1. science
2. social science
3. linguistics

Hurford disjunctions and the nature of alternatives∗
Ivano Ciardelli and Floris Roelofsen
April 16, 2015
Abstract
This paper brings together two recent lines of work in which disjunction plays a prominent role,
namely, on the one hand, work on so-called Hurford disjunctions, i.e., disjunctions where one
disjunct entails another, and on the other hand, work in alternative and inquisitive semantics
where disjunction has been argued to generate multiple propositional alternatives. We show
that each of these lines of research is relevant to the other.
On the one hand, we point out that Hurford effects are found not only in disjunctive declaratives, but also in disjunctive questions. These cases are not covered by the standard accounts
of Hurford phenomena, which crucially rely on a truth-conditional account of disjunction. We
show that inquisitive semantics makes it possible to generalize these accounts, yielding a unified
explanation of Hurford phenomena as they occur both in declaratives, and in questions—an
advantage which is not shared by other theories of disjunction.
On the other hand, we discuss how Hurford phenomena provide an empirical handle on
the subtle but fundamental differences existing between inquisitive semantics and alternative
semantics, providing insight into the notion of meaning and the nature of alternatives in the
two frameworks.
1
Introduction
Disjunction has played a prominent role in two recent lines of work in natural language semantics.
In each case, the focus has been on certain phenomena that are not directly captured by the
classical semantic treatment of disjunction, and these phenomena have been used to motivate
rather fundamental revisions/refinements of the standard view on meaning and the division of
labor between semantics and pragmatics.
The first line of work (e.g., Chierchia et al., 2009, 2012; Katzir and Singh, 2013; Meyer, 2014; Fox
and Spector, 2015) has paid special attention to disjunctions where one disjunct entails another.
Hurford (1974) suggested, based on example like those in (1) below, that such disjunctions are
infelicitous, a generalization that is now generally referred to as Hurford’s constraint.
(1)
a. #John is an American or a Californian.
b. #That painting is of a man or a bachelor.
c. #The value of x is different from 6 or greater than 6.
Gazdar (1979) noted, however, that apparent counterexamples to Hurford’s constraint can be generated in a systematic way, by disjoining two scalar items, one stronger than the other:
(2)
∗
a.
Mary read most or all of the books on this shelf.
This is work in progress. Comments more than welcome!
1
b.
c.
John and Mary have three or four kids.
Mary is having dinner with John, with Bill, or with both.
Chierchia et al. (2009, 2012) argue that Gazdar’s observation may be seen as evidence for the
existence of a local exhaustive strengthening operator, an operator that can in principle be inserted
at any node in the logical form of a sentence and which roughly has the semantic effect of only. For
instance, when applied to a scalar item like most it yields most but not all and when applied to three
it yields three but not more, i.e., exactly three. If such an operator is indeed part of the grammar,
the contrast between Hurford’s and Gazdar’s observations can be given a principled explanation:
Hurford’s constraint is generally in force, but can in some cases be obviated by local exhaustification
of the disjuncts. This account has been further refined by Fox and Spector (2015), who propose
certain constraints on the insertion of exhaustive strengthening operators in order to account for a
number of further empirical issues concerning Hurford disjunctions, originally put forth by Singh
(2008) and Gajewski and Sharvit (2012).
An important remaining question is why Hurford’s constraint would be in force to begin with.
Katzir and Singh (2013) and Meyer (2014) offer a natural explanation in terms of redundancy: in
the absence of exhaustive strengthening, a Hurford disjunction is inevitably equivalent with one of
its disjuncts, rendering the other disjunct, and the disjunction operation as such, redundant.
The second line of work that we will consider has focused on disjunctions appearing in nonassertive contexts, e.g., in questions, imperatives, conditional antecedents, and in the scope of modal
operators, and has come to the general conclusion that disjunction has the potential to introduce
sets of propositional alternatives. In a question, these alternatives are needed to capture the answerhood/resolution conditions of the question (e.g. von Stechow, 1991; Roelofsen and van Gool,
2010; Biezma and Rawlins, 2012), in the antecedent of a conditional they determine against which
backgrounds the consequent needs to be evaluated (e.g. Alonso-Ovalle, 2009), and in imperatives
and modal contexts they are needed to account for so-called free choice effects (e.g. Simons, 2005;
Alonso-Ovalle, 2006; Aloni, 2007; Aloni and Ciardelli, 2013).
This has motivated a move from the classical truth-conditional notion of meaning to a more finegrained notion, which does not associate sentences with single propositions, but rather with sets of
propositions. Within this general line of work, two more specific approaches have been taken. On
the first approach, originating in Hamblin (1973), the notion of a propositional alternative is taken
to be a primitive notion, i.e., it is not characterized in terms of yet more basic notions. For any
given sentence, the propositional alternatives associated with that sentence are simply determined
by the rules of the compositional semantics that is given for (the relevant fragment of) the language.
