ABSTRACT
Title of Dissertation: NEW PERSPECTIVES ON INQUISITIVE
SEMANTICS
Yichi Zhang, Doctor of Philosophy, 2022
Dissertation directed by: Associate Professor Eric Pacuit
? Department of Philosophy
Associate Professor Paolo Santorio
? Department of Philosophy
Inquisitive semantics offers a unified analysis of declarative and interrogative sentences
by construing information exchange as a process of raising and resolving issues. In this
dissertation, I apply and extend inquisitive semantics in various new ways. On the one
hand, I build upon the theoretical insight of inquisitive semantics and explore the prospect
of incorporating other types of content into our conception of information exchange. On the
other hand, the logical framework underlying inquisitive semantics is also of great interest
in itself as it enjoys certain unique properties and is thus worth further investigation. In
the first paper, I provide an account of live possibilities and model the dynamics of bring-
ing a possibility to salience using inquisitive semantics. This account gives rise to a new
dynamic analysis of conditionals, which is capable of capturing what I call the Extended
Sobel Inference. In the second paper, drawing on the fact that disjunction in inquisitive
semantics is understood as introducing a set of alternative answers to a question, I propose
a Questions-Under-Discussion-based account of informational redundancy to tackle various
Hurford sentences. In the third paper, I explore the prospect of cashing out the theoretical
intuition behind inquisitive semantics using a non-bivalent framework. I develop a new logic
which invalidates the Law of Excluded Middle just like inquisitive logic, but unlike inquisitive
logic, it employs a negation that vindicates Double Negation Elimination.
NEW PERSPECTIVES ON INQUISITIVE SEMANTICS
by
Yichi (Raven) Zhang
The Department of Philosophy
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
June, 2022
Advisory Committee:
Associate Professor Eric Pacuit, Chair
Associate Professor Paolo Santorio, Co-Chair
Associate Professor Fabrizio Cariani
Professor Valentine Hacquard
Associate Professor Alexander Williams, Dean?s Representative
? Copyright by
Yichi Zhang
2022
Acknowledgement
First and foremost, I would like to extend my deepest gratitude to my supervisors: Eric
Pacuit and Paolo Santorio. Without their guidance, extensive feedback, and unwavering
support, this dissertation would not have been possible. I am also immensely grateful to my
other committee members: Fabrizio Cariani, Alexander Williams, and Valentine, for their
feedback, encouragement and support.
Additionally, I would like to thank Steven Kuhn for providing me with very helpful
comments, and members of the logic group and the MEANING group at the University of
Maryland for valuable discussion.
Lastly, I am thankful to my colleagues, faulty and staff at the department of philosophy
for their support.
ii
Table of Contents
Acknowledgement iii
Table of Contents iii
Chapter 1: Introduction 1
Chapter 2: Weak Persistence 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 The Extended Sobel Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Weak Persistence: An Informal Sketch . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Weak Persistence Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 New Dynamic Strict Analysis of Conditionals . . . . . . . . . . . . . . . . . . . . 33
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Appendix: Formal Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 3: A QUD-Based Account of Redundancy 42
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Empirical Landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 A New QUD-Based Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Open Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 4: A Non-Bivalent Approach to Inquisitive Logic 72
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
iii
4.3 A Non-Bivalent Approach to Inquisitiveness . . . . . . . . . . . . . . . . . . . . . 88
4.4 The Logic of Pseudo-Complemented Propositions . . . . . . . . . . . . . . . . . 90
4.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography 113
iv
Chapter 1
Introduction
We use language for a multitude of purposes, and one common use is to exchange informa-
tion. Among the types of information we normally exchange, there are those that concern
what the world is like and the ways things are. In asserting ?Alice is at the party,? the
speaker proposes to make it a piece of common ground information that what the sentence
says is true (cf. Stalnaker, 1978). Whereas the focus on truth-conditional information re-
flects historical emphasis on declarative sentences and assertions, linguistic investigation into
questions (Hamblin, 1973; Karttunen, 1977; Groenendijk & Stokhof, 1984) and, in particu-
lar, the recent development of inquisitive semantics (Ciardelli & Roelofsen, 2011; Ciardelli
et al., 2013; 2018) have contributed to expanding the conception of information exchange.
According to inquisitive semantics, in uttering an interrogative sentence, the speaker is taken
not as suggesting that the world is in any particular way but rather as raising an issue by
putting forward alternative ways for it to be resolved, thereby orienting future exchange in
a certain direction. In uttering ?Is Alice at the party?? for instance, the speaker proposes
two different ways to update the common ground information: either Alice is at the party,
or she is not. It then invites the interlocutor to resolve this issue by settling on one of the
two alternatives. Information exchange, under inquisitive semantics, can thus be viewed as
a process of raising and resolving issues.
The main aim of this dissertation is to apply and extend inquisitive semantics in various
new ways. On the one hand, I build upon the theoretical insight of inquisitive semantics
and explore the prospect of incorporating other types of content into our conception of
1
information exchange. On the other hand, the logical framework underlying inquisitive
semantics is also of great interest in itself as it enjoys certain unique properties and is thus
worth further investigation.
Before describing my work in some more detail, let me provide a brief overview of inquis-
itive semantics. Formally, inquisitive semantics construes the semantic content of a question
as a set of Stalnakerian propositions (i.e., sets of possible worlds) each of which represents
a body of information that is capable of resolving the question under discussion. The idea
that the semantic content of a question is identified as a set of propositions is not unique
to inquisitive semantics. For instance, Hamblin (1973) takes a question to denote a set of
propositions each of which represents a possible answer to the question; similarly for Kart-
tunen (1977), each proposition represents a true answer to the question. While these other
accounts spell out the semantics of a question in terms of its answerhood, inquisitive seman-
tics does so in terms of resolution conditions. A set of worlds can contain enough information
so as to resolve a question even if the corresponding proposition is not normally viewed as
an answer to the question.
To elucidate, consider a universe that contains four worlds: AB,AB?, A?B, A?B?. In AB,
both Alice and Bob are at the party; in A?B?, neither of them is; in AB? and A?B, one but
not the other is. Under the Hamblin-style alternative semantics, the question ?Is Alice at
the party?? denotes the set {{AB,AB?},{A?B, A?B?}}. This set contains two propositions:
the proposition that Alice is at the party (i.e., {AB,AB?}), and the proposition that she
isn?t (i.e., {A?B, A?B?}). By contrast, under inquisitive semantics, in addition to the two sets
{AB,AB?} and {A?B, A?B?}, the set of propositions the question denotes also contains all
the subsets of these two sets such as {AB}, {AB?}, and so on. Given that the set {AB}
embodies the information that Alice and Bob are both at the party, the information contained
is enough to resolve the question ?Is Alice at the party??. Hence, this set is included in the
denotation of the question, even if ?Alice and Bob are both at the party? is not normally
conceived of as an answer to ?Is Alice at the party?? under alternative semantics. By taking
the resolution condition instead of answerhood as its central notion, inquisitive semantics
places additional constraints on what kind of set-theoretic entities can serve as the proper
denotation of questions: only sets of sets of worlds that are closed under the subset relation
2
can serve this purpose.
As a consequence of making the semantic denotation satisfy this closure property, inquis-
itive semantics imparts a well-behaved algebraic structure that facilitates finding a suitable
notion of entailment as well as defining other logical operations. In inquisitive semantics,
semantic entailment can simply be defined via the subset relation: ? ? ? iff J?K ? J?K, where
J?K and J?K are the denotation of ? and ?. For example, consider the question ?Who (of Alice
and Bob) is at the party??. With respect to the aforementioned universe consisting of the four
worlds AB,AB?, A?B, and A?B?, this question denotes the set {{AB},{AB?},{A?B},{A?B?},?}.1
Now recall that the question ?Is Alice at the party?? denotes the set that contains {AB,AB?},
{A?B, A?B?}, and all of their subsets. Given this, the denotation of ?Who (of Alice and Bob)
is at the party?? is indeed a subset of ?Is Alice at the party??. Thus, under inquisitive
semantics, the former question entails the latter. This is a desirable result since intuitively
what this entailment conveys is that every complete answer to ?Who (of Alice and Bob) is
at the party?? also provides an answer to ?Is Alice at the party?? (cf. Roberts, 1996).
In a similar vein, inquisitive semantics manages to provide simple set-theoretic definitions
for logical operations on pairs of questions. For instance, conjunction is simply defined as
set-intersection. The denotation of ?Who (of Alice and Bob) is at the party?? can be derived
from taking the intersection of the denotation of ?Is Alice at the party?? and that of ?Is
Bob at the party??; intuitively, the former wh-question is equivalent to conjoining the two
latter polar questions. Analogously, disjunction is defined as set-union. As such, to resolve a
disjunctive question such as ?Where can we rent a car, or who might have one that we could
borrow?? (Ciardelli et al., 2018, p.16.), it is enough to resolve one of the disjoined questions.
This ability to properly define entailment and other logical operations is one main advantage
of inquisitive semantics over other Hamblin-style alternative-based semantics (see Ciardelli
et al. 2017 for further discussion).
What the formal architecture of inquisitive semantics further enables is a unified analysis
of interrogative and declarative sentences. Both interrogative and declarative sentences
now denote a set of sets of worlds. For declarative sentences, this amounts to taking the
1The empty set is included because it is a subset of every set. Conceptually, the empty set represents
a body of information that is inconsistent. Assuming that the principle of explosion holds, an inconsistent
body of information will entail everything and as such will resolve every question.
3
Stalnakerian proposition that the sentence standardly denotes and then forming the set
that contains it and all of its subsets. By unifying the semantic denotation of declarative
and interrogative sentences, inquisitive semantics offers a straightforward way to capture
embeddings of interrogatives under propositional attitudes (Ciardelli & Roelofsen, 2015;
Ciardelli et al., 2018), as in (1) and (2):
(1) John knows who is at the party.
(2) John wonders whether Alice is at the party.
Furthermore, a unified account allows inquisitive semantics to supply an enriched notion
of conversational contexts. By construing conversational contexts as a set of sets of worlds
instead of a single set of worlds as under the more traditional analysis (e.g., Stalnaker, 1978;
Veltman, 1996), inquisitive semantics enables contexts to encode not only what facts have
been settled but also what issues have been brought up. As such, inquisitive semantics can
model information exchange that goes beyond the exchange of truth-conditional content.
The current project of exploring new perspectives on inquisitive semantics has yielded
three independent papers. In Weak Persistence, I articulate a notion of live possibilities
with the help of inquisitive semantics. In addition to conveying truth-conditional informa-
tion and raising issues, sentences in natural language can also draw attention to certain
possibilities. As a prominent example, epistemic might-claims, such as ?Alice might be at
the party?, can be conceived of as having the discourse function to highlight a possibility
or entertain a new one. Conversational contexts can likewise be enriched to represent what
possibilities have become salient in discourse. While similar ideas have been explored in the
past (e.g., Willer, 2013; see also, Westera, 2017), my current aim is to cash out this dy-
namics of bringing a possibility to salience in an inquisitive framework in the hope of better
integrating this notion of live possibilities with inquisitive content in the future.
In this paper, I present a framework that is capable of modeling the dynamics of bringing
a possibility to salience with the aim of capturing what I call the Extended Sobel Inference
(ESI) as illustrated by (3):
(3) If Alice comes, the party will be fun. But if Alice and Bob both come, the party won?t
be fun. ? Therefore, if Alice but not Bob comes, the party will be fun.
4
Despite the intuitiveness of this inference, ESI poses a challenge for two leading accounts
of conditionals: the variably strict analysis (Lewis, 1973; Stalnaker, 1968) and the dynamic
strict analysis (von Fintel, 2001; Gillies, 2007; Willer, 2017). While the latter fails to vin-
dicate ESI, the former licenses the additional inference to ?If Alice comes, then Bob won?t
come?, which does not seem to be a natural inference that people will normally draw. By
contrast, my account is able to validate ESI without validating the additional inference. I
adopt an enriched notion of conversational contexts that is capable of distinguishing live
possibilities that have become salient from plain possibilities. I then postulate the following
principle governing the dynamics of raising a possibility to salience:
Weak Persistence: When a plain possibility ? is brought to salience, past information
is preserved either in all the ?-possibilities or in all the ??-possibilities.
In a nutshell, when the possibility of Alice and Bob both coming to the party is brought to
salience by the second conditional in the sequence?i.e., ?If Alice and Bob both come, the
party won?t be fun??the past information embodied by ?If Alice comes, the party will be
fun? is preserved in all the possibilities where Alice but not Bob comes. This means that in
all the possibilities where Alice but not Bob comes, the party will still be fun, which thereby
vindicates ESI. Moreover, it does not validate the further inference to ?If Alice comes, then
Bob won?t come? and thus avoids the drawback of the variably strict analysis.
The second paper A Question Under Discussion Based Account of Redundancy
is focused on providing an account of informational redundancy that can adequately predict
the infelicity of various Hurford sentences. Hurford (1974) observed that disjunctions where
one disjunct entails the other are generally infelicitous:
(4) #John was born in Paris, or he was born in France.
To explain their infelicity, one common approach is to take Hurford disjunctions as in-
volving a truth-conditionally redundant constituent whose deletion has no effect on the
truth-conditional content of the whole sentence. Such a redundancy account has been of-
fered within an inquisitive framework (Ciardelli & Roelofsen, 2017; Anvari, 2021). To ex-
plain in brief, recall that under inquisitive semantics, declarative and interrogative sentences
uniformly denote sets of Stalnakerian propositions that are closed under the subset rela-
tion. Given that disjunction is viewed as a device to introduce alternatives and defined
5
via set-theoretic union, (4) denotes the set that contains the two Stalnakerian propositions
JJohn was born in ParisK and JJohn was born in FranceK as well as all of their subsets. But
give that JJohn was born in ParisK is itself a subset of JJohn was born in FranceK, what (4)
denotes in fact is just the set that contains JJohn was born in FranceK and all of its subsets.
In other words, (4) is equivalent to the simplification ?John was born in France?. As a result,
the first disjunct ?John was born in Paris? turns out to be redundant.
However, this analysis fails to explain why the disjunction in (5), which is an instance of
what I call conjunctive Hurford disjunctions, is perceived defective:
(5) #John was born in Paris, or he was born in France and Mary was born in London.
Indeed, no theory on the market can adequately capture the infelicity of conjunctive Hurford
disjunctions. To address this challenge, I follow Simons (2001) and propose an analysis that
utilizes the notion of questions under discussion (van Kuppevelt 1995; Roberts, 1996; Bu?ring,
2003). Similar to the inquisitive approach sketched above, I take disjunction as a device to
introduce alternatives in the sense that each disjunct is supposed to provide an answer,
and moreover a distinct answer, to some discourse question. But disjunction also embodies
additional information that is not covered by the standard inquisitive approach, namely that
each disjunct should also answer the discourse question ?in the same way?. I will show how
this analysis can capture the infelicity (and felicity) of a wide range of Hurford sentences.
Finally, as previously mentioned, the formal architecture and algebraic structure under-
lying inquisitive semantics is also of great interest in itself. In A Non-Bivalent Approach
to Inquisitive Logic, I aim to extend the study of inquisitiveness in this direction. The
central question here is whether there are other formal frameworks that can cash out the
theoretical intuition behind inquisitive semantics. One essential feature of inquisitive seman-
tics is failure of the Law of Excluded Middle (LEM)?that is, A ? ?A is no longer valid. In
inquisitive semantics, A ? ?A serves as the logical form of a polar question like ?Is Alice at
the party??. In the standard setting, what it means for a sentence to be valid is for it to
be completely informationally trivial: an utterance of it will have no effect on the conver-
sational context. But since A ? ?A raises a question, it is not completely informationally
trivial. Consequently, LEM is not valid in inquisitive semantics.
Standard inquisitive semantics invalidates LEM by employing an intuitionistic negation,
6
which as a consequence also invalidates Double Negation Elimination (DNE)?that is, ??A
no longer entails A. By contrast, I show that there is a different route adopters of the
inquisitive approach can take. I develop a new logic for modeling inquisitiveness which
employs a negation that validates DNE but not LEM. The framework is non-bivalent in the
sense that it simultaneously utilizes two non-complementary notions when defining semantic
satisfaction conditions: support and rejection. For instance, given a body of information,
an assertion is supported if it is settled true and rejected if it is settled false; a question
is supported if it is resolved by the body of information and rejected if it is incompatible
with the body of information, and in the latter case, asking the question will be deemed
infelicitous in the first place. For example, if it is already common ground that neither Alice
nor Bob is at the party, then the following alternative question will be rejected and deemed
infelicitous.
(6) Is Alice at the party?, or is Bob at the party??2
The two notions above are non-complementary because there can be sentences that are nei-
ther supported nor rejected by a given body of information. I present an algebraic semantics
for this logic via the so-called twist-structures. As a generalization, I show how this method
of constructing twist-structures can be systematically employed to convert bivalent systems
to many-valued ones. As such, it serves as a valuable tool in a philosopher?s toolbox.
2The upward and downward arrows are used to indicate intonation. An interrogative sentence can receive
different interpretations depending on the intonation pattern (see e.g., Roelofsen, 2015)
7
Chapter 2
Weak Persistence
This paper investigates an inference pattern that is intuitively valid in our ordinary reasoning
with conditionals. I call this inference the Extended Sobel Inference (ESI). I show that the
validity of ESI poses a problem for both the standard variably strict analysis of conditionals
(VSA) and the dynamic strict analysis of conditionals (DSA). Whereas DSA fails to vindi-
cate ESI, VSA, albeit validating ESI, validates another inference which does not appear to
hold under all circumstances. In response, I propose a new dynamic analysis of conditionals
by first devising a novel dynamic inquisitive framework that enables the modeling of infor-
mation dynamics associated with entertaining a new possibility. In particular, I formulate a
notion of weak persistence, which captures how past information is preserved in light of new
possibilities. Combined with a notion of postsuppositions, the current framework manages
to vindicate ESI without incurring the drawback of VSA.
Keywords: Conditionals, Sobel sequences, Dynamic strict analysis, Inquisitive semantics,
Update semantics, Postsupposition
8
2.1 Introduction
Consider the following inferences:
(a) If Alice comes, the party will be fun.
(1) (b) But if Alice and Bob both come, the party won?t be fun.
(c) If Alice but not Bob comes, the party will be fun.
(a) If Alice had come, the party would have been fun.
(2) (b) But if Alice and Bob both had come, the party wouldn?t have been fun.
(c) If Alice but not Bob had come, the party would have been fun.
Intuitively, the first two conditionals in each sequence entail the last one. Moreover, the
naturalness of such inferences appears unaffected by whether the conditionals involved are
indicatives, as in the case of (1), or counterfactuals, as in the case of (2). Let us call this
general form of inferences in our conditional reasoning the Extended Sobel Inference (ESI),
which we can schematically represent as ? > ?, (? ? ?) > ?? ? (? ? ??) > ?, where ?? > ??
abbreviates ?if ?, then (would) ??.
The inference is so called because its two premises constitute a Sobel sequence, that is,
a sequence consisting of, in the following order, two conditionals ? > ? and (? ? ?) > ??, as
exemplified by the two premises from (1) and (2). The fact that Sobel sequences are felicitous
suggests that antecedent strengthening?that is, the inference from ? > ? to (???) > ??fails
as a natural inference for conditionals (Lewis, 1973; Stalnaker, 1968; Willer, 2017).
The current observation concerning ESI suggests that although, given the failure of an-
tecedent strengthening, ? > ? alone entails neither (???) > ? nor (????) > ?, nonetheless,
by extending the premise into a Sobel sequence, we are warranted to infer either (???) > ?
or (? ? ??) > ? depending on how ? > ? is extended. For instance, if we follow up ? > ?
with (? ? ?) > ??, as in the case of (1) and (2), we can infer (? ? ??) > ?. Alternatively, if
we follow up ? > ? with (? ? ??) > ??, as in the case of (3) below, we can infer (? ??) > ?.
(3) (a) If Alice comes, the party will be fun. (b) If Alice comes but she isn?t in a good
mood, the party won?t be fun. ? (c) If Alice comes and she is in a good mood, the
party will be fun.
As it turns out, vindicating ESI is not an easy task. It causes different problems for two
9
leading accounts of conditionals, namely the variably strict analysis (VSA) and the dynamic
strict analysis (DSA). After briefly introducing these two accounts in ?2.2, I argue in ?2.3
that, on the one hand, DSA fails to validate ESI; on the other hand, although VSA validates
ESI, it further licenses an additional inference?i.e., the inference from ? > ? and (???) > ??
to ? > ???which, as (4) demonstrates, does not appear to hold universally:
(4) (a) If Alice comes, the party will be fun. (b) But if Alice and Bob both come, the
party won?t be fun. ? (c) If Alice comes, Bob won?t come.
Drawing inspiration from DSA, inquisitive semantics, and update semantics, I propose a
new dynamic account of conditionals that vindicates ESI without incurring the drawback of
VSA. The details of this framework need to wait for the second half of this paper, but here
is the general idea.
Standardly, the effect of assertions is to remove certain possibilities from a stock of shared
information (Stalnaker, 1978), and whenever a possible world is removed, it can never be
brought back. By contrast, one distinguishing feature of this new framework is that it allows
updates to be genuinely non-eliminative. This is meant in the following sense: when we
entertain a new possibility, worlds that have been previously eliminated can be resurrected.
For instance, in the case of (1), the utterance of the first conditional (1a) A > F eliminates
all worlds where Alice comes to the party and the party is not fun. As we entertain the new
possibility that Alice and Bob both come to the party with (1b), some of those previously
eliminated worlds can be resurrected. As a result, it is no longer the case that in all worlds
wherein Alice comes to the party, the party is fun. This in turn allows us to capture failure
of antecedent strengthening.
At the same time, the revival of worlds needs to be constrained in a way such that ESI
comes out valid. To this end, I postulate a constraint called weak persistence, which, for the
time being, can be characterized rather informally as follows:
Weak Persistence: When a hitherto unentertained possibility ? is introduced, past
information is preserved either in all the ?-worlds or in all the ??-worlds.
Essentially, what weak persistence does is afford two alternative ways for past information
(in particular, the information embodied by the first conditional in a Sobel sequence) to
10
be preserved in light of new possibilities. Thus, when the second conditional in a Sobel
sequence subsequently eliminates one of the two alternatives, we are warranted to either
infer (? ? ?) > ? or infer (? ? ??) > ?. In the case of (1), since the second conditional
?if Alice and Bob both come, the party won?t be fun? rules out the option of preserving
the information embodied by (1a) (viz., that all worlds where Alice comes are worlds where
the party is fun) in all worlds where Alice and Bob both come, the information must be
preserved in the other alternative, thereby licensing the inference to (1c).
2.2 Preliminaries
2.2.1 Variably Strict Analysis of Conditionals
The observation that Sobel sequences are felicitous causes trouble for any analyses of con-
ditionals that validate antecedent strengthening. One such prominent theory is the strict
material analysis of conditionals, according to which ? > ? is interpreted as ?(? ? ?) and
is thus evaluated true at a world w iff in all accessible worlds from w where ? is true, ? is
true. But since all the worlds where ? and ? are both true must be worlds where ? is true,
antecedent strengthening holds. Hence, as the folklore goes, conditionals cannot receive a
strict interpretation.
