Bilattices and the Semantics of Natural Language Questions

R. NELKEN and N. FRANCEZ

BILATTICES AND THE SEMANTICS OF NATURALLANGUAGE QUESTIONS �

ABSTRACT. In this paper we reexamine the question of whether questions are inher-ently intensional entities. We do so by proposing a novel extensional theory of questions,based on a re-interpretation of the domain of type t as a bilattice rather than the usualboolean interpretation. We discuss the adequacy of our theory with respect to the adequacycriteria imposed on the semantics of questions by (Groenendijk and Stokhof 1997). Weshow that the theory is able to account in a straightforward manner for some complexissues in the semantics of questions including coordinated questions, combined indicativeand interrogative sentences, questions with quantifiers, and the impossibility of negatingquestions.

1. INTRODUCTION

The semantics of natural language interrogative sentences, henceforth ab-breviated to questions, has received considerable attention in the linguisticand philosophical logic literature. See (Groenendijk and Stokhof 1997;Harrah 1984) for surveys. One important view in the linguistic literatureis that the semantics of questions should provide answerability conditionsfor them in direct analogy with truth conditions for indicative sentences.Intuitively, answerability conditions should determine what situations arerequired to hold in order to answer a question. There is less agreementon how this intuitive notion should be implemented in a formal theory,with different researchers interpreting it in different ways. In (Groenendijkand Stokhof 1997), the linguistic approaches are divided into three variet-ies: partition theories, sets of proposition theories and categorial theories.Partition theories view the meaning of questions as partitioning the set ofpossibilities into mutually exclusive subsets, which jointly exhaust the full

� We wish to thank Jonathan Ginzburg, Jeroen Groenendijk, Ian Pratt, Yoad Winterand the anonymous referees for illuminating comments on previous versions of this work.This work was carried out as part of the research project ‘Semantics of Natural LanguageTemporal Questions and Interfaces to Temporal Database Systems’ sponsored by the Fundfor Interdisciplinary Research, administered by the Israeli Academy of Science. The workof the second author was partially supported by the fund for the promotion of research inthe Technion.

Linguistics and Philosophy 25: 37–64, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

38 NELKEN AND FRANCEZ

set of possibilities. A major proponent of this view is the work of Groen-endijk and Stokhof (henceforth G&S) (1982, 1984, 1989, 1997). Sets ofproposition theories view the meaning of a question as a set of propositionsthat answer the question, as in the work of (Karttunen, 1977; Karttunen andPeters 1980). Finally, categorial theories (e.g., Hausser 1983; Scha 1983)also analyze the meaning of questions in terms of their answers, but theseanswers are not required to be propositions. Instead, they may be elementsof other non-sentential categories, depending on the question. In order tobe able to evaluate the different varieties, G&S impose a set adequacycriteria, which any theory of the semantics of questions should satisfy. Asshown by (Groenendijk and Stokhof 1997), it is only the partition approachand in particular their own theory that satisfies these criteria. The partitiontheory of G&S is intensional, in the sense that the meaning of a question,even in a fixed possible world, is a class of possible worlds. According to(Groenendijk and Stokhof 1997), this is not accidental, as they claim thatany theory satisfying the adequacy criteria must be intensional.

In this paper we wish to question this claim by presenting a novel viewon the semantics of questions. We show that our account can be made tosatisfy the adequacy criteria while remaining extensional. This is achievedby interpreting the meaning of questions as elements of type t, and re-interpreting the domain of type t as a bilattice (Fitting 1991; Ginsberg1988; Ginsberg 1990). A bilattice is an algebraic structure which containstwo partial order relations. Bilattices provide a framework for multi-valuedlogic, incorporating reasoning with conflicting or partial information. Theyhave been successfully applied to several domains, most notably to thesemantics of logic programming. In linguistics, they have been used tohandle propositional attitudes (Muskens 1989) and to account for prag-matic phenomena (Schoeter 1996). Their application to the semantics ofinterrogatives is new.

We focus on extensional (rather than intensional), matrix (rather thanembedded) questions in a static setting (rather than the dynamics ofquestions and answers in dialog), including both yes/no questions andwh-questions of verb complements but not adjuncts. We have chosen thisfocus, as we believe these issues form the core of the understanding of thesemantics of questions, and that our analysis may be extended to cover theremaining issues as well.

Interpreting questions as type t entities opens the way to a simple ana-lysis of several problems in the semantics of questions. This is due to thefact that once questions are seen as type t elements, boolean operators andquantifiers may freely apply to them. This is a sharp contrast from existing

BILATTICES AND THE SEMANTICS OF NATURAL LANGUAGE QUESTIONS 39

theories which assume a more complex interpretation of questions, makingthe application of such operators and quantifiers extremely difficult.

We represent sentence meanings using a simple language called Lo-gic of Questions (LQ), similar to (but more expressive than) that used by(Groenendijk and Stokhof 1997). In (Nelken and Francez 1998, 1999) weshow how our account leads to a straightforward compositional methodfor constructing question meanings. Consequently, we use a richer typedhigher-order language there, related to Montague’s IL. Since we willconcentrate only on the interpretation here, we can use a simpler language.

This paper is structured as follows. In Section 2 we provide the intuitivemotivation for our approach. In Section 3 we formalize it. In Section 4 wediscuss the adequacy of our approach. Finally, in Section 5 we discussseveral applications of our approach to the semantics of questions.

2. MOTIVATION

2.1. G&S’s Adequacy Criteria

An important consideration for the semantics of questions is the followingset of adequacy criteria introduced by (Groenendijk and Stokhof 1997).

− Material adequacy: A model-theoretic interpretation of questionsshould specify notions of answerhood, entailment and equivalencebetween the meanings of either indicative or interrogative sentences:

• Answerhood is a relation between the meanings of an indicativeand an interrogative sentence. An adequate semantics of questionsshould define such a relation as a formal counterpart of the intuit-ive notion of an indicative sentence being a possible answer to thequestion.

• Entailment between the meanings of questions should be definedaccording to the following criterion: the meaning of one ques-tion should entail the meaning of another if whenever the first isresolved, so is the second.

• Equivalence between the meanings of two questions should bedefined so that the meanings of two questions are equivalent iffthey entail each other.

− Formal adequacy: Equivalence of meaning should reduce to iden-tity of the semantic values of questions. Entailment should reduce to‘inclusion’ (or more generally a partial order) between the semanticvalues. Similarly, conjunction and disjunction of interrogatives shouldbe defined standardly in terms of set intersection and set union.


− Empirical adequacy: The semantic notions of answerhood, entail-ment and equivalence should correspond to native speaker intuitionsregarding “test cases”.1

Of the semantic relations required by the material adequacy require-ment, interrogative entailment is perhaps the least intuitive notion, sincein everyday parlance, we do not speak of such entailment. However, thisrelation actually crops up in many contexts. For instance, it is exactly therelation used in the common scientific endeavor of reducing one scientificquestion to another.

