Redundancy and Exhaustification in Disjunctive Questions

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Redundancy and Exhaustification in

Disjunctive Questions

Casper van Houten 10194355

Bachelor thesis Credits: 18 EC

Bachelor Opleiding Kunstmatige Intelligentie University of Amsterdam Faculty of Science Science Park 904 1098 XH Amsterdam Supervisor dr. Floris Roelofsen ILLC, University of Amsterdam

Science Park 107, 1098 XG Amsterdam, The Netherlands

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1 Introduction

In natural language, well-formed and meaningful sentences can be constructed that odd in some manner. Disjunctive sentences in particular (or in natural language) are suspect to this oddness:

(1) a. Bruce was born in Paris or France. b. Alfred grows crops or he is a farmer. c. Selina will take the train or travel by rail

Intuitively, these sentences seem odd because the disjunction is unnecessary: simply stating that Bruce was

born in Paris entails that he was born in France. Hurford [1974] attempted to capture this intuition, by

describing a set of disjunctions that are odd because of the entailing disjuncts. (2) Hurford’s constraint

A disjunction of the form X1or X2is odd if X1entailsX2or vice versa.

However, there are some sentences in which a disjunct entails another, while the disjunction seems accept-able:

(3) a. Alfred ate some of the ice-cream, or all of it.1 b. Bruce corrected most of the homework or all of it. c. I will try to jump, or I will succeed.

Gazdar [1979] observed that these sentences all include scalars (5) in front of the disjunction, which leads to a disjunct entailing another. In natural language, we would conclude that only some of the ice-cream must have been eaten, otherwise part of the sentence would have been redundant. This is different than the example of France and Paris, as a similar disambiguation is not possible there.

(4) Gazdar’s observation

If the weaker proposition of a disjunction contains a scalar, it has influence on its acceptance. The scalars that Gazdar refers to are a theory of Hirschberg [1985], that outlines categories of words that are semantically related to each other: they all describe the same meaning, but in different strengths. A few examples are listed below, and they reveal that scalars are not limited to a certain part of speech or domain. (5) shows the entire list Hirschberg adapted from Levinson [1983].

1

Some might argue that this sentence does not lead to entailment if the semantics of ’some’ do not include ’all’, like it does in first order logic. This is true, but later in this thesis arguments will be made why this is not the correct interpretation

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(5) Scales

•all, most, many, some, few •and, or

•n, . . . , 5, 4, 3, 2, 1 •excellent, good •hot, warm

•always, often, sometimes

•succeed, Ving, try to V, want to V •necessarily p, p, possibly p

•certain that p, probable that p, possible that p •must, should, may

•cold, cool •love, like •none, not all

To generalise these observations, a theory can be formulated. Katzir and Singh attempted to describe a general notion of redundancy, that could be able to predict when a sentence is judged odd in natural language. They found a general theory that could predict the oddness of sentences like 1(a), but failed on Gazdar’s examples like (3-a). The observations of Gazdar where then formalized by Chierchia et al. [2009], who described an operator that could strengthen scalar propositions. Even with this addition, the theory still has limits. For instance, it can not predict the oddness of questions, because FOL, for which it was built, does not have a representation for them. To formulate a theory that predicts the oddness of a question, we will need a formal representation for them. Two theories have been chosen due to their seeming similarity but profound theoretical differences. Both alternative and inquisitive semantics will be discussed as formal representations for questions, and a theory similar to that of Chierchia et al. [2009] and Katzir and Singh [2013] will be constructed for each representation. Then, these constructed theories will be tested on odd and acceptable sentences to see if the predictions are similar, and any differences will be addressed and discussed. This might improve our insight on the differences between alternative and inquisitive semantics.

2 Background

To formalise the intuitions on oddness in sentences, first we will need to understand the phenomena that seem to cause them. The most apparent of these is redundancy, the notion that information could be ex-pressed in fewer words than used. Katzir and Singh offer multiple theories that could describe this occur-rence. The other discussed concept related to oddness are propositions with scalars. Chierchia et al. [2009] present a strengthening of these propositions that can explicitly remove the prediction of oddness, so that the predictions of the theory match with our judgement.

2.1 Redundancy

Redundancy in the most general sense is any information that is expressed more than once. On a syntactic level, redundancy would be any statement in which you can remove a proposition, without changing the meaning of it (6).

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(6) Syntactic redundance

A sentence is redundant if the information it expresses can be expressed in a shorter sentence. Redundant sentences are very apparent, and will often look odd.

(7) a. Alfred is in his bedroom and he is at home. b. If Bruce lives in France, then he lives in France. c. Bruce is Bruce.

Given this definition of redundancy, and these examples, it seems like redundancy leads directly to oddness. Because of this, Katzir and Singh attempt to ban any of such redundant material.

2.1.1 General economy attempt

The intuition that redundant material directly leads to oddness seems logical when observing (7), and Katzir and Singh propose exactly that:

(8) Do not use redundant material A sentence is odd if it fits with (6)

Sentences like 1(a) are correctly predicted as odd, as: Bruce was born in Paris or France

p = Bruce was born in France. q = Bruce was born in Paris.