By contrast, antecedent strengthening fails under the Lewis-Stalnaker style variably strict
analysis (Lewis, 1973; Stalnaker, 1968, 1981). Under VSA, roughly, ? > ? is true at w iff ?
is true at all the closest accessible world(s) to w where ? is true. Following Lewis, we use
a system of similarity spheres to model closeness among worlds. A system of spheres is a
collection of nested sets of possible worlds centered on a world w at which the utterance is
evaluated. One world w? is closer to w than another world w?? iff w? is located in a sphere
that is a proper subset of the sphere in which w?? is located. As Figure 2.1 shows, since it
is possible that the closest worlds where Alice comes to the party (i.e., the A-worlds in S2)
are distinct from the closest worlds where Alice and Bob both come to the party (i.e., the
(A ?B)-worlds in S3), we do not have the entailment from A > F to (A ?B) > F . Hence,
antecedent strengthening fails under VSA, as desired.
11
B
S3 S2 S1 A
F
Figure 2.1: A system of spheres
2.2.2 Dynamic Strict Analysis of Conditionals
Within the dynamic tradition, conversation is understood as a process of updating com-
mon ground information. The meaning of a sentence is taken to be its update potential
on information states, and hence the name context change potential (CCP). A context or
information state is standardly construed as a set of worlds, and uttering a sentence has the
effect of eliminating from it worlds that are incompatible with the information embodied by
the sentence. For instance, uttering ?Alice is at the party? will eliminate every world where
Alice is not at the party from its input information state.
Under DSA, conditionals receive a strict interpretation with respect to a constantly evolv-
ing modal domain (von Fintel, 2001; Gillies, 2007; Willer, 2017, 2018). Gillies represents this
background modal domain via a system of spheres?just like the one depicted in Figure 2.1?
and he calls it a hyper-domain. Spheres in a hyper-domain represent information states, and
the innermost sphere represents the current modal center over which the universal quantifier
of the strict conditional ranges.1
The CCP of a conditional ?? > ?? comprises two parts: the CCP of its entertainability
presupposition ????, and that of its asserted content ??(? ? ?)?. The entertainability
presupposition ?? requires that the conditional?s antecedent is possible with respect to the
current modal center, i.e., the innermost sphere of a hyper-domain. As such, updating with
1The types of ordering used to generate systems of spheres depend on the type of conditionals under
evaluation: counterfactuals come with a standard similarity ordering whereas indicatives come with some
kind of relevance ordering (cf. Willer, 2017).
12
the entertainability presupposition expands the innermost sphere to one that contains a ?-
world. If the input modal center already contains a ?-world, then the update has no effect,
or as we will say idles. On the other hand, if the whole hyper-domain does not contain any
?-world, then the update returns the empty set, thereby signaling presupposition failure.
Next, updating on this new hyper-domain with the asserted content of ? > ?, namely
with the strict conditional ?(? ? ?), boils down to checking whether all the ?-worlds in
the current modal center are ?-worlds. If so, then the update idles; otherwise, the update
returns the empty set, thereby indicating discourse anomaly.
To illustrate how antecedent strengthening fails under DSA, consider the hyper-domain
depicted in Figure 2.2(a). Suppose the initial modal center is S1. When A > F is produced,
given that S1 does not contain any A-world, the modal center expands to S2 so as to satisfy
the entertainability presupposition ?A. Since all the A-worlds in S2 are F -worlds, updating
with the asserted content ?(A ? F ) idles. Since S2 does not contain any (A ? B)-world,
the modal center needs to expand further upon the utterance of (A ? B) > ?F , thereby
establishing S3 as the new center as shown in Figure 2.2(b). (The dashed lines represent
spheres that have been eliminated.) Given that all the (A ?B)-worlds in S3 are ?F -worlds,
updating with the asserted content ?((A ? B) ? ?F ) idles. The Sobel sequence is thus
predicted to be consistent, which means antecedent strengthening also fails under DSA.2
B B
S3 S2 S1 A S3 S2 S1 A
F F
(a) (b)
Figure 2.2: Domain expansion under DSA
2Additionally, DSA accounts for the infelicity of the so-called reverse Sobel sequence (von Fintel 2001):
#If Alice and Bob both come, the party won?t be fun. But if Alice comes, the party will be fun.
13
2.3 The Extended Sobel Inference
2.3.1 The Problem of Unwanted Worlds
Recall that the extended Sobel inference, as exemplified by (5), is of the form:
? > ?, (? ? ?) > ?? ? (? ? ??) > ?
(5) (a) If Alice comes, the party will be fun. (b) But if Alice and Bob both come, the
party won?t be fun. ? (c) If Alice but not Bob comes, the party will be fun.
The validity of ESI poses a serious challenge to DSA as it fails to vindicate this inference.
Consider the hyper-domain depicted in Figure 2.3 below, which, as we have seen previously,
is the output hyper-domain from the update with the first two conditionals in (5), i.e., A > F
and (A ?B) > ?F .
The problem concerns the second update with (5b). When the modal center expands
from S2 to S3 so as to satisfy the entertainability presupposition of (5b), some unwanted
worlds are absorbed into the modal center. In particular, there are worlds in S3, designated
by the shaded area in Figure 2.3, that make A true but both B and F false. Given the
presence of these worlds, updating with the asserted content of (5c), i.e., ?((A ? ?B) ? F ),
B
S3 S2 S1 A
F
Figure 2.3: A countermodel to ESI under DSA
When (A?B) > ?F becomes the first conditional, we instantly expand to S3 to accommodate the entertain-
ability presupposition ?(A?B). But since S3 contains some (A??F )-worlds, A > F can longer be satisfied.
Hence, reverse Sobel sequences are predicted to be inconsistent under DSA. For some other approaches to
reverse Sobel sequences, see, e.g., Moss, 2012; Lewis, 2018; Ippolito, 2020.
14
will eliminate S3, thereby yielding the empty set as its output. Given the existence of a
countermodel, ESI is not valid under DSA.3
That being said, one may wonder whether there is an easy fix for the existing dynamic
account. For instance, one could suggest that when the modal center expands to accommo-
date (5b)?s entertainability presupposition ?(A ?B), we do not simply expand from S2 to
S3 but rather to S2 ? (S3 ? JA ?BK). Perhaps, the thought is that the modal center should
expand in a minimal fashion such that only those worlds that contribute to fulfilling the en-
tertainability presupposition are added to the center. And since no new (A??B)-worlds are
included during expansion, all the (A ? ?B)-worlds in the modal domain are still F -worlds,
thereby vindicating ESI.
This solution, however, immediately encounters another difficulty. If the modal center
expands in the way just described, then we should expect (5a) and (5b) together to entail
the conditional ?if Bob comes to the party, then Alice will come? given that all the B-worlds
in S2 ? (S3 ? JA ?BK) are A-worlds. But this further inference is hardly intuitive. To avoid
this, perhaps one could further suggest that we should also add some (?A ? B)-worlds in
addition to the (A ?B)-worlds to the expanded modal center. But now it becomes rather
unclear what the criterion is for deciding which worlds should get added. Why is it the case
that only some (?A?B)-worlds but no (A??B)-worlds are added? There does not seem to
be a straightforward answer where one can simply read off the logical form of the conditional
to decide how the modal center expands.
Another rescue strategy is to attempt at capturing the ESI in (5) via some pragmatic
means, for example, as a scalar implicature. Since a conjunction and its conjuncts form
a scale (cf. Sauerland, 2004), the weaker alternative ?Alice comes to the party? can be
exhaustified to convey that the stronger alternative ?Alice and Bob both come? is false.
In other words, ?Alice comes to the party? can be strengthened to mean ?Alice but not
Bob comes to the party?. Now, if we postulate that exhaustification can occur locally (e.g.,
3This problem of unwanted worlds is related to the problem of intermediate worlds pointed out by Nichols
(2017). Nichols?s criticism is based on the observation that DSA has difficulty rendering two counterfactuals
of the form (A?B) > ?F and (A??B) > F simultaneously true (or assertible) given a particular setup, that
is, when ?more departure from actuality is required to make the antecedent of an earlier counterfactual true
than to make a subsequent counterfactual?s antecedent true but its consequent false? (p. 627). By contrast,
my current focus is on DSA?s inability to vindicate a desirable general inference with conditionals.
15
Chierchia, 2012)?more specifically, within the antecedent of a conditional?then we can
view the inference to ?If Alice but not Bob comes, the party will be fun? as a case scalar
inference.
The problem with this strategy is that it fails to generalize to ESIs that invoke two
alternatives where the weaker one cannot be exhaustified to mean the denial of the stronger
alternative. For example, consider (6):
(6) (a) If Alice is from France, the party will be fun. (b) But if she is from Paris, the party
won?t be fun. ? (c) If Alice is from France but not Paris, the party will be fun.
Although the inference still holds, the weaker alternative ?Alice is from France? cannot be
exhaustified to mean ?Alice is from France but not Paris?. This is attested by the Hurford
disjunction in (7a):
(7) (a) #Either Alice is from France, or she is from Paris.
(b) Either Alice comes to the party, or Alice and Bob both come to the party.
Compare (7a) to (7b). Disjunctions where one disjunct contextually entails the other are
generally infelicitous (Hurford, 1974). But when the weaker disjunct can be exhaustified so
that it no longer entails the stronger alternative, the disjunction becomes felicity (Chierchia,
2009). This is the case for (7b), but not for (7a). Hence, resorting to scalar implicature
alone cannot explain why ESI holds in full generality.
Here is a brief diagnosis of the problem of unwanted worlds encountered by the existing
dynamic account. DSA models incorporation of new possibilities via expansion of the modal
center. But given a lack of constraint on expansion, when the center expands, it may
absorb unwanted worlds, and as a result, too much old information becomes lost. One
the one hand, allowing for discarding some old necessity enables DSA to capture failure of
antecedent strengthening. As in the case of (5), discarding the old necessity A ? F (viz., the
necessity that all A-worlds are F -worlds) enacted by (5a) leaves room for the subsequent
utterance of (5b) to make (A ?B) ? ?F a new necessity. On the other hand, the validity of
ESI shows that when the center expands, it is hardly the case that all past information is
lost. The old necessity A ? F should somewhat persist even in light of the newly entertained
possibility. When the antecedent of (5b) introduces the new possibility A ?B, although the
16
previously established necessity A ? F no longer holds in all the (A ?B)-worlds, it should
nevertheless still persist in all the ?(A ?B)-worlds. And to say that A ? F holds in all the
?(A ?B)-worlds, given basic propositional reasoning, is to say that all the (A ? ?B)-worlds
are F -worlds,4 thereby licensing the inference to (5c).
Hence, to vindicate ESI, we need a principle governing information preservation in light
of new possibilities. This principle, as we now recall, is weak persistence:
Weak Persistence: When a hitherto unentertained possibility ? is introduced, past
information is preserved either in all the ?-worlds or in all the ??-worlds.
To illustrate, consider (5) again, repeated below:
(5) (a) If Alice comes to the party, it will be fun. (b) But if Alice and Bob both come, it
will not be fun. ? (c) If Alice but not Bob comes, the party will be fun.
Suppose we are speculating whether this weekend?s party will be fun. We know that whenever
Alice is at the party, the party is almost always fun. Hence, we establish (5a). So far, the
possibility that Alice and Bob both come to the party remains not salient to us, since, say,
Bob travels a lot. But as we remind ourselves that Bob is in town this week and become
aware that he might also come, we no longer have to fully endorse (5a) given that we can
establish (5b) as a new necessity. However, this is not to say that the past information,
namely, the necessity established by (5a), is completely lost. By weak persistence, when the
possibility that Alice and Bob both come becomes salient, our previous conclusion that the
party will be fun if Alice comes should still hold either in all the worlds where Alice and
Bob both come or in all the worlds where that is not the case. But since (5b) eliminates the
first option, it must be that in all the ?(A?B)-worlds, the party will be fun if Alice comes.
Hence, we can draw the conclusion that if Alice but not Bob comes, the party will be fun.
2.3.2 The Problem of Unwarranted Inference
The variably strict analysis does not face the problem of unwanted worlds which plagues
DSA, as the former only quantifies over the closest worlds where the antecedent is true.
4To see this, note that ?(A ?B) ? (A ? F ) is equivalent to (A ? ?B) ? F .
17
Consider Figure 2.3 again. Since the closest (A ? ?B)-worlds are all F -worlds, ESI holds
under VSA. Crucially, what renders ESI valid under VSA is the fact that the truth of the
two conditionals in a Sobel sequence forces a particular ordering: any system of spheres that
makes A > F and (A?B) > ?F true must force the closest A-and-B-worlds to be closer than
the closest B-worlds. Consequently, all the closest A-worlds must be ?B-worlds. And since
all the closest A-worlds are also F -worlds, VSA predicts the entailment to (A ? ?B) > F .
However, the particular ordering invoked by VSA also happens to vindicate the following
additional inference:
? > ?, (? ? ?) > ?? ? ? > ??
This additional inference is good for some counterfactuals, though not for all (see Klecha
2015, 2021). Moreover, it is clearly too strong for indicative conditionals. As I have men-
tioned in ?2.1, the inference from (4a) and (4b) to (4c) seems quite bad.
(4) (a) If Alice comes, the party will be fun. (b) But if Alice and Bob both come, the
party won?t be fun. ? (c) If Alice comes, Bob won?t come.
It has been suggested by Klecha (2015, 2021) that there are two types of Sobel sequences:
those that do license the inference to ? > ??, which Klecha deems as genuine Sobel sequences,
and those that do not, which he names Lewis sequences. For Klecha, all indicative versions
of Sobel sequences are indeed Lewis sequences as they do not license the further entailment
to ? > ??.5
The current observation is that the validity of ESI does not hinge on whether the Sobel
sequences under discussion are genuine Sobel sequences or are in fact Lewis sequences; ESI
is valid regardless. If so, then a unified explanation of the validity of ESI should not rest
on the particular ordering that is induced by the first two conditionals as in a genuine Sobel
sequence.6 The main aim of this paper is thus to explore such a unified explanation.
5As a side note, Klecha (2015, 2018) views Lewis sequences as instances of a more general phenomenon
of precisification. It remains to be seen whether his account can adequately capture the validity of ESI.
6This is not to say that I wish to completely dispense with any ordering, since as we shall see in ?2.5.4,
ordering can help in supplying a reservoir context from which previously eliminated worlds can be selected
to be revived. My contention is that an adequate analysis that captures ESI should not merely appeal to
some particular ordering alone.
18
2.3.3 Interim Conclusion
To take stock, the standard dynamic strict account fails to validate ESI because modal
expansion can potentially assimilate unwanted worlds; the variably strict analysis, albeit
validating ESI, also vindicates the additional inference to ? > ??, which intuitively does
not hold under all circumstances. In response, I aim to validate ESI by developing a more
sophisticated update system that can capture bringing a hitherto unentertained possibility
to salience without explicitly relying on any particular ordering. This update framework
enables us to formally cash out weak persistence, which in turn allows us to capture the
validity of ESI.
A brief note on presentation of examples in what follows: since my goal is to validate
ESI in cases where the additional inference to ? > ?? cannot be drawn, I will focus my
discussion on the indicative versions of ESI. I leave it open how to extend this analysis
to counterfactuals, which requires a detailed discussion of how similarity ordering can be
incorporated into my update framework.
2.4 Weak Persistence: An Informal Sketch
Before delving into details of my framework, I informally sketch how weak persistence will
be realized. To capture weak persistence, the formal apparatus needs to do two things:
Requirement A: It should afford a principled way to distinguish live possibilities that
have become salient from plain possibilities that have yet to be actively entertained;
Requirement B: It should be able to encode a set of alternatives, each of which provides
a complete description of the space of possibilities at the current stage of discourse. Such
a possibility space contains information about what has been settled, what possibilities
have become live, and how live possibilities are related.
The first requirement is needed for modeling the process of bringing a hitherto unentertained
possibility to salience, while the second is needed for codifying alternative ways to preserve
past information in light of new possibilities.
19
A formal framework that fulfills both requirements can then capture weak persistence.
Consider the party example again. Let Figure 2.4(a) represent the total body of informa-
tion after the utterance of the first conditional ?if Alice comes to the party, it will be fun?.
The outermost rectangle with rounded corners represents the total logical space. The inner
rectangle, which is partitioned into some A-worlds and some ?A-worlds, represents the cur-
rent space of possibilities. The dotted region represents the set of A-worlds that are also
F -worlds.7 After the utterance of the first conditional A > F , it becomes common ground
that all worlds where Alice comes are worlds where the party is fun. Hence, all the A-worlds
in 4(a) are in the dotted region.
As the possibility of Alice and Bob both coming to the party has yet to become salient in
4(a), accommodating the entertainability presupposition of the conditional ?if Alice and Bob
both come, the party won?t be fun? will resurrect some (A ? ?F )-worlds. This is reflected
in Figure 2.4 by the fact that in both 4(b) and 4(c), the total possibility space is extended
with an additional rectangle which comprises A-worlds but is nonetheless not contained in
the dotted region. Also notice that by making a new partition, both 4(b) and 4(c) are now
B
?B
A
?(A ?B) (b)
or
A
B
(a)
?B
A
(c)
Figure 2.4: A sketch of weak persistence
7To clarify, the dotted region does not represent the set of all F -worlds.
20
actively distinguishing between B and ?B. The difference between 4(b) and 4(c) concerns
whether the resurrected worlds are B-worlds or ?B-worlds. In 4(b), entertaining the new
possibility that Alice and Bob both come brings back some worlds where Alice comes but
the party is not fun, all of which are worlds where Bob also comes. By contrast, in 4(c),
entertaining the same antecedent again brings back some worlds where Alice comes but the
party is not fun, but this time, all of them are worlds where Bob does not come. As such,
we are presented with two alternative ways to preserve the information that all A-worlds are
F -worlds: the information is preserved either in all the B-worlds or in all the ?B-worlds.
This is how weak persistence can be realized.
Given weak persistence, here is how we can vindicate ESI without also vindicating the
additional inference from VSA. As it turns out, 4(c) is indeed incompatible with the asserted
content of the second conditional ?if Alice and Bob both come, the party won?t be fun?
because all the worlds where Alice and Bob both come in 4(c) are in the dotted region and
are thus worlds where the party is fun. Consequently, this alternative is discarded, which
leaves us with 4(b). Given that in 4(b), all the worlds where Alice comes to the party but
Bob does not are worlds where the party is fun, we successfully predict the entailment to
(A??B) > F . On the other hand, since 4(b) still contains some worlds where Alice and Bob
both come to the party, we do not get the additional entailment to A > ?B.
Now, both Requirement A and Requirement B can be fulfilled in a framework that enables
representation of alternatives. The connection to the second requirement is obvious. As for
the first requirement, we can avail ourselves of alternatives to differentiate between live and
plain possibilities in the following way. We can construe ? as a live possibility with respect to
a given body of information just in case the body of information makes an active distinction
between two alternatives: namely, ? and ??.
One such framework is inquisitive semantics (Ciardelli & Roelofsen, 2011; Ciardelli et al.,
2015, 2018). In inquisitive semantics, disjunction functions to supply a set of alternatives.
While inquisitive semantics was originally designed to provide a uniform analysis for both
declarative and interrogative sentences rather than modeling live and plain possibilities, we
can readapt the framework for our current purposes. More specifically, given that the Law
of Excluded Middle (? ? ??) is not valid under inquisitive semantics, we can exploit this
21
feature to distinguish a body of information where ? has become salient from one where ?
has yet to be entertained.
Be that as it may, we still need to make some significant modifications to the existing
inquisitive framework for present purposes. In particular, since alternatives will be used for
the dual purpose of representing live possibilities and providing alternative ways to preserve
past information, we need to employ alternatives at two different levels without conflating
them. This will then require us to move one level up on the set-theoretic hierarchy.
2.5 Weak Persistence Semantics
Here is a quick breakdown of this section. ?2.5.1 introduces some basic notions from standard
inquisitive semantics. Each of the subsequent subsections motivates and introduces one piece
of the machinery, which eventually leads us to a dynamic inquisitive framework capable of
capturing weak persistence. First of all, ?2.5.2 introduces dynamic modals alongside a notion
of refinement. Together, they allow us to model the dynamics of bringing a plain possibility
alive. In ?2.5.3, I ascend one level up on the set-theoretic hierarchy and introduce hyper-
contexts. Finally, ?2.5.4 defines a refinement operation that captures weak persistence at
the level of hyper-contexts. Throughout ?2.5, my presentation will be semi-formal and rely
heavily on diagrams. The formal details are left to the appendix. In ?2.6, I present a
new dynamic strict analysis of conditionals within this framework and elucidate how ESI is
validated.
2.5.1 Basic Inquisitive Semantics
We begin by introducing some basic notions from inquisitive semantics particularly pertinent
to us. First, we have the familiar notion of information states (e.g., Veltman, 1996). An
information state s is a set of possible worlds compatible with a certain body of information.
In standard possible world semantics, formulas are evaluated at worlds in terms of their
truth and falsity. In inquisitive semantics, formulas are evaluated at information states in
terms of support.
An information state s supports an atomic formula p iff p is settled true in s, that is,
22
p is true at every world in s. As an example, suppose our logical space W contains four
worlds: one where A and B are both true, one where A is true but B is false, one where
B is true but A is false, and one where neither is true. Among the four information states
depicted in Figure 2.5, only 5(a) supports A as well as B, since both formulas are true at
every world in the state {AB}; by contrast, 5(b) supports A but does not support B whereas
5(c) and 5(d) support neither A nor B. Additionally, the empty set which represents the
absurd information state is taken to support everything.
A state s supports a negation ?? iff no subset of s except for the empty set supports ?.
For example, 5(b) supports ?A, but it does not supports ?B since it contains a non-empty
subset that supports B, namely {AB}.
A state supports a conjunction iff it supports both of its conjuncts, and a disjunction
iff it supports either of its disjuncts. For example, 5(b) supports A ? ?A by supporting A,
but it does not support B ? ?B since it supports neither B nor ?B individually. As for
implication, a state s supports A ? B iff every subset of s that supports A also supports
B; hence, only 5(a) and 5(c) support A ? B since only for these two states do all of their
subsets that support A also support B.
Next, let us define a technical notion of contexts. Standardly, contexts are often construed
as sets of worlds (Stalnaker, 1978). However, in order to separate a context where A is a
live possibility from one where A is a mere plain possibility, additional structure is needed.
In inquisitive semantics, a context C is defined as a non-empty downward closed set of
information states?that is, for any information state s that belongs to C, any subsets of s
must also belong to C. We call a maximal element in C (i.e., an information state that is not
AB AB? AB AB? AB AB? AB AB?
A?B A?B? A?B A?B? A?B A?B? A?B A?B?
(a) (b) (c) (d)
Figure 2.5: Some examples of information states
23
a proper subset of any other states in C) an alternative in C. We can then use alternatives
to represent salient possibilities in a context.
Consider the four contexts depicted in Figure 2.6. For each context, only its alternatives
are explicitly drawn: for example, given that contexts are downward closed, 6(c) represents
the context {{AB},{AB?},?}, and 6(b) represents the context {{AB,AB?},{AB},{AB?},?}.