Of existing theories, it is only that of G&S themselves that satisfiesthe adequacy criteria. G&S claim that unlike indicatives, interrogatives areinherently intensional. They construe a question as a partition on the setof possible worlds into equivalence classes, determined by the mutuallyexclusive possible answers to the questions. Even in a fixed possible world,a question denotes a set (equivalence class) of possible worlds. Moreover,G&S claim that no extensional treatment that satisfies the adequacy criteriais possible. We review their argument in the following section.

2.2. Does Adequacy Entail Intensionality?

The G&S argument according to which adequacy entails intensionalityruns roughly as follows. Assume some extensional account of the se-mantics of questions, which interprets questions as extensional entities.Consider yes/no questions. What kind of extensional entity would be as-signed to such questions? A natural choice is to assign them the abstractsemantic entities yes if the answer is positive and no if the answer isnegative.

Now consider a pair of questions such as (1) and (2) that have the samecontent only one is phrased in the positive and the other in the negative.

Did Mary kiss John?(1)

Did Mary not kiss John?(2)

Naturally, if one of these questions is assigned the value yes then theother one is automatically assigned the value no. However, by materialadequacy, the two questions are equivalent, since knowing the answer toone is the same as knowing the answer to the other. Hence, by formaladequacy, they must be assigned the same value. This rules out the aboveinterpretation, since yes and no cannot be the same entity.

1 This criterion is not explicitly introduced by G&S, but is implied by their discussionof such test cases.


By deriving this contradiction, G&S argue that no adequate semanticsof questions may be extensional, and therefore, they argue that intension-ality is necessary in the handling of questions.

However, this result is somewhat surprising. In particular, it leads toan a-priori unexpected asymmetry between indicative and interrogativesentences. Whereas in a fixed possible world we can analyze an indicativesentence just by examining its truth value, for an interrogative sentence,we must look at an intensional object, i.e., take into account the meaningof the question under different sets of situations (possible worlds). Notonly is this result unexpected, it also leads to many complications in try-ing to apply natural operations such as coordination and quantification tointerrogatives. This is witnessed for example by (Szabolcsi 1997) who, indiscussing quantification over questions notes:

The crux of the matter is that quantification is defined for domains of type t (expressionsthat can be true or false), and interrogatives are not such.

It is our purpose in this paper to reopen the question of whether ad-equacy entails intensionality. We do so by introducing an extensionaltheory of the semantics of questions, which satisfies the adequacy criteria.We wish to try and follow this approach to the limit, in order to see justhow far we can go with an extensional analysis. In the next section, wepresent the intuitive motivation behind our new interpretation.

2.3. Bilattice Interpretation – Intuitive Motivation

In our attempt to define an extensional interpretation of questions, we wishto assign questions truth values, thereby restoring the symmetry betweenindicative and interrogative sentences. Of course, it would not be wise touse the values true (t) or false (f ). For yes/no questions, this would lead tothe problem described by G&S’s argument above. For wh-questions, thisis even less reasonable since it is meaningless to say such a question is trueor false. Instead, we wish to add another pair of values, resolved (r) andunresolved (ur). In the resulting four-valued system, indicative sentencesare assigned either t or f and interrogatives are assigned one of r or ur,depending on whether they are resolved or not. See (Ginzburg 1995) for adiscussion of the role of resolvedness in the semantics of questions.

These two dichotomies yield symmetric notions of truth and answer-ability conditions. Truth conditions for an indicative sentence are theconditions defining the situations in which it is assigned t . Similarly, wedefine the answerability conditions of an interrogative sentence as thesituations in which it is assigned r.

Perhaps not surprisingly, this simple system already partially satisfiesthe adequacy criteria. It yields natural relations of interrogative entailment


and equivalence. One question entails another if whenever the former isassigned r, so is the latter. Two questions are equivalent if whenever eitheris assigned r, so is the other. However, what is lacking from this picture isthe answerhood relation. In order to add it, we must link the truth valuesassigned to interrogatives with those assigned to indicatives.

A recurring theme underlying many previous theories in all threeclasses of theories (sets of propositions, categorial, and partitions) is thatthe meaning of a question somehow “encodes” the meanings of its an-swers. However, this feature is not actually necessary. This is advantageousfor our approach since it is clearly impossible to encode the answer toa wh-question as a truth value. All that is required by the adequacy cri-teria is an answerhood relation, which in turn suggests some link betweenthe semantic value assigned to a question and the values assigned to itsanswers.

For our link, we employ an idea of (Ginsberg 1990), who shows howa Moore-style auto-epistemic modal operator, L, may be encoded usingtruth values. Given a valuation v that may assign sentences the value t , for unknown (uk), the truth value of a formula Lϕ (“it is known that ϕ”) isdefined as:

v(Lϕ) ={t if v(ϕ) = t

f if v(ϕ) ∈ {uk, f }Employing a similar strategy, we add to our four existing truth values a

final fifth truth value, uk, and add the interrogative operator ©? , defining:

v(©? ϕ) ={r if v(ϕ) ∈ {t, f }ur if v(ϕ) = uk

This operator will be used in computing the truth value of a yes/noquestion such as (1). The truth value of this question is r just in case thecorresponding indicative sentence (3) is either known to be true or knownto be false. Otherwise, it is ur.

Mary kissed John(3)

We thus have a final total of five truth values, separated into two classes,three indicative truth values: {f, uk, t} and two interrogative truth values:{ur, r}. Notice that this strategy makes the addition of the third truth valuefor indicatives, uk, necessary, since otherwise all questions would be re-solved. The addition of uk implies that t and f are now interpreted as“known to be true” and “known to be false”, respectively.2

2 A more economical system using only four values ensues if we identify ur and uk.However, for interpreting more complex phenomena, as discussed in Section 5.2, these twovalues must remain distinct.


We are thus adding a knowledge ingredient to our system. In fact, know-ledge seems an important part of the intuitive interpretation of questions. Inparticular, it seems to underly the semantic relations required by adequacy.For instance, in order for a question to be answerable, its answer not onlyhas to be true, it must also be known. Similarly, the definition of interrogat-ive entailment directly refers to knowledge. The addition of the knowledgeingredient is a critical point with regard to the question of whether inten-sionality is a necessary component of the interpretation of questions. Thestandard way to analyze knowledge resorts to intensionality (or modality).However, we believe that it is possible to interpret questions by addingjust a “dab” of knowledge.3 By employing (Ginsberg 1990)’s technique,as described above, we are able to encode a bit of knowledge in an ex-tensional manner using just truth values, without resorting to “full-blown”intensionality.