11 10 01 00 (a) p 11 10 01 00 (b) q 11 10 01 00 (c) p∨ q

Figure 1: Bruce was born in Paris or France

[[p]]≡ [[p ∨ q]], and p is shorter than p ∨ q, so p ∨ q will correctly be predicted as odd.

This definition predicts any redundancy to lead to oddness, while in natural language, some theoretical redundancy might not lead to a sentence being judged as odd:

(9) a. Why doesn’t Mary pick up the phone?

b. She is not home, or she is and she doesn’t want to talk. c. She is not home, or she doesn’t want to talk.

The sentence (9-b) is predicted to be odd because the information that she is home is redundant if she does not want to talk. The other sentence (9-c) is predicted to be correct, while we would not judge it more correct than b. It seems that not all redundancy is unacceptable. To account for this, Katzir and Singh

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propose a local, more fine-grained check that decides whether operators are redundant rather than entire sentences.

2.1.2 Local redundancy check

Checking redundancy at sentence level does not work properly, but perhaps a more local check at operator level would. Katzir and Singh suggest that a binary operator is odd if the output of the operator could be expressed by either of the input propositions on its own.

(10) Local redundancy check An operator O applied to alpha and beta is odd if the result of the operator O(α, β) is equivalent to either α of β itself.

This check works for 1(a) and most notably, unlike the general economy for (9-b). The predictions in these types of sentences are correct, and it seems it provides a good handle to describe redundancy with.

Mary is not home, or she is, and does not want to talk to you p = Mary is home.

q = Mary wants to talk to you.

This sentence is built up out of more than just two propositions, where ϕ =¬p and ψ = p ∧ q

11 10 01 00 (a)¬p 11 10 01 00 (b) q 11 10 01 00 (c) q 11 10 01 00 (d)

Figure 2: Mary is not home, or she is, and does not want to talk to you.

Gazdars observation regarding the influence of scalars on oddness is still left unchecked. The next section will introduce an enrichment proposed by Chierchia et al. [2009], that provides a way to express this influence.

2.2 Exhaustification

Gazdars observation regarding the influence of scalars on the acceptance of Hurford sentences can give us some insight on how we might filter these sentences out when predicting the oddness of a sentence.

The general observation Gazdar [1979] on sentences such as (3-a) if we assume some to mean only

some. However, in natural language, this is not necessarily correct. In minimal sentences one might assume

this more narrow meaning, but when embedding scalars in conditionals, the problem becomes clear: (11) If some of you want ice-cream, you can take it from the freezer.

If everyone at the table wants some ice-cream, one would not expect no one to get up just because all of them wanted ice-cream, and not just some of them. So just assuming some to mean some and not all unconditionally does not yield the required results either. These scalars seem to change meaning without

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any syntax to signal it. Chierchia et al. [2009] propose an operator to pick up on this more specific meaning and add it to the syntax. This strengthening is called exhaustification:

(12) Exhaustive operator (exh())

[[exhALT(S)]] ={w|w ∈ [[s]]} and w ̸∈ [[S′]] for all S′∈ ALT such that S ̸⊆ S′.

This definition states that when exhausting a sentence S, any alternatives that are not contained in the ALT used in S are negated. So, for example:

(13) S = Alfred ate some of the ice-cream a. exhsome= some and not all.

b. exhsome(S) = Alfred ate some of the ice-cream, and not all of it.

The meaning of (13-b) is judged equivalent to (13), so it seems redundant to add the word in natural lan-guage. This operator will add the meaning of this to the formal translation of sentences like (13), which will ensure that the disjuncts in sentences like (3-a) do not entail one another. Instead of the conjunction like shown above, the exh() can be viewed as a silent only:

(14) Alfred ate only some of the ice-cream, or all of it.

In total, the chosen definition of redundancy (10), and the definition of exhaustification (12) form a Redun-dancy and Exhaustification Theory (RET). This theory, referred to from now as Classical RET will predict the acceptability of a sentence when applied to it.

(15) Classic RET

a. A sentence S is unacceptable if it contains an operator O that is unacceptable.

b. An operator O applied to α and β is unacceptable if O(exh(α), exh(β)) ≡ exh(α) or

O(exh(α), exh(β))≡ exh(β)

Applying this theory to (3-a) will exemplify its correctness: Alfred ate some of the ice-cream, or all of it

If Alfred has two buckets of ice-cream: p = Alfred ate the one bucket of the ice-cream. q = Alfred ate the other bucket of the ice-cream.

ϕ = Alfred ate some of the ice-cream. ψ = Alfred ate all of the ice-cream.

exh(ϕ) = Alfred ate only some of the ice-cream. exh(ψ) = Alfred ate all of the ice-cream.

exh(ψ) = ψ because there are no scalars that are not contained in the universal quantifier.

[[p]]̸≡ [[p ∨ q]] and [[q]] ̸≡ [[p ∨ q]], so this sentence will correctly be predicted as acceptable.

The above sentences are all predicted correctly, but this classic theory still has some limitations. For instance, it can not predict the oddness of questions.

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11 10 01 00 (a) exh(ϕ) 11 10 01 00 (b) exh(ψ) 11 10 01 00 (c) exh(ϕ)∨ exh(ψ)

Figure 3: Alfred ate some of the ice-cream, or all of it

2.3 Limitations of the Classic RET

However, the formulated theory only works for declarative sentences, while there are also odd sentences that are not declarative, like questions:

(16) a. Was Bruce born in Paris or in France?

b. Did Alfred eat some of the ice-cream, or all of it? c. Did Bruce correct most of the homework, or some of it?