We say that a context is inquisitive iff it contains more than one alternative; 6(a) and 6(c) are
inquisitive whereas the other two are not. In standard inquisitive semantics, an inquisitive
context raises an issue by putting forth alternative ways for the issue to be resolved. Under
the current setting, an inquisitive context indicates awareness of the distinctions among
different alternative possibilities. Henceforth I will just describe the context itself as being
aware of these possibilities. Compare 6(b) and 6(c): although both contexts are made up of
the same two worlds, 6(c) is actively aware of the distinction between B and ?B whereas
6(b) is oblivious to this distinction.
We highlight two special contexts: C? and C?. For any given logical space W , the initial
context C? which represents a total lack of information and awareness is given by the power
set ?(W ). The absurd context C? which serves to signal discourse anomaly is identified with
{?}.
Contexts are connected via updates. I write C[?]u to denote the context resulting from
updating C with the formula ?.8 Updating C with ? amounts to collecting only those
information states in C that support ?, or equivalently, eliminating all states in C that do
not support ?. For example, updating the above context 6(b) with [B ??B]u eliminates the
AB AB? AB AB? AB AB? AB AB?
A?B A?B? A?B A?B? A?B A?B? A?B A?B?
(a) (b) (c) (d)
Figure 2.6: Some examples of contexts
8The subscript u is used to differentiate the present type of proper-updates from the type of refinement-
updates to be introduced in ?2.5.2.
24
only state in it that does not support B ? ?B, namely {AB,AB?}, thereby outputting 6(c)
wherein the distinction between B and ?B now becomes salient.
2.5.2 Introduce Dynamic Modals and Refinement
We introduce dynamic modals and a notion of refinement next. Together, they enable
us to model the dynamics of transforming a plain possibility into a live possibility. In
accordance with the existing dynamic approach to modals (cf. Veltman 1996, 2005), we
construe possibility and necessity modals (i.e., ? and ?) as tests that examine whether
certain global conditions are satisfied by the current body of information. We define the
support condition of a modal formula by an information state s relative to a context to which
s belongs. Given a context C, a state s in it supports ?? iff there exists an alternative in
C that supports ?; analogously, s supports ?? iff every alternative in C supports ?. Since
alternatives serve to represent live possibilities, what the above definitions amount to is
simply this: s supports ?? iff ? is a live possibility in C; s supports ?? iff every live
possibility in C is a ?-possibility.
To illustrate, let s be the state {AB}. Consider whether s supports ?B with re-
spect to the following two contexts from Figure 2.6: 6(b), i.e., {{AB,AB?}}, and 6(c), i.e.,
{{AB},{AB?}}.9 Given that there exists an alternative in (6c), namely {AB}, such that it
supports B, s supports ?B with respect to 6(c); given that the only alternative in 6(b),
namely {AB,AB?}, does not support B, s does not support ?B with respect to 6(c). On
the other hand, s supports ?A with respect to both 6(b) and 6(c) since every alternative in
them supports A.
To elucidate why we need a new type of update operation which I call refinement, let
us first try to define a notion of support at the level of contexts. As our first pass, we say
that a context C supports ? iff C[?]u = C, or equivalently, iff every state s in C supports ?.
The problem with this definition is that it fails to satisfy the following intuitively desirable
principle governing awareness.
Principle of Uncertainty: For any context C and formula ?, if C supports ?? but
9Here and hereafter, I will abbreviate a context using its set of alternatives.
25
does not support ?, then it must support ???.
To put it in plain language, if we deem ? possible but do not have enough information to
establish ?, then we must also deem ?? possible. However, the principle of uncertainty is
violated given the present setup. Let C be the following context: {{AB},{AB?, A?B, A?B?}}.
It is easy to verify that C supports ?A but not A, yet it fails to support ??A.
To modify the framework so as to satisfy the principle of uncertainty, I introduce a notion
of general updates [?] and conceive them as proceeding in two separate steps: C[?] ?=
C[?]r[?]u. Updating C with [?] first refines the input context with [?]r, which boils down
to updating C successively with [p??p]u for every atomic proposition p in ?. In inquisitive
semantics, formulas of the form ?p??p?, often abbreviated as ??p?, are understood as asking
the question about whether or not p. Analogously, under the current setting, refining C with
p amounts to entertaining whether or not p, thereby making a distinction between p and
?p. Refining C with [?]r is then tantamount to bringing every atomic proposition in ? to
salience in the post-refinement context.10 This post-refinement context is in turn updated
with [?]u as usual. As a consequence, we now define context-level support in terms of general
updates: C supports ? iff C[?] = C.
To illustrate, consider updating the first context in Figure 2.7 with ?(A ? B). The
general update with [?(A ?B)] is divided into two steps: first, the refinement on 7(a) with
[?(A ? B)]r, and then the update on the post-refinement context with [?(A ? B)]u. The
refinement with [?(A ?B)]r equates to the sequential refinement with [A]r and then [B]r,
each of which is then reduced to updating with a disjunction of the form [p??p]u. Updating
AB AB? [A]r AB AB? [B]r AB AB? [?(A ?B)]u AB AB?
A?B A?B? [A ? ?A]u A?B A?B? [B ? ?B]u A?B A?B? A?B A?B?
(a) (b) (c) (d)
Figure 2.7: An example of the general update procedure
10The version of refinement proposed here provides one rather simple way to satisfy the principle of
uncertainty, and as such it may need further fine-tuning. I shall leave exploring a more sophisticated
definition for refinement to another occasion.
26
the post-refinement context 7(c) with [?(A?B)]u eliminates the world AB and returns 7(d).
As an upshot of this modification, not all non-empty downward closed sets of information
states will be considered as proper contexts. For example, there is no update procedure that
can derive the purported context {{AB},{AB?, A?B, A?B?}}, which, as we have seen, violates
the principle of uncertainty. To amend this, we revise the definition of contexts as follows:
Contexts: Given a logical space W , a context C is a non-empty downward closed set of
information states that can be derived from performing certain general updates on the
initial context C? = ?(W ).
At the level of contexts, we can now distinguish live possibilities from plain possibilities: ?
is a live possibility in C if C[??] = C;11 ? is a plain possibility if C[??] ? C and C[??] ? C?.
Consider once more these two contexts from Figure 2.6: 6(b), i.e., {{AB,AB?}}, and 6(c),
i.e., {{AB},{AB?}}. While A is a live possibility (and indeed a necessity) in both of them
given that the update with [?A] idles for both contexts, B is only a live possibility in 6(c).
Updating 6(c) with [?B] idles; by contrast, updating 6(b) with [?B] returns 6(c). Hence,
B is merely a plain possibility in 6(b).
The current framework?s ability to represent live possibilities further addresses a puzzle
typically associated with the traditional dynamic approach to modals. As noted by Yalcin
(2007), if updating with ?A only returns either the input state or the absurd state, then
an utterance of ?A can never be truly informative, since either it does not provide any new
information as the update produces no effect, or the information carried by the sentence
cannot be consistently integrated. However, on many occasions, a might-claim is prima
facie non-trivial and appears to communicate new information, as the following examples
demonstrate (Yalcin, 2007, p. 1012):
(6) Cheerios may reduce the risk of heart disease.
(7) Late Antarctic spring might be caused by ozone depletion.
Whereas Yalcin proffers a pragmatic analysis for the seeming informativeness of these sen-
tences, the current framework is able to supply a semantic story (see also Willer, 2013),
11We could further impose the condition that C[??] ? C? if we want to avoid considering every sentence
a live possibility in the absurd context.
27
that is, sentences with an epistemic possibility modal like (6) and (7) can convey non-trivial
information by raising some hitherto overlooked possibility to salience and thereby refining
the common ground. Before the utterance of (6), the context is oblivious to the possibility
that Cheerios may reduce the risk of heart disease. The possibility is a plain possibility as
the common ground does not contain an alternative that supports ?Cheerios reduce the risk
of heart disease?, nor an alternative that supports its negation. After the utterance, the
possibility at issue becomes live as the updated context now contains both alternatives.
The refinement operation considered thus far is strongly persistent in the sense that it can
only raise a possibility to salience but can never bring back any previously eliminated worlds.
To capture failure of antecedent strengthening and weak persistence, our next step is to allow
refinement to revive ?dead? worlds. To do so, we will need to move to hyper-contexts.
2.5.3 The Lift to Hyper-Contexts
Recall our preliminary definition of weak persistence, which states that when a hitherto unen-
tertained possibility ? is introduced, past information is preserved either in all ?-possibilities
or in all ??-possibilities. To preserve past information, under the current setting, is for the
refinement operation not to reintroduce worlds that have been previously eliminated. Sup-
pose ? has already been established; then while entertaining ? can resurrect some ??-worlds,
either all of them need to be ?-worlds, or they all need to be ??-worlds. Hence, entertaining
a new possibility should create two alternatives, each of which embodies information about
how live possibilities are related?that is, either all ?-possibilities are still ?-possibilities, or
all ??-possibilities are still ?-possibilities.
At the level of contexts, however, we cannot adequately model relations between live
possibilities, since alternatives in contexts are already used to encode information about what
possibilities have become live. Say we wish to model a case where either all B-possibilities are
A-possibilities or all ?B-possibilities are A-possibilities but not both. However, no contexts
can encode such information. For instance, the context {{AB},{AB?},{A?B?}} is such that
all the B-possibilities in it are A-possibilities, and the context {{AB},{AB?},{A?B}} is such
that all the ?B-possibilities in it are A-possibilities; but if we simply take the union of the
two sets, the resulting context, i.e., {{AB},{AB?},{A?B},{A?B?}}, is such that neither all the
28
B-possibilities nor all the ?B-possibilities are A-possibilities. What we want instead is a set
where {{AB},{AB?},{A?B?}} and {{AB},{AB?},{A?B}} are its two alternatives. Therefore,
to enable live possibilities to be related in multiple ways, we need to move one level up on
the set-theoretic hierarchy from contexts to what I call hyper-contexts.
A hyper-context ? is a non-empty restrictively downward closed set of contexts, that is,
for any context C in ?, if x is a subset of C and x is itself also a context, then x is also in
?. Furthermore, as with (proper) contexts, not all non-empty restrictively downward closed
sets of contexts will be regarded as (proper) hyper-contexts because the update procedure,
to be defined below, will not always produce every such set. Hence, we define hyper-contexts
as follows, analogous to the revised definition of contexts:
Hyper-contexts : A hyper-context ? is a non-empty restrictively downward closed set
of contexts that can be derived from performing certain updates on the initial hyper-
context ??.12
We define alternatives in a hyper-context as maximal elements in it. While an alternative in
a context represents a single salient possibility, an alternative in a hyper-context represents
a set of salient possibilities, viz., a possibility space. Hence, a hyper-context that contains
more than one alternative indicates that there is more than one way for live possibilities to
be related.
As with contexts, general updates on hyper contexts proceed in two steps: ?[?] =
?[?]r[?]u. The (strongly persistent) refinement operation is the same as before: refin-
ing with [?]r amounts to refining with every atomic formula p in ?, which has the effect
of bringing every p to salience. As for proper updates, updating ? with [?]u amounts to
updating each individual context C in ? with [?]u and then taking the restrictive downward
closure of the resulting set. We say a hyper-context ? supports ? iff ?[?] = ?.
To adduce two concrete examples, first consider the update with [?A] on ?1 as shown
in Figure 2.8.13 This update consists of first refining ?1 with [?A]r?which is reduced to
12The initial hyper-context ?? is defined as the restrictive downward closure of the initial context C? with
respect to a given logical space W (see Appendix, Definition 13).
13The outermost circle in the diagram represents the hyper-context; the middle circles/ellipses represent
alternatives in the hyper-context; and the innermost circles represent alternatives in their respective contexts.
For example, ?1 contains C1 as its sole alternative, which in turn contains {A} and {A?} as its two alternatives.
29
[?A]u
A A? A A?
C1 ? {{A},{A?}} C1 ? {{A},{A?}}
[?A]u
A A
C2 ? {{A}} C2 ? {{A}}
A A? A A?
[?A]u
A? C? ? {?}
C ? {{A?}}
?1 ? {C1,C2,C3,C?}
3 ?1 ? {C1,C2,C3,C?}
[?A]u
C? ? {?} C? ? {?}
Figure 2.8: Updating ?1 with [?A]u
updating it with [?A]u?and then updating the post-refinement hyper-context with [?A]u.
Since ?1 already distinguishes between A and ?A, the refinement idles. Figure 2.8 then
details what the subsequent proper update looks like. Since only C1 and C2 contains A as a
live possibility, the updates on C1 and C2 idle whereas the updates on the other two contexts
return the absurd context. Collective the resulting contexts {C1,C2,C?} and then taking
the restrictive downward closure give back ?1. Therefore, since ?1[?A] = ?1, ?1 already
supports ?A.
Next, consider the update with [?A] on the same hyper-context ?1. As before, the
refinement with [?A]r idles. Figure 2.9 depicts the subsequent update with [?A]u. Among
the four contexts contained in ?1, only the update on C2 returns a non-absurd context.
Collecting all the output contexts returns the posterior hyper-context ? 143.
14Note that we do not output the absurd hyper-context ?? = {{?}} in this case. As such, the current
framework manages to address one irregularity commonly associated with the dynamic approach to necessity
modals. Since updating with [?A] on any contexts wherein some but not all information states support A
always returns the absurd state, it means that, whenever we are uncertain about the truth of some proposition
A, uttering something like ?must A? would be regarded as inconsistent. But this can hardly be the case.
Suppose we are uncertain about whether Alice is at the party. In this scenario, we should not expect ?Alice
must be at the party? to be inconsistent with the common ground information such that the update winds
up outputting the absurd state. On my account, we can thus accommodate this intuition at the level of
hyper-contexts.
30
[?A]u
A A? C? ? {?}
C1 ? {{A},{A?}}
[?A]u
A A
C2 ? {{A}} C2 ? {{A}}
A A? A
[?A]u
A? C? ? {?}
C3 ? {{A?}}?1 ? {C1,C2,C3,C?} ?3 ? {C2,C?}
[?A]u
C? ? {?} C? ? {?}
Figure 2.9: Updating ?1 with [?A]u
2.5.4 Weakly Persistent Refinement
At the level of hyper-contexts, we can now cash out weak persistence induced by the con-
ditional?s antecedent in terms of weakly persistent refinement. The refinement associated
with the entertainability presupposition of a conditional [??]r becomes weakly persistent
when the input hyper-context ? does not actively distinguishes ? from ??, that is, when
? does not support ??. When this happens, certain previously eliminated possibilities can
be reintroduced from a reservoir context CR, which I shall return to shortly; meanwhile, for
any alternative C in ??which, as we recall, represents a particular way of how live pos-
sibilities are related in ??if C supports ?, then refining with [??]r will split C into two
separate possibility spaces: one where all the ?-possibilities are still ?-possibilities, thereby
supporting ?? ?, and the other where all the ??-possibilities are still ?-possibilities, thereby
supporting ??? ? (see Appendix, Definition 19 for details).
The reservoir context CR is a contextually supplied set that may contain worlds that
have already been removed from ?. For the time being, I remain rather agnostic about how
CR is determined. But what CR essentially does is to supply a reservoir of worlds to be
reintroduced upon entertaining a new possibility. The intuitive appeal behind postulating
31
such a fallback position is that raising a new possibility should only allow reviving ?dead?
worlds that are in some sense relevant to the new possibility being entertained. In the party
example, for instance, while entertaining the possibility of Bob?s coming to the party allows
us to bring back some worlds wherein the party is not fun, it should not bring back worlds
wherein the basic laws of physics are different.15
Now, to illustrate this dynamics of raising a possibility to salience, consider the refinement
depicted in Figure 2.10.16 Suppose our input hyper-context is ?1 where it is established that
Alice will come to the party but the possibility of Bob?s coming to the party has yet to
be entertained. Since ?1 does not support ?B, the refinement with [?B]r will be weakly
persistent. That is, upon accommodating the entertainability presupposition of ?if Bob
comes to the party?, the sole alternative in ?1, namely C1, will split into two contexts C2
and C3, each of which not only becomes aware of B but also brings back a world that
was previously eliminated. Assume that the reservoir context CR in this case is simply
{{AB,AB?, A?B, A?B?}}; then both C2 and C3 will bring back a ?A-world. At the same time,
to satisfy weak persistence, the resurrected ?A-world must be either a B-world as in C2, or
a ?B-world as in C3. Taking the restrictive downward closure of the set containing C2 and
C3 yields the post-refinement hyper-context ?2. We thus obtain two ways of preserving past
information as witnessed by the two alternatives of ?2: either all the ?B-possibilities are
AB AB? A?B
AB AB? A?B
[?B]r C2 ? {{AB},{AB?},{A?B}}
AB AB? AB AB?
C ? {{AB,AB?}} AB AB? A?B? AB AB? A?B?1
C3 ? {{AB},{AB?},{A?B?}}
? ?1 ? {C1} ?2 ? {C2,C3}
?
Figure 2.10: An example of weakly persistent refinement.
15While I do not intend to provide a detailed account of how CR is determined in this paper, I believe this
is where invoking some sort of ordering?e.g., relevance ordering as employed by Willer (2017)?can help.
16The symbol ? in the figure represents taking the restrictive downward closure of a set. See Appendix,
Definition 12.
32
still A-possibilities as witnessed by C2, or all the B-possibilities are still A-possibilities as
witnessed by C3.
As it turns out, this post-refinement hyper-context ?2 from the refinement with [?B]r
is indeed identical to the posterior hyper-context after the general update [?B]r[?B]u.
This is so because, similar to what happens with the proper update depicted in Figure 2.8,
the proper update with [?B]u on each alternative in ?2 idles, which means applying the
restrictive downward closure at the end will give back the input hyper-context.
Before ending this section, let me mention one more observation we can account for by
appealing to the weakly persistent refinement induced by the antecedent of a conditional.
Consider the following conversation:
(8) John: Alice will come to the party.
Mary: But what if Bob comes?
Mary appears to be neither completely agreeing nor outright disagreeing with what John
just said. Instead, she seems to be asking John to reconsider his assertion in light of a new
possibility without explicitly affirming or denying it. We can account for this intuition as
follows: on the one hand, since entertaining a new possibility allows for revival of ?dead?
worlds, it explains why Mary?s utterance is deemed as urging John to evaluate his claim;
on the other hand, since John?s claim should somewhat persist given weak persistence, it
also explains why we tend to see Mary not as straightforwardly voicing a disagreement.
Here, we restrict attention to the weakly persistent refinement induced by the antecedent
of a conditional, but similar discourse effects can be produced by other expressions?e.g.,
by an epistemic might-claim. I shall leave exploring other triggering conditions for weakly
persistent refinement to future work.
2.6 New Dynamic Strict Analysis of Conditionals
2.6.1 Introduce Postsuppositions
With a formal framework which enables us to materialize weak persistence at hand, I present
a new dynamic strict analysis of conditionals in this section. In short, I construe a conditional
33
? > ? as inducing the following sequential update: [??][?(? ? ?)][??].17 The first two
updates resemble components from the existing dynamic strict analysis of conditionals, that
is, updating with ? > ? amounts to first updating with its entertainability presupposition
[??] and then with its asserted content [?(? ? ?)]. The last update with [??] involves
what I call the postsupposition of a conditional. It functions as a delayed test whose purpose
is to ensure that the posterior hyper-context after the update with the asserted content still
satisfies the entertainability presupposition of the very conditional.
This idea that a sentence carries a postsupposition has been invoked in the past for
various reasons to specify conditions that a relevant body of information needs to satisfy
only after it has been updated with the at-issue content of the sentence. (Brasoveanu, 2012;
Brasoveanu and Szabolsci, 2013; Condoravdi, 2015; Constant, 2012; Henderson, 2014). With
respect to the current setting, the idea that the posterior hyper-context should still satisfy
the entertainability presupposition of a conditional resonates with the idea that, in general,
update procedures should be more or less idempotent (Veltman, 1996; Yalcin, 2015). More
specifically, if one can utter a conditional to update a body of information consistently, then
one should also be able to utter the same conditional again immediately afterwards, and
the update should idle since the conditional should have already been supported. Hence,
if updating a hyper-context first with the conditional?s presupposition [??] and then with
its asserted content [?(? ? ?)] makes the entertainability presupposition ?? no longer
satisfiable, this suggests that the conditional cannot be felicitously uttered in the first place.
To guarantee that this does not happen, we attach a postsupposition [??] to the first two
updates.18
We can marshal some support for the idempotence of conditionals from the observation
that conditionals of the form A > ?A (as well as ?A > A) sound contradictory (cf. Wansing,
2014). This anomaly can be captured in the current framework with the aid of postsup-
position. The conditional A > ?A sounds contradictory because for any hyper-context ?,
17Although strictly speaking, the implication ? employed in the current framework is not the material
implication ?, I still views this analysis as a form of dynamic strict analysis since we are still quantifying
over all information states that are subsets of s when evaluating ?? ? at s.
18Additionally, it is worth mentioning that construing update idempotence as a postsupposition rather
than some precondition for successful update can address the overgeneration problem pointed out by Man-
delkern (2019). I leave a detailed discussion of this problem to another occasion.
34
?[?A][?(A ? ?A)][?A] = ??, where ?? represents the absurd hyper-context {{?}}. This
is so because for any ?, the update with [?(A? ?A)] necessarily eliminates every A-world
from it, thereby making the subsequent update with the postsupposition [?A] output the
absurd hyper-context.
2.6.2 Vindicate ESI
We can finally put different pieces of the formal machinery together to show how ESI is
vindicated. First, we define semantic consequence ??? as follows:
Semantic Consequence: ?1, . . . , ?n ? ? iff for any hyper-context ?, the sequential
update with [?1], . . . , [?n] yields a hyper-context that supports ?.19
To elucidate how ESI?viz., ? > ?, (? ? ?) > ?? ? (? ? ??) > ??is validated, a simple
example should suffice for present purposes. Consider the following simplified variant of
ESI:
?,? > ?? ? ?? > ?
What we have above is a special instance of ESI derived from substituting a tautology ? for
the first antecedent ?. As a concrete example, (9a) and (9b) together entail (9c):
(9) (a) Alice will come to the party, (b) although she won?t if Bob comes. ? (c) If Bob
doesn?t come, Alice will come to the party.
Here is how this inference can be captured on my account. Suppose our initial hyper-
context ??, as shown in Figure 2.11, is the set {{{AB,AB?, A?B, A?B?}}}.20 Updating ?? with
(9a) eliminates all the ?A-worlds in ?? and outputs ?1.
Since ?1 does not support ?B, updating with the entertainability presupposition of (9b),
namely [?B], triggers the weakly persistent refinement with [?B]r. This refinement is
exactly the same as the one we just saw in Figure 2.10, where the only alternative in ?1
is split into two contexts: {{AB},{AB?},{A?B}}, and {{AB},{AB?},{A?B?}}. The post-
refinement hyper-context is ?2, upon which the proper update with [?B]u idles.
19This is commonly known as the update-to-test consequence in the dynamic literature (cf. Veltman,
1996).