We now define the final relation required by adequacy, answerhood. Anindicative sentence answers an interrogative one if whenever the former isassigned t , the latter is assigned r. It is easy to see that both the positiveand negative answers, (3) and (4) stand in this relation to (1).

Mary did not kiss John.(4)

For wh-questions, we use a generalization of the strategy for yes/noquestions. We note that the structure of wh-questions is similar to that ofquantified sentences. We can compute the truth value of the quantified sen-tence (5) by taking the infinitary conjunction of the truth values assignedto the open sentence (6) for any possible value of the variable x.

Every woman kissed John.(5)

x is a woman who kissed John.(6)

Likewise, we define the truth value of a wh-question (7) by applying aninfinitary version of the interrogative operator to the values assigned to theopen sentence (6) for any possible value of x.

Which woman kissed John?(7)

We define the infinitary version of the operator ©? (also denoted ©? ) asan operator that applies to a set of indicative values and yields ur iff one

3 In particular, note that we have introduced knowledge as an objective measure. Furtherrefinement of our approach may relativize knowledge with respect to different agents.


of the values in this set is uk, otherwise it yields r. Formally, if S is such aset, then

©? S ={ur if uk ∈ S

r otherwise.

A wh-question is assigned the truth value r just in case for any valueof x, the corresponding open sentence is known to be true or known tobe false. Otherwise, if there is at least one value of x for which the truthvalue of the open sentence is uk, then the question is assigned ur. Forinstance, (7) is resolved just in case that for each woman it is knownwhether she kissed John or not.4 Consequently, any sentence that providesan exhaustive specification (cf. G&S(1997)) of those women that did kissJohn is an answer to the question, since whenever such a specification istrue, the question is resolved.

This concludes the intuitive basis of our account. In the next section,we formalize it.

3. FORMALIZATION

In formalizing our interpretation, we will use the simple language LQ, thesyntax of which is defined as follows.

3.1. LQ Syntax

LQ consists of first order logic enriched with a pair of new operatorsdefined on formulae: the interrogative operator (?), used for yes/no ques-tions and the binding interrogative operator (?x), used for wh-questions.These are syntactic counterparts of the unary and infinitary versions of thesemantic interrogative operator ©? .

DEFINITION 1. (LQ syntax) The set of LQ formulae is the minimal set offormulae satisfying the conditions:

1. Any atomic FOL formula is an LQ formula.2. If ϕ,ψ ∈ LQ then the following are also in LQ: ¬ϕ, ϕ ∨ ψ, ϕ ∧

ψ, ϕ → ψ, ?[ϕ].3. If ϕ ∈ LQ and x is a variable, then the following are also LQ formulae:

∃x[ϕ], ∀x[ϕ], ?x[ϕ].4 For the sake of simplicity, we ignore any uniqueness presuppositions usually attrib-

uted to questions of the form: which + singular N.


Strictly speaking, LQ is untyped, but there is a natural type disciplinethat is implicit in its definition. All formulae may be thought of as being oftype t, and all constants and variables may be thought of as being of typee. It is in this sense that we interpret questions as type t entities, assigningthem a truth value.5

We distinguish between two flavors of LQ formulae. A formula is calledinterrogative iff it contains an occurrence of one of the interrogative oper-ators (?) or (?x). Any other formula is called indicative. Note that the LQoperators and quantifiers may apply to formulae of either flavor, as wellas to combinations such as an implication between an indicative and aninterrogative. This is the main syntactic difference from the language usedby (Groenendijk and Stokhof 1997).

Free variables are defined as for FOL, with the addition that the bindinginterrogative operator is true to its name, i.e., FV (?x[ϕ]) = FV (ϕ) \ {x}.When iterating quantifiers or interrogative operators, we freely omit theextra square brackets, writing e.g., ?x?y[ϕ] instead of ?x[?y[ϕ]].

Here are some examples of the representations of a couple of basicquestions. Formulae for more complex questions are given below.

[|Did Mary kiss John?|]=?[KISS(MARY,JOHN)](8)

[|Which woman kissed John?|] =?x[WOMAN(x) ∧ KISS(x, JOHN)].

(9)

We now turn to the semantics of LQ.

3.2. LQ Semantics

3.2.1. BilatticesThe semantics of LQ is based on reinterpreting type t as a bilattice ratherthan the classical boolean interpretation. A bilattice is a set B of truth val-ues, which is simultaneously viewed as two complete lattices and containsa single negation operation. Each lattice induces a separate partial orderon B. These are usually referred to as the order of truth (≤t ) and the orderof knowledge (≤k). Assigning a formula a value taken from a bilatticesimultaneously reflects the amount of truth and the amount of knowledgeit is assigned. The minimal non-trivial bilattice contains four values. Usingit, a formula may be either known to be false (f ), known to be true (t),unknown (⊥), or inconsistently known to be both true and false (�). Thesefour values constitute the bottom and top elements of the two complete

5 In (Nelken and Francez 1998, 1999), where we use a richer typed higher-orderlanguage, this typing is made explicit.


lattices and are therefore present in any bilattice. Larger bilattices containadditional values as well. Bilattices are formally defined as follows:

DEFINITION 2. (Bilattice) A bilattice is a structure:

(B,∧t ,∨t ,∧k,∨k,©¬ , f, t,⊥,�)such that:

1. (B,∧t ,∨t , f, t) and (B,∧k,∨k,⊥,�) are both complete lattices, inwhich f, t and ⊥,� are respectively the bottom and top elements ofthe two lattices, and

2. ©¬ : B → B is a mapping with:

a) For any b ∈ B, ©¬ (©¬ b) = b, andb) ©¬ is a lattice homomorphism from (B,∧t ,∨t , f, t)

to (B,∨t ,∧t , t, f ) and from (B,∧k,∨k,⊥,�) to itself.

For our interpretation of LQ, we use a particular bilattice called FIVE,which contains the five different truth values described in Section 2. Sincewe are dealing with questions, we adopt a somewhat different view onbilattices than the one just presented. For our purposes here, we preferto think of the second dimension, usually viewed as the dimension ofknowledge, as one of resolvedness. We change the subscripts on the partialorder relation and the meet and join operators accordingly, yielding ≤r , ∧r ,and ∨r . We also change the names of the bottom and top elements of thisdimension to be unresolved (ur) and resolved (r), instead of ⊥ and �,respectively.

The resulting bilattice is depicted in the double Hasse diagram of Fig-ure 1. The two induced partial orders are reflected by the horizontal andvertical arrows. The meet and join of each dimension can be computed inthe usual manner according to the ordering relations. It is easy to verifythat FIVE is indeed a bilattice.