This take on propositional alternatives is characteristic for the framework of alternative semantics
(Hamblin, 1973; Kratzer and Shimoyama, 2002; Alonso-Ovalle, 2006, among others).
On the second approach, developed in recent work on inquisitive semantics (e.g., Ciardelli et al.,
2012, 2013), the meaning of a sentence is identified with the conditions under which it is supported
by a given information state. For instance, a declarative sentence like Bill sang is supported by an
information state s in case it follows from the information available in s that Bill sang. On the other
hand, a question like Did Bill sing? is supported in s in case it either follows from the information
available in s that Bill sang, or it follows from the information available in s that Bill didn’t sing.
That is, a declarative sentence is supported in an information state if the information available in
that state establishes that the sentence holds, while an interrogative sentence is supported if the
information available resolves the issue that the sentence raises. The meaning of a sentence is the
set of all information states in which the sentence is supported. Information states, in turn, are
modeled as sets of possible worlds, i.e., propositions, but now regarded as representing a body of
information rather than the content of a sentence. Thus, technically, the meaning of a sentence
2
in inquisitive semantics is a set of propositions, just like in alternative semantics. However, these
propositions are not directly viewed as the propositional alternatives associated with the sentence.
Rather, the propositional alternatives associated with a sentence are those information states that
contain just enough information to support the sentence, i.e., information states that support the
sentence and cannot be weakened without losing support. As a consequence, not just any set
of propositions can be the set of alternatives associated with a sentence. In particular, suppose
p contains strictly more information than q, i.e., p ⊂ q: then p and q cannot both contain just
enough information to support a given sentence, and thus, they cannot both belong to the set of
alternatives associated with it.1
The goal of this paper is to connect these two general lines of work, showing how each is relevant
to the other. First, we note that the effects that Hurford and Gazdar observed do not only obtain
in declarative sentences but also in questions. For instance, the question in (3a) below is just as
infelicitous as Hurford’s declarative (1a), and (3b) is as felicitous as its declarative counterpart (2a).
(3)
a. #Is John an American, or a Californian?
b. Did Mary read most of the books on this shelf, or all of them?
Clearly, we would hope—and expect—that these observations could be explained by the same
general principles that have been proposed to explain the original observations, namely, a ban
against redundant operations and the availability of local strengthening as a way of obviating
such redundancy. However, whether such an explanation is possible crucially depends on the way
disjunction is taken to operate in forming alternative questions. We will see that under the most
widely adopted assumptions about disjunction in alternative questions, as found in alternative
semantics and also in Karttunen’s (1977) classical account, the explanation does in fact not go
through. By contrast, we will see that inquisitive semantics provides a setting in which it is
possible to naturally extend the given explanation of Hurford effects to questions.
In the second part of the paper, we will explore whether this empirical discrepancy between
inquisitive and alternative semantics is connected to more fundamental differences between the
two frameworks. We will show that there is indeed a very direct connection between the distinct
predictions made in the domain of Hurford disjunctions and differences in the basic notions of
meaning and propositional alternatives that underly alternative and inquisitive semantics. In doing
so we hope to illustrate that the abstract logical features of a semantic framework can have very
concrete empirical repercussions.
Thus, on the one hand we will show that an inquisitive account of disjunction makes it possible
to obtain a more comprehensive account of Hurford phenomena, as they occur not only in declaratives, but also in questions. On the other hand, we will find that Hurford disjunctions provide an
empirical handle on the subtle but fundamental difference between the notions of meaning adopted
in alternative and inquisitive semantics.
The paper is organized as follows. First, in Section 2, we discuss the account of Hurford
disjunctions that has emerged from the work of Chierchia et al. (2009), Katzir and Singh (2013),
and others in somewhat more detail. Then, in Section 3, we show that in inquisitive semantics this
account can be extended in a natural way to the domain of questions, while in alternative semantics
such an extension is problematic. In sections 4 and 5 we connect this empirical difference between
inquisitive and alternative semantics to more fundamental features of the two frameworks. Finally,
Section 6 concludes.
1
A note is in place here to avoid confusion: the notion of semantic alternatives that is at stake here is different
both from the notion of focus alternatives (cf., Rooth, 1985) and from the notion of scalar alternatives (e.g., Horn,
1972; Chierchia et al., 2012).
3
2
Background on Hurford’s constraint
As mentioned above, Hurford (1974) observed that disjunctions where one disjunct entails the other
are generally infelicitous, based on examples like the following:
(4)
a. #John is an American or a Californian.
b. #That painting is of a man or a bachelor.
c. #The value of x is different from 6 or greater than 6.
Gazdar (1979) remarked that there is a certain recipe to systematically generate counterexamples to
Hurford’s generalization. Namely, if a phrase A[t] contains a certain scalar item t, then disjoining it
with the phrase A[t0 ], where t0 is a stronger scalar alternative for t, results in a felicitous disjunction.
(5)
a.
b.
c.