20I abbreviate this hyper-context using its set of alternatives.
35
AB AB? A?B
AB AB? [A] [?B]
AB AB?
A?B A?B?
AB AB? A?B?
??: ?1: ?2:
[?(B ? ?A)]
AB? A?B
[?B]
[??B][?(?B ? A)][??B] AB? A?B
AB? A?B?
?4: ?3:
Figure 2.11: An example showcasing how ESI is vindicated
The next step is to update ?2 with the asserted content of (9b), namely with [?(B ?
?A)]. Since ?2 is already aware of both A and B, the refinement with [?(B ? ?A)]r idles;
the update with [?(B ? ?A)]u then eliminates every context in ?2 that contains the world
AB. The posterior hyper-context is ?3.
Next, updating with the postsupposition of (9b), namely with [?B], keeps only the first
alternative {{AB?},{A?B}} but eliminates the second alternative {{AB?},{A?B?}}. This is so
because whereas B is a live possibility in the former alternative, it is deemed impossible in
the latter one. Hence, we obtain ?4 as our posterior hyper-context after the updates with
(9a) and (9b).
Finally, updating ?4 with the conditional (9c), that is, with the sequential update
[??B][?(?B ? A)][??B], idles. To elaborate, since ?4 already actively distinguishes be-
tween B and ?B, the update with [??B] idles; and since ?4 does not contain any (?A??B)-
worlds, the update with [?(?B ? A)] also idles. What this means is that updating ?? with
(9a) and (9b) yields a hyper-context that already supports (9c). Hence, ESI holds under the
36
current analysis.
Moreover, since ?4 still contains some B-world, it does not support ?B, which means the
additional inference from ? > A and (? ?B) > ?A to ? > ?B cannot be drawn. Therefore,
the current framework also manages to avoid the shortcoming of VSA.
2.7 Conclusion
I have argued that the existing dynamic strict account fails to validate the extended Sobel
inference due to a lack of constraints on how the modal center expands. The traditional
variably strict analysis, by directly appealing to a particular ordering, is capable of validat-
ing ESI, but in doing so, it also vindicates an additional unwarranted inference. In response,
I propose a dynamic inquisitive framework which is capable of modeling the operation of
raising a hitherto unentertained possibility to salience without directly relying on any par-
ticular ordering. Within this new framework, we are able to cash out a notion of weak
persistence, which I consider as an important property governing how information evolves
in light of new possibilities. By incorporating a notion of postsuppositions, the modified
dynamic strict analysis of conditionals presented here manages to vindicate ESI without
incurring the drawback associated with the variably strict account.
37
2.8 Appendix: Formal Definitions
Definition 2.8.1 (Information states). An information state s is a set of possible worlds
and a subset of the whole logical space s ?W .
Definition 2.8.2 (Support). We use ??? to denote support, specified as follows:
? s? p iff ?w ? s ? p is true at w, where p is atomic;
? s? ? ? ? iff s? ? and s? ?;
? s? ? ? ? iff s? ? or s? ?;
? s? ?? ? iff ?t ? s ? if t? ?, then t? ?;
? s? ?? iff ?t ? s ? if t ? ?, then t? ?.
Definition 2.8.3 (Downward closed). A set of states S is downward closed iff ?s?t ? if s ? S
and t ? s, then t ? S.
Definition 2.8.4 (Contexts). Given a logical spaceW , a context C is a non-empty downward
closed set of information states that can be derived from performing certain general updates
on the initial context C? ?= ?{W}.
Definition 2.8.5 (Alternatives in a context). An alternative in a context C is a maximal
element in C (i.e., an information state such that it is not a proper subset of any other states
in C). We use Alt(C) to denote the set of all alternatives in C.
Definition 2.8.6 (Support of modals by a state with respect to a context). Let sc be an
information state that is contained in the context C. Then,21
sc ??? iff ?s? ? Alt(C) ? s? ? ?
sc ? ?? iff ?s? ? Alt(C) ? s? ? ?
Definition 2.8.7 (General updates). C[?] ?= C[?]r[?]u.
21Alternatively, we could define support of modals directly at contexts instead of at information states
relative to a context. I adopt the above definition so that we can have a uniform definition of proper updates
for both modal and non-modal formulas, as given by Definition 8.
38
Definition 2.8.8 (Proper updates). C[?]u ?= {s ? C ? s? ?}.
Definition 2.8.9 (Refinement). We define the refinement operation recursively in terms of
the update operation as follows:
? C[p]r ?= C[p ? ?p]u, where p is atomic;
? C[??]r ?= C[?]r;
? C[? ? ?]r/[? ? ?]r/[?? ?]r ?= C[?]r[?]r.
? C[??]r/[??]r ?= C[?]r
Definition 2.8.10 (Context-level support). A context C supports a sentence ?, notated as
C ? ?, iff C[?] = C.
Definition 2.8.11 (Live and plain possibilities).
? ? is a live possibility in C if C[??] = C;
? ? is a plain possibility in C if C[??] ? C and C[??] ? C? (viz., C[??] ? {?});
? Otherwise, that is, when C[??] = C?, ? is an impossibility in C.
Definition 2.8.12 (Restrictive downward closed). A set of contexts X is restrictively down-
ward closed iff for all C ?X and for all x, if x ? C and x is a context, then x ? X. For any
given context C, ?C denotes the restrictive downward closure of C, i.e., a set of contexts
containing C and every sub-context of C. For any given set of contexts X, we write X? to
denote the restrictive downward closure of X, that is, ? ?C.
C?X
Definition 2.8.13 (The initial and absurd hyper-contexts). For any given logical space W ,
? The initial hyper-context ?? ?=?C?, where C? is the initial context w.r.t. W ;
? The absurd hyper-context ?? ?= {{?}}.
Definition 2.8.14 (Hyper-contexts). Given a logical space W , a hyper-context ? is a non-
empty restrictively downward closed set of contexts that can be derived from performing
certain updates on the initial hyper-context ??.
39
Definition 2.8.15 (Alternatives in a hyper-context). An alternative in a hyper-context ?
is a maximal element in ?, and we use Alt(?) to denote the set of alternatives in ?.
Definition 2.8.16 (General updates on hyper-contexts). ?[?] = ?[?]r[?]u.
Definition 2.8.17 (Proper updates on hyper-contexts). ?[?]u = {C[?]u ? C ? ?}?.
Definition 2.8.18 (Strongly persistent refinement on hyper-contexts). As before, we define
refinement recursively in terms of proper updates as follows:
? ?[p]r ?= ?[p ? ?p]u, where p is atomic;
? ?[??]r ?= ?[?]r;
? ?[? ? ?]r/[? ? ?]r/[?? ?]r ?= ?[?]r[?]r;
? ?[??]r/[??]r ?= ?[?]r.
Definition 2.8.19 (Weakly persistent refinement). Refining ? with the entertainability pre-
supposition of a conditional [??]r becomes weakly persistent when ????.22 The refinement
proceeds as follows:
? ?[??]r ?= ({C[?]r ? CR[pc]r[?] ? C ?Alt(?)} ? {C[?]r ? CR[p ?c]r[??] ? C ?Alt(?)}) ,
where [pc]r = [p1]r, . . . , [pn]r for every p such that C ??p, and CR is a contextually
provided backup context.
Remark. To unpack this definition, we split every context C belonging to the set of alterna-
tives Alt(?) into two separate possibility spaces: C[?]r?CR[pc]r[?] and C[?]r?CR[pc]r[??],
each of which represents a way of how past information is preserved. Take the alter-
native C[?]r ? CR[pc]r[?] as an example. Its first component C[?]r simply brings ? to
salience in the resulting context. The second component CR[pc]r[?] then reintroduces
some previously eliminated possibilities from the reservoir context CR. This is achieved
by first refining CR with [pc]r, which looks at every atomic proposition the input con-
text C is actively aware of and then makes CR also come to aware of these propositions.
22Here we restrict attention to weakly persistent refinement induced by the antecedent of a conditional.
I leave exploring other triggering conditions for weakly persistent refinement to future work.
40
As such, we do not lose any distinctions the input context C is already making. The
second update with [?] ensures that all the worlds from CR we are bringing back are
indeed ?-worlds. As a consequence, we preserve past information in all the ??-worlds.
Likewise, for the second alternative C[?]r ? CR[pc]r[??], past information is preserved in
all the ?-worlds. By conjoining the two sets and taking the restrictive downward closure
({C[?]r ?CR[pc]r[?] ? C ? Alt(?)}? {C[?]r ?CR[pc]r[??] ? C ? Alt(?)})?, we obtain a new
hyper-context such that every alternative in the original hyper-context ? is split into two:
one wherein past information is preserved in all the ?-worlds, thereby supporting ? ? ?,
and the other wherein past information is preserved in all the ??-worlds, thereby supporting
??? ?.
Definition 2.8.20 (Hyper-context support). ?? ? iff ?[?] = ?.
Definition 2.8.21 (Dynamic consistency). An update sequence [?1], . . . , [?n] is dynamically
consistent iff there exists a hyper-context ? such that ?[?1], . . . , [?n] ? ??.
Definition 2.8.22 (Dynamic consequence). ?1, . . . , ?n ? ? iff ?? ? ?[?1], . . . , [?n]? ?.
41
Chapter 3
A Questions-Under-Discussion-Based
Account of Redundancy
Hurford (1974) observed that disjunctions where one disjunct entails the other (e.g., #John
was born in Paris or in France) are generally infelicitous. To explain their infelicity, one
common approach is to take Hurford disjunctions as involving a truth-conditionally redun-
dant constituent whose deletion has no effect on the truth-conditional content of the whole
sentence. However, this approach faces difficulty in light of other variants of Hurford dis-
junctions (Marty & Romoli, 2022). Drawing on Simons (2001), I present a new account
of redundancy which utilizes questions under discussion (Roberts, 1996) and discourse trees
(Bu?ring, 2003). By postulating general constraints on the structure of discourse trees, I show
how the (in)felicity of various Hurford sentences can be accounted for.
Keywords: Hurford disjunction, redundancy, question under discussion, discourse tree
42
3.1 Introduction
Consider an example of so-called Hurford Disjunctions (HDs):
(1) #John was born in Paris, or he was born in France.
In a context where the interlocutors know that Paris is located in France, ?John was born in
Paris? contextually entails ? John was born in France?, and as (1) demonstrates, disjunctions
where one disjunct contextually entails the other sound odd. One explanation pins this
oddness down on the apparent redundancy HDs involve. Since ?John was born in Paris?
contextually entails ?John was born in France?, (1) is truth-conditionally equivalent to ?John
was born in France?. As a result, the first disjunct of (1) comes out as truth-conditionally
redundant: deleting it would have no effect on the sentence?s truth condition. Consequently,
by appealing to Grice?s (1975) maxim of manner?to be more specific, the submaxim of
brevity?we can attribute the oddness of HDs to the presence of apparently redundant
material.
A general notion of redundancy also helps to explain why certain HDs are nonetheless
felicitous. Such examples were first noted by Gazdar (1979). For example, consider (2):
(2) John read some or all of the books .
In (2), although ?John read all of the books? entails ?John read some of the books?, the
disjunction is nonetheless felicitous. One common explanation as to why disjunctions like (2)
remain unmarked is that the first disjunct is locally exhaustified to receive a strengthened
reading so that it is no longer entailed by the second disjunct (cf. Chierchia et al., 2012).
For example, ?John read some of the books? in (2) is exhaustified to mean that John read
some but not all of the books, and as such it is no longer entailed by ?John read all of the
books?. Given that the first disjunct is no longer entailed by the second, its deletion would
have an impact on the sentence?s truth condition. Consequently, disjunctions like (2) are no
longer regarded as containing any redundant material and are thus deemed felicitous.
Although the suggestion of invoking redundancy to explain the oddness of HDs appears
promising, working out the details is not an easy task. A general theory of redundancy
needs to capture not only the infelicity of HDs but also the infelicity (or felicity) of other
Hurford sentences in the vicinity. To illustrate, consider an example of what Marty and
43
Romoli (2022) call Quasi Hurford Disjunctions (QHDs):
(3) John was born in Paris, or he was born in France but not in Paris.
For some concreteness, let us evaluate (3) against the following constituent-based non-
redundancy account (Katzir & Singh, 2013; Marty & Romoli, 2022).
Constituent-based non-redundancy A sentence S cannot be used in context c if
there is any constituent X in S that is contextually equivalent to one of X?s subcon-
stituents.
On this account, (3) is predicted to be just as infelicitous as (1) because both sentences
contain ?John was born in France? as a subconstituent which is contextually equivalent
to the original disjunction. Hence, both (1) and (3) are considered containing redundant
material and thus deemed infelicitous. This prediction is contrary to our intuitive judgment:
unlike (1), (3) sounds perfectly natural. The constituent-based non-redundancy account fails
to distinguish the felicitous QHDs from the infelicitous HDs.
Moreover, as we shall see in ?3.2, other accounts on the market also run into trouble in
light of a wide variety of Hurford sentences. In particular, I will draw attention to disjunctions
like (4) which I call Conjunctive Hurford Disjunctions (CHDs):
(4) #John was born in Paris, or he was born in France and Mary was born in London.
Since (4) is not contextually equivalent to any of its subconstituents and nor does it contain
any constituent that is contextually equivalent to any of its subconstituents, it is not ruled
out on the constituent-based non-redundancy account. Thus, it is predicted to be felicitous,
contrary to our intuitive judgement. In fact, as we shall see, CHDs pose a thorny challenge
for almost all existing theories of redundancy.
To account for the (in)felicity of various types of Hurford sentences, I present an anal-
ysis that utilizes the notion of Questions Under Discussion (QUDs) (van Kuppevelt 1995;
Roberts, 1996; Asher and Lascarides, 2003) while still retaining the spirit of the redundancy
account. Under this approach, we understand assertions as responding to some discourse
question. For example, the assertion ?John was born in FranceF? with a focus marking on
?France? is normally understood as responding to the question ?Where was John born??.
Given this relation, we can recast the ban against redundant material as a general constraint
44
on how one should respond to discourse questions. When a sentence purports to offer multiple
answers to the same question, these answers must be distinct. In fact, an account along this
line has been suggested by Simons (2001). According to her, each disjunct in a disjunction
should provide a distinct answer to the discourse question so that none should ?contextually
entail? another. This paper aims to further develop such a QUD-based approach so that the
(in)felicity of various Hurford sentences can be properly captured. In particular, I propose
that the disjuncts in a disjunction are not only taken to provide distinct answers to the
same question?call this the distinctness constraint?but also understood as engaging the
question ?in the same way??call this the uniformity constraint. I will explicate this idea of
answering a question ?in the same way? using Bu?ring?s (2003) notion of discourse-trees.
Additionally, my account can be extended to explain why certain particles such as ?at
least? appear to repair an otherwise infelicitous HD, as (5) demonstrates:
(5) John was born in Paris, or at least he was born in France.
Past literature (e.g., Singh, 2008) often sets aside disjunctions like (5) as special cases that
do not contain a genuine disjunction and thus falls outside the explananda to be accounted
for. In this paper, I argue for a more unified analysis and show how such an account is
possible under the current QUD-based approach.
This paper proceeds as follows. In ?3.2, I review the literature on Hurford disjunction,
lay out the data points I plan to capture, and highlight the difficulties faced by the existing
redundancy accounts. I then offer my QUD-based analysis in ?3.3. By postulating general
constraints on the structure of discourse-trees, I shall demonstrate how the (in)felicity of
various Hurford sentences can be properly captured: ?3.3.1 focuses on the standard HDs
and other infelicitous variants, ?3.3.2 discusses the felicitous ones, namely QHDs, and ?3.3.3
extends the analysis to accommodate the repairing effect of scalar particles such as ?at least?.
?3.4 briefly discusses one open issue concerning Hurford questions. ?3.5 concludes.
45
3.2 Empirical Landscape
3.2.1 Quasi Hurford Disjunctions
Recall that according to the constituent-based account of redundancy, a sentence contains
redundant material if it contains a constituent that is contextually equivalent to one of its
subconstituents. This account, though simple and intuitively appealing, fails to explain the
contrast between the infelicitous HD in (1) and the felicitous QHD in (3). Both sentences
contain as a subconstituent ?John was born in France? which is contextually equivalent to
the original Hurford sentence. To explain this contrast, Marty and Romoli (2022) develop
a more sophisticated redundancy account which takes a sentence?s exhaustified meaning
into consideration. More specifically, they make use of the exhaustified interpretation a
disjunction ?p or q? can receive, namely the exclusive reading ?p or q, but not both?. To
explain why QHDs are felicitous with the help of exhausitification, they draw on Mayr and
Romoli?s (2016) explanation as to why a disjunction like (6) is felicitous even though it
also has a seemingly truth-conditionally redundant component and thus could potentially
be blocked by its simplification in (7):
(6) Mary isn?t pregnant, or she is (pregnant) and she is happy.
(7) Mary isn?t pregnant, or she is happy.
According to Mayr and Romoli, in order to determine redundancy, we should also compare
the exhaustified meaning of a sentence containing an allegedly redundant component to that
of its simplification without it, and only when the allegedly redundant component contributes
nothing to altering the sentence?s exhaustified meaning can it be truly deemed redundant.
Since the exhaustified interpretation of a disjunction (A ? B) is given by the exclusive
disjunction (A ?B) ? ?(A ?B), (6) receives the following strengthened reading:
(8) (Mary isn?t pregnant, or she is pregnant and she is happy) ??(Mary isn?t pregnant,
AND she is pregnant and she is happy) ? (6).
Given that the conjunction ?Mary isn?t pregnant, AND she is pregnant and she is happy?
is a contradiction and thus its negation a tautology, the exhaustified interpretation given by
(8) turns out to be equivalent to (6). In other words, exhaustification is vacuous in the case
46
of (6). By contrast, exhaustifying (7) yields:
(9) (Mary isn?t pregnant, or she is happy) ??(Mary isn?t pregnant, AND she is happy) ?
Either Mary isn?t pregnant, or she is happy, but not both.
What (9) rules out is the possibility that Mary is happy but not pregnant. This possibility is
not explicitly ruled out by the more complex disjunction in (6). As (6) and (7) differ in their
exhaustified meaning, the constituent ?she is (pregnant)? in (6) is deemed non-redundant.
Now, since ?Mary isn?t pregnant, or she is and she is happy? shares a similar structure
with the QHD ?John was born in Paris, or he was born in France but not in Paris?, Marty
and Romoli contend that we can as well appeal to exhaustification to explain why QHDs are
felicitous. Compare the exhaustified reading of the QHD in (10) to that of its simplification
in (11):
(10) John was born in Paris, or he was born in France but not in Paris.
(11) John was born in Paris, or he was born in France.
As with (6), exhaustifying (10) has no effect since the conjunction ?John was born in Paris,
AND he was born in France but not in Paris? is a contradiction. On the other hand, the
exclusive interpretation of (11) is given by (12):
(12) (John was born in Paris, or he was born in France) ??(John was born in Paris, AND
he was born in France) ? John was born in France but not in Paris.
Since (10) and (11) differ in their exhaustified meaning, the additional conjunct in (10) is
not redundant, and (10) is predicted to be unmarked.
However, Marty and Romoli?s solution faces several problems. First, unlike ?Mary isn?t
pregnant, or she is happy? which can be strengthened to mean ?Mary isn?t pregnant, or she
is happy, but not both?, ?John was born in Paris, or he was born in France? cannot be taken
to mean ?John was born in France but not in Paris?. On one hand, it is highly doubtful that
the strengthened interpretation is empirically attested; on the other hand, the strengthened
interpretation would render the standard HD ?John was born in Paris, or he was born in
France? no longer infelicitous on Mayr and Romoli?s modified redundancy account. To see
this latter point, note that if we were to allow ?John was born in Paris, or he was born in
France? to mean ?John was born in France but not in Paris?, then ?John was born in Paris,
47
or he was born in France? and its simplification ?John was born in France? would differ in
their exhaustified meanings. As a result, ?John was born in Paris, or he was born in France?
would no longer be predicted to be defective.
Furthermore, even if we grant that the constituent ?but not in Paris? in (10) is not
redundant, we still have not adequately explained why (10) is felicitous. This is so because
even if (10) and (11) differ in their exhaustified meaning, (10) still has the same exhaustified
meaning as the further simplification ?John was born in France?. Even after exhaustification,
(10) and ?John was born in France? still share the same truth condition, that is, true iff
John was born in France. Hence, Marty and Romoli?s account would still predict (10) to be
infelicitous because of the availability of a simpler truth-conditionally equivalent sentence.
In short, while appealing to exhaustification may offer a solution to the puzzle concerning
why disjunctions such as ?Mary isn?t pregnant, or she is (pregnant) and she is happy? do
not involve redundancy, the same strategy cannot be replicated so as to capture the contrast
between felicitous QHDs and infelicitous HDs.
3.2.2 Long Distance Hurford Disjunction
A different way to modify the redundancy account is to check redundancy not at the global
level but at the molecular level (Katzir & Singh, 2013; Ciardelli & Roelofsen, 2017):
Molecular Non-Redundancy: A sentence S is deviant in a context c if S?s logical
form contains a node O(X,Y ) where O is a binary connective and X and Y are its two
arguments such that O(X,Y ) is contextually equivalent to either X or Y .1
On this account, HDs like (1) are still correctly predicted to be deviant: ?John was born in
Paris or in France? is contextually equivalent to ?John was born in France?. On the other
hand, it also predicts the QHD in (2) to be felicitous as desired: the whole disjunction ?John
was born in Paris, or he was born in France but not in Paris? is not contextually equivalent
to either of its disjuncts; likewise, the embedded conjunction ?he was born in France but not
in Paris? is not contextually equivalent to either of its conjuncts.
1Here, I follow Ciardelli and Roelofsen (2017) in formulating the constraint at the level of logical forms.
And to further simplify, let us assume that X and Y are both sentential/propositional constituents of S.
48
The molecular non-redundancy account, however, fails to explain why (13), which is an
example of what Marty and Romoli call Long Distance Hurford Disjunctions (LDHDs), is
infelicitous.
(13) #John was born in London or in Paris, or he was born in France.
Since none of the binary nodes can be replaced with one of its immediate constituents, the
molecular non-redundancy account incorrectly predicts it to be felicitous. One immediate
response is to introduce n-ary connectives:
Non-Redundancy with n-ary Connectives: A sentence S is deviant in a context c
if S?s logical form contains a node O(X1,X2, . . . ,Xn) where O is an n-ary connective
(n ? 2) such that O(X1,X2, . . . ,Xn) is contextually equivalent to a sentence derived
from deleting any constituent from X1,X2, . . . ,Xn.
Under the n-ary version, for example, since (13) is truth-conditionally equivalent to the
simpler ?John was born in London, or he was born in France?, the extra disjunct ?John
was born in Paris? now becomes redundant. One way to formally realize such an account
is via inquisitive semantics (Ciardelli & Roelofsen, 2017; Anvari, 2021; see also Ciardelli et
al., 2018). In brief, under inquisitive semantics, disjunction introduces a set of alternatives.