In (Arieli and Avron 1996), bilattices are used to construct logics, in-cluding consequence relations and proof systems. In order to be able todefine a consequence relation for multi-valued systems, a subset of thevalues must be defined as designated. We follow (Arieli and Avron 1996)in choosing the designated values of FIVE to be D = {t, r}.3.2.2. Truth TablesWe now wish to see how we can use FIVE’s operators for interpretingoperations on indicative or interrogative sentences. We will first discuss theunary operations, negation and the formation of a yes/no-question. We then


Figure 1. The bilattice FIVE.

discuss the binary operations including conjunction and disjunction andfinally the infinitary operations, including quantification and the formationof wh-questions.

We begin by giving the truth tables for the unary operators in Table I. Init, the horizontal line separates the indicative values from the interrogativeones.

TABLE I

Truth tables for unary operations on FIVE

A ©¬ A ©? A

uk uk ur

f t r

t f r

ur ur ur

r r r

Consider negation first. By definition, bilattices come equipped with anegation operation that switches the amount of truth but does not affectthe amount of knowledge (resolvedness). Consequently, applying negationto any of the interrogative values is well-defined, and has a null effect.However negation of interrogative values seems to have no natural lan-guage counterpart, as it is impossible to negate questions. From a technicalstandpoint, this just means that our model-theoretic interpretation has thepower to handle phenomena that go beyond the range of those appearingin natural language. However, the impossibility of negating questions is


an interesting generalization about natural language. In Section 5.4, wepropose a speculative explanation for this phenomenon.

A similar situation holds for the interrogative operator. As definedabove, this operator applies to an indicative value and yields an interrogat-ive one, and thus corresponds to the formation of a yes/no question. For thesake of completeness, we allow this operator to also apply to interrogativevalues, but having a null effect. This operation too seems to have no naturallanguage counterpart, since yes/no questions cannot be formed from otherquestions. We believe that this stems from the fact that questions maynot be negated. A yes/no question takes an indicative sentence and askswhether it or its negation is true. Since a question cannot be negated, theyes/no question cannot offer a real choice. It is therefore not surprising thatnatural language does not allow such a degenerated choice.

Turning to binary operations, we note that whereas FIVE has just asingle negation operation, it has two kinds of meet and join. It is not clearwhich of these semantic operations to use in interpreting natural languagecoordination. For example, should we interpret conjunction using ∧t or∧r? As it turns out, to capture speaker intuitions, the choice depends onwhether the conjuncts are indicative or interrogative. Intuitively, for indic-ative sentences, conjunction should operate on the measure of truth. So,when conjoining a pair of indicative values, we use ∧t . The only deviationfrom the standard interpretation is the addition of the third indicative value,uk. The interpretation of the truth-dimension operations on the indicativevalues yields a strong Kleene three-valued system. For interrogative val-ues, it is appropriate to use the operations of the resolvedness dimension.This is discussed in Section 5.1. Intuitively, a conjunction of questionsis resolved in case both questions are resolved. This is exactly how ∧r

operates.When considering binary operations we have so far only considered

operations involving either purely indicative values or to purely interrogat-ive values. However, since both indicative and interrogatives are uniformlyinterpreted as type t elements, the binary operations may apply to “mixed”values. In fact, we will use this capability in interpreting constructions suchas coordination of an indicative and an interrogative in Section 5.2. As itturns out, the interpretation of such mixed sentences will require a thirdversion of the boolean operators. At this point we have three differentinterpretations of each boolean operator, depending on its operands. Forinstance for conjunction, we have a version for indicative values, ∧t , asecond version for interrogatives, ∧r , and a third version for mixed pairs.Rather than maintain these three variants, we will just keep one generalizedform of conjunction, denoted ©∧ , which simulates the appropriate behavior


depending on its operands. Truth tables for ©∧ and generalized disjunction,©∨ , are given in Table II. For the sake of simplicity, we first give the truthtables without the mixed values, which are completed in Table IV. Wedefine implication, ©→, in the usual way using ©¬ and ©∨ .

TABLE II

Truth tables for binary operations on FIVE

A B A©∨ B A©∧ B

uk uk uk uk

uk f uk f

uk t t uk

f uk uk f

f f f f

f t t f

t uk t uk

t f t f

t t t t

ur ur ur ur

ur r r ur

r ur r ur

r r r r

Finally, we turn to the infinitary operations. We define the operationsas infinitary versions of the generalized operations, which may be appliedto either indicative or interrogative values. Note that this is possible onlybecause the generalized binary operations of disjunction and conjunctionare idempotent, commutative and associative. Formally, if S is a set ofvalues of FIVE, we define:

©? S ={ur if uk ∈ S or ur ∈ S

r otherwise

Similarly, universal quantification is defined as an infinitary version of©∧ . This will allow us to interpret regular quantification over indicative sen-tences but also quantification over interrogative sentences, as discussed inSection 5.3. Likewise, existential quantification is defined as an infinitaryversion of ©∨ .


3.3. Interpretation of LQ

A structure M for LQ consists of a non-empty domain D, and an inter-pretation function I . I maps constants to domain elements and functionsymbols to functions of the appropriate arity on D. Unlike the standardinterpretation, where the interpretation function maps each n-ary predic-ate symbol to a relation on Dn, here we must change the definition toincorporate the third indicative value, uk. Note that in FOL we can viewrelations on Dn as binary partitions of Dn into the set of domain elementsthat are in the relation and those that are not. Here we must look at ternarypartitions of Dn into those tuples that are known to be in the relation, thosethat are known not to, and those for which it is unknown whether they arein the relation or not. Thus, I maps each n-ary predicate symbol p to aternary partition of Dn as follows: Dn = I t (p) ∪ I f (p) ∪ I uk(p). We useσ to denote an assignment function from variables to domain elements. Byabuse of notation, we extend σ to an assignment from arbitrary terms todomain elements.

In Definition 3, we use the following notation. We let ϕ,ψ range overLQ formulae. If ‘·’ is an LQ operator, we denote by ‘�’ the correspondingsemantic operation on FIVE. Given an assignment function σ , we denoteby σ [d/u] the assignment that agrees with σ on any variable except maybeon u and assigns u the value d.

DEFINITION 3. (LQ Extension) Let σ be an assignment function. Theextension of an LQ formula ϕ in a structure M, denoted [[eϕ|]M,σ is:

1. If ϕ is an atomic formula, then

[[ep(t1, . . . , tn)|]M,σ =t if 〈σ (t1), . . . , σ (tn)〉 ∈ I t (p)

f if 〈σ (t1), . . . , σ (tn)〉 ∈ I f (p)

uk if 〈σ (t1), . . . , σ (tn)〉 ∈ I uk(p)

2. If ‘·’ is a unary operator, then [[e · ϕ|]M,σ = �[[eϕ|]M,σ

3. If ‘·’ is a binary operator, then [[eϕ · ψ |]M,σ = [[eϕ|]M,σ � [[eψ |]M,σ

4. [[e∃u[α]|]M,σ = ©∨ {[[eα|]M,σ [d/u]|d ∈ D}5. [[e∀u[α]|]M,σ = ©∧ {[[eα|]M,σ [d/u]|d ∈ D}6. [[e?u[α]|]M,σ = ©? {[[eα|]M,σ [d/u]|d ∈ D}

We now turn to examine the adequacy of our interpretation.