Mary read most or all of the books on this shelf.
John and Mary have three or four kids.
Mary is having dinner with John, or with Bill, or with both.
To account for these cases, Gazdar proposed to weaken Hurford’s generalization in a rather adhoc way. Recently, however, Chierchia et al. (2009, 2012) pointed out that in the context of the
so-called grammatical view on scalar implicatures that they advocate, the contrast can be given a
more insightful explanation. According to this view, effects that are usually attributed to scalar
implicatures are in fact not implicatures at all, in the standard, Gricean sense. Rather, they come
about because the logical form of a sentence contains occurrences of an operator exh, which behaves
very similar to the particle only: it strengthens the meaning of the node to which it applies, making
it exhaustive relative to the scalar alternatives for that node.
Now, notice that there is a systematic connection between the felicity of a Hurford disjunction,
and the availability of a scalar implicature/exhaustive interpretation for the weak disjunct. In the
disjunctions in (5), the weak disjunct may be given—be it by semantic or by pragmatic means—a
strong interpretation which no longer entails, and in fact rules out, the strong disjunct.
(6)
a.
b.
c.
Mary read most of the books.
not all
John and Mary have three kids.
not four
Mary is having dinner with John.
not with Bill, Sue, . . .
By contrast, the weak disjuncts in (4) cannot receive such a strengthened interpretation.
(7)
a.
b.
c.
John is an American. 6 not a Californian
That painting is of a man. 6 not a bachelor
The value of x is different from 6. 6 not greater than 6
Now suppose that Hurford’s original generalization was right, i.e., that there is indeed a general
ban against the application of the disjunction operator to two arguments one of which entails the
other. Then, the theory of Chierchia et al. (2009) gives us a perspicuous explanation of the contrast
between (4) and (5).
To see this, consider first a Hurford disjunction A or B of the felicitous kind exemplified in (5),
where A is the weak disjunct, and B the strong one. One possible logical form for the disjunction
is simply or(A, B), which is ruled out by Hurford’s constraint. However, the grammatical theory of
scalar implicatures allows for another logical form, namely, or(exh(A), exh(B)) in which the disjuncts
are exhaustified prior to the application of disjunction. Now, in each of the examples in (5), exh(A)
is incompatible with B, and thus, a fortiori, it is incompatible with exh(B). Since both exh(A) and
4
exh(B) are consistent, this ensures that there is no entailment between exh(A) and exh(B). Thus,
the logical form or(exh(A), exh(B)) satisfies Hurford’s constraint, which accounts for the fact that
the sentence is perceived as felicitous.2
Now consider a Hurford disjunction A or B of the infelicitous kind exemplified in (4). Again,
the basic logical form or(A, B) is ruled out by Hurford’s constraint. Moreover, in this case exhaustification does not improve the situation, leaving the meanings of the disjuncts unchanged.3 Thus,
the second logical form available for the sentence, or(exh(A), exh(B)), violates Hurford’s constraint
as well. Thus, in this case all logical forms for the sentence violate Hurford’s constraint, which
explains why the sentence is perceived as infelicitous.
Notice that, besides discriminating in a principled way between felicitous and infelicitous Hurford disjunctions, the theory of Chierchia et al. (2009) also makes a particular prediction about
the interpretation of felicitous Hurford disjunctions. Namely, it predicts that the only available interpretation for such a disjunction is one in which the disjuncts are interpreted exhaustively. This
turns out to be correct. Consider the following example from Chierchia et al. (2009).
(8)
Either John solved two exercises, or he solved all of them.
This sentence is unambiguously false if John solved exactly three out of five exercises. This witnesses
that, indeed, only the reading or(exh(A), exh(B)) is available for the sentence, and not the reading
or(A, B), under which the sentence would be true if John solved three out of five exercises.
So, the seemingly puzzling data concerning the felicity and interpretation of Hurford disjunctions
have a clear explanation under the assumption that (i) local exhaustification is possible, and (ii)
there is a ban against the application of disjunction to two arguments one of which entails the
other.4
Of course, given this explanation, the question that naturally arises is why such a ban would
be in force at all. An appealing hypothesis that has been proposed by Katzir and Singh (2013)
(see also Simons, 2001; Meyer, 2014) is that this has to do with structural redundancy, that is, with
the logical form of the sentence containing unnecessary material. More precisely, Katzir and Singh
(2013) propose the following principle:
Local redundancy: a sentence is deviant if its logical form contains a node O(A, B)
which is obtained by application of a functor O to two arguments A, B, and the outcome
is semantically equivalent to one of the arguments.5
Let us examine how this principle predicts Hurford’s constraint. Recall that, in classical semantics,
the meaning of a sentence A is taken to be a proposition |A|, which amounts to the set of worlds
where the sentence is true. A sentence A entails a sentence B in case B is true whenever A is
2
It is not clear to us whether the logical forms or(exh(A), B) and or(A, exh(B)) are also allowed by Chierchia
et al. (2009). For concreteness we will assume that they are not allowed, but our discussion will not hinge on this
assumption in any way.