The pattern that emerges from these sentences seems similar to that of the informative sentences. Hurford’s and Gazdar’s generalisations seem to apply to questions as well, but entailment and exhaustification that followed from those generalisations might be harder to capture. To start, we will need a representation for these questions, as first order logic can only handle purely informative sentences. The next section will introduce two representations of questions, and exemplify their differences.

3 Redundancy and exhaustification for disjunctive questions

3.1 Questions

Until now, all examples have been declarative sentences. Declarative sentences are combinations of propo-sitions and operators, that together describe a set of worlds in which the sentence is true. There are other types of sentences, such as questions. A question is a request for information, rather than a declaration of it. Questions often include disjunction, for example:

(17) a. Would you like coffee, or tea? b. Would you like coffee (or not)?

These examples illustrate that a disjunction in a question is different than in a declarative sentence. Rather than stating that any world in which either of the disjuncts is true satisfies the sentence, it seems to provide to alternatives that satisfy the sentence. An answer to the question would provide information about which of the alternatives is the correct one. To represent questions or interrogatives, we will need to formalise this concept of representing a disjunction as a set of alternatives. There are two main theories regarding this representation. Alonso-Ovalle [2007] formulated a theory called Alternative semantics, that redefined the disjunction entirely as an operator that provided alternatives. Roelofsen [2015] describes a different theory referred to as Inquisitive semantics, that attempts to merge the original definition of the disjunction with the definition of alternative semantics. In the following sections, these two theories will be described extensively.

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For both theories, four concepts will be defined on which the RET will be based. First, a definition of a proposition will be given, as that is the most basic idea of semantics: what do the symbols mean? Then, to represent disjunctive sentences, we will need to define the logical disjunction∨. In FOL, a disjunction of two propositions is a set of worlds described by: [ϕ∨ ψ] = [ϕ] ∪ [ψ]. In the representation for inquisitive content, this definition will have to be reviewed to account for the concept of alternatives. Subsequently, to disambiguate between informative and inquisitive content, two operators ! and ? will be defined to represent their respective content.

3.2 Alternative semantics

The observation that a disjunction used in combination with an interrogative leads to two alternative possi-bilities was formalised by Alonso-Ovalle [2007]. This differs from an informative sentence because rather than providing information, it requests information. As described in the previous section, we will need to define four concepts to outline the semantics in such a way that we can describe it formally. First, a general introduction to informative and inquisitive content will be given, and from that introduction, propositions will be defined. Subsequently, we will look how these operators interact with each other when applied to the disjunction operator. Finally, ! and ? will be defined so that we can translate sentences properly. 3.2.1 Informative and inquisitive sentences

11 10

01 00

(a) Bruce lives in France

11 10

01 00

(b) Does Bruce live in France?

The informative sentence 4(a) is informative. It describes a single subset of all possible worlds. Con-trarily, 4(b) is not informative. It provides two possibilities, and does not exclude any worlds. From this example, we can define the differences between informative and inquisitive content.

All of rectangles in the figures above can be viewed as information states. An information state is a non-empty set of possible worlds. An information state is trivial if it contains all possible worlds, as it does not add any new information. Using this definition of an information state, we can describe informative and inquisitive content formally:

(18) Informative and inquisitive content

A sentence that describes a single information state is a purely informative sentence. Any sen-tence that has all possible worlds contianed in different alternatives is purely inquisitive. All other sentences are a combination of informative and inquisitive content.

From this definition, we can conclude that indeed 4(a) is informative, as it excludes two possibilities (those where Bruce does not live in France) and 4(b) is inquisitive as it contains two alternatives (the alternative where Bruce lives in France, and the one where he does not). Moreover, 4(a) is not inquisitive as it only contains one alternative, and 4(b) is not informative as it does not exclude any possible worlds.

It is important to note that any proposition [p] in alternative semantics is a set of information states. So even purely informative sentences, such as p, is a set with one information state. An inquistive sentence will

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be a set of multiple information states, so it will be a set of embedded sets including possible worlds. (19) [p] ={{w|w(p) = 1}}

There are sentences that have an informative as well as inquisitive component, but operators are required to construct these sentences. Now that we have a notion of informative and inquisitive sentences, we will construct the theory surrounding the disjunctive operator.

3.2.2 The disjunctive operator

In first order logic, the disjunctive operator (∨) unifies two propositions ϕ and ψ, creating one new set of worlds [ϕ∨ ψ]. ∨ has some remarkable properties in FOL, some of which have already been discussed, and some of which will be shown later.

In alternative semantics∨ does a similar unification, but due to the possibility of alternatives, it will have more possible results.

(20) [ϕ∨altψ] = [ϕ]∪ [ψ]

In practice, a disjunctive proposition such as (21) p∨ q

is described by two alternatives:

{{{p, ¬q}, {p, q}}} ∪ {{{¬p, q}, {p, q}}}

The result is one set consisting of two information states:

{{{p, ¬q}, {p, q}}, {{¬p, q}, {p, q}}}

Without expressions to disambiguate between informative and inquisitive content however, the sentence would look exactly the same for the interrogative equivalent of this question. Because of this, ! and ? will be introduced.