The LDHD in (13), when taken as a whole, introduces three alternatives: [John was born
in London], [John was born in Paris], and [John was born in France]. But since ?John
was born in Paris? entails ?John was born in France?, the alternative [John was born in
Paris] collapses into [John was born in France] and thus becomes redundant. That being
said, simply making disjunction n-ary and adopting inquisitive disjunction is still inadequate
because similar oddness resurfaces in sentences that contain a combination of disjunction and
conjunction, as in the case of conjunctive Hurford disjunctions.
3.2.3 Conjunctive Hurford Disjunctions
Recall the CHD from (4), repeated below as (14):
(14) #John was born in Paris, or he was born in France and Mary was born in London.
The infelicity of CHDs is problematic because, unlike HDs and LDHDs, there does not
appear to be any constituents that are truth-conditionally redundant at first glance. None
49
of the constituents in (14) appears deletable without affecting the truth condition of the
whole sentence. Nevertheless, the sentence is still perceived as defective.
To my knowledge, the only existing account that addresses this issue to some degree
comes from Simons (2001). She discusses the following example:
(15) Q: What kind of car does Jane drive?
A: #Either she drives a Subaru station wagon, or George drives a Toyota and she
drives a Subaru.
The answer takes the form of a CHD. According to Simons, although (15A) is not truth-
conditionally equivalent to any of its simplifications, it still in a sense contains redundant
material when viewed as an answer to the question in (15Q). Since the question concerns
the kind of car that Jane drives, the subconstituent ?George drives a Toyota? in (15A) con-
tributes nothing to answering this question. Therefore, with respect to the question (15Q),
(15A) is indeed as informative as its simplification ?Jane drives a Subaru?.2 Consequently,
(15A) is deemed infelicitous as an answer to (15Q).
To put it another way, what Simons essentially postulates is a distinctness constraint on
disjunctive answers. That is, each disjunct in a disjunction should provide a unique answer
to the topic question. Since the answers provided by the two disjuncts in (15A) with respect
to the question in (15Q) are not non-entailing, infelicity ensues.
However, Simons?s account only partially addresses the challenge posed by CHDs because
it fails to generalize to other discourse questions. For example, suppose that the discourse
question in (15) is ?Who drives what?? instead. Then, since ?George drives a Toyota? does
contribute as a partial answer to this question, (15A) is no longer as informative as the
simplification ?Jane drives a Subaru? with respect to this question. As a result, contrary to
our intuitive judgement, (15A) would be regarded as a perfectly natural answer. Similarly,
on Simons?s account, although the CHD in (14) is deemed infelicitous as an answer to the
question ?Where was John born??, it is again incorrectly predicted as felicitous with respect
to the question ?Who was born where??.
To address this inadequacy, I propose to modify Simons?s account in the following way.
2This is cashed out formally in Simons (2001) using Groenendijk and Stokhof?s (1984) partition semantics.
The detail does not concern us too much here.
50
The inability to account for the infelicity of (15A) with respect to the discourse question
?Who drives what?? stems from the fact that, on Simons?s account, the distinctness con-
straint only applies at the level of discourse questions. Given this, one solution is thus to
make the constraint also apply at a more local level. More specifically, ?Jane drives a Sub-
aru? and ?Jane drives a Subaru station wagon? can still be regarded as responding to the
same QUD (e.g., ?What does Jane drive??) even if this QUD is not the discourse question.
As such, we can still obtain a violation of the distinctness constraint. Moreover, the reason
why distinctness can be checked with respect to the question ?What does Jane drive?? even
if the discourse question is ?Who drives what?? instead is that, assuming Jane is a rele-
vant individual, the question ?What does Jane drive?? is inherently related to the discourse
question given that an answer to the former provides a partial answer to the latter. Hence,
answering the question ?What does Jane drive?? can be regarded as part of the inquiry
strategy to address the discourse question ?Who drives what??. On my account, the dis-
tinctness constraint applies not only at the level of the discourse question but also at the
level of subquestions contained in the strategy of inquiry that is used to tackle the discourse
question. To materialize this picture, I will make use of Bu?ring?s (2003) notion of discourse
trees and postulate an additional constraint called uniformity.
To take stock, existing accounts of redundancy fail to capture the whole range of data
concerning Hurford disjunctions. The table below summarizes the relevant data points dis-
cussed so far alongside how well the existing redundancy accounts handle these data.
Types of Hurford Disjunctions Are they felicitous? Constituent Molecular N -ary
Simple Hurford No 3 3 3
Quasi Hurford Yes 7 3 3
Long Distance Hurford No 7 7 3
Conjunctive Hurford No 7 7 7
As the table shows, none of the accounts on the market can adequately handle the full
range of Hurford disjunctions; in particular, they all fail to explain why conjunctive Hurford
disjunctions are infelicitous. Building upon Simons (2001), I devise a more sophisticated
QUD-based account that manages to capture these data. As we shall see in ?3.3.1, I explain
51
why simple Hurford, long-distance Hurford and conjunctive Hurford disjunctions are all
infelicitous by appealing to two constraints on the structure of discourse trees: uniformity and
distinctness. These two constraints together help to cash out the thought that a disjunction?s
disjuncts should tackle a given QUD in the same way and each provide a distinct answer to
it. The felicity of quasi Hurford disjunctions will be dealt with in ?3.3.2 which requires the
postulation of additional constraints.
3.2.4 Hurford Disjunction with a Scalar Particle
I conclude ?3.2 by introducing another piece of data that I wish to capture. As mentioned,
it has been observed that certain scalar particles such as ?at least? (Singh, 2008) and ?even?
(Westera, 2020) can repair an otherwise infelicitous Hurford disjunction:
(16) John was born in Paris, or at least in France.
(17) John was born in France, or even in Paris.
Sentences like (16) and (17) have often been set aside in the literature on Hurford disjunction.
For example, in his brief explanation as to why ?at least? fixes an otherwise infelicitous HD,
Singh (2008) comments that the ?or? in ?or at least? is not a genuine disjunction; rather,
?or at least? should be better viewed as a form of correction: the speaker retracts what had
just been said proceeding it.
I disagree with this diagnosis as I think the ?or? in (16), as well as in (17), retains
its disjunctive force. For one thing, note that we can embed (16) in the antecedent of a
conditional and the resulting conditional still licenses the inference commonly known as the
simplification of disjunctive antecedent:
(18) Mary will date John if he was born in Paris or at least in France. ?
(19) Mary will date John if he was born in Paris.
(20) Mary will date John if he was born in France.
The inferences from (18) to (19) and (20) seem natural. However, if we were to construe
?or at least? simply as a device of retraction which renders (18) come out expressing the
same as (20), we would have difficulty explaining the inference from (18) to (19). This is so
because conditionals are not monotonic in the antecedent place as attested by the so-called
52
Sobel sequences (Lewis, 1973; Stalnaker, 1968; Willer, 2017). For instance, (21) does not
feel inconsistent:
(21) Mary will date John if he was born in France. But she won?t if he was born in Paris.
Since we cannot view (19) as being entailed by (20), the inference from (18) to (19) would
be left unexplained on the retraction account.
A more plausible approach, I think, is to treat the apparent disjunction in ?or at least?
as genuine disjunction with its disjunctive force intact. I will provide such an account in
?3.3.3. To explain the reason why the presence of ?at least? repairs what would otherwise
be an infelicitous Hurford disjunction, I shall once more invoke QUDs. In a nutshell, scalar
particles such as ?at least? and ?even? introduce alternative strategies of inquiry such that
the two answers ?John was born in Pairs? and ?John was born in France? will no longer
respond to the same QUD. Hence, even though ?John was born in Paris? entails ?John was
born in France?, no violation of the distinctness constraint occurs as they do not answer the
same QUD. Consequently, (16) and (17) are judged as felicitous.
3.3 A New QUD-Based Analysis
3.3.1 D-Trees and Core Constraints
In this section, I will show how the infelicity of HDs, LDHDs, and CHDs can be captured on
a QUD-based account of redundancy. As mentioned in ?3.2.3, in order to explain why CHDs
are infelicitous, I will modify Simons?s account by having the distinctness constraint checked
not only at the level of discourse questions but also locally at the level of subquestions.
According to Roberts (2012), discourse participants devise strategies to answer a given topic
question by breaking it down to more specific subquestions. Let us call Q? a subquestion of
Q iff Q entails Q?, that is, iff every complete answer to Q also provides a complete answer
to Q?.3 A subquestion can be broken down into further subquestions. Thus, QUDs enjoy
a hierarchical structure, and one convenient way to visually represent such a hierarchical
structure is to organize QUDs into a d(iscourse)-tree (Bu?ring, 2003).
3Alternatively, we can view Q? as a subquestion of Q iff a complete answer to Q? provides a partial
answer to Q. The detail does not concern us greatly here.
53
We define a d-tree as a partially ordered set of nodes, each representing a declarative or
an interrogative sentence. For present purposes, we may simplify by assuming that these
nodes stand in one of the two relations: (a) the question-and-subquestion relation, which
relates two interrogative sentences, and (b) the question-and-answer relation, which relates
an interrogative sentence to a declarative sentence.4 If an interrogative sentence dominates a
declarative sentence, then the latter must answer the former. In other words, we shall enforce
the question-answer congruence constraint as standardly postulated (Roberts, 1996; Onea
& Zimmermann 2019). Next, let us call any sub-tree of a d-tree that consists exclusively
of questions a QUD-tree. A QUD-tree represents a strategy of inquiry which breaks down
a superquestion into more specific subquestions, thereby laying out a particular way of
tackling the superquestion. To predict the infelicity of various Hurford sentences, I posit the
following core constraints (as for capturing the felicity of QHDs, additional constraints are
needed which will be laid out in ?3.3.2):
Uniformity: A disjunction?s disjuncts must evoke the same strategy of inquiry with
respect to the QUD answered by the whole disjunction.
Distinctness: Answers to the same question must be distinct in terms of non-entailment.
Consider the uniformity constraint first. It is supposed to capture the idea that the
disjuncts of a disjunction should answer a question in the same way. Given that strategies
of inquiry are represented by QUD-trees, uniformity says that the root question Q of a
disjunction should branch into two identical QUD-trees each of which is dominated by a
copy of Q. To illustrate with a simple example, consider the HD ?John was born in Paris,
or he was born in France? in (1). Suppose that this disjunction is uttered as a response
to the question ?Where was John born??. We can then depict its corresponding d-tree as
shown in Figure 3.1. The root question to which the disjunction responds in this case is
?Where was John born??. By uniformity, it branches into two copies of itself each of which
immediately dominates one disjunct of the whole disjunction. Since the two disjuncts fail
to provide non-entailing answers to the same question, distinctness is violated, and thus we
4Recent work on d-trees have postulated other types of relations such as dependent questions and po-
tential questions (see, e.g., Onea & Zimmermann 2019). It is possible to modify the current framework to
allow for other types of relations between nodes.
54
Where was John born?
Where was John born? Where was John born?
John was born in Paris. John was born in France.
Figure 3.1: D-tree for the Hurford disjunction in (1).
predict (1) to be infelicitous. In this example, each disjunct does not invoke complex inquiry
strategies; hence, only the root question itself is copied on each branch. In other cases where
complex strategies are evoked as in the case of the CHD shown in Figure 3.4 below, the
entire QUD-tree will be duplicated on each branch to observe uniformity.
To adduce a different example where the uniformity constraint is violated, consider (22A)
which is infelicitous as a response to (22Q).
(22) Q: Where was John born? And where was Mary born?
A: #John was born in Paris or Mary was born in London.
Uniformity straightforwardly explains why (22A) feels odd: since the first disjunct purports
to answer the first question whereas the second disjunct a completely different one, uniformity
is violated.
The current implementation of d-trees also provides a convenient way to tease apart trees
that correspond to disjunction and conjunction. Conjunctions such as (23) and (24) where
the two conjuncts differ in only one dimension are conceived of as generating a d-tree where
the two conjuncts are immediately dominated by the root question.
(23) John likes raspberries and strawberries.
(24) #John likes raspberries and berries.
For example, suppose that (24) is uttered as a response to the discourse question ?what fruit
does John like??; we can then draw the d-tree for (24) as shown in Figure 3.2. Intuitively,
when the two conjuncts of a conjunction differs only in one dimension, they each purport to
provide a partial answer to the QUD answered by the whole conjunction (cf. Roberts, 1996;
Jasinskaja & Zee, 2008; Riester, 2019). But since one conjunct in (24) entails the other, the
two answers fail to be distinct, and thus (24) is regarded as defective. While this simple
55
What fruit does John like?
John likes raspberries. John likes berries.
Figure 3.2: D-tree for the Hurford conjunction in (24).
treatment of conjunctive answers is not without any criticism (e.g., Onea, 2019), I shall stick
to it in this paper for it offers an elegant way to reflect the difference between disjunction
and conjunction in d-trees.
One last caveat regarding uniformity is that a disjunction will not always require branch-
ing. When branching is involved, each disjunct is considered as a separate answer to the
QUD addressed by the whole disjunction. But branching does not always occur. For exam-
ple, when the QUD is a polar question such as ?Does John like raspberries or strawberries???,
a positive answer ?yes? to this question will be immediately dominated by it without the
polar question splitting into further questions.
Now, as for the distinctness constraint, it is modeled after Simons (2001) and requires that
answers to the same question be distinct and non-entailing. The difference is that distinctness
is no longer just a constraint on disjunction with respect to the topic question but intended
as a general constraint on all question-answer pairs in a given d-tree. It dictates that no two
question-answer pairs in a d-tree can share the same question but have two answers where
one entails the other. This includes cases where the two answers are dominated by the same
node as in the tree from Figure 3.2 as well as cases where the two answers are dominated
by two different nodes where the two nodes just happen to denote the same question as in
the tree from Figure 3.1. Since both of them violate distinctness, we correctly predict the
Hurford disjunction in (1) and the Hurford conjunction in (24) to be infelicitous.
A similar story can be told as to why long-distance Hurford disjunctions are infelicitous.
This is where a QUD-based analysis starts to pay off. Consider the LDHD in (13), repeated
below as (25):
(25) #John was born in Paris or in London, or he was born in France.
Since the first disjunct of (25) is itself a disjunction, the question to which it responds will
split once more into two copies of itself. By uniformity, the QUD-tree on the left branch
56
Where was John born?
Where was John born? Where was John born?
Where was John born? Where was John born? Where was John born? Where was John born?
John was born in Paris. John was born in London. John was born in France.
Figure 3.3: One tentative d-tree for (25).
is again duplicated on the right, thereby yielding the QUD-tree in Figure 3.3. One small
wrinkle here concerns how we should connect the second disjunct ?John was born in France?
to the QUD-tree on the right branch, since there are two question nodes but only one answer
node. A full resolution of this issue is not required for our current purposes as it does not
play a significant role in explaining the infelicity of (25). For example, as Figure 3.3 shows,
we could leave the last question node unanswered, which in turn can be used to signify
that there isn?t a fourth possibility regarding John?s birthplace. Alternatively, if we consider
disjunction as n-ary rather than binary, we can split the root question into three copies of
itself and thereby circumvent the problem. In any case, since ?John was born in Paris? and
?John was born in France? respond to the same question, (25) is predicted to be infelicitous
as desired.
Move on to conjunctive Hurford disjunctions. We can associate different d-trees with the
CHD in (26) depending on how the discourse question is construed.
(26) #John was born in Paris, or he was born in France and Mary was born in London.
Suppose that (26) is uttered as a response to the question ?Who was born where??. In regard
to this kind of multiple wh-question, it is natural to treat the second disjunct ?John was born
in France and Mary was born in London? where the two conjuncts differ in two dimensions
as invoking contrastive topics. Following Bu?ring?s (2003) analysis of contrastive topics, we
can treat the two conjuncts of ?John was born in France and Mary was born in London? as
each responding to a different subquestion of ?Who was born where??. The resulting d-tree
is depicted in Figure 3.4. On the right branch ?John was born in France? and ?Mary was
born in London? are dominated by two separate subquestions of ?Who was born where??,
57
Who was born where?
Who was born where? Who was born where?
Where was John born? Where was Mary born? Where was John born? Where was Mary born?
John was born in Paris. John was born in France. Mary was born in London.
Figure 3.4: A d-tree for (26) with contrastive topics; note that the QUD-tree on the right is copied to the
left branch.
namely ?Where was John born?? and ?Where was Mary born?? respectively. By uniformity,
the QUD-tree on the right branch is copied to the left. Since the two occurrences of ?Where
was John born?? are associated with two answers where one entails the other, distinctness
is violated and (26) is predicted to be infelicitous.
On the other hand, we can also construct an alternative d-tree for (26) that does not
invoke contrastive topics. This is more natural when (26) is used as an answer to a single
wh-question such as ?Under what conditions do I win the bet??. This time, as Figure 3.5
shows, the question node ?Under what conditions do I win the bet?? that appears on the
right branch no longer further splits into additional subquestions but instead immediately
dominates the two conjuncts of the conjunctive answer ?John was born in France and Mary
was born in London?. Nonetheless, since ?John was born in Paris? and ?John was born in
France? are still two non-distinct answers to the same question, we correctly predict (26) to
be infelicitous with respect to this new discourse question as well.
Under what conditions do I win the bet?
Under what conditions do I win the bet? Under what conditions do I win the bet?
John was born in Paris. John was born in France. Mary was born in London.
Figure 3.5: A d-tree for (26) with a different discourse question.
58
3.3.2 Additional Constraints
The QUD-based analysis so far successfully captures the infelicity of simple Hurford disjunc-
tions, long-distance Hurford disjunctions, and conjunctive Hurford disjunctions. However, it
does not immediately explain why quasi Hurford disjunctions are felicitous. Consider again
the felicitous QHD in (3), repeated below as (27):
(27) John was born in Paris, or he was born in France but not in Paris.
To correctly predict (27) as felicitous, we need to block the d-tree depicted in Figure 3.6.
Where was John born?
Where was John born? Where was John born?
John was born in Paris. John was born in France. John wasn?t born in Paris.
Figure 3.6: A potential d-tree for (27) that needs to be ruled out.
Since this d-tree violates distinctness, were it to be allowed, we would incorrectly predict (27)
as infelicitous. One way to rule it out as an admissible representation for (27) is to constrain
how implicit QUDs can be reconstructed. Although the issue about how to reconstruct the
implicit QUD for a target sentence remains rather elusive, some headway has been made
(e.g., Riester 2019; Onea 2019). One constraint that I will appeal to in particular is the
following one postulated by Riester (2019):
Maximize-Q-Anaphoricity: Implicit QUDs should contain as much given (or salient)
material as possible.
The relevant notion of givenness is from Schwarzschild (1999) (see also Rochemont, 2016). To
give an example from Riester, among the three candidates QUDs in (29), (29c) is preferred
as the implicit QUD for (28) over (29a) and (29b) because it maximizes the given/salient
material in (28).
(28) He literally suffocated.
(29) (a) What happened?
(b) What about him?
59
(c) How bad was his condition?
The underlying intuition behind this constraint is that an implicit QUD reconstructed from
an answer should enjoy a certain degree of specificity with respect to its answer: an implicit
QUD should be maximally specific granted that it does not contain any new non-salient
material. To provide another example, between the two sentences in (31), (31b) seems to
serve as a better answer to (30) than (31a).
(30) John was born in Paris. What about Mary?
(31) (a) She was born in the UK.
(b) She was born in London.
This is so because the implicit QUD answered by ?John was born in Paris? can be readily
reconstructed as ?Which city was John born in??. As such, the what-about question will
be interpreted as asking ?Which city was Mary born in??, thereby making (31b) the more
suitable answer. Of course, this is not to say that a suitable answer to (30) will always
specifies a city. When the speaker cannot specify the city or when the city in which Mary
was born is not well-known, an answer that specifies a country would be more appropriate
due to other general Gricean considerations. Setting these cases aside, it does seem that a
proper answer to ?What about Mary?? should provide the same degree of specificity as that
of ?John was born in Paris?.
Imposing Maximize-Q-Anaphoricity now blocks the problematic d-tree in Figure 3.6 be-
cause ?Where was John born?? fails to maximize salient material when taken as an implicit
QUD for ?John was born in France? and ?John was born in Paris?. As the first answer
mentions a country while the second one a city, a better d-tree that accommodates this is
shown in Figure 3.7. The question ?Where was John born? is divided into ?Which country??
and ?Which city??, and by uniformity, this happens on both branches. As a result, the two
entailing answers ?John was born in Paris? and ?John was born in France? no longer re-
spond to the same QUD, which means distinctness is no longer violated. Hence, we correctly
predict the QUD in (27) to be felicitous.
One small complication concerns the leftmost answer node in Figure 3.7. It is enclosed in
a pair of parentheses because although the question node that dominates it is not explicitly
60
Where was John born?
Where was John born? Where was John born?
Which country? Which city? Which country? Which city?
(John was born in France.) John was born in Paris. John was born in France. John wasn?t born in Paris.
Figure 3.7: A d-tree for (27). The ?John was born in France? on the left is enclosed in parentheses as this
answer is not explicitly given but only contextually entailed by ?John was born in Paris?.
answered by the first disjunct of (27), an answer to this question, namely that John was
born in France, is entailed by ?John was born in Paris?. We could have alternatively left the
?Which country?? node on the left unanswered without affecting the core prediction of the
current analysis, but enclosing a contextually entailed yet not explicitly mentioned answer
in parentheses enables us to maximize the information embodied by d-trees. Also with this
change, answer nodes that are not explicitly mentioned do not figure into deciding whether
a violation of distinctness has occurred.
It is worth noting that the d-tree in Figure 3.7 is not the only available representation
for (27). For example, this tree can be further refined to be made compatible with existing
accounts regarding the implicit QUDs introduced by negated sentences. According to Tian
et al. (2010, 2016), a simple negated sentence such as ?John wasn?t born in Paris? assumes
a QUD that takes the form of a positive polar question, i.e., ?Was John born in Paris??, in
absence of other contextual information. By contrast, a cleft negation sentence such as ?It is
John who wasn?t born in Paris? assumes a negative wh-question, i.e., ?Who wasn?t born in
Paris?. They appeal to this difference to explain why the positive component of a negative
sentence appears to be represented only for simple negations but not for cleft negations. The
simple negative sentence but not the cleft sentence triggers the representation of the positive
component because only the former evokes a QUD that takes the form of a positive polar
question.
Assuming that a simple negative sentence invokes a positive polar question as its im-
mediate QUD, we can provide a corresponding d-tree as shown in Figure 3.8. Here, the
61
Where was John born?
Where was John born? Where was John born?
Was John born in France? Was John born in Paris? Was John born in France? Was John born in Paris?
(John was born in France.) John was born in Paris. John was born in France. John wasn?t born in Paris.
Figure 3.8: A different d-tree for (27) where the QUD that dominates the negated sentence ?John wasn?t
born in Paris? is construed as the polar question ?Was John born in Paris?.
QUD that dominates ?John wasn?t born in Paris? is now the polar question ?Was John born
in Paris??. The resulting d-tree still observes Maximize-Q-Anaphoricity. Additionally, the
implicit QUDs are made more specific compared to the ones from Figure 3.7. Despite this
difference in details, since ?John was born in Paris? and ?John was born in France? still
respond to two different QUDs, this new d-tree also predicts (27) to be felicitous.