3
This may be due to the fact that the relevant items are not scalar items, or it may be due to the particular structure
of the space of scalar alternatives. For instance, if the stronger alternatives to American include Californian, Texan,
etc., then these alternatives fully exhaust the denotation of the predicate. It is thus impossible to conjoin the predicate
with the negation of the stronger alternatives without ending up with a contradiction (cf., Fox, 2007).
4
As mentioned in the introduction, the theory of Chierchia et al. (2009) has been extended in Fox and Spector
(2015) in order to account for some additional empirical issues concerning Hurford disjunctions, first discussed by
Singh (2008) and Gajewski and Sharvit (2012). However, since these issues are orthogonal to our concerns in the
present paper, we will not discuss this work in detail.
5
Katzir and Singh (2013)’s proposal is relativized to a context of utterance c. Since context-dependency plays no
role in our discussion, we omit reference to contexts for ease of exposition.
5
true, i.e., in case |A| ⊆ |B|. Moreover, sentential disjunction is taken to yield the union of two
propositions, that is, |or(A, B)| = |A| ∪ |B|.6
Now, suppose that the logical form of a sentence contains a node at which disjunction applies
to two arguments A and B, where |A| ⊆ |B|: then, |or(A, B)| = |A| ∪ |B| = |B|. So, we have a node at
which a binary operator yields an outcome which is semantically equivalent with one of the inputs.
Thus, the given logical form fails the local redundancy test. In this way, Hurford’s constraint is
explained as a particular consequence of a more general ban against local redundancies.7
3
Hurford effects in questions
Work on Hurford effects has focused so far on declarative sentences. However, the same effects occur
in questions as well. Alternative questions corresponding to the infelicitous disjunctive declaratives
in (4) are equally infelicitous:8
(9)
a. #Is John an American, or a Californian?
b. #Is that painting of a man, or of a bachelor?
c. #Is the value of x different from 6, or greater than 6?
On the other hand, alternative questions corresponding to the acceptable disjunctive declaratives
in (5) are acceptable as well:
(10)
a.
b.
c.
Did Mary read most of the books on this shelf, or all of them?
Do John and Mary have three kids, or four?
Is Mary having dinner with John, with Bill, or with both?
Thus, the contrast exhibited by disjunctive declaratives extends to questions: a Hurford-type alternative question is felicitous only if the weak disjunct may be given an exhaustive interpretation
so as to break the entailment.
Moreover, recall from above that if a Hurford disjunction is felicitous, its unique interpretation
is the one resulting from an exhaustive interpretation of the disjuncts. This is also true for Hurfordtype alternative questions. For instance, the question in (11) presupposes that John solved either
exactly two exercises, or all of the exercises, and it asks which of these two possibilities holds.
(11)
Did John solve two of the exercises, or all of them?
6
We will focus on the sentential case for simplicity, but the explanation carries over to disjunctions of arbitrary
conjoinable types.
7
Meyer (2014) argues that it is also possible to derive Hurford’s constraint from a more global redundancy constraint. However, her proposal crucially relies on the assumption that every declarative sentence involves a covert
modal operator K, which is interpreted by default relative to the speaker’s epistemic state. For instance, if Bill says
“John left”, then this sentence is parsed as [KBill John left] and interpreted as saying that Bill believes that John
left. This assumption clearly raises several thorny issues, e.g., concerning embedding and propositional anaphora. In
particular, it is not clear how the account would extend to questions, which will be our main concern in the present
paper. We therefore focus our attention on Katzir and Singh’s local redundancy account.
8
Disjunctive questions in English can be interpreted in various ways, depending on how they are pronounced.
We will only be concerned with disjunctive questions that are pronounced with a prosodic phrase break before the
disjunction word, and a falling pitch contour on the final disjunct. Such questions are referred to as alternative
questions, in contrast with polar disjunctive questions which have no prosodic phrase break before the disjunction
word and a rising pitch contour on the final disjunct (see, e.g., Bartels, 1999; Pruitt and Roelofsen, 2013) and open
disjunctive questions which have a prosodic phrase break before the disjunction word and a rising contour on both
disjuncts (Roelofsen and van Gool, 2010).
6
What these data show is that Hurford’s constraint, i.e., the ban against application of disjunction
to two arguments one of which entails the other, concerns disjunction in general : not only when
it occurs in declaratives, but also when it is used to form questions. The assumption that such
a general ban is in force, together with the theory of Chierchia et al. (2009), accounts for the
observations in (9)–(11).
However, the important question that remains to be addressed is whether the general redundancy principle which was taken to explain the existence of Hurford’s constraint in declaratives
carries over to alternative questions as well. It is not obvious that it does, because the redundancybased explanation of Hurford’s constraint in declaratives crucially relied on disjunction being interpreted as the operation that yields the union of two propositions. Clearly, this cannot be the
role that disjunction plays in the formation of alternative questions: in this case, the propositions
expressed by the two disjuncts are not merged into one, but rather—it seems—they are kept apart,
each contributing a separate proposition to the meaning of the question.