3.2.3 ! and ?

Because more types of sentences need to be represented, we will also need more symbols to disambiguate them. ! signifies that a sentence is a declaration and thus, as shown in (18), contains only one information state. This information state can be obtained by unifying all alternatives stated in that sentence.

(22) [!ϕ] =∪[ϕ]

To exemplify this rule, a simple disjunctive declaration will be used: (23) !p∨ q

Like a traditional disjunction, this declaration is satisfied by either p or q being true, or both.

With this expression, we can reproduce all disjunctive sentences we could represent in FOL. To express the inquisitive sentences, ? is required.

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generates two sets of alternatives for a proposition ϕ, one set where ϕ is true, and one where ϕ is false. This is equivalent to a polar question, such as (17-b). Formally, this makes ?:

(24) [?ϕ] = [ϕ]∪ (∪[ϕ]c)

Using ? on a single propositional letter will yield:

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(a) ?p

Comparing the examples given above, ? and ! might seem mutually exclusive. Even though seem to describe the exact opposite, they can be used in conjunction to enrich the language. Let us first summarise all definitions of ALT given in the previous sections:

(25) ALT for disjunctive questions a. [p] ={{w|w(p) = 1}} b. [ϕ∨altψ] = [ϕ]∪ [ψ]

c. [!ϕ] =∪[ϕ]

d. [?ϕ] = [ϕ]∪ (∪[ϕ]c) 3.2.4 Combining ! and ?

Most sentences are not as extreme as the simple examples in the previous section; often, embedded structures will be found. Combining ! and ?, these sentences can be expressed.

Intuitively, ! will always go last, as it unifies all alternatives gathered until then. Because ? generates alternatives that span the entire domain, ! will then always result in a trivial sentence (18). An intuitive example is 4(a), which is fundamentally different from 4(b).

11 10 01 00 (a) ?!p∨ q 11 10 01 00 (b) ?p∨ q

Now that we have a full notation for alternative semantics, we will construct a similar notation for inquisitive semantics.

3.3 Inquisitive Semantics

Inquisitive semantics are proposed by ?, as an enhancement of the alternative semantics described by ?. As this framework is very similar to the previously discussed one, the purpose of this section is mainly to high-light the differences between the two, rather than describing the framework from scratch. The definitions will still be given explicitly, but they follow from the definitions of alternative semantics rather intuitively.

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3.3.1 Propositions

Inquisitive semantics uses alternatives similar to alternative semantics. Rather than stating any element in an alternative will satisfy the alternative, it explicitly states them using power sets:

(26) [p] =P{w|w(p) = 1}

A proposition in inquisitive semantics expresses all worlds which it satisfies, and any subset of them. This is because inquisitive semantics describes a proposition as a set of solutions that resolve the sentence.

A sentence true in {{p, ¬q}, {p, q}} is enhanced in inquisitive semantics to the specific answers that would satisfy it:

{{p, q}, {p, ¬q}}, {{p, ¬q}}, {{p, q}}, ∅}

Intuitively, this is an enumeration that lists all possible answers that would satisfy the sentence. This enumeration gives INQ another interesting property; downward closure.

(27) Downward Closure

a sentence S is downward closed if: S = {β|β ⊆ α for someα ∈ S}. So, if every subset of the elements of S are also included in S. All representations in INQ share this property. They are not shown on graphical examples, but always included.

3.3.2 Disjunctions in Inq

Now that propositions are defined, we can define how the disjunction operator works for inquisitive seman-tics. The definition is exactly the same as for alternative semantics; it takes the union of two propositions. Any resolution to ψ or ϕ is also a resolution to ψ∨ ϕ:

(28) ψ∨ ϕ = [ψ] ∪ [ϕ]

A simple example p∨ q will yield:

{{{p, q}}, {{p, ¬q}}, {{¬p, q}}, {{p, q}, {p, ¬q}}, {{p, q}, {¬p, q}}}

One difference to note is that because of the inclusion of subsets by the powerset operator., a disjunction of two propositions will have more elements than either of the propositions by itself. Similar to ALT, we can again not disambiguate between informative and inquisitive content. We will define the same expressions as for ALT, ! and ?.

3.3.3 ! and ?

As the definition of declarative sentences and inquisitive sentences has not changed from ALT, ! and ? will need to do the exact same thing as for ALT. However, to account for the downward closure of INQ, the definitions will have to be changed slightly.

Recall that !ϕ unifies all propositions in ϕ. For INQ, this will defined as: (29) !ϕ =P(∪[ϕ])

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As discussed, this definition is the same as for ALT, expect for the inclusion of the powerset that accounts for the downward closure. Similar to ALT, ! provides one information state that merges all alternatives into one larger one. Subsequently though, the powerset adds all subsets of the set as possible alternatives back into the set, making it downward closed again.