Now, allowing the superquestion ?Where was John born?? to be divided into the sub-
questions ?Which country?? and ?Which city?? as in Figure 3.7 raises an additional problem.
If such an inquiry strategy is viable, then why doesn?t the vanilla HD ?John was born in
Paris or he was born in France? in (1) evoke this strategy? After all, if we were to adopt this
strategy and construct a tree accordingly as in Figure 3.9, we would incorrectly predict (1)
to be felicitous since distinctness would no longer be violated as ?John was born in Paris?
and ?John was born in France? now respond to different questions.
In order to rule out the d-tree in Figure 3.9 as an admissible representation for (1), I
Where was John born?
Where was John born? Where was John born?
Which country? Which city? Which country? Which city?
(John was born in France.) John was born in Paris. John was born in France.
Figure 3.9: A d-tree for (1) which would incorrectly render the Hurford disjunction felicitous.
62
postulate the following constraint:
Q-Quantity: Implicit QUDs should make the answer(s) appear as informative as pos-
sible. In other words, a d-tree that has fewer unanswered questions is preferable to one
that has more.
Q-quantity is reminiscent of Grice?s (1975) maxim of quantity but is now applied to implicit
QUDs. The maxim of quantity (or to be more precise, its first submaxim) demands con-
versational participants be as informative as possible. But unlike in Grice?s case, given that
we do not begin with an already fixed QUD, we do not infer based on what is not said in
addressing some conversational goal to the conclusion that the speaker is not in a position
to assert it. Rather, according to Q-quantity, assuming that the speaker is being maximally
informative, when we try to select a QUD-tree among its competing alternatives, we will
take the lack of an answer to a particular question to indicate that the question is not part
of the current QUDs. For example, an utterance of ?John was at the party last night? in an
out of the blue context will invoke ?Was John at the party last night?? rather than ?Was
John at the party, and was Mary at the party last night?? as its implicit QUD. The lack of
an answer to Mary?s presence at the party is used to signify that the corresponding question
is not part of the current QUD.
This is not to say that any d-trees that contain an unanswered question are automatically
discarded. Q-quantity helps to select an optimal d-tree from a set of candidates only after
other constraints on d-trees such as those imposed by contrastive topics, negations, and other
discourse markers have been taken into account. For example, the d-tree for ?John was born
in Paris, or he was born in France and Mary was born in London? in Figure 3.4 contains
an unanswered question given that the first disjunct does not provide an answer to ?Where
was Mary born??. Nonetheless, this d-tree is deemed admissible since the existence of the
unanswered question is largely due to the use of contrastive topics. By contrast, since the
d-tree in Figure 3.9 has a proper competitor, namely the d-tree in Figure 3.1, which does
not contain any unanswered questions, it is dispreferred.
On the other hand, what indeed are ruled out automatically are cases where the recon-
struction of implicit QUDs includes a question that is left completely unaddressed. To use an
earlier example, with respect to ?John was at the party last night?, the QUD ?Was John at
63
the party, and was Mary at the party last night?? leaves the subquestion ?Was Mary at the
party last night?? completely unaddressed. Hence, this QUD is automatically eliminated.
By contrast, the question ?Where was Mary born?? in Figure 3.4 does receive an answer?
i.e., ?Mary was born in London??on the right branch. Hence, there is no hard violation of
Q-quantity. For a ban against this type of hard violation of Q-quantity, it is crucial that we
are only concerned with the reconstruction of implicit QUDs. Q-quantity is by no means
intended as a general constraint on explicit discourse questions, since otherwise we would be
prohibited from constructing d-trees where the interlocutor is unable to provide an answer
to some discourse question.
One final comment regarding the two constraints discussed in this section: We can at-
tempt to unify Maximize-Q-anaphoricity and Q-quantity since they can be viewed as placing
an upper and lower bound on how specific implicit QUDs should be:
Q-Specificity: Implicit QUDs should be maximally specific (as per Maximize-Q-anaphoricity)
but not overly specific so that they are left unanswered (as per Q-quantity).
The reconstruction of implicit QUDs can then be viewed as a matter of finding the opti-
mal candidate that manages to balance conflicting constraints. As such, future research
may explore the prospect of further theorizing QUD-reconstruction from the perspective of
Optimality Theory (Prince & Smolensky, 1993; Blutner, 2000, 2013).
3.3.3 Markers of Alternative Strategies of Inquiry
In ?3.2.4, I argued in favor of a uniform analysis under which the apparent disjunction
appearing in ?or at least? retains its disjunctive force. I shall provide an analysis along
this line here. I submit that scalar particles such as ?at least? and ?even? can repair an
otherwise infelicitous HD because they function to evoke alternative inquiry strategies so
that the resulting d-trees no longer violate the distinctness constraint.
For concreteness, let us focus on ?at least?. Although the exact lexical entry for ?at least?
is still up for debate and I do not intend to settle the issue here, several recent proposals
have argued for the idea that ?at least? invokes an ordering on the set of alternative answers
to some current QUD, ranked by the pragmatic strength of the answers (Coppock & Beaver,
64
2011, 2013; Biezma, 2013). For instance, consider the following lexical entry taken from
Coppock and Beaver (2011):
? Jat leastK = ?p.?w.?p? ? CQ ?S[p (w) = 1 ? p? ?S p].
In words, ?at least p? states that there exists an proposition p? in the set of answers to the
current QUD (abbreviated as CQ) such that p? is true and is ranked not lower than p in
terms of pragmatic strength. Pragmatic strength subsumes semantic entailment but also
goes beyond it. For example, consider (32), uttered as a response to an inquiry about the
academic position that John holds:
(32) John is at least an assistant professor.
The set of answers to the current QUD includes propositions such as ?John is an assistant
professor? and ?John is an associate professor?, and they are ordered by pragmatic strength
such that ?John is an associate professor? is ranked higher than ?John is an assistant pro-
fessor? even though the former does not entail the latter. Given such an ordering, (32) is
true just in case either ?John is an assistant professor? is true or a higher-ranked alternative
is true.
Now, when trying to explain the repairing effect of ?at least? in HDs, if we were to
similarly treat the ordering induced by ?at least? as placed on the set of answers to the
current QUD, we would run into a problem. Consider again the use of ?at least? in (33A):
(33) Q: Where was John born?
A: He was born in Paris or at least in France.
What are the stronger alternatives to the prejacent of ?at least? in (33A)? Since the prejacent
of ?at least? is ?John was born in France?, one could plausibly take the stronger alternatives
as consisting of answers such as ?John was born in Paris?, ?John was born in Marseille?,
and so on. This would in turn force ?John was born in France? and ?John was born in
Paris? to respond to the same QUD. But this is exactly what renders the standard HD in
(1) infelicitous in the first place. In other words, if ?at least? requires the two disjuncts in
(33A) to be viewed as addressing the same QUD, then the reconstructed d-tree for (33A)
will inevitably violate distinctness, and as a result we will incorrectly judge it as infelicitous.
On the other hand, if we choose not to treat answers such as ?John was born in Paris?
65
and ?John was born in Marseille? as stronger alternatives to ?John was born in France?, it
becomes rather unclear what other options remain.
In response, I propose that the use of ?at least? in (33A) does not invoke an ordering on
the set of alternative answers to the current QUD addressed by the prejacent but instead
on a set of alternative QUDs among which the current QUD belongs. More specifically,
assuming that the set of alternative QUDs is not vacuous in the sense that it contains at
least one QUD that is different from the current QUD, we can treat ?at least? as signaling
the existence of at least one alternative QUD in the d-tree such that it is a sister of the
current QUD and is pragmatically stronger than it.
The use of ?at least? in (33A) signifies that the QUD answered by the prejacent ?John
was born in France? has a sister node that is stronger. This requirement rules out the d-
tree depicted back in Figure 3.1 where the question ?Where was John born?? on the right
branch immediately dominates the answer ?John was born in France? which is the prejacent
of ?at least?. Instead, the question ?Where was John born?? on the right should be further
divided into (at least) two subquestions such that one is pragmatically stronger than the
other. One straightforward way to cash out strength for the current purpose is simply to
use the entailment relation. For example, consider the d-tree in Figure 3.10. Assuming that
the name of the country where a given city is located is common ground knowledge, then
the question ?Which city?? contextually entails ?Which country?? since an answer to the
first question also answers the second question. Hence, ?Which city?? serves as a stronger
alternative QUD to ?Which country??, thereby satisfying the demand from the use of ?at
Where was John born?
Where was John born? Where was John born?
Which country? Which city? Which country? Which city?
(John was born in France.) John was born in Paris. John was born in France.
Figure 3.10: A d-tree for (33A). The presence of ?at least? signals that the QUD ?Which country?? on
the right branch has ?Which city?? as a stronger sister.
66
least?. And by uniformity, the same QUD-tree is duplicated on the left. As a result, ?John
was born in Paris? and ?John was born in France? now respond to two different QUDs.
Since there is no violation of distinctness, (33A) is judged as felicitous.
Note that the d-tree in Figure 3.10 is in fact identical to the one from Figure 3.9. Recall
that, in ?3.3.2, this particular tree is ruled out as an optimal representation for the HD in
(1) because there exists a competing d-tree that does not contain any unanswered question,
namely the one in Figure 3.1 where ?John was born in France? on the right branch is
immediately dominated by the discourse question ?Where was John born??. Thus, by Q-
quantity, the d-tree in Figure 3.9 is dispreferred. The situation is different here. The addition
of ?at least? signals the need for a more complex inquiry strategy which eliminates the
simpler d-tree as a potential competitor. In addition, although the above d-tree contains
an unanswered question on the right branch, the same question (i.e., ?Which city??) does
receive an answer on the left branch. Hence, there is no hard violation of Q-quantity. As a
result, the d-tree in Figure 3.10 now becomes admissible with respect to (33A).
If the current analysis is on the right track, then we should expect the repairing effect
of ?at least? to more or less diminish in cases where it is harder to find a stronger sister of
the QUD that dominates the prejacent of ?at least?. For instance, the answer in (34A) still
sounds odd even with the presence of ?at least?.
(34) Q: Where was John born?
A: ??He was born in France, or at least he was born in France or Germany.
Where was John born?
Where was John born? Where was John born?
Which country? Which city? Which country? Which city?
Which country? Which country? Which country? Which country?
France France Germany
Figure 3.11: A potential d-tree for (34A) which is blocked by Q-quantity since ?Which city?? does not
receive an answer on either branch.
67
What we want to know is whether we can generate an admissible d-tree for (34A) such that
the QUD that dominates the prejacent of ?at least? has a stronger sister. One candidate is
given in Figure 3.11, where just like in Figure 3.10, the discourse question is split into ?Which
country?? and ?Which city??. The ?Which country?? on the right is further divided into
two questions each dominating a disjunct of the prejacent of ?at least? in (34A). However,
unlike in Figure 3.10 where the first disjunct of (33A), namely ?John was born in Paris?,
supplies an answer to ?Which city??, this question is left completely unaddressed in Figure
3.11: neither of the two nodes representing this question receives an answer. As such, the
d-tree involves a hard violation of Q-quantity and thus is considered inadmissible.
By contrast, the above answer does appear to fit better as a response to the following
alternative question:
(35) Q: Was John born in France, Germany, or Belgium?
A: He was born in France, or at least he was born in France or Germany.
Figure 3.12 shows a possible d-tree for (35A). The QUD that dominates the prejacent of
?at least? is conceived of as the polar question ?Was John born in France or Germany??
this time around. As such, the question ?Which country?? constitutes a stronger sister of
it, since a complete answer to the former question also provides an answer to the latter.
Moreover, since none of the questions in Figure 3.12 is left completely unaddressed, there is
no hard violation of Q-quantity. And as there is no violation of distinctness in Figure 3.12,
we predict (35A) to be an unmarked answer to (35Q).
Was John born in France, Germany, or Belgium?
Was John ... or Belgium? Was John ... or Belgium?
Which country? Was John born in France or Germany? Which country? Was John born in France or Germany?
France (Yes) He was born in France or Germany.
Figure 3.12: A d-tree for (35A). The prejacent of ?at least? on the right branch is dominated by the polar
question ?Was John born in France or Germany?? which has ?Which country?? as its stronger sister. (A
positive answer to the polar question on the left branch is entailed by the disjunct ?He was born in France?).
68
To recapitulate the difference between (34) and (35), the reason why (35A) feels more
natural than (34A) is that given the discourse question in (35Q), it is easier to construct a
QUD like ?Was John born in France or Germany??. When the discourse question is simply
?Where was John born??, it is hard to envision why such a polar question would figure into a
reasonable inquiry strategy. By contrast, when the discourse question has already mentioned
?France?, ?Germany?, and ?Belgium? as three candidate answers, the question ?Was John
born in France or Germany?? can be conceived of as a sensible way to narrow down and
develop the original discourse question.
Besides ?at least?, we can explain the repairing effect of ?even? along the same lines. A
common recipe for the lexical analysis of ?even? is an ordering on the set of focus alternatives
to the prejacent of ?even? (see, e.g., Rooth 1992; Chierchia 2013; Greenberg 2016). Roughly,
?even? presupposes that its prejacent should be stronger than any of its alternatives. Since
the focus alternatives of a proposition go hand in hand with the QUD addressed by it, we can
provide a similar analysis by treating ?even? as inducing an inquiry strategy which requires
that the QUD immediately dominates the prejacent of ?even? has a sister that is weaker
than it (see Figure 3.13). Just as in the case of ?at least?, this more complex inquiry strategy
evoked by the use of ?even? yields a d-tree where ?John was born in France? and ?John
was born in Paris? respond to two different QUDs, thereby capturing the repairing effect of
?even?.
Lastly, it is worth noting that not all scalar particles seem capable of fixing HDs. For
example, the following disjunction still sounds odd with the addition of ?at most?:
Where was John born?
Where was John born? Where was John born?
Which country? Which city? Which country? Which city?
John was born in France. (John was born in France.) John was born in Paris.
Figure 3.13: A d-tree for ?John was born in France or even in Paris?. The QUD ?Which city?? on the
right branch which dominates the prejacent of ?even? has ?Which country?? as its weaker sister.
69
(36) #John was born in France or at most in Paris.
This seems to indicate that scalar particles such as ?at least? and ?even? have a special status.
For one thing, it has been suggested that ?at least? is indeed associated with multiple distinct
lexical entries (Kay, 1992; Nakanishi & Rullmann, 2009; Cohen & Krifka, 2011; Coppock &
Brochhagen, 2013). On the other hand, there have also been attempts to provide a uniform
lexical entry for ?at least? (see, e.g., Biezma, 2013). How the current observation figures
into this debate is a question to be explored in future research.
3.4 Open Issue
One remaining issue I want to briefly highlight concerns Hurford questions. It has been
observed that Hurford disjunctions can also take the form of an interrogative sentence (Cia-
rdelli & Roelofsen, 2017). For instance, just as its declarative counterpart, the alternative
question in (37) is deemed infelicitous:5
(37) #Was John born in Paris?, or was he born in France??
Ciardelli and Roelofsen (2017) combine a molecular redundancy account with inquisitive
semantics to explain why questions such as (37) are infelicitous. Since (37) introduces two
alternatives such that one entails the other, the alternative question is defective. However,
their account is unable to account for the interrogative variants of conjunctive Hurford
disjunctions. For instance, since the two alternatives introduced by (38) are non-entailing,
resorting to inquisitive semantics alone cannot explain its oddness.
(38) #Was John born in Paris?, or was he born in France and Mary in London??
Neither does my current analysis offer an immediate solution to the puzzle raised by Hur-
ford questions. We could, as a makeshift solution, try to account for the data by examining
the d-tree associated with the assertion that corresponds to the sentence-radical of a given
question. For example, under inquisitive semantics, the alternative question in (38) shares
the same sentence-radical as the assertion ?John was born in Paris, or he was born in France
and Mary in London?. As such, we can explain why (38) is bad by appealing to the same
d-tree that explain why the matching CHD is bad.
5The upward and downward arrows are used to indicate intonation.
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Additionally, by taking a simple negative sentence as assuming a positive polar question
as its QUD, we can also explain why Hurford disjunctions are infelicitous when they are
embedded under negation in a similar fashion. For example, the negation in (39) is perceived
as odd because the question it allegedly answers in (40) is defective in the first place.
(39) John wasn?t born in Paris or in France.
(40) #Was John born in Paris or in France??
Although this roundabout strategy yields the right predictions, a more preferable ap-
proach would be to give a theory of how implicit QUDs can be reconstructed from explicit
questions. This would enable us to generate a d-tree for the alternative question in (38) in
a more straightforward manner. Such an account is worth exploring in the future.
3.5 Conclusion
This paper presents a QUD-based analysis of informational redundancy that is capable of
capturing the (in)felicity of a wide range of Hurford disjunctions. In particular, it readily
predicts the infelicity of conjunctive Hurford disjunctions, which is left unaccounted for under
existing redundancy theories. To accomplish this, I make use of a notion of discourse trees
and postulate several constraints governing their structures. More specifically, I propose that
the disjuncts of a disjunction should evoke the same inquiry strategy and that answers to
the same question in a d-tree should be distinct and non-entailing; additionally, the implicit
QUDs reconstructed from a target sentence should be maximally but nor overly specific. I
have further illustrated how this analysis can be extended to elucidate the repairing effect
of scalar particles such as ?at least?, thereby enabling a uniform analysis of disjunction in
discourse.
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Chapter 4
A Non-Bivalent Approach to
Inquisitive Logic
In this paper, I explore a non-bivalent approach to inquisitiveness and develop a new logic
called the logic of pseudo-complemented propositions (LPP). Standard inquisitive logic, which
takes support as its central notion, rejects the Law of Excluded Middle (LEM) by employ-
ing an intuitionistic negation. Under the current approach, I take support and rejection as
two separate central notions and reject LEM on the ground that the two central notions,
that of support and that of rejection, do not fully complement each other and thus allow
gaps between them. As such, LPP incorporates a negation that does not validate LEM
but does validate Double Negation Elimination. I supply an algebraic semantics for LPP
via the so-called twist-structures. I define a new class of twist-structures, call them pseudo-
complemented twist-structures (PTS), and prove the completeness of LPP with respect to
the class of PTS. The utility of this method of constructing twist-structures will be further
highlighted via exploration of some generalizations of PTS.
Keywords: Inquisitive logic, Law of Excluded Middle, Constructive logic, Algebraic seman-
tics, Twist-structures
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4.1 Introduction
Inquisitive semantics (Groenendijk, 2009; Ciardelli & Roelofsen, 2011; Ciardelli et al., 2015,
2018) offers a unified formal framework for analyzing both declarative and interrogative
sentences. In doing so, it provides a new conception of information exchange that goes beyond
the exchange of truth-conditional content typically associated with declarative sentences.
Under inquisitive semantics, information exchange is construed as a process of raising and
resolving issues: issues are raised by asking a question, which can then be subsequently
resolved by a declarative sentence. As such, asking a question can be viewed as proposing a
set of alternative ways for the common ground information to be updated. For example, a
polar question like ?Is Alice at the party?? puts forward two alternatives which correspond
to the positive and negative answer to this question.
The central component of inquisitive semantics is an alternative-based account of dis-
junction: disjunction is interpreted as presenting a set of alternatives. For example, we can
symbolize the aforementioned polar question using the disjunction A??A which introduces
two alternative possibilities, namely, one where Alice is at the party and one where she is
not. Given this treatment of disjunction, the Law of Excluded Middle (LEM) is no longer
valid:
(LEM) ? ? ??
The loss of LEM has an intuitive explanation. For a sentence to be valid in inquisitive se-
mantics, it needs to be accepted or evaluated true with respect to every body of information,
which means that the sentence must be informationally trivial. But since ???? symbolizes
a question in inquisitive semantics, the sentence is not completely trivial as it raises the
issue of whether ?, thereby making a significant semantic contribution in terms of its update
effect on the conversational context. Given this, LEM is rejected as a validity in inquisitive
semantics.
Now, if we assume a standard semantic clause for disjunction, that is, a disjunction is
evaluated true whenever one of its disjuncts is evaluated true, then we can choose between
two broad strategies to reject LEM: either we abandon valuation bivalence and adopt a non-
bivalent valuation function, or we stick to a bivalent valuation function but instead resort
73
to an account of negation (e.g., intuitionistic negation) that does not vindicate LEM.
To elaborate, let us assume a possible world semantics with the following satisfaction
clause for disjunction: w ?1 ? ? ? iff w ?1 ? or w ?1 ?. That is, ? ? ? is assigned the value
1 at a world w iff either ? is assigned 1 at w or ? is assigned 1 at w. (We shall come back
to the exact interpretation of the value 1 shortly.) To invalidate LEM, we need to make it
happen that w ?1 ? and w ?1 ??. Here are the two general strategies mentioned above:
Non-bivalent valuation: w ?1 ? ?/ w ?0 ?; but w ?0 ? ? w ?1 ??.
Non-classical negation: w ?1 ? ? w ?0 ?; but w ?0 ? ?/ w ?1 ??.
Both proposals, by rejecting one of the two equivalences, allow w ?1 ? and w ?1 ??, thereby
invalidating LEM, but they achieve this through different means.1 According to the first
approach where valuation bivalence is abandoned, ? can receive a value that is neither 1 nor
0 (including the possibility of not receiving a value at all in the case of partial valuation
functions); consequently, not assigning 1 to ? is not equivalent to assigning 0 to ?. The
second equivalence w ?0 ? ? w ?1 ?? then construes negation as a toggle operation that
connects the two values: ?? is assigned 1 at w just in case ? is assigned 0 there. As for
the other option, whereas valuation bivalence is upheld, assigning 1 to ?? is not equivalent
to assigning 0 to ?, and thus, in presence of the first equivalence w ?1 ? ? w ?0 ?, not
equivalent to not assigning 1 to ?. As such, this second approach needs to find a suitable
non-classical negation that denies the second equivalence.
Whereas the existing inquisitive semantics, laid out in Ciardelli et al. (2018), chooses
the second route, I will explore the non-bivalent approach in this paper. Generally speaking,
given that there are many reasons why one may want to reject LEM, which of the two
approaches one choose largely comes down to one?s motivation, and more specifically, how
one interprets ?1 and ?0. This is not to say that given any particular motivation, we must
always settle on one approach. In fact, for some (and perhaps many) cases, we can motivate
both a bivalent and a non-bivalent approach.
1Of course, there could be a third option where we reject both equivalences. Moreover, even if we adopt
a non-bivalent approach, we could still incorporate another negation that does not reinforce the second
equivalence. But for the time being, let us focus on the two minimalist solutions given above.
74
To help illuminate this point, let us consider a different case. Take constructive mathe-
matics as an example. According to constructivism, mathematical objects are mental con-
structs (cf. Iemhoff, 2019). Thus, for any mathematical statement ? to be true, there needs
to be a mental construct that verifies ? via a proof of ?. Likewise, for ?? to be true, there
needs to be a proof of not-?, which under the traditional Brouwer-Heyting-Kolmogorov-
interpretation, is understood as having a way to turn any proof of ? into a proof of a
manifestly false claim such as 1 = 0. Given this interpretation, LEM is rejected on the
ground that there are certain mathematical statements, e.g., Goldbach?s conjecture, that are
neither provably true nor provably false.