In most accounts of alternative questions, this intuition is cashed out by letting the disjunction
in an alternative question pull together the propositions expressed by the two disjunctions into a
set of propositions. This idea can be implemented in a straightforward way in the framework of
alternative semantics (Hamblin, 1973). In this framework, a basic clause denotes a singleton set
consisting of the proposition that is classically associated with the clause: [[A]] = {|A|}. Disjunction
is still taken to perform union (see Alonso-Ovalle, 2006), but now at the level of sets of propositions.
So, for a disjunction of two basic clauses A and B, we get:9
[[A or B]] = {|A|} ∪ {|B|} = {|A|, |B|}
However, if this is indeed the role played by disjunction in alternative questions, the existence
of Hurford’s constraint in questions cannot be explained in terms of redundancy. To see this,
consider a disjunction A or B where B strictly entails A, that is, |B| ⊂ |A|. Then the meaning of the
disjunction, {|A|, |B|}, contains two alternatives, and is therefore distinct from the meaning of each
disjunct. Thus, even if one disjunct entails the other, this does not make either disjunct redundant.
One may perhaps suspect that in the particular domain of questions, Hurford effects could
result from a different principle, having to do with the interaction between information asked
and information presupposed. For, it is generally acknowledged that a question comes with the
presupposition that one of its possible answers is true (Belnap, 1966, among many others). In
alternative questions like those in (9), this presupposition is enough to actually establish one of the
answers to the question. Thus, a speaker asking such a question would be presupposing enough
information to actually resolve the question, which seems to be sufficient reason to regard the
question as deviant. However, this explanation does not carry over to Hurford-type alternative
questions with more than two disjuncts, such as (12).
(12) #Is John Russian, American, or Californian?
This question is as odd as its two-disjunct counterpart, (9a). However, in this case, presupposing
that one of the answers is true amounts to presupposing that John is either Russian or American.
This presupposition does not establish any answer to the question. Thus, the above argument
would not explain the infelicity of this question.
It is also often assumed that an alternative question does not only presuppose that at least one
of its answers is true, but that exactly one is (see, e.g., Biezma and Rawlins, 2012). But even this
9
For concrete accounts of alternative questions based on this treatment of disjunction, see for instance Roelofsen
and van Gool (2010); Biezma and Rawlins (2012); Uegaki (2014). A similar strategy also underlies the accounts of
Karttunen (1977); Karttunen and Peters (1979); Aloni and van Rooij (2002); Beck and Kim (2006).
7
stronger presupposition does not help with explaining the infelicity of (12). Assuming for simplicity
that American and Russian are mutually exclusive properties, the question would presuppose that
John is either a Russian, or a non-Californian American, and it would ask the addressee to establish
one of these two options. Thus, (12) would be predicted to be equivalent to the following question,
which is felicitous.10
(13)
Is John Russian, or a non-Californian American?
We are left wondering, then, why Hurford’s constraint should be in force in questions. This, in
turn, casts doubt on Katzir and Singh’s redundancy-based explanation of Hurford’s constraint in
declaratives as well: if this explanation were right, then we would not expect to find Hurford effects
in questions, contrary to what we observed.
We should not forget, however, that this skeptical conclusion only follows given a specific account
of the role of disjunction in alternative questions. A closely related but subtly different treatment
of disjunction (in alternative questions and more generally) has been proposed in the recently
developed framework of inquisitive semantics (Ciardelli et al., 2012, 2013, among others). As
mentioned above, in inquisitive semantics the meaning of a question is taken to lie in its support
conditions, that is, in the conditions under which the question is resolved in a given information
state.
For an illustration, consider a polar question whether A, denoted ?A: this question is resolved
in a state s if it follows from the information available in s that A is true, i.e., s ⊆ |A|, or if
it follows from the information available in s that A is false, i.e., s ⊆ |A|, where |A| is the settheoretic complement of |A|. Thus, the meaning of the polar question amounts to the following set
of information states:
[[?A]] = {s | s ⊆ |A| or s ⊆ |A|}
The meaning of a declarative sentence A is also taken to be a set of information states. In this
case, the set contains precisely those information states which establish A, i.e., those states which
imply that A is true. Thus, the meaning of a declarative A in inquisitive semantics, [[A]], is related
to the proposition that it is standardly taken to express, |A|, in the following way:
[[A]] = {s | s ⊆ |A|}
In inquisitive semantics, like in alternative semantics, sentences are also associated with a set of
propositional alternatives. However, in this case, the meaning of a sentence is not identified with
this set of alternatives; rather, the alternatives associated with a sentence A are defined as the
maximal elements of its meaning. Formally:
alt(A) = {s | s ∈ [[A]] and there is no t ∈ [[A]] such that t ⊃ s}
If A is a declarative sentence, the unique alternative associated with it will be the proposition |A|.