One of the most basic examples:

11 10 01 00 (a) p 11 10 01 00 (b) q 11 10 01 00 (c) !p∨ q

The only difference between the result in alternative semantics ?? and 4(c) is the downward closure of 4(c). Now that declarations can be made, inquisitions will have to be defined to translate all disjunctive questions. The definition will again be similar to ALT, but with an addition of downward closure property. (30) ?ϕ = [ϕ]∪ P(∪[ϕ]c)

This definition will result in two or more downward closed sets: [ϕ] is downward closed by (26), and

P (∪[ϕ]c) will always be downward closed because of the power set operator. The results of a basic sentence such as ?p will be:

11 10

01 00

(a) ?p

We have now defined all four concepts required to represent disjunctive questions in INQ. Note that INQ has definitions for implications and conjunctions, but these are not necessary to represent our examples. Before given an overview of more complex sentences, let us review the definitions:

(31) INQ for disjunctive questions a. [p] =P{w|w(p) = 1} b. ψ∨ ϕ = [ψ] ∪ [ϕ]

c. !ϕ =P(∪[ϕ]) d. ?ϕ = [ϕ]∪ P(∪[ϕ]c) 3.3.4 Combining ! and ?

Until now, all examples for INQ have been entirely inquisitive informative. Most sentences however, will have a combination of these two components. The next sentences will cover most combinations of the expressions, to give a good intuition in to how they function.

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11 10 01 00 (a) p 11 10 01 00 (b) p∨ q 11 10 01 00 (c) !p∨ q 11 10 01 00 (d) ?p∨ q 11 10 01 00 (e) ?!p∨ q

3.4 Alternative vs. Inquisitive Semantics

From the definitions and examples given of both alternative and inquisitive semantics, conclusions can be drawn regarding their differences. Evidently, the translations of the semantics describe the same worlds. This seems intuitive, as a sentence in natural language will have a meaning, and that meaning (a set of information states) is described by our formal semantics. Having different translations would by default disqualify one, as the translation would simply be incorrect. However, The theoretical assumption the semantics function under can differ. Alternative semantics provides alternatives which describe answers to an enquiry in the most general form (ref?), whereas inquisitive semantics makes an explicit alternative for any specific possibility that would resolve an issue. When translating sentences, this difference does not surface often, but it might when attempting to classify sentences as redundant. In the next sections, RETs will be constructed based on the previously discussed classic RET (RETc) (15) to investigate this. As the

two RETs might differ, both will be separately constructed.

3.5 RET for Alternative Semantics

Constructing an RET will be done in the same fashion as (15). First, redundancy will be described in terms of the semantics. With this notion we will review which sentences are predicted correctly, and which are not. Deriving from Gazdar (4) the leftover sentences could be predicted incorrectly due to scalars in the disjuncts. To account for these sentences, an exhaustive operator will be defined to strengthen the meaning of propositions, this time in alternative semantics. With notions of redundancy and exhaustification, the RET should be able to predict the oddness of informative and inquisitive sentences in alternative semantics correctly.

3.5.1 Redundancy

In the introduction, many examples have been given of odd declarative sentences, like the staple 1(a). To assess redundancy in inquisitive sentences, similar examples need to be acquired for that type of sentence. First, the previous examples can be transformed to inquisitive sentences to see whether the oddness persists. (32) a. Does Bruce live in Paris, or in France?

b. Does Alfred grow crops, or is he a farmer?

c. Does Selina travel by rail, or will she take the train?

Unlike declarative sentences, these sentences have multiple interpretations [Farkas and Roelofsen, 2011]. These interpretations have different translations into formal representations, as they differ in meaning. To disambiguate these interpretations in natural language, intonation is used.

These intonations can be described by the pitch in which they are uttered. These pitches will be repre-sented by↑ and ↓, signifying a rising and falling pitch respectively. The only intonation we will use in this paper is the closed interrogative intonation, which looks (and sounds like):

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(33) a. Does Bruce speak English↓? (?p)

b. Does Bruce speak English↑ or French↓? (p ∨ q)

This intonation provides an alternative for each proposition, which is the most relevant intonation for the redundancy check, as it can provide the most critical examples.

When attempting to describe redundancy formally for ALT, the intuition should be comparable to (10). In essence, that notion is sufficiently general to copy directly.

(34) Redundancy in ALT

a. A sentence S is odd if contains a redundant operator O.

b. An operator O applied to arguments α and β is redundant if O(α, β)≡ α or O(α, β) ≡ β Note that the operators ! and ? can be redundant as well, but like other operators on a single proposition (¬ϕ,ϕc), they can be overlooked and removed without question. This is not possible on an operator with two arguments, as it can not be predicted what letter to replace the expression with.

We have once again captured the observation of Hurford regarding the oddness of sentences, but will still need to account for Gazdars observations regarding scalars. As before, we will use the operator exh(). 3.5.2 exh()

Similar to FOL, using only the notion of redundancy does not take into account gazdars observation regard-ing scalars. Before discussregard-ing this topic, a disambiguation will need to be made between scalar alternatives and propositional alternatives. The first are the scalars as described in (5), and the latter are the alternatives used in alternative semantics as answers to inquisitive sentences.

In RETc, exh(ϕ) selects a set of worlds α out of all worlds, where α⊆ ϕ, by checking a requirement:

the world is select iff the alternative it describes is a subset of the alternative in the sentence. A similar approach is not possible in alternative semantics however, due to the alternatives being embedded in sets. Only comparing the alternatives does not result in correct exhaustification.