4. ADEQUACY

4.1. Material Adequacy

The adequacy criteria require us to define semantic notions of answer-hood, interrogative entailment and equivalence. Our interpretation not onlyprovides the required semantic relations, but offers a unifying view of theserelations as instances of a more general consequence relation as defined by(Arieli and Avron 1996, 1998). In the following definitions, let ϕ be anLQ formula and let M be a structure. A structure satisfies a formula if itassigns it a designated value. Formally:

DEFINITION 4. (Satisfaction) M satisfies ϕ, denoted |=M ϕ, iff for anyassignment function σ , [[eϕ|]M,σ ∈ D (= {t, r}.)

Satisfaction is extended to a set of formulae, ! in the standard way.

DEFINITION 5. (Consequence) ϕ is a consequence of !, relative to M,denoted ! |=M ϕ, iff |=M ! entails |=M ϕ

As usual, we define ! |= ϕ iff ! |=M ϕ for any structure M. Wecan now see that the required relations are just particular instances of theconsequence relation, depending on the flavor of the formulae involved, asdepicted in Table III. Consequence between a pair of indicative formulaeyields indicative entailment. Consequence between a pair of interrogat-ive formulae yields interrogative entailment. Furthermore, consequencebetween an indicative formula and an interrogative formula yields the an-swerhood relation. Finally, equivalence can be defined as bi-directionalconsequence.

TABLE III

Generalized entailment

ϕ ψ Entailment

Indicative Indicative Indicative

Interrogative Interrogative Interrogative

Indicative Interrogative Answerhood

For some examples of these relations see Section 4.3.Notice that Table III allows a fourth kind of entailment relation, the

mirror image of answerhood, i.e., entailment between an interrogative


sentence and an indicative one. Such a relation holds if whenever theinterrogative is resolved, the indicative is true. While such entailment ispossible, it seems to be of a rather technical logical nature and not partic-ularly relevant for natural language. We believe that this is because suchentailment reflects a kind of higher level reasoning. It allows us to deducea proposition from the fact that a particular question is resolved.

4.2. Formal Adequacy

Recall that formal adequacy requires the semantic notions to be interpretedas the appropriate set-theoretic operations. In order to provide such a set-theoretic interpretation, we have to associate a set with each formula. In theclassical boolean interpretation, we may naturally view the characteristicfunction from structures to truth values as a set, i.e., the set of structuresin which the formula is assigned the truth value t . In our setting, we mayextend this view by looking at the set of structures in which a formulais assigned a designated value. In particular, for an interrogative formula,this set will be the set of structures in which it is assigned the truth valuer. This yields the required relations. Entailment between two formulaereduces to inclusion between the two sets of structures associated withthem. Likewise, equivalence reduces to equality between these sets.

As will be shown in Section 5.1, formal adequacy also holds for ourtreatment of coordination. Basically, since we interpret e.g., conjunctionof a pair of questions as the conjunction of their truth values, the set ofstructures associated with the conjunction will be the intersection of thesets associated with the conjuncts. Similarly for disjunction with respectto union.

An alternative way of associating a set with each formula, which we fol-low in (Nelken and Francez 1999), is to interpret LQ relative to structurescontaining a set of possible worlds. We define the intension of a formula asthe set of possible worlds in which it is assigned a designated value. Onceagain, this is just an extended view of a characteristic function as a set.This alternative “internalizes” the different structures as possible worlds,belonging to a single structure. The choice between these alternatives doesnot appear to be essential.

4.3. Empirical Adequacy

4.3.1. ExamplesWe now turn to discuss the empirical adequacy of our approach. We beginby illustrating the semantic notions with a few simple examples.

FACT 1. ?x[ϕ(x)] |=?[∃x[ϕ(x)]].


By way of proof, let M be a structure and w a world such that[[e?x[ϕ(x)]|]M,σ = r, then for all d ∈ D, [[eϕ(x)|]M,σ ∈ {f, t}. Hence,[[e∃x[ϕ(x)]|]M,σ ∈ {f, t}, and therefore [[e?[∃x[ϕ(x)]]|]M,σ = r.

FACT 2. [|Only Mary and Sue kissed John|] =∀x[KISS(x, JOHN) ↔ (x = MARY ∨ x = SUE)] |=[|Which woman kissed John?|] =?x[WOMAN(x) ∧ KISS(x, JOHN)].

For this fact, assume a structure M in which the left-hand side is sat-isfied (t). It is easy to check that6 for any value of x, [[eWOMAN(x) ∧KISS(x, JOHN)|]M,σ ∈ {f, t}. Hence the right-hand side is satisfied (r).

FACT 3. ?[ϕ] ≡?[¬ϕ].Both formulae are assigned r iff ϕ is either true or false. Otherwise,

both formulae are assigned ur. This example shows that our account doesnot fall prey to G&S’s argument against an extensional account.

Notice that our account also provides a natural notion of rhetoricalquestion. A rhetorical question is one that is always assigned the valuer. Such a question needs no answer, since it is always resolved.

These examples would seem to indicate that our account is empiricallyadequate. Unfortunately, in the current formulation, it is not.

4.3.2. A Problem for Our AccountAs it turns out, our answerhood relation is in fact too weak as we now il-lustrate. In particular, our interpretation predicts the following answerhoodpattern, where ϕ is any indicative formula.

ϕ ∨ ¬ϕ |=?ϕ.(10)

To see why this is so, first notice that in our system, such a disjunction,ϕ ∨ ¬ϕ, is not tautological. In case ϕ is assigned the value uk, then so is¬ϕ, and consequently the whole disjunction is also assigned uk. However,consider those cases in which the disjunction is assigned t . By examiningthe truth tables in Table II we see that in those cases one of the disjunctsmust be assigned t . Thus, either ϕ is assigned t or ¬ϕ is assigned t , inwhich case ϕ is assigned f . In both cases ?ϕ is assigned r. Consequently,in those cases in which the disjunction is assigned t , then the yes/no ques-tion is assigned r. However, one could hardly say that in natural languagethe corresponding disjunction answers the question. Similar argumentation

6 Assuming Mary and Sue are both known to be women.


shows that our account wrongly predicts that (11) is answered by (12) oreven (13).7

Is John asleep?(11)

Everyone is either asleep or not asleep.(12)

Exactly three people are asleep (and everyone else is notasleep).

(13)

This is a genuine empirical problem for our analysis as formulatedabove. There are two possibilities of fixing this problem, both of whichcome with a price.