For a question, instead, the alternatives are those states which contain just enough information to
resolve the question. In the case of ?A, these states are |A| and |¬A|.
´ e, 2010; Roelofsen and van Gool, 2010) implement
Some accounts of alternative questions (e.g., Aloni and Egr´
the ‘exactly one true answer’ implication somewhat differently. For instance, Roelofsen and van Gool (2010) predict
that (12) is not equivalent with (13) but rather with (i) below:
10
(i)
Is John Russian, Californian, or a non-Californian American?
Despite this slightly different prediction, the oddness of (12) is left unexplained on this account as well.
8
Just like in classical semantics and in alternative semantics, disjunction is taken to perform the
union operation in inquisitive semantics as well. Suppose that A and B are two basic clauses such
that [[A]] = {s | s ⊆ |A|} and [[B]] = {s | s ⊆ |B|}. Then we have that:
[[A or B]] = [[A]] ∪ [[B]] = {s | s ⊆ |A|} ∪ {s | s ⊆ |B|} = {s | s ⊆ |A| or s ⊆ |B|}
Now, suppose A and B are logically independent, that is, neither of |A| and |B| is included in the
other. Then, each of |A| and |B| is a maximal element in [[A or B]], and we have that:
alt(A or B) = {|A|, |B|}
This shows that inquisitive disjunction, just like disjunction in alternative semantics, has the potential to ‘pull together’ the alternatives associated with the two disjuncts without collapsing them
into one, which makes it possible to account for the role of disjunction in alternative questions.
However, suppose now that B entails A, that is, |B| ⊂ |A|. In this case, the proposition |B| is
not a maximal element of [[A or B]], since |B| ⊂ |A| and |A| ∈ [[A or B]]. So, in this case, one of the
two disjuncts fails to contribute an alternative to the meaning of the disjunction as a whole. As a
result, the meaning of A or B has only one alternative, |A|, thus coinciding with the meaning of A
itself. More formally, in this case we have:
[[A or B]] = {s | s ⊆ |A|} ∪ {s | s ⊆ |B|} = {s | s ⊆ |A| or s ⊆ |B|} = {s | s ⊆ |A|} = [[A]]
This shows that, in inquisitive semantics, disjoining a clause with one that is strictly stronger is a
redundant operation. As a consequence, Katzir and Singh’s ban against redundant operations can
now be invoked to explain the presence of Hurford’s constraint in questions.
The conceptual reason behind this is that, in inquisitive semantics, the meaning of a question
is not identified with a set of alternatives, but rather with the set of all information states in which
the question is resolved. It is at this more basic level that disjunction operates, rather than at the
level of alternatives. What the clause for disjunction says is that, in order to resolve A or B, a state
must either establish A, or it must establish B. But if B entails A, then this just boils down to
saying that, in order to resolve A or B, a state must establish A. In this case, establishing B is not
really an ‘alternative’ to establishing A, because establishing B inevitably amounts to establishing
A as well. This is formally reflected by the fact that |B| is not an alternative associated with the
disjunction as a whole.
The ways in which classical semantics, inquisitive semantics, and alternative semantics differ
in their treatment of disjunction can be illustrated pictorially as follows. Suppose that A, B, and
C are basic clauses expressing the following propositions (in each case, the outer square represents
the entire logical space, and the shaded area is the set of possible worlds that the proposition
consists of):
B
A
C
Notice that A and B are logically independent, while C entails A and is inconsistent with B. Now
consider the disjunctions (A or B) and (A or C). These are treated in classical semantics, alternative semantics, and inquisitive semantics as depicted below. In the case of classical semantics, we
9
depict the propositions expressed by each disjunction, while in the case of alternative semantics
and inquisitive semantics, we depict the set of alternatives associated with each disjunction.
Classical semantics
Alternative semantics
Inquisitive semantics
A or B
A or C
In the case of (A or B), alternative semantics and inquisitive semantics yield the same result, differing from classical semantics: the propositions expressed by the two disjuncts surface as distinct
alternatives. On the other hand, in the case of (A or C), inquisitive semantics patterns with classical semantics rather than with alternative semantics: the proposition expressed by the stronger
disjunct, C, does not surface as a separate alternative for the disjunction as a whole.
Thus, inquisitive semantics strikes a balance between classical semantics and alternative semantics. On the one hand, if we disjoin two logically independent sentences in inquisitive semantics,
the disjunction does not conflate the propositions expressed by the two, but it keeps them apart as
distinct alternatives. This feature, shared by alternative semantics but not by classical semantics,
is necessary to account for the role of disjunction in alternative questions (as well as in imperatives,
conditional antecedents, and under modals, see the references in the introduction). On the other
hand, if we disjoin two clauses one of which entails the other, the disjunction simply boils down to
the weaker disjunct. This feature, shared by classical semantics but not by alternative semantics,
is necessary to explain the existence of Hurford’s constraint in terms of a ban against redundant
operations, and thus to extend the account of Katzir and Singh to cover a broader range of Hurford
effects, not only in declaratives but also in questions.