Instead, it is more logical to define all scalar alternatives, and subtract those that need to be removed. (35) exh() for ALT

exhALT(ϕ) ={α|∃β[ϕ] and α = β−

∪

{(info(ψ)|info(ψ) ̸⊆info(ϕ)}}

Where info(ϕ) describes all information in ϕ in one alternative: info(ϕ) =∪(ϕ)

For a set of propositional alternatives ϕ we describe a strengthened version of it, in which every element has been strengthened by subtracting every stronger alternative from it. Reviewing the basic exhaustification example (3-a).

(36) Alfred ate some of the ice-cream, or all of it. a. p∨ q

Because no quantifiers have been defined, we will define p as Alfred ate some of the ice-cream and q as

Alfred ate all of the ice-cream, and assume four worlds, in three of which some ice-cream has been eaten,

and one where it has all been eaten.

The only element in p:{{p,¬q},{¬p,q},{p,q}} will have subtracted from it the set of all stronger scalar alternatives, in this case the set {{p,q}}. This will result in the set {{{p,¬q},{¬p,q}}}, which translates

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back to the correct Alfred ate only some of the ice-cream. The result is equivalent to the same exhaustification in FOL:

11 10 01 00 (a) exh(ϕ) 11 10 01 00 (b) exh(ψ) 11 10 01 00 (c) !exh(ϕ)∨exh(ψ)

Figure 4: Alfred ate some of the ice-cream, or all of it

For a set with multiple propositional alternatives (any interrogative sentence), this process will be re-peated multiple times, once for every propositional alternative.

3.6 RET for Inquisitive Semantics

The RET for INQ will be designed in the same structure as the two before, whilst trying to stay as close as possible to the definitions of alternative semantics, to make the comparison as easy as possible. This section will once again only highlight the differences with alternative semantics.

3.6.1 Redundancy

As shown in alternative semantics, the notion of redundancy by Katzir and Singh is generally applicable, and it does not have to be altered to describe redundancy for INQ:

(37) Redundancy in ALT

a. A sentence S is odd if contains a redundant operator O.

b. An operator O applied to arguments α and β is redundant if O(α, β)≡ α or O(α, β) ≡ β 3.6.2 Exh() for INQ

The exhaustification operator INQ will again have to be different from RETc. The selection mechanism

from exhc() can lead to a non downward closed set, because it does not always select all the worlds inside

a propositional alternative. A solution to this would be to combine it with a power set operator, but then all propositions exh() did not select will become part of the set again. The approach used for RETALT does

work, and can be copied exactly. (38) exh() for INQ

exhIN Q(ϕ) ={α|∃β[ϕ] and α = β−

∪

{(info(ψ)|info(ψ) ̸⊆info(ϕ)}}.

Incidentally, this notion of exh() always leads to a downward cosed set because it will cut the selected worlds to be subtracted in all subsets of a propositional alternative.

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3.7 Comparing the RETs

To investigate whether INQ and ALT have practical differences regarding the prediction of oddness, signif-icant test phrases will used, most of which have been displayed throughout the thesis already. For every example, the predictions of both RETs will be given with a short explanation, and those predictions will be compared to our judgement in natural language. Since there is no objective measure to base these judge-ments on, any doubt regarding the judgejudge-ments will be expressed explicitly.

4 Results

4.1 Predictions vs. Judgements

4.1.1 Declarative sentences 1. Bruce lives in France.

p = Bruce lives in France As there are no operators, this sentence is automatically predicted to be

11 10

01 00

(a) p

acceptable by both RETs. It is also judged acceptable. 2. Bruce speaks French or Spanish.

p = Bruce speaks French q = Bruce speaks Spanish2

As p̸≡ !p ∨ q in alternative semantics, this sentence will be correctly predicted to be acceptable.

11 10 01 00 (b) p 11 10 01 00 (c) q 11 10 01 00 (d) !(p∨ q)

This is similar to the result for inquisitive semantics, only differing in the inclusion of the subsets in INQ. RETInqwill also correctly predict this sentence to be acceptable.

2

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3. Bruce lives in Paris or in France. p = Bruce lives in Paris.

q = Bruce lives in France.

In alternative semantics, p̸≡ p ∨ q and q ̸≡ p ∨ q, so this sentence will be incorrectly predicted to

11 10 01 00 (a) p 11 10 01 00 (b) q 11 10 01 00 (c) p∨ q 11 10 01 00 (d) !p∨ q

be acceptable. Note that q≡!(p ∨ q), but the local redundancy check, which has shown to make the most accurate predictions does not check this.

To exemplify the differences with AS, the graphical representation of INQ here will show all subsets that are included in the propositions.