4.3.3. Two Possible SolutionsThe first solution to the empirical problem encountered above is to adoptan intuitionistic interpretation. Notice that our interpretation of the booleanoperators on the indicative values has an intuitionistic flavor. For instance,for a disjunction to be assigned t , at least one of the disjuncts must beassigned t , i.e., known to be true. If we accept this interpretation of theboolean operators, then it follows that the answerhood pattern above isjustified, since if the answer ϕ ∨ ¬ϕ is assigned t , then one of its disjunctsis indeed known, and thus it is justified to say that the yes/no question isresolved. An approach along these lines is offered by (Ranta 1994). Ranta,in the course of developing a type-theoretical interpretation of naturallanguage, including questions, assumes a principle of fairness of dialog.According to this principle, one may ask the question ?ϕ in a fair dialogonly if the hearer knows that ϕ∨¬ϕ is actually true. Ranta notes that in anintuitionistic interpretation, this means that the hearer knows that one ofϕ or ¬ϕ is true (i.e., knows a proof of one of them). Thus the hearer mayanswer with the disjunction only if she knows which disjunct is true. Whilethe interpretation of questions in an intuitionistic setting is an interestingdirection for study, it seems to provide an unsatisfactory interpretation ofthe semantics of natural language. This is because the interpretation ofnatural language operators is not necessarily intuitionistic, and intuitivelythe disjunction above is understood as tautological.

The second solution we propose is more suitable for the interpretationof regular natural language. It involves a change in the way extensionsof LQ formulae are defined. Recall that we have assigned formulae truthvalues according to Definition 3 in a strictly “bottom-up” fashion. Forexample, the truth value of a disjunction is defined as the disjunction of

7 We thank an anonymous referee for drawing our attention to this problem.


the truth values of the disjuncts. This of course is the classical way todefine truth values of formulae. However, it appears to be insufficient onceknowledge is taken into account. Intuitively, one may know that a cer-tain formula is true without knowing exactly whether its sub-formulae aretrue or not. The tautological disjunction is a case in point. We understandit as tautologically true without having to know (as in the intuitionisticinterpretation) which disjunct is actually true. Ginsberg deals with a sim-ilar problem (Ginsberg 1990). He wishes to assign formulae truth valuestaken from a bilattice, but to keep classical tautologies true and classicalcontradictions false. His solution abandons the strict “bottom-up” fash-ion of computing truth values of a complex formula from those of itssub-formulae. Instead, he defines truth assignments as mappings fromall sentences to the bilattice, satisfying certain conditions. Importing hisdefinition to LQ yields the following revised definition of extension.

DEFINITION 6. (LQ Revised Extension) Let σ be an assignment function.Let |=C denote classical entailment. The extension of any LQ formula ϕ,relative to M is a truth value of FIVE, subject to the following conditions:

1. If ϕ is an atomic formula, then

[[ep(t1, . . . , tn)|]M,σ =t if 〈σ (t1), . . . , σ (tn)〉 ∈ Iw(p)

t

f if 〈σ (t1), . . . , σ (tn)〉 ∈ Iw(p)f

uk if 〈σ (t1), . . . , σ (tn)〉 ∈ Iw(p)uk

2. If ‘·’ is a unary operator, then [[e · ϕ|]M,σ ≥r �[[eϕ|]M,σ

3. If ‘·’ is a binary operator, then [[eϕ · ψ |]M,σ ≥r [[eϕ|]M,σ � [[eψ |]M,σ

4. [[e∃u[α]|]M,σ ≥r ©∨ {[[eα|]M,σ [d/u]|d ∈ D}5. [[e∀u[α]|]M,σ ≥r ©∧ {[[eα|]M,σ [d/u]|d ∈ D}6. [[e?u[α]|]M,σ ≥r ©? {[[eα|]M,σ [d/u]|d ∈ D}7. If ϕ is indicative then [[eϕ|]M,σ ∈ {f, uk, t}; if ϕ is interrogative then

[[eϕ|]M,σ ∈ {ur, r}8. If ϕ |=C ψ , then [[eψ |]M,σ ≥t [[eϕ|]M,σ

In this definition truth values are no longer constructed in a bottom-up fashion. The interpretation simultaneously determines a truth value forall formulae. The truth values assigned must satisfy a set of plausiblerestrictions, in such a way as to make the values assigned to related for-mulae sensible. The clause for atomic formulae remains the same as inthe original definition. However, by Clauses 2 to 6, when applying any ofthe operations the result is not necessarily the one given by applying thesemantic operation to the extensions of the sub-formulae, but may be avalue that is ≥r than it. Clause 7 guarantees that the resulting truth valueremains indicative for indicative formulae.


For some formulae, this definition uniquely determines a truth value.For instance, consider the tautological disjunction ϕ∨¬ϕ in the case whereϕ is assigned uk. By Definition 3, the formula is assigned uk. By contrast,according to clause 3 of Definition 6, this value may be a value ≥r uk.Clause 7 ensures that this value is indicative. Finally, Clause 8 ensures thatthis value is t . To see this, let ψ be any formula such that [[eψ |]M,σ = t .Thus ψ |=C ϕ ∨ ¬ϕ. And by Clause 8, [[eϕ∨ ¬ϕ|]M,σ ≥t [[eψ |]M,σ . Hence,[[eϕ ∨ ¬ϕ|]M,σ = t .8

The revised definition solves the problem of the spurious answerhoodin (10). This is because the tautology is true even in case ϕ is assigneduk, in which case ?ϕ is assigned ur. Similar argumentation shows that thissolution also eliminates the spurious answerhood relations in (12) or (13).This approach solves the above empirical problem. However, there is aprice-tag attached to this type of solution. The truth value of a complexformula can no longer be computed compositionally by applying opera-tions on the model-theoretic denotations of its components. Instead, thestructure of the complex formula has to be taken into account as well. Aformula of the form ϕ ∨ ¬ϕ may be assigned a different truth value thanϕ ∨ ψ even if ¬ϕ and ψ are assigned the same truth value. The priceof adopting the account, is that the strict compositional model-theoreticinterpretation is lost in favor a representational account.

In (Nelken and Francez 1998), we show that our answerhood relationis more inclusive than that of G&S, i.e., if an indicative ψ is an answer toan interrogative ϕ according to G&S, then it is also an answer according toour account, but not vice-versa. The essential observation is that accordingto the G&S interpretation, in order for ψ to answer ϕ, the intension of ψmust be wholly included in one of the blocks of the partition induced byϕ. In each possible world in the intension of ψ , the answer must be thesame. By contrast, on our account if ψ answers ϕ, then it is sufficient thatϕ is resolved in any structure in which ψ is true. In particular, it may beresolved differently in different structures.