4
Hurford disjunctions and semantic structure
We have seen that both in alternative semantics and in inquisitive semantics, the meaning of a
sentence is a set of propositions. However, in alternative semantics, the propositions in this set are
directly thought of as the alternatives that the sentence introduces, while in inquisitive semantics
they are thought of as the information states where the sentence is supported. Thus, for instance, a
basic clause A has meaning {|A|} in alternative semantics, and {s | s ⊆ |A|} in inquisitive semantics.
For some applications, the difference between the two frameworks is immaterial. However, in the
previous section we saw that in the case of Hurford disjunctions, the predictions do come apart. In
this section, we will connect this concrete empirical discrepancy to a somewhat more abstract (and
more fundamental) difference between the two frameworks.
An crucial fact about inquisitive semantics is that it comes with a cross-categorial notion of
entailment. Just as in classical semantics, entailment simply amounts to inclusion: A |= B ⇐⇒
10
[[A]] ⊆ [[B]]. Moreover, again just like in classical semantics, the space of all meanings in inquisitive
semantics forms a lattice. That is, every two meanings [[A]] and [[B]] have a greatest lower bound
(meet) with respect to entailment, namely their intersection [[A]] ∩ [[B]], and a least upper bound
(join), namely their union [[A]] ∪ [[B]].11 In classical semantics, this structural feature of the space
of meanings is crucially exploited in the treatment of conjunction and disjunction. Namely, a
conjunction A and B is taken to express the meet of [[A]] and [[B]], and a disjunction A or B is taken
to express the join of [[A]] and [[B]]. Since the space of meanings in inquisitive semantics still forms
a lattice, precisely the same treatment of conjunction and disjunction can be given in this setting.
So, while inquisitive semantics enriches the classical semantic framework, it retains its fundamental
structural features, and therefore also allows us to preserve the essence of the classical treatment
of conjunction and disjunction.
Now, as discussed in detail in Roelofsen (2013); Ciardelli and Roelofsen (2014), this is not so for
alternative semantics, where the standard notion of entailment as inclusion does not give the right
results, and has not been replaced by a suitable alternative notion. As a consequence, the principled
treatment of conjunction and disjunction as expressing meet and join operations with respect to
entailment is also lost. While disjunction is still taken to yield the union of the alternative sets
associated with the two disjuncts, this operation no longer has the same status and the same logical
properties as in classical and in inquisitive semantics. In the case of conjunction, the situation is
even more problematic—taking conjunction to express intersection leads to undesirable results even
for the most basic cases (see Ciardelli and Roelofsen, 2014, for concrete examples).
These considerations are of a more abstract nature than the ones that we have been concerned
with above. Upon closer examination, however, they bring out precisely those structural features
of inquisitive semantics that are responsible for its success in accounting for Hurford’s constraint
in terms of redundancy. Let us see why.
Consider an arbitrary space of meanings M ordered by a suitable relation of semantic strength ≤.
Suppose the space hM, ≤i is a lattice, that is, suppose any two meanings M, M 0 ∈ M have a meet
and a join, which we will denote by M ∧ M 0 and M ∨ M 0 respectively. If our semantics is based on
such a space of meanings, then we can say that a sentence A entails a sentence B just in case the
meaning of A is at least as strong as the meaning of B:
A |= B ⇐⇒ [[A]] ≤ [[B]]
Moreover, we can associate conjunction and disjunction with the two lattice operations:
[[and]] = λM.λM 0 . M ∧ M 0
[[or]] = λM.λM 0 . M ∨ M 0
Now, it follows from the very definition of the join operation that, if M ≤ M 0 , then M ∨ M 0 = M 0 .
Thus, if our semantics is based on a space of meanings that constitutes a lattice, and if disjunction is
taken to express the join operation in this space, this is sufficient to predict redundancy in Hurford
configurations:12
if A |= B, then [[A or B]] = [[A]] ∨ [[B]] = [[B]]
Both classical semantics and inquisitive semantics instantiate this general scheme: in both cases,
semantic strength is captured by inclusion, and the join operation amounts to set-theoretic union.
11
The greatest lower bound of two meanings with respect to entailment is the weakest meaning that entails both,
and their least upper bound is the strongest meaning that is entailed by both. For more elaborate discussion of these
notions in the context of inquisitive semantics, see Roelofsen (2013).
12
Likewise, if conjunction is taken to express the meet operation, redundancy is automatically predicted when one
conjunct is stronger than the other, e.g., John lives in Paris and in France (see, e.g., Schlenker, 2008; Katzir and
Singh, 2013). As a consequence, the infelicity of such cases can also be explained in terms of redundancy in any
lattice-based semantics.