11 10 01 00 (a) p 11 10 01 00 (b) q 11 10 01 00 (c) p∨ q 11 10 01 00 (d) !(p∨ q)

Here, q≡ p ∨ q, so this sentence will be correctly predicted to be odd. 4. Alfred ate some of the ice-cream, or all of it

As no quantifiers have been defined for our languages, a workaround will have to be constructed to translate this sentence properly. We will use the same 4 worlds as we have before, but will use p and

q instead of the binary digits in the other graphics.

p = Alfred ate some of the ice-cream. q = Alfred ate all of the ice-cream.

p q p¬q

p¬q ¬p ¬q

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Alternative Semantics

To display how the operations work on the sets, rather than just showing graphics, the propositions below have been defined in terms of the sets they are represented by:

[p] ={{{p, q}, {p, ¬q}}}

[exh(p)] ={{{p, q}, {p, ¬q}} − {{p, q}}} = {{{p, ¬q}}} [q] ={{{p, q}}}

[exh(q)] ={{{p, q}}}

[exh(p)∨ exh(q)] = {{{p, ¬q}}} ∪ {{{p, q}}} = {{{p, q}}, {{p, ¬q}}} Back in the graphic representation, the above would look like:

p q p¬q p¬q ¬p ¬q (a) exh(p) p q p¬q p¬q ¬p ¬q (b) exh(q) p q p¬q p¬q ¬p ¬q (c) exh(p)∨ exh(q) p q p¬q p¬q ¬p ¬q (d) !(exh(p)∨exh(q))

As exh(p)̸≡ exh(p)∨exh(q) or exh(q) ̸≡ exh(p)∨exh(q), this sentence will be correctly predicted to be acceptable by RETALT

Inquisitive semantics

For inquisitive semantics, the exhaustion works the same as for ALT, but the stronger scalar alternative will have to be subtracted from multiple propositional alternatives, the defined exh() does this well, so the result (and prediction) will be equivalent to that of RETALT; this sentence will be correctly

predicted as acceptable by RETIN Q

4.1.2 Inquisitive sentences 1. Does Bruce live in France?

p = Bruce lives in France.

There are no (two-place) operators in this sentence, so it will correctly predicted as acceptable.

11 10

01 00

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2. Does Bruce speak French↑ or Spanish↓? p = Bruce speaks French.

q = Bruce speaks Spanish. Alternative Semantics

Recall that alternative semantics creates an information state for each proposition, and that a disjunc-tion creates two alternative informadisjunc-tion states:

[p] ={{{p, q}, {p, ¬q}}} [q] ={{{p, q}, {¬p, q}}}

[p∨ q] = {{{p, q}, {p, ¬q}}, {{p, q}, {¬p, q}}} represented graphically, this will look like:

11 10 01 00 (a) p 11 10 01 00 (b) q 11 10 01 00 (c) p∨ q

p̸≡ p ∨ q and q ̸≡ p ∨ q, so RETALT will correctly predict this sentence to be acceptable.

Inquisitive Semantics

In inquisitive semantics, the graphical representation remains the same, so the prediction does as well. The sets slightly differ due to the downward closure property:

[p] ={{{p, q}, {p, ¬q}}, {{p, ¬q}}, {{p, q}}} [q] ={{{p, q}, {¬p, q}}, {{¬p, q}}, {{p, q}}}

[p∨ q] = {{{p, q}, {p, ¬q}}, {{p, ¬q}}, {{p, q}}, {{p, q}, {¬p, q}}, {{¬p, q}}}

Note that the disjunction has 5 elements while both propositions have 3 elements, because the element

{{p, q}} appears in both propositions. As said, the graphical display of this is equivalent to that of

ALT, so RETIN Qwill also correctly predict this sentence to be acceptable.

3. Does Bruce live in Paris↑ or France↓? p = Bruce lives in Paris.

q = Bruce lives in France. Alternative Semantics

The worlds that these sentences describe are similar to the previous example, the only difference is that [p] does not include a world in which Bruce lives in Paris but not in France, as that is not possible. [p] ={{{p, q}}}

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[p∨ q] = {{{p, q}}, {{p, q}, {¬p, q}}}

In the graphical representation, the sentence would be displayed as:

11 10 01 00 (a) p 11 10 01 00 (b) q 11 10 01 00 (c) p∨ q

From both representations, it is evident that p ̸≡ p ∨ q and q ̸≡ p ∨ q, so this sentence will be incorrectly judged as acceptable by RETALT. Note that unlike in the declarative sentence, looking

at equivalence on sentence level rather than operator level does not yield a different (and correct) result.

Inquisitive Semantics

The propositions in inquisitive semantics are similar to alternative semantics, with the addition of downward closure.

[p] ={{{p, q}}}

[q] ={{{p, q}, {¬p, q}}, {{p, q}}, {{¬p, q}}} [p∨ q] = {{{p, q}, {¬p, q}}, {{p, q}}, {{¬p, q}}}

Here, q≡ p ∨ q, so RETIN Qwill correctly predict this sentence to be odd.

4. Did Alfred eat some of the ice-cream, or all of it?

This sentence is very similar to its declarative form in both alternative and inquisitive semantics, so only the resulting graphical representation will be shown.

Alternative Semantics p q p¬q p¬q ¬p ¬q (a) exh(p) p q p¬q p¬q ¬p ¬q (b) exh(q) p q p¬q p¬q ¬p ¬q (c) exh(p)∨ exh(q)

RETALT will correctly predict this sentence to be acceptable, as exh(p) ̸≡ exh(p) ∨ exh(q) and

exh(q)̸≡ exh(p) ∨ exh(q)

Inquisitive semanticsThis sentence, just like the declarative form, functions equivalently to alterna-tive semantics. The resulting graph is the same, apart from the inclusion of the downward closure.

