as well (Ciardelli, 2010).

It also models the dynamic aspect of language through the changes in the common ground,

which enables this framework to be used in analysing longer, continuous texts and the effect

each sentence has on the interpretation as a whole. The innovation of inquisitive semantics

that are the possibilities are also invaluable in an adequate analysis of ambiguity.

This framework is capable of appropriately modelling sentences with collective -

distributive ambiguity with the help of possibilities, which are built in the system. This

enables the model to keep the alternative readings separate, but within one representation. The

possibilities each represent one of the readings of the sentence. These possibilities are

maximal states, so they remain separate in this framework. The whole, combined meaning of

every interpretation of a sentence is also available as the proposition expressed by the

sentence, the set of all the possibilities making it up.

As inquisitive semantics is a dynamic semantic model, it can be used to represent the

changes in the common ground throughout longer pieces of text, like the everyday

communication between people. In the inquisitive framework, these ambiguous elements no

longer create the problems caused by dealing with alternative versions of the common ground.

Each sentence consists of a single proposition, it does not matter whether they are ambiguous

or not. These propositions may be made up of more than one possibilities, but this is not a

distinctive attribute of ambiguous statements either, as sentences containing disjunction are

also represented this way. This means that these ambiguous sentences do not make the

analysis overly complicated.

28

6 Conclusion

The aim of this paper was to find a framework that can adequately represent sentences with

collective - distributive ambiguity. Groenendijk and Roelofsen’s inquisitive semantics model

seems to be capable of achieving this goal due to its built-in system of possibilities.

The first half of the paper presents the established classification of ambiguity,

distinguishing lexically, structurally and semantically ambiguous sentences. The focus of this

paper is the collective - distributive ambiguity, a type of semantic ambiguity, with two

alternative readings: one looking at the individuals in the proposition as a group, the other

interpreted for the individuals separately. Landman’s model for the interpretation of this

phenomenon was presented together with a way of modelling that takes us closer to the

logical representation of sentences.

The second half of the paper introduces the framework of inquisitive semantics, the

models it is based on and its most important concepts and workings and their formal

definitions. The model is then used to analyse a sentence with collective - distributive

ambiguity.

The inquisitive model is very useful in dealing with sentences with collective -

distributive ambiguity as it is capable of managing the alternative readings of a sentence

separately, preserving its different interpretations, while at the same time keeping these

alternatives in one representation, facilitating their semantic analysis.

Inquisitive semantics is a framework that may be helpful in the semantic representation

of a wide range of problematic sentence types. It is based on formal logic and provides

numerous tools enabling detailed representation of natural language sentences.

Further studies may find this framework suitable for the analysis of the other types of

ambiguity or in the modelling of other linguistic phenomena as it is a relatively recent

approach of semantics which means that there are a lot of areas where it could be useful but

has not been implemented yet.

29

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