Now, depending on how one chooses to interpret ?1 and ?0, we arrive at different methods
to formally cash out constructivism. By taking proof as the sole central notion?that is, ?
is assigned 1 at w iff there exists a proof of ? at w?and interpreting the other value (i.e.,
0) as derivative of that of the first one, namely, as lack of proof, standard intuitionistic logic
counts as a bivalent account where LEM is rejected via the intuitionistic negation. On the
other hand, we can adopt a non-bivalent approach by bringing in another central notion,
namely counterexample (cf. Nelson, 1949). We can then interpret 0 as standing for the
existence of a counterexample. As such, LEM is rejected on the ground that it is not the
case that for any mathematical statement ?, we can always either construct a proof of ? or
find a counterexample to ?. This change of perspective results in a different logic, Nelson?s
three-valued logic N3, which incorporates a toggle negation ? that invalidates LEM while
still vindicating Double Negation Elimination (DNE), viz., ? ??? ? ?.2 It demonstrates
that intuitionistic logic is not the only formalism available for constructivists.
In this paper, I will apply this lesson to inquisitive semantics and explore a non-bivalent
counterpart to inquisitive logic, just as N3 can serve as a non-bivalent counterpart to intu-
itionistic logic. Whereas the existing inquisitive semantics follows a bivalent approach by
taking support as its sole central notion and opting for an intuitionistic negation to account
for the failure of LEM, I entertain a different approach that takes support and rejection as
two central notions that do not necessarily complement each other. I shall leave a more de-
tailed discussion of how the additional notion of rejection can be understood in an inquisitive
2To disambiguate, I will reserve ? for the intuitionistic negation and use ? for the toggle negation.
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setting to ?4.3, but here is the basic idea. In standard inquisitive semantics (Ciardelli et al.,
2018), supporting a question amounts to possessing enough information to be able to answer
the question. Analogously, we can then understand what it means to reject a question as
having enough information to determine that the question cannot be felicitously asked in the
first place?for example, because the question presupposes something that is already known
to be false. Since it is not the case that for any question, we are able to either answer it or
decide that it cannot be felicitously asked in the first place, the two notions of support and
rejection do not necessarily complement each other.
The resulting logic of the current non-bivalent approach, which I name the Logic of
Pseudo-complemented Propositions (LPP), resembles N3 in the sense that they both invali-
date LEM without simultaneously invalidating DNE. But unlike N3, the rejection conditions
of complex formulas are interpreted non-constructively, and unlike N3 but like the standard
inquisitive logic (Ciardelli & Roelofsen, 2011), LPP is not closed under uniform substitution.
My main objective here is not to argue in favor of LPP over the existing inquisitive logic;
rather the aim is to explore new possibilities by showing that the existing bivalent approach
(e.g., Ciardelli & Roelofsen, 2011) is not the only option for theorizing about inquisitiveness.
I present an algebraic semantics for LPP via the so-called twist-structures. Twist-
structures are algebras whose carrier set consists of ordered pairs which are capable of cod-
ifying two separate bodies of information?for example, information regarding the support
and rejection conditions of a sentence.3 As such, they fit well with logics that adopt a non-
bivalent perspective. This method of constructing twist-structures has been previously used
to devise algebraic semantics for N3 as well as other Nelson?s logics (see, e.g., Vakarelov,
1977; Kracht, 1998; Odintsov, 2004). A second goal of this paper is thus to further popu-
larize this method by showing that by varying certain parameters of the twist-operation, we
can derive a wide range of twist-structures. Among these twist-structures are the pseudo-
complemented twist-structures (PTS) which will supply us with an algebraic semantics for
LPP.
This paper proceeds as follows. In ?4.2, I review the basic inquisitive logic and the
3A note on terminology. The name ?twist-structure? comes from Kracht (1998); Vakarelov (2005) calls
this method ?counterexample-construction?. But since the current paper explores generalizations of this
method beyond constructive mathematics, I will adopt Kracht?s terminology.
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logic N3. Since I intend to extend existing inquisitive logic in the same fashion as how
N3 extends the standard intuitionistic logic?that is, by incorporating a toggle negation?I
shall introduce the algebraic semantics for N3 based on basic twist-structures (BTS) to set
up the stage for my own use of the twist-construction method. In ?4.3, I first motivate a non-
bivalent approach to inquisitiveness and then elucidate why the existing logic N3, albeit being
non-bivalent, is unsuitable for this purpose. This prompts us to devise a new non-bivalent
framework along the lines of N3. In ?4.4, I present LPP together with an algebraic semantics
for it via PTS and prove completeness. In ?4.5, I move on to the aforementioned second
objective by exploring several generalizations of the particular twist-operation proposed in
?4.4. And ?4.6 concludes.
4.2 Preliminaries
4.2.1 Basic Inquisitive Semantics and Inquisitive Logic
Let us begin by defining the language for basic inquisitive logic. The language is given as
follows, where p is a member of a countable set of atomic propositional formulas:
Definition 4.2.1 (Logical syntax). ? ??= p?????? ? ??? ? ???? ???????
The last expression ???? abbreviates ?? ? ???.
As mentioned previously, under inquisitive semantics, a question proposes alternative
ways to update common ground information. Each alternative introduced by a question is
formally represented as a set of possible worlds that are compatible with the information
embodied by the corresponding answer to the question. For example, the alternative ?Alice is
at the party? can be represented as a set of worlds where Alice is at the party. Since a question
introduces multiple alternatives, the propositional content associated with a question is taken
to be a set of sets of possible worlds under inquisitive semantics. To obtain uniformity, we can
then construe the propositional content of any sentence, be it declarative or interrogative, as
a set of sets of possible worlds, that is, as a set of information states. We define information
states, propositions, and alternatives as follows:
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Definition 4.2.2 (Information states). An information state s is a set of possible worlds
and a subset of the total logical space (i.e., the set of all worlds) s ?W .
Definition 4.2.3 (Propositions). A proposition P is a non-empty downward closed set of
information states, that is, for all information states s and t, if s ? P and t ? s, then t ? P .
Definition 4.2.4 (Alternatives in a proposition). The maximal elements of a proposition P
are the alternatives in P . We use Alt(P ) to denote the set of alternatives in P .
An information state is a set of worlds that are compatible with a certain body of information.
The proposition expressed by a sentence is then identified with those information states that
either already accept the assertion of it if the sentence is declarative, or resolve the question
it raises if the sentence is interrogative. And since whenever an information state already
contains enough information to accept an assertion or resolve a question, any subset of it,
which contains more information by excluding certain worlds, must also be capable of doing
so, propositions are construed as downward closed sets of information states.
Figure 4.1 displays some examples of propositions all generated from the same logical
space {AB,AB?, A?B, A?B?}, where AB stands for a world where Alice and Bob are both at
the party, AB? a world where Alice but not Bob is at the party, and so on. I use J K to denote
propositions, and for readability, I will represent a proposition using its alternatives. For
instance, Figure 4.1(a) depicts the proposition JA??AK, which will be abbreviated as J?AK,
and as the figure shows, this proposition is represented by its two alternatives {AB,AB?} and
{A?B, A?B?}. We call a proposition that contains more than one alternative inquisitive. J?AK
is inquisitive: it raises the question of whether or not A, with each alternative representing a
way to answer the question, that is, by either affirming or denying A. Similarly, the proposi-
tion JA?BK depicted by 4.1(b) is also inquisitive: it corresponds to the alternative question
AB AB? AB AB? AB AB? AB AB?
A?B A?B? A?B A?B? A?B A?B? A?B A?B?
(a): J?AK (b): JA ?BK (c): J?(A ?B)K (d): J??(A ?B)K
Figure 4.1: Some examples of propositions
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?Is Alice at the party?, or is Bob at the party???.4 By contrast, the propositions depicted
by 4.1(c) and 4.1(d) are not inquisitive as they contain only one alternative. Addition-
ally, there are two special propositions for any given logical space: the minimal proposition
? ?= ?(W )?i.e., the power set of the logical space W?and the absurd proposition ? ?= {?}.
Now, given that propositions under inquisitive semantics are sets of information states, we
can apply basic algebraic operations to them. More specifically, the set of all propositions,
call it ?, ordered by the subset relation ? forms a complete Heyting algebra ??,?,?,?
,?,?,??, where the set intersection ? and the set union ? denote its meet and join, re-
spectively, and ? denotes the relative pseudo-complement of this algebra (Roelofsen, 2013;
Ciardelli et al., 2018).5 Propositions are then connected via the following algebraic opera-
tions:
? ?= ?(W );
? ?= {?};
J? ? ?K ?= J?K ? J?K;
J? ? ?K ?= J?K ? J?K;
J?? ?K ?= J?K? J?K;
J??K ?= J?K? ?.
Traditionally, sentences are evaluated at worlds, and a sentence ? is evaluated true at a
world w iff w belongs to the proposition expressed by ?, that is, the set of ?-worlds. Under
inquisitive semantics, sentences are evaluated at information states in terms of support. As
mentioned earlier, to support a sentence is to accept its assertion when the sentence is
declarative or to resolve the issue it raises when the sentence is interrogative. We then define
semantic consequence in terms of preservation of support.
Definition 4.2.5 (Support). An information state s supports ?, notated as s ?1 ?, iff
4The upward and downward arrow represent raising and falling intonation, respectively. Varying the
intonation will also vary the type of questions raised by this sentence (cf. Ciardelli et al., 2018).
5We can be more specific. The set of all propositions under inquisitive semantics indeed forms a special
type of Heyting algebras that satisfy certain additional properties (see Bezhanishvili et al., 2020, and ?5.1
below).
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s ? J?K.6
Definition 4.2.6 (Consequence via support). ? ? ? iff for all logical spaces W and all
s ?W , if s ?1 ? for every ? ? ?, then s ?1 ?.
As an essential feature of inquisitive semantics, LEM is invalid. Figure 4.1(a) provides
a counterexample to the validity of ?A, since, given the setup, it is not the case that all
information states support ?A: the information state {AB,AB?, A?B, A?B?} does not, as it
does not belong to J?AK. Also, since inquisitive semantics employs an intuitionistic negation,7
neither is DNE valid in general: as Figure 4.1(d) and 4.1(b) demonstrate, the information
state {AB,AB?, A?B} belongs to J??(A ?B)K but does not belong to JA ?BK. That being
said, DNE does obtain for atomic propositions.
The above semantics gives rise to inquisitive logic which can be axiomatized by a system
that contains the following set of axiom schemas along with the inference rule of modus
ponens (Ciardelli & Roelofsen, 2011; see also Ciardelli, 2016):
IPL. All axiom schemas of intuitionistic propositional logic.
KP. (??? (? ? ?))? ((??? ?) ? (??? ?)).8
DNE. ??p? p, for every atomic proposition p.
As a noteworthy feature of inquisitive logic, since DNE only obtains for atomic propositions,
the logic is not closed under uniform substitution.
Lastly, given that propositions under inquisitive semantics form a complete Heyting
algebra, we can provide an algebraic semantics for inquisitive logic via Heyting algebra
(see Bezhanishvili et al., 2019). Given a Heyting algebra H, let us first consider a spe-
cial class of valuation functions that assign every atomic proposition to an element of
H?? = {??x ? x ? H}; call such valuations inquisitive valuations. Since, as Johnstone (1982)
6Given the algebraic foundations of inquisitive propositions, this definition coincides with the recursive
definition of support conditions (Ciardelli et al., 2018). Additionally, although inquisitive semantics is not
originally formulated using Kripke models, we can convert it into a Kripke semantics using the following
Kripke frame ??(W )?{?},?? (Ciardelli & Roelofsen, 2011). Hence, we can use the same Kripkean satisfaction
relation ?1 for the support relation.
7The name ?intuitionistic negation? might be a slight misnomer in the inquisitive setting. Nonetheless,
due to the lack of a catchier name, I will continue using it to refer to the negation employed in standard
inquisitive semantics.
8KP stands for ?Kreisel-Putnam?.
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shows, H?? constitutes a Boolean algebra, it follows that for all y ? H??, the operation
??y ? y returns the top element. Now, we say that a formula ? is inquisitively valid in H
iff for every inquisitive valuation v on H: v(?) = 1; and ? is inquisitively valid simpliciter iff
it is inquisitively valid in every Heyting algebra. Hence, given that atomic propositions are
assigned to H??, they come to satisfy DNE. Additionally, if we further require that the class
of Heyting algebras satisfy the KP axiom, we derive the exact set of validities that are valid
in inquisitive logic.
4.2.2 A case study of the non-bivalent approach: N3
In the following two subsections, I will describe the logic N3 in some detail. The reason for
investigating N3 is mainly two-fold. On one hand, it helps us to see why N3, albeit being
a non-bivalent framework, is unsuitable for the current purpose of modeling inquisitiveness,
which I shall further explicate in ?4.3.2. On the other hand, since the framework I will
eventually propose employs a toggle negation in the spirit of N3 and comes with an algebraic
semantics via twist-structures, an examination of N3 is conducive to understanding my
proposal.
The logic of N3 originates from considering two different ways of proving the negation
of ?: we can do so by either applying reductio ad absurdum or directly constructing a
counterexample of ? (Nelson, 1949; see also Kapsner, 2014). The former corresponds to the
intuitionistic negation ??, as ?? can be defined as ?? ? which conveys that any proof of ?
leads to a proof of absurdity. On the other hand, standard intuitionistic logic is unable to
capture the latter idea of refuting a sentence by constructing a counterexample, as the same
intuitionistic negation cannot fulfill this additional purpose. To elaborate, in intuitionistic
logic, ?(? ? ?) does not entail ?? ? ??; but if we were to interpret negation as expressing
the presence of a counterexample, we should expect the entailment to hold, since for us to
be able to construct a counterexample of a conjunction, we must be able to construct a
counterexample to one of its conjuncts.
The search for an alternative negation leads us to the aforementioned non-bivalent picture
where we take proof and counterexample as two non-complementary central notions and use
them to interpret ?1 and ?0, respectively.
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To explain, let us first consider a Kripke semantics for N3. A Kripke model for N3 is
a tuple ?W,?, v?, where W is a non-empty set of worlds (or information states), ? is as a
partial order (i.e., a binary relation that is reflexive, transitive, and antisymmetric), and v is
a partial function from pairs of an atomic proposition and a world to 1 and 0. Given a Kripke
model, we define two separate satisfaction relations ?1 and ?0 that extend the valuation v
to every formula in the language:
w ?1 p iff v(p,w) = 1, when p is atomic;
w ?0 p iff v(p,w) = 0, when p is atomic;
w ?1 ? ? ? iff w ?1 ? and w ?1 ?;
w ?0 ? ? ? iff w ?0 ? or w ?0 ?;
w ?1 ? ? ? iff w ?1 ? or w ?1 ?;
w ?0 ? ? ? iff w ?0 ? and w ?0 ?;
w ?1 ?? ? iff ?w? ? w: if w? ? ? then w?1 ?1 ?;
w ?0 ?? ? iff w ?1 ? and w ?0 ?;
w ?1 ?? iff w ?0 ?;
w ?0 ?? iff w ?1 ?.
As with the Kripke semantics for intuitionistic logic, we interpret worlds in the set W as
proof stages : w ? w? just in case w? is a stage no earlier than w in our mathematical inquiry.
Since proofs and counterexamples are supposed to be conclusive, we posit the following two
heredity conditions:
For all w and w?, if w ?1 p, where p is atomic, and w ? w?, then w? ?1 p
For all w and w?, if w ?0 p, where p is atomic, and w ? w?, then w? ?0 p
That is, if p is either proved (viz., evaluated as 1) or refuted by a counterexample (viz.,
evaluated as 0) at an earlier stage w, then it must remain proved or refuted and receive the
same value at any later stage w?.9
9The heredity constraints on atomic propositions together with the above satisfaction clauses for complex
formulas ensure that the heredity conditions hold for all formulas (see Priest, 2001).
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Note that since the valuation functions are allowed to be partial, p may receive neither
1 nor 0 at w, and in that case, it means that, at w, we can neither prove p nor construct a
counterexample of p. The satisfaction conditions for complex formulas are defined separately
for 1 and 0. The former very much resemble the satisfaction clauses under intuitionistic logic.
In particular, the satisfaction clause for implication states that for there to be a proof of
?? ? at w, we need to be able to convert any proof of ? into a proof of ? at any stage w?.
As for the satisfaction conditions for the other value 0, connectives are again defined
constructively. For ? ? ? to receive 0 at w?that is, for there to be a counterexample of
it at w?either ? must receive 0 at w or ? must receive 0 at w?that is, we must be able
to construct a counterexample to at least one of the conjuncts. And for there to be a
counterexample of ?? ? at w, we must be able to obtain, simultaneously, a proof of ? and
a counterexample of ? at w.
In N3, the negation ? toggles between the satisfaction conditions associated with the two
values. For ?? to receive 1 at w?that is, for the negation of ? to be provable at w?? needs
to receive 0 at w?that is, we need to be able to construct a counterexample of ? at w, and
vice versa.
Lastly, we define semantic consequence in terms of preservation of the value 1 at every
world in every model:
? ? ? iff for every model M and every world w, if w ?1 ? for every ? ? ?, then w ?1 ?.
And in the case ? is empty, that is, ? ?, we say ? is valid in N3.
A complete axiomatization of N3 is provided below, which contains the following axiom
schemas complemented with the inference rule modus ponens. The biconditional ? ? ? is
defined as (?? ?) ? (? ? ?) as usual.
AS1. ?? (? ? ?)
AS2. (?? (? ? ?))? ((?? ?)? (?? ?))
AS3. ?? (? ? ?)
AS4. ? ? (? ? ?)
AS5. (?? ?)? ((? ? ?)? (? ? ? ? ?))
AS6. (? ? ?)? ?
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AS7. (? ? ?)? ?
AS8. ?? (? ? (? ? ?))
AS9. ???? ?
AS10. ??? (?? ?)
AS11. ?(? ? ?)? (?? ? ??)
AS12. ?(? ? ?)? (?? ? ??)
AS13. ?(?? ?)? (? ? ??)
The first eight axiom schemas come from the positive fragment of intuitionistic logic. AS9
specifies how proof-construction and counterexample-construction are connected via toggle
negation, that is, constructing a counterexample of ?? essentially amounts to having a proof
of ?. Hence, unlike intuitionistic negation, toggle negation vindicates DNE. AS11?13 capture
the constructiveness of counterexample-construction. Lastly, AS10, the principle of ex falso
quodlibet, ensures that a proof of any contradiction leads to explosion.10 Without it, the
resulting logic becomes Nelson?s paraconsistent logic N4. Additionally, given the presence
of ex falso, we can define intuitionistic negation ?? in N3 as ?? ??.11 Defined as such, the
intuitionistic negation ?? is entailed by the toggle negation ??.12 This means that in N3,
we can capture the two distinct conceptions of negation delineated above, that is, negation
via proving reductio and negation via constructing a counterexample.
4.2.3 An algebraic semantics for N3 via twist-structures
An algebraic semantic for N3 has been provided by Rasiowa (1958, 1974) using the so-
called N -lattices (i.e., quasi-pseudo-Boolean algebras). An important method to generate
10To disambiguate, I shall use ex falso quodlibet to refer to the principle of explosion stated in the axiom
form while using ex contradictione quodlibet to refer to the inference ??,? ? ?. The two are nonetheless
identical in presence of the deduction theorem.
11To spell out, we first define ?? as ?? ? as usual. In N3, ? can be defined as ? ? ??, which means that
?? can be defined as ? ? (? ? ??). Given that we have ? ? ? as a theorem, ?? can simply be defined as
?? ??.
12Due to the fact that ?? entails ?? in N3, the toggle negation ? is also often termed ?strong negation?
(e.g., Rasiowa, 1958; Vakarelov, 1977); another name for this negation is ?constructive negation? (e.g., Priest,
2001). However, since neither the entailment to intuitionistic negation nor constructiveness are necessary
features of ? (see, e.g., Vakarelov, 2005), I shall stick to the name of ?toggle negation? as used by Kapsner
(2014).
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N -lattices is via the so-called twist-structures (Vakarelov, 1977; Kracht, 1998), as twist-
structures provide a much more intuitive account of the meaning of the logical operators
employed in N -lattices, which, as we shall see, is closely related to the interpretation of
these operators given by the above Kripke semantics.
Since this paper will explore different generalizations of twist-structures, I shall call the
particular kind of twist-structures proposed in Vakarelov (1977) basic twist-structures (BTS).
Given a Heyting algebra (i.e., an implicative lattice with a distinguished bottom element)13
H = ?H,?,?,?,?,??, we define a BTS as H& = ?H&,?&,?&,?&,?,1,0?. Its carrier set is:
H& = {?x,x?? ? H ?H? x ? x? = ?}
And we define the operators as follows:
?x,x?? ?& ?y, y?? ?= ?x ? y, x? ? y??
?x,x?? ?& ?y, y?? ?= ?x ? y, x? ? y??
?x,x???& ?y, y?? ?= ?x? y, x ? y??
??x,x?? ?= ?x?, x?
1 ?= ??,??
0 ?= ??,??
Alternatively, we can also construct the BTS of a Heyting algebra H induced by the ordering
? by first generating the product lattice H?H with the following new ordering: ?a, b? ? ?c, d?
iff a ? c and d ? b, and then restricting the carrier set to H&. It can then be easily shown
that the resulting algebra is closed under the set of operations {?&,?&,?&,?}.
To adduce a concrete example, consider the BTS in Figure 4.2(b), which is generated
from the Heyting algebra ?{a, b, c},?,?,?,?,?? depicted in 2(a). For every pair ?x, y? in
the resulting BTS, x ? y is always the bottom element in the original Heyting algebra. The
13An implicative lattice is a distributive lattice ordered by ???such that for any two elements x and y,
there exists a unique greatest element x? y such that (x? y)?x ? y. We call x? y the pseudo-complement
of x relative to y. And when the implicative lattice comes with a bottom element, we call x ? ? the
pseudo-complement of x.
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??,??
? ?c,??
c ?a,?? ?b,??
a b ?a, b? ??,?? ?b, a?
? ??, b? ??, a?
??, c?
??,??
(a) (b)
Figure 4.2: An example showcasing twist-construction
new meet, join, and weak relative pseudo-complement in the resulting BTS are given by
?&,?&, and?&, respectively.
A valuation v in a twist-structure is defined in the standard algebraic way, that is, as a
function from formulas to elements in the carrier of the twist-structure while observing the
definitions of all the logical operators for all complex formulas.
Given this, twist-structures provide an intuitive algebraic semantics for N3. Each pair
from the carrier of BTS can be construed as codifying positive and negative information
about the formula assigned to the given pair.14 In the present case, the positive information
14This interpretation of such order pairs is also common in the literature on bilattices (see, e.g., Fitting,
1989; Gargov, 1999).