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4.2 Conclusion

Starting with the observations of Hurford and Gazdar, an intuition regarding oddness was described, us-ing concepts such as entailment and Hirschberg’s scalars. Subsequently, a theory by Katzir and Sus-ingh was used to formalise this intuition with a theory that could predict oddness on operator level. This theory was tested on sentences in FOL, resulting in some incorrect predictions on sentences with scalars. To solve this, exh() was introduced, an operator that strengthens scalar propositions by removing some possible worlds. The theory regarding oddness and exh() were combined to form a redundancy and exhaustification theory (RET), that could correctly predict the oddness in a variety of disjunctive FOL sentences. This theory could not make any predictions regarding questions, as FOL has no representation for them. Two semantics were introduced: alternative semantics and inquisitive semantics. These semantics are similar as they both view the disjunctive operator (∨) as a generator of alternatives. These alternatives make it possible to represent questions. For each of these semantics, an RET was formulated, as similar as possible to the RET defined for FOL. With these RETs, we could also predict the oddness of inquisitive sentences, and could compare which of the semantics was more suited to make these predictions correctly.

In alternative semantics, it proved difficult to compare sentences by comparing the input of a binary operator to the output, as applying a disjunction to two alternatives will almost always lead to a differing set of alternatives. We can therefore conclude that alternative semantics is not particularly suited to predict the oddness of sentences in a similar manner to first order logic.

Inquisitive semantics generates all possible information states that lead to a proposition being true in a separate alternative. This leads to a disjunction of two propositions generating less extra worlds than in alternative semantics, as some of the worlds are likely to already be in both sets. This is a very relevant feature of the semantics when wanting to compare the input of an operator to the output, as they are more likely to be the same than in alternative semantics. This feature of the language, which sprouted from the downward closure property that defines inquisitive semantics, makes it an accurate predictor of oddness in disjunctive sentences.

4.3 Discussion

4.3.1 Relation between conjunction and disjunction

In this paper, the conjunction is completely overlooked. The reason for this is that conjunctions are not re-quired to discuss the theory surrounding questions formally. When translating sentences, a good definition of conjunction will yield more elegant results, but the disjunction is far more important due to its ability to generate alternatives. In FOL, the relation between conjunction and disjunction is different; they are both basic operations between two sets and represent the set operations meet and join respectively. The meet operator (∩) does not yield desired results in alternative semantics:

(39) Bruce speaks French and Spanish a. p = Bruce speaks French. b. q = Bruce speaks Spanish.

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When defining land as the intersection of two sets:

ϕ∩ ψ = {α|α ∈ ϕ ∧ α ∈ ψ}

[p∧ q] will yield ∅, as no element is in both sets. Alternative semantics does have work-arounds for this, but the elegance of a simple set operation does not hold. Inquisitive semantics can define∧ as the inter-section of the two sets, because it will always find elements that are in both sets due to its downward closure.

4.3.2 Scalar structure

Given the diverseness of scalars in (5), an argument could be made to include a scalar such as: living somewhere (on Earth, in Europe, in France, in Paris), and the exhaustification of Bruce lives in Paris to be: Bruce lives in Paris and not somewhere else in France or in France. Due to the structure of this scale however, this leads to an odd enumeration.

d.

(40) Bruce lives in France, or in Paris and not in Bordeaux, Lyon, Lille...

The other example of exhaustification Alfred ate only some of the ice-cream, or all of it remains acceptable after the exhaustification has been applied. Bruce lives in France, or in Paris and not somewhere else in

France still seems odd after the attempted exhaustification, and seems to have changed in meaning. From

this we might conclude that the exhaustification was incorrect, and that this geographical scalar does not function like some of the other scales.

4.4 Further research

Alternative semantics and inquisitive semantics can describe more types of non-assertive sentences than the questions described in this thesis. ? describes counterfactual sentences such as:

(41) If Bruce lives in Paris, then he lives in France.

which might also be subject to oddness, especially when combined with disjunctions: (42) If Bruce lives in Paris or Barcelona, then he lives in France.

Further research could extend the RETs to account for these sentences, and test whether the predictions of alternative and inquisitive semantics differ in the same way as for the questions described here.

References

Luis Alonso-Ovalle. Alternatives in the disjunctive antecedents problem. In Proceedings of WCCFL, vol-ume 26, 2007.

Gennaro Chierchia, Danny Fox, and Benjamin Spector. Hurfords constraint and the theory of scalar impli-catures. Presuppositions and implicatures, 60:47–62, 2009.

Donka Farkas and Floris Roelofsen. Polarity particles in an inquisitive discourse model. Manuscript,

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Gerald Gazdar. Pragmatics: Implicature, presupposition, and logical form. Academic Press New York, 1979.

Julia Linn Bell Hirschberg. A theory of scalar implicature. University of Pennsylvania, 1985.

James R Hurford. Exclusive or inclusive disjunction. Foundations of Language, pages 409–411, 1974. Roni Katzir and Raj Singh. Hurford disjunctions: embedded exhaustification and structural economy. In

Proceedings of Sinn und Bedeutung, volume 18, 2013.

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Redundancy and Exhaustification in Disjunctive Questions