5. APPLICATIONS

After having presented our account, we briefly review some interestingapplications of it to notorious problems in the semantics of questions. We

8 There are however other formulae for which this definition does not uniquely determ-ine a truth value, even in case the truth values of the atoms are fixed. It is possible to reducethis non-determinism by further restricting the value assigned to a formula. For instance, itis possible to choose the minimal value that satisfies these restrictions.


discuss interrogative coordination, combined indicative and interrogativesentences, and questions with quantifiers. The underlying theme is that ournew interpretation of questions as entities of (the newly interpreted) type tgreatly simplifies the analysis of these issues.

5.1. Interrogative Coordination

Consider coordination of interrogative sentences as in:9

Does John walk? And/Or does Mary walk?(14)

Who walks? And/Or who talks?(15)

The analysis of such questions is notoriously difficult. In particular, theanalysis of (Groenendijk and Stokhof 1989) shows that while conjunctionmay be treated directly, disjunction leads to a complication. Since the G&Saccount takes questions to denote partitions of the set of possible worlds,a conjunction of questions is defined as the point-wise intersection of thepartitions. However, the same approach does not apply to disjunction, sincethe point-wise union of a pair of partitions does not yield a partition. G&Ssolve this by using a type-shifting operation. The meanings of the disjunctquestions in a possible world are shifted from equivalence classes to sets ofproperties of equivalence classes on worlds and taking the union of thesesets. However, on this account the type-shifted meaning of a question evenin a fixed possible world is of a formidably complex type.

By contrast, on our account, since questions denote type t elements,interrogative conjunction and disjunction can be modeled by operationson the truth values assigned to the questions. In fact, the meet and joinoperators of the resolvedness dimension, ∧r and ∨r , provide the requiredoperations. These operators yield the appropriate resolvedness conditionssince in order for a conjunction of interrogatives to be resolved, bothconjuncts must be resolved. Likewise, in order for a disjunction of inter-rogatives to be resolved, at least one disjunct should be resolved. Thusour account yields a much simpler direct interpretation.10 Thus, whereas

9 These examples should be distinguished from sentences in which the interrogativeoperator has higher scope than the coordination operator. The meaning of these will berepresented by formulae of the form ?[ϕ∧ψ] (or ?[ϕ∨ψ]) and poses no particular problem.

10 Szabolcsi (1997), citing Hungarian evidence, claims, contrary to G&S, that the or inthese examples should be understood not as disjunction, but as an idiomatic device that can-cels the first question, replacing it by the second. We believe that there is a non-idiomaticreading of or (although perhaps not in Hungarian), since one way of answering (15) (withor) is by answering the first question (as predicted by our answerhood relation). This wouldbe impossible had the first question been canceled.


coordination of indicative sentences is modeled using the truth-dimensionoperators, coordination of interrogatives is modeled using the operatorsof the resolvedness dimension. Recall that we use a generalized versionof each operator that simulates the behavior of the appropriate operator,depending on whether its arguments are indicative or interrogative.

As we noted above, a third variant arises in “mixed” coordination of anindicative sentence with an interrogative one. These are discussed in thefollowing section.

5.2. Mixed Coordination

As mentioned by (Harrah 1984), examples involving mixed coordinationas in (16) have received only limited previous attention. Such examples areperhaps most acceptable with implication as in (17).

The machine is broken; or does it just need fuel? (Harrah 1984)(16)

If Mary kissed John then who kissed Bill?(17)

By uniformly interpreting both indicatives and interrogatives as typet elements, we are now able to account for such examples as well. Ad-mittedly, such sentences are perhaps not amongst the foremost desiderataof the semantics of questions, but their mere acceptability lends supportto our hypothesis that indicatives and interrogatives are to be treated asentities of the same type. Moreover, the treatment of these examples is astepping-stone for our analysis of questions with quantifiers in Section 5.3.

In order to account for these hybrid sentences, we first note that theyare interrogative in nature. Such a sentence may be either resolved or un-resolved, but it makes little sense to say it is true or false. When applyingto a mixed pair of values, the coordination operators must always yield aninterrogative value. Therefore neither the operators of the truth dimensionnor those of the resolvedness dimension are appropriate. Instead, we needa third variant of the disjunction/conjunction operator to be incorporatedinto the unified ©∨ and ©∧ operators. Intuitively, a disjunction of an in-dicative and an interrogative yields r just in case one of the disjuncts isdesignated, i.e., if the indicative is known to be true or the interrogativeis resolved. Otherwise, the disjunction is unresolved. Hence, an indicativesentence stands in the answerhood relation to such a mixed question ifit either asserts the indicative disjunct or answers the interrogative one.Similarly, a mixed conjunction yields r just in case both conjuncts aredesignated. Mixed implication is defined as usual using ©∨ and ©¬ . We maythus complete the truth tables of Table II as in Table IV. We define ©∨ and


©∧ as symmetric operations. In the table, each symmetric pair appears onlyonce for conciseness.11

TABLE IV

Truth tables for “mixed” operations on FIVE

A B A©∨ B A©∧ B

uk ur ur ur

uk r r ur

f ur ur ur

f r r ur

t ur r ur

t r r r

The need to cater for mixed coordination is one of the reasons we needto use a bilattice containing five values, and cannot use the simplest bilat-tice containing only four values. For instance, if we were to try and identifyuk and ur, we would get a conflict between t ∨ ur = r and t ∨ uk = t .

5.3. Questions with Quantifiers

We now turn to show how our theory can deal in a straightforward man-ner with wh-questions containing quantifiers or multiple wh-terms. Forexample, consider the following question:

Which professor recommends each candidate?(18)

Such questions are traditionally seen as having two different readings(see e.g. (Chierchia 1993)). These are the individual reading, which askswhich professor is such that she recommends every candidate, and the pair-list answer which asks for each candidate which professor recommendsher.12

Whereas the individual reading is derived directly by allowing the wh-term to have higher scope over the quantified NP, the pair-list reading is

11 Notice that while the operations are defined to be symmetric, reversing the orderof the indicative and the interrogative sometimes yields ungrammaticality as in: *If whokissed Bill? then Mary kissed John. A similar phenomena seems to happen with (regular)indicative implication, which may appear in the consequent of a second implication as inIf A then (if B then C), but not as its antecedent as in *If (if A then B) then C. We are notaware of a semantic or syntactic explanation of this phenomenon.

12 We ignore functional readings in this paper.


notoriously difficult to derive compositionally in previous theories. Intu-itively, the pair-list reading may be obtained by allowing the universalquantifier to have higher scope than the interrogative operator. However,since in previous theories questions are of types that do not easily lendthemselves to quantification, it is not clear how to derive this reading. Forinstance, (Karttunen and Peters 1980) get the pair-list reading by usinga double negation operation, which seems rather unmotivated. In G&S’stheory, pair-list readings are initially available just for universal quantifi-ers. To allow pair-list readings associated with other quantifiers they onceagain use a type-shifting operation. This operation shifts the meanings ofquestions into sets of sets of partitions on the set of possible worlds. Theresulting type is even more complex than in the case of coordination.