11
Thus, in classical and inquisitive semantics, Hurford disjunctions are redundant for the same fundamental reason: they involve a join operation on two arguments one of which is stronger than the
other.
This does not hold for alternative semantics. As we saw, the most fundamental problem is that
it is not clear how meanings should be compared in this setting, i.e., how they should be ordered in
terms of semantic strength. But suppose we manage to define a suitable ordering ≤ after all. For
basic clauses, whose meaning is a singleton set of propositions, we would want entailment to reduce
to the classical notion: {|A|} ≤ {|B|} ⇐⇒ |A| ⊆ |B|. But then, alternative semantics disjunction
cannot be a join operation. For, suppose that |A| ⊂ |B|. Then the least upper bound of {|A|} and
{|B|} would be {|B|}, but alternative semantics disjunction yields {|A|} ∪ {|B|} = {|A|, |B|} =
6 {|B|}.
Thus, the fact that alternative semantics does not predict redundancy in Hurford disjunctions is
directly connected to the fact that the account of disjunction as a join operation is not preserved.
5
Hurford disjunctions and the nature of alternatives
While both alternative semantics and inquisitive semantics associate sentences with sets of propositional alternatives, the assumed notion of alternatives is conceptually quite different in the two
cases.
In alternative semantics, the notion of a propositional alternative is a primitive notion and
there are no constraints on which kinds of sets of propositions count as proper sets of alternatives.
In inquisitive semantics, alternatives are defined in terms of the more basic notion of support:
the alternatives associated with a sentence are those propositions that support the sentence in a
minimal way. This more concrete characterization implies that sets of alternatives have to be of a
particular form: two alternatives are always logically independent, that is, one is never contained
in the other.
Let us refer to sets of propositions whose elements are pairwise logically independent as nonnested sets. In inquisitive semantics, then, unlike in alternative semantics, only non-nested sets of
propositions are regarded as proper sets of alternatives. Thus, alternative semantics makes more
meanings available than inquisitive semantics. However, these additional meanings, i.e., nested
sets of alternatives, seem impossible to express in languages like English. In principle, a Hurford
disjunction would be exactly the right kind of construction to express a nested set of alternatives.
But we have seen above that such disjunctions are felicitous only if they can be re-interpreted
in such a way that their set of alternatives actually becomes non-nested, and that, if such reinterpretation is possible, the resulting non-nested set constitutes the only available interpretation
for the disjunction.
This seems to indicate that there is something wrong with nested sets of alternatives as meanings: the sentences that are supposed to express such sets are either infelicitous, or re-interpreted
as expressing non-nested sets of alternatives. From the perspective of alternative semantics, this is
puzzling, since nested sets of alternatives are just as good as non-nested sets.
In inquisitive semantics, the puzzle does not arise, because nested sets of alternatives are not
available in the first place. Importantly, notice that such sets are not ruled out by some special
purpose constraint: rather, it simply follows from the way alternatives are construed in inquisitive
semantics that they are never nested. This means that from the perspective of inquisitive semantics,
what is special about Hurford disjunctions is not that they express some distinguished class of
meanings, but rather that they involve redundant disjuncts, which fail to contribute an alternative
to the meaning of the disjunction. As we discussed, this is precisely what makes it possible to explain
the need for re-interpretation and, in case this is not possible, the infelicity of such disjunctions.
12
6
Conclusion
We hope to have achieved two things in this paper. First, we hope to have shown the relevance of
inquisitive semantics to the theory of Hurford disjunctions, showing that a more general account of
Hurford effects, as they occur both in declaratives and in questions, becomes available if we combine
the theories of Chierchia et al. (2009) and Katzir and Singh (2013) with an inquisitive account of
disjunction—a result that seems difficult to obtain based on other existing accounts of disjunction.
Of particular importance here is that, in its treatment of disjunction, inquisitive semantics strikes
a subtle balance between classical semantics and alternative semantics.
Second, we hope to have shown how the concrete empirical discrepancy between inquisitive semantics and alternative semantics in the domain of Hurford disjunctions is connected to some more
abstract and more fundamental differences between the two frameworks. One of these differences
concerns the structure of the underlying semantic space: the essential structural features of the
semantic space assumed in classical semantics are preserved in inquisitive semantics, but not in
alternative semantics. The other difference that we discussed concerns the notion of propositional
alternatives: in alternative semantics this is a primitive, unconstrained notion; on the other hand,
in inquisitive semantics, alternatives are characterized in terms of the more basic semantic notion
of support, and it follows from this characterization that sets of alternatives have to have a certain
form—namely, one alternative can never be nested in another. We argued that both of these more
abstract differences between the two frameworks are directly connected to the different predictions
that they yield about Hurford disjunctions. Thus, besides contributing to the theory of Hurford
disjunctions as such, we hope to have illustrated that more abstract features of a semantic framework are not just important from a logical and philosophical point of view, but also crucial to its
empirical success.
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