By contrast, since in our account questions are interpreted as type t en-tities, we can straightforwardly quantify over questions. Thus, we give thetwo readings of (18) as (19) and (20). For the compositional constructionof these meanings see (Nelken and Francez 1998).

?x[P(x) ∧ ∀y[C(y) → R(x, y)]].(19)

∀y[C(y) →?x[P(x) ∧ R(x, y)]].(20)

To see that these formulae reflect the intuitive interpretation of the tworeadings, note that (19) may be answered by an exhaustive specificationof the possible values of x such that x is a professor and x recommendsall the candidates. Conversely, (20) requires for each value of y that is acandidate, a complete specification of the professors x that recommend y.Note that in (20) we use a mixed implication, as developed in the previoussection, between an indicative formula (C(y)) and an interrogative one(?x[P(x) ∧ R(x, y)]).

This kind of analysis seems to imply that pair-list readings are availablefor any quantifier. In fact, the precise pattern of the acceptability of pair-list readings is quite complex, depending not only on the quantifier, butalso on whether the question is a matrix question or an embedded onewhether the embedding verb is “extensional” or “intensional” (Szabolcsi1997). See also (Pafel 1999). One generalization that can be made is thatpair-list readings are impossible with downwards monotonic quantifiers.This is illustrated by the observation that (21) may not be answered by alist of pairs of professors and the candidates they do not recommend.

Which professor recommends no candidate?(21)

As noted by (Szabolcsi 1997), there are no known compelling syntacticor semantic explanations for this phenomenon. A pragmatic explanation is


offered by G&S. According to their claim, assigning the NP no candidatehigher scope means asking for no candidate which professor recommendsher. Thus the question asks the replier to remain silent, and is thereforepragmatically unacceptable.

Note however that the fact that downwards monotonic quantifiers donot admit pair-list readings is actually an instance of the more general factthat questions may not be negated. We suggest a semantic explanation forthis fact in Section 5.4.

Questions with multiple wh-terms such as (22) receive a similar treat-ment, yielding either (23) or (24) depending on the relative scopes assignedto the two wh-terms.

Which professor recommends which candidate?(22)

?x[P(x)∧?y[C(y) ∧ R(x, y)]](23)

?y[C(y)∧?x[P(x) ∧ R(x, y)]].(24)

It is easy to verify that both resulting formulae are equivalent. Anypair-list specification linking professors to the candidates they recommendanswers this question.

5.4. Negation

We have noted above that natural language provides no means of negatingquestions. We also noted that this entails additional generalizations suchas the fact that yes/no-questions may not be formed from another questionand that downwards monotonic quantifiers do not admit pair-list readings.Note that these empirical observations are independent of our particulartheory. We now proceed to propose an explanation for this.

Since natural language does not provide a negation operation on ques-tions, we can only speculate as to what this operation could do. We doso by looking at two relevant generalizations we can draw from the otheroperations on questions considered in this paper. First, all operations onquestions yield questions. Thus, it is reasonable to assume that the resultof negating a question would be a question too. Second, all these operationsapply to the resolvedness dimension rather than the truth dimension. It istherefore plausible that a negation operation on questions would reversethe resolvedness value, switching r to ur and ur to r. In fact, the bilatticeliterature contains such an operation, called conflation.

Accordingly, assume that questions could be negated. The result wouldbe once more a question, with a truth value that is the conflation of thetruth value of the original question.


What would an answer to such a question be? It would be a propositionthat makes the negated question resolved, and therefore makes the originalquestion unresolved. Such an “anti-answer”, as we may call it, would haveto make all the possible answers of the original question either false orunknown. Note that it is not enough for it to eliminate just some of thepossible answers, since there may be a structure in which an alternat-ive answer that was not eliminated is known to be true. Relative to thatstructure, the anti-answer would fail to make the original question unre-solved. On the contrary, eliminating just a subset of the possible answersto the original question just makes it partially resolved. It seems that theonly way in which a proposition may make the original question unre-solved is by denying that the question is possible, e.g., by contradicting itspresuppositions.

According to this view, the negation of a question would be a questionintended to elicit from the hearer a proposition denying the very validity ofthe original question. This appears to be an extremely strange and awkwardconstruct to imagine. It is thus perhaps not surprising that it does not appearto be available in natural languages.

This explanation may be used in deriving the unavailability of pair-list readings for downwards monotonic quantifiers. For instance, for (21),allowing the quantifier no candidate to have scope over the question wouldlead to a question, requiring the hearer to provide a proposition that wouldensure that for no candidate it is known which professor recommends her.It is hard to imagine such an answer. A similar explanation also works forother downwards monotonic quantifiers.13

6. CONCLUSION

We opened this paper with the question of whether questions are inherentlyintensional. In attempting to answer this question, we have developed anextensional theory of questions. As we have seen it is possible to interpretthis theory so that is satisfies G&S’s adequacy criteria. In particular, itprovides a unified view of the semantic notions required by adequacy asinstances of a more general consequence relation. Our account is basedon rejecting the assumption of many previous theories that the meaningof a question should “encode” the meaning of its answers. Instead, wereinterpret the domain of type t as a bilattice, and assign questions a truthvalue.

13 As mentioned above, the full picture regarding the empirical availability of pair-listreadings is considerably more complex.


Interpreting questions as entities of type t also offers a simplifying viewon many problems in the semantics of questions including coordination,mixed indicative and interrogative sentences and quantification over ques-tions. We have also suggested a speculative explanation for the fact thatquestions may not be negated and some of its consequences, including thefact that downwards monotonic quantifiers do not admit pair-list readings.

So, are questions inherently intensional? As we have seen, our ex-tensional theory is able to cover a wide-range of phenomena. However,the problem of spurious answerhood we encountered forces us to choosebetween either adopting an intuitionistic interpretation or rejecting thestrict compositional model-theoretic interpretation, in favor of a repres-entational approach. This seems to us to imply that questions requiresomething “extra”, which may be manifested either as intensionality, oras an intuitionistic interpretation, or as a loss of strict compositionality.The question of what this “extra” ingredient really is remains open.

The semantics of questions is a very wide field. We believe our theoryoffers an interesting framework for exploring additional issues in the se-mantics of questions including embedded questions, intensional questionsand the dynamics of question answering in dialog. A thorough understand-ing of these issues may provide a better answer to the question of whatdegree of intensionality is actually needed.

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Department of Computer ScienceThe Technion, Israel Institute of TechnologyHaifa 32000, IsraelE-mail: {nelken,francez}@cs.technion.ac.il

Bilattices and the Semantics of Natural Language Questions

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