86
is the formula?s provability condition, and the negative information is its disprovability con-
dition. The definitions of the logical operators, which go hand in hand with the definitions
provided via Kripke semantics, then tell us how the provability and disprovability conditions
of complex formulas are derived from those of their constituents. For instance, let ? denote
the pair ?x,x?? and ? denote the pair ?y, y??. The pair denoted by the conditional formula
? ? ? is ?x ? y, x ? y??. The first coordinate x ? y tells us what it means to prove this
formula, that is, by being able to convert any positive information about ? (i.e., a proof of
?) into positive information about ? (i.e., a proof of ?); the second coordinate x ? y? tells
us what it means to disprove it, that is, by being able to obtain both positive information
about ? (i.e., a proof of ?) and negative information about ? (i.e., a counterexample of ?).
Lastly, validity and consequence are defined equationally in the usual way. For any given
algebra A, ? is valid in A, notated as ?A ?, iff for all valuations v: F ?A, v(?) = 1. And
for any given class of algebras A, ?A ? iff ?A ? for every A in A.15
For any given algebra A, ? is a semantic consequence of a set of sentences ? in A, notated
as ? ?A ?, iff for all valuations v: F?A, if v(?) = 1 for every ? ? ?, then v(?) = 1. And for
any given class of algebras A, ? ?A ? iff ? ?A ? for every A in A.
A distinguishing feature of N3 is that ?? ? alone does not define a congruence relation
(i.e., an equivalence relation that is preserved under all logical operations). Here is a coun-
terexample using the BTS depicted in Figure 4.2(b). Put v(?) = ?b,?? and v(?) = ?b, a?.
Given this, v(?? ?) = v((? ? ?) ? (? ? ?)) = (?b,?? ?& ?b, a?) ?& (?b, a? ?& ?b,??) = ??,??.
However, v(??? ??) = v((?? ? ??) ? (?? ? ??)) = (??, b? ?& ?a, b?) ?& (?a, b? ?& ??, b?) =
?b,?? ? ??,??. Instead, in order for ? and ? to be considered congruent, both ? ? ? and
??? ?? should obtain. As we shall see, this feature of N3 is also shared by the logic LPP
to be developed in this paper.
15Hereafter, when the relative class of algebras is clear, I shall omit the subscript ?A?.
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4.3 A Non-Bivalent Approach to Inquisitiveness
4.3.1 Support and Rejection
To develop an inquisitive framework that rejects LEM from a non-bivalent perspective, first
we need to spell out what the dual notion of support ??1? is. There is one immediate
candidate for the interpretation of ??0?, namely, the notion of rejection.
If ? is a declarative sentence, then for an information state s to reject ? is for it to already
contain enough information to settle the sentence false, just as for s to support ? is for it
to already contain enough information to settle the sentence true. Since s may not contain
enough information to either settle ? true or settle it false, the equivalence s ?1 ? ? s ?0 ?
does not hold. And given that the negation ? is now interpreted as a toggle operation
between the support and rejection conditions of a sentence?that is, to support ?? is to
reject ?, and vice versa?LEM is rejected for the very same reason.
On the other hand, when ? is an interrogative sentence, one natural way to interpret the
rejection of ? by an information state s is to consider ? as making a presupposition that
is already settled false in s. For example, an alternative question such as ?is Alice at the
party? or is Bob at the party??? is often taken to presuppose that either Alice or Bob is at the
party (cf. Karttunen & Peters, 1976; Biezma & Rawlins, 2012). When it is common ground
that neither Alice nor Bob is at the party, the presupposition is already settled false and
thus cannot be accommodated. As a result, the alternative question cannot be felicitously
uttered in this context, which leads to its rejection.16
Now, since it is not the case that every information state must either resolve a question
or directly contradict the presupposition it carries?e.g., the state {AB,AB?, A?B} is unable
to answer the question A ?B but nevertheless satisfies its presupposition?the equivalence
s ?1 ? ? s ?0 ? does not hold. Consequently, LEM is rejected again.
As we have seen with N3, the toggle negation ?, unlike the intuitionistic negation ?,
16Alternatively, we may hold a view according to which a question can be justifiably rejected as long
as its presupposition is not settled true. This could happen in cases where accommodating a question?s
presupposition turns out to be difficult or costly. I will consider both interpretations of rejection as available
options. And the difference here will not affect what follows. For example, if we adopt the second view,
then DNE can be motivated on the ground that denying that the presupposition of a question is not already
settled true amounts to affirming that the presupposition is already satisfied.
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vindicates DNE. In N3, the intuitiveness of DNE is explained based on the thought that to
find a counterexample to a counterexample of ? is to prove ?. Under the inquisitive setting,
we can illuminate the validity of DNE in the following way. For declarative sentences, DNE
amounts to saying that to object to a rejection of ? is to support ?. As for interrogative
sentences, as mentioned above, to reject a question can be understood as saying that the
presupposition of the question is already settled false, and when this happens, the question
fails to make an inquiry and does not request future discourse to supply an answer to it.
Given this, to object to the rejection of a question can be understood as rejecting the
claim that the presupposition of the question is already settled false, thereby reinstating the
question alongside its inquisitive content. Hence, after rejecting an objection to a question,
the original question should still solicit an answer to it. As a paradigmatic example of this
dynamics, consider cross examination in court. If the judge rules that the objection to a
question having posed to the witness is ?overruled?, then the original question still stands,
and the witness still needs to answer the question.
As such, we can make sense of the rejection of LEM from a non-bivalent perspective as
well as the validity of DNE associated with the toggle negation.
4.3.2 Why N3 is Unsuitable
Since N3 is already a non-bivalent account that employs a constructive disjunction that does
not vindicate LEM, we may wonder whether we can repurpose it to model inquisitiveness.
However, the reason why N3 alongside BTS is unsuitable for this purpose is this. The basic
twist-structure makes the rejection condition of a proposition more stringent than what we
would normally expect. In particular, it requires the rejection of a conjunction to also be
constructive.
For instance, let v(A) = ?a, a?? and v(B) = ?b, b??. Then, according to BTS, v(A ?B) =
?a ? b, a? ? b??. Under the inquisitive setting, we can interpret the two coordinates of a given
pair as codifying positive and negative information about what it takes for the assigned
formula to be supported and rejected. The support condition for A ?B provided by BTS is
fairly intuitive, since in order to settle ?Alice and Bob are both at the party? true, we need
to both know that Alice is at the party and know that Bob is at the party. The rejection
89
condition for A ?B, on the contrary, appears too strong, since it commands that in order
to reject the above assertion, we must either know that Alice is not at the party or know
that Bob is not at the party. But if we know for sure that either Alice or Bob is out of town
but cannot remember exactly who it is, then, arguably, we have already obtained enough
information to reject the conjunction.
The above claim that the rejection condition of a sentence should not be constructive
squares with the existing inquisitive semantics. Although inquisitive semantics does not em-
ploy this notion of rejection, it does construe the negation of a conjunction as non-inquisitive.
Hence, unlike ?A ? ?B, ?(A ? B) does not introduce any alternatives. Consequently, any
information state that only eliminates all A-and-B-possibilities (e.g., {AB?, A?B, A?B?}) will
come to support ?(A ?B).
To take stock, whereas N3 implements constructivism globally, we want, for the current
purpose of devising a non-bivalent account of inquisitiveness, a framework that binds con-
structiveness more closely to the support condition for disjunction as disjunction uniquely
plays the role of introducing alternatives. In other words, we want to introduce asymmetry
between the support condition and the rejection condition of a sentence in respect of their
constructiveness so as to make the support condition of a disjunction but not the rejection
condition of a conjunction constructive. In the next section, I propose a framework along
this line.
4.4 The Logic of Pseudo-Complemented Propositions
4.4.1 Pseudo-complemented twist-structures
We begin by exploring how we can modify BTS so as to vary the constructiveness of support
and rejection conditions. In devising such a framework, I shall stick to the following two
guiding principles:
Principle 1: The modified twist-structure still employs a toggle negation ? that switches
between the support and rejection conditions of a sentence.
Principle 2: The rejection condition of a sentence ? is given, whenever possible, by the
90
support condition of its intuitionistic negation ?? (viz., ?? ?) as provided in the original
inquisitive semantics.
These two guiding principles together give rise to a class of twist-structures, i.e., pseudo-
complemented twist-structures (PTS), that, on the one hand, incorporate a toggle negation
and thus reject LEM from a non-bivalent standpoint, and on the other hand, by identifying
the rejection condition of a sentence, whenever possible, with the support condition of its
intuitionistic negation, render the rejection conditions of complex formulas non-constructive
in a way that deviates minimally from the original inquisitive semantics. Before giving the
definition for PTS, I want to emphasize that it is not my contention that PTS is the sole
available non-bivalent treatment of inquisitiveness and we must adhere to the aforementioned
two guiding principles. In particular, although Principle 2 affords a convenient way to specify
the rejection conditions of complex formulas, we might have reasons to reject it based on
other empirical motivations. Consequently, we may adopt a less uniform approach when
defining the rejection condition for each connective. Be that as it may, in this paper, let us
explore a framework that fulfills these two principles.
Given a Heyting algebra H = ?H,?,?,?,?,??, we define a PTS, H&? = ?H&?,?? ?R,?R,??R
,?,1,0?, as follows:
H&? = {?x,x??? either x? = x?,or x = x??}
where x? abbreviates x? ? for any x, and the logical operators are defined as follows:17
?x,x?? ??R ?y, y?? ?= ?x ? y, (x ? y)??
?x,x?? ??R ?y, y?? ?= ?x ? y, (x ? y)??
?x,x????R ?y, y?? ?= ?x? y, (x? y)??
??x,x?? ?= ?x?, x?
1 ?= ??,??
0 ?= ??,??
17The subscript ?R? indicates that the operator yields pairs whose second coordinate is the pseudo-
complement of the first one. This is to be contrasted with a variation of PTS explored in ?5.3 below.
91
? ??,??
c ?c,??
a b ?a, b? ?b, a?
? ??, c?
??,??
(a) (b)
Figure 4.3: An example of PTS.
Again, we define valuation functions inquisitively by mapping every atomic proposition
to an element of H&??? = {(?x,x?? ??R 0) ??R 0 ??x,x?? ? H&?}. Semantic consequence and
validity are defined in terms of inquisitive valuations as before.
Figure 4.3(b) depicts the PTS generated from the Heyting algebra as shown in 3(a).
For the twist-operations associated with the three binary connectives, the operations are
defined component-wise only for the first coordinate; the second coordinate is instead always
the pseudo-complement of the first one. Defined as such, rejection conditions for complex
formulas come to fulfill Principle 2. The definition for toggle negation remains unchanged,
thereby fulfilling Principle 1. As a consequence, the carrier of PTS consists exclusively of
pairs such that either the first component is the pseudo-complement of the second one, or
the second component is the pseudo-complement of the first one.
As a notable feature of PTS, the two operations ??R and ??R defined above do not always
form the meets and joins of the new algebra. For instance, the meet of the two pairs ?a, b?
and ?b, a? in the PTS as shown in Figure 4.3(b) is ??, c?; however, applying the operation
92
?a, b? ??R ?b, a? gives us ??,??. Likewise, the join of ??, c? and itself is itself, whereas ??, c? ??R
??, c? gives us ??,?? again.
As it turns out, any two elements in PTS do not always have a greatest lower bound and
a least upper bound; in other words, PTS does not form a lattice. To illustrate, consider the
PTS generated from the Heyting algebra whose carrier set is set of all non-empty downsets
in ?{AB,AB?, A?B, A?B?}; in other words, we are considering a concrete Heyting algebra where
each element from its carrier stands for some proposition understood as per inquisitive se-
mantics. Now, let us consider the following two pairs ?a, a?? and ?b, b?? whose components
are shown in Figure 4.4 (as with Figure 4.1, we represent each proposition using its set of
alternatives). For each pair, the first component is the pseudo-complement of the second
one but not vice versa.
Combining these two pairs via the old operation ?& from BTS yields ?a ? b, a? ? b??, that
is, ?c, d?, which is in fact the join of the two pairs in BTS. However, since neither component
in ?c, d? is the pseudo-complement of the other, this pair does not belong to the carrier of
PTS. What we do have in PTS are the two pairs ?c, c?? and ?d?, d?, which are derived from
?a, a?? ?? ?b, b?? and ?(??a, a?? ??R R ??b, b??), respectively.
Recall that PTS, as with BTS, is induced by an ordering such that for any two pairs ?x,x??
and ?y, y??, ?x,x?? ? ?y, y?? in PTS iff x ? x? and y ? y? in the original Heyting algebra. It
follows that, given the absence of ?c, d?, the two pairs ?c, c?? and ?d?, d? are indeed competing
AB AB? AB AB? AB AB? AB AB?
A?B A?B? A?B A?B? A?B A?B? A?B A?B?
a a? b b
?
AB AB? AB AB? AB AB? AB AB?
A?B A?B? A?B A?B? A?B A?B? A?B A?B?
c c? d d?
Figure 4.4: An illustration of why PTS does not form a lattice.
93
for the greatest lower bound of ?a, a?? and ?b, b??. Hence, ?a, a?? and ?b, b?? do not have a
unique greatest lower bound. By parity of reasoning, any two elements in PTS also do not
always have a least upper bound.
Algebraically, the two operators ??R and ??R satisfy commutativity (viz., X?? Y = Y ??R RX,
andX??RY = Y ?? 18RX) , associativity (viz., X??R(Y ??RZ) = (X??RY )??Z, andX?? (Y ??R R RZ) =
(X ?? Y ) ??R R Z), and distributivity (viz., X ??R (Y ??R Z) = (X ??R Y ) ??R (X ??R Z), and
X ??R (Y ??RZ) = (X ??RY )??R (X ??RZ))19, but they do not satisfy the principle of absorption
which is characterized by the two equations: X ??R (X ??R Y ) = X, and X ??R (X ??R Y ) = X.
Take the former equation as an example, and let X be a pair wherein the first component is
the pseudo-complement of the second one but not vice versa. Since ?? and ??R R necessarily
produce a pair whose second component is the pseudo-complement of the first, the left hand
side and the right hand side of the equation cannot be the same.
For the same reason, idempotence also fails. That is, the following two equations do not
hold in PTS: X ??R X = X, X ??R X = X?specifically, when the second component of X is
not the pseudo-complement of the first one. The loss of idempotence for ??R and ??R may at
first appear problematic. While that being the case, ? and ? ? ?, as well as ? and ? ? ?,
nonetheless semantically entail each other for any formula ?. This is so because given how
semantic consequence is defined, when the premise?be it ?, ? ? ?, or ? ? ??is assumed
to take the top element ??,?? as its value, its first and second components are forced to be
the pseudo-complements of each other, thereby forcing the conclusion to also take the top
element as its value. Additionally, both ? (? ? ?) ? ? and ? (? ? ?) ? ? obtain in PTS.
Therefore, potential negative impact from the loss of idempotence is largely mitigated.20
18Here and hereafter, I will use the uppercase X, Y , and Z to stand for elements from the carrier set of
a twist-structure.
19The proofs of these results follow from the fact that, for each equation, the resulting pairs on each side
are such that their first coordinates are identical given that the Heyting algebra forms a distributive lattice
and thus satisfies commutativity, associativity and distributivity. Given that the second coordinate of each
pair is the pseudo-complement of its first coordinate, which, as we have just said, are identical on both sides
for each equation, these equations hold in PTS.
20What we do lose in PTS are the following: ? ?(???)? ?? and ? ?(???)? ??. How troublesome this
is remains to be further investigated.
94
4.4.2 The logic of pseudo-complemented propositions
I provide an axiomatization of the corresponding logic of PTS here and prove completeness
in ?4.4.3. The logic of pseudo-complemented propositions (LPP) contains, in addition to the
axiom schemas AS1?10 from N3 (viz., all axiom schemas from the positive logic plus ???? ?
and ?? ? (? ? ?)), the axiom schemas listed below complemented with the inference rule
of modus ponens.
AS14. ?? ?
AS15. ((p? ?)? ?)? p, where p is an atomic formula.
AS16. (? ? ?)?(? ? ??)? ??, where ? is a non-negated formula, i.e., a formula whose
main connective is not ?, and ? is any arbitrary formula.
In LPP, we introduce a propositional constant ? along with AS14. Together with all axiom
schemas of the positive logic, it follows that LPP is an extension of IPL. The addition of
? turns out to be useful later in characterizing certain formal results and in our proof of
completeness. AS15 amounts to the restricted version of DNE found in inquisitive logic,
where DNE is valid with intuitionistic negation but only for atomic propositions. Since
the operations ?? , ?? , and ??R R R do not define their second coordinates component-wise, the
AS11-AS13 from N3 are not valid. In their place, we have AS16 which embodies a weakened
version of negation introduction found in intuitionistic logic and classical logic with the
additional restriction that ? needs to be a formula whose main connective is not ?. Note
that this restriction does allow negation to appear when embedded as in ? ? ??.
As a noteworthy feature of this logic, the validity of AS16 hinges on the fact that given
how logical operators are defined in PTS, operations other than negation (i.e., ?? ?R, ?R, and
??R) will always result in a pair wherein the second component is the pseudo-complement of
the first one; additionally, since valuation functions assign atomic propositions to the regular
elements of PTS, every pair that interprets an atomic formula will again be such that its
second component is the pseudo-complement of the first one.
Given this, we can prove the validity of (? ? ?)? (? ? ??)? ?? as follows. Let v be an
arbitrary valuation; put v(?) = ?x,x?? and v(?) = ?y, y??. Then v(? ? ?) = ?x? y, (x? y)??
and v(? ? ??) = ?x? y?, (x? y?)??. It follows that v((? ? ?)? (? ? ??)) = ?(x? y)? (x?
95
y?), ((x ? y) ? (x ? y?))?? = ?x ? (y ? y?), (x ? (y ? y?))?? = ?x ? ?, (x ? ?)??. Hence,
v((? ? ?)?(? ? ??)? ??) = ?(x? ?)? x?, ((x? ?)? x?)??. Now, because the pair ?x,x??
that interprets ? is such that x? is the pseudo-complement x, (x ? ?) ? x? = x? ? x? = ?.
Thus, v((? ? ?) ? (? ? ??)? ??) = ??,?? = 1.
By contrast, when ?x,x?? is such that its first component is the pseudo-complement of
the second component but not vice versa, we only have (x ? ?) ? x? = ((x? ? ?) ? ?) ? x?,
which is not always equal to the top element in a Heyting algebra.
Additionally, note that by replacing AS16 with the stronger negation introduction (NI)
which is just like AS16 but does not contain any additional restriction on the structure of
?, we would turn LPP into classical propositional logic (CPL).21 In order to validate the
unconstrained NI, the second coordinate of any pair that interprets ? needs to always be
the pseudo-complement of its first coordinate, even if ? is led by a single ?. To satisfy
this, we would need to make the pseudo-complement relation between x and x? symmetric.
This would in turn generate a new class of twist-structures, call them complemented twist-
structures (CTS). Indeed, as we shall see in ?4.5.2, the class of CTS is isomorphic to the
class of Boolean algebras.
Let me highlight some other properties of LPP which we will come back to later in our
proof of completeness.
Proposition 4.4.1. ?LPP (? ? ?)? (??? ??), provided that ? is a non-negated formula.
Proof. Since we have (? ? ?) ? (? ? ??) ? ?? from AS16 and ?LPP ?? ? (? ? ??) from
AS1, we can easily derive ?LPP (? ? ?)? (??? ??).
Corollary 4.4.1.1. ?LPP (?? ?)? (??? ??), provided that both ? and ? are non-negated
formulas.
Remark. As with N3, since (? ? ?) ? (?? ? ??) does not hold generally, ?LPP ? ? ?
alone does not define a congruence relation; instead, in order for ? and ? to belong to the
same congruence class, we need to have both ?LPP ?? ? and ?LPP ??? ??.
Proposition 4.4.2. ?LPP ??? (? ? ?), provided that ? is a non-negated formula.
21More specifically, AS1?8 + NI give rise to minimal logic, upon which the addition of DNE yields CPL
(See Odintsov, 2008).
96
Proof. Left-to-right follows directly from AS10. Right-to-left follows from AS16 together
with the fact that ?LPP ? ? ??, which is derivable from applying contraposition to ?LPP
?? ??, given that ? is a non-negated formula.
Remark. Note that we do not have the contrapositive ?LPP ? ? ?(? ? ?) as ?? is not a
non-negated formula.
Proposition 4.4.3. The following bi-implications are theorems of LPP, provided that ? is
a non-negated formula:
?LPP (? ? ??)? (? ? (? ? ?))
?LPP ?(? ? ??)? ?(? ? (? ? ?))
?LPP (? ? ??)? (? ? (? ? ?))
?LPP ?(? ? ??)? ?(? ? (? ? ?))
?LPP (?? ??)? (?? (? ? ?))
?LPP ?(?? ??)? ?(?? (? ? ?))
?LPP (?? ? ?)? ((? ? ?)? ?)
?LPP ?(?? ? ?)? ?((? ? ?)? ?)
Proof. The four bi-implications where both sides of? are non-negated formulas follow easily
from Proposition 4.4.2. As for the four bi-implications of the negative form, they follow from
their respective positive form by Corollary 4.4.1.1.
Remark. What Proposition 4.4.3 amounts to is that whenever a toggle negation appears
alone embedded below a binary connective, it becomes equivalent to the intuitionistic nega-
tion. This allows us to reduce embedded toggle negations to intuitionistic negations. As an
example, we can convert ?(A ? ?A) to ?(A ? (A? ?)).
Proposition 4.4.4 (Normal form). Formulas of LPP can be converted into either a positive
formula or a positive formula prefixed by a single negation ?, where a positive formula is
defined as a formula that does not contain any ? (but may contain ?).
97
Proof. We prove this inductively. First, all literals apparently satisfy this condition. Next,
we want to show that for any formulas ? and ? that satisfy this condition, any complex
formula that contains ? and ? as its immediate constituents can be transformed into either
a positive formula or a positive formula headed by a single ?. Consider negation first, given
the inductive hypothesis, ? can take the form of either ? or ??, where ? is a positive formula.
If ? takes the form of ?, then ?? becomes ??, which is a positive formula prefixed by a single
?; if ? takes the form of ??, then ?? becomes ???, which given DNE, is reduced to a positive
formula ?. For conjunction, to simplify, let us focus on the structure of ? by granting that
? is a positive formula. If ? takes the form of ?, then ? ?? is just ? ??, which is a positive
formula; if ? takes the form of ??, then ??? becomes ????, which given Proposition 4.4.3,
can be converted to ??(? ? ?), which is again a positive formula. The same would apply to
? if it was a positive formula prefixed by a single ?. Hence, conjunction preserves the above
condition. By similar reasoning, disjunction and implication also produce either a positive
formula or one that is prefixed by a single ?.
Remark. As it turns out, the toggle negation in LPP can only appear in one place where
it makes substantial semantic contribution, that is, at a sentence?s widest scope. As such,
? can be viewed as a global rejection operator under LPP. This result appears to fare well
with our previous characterization of ? as an operator that toggles between the support and
rejection conditions of a sentence.
Corollary 4.4.4.1. LPP can be axiomatized alternatively by replacing AS16, i.e., (? ?
?) ? (? ? ??)? ??, with an axiom that further restricts ? to only positive formulas.
4.4.3 Completeness
Let me first provide a sketch of the proof. We prove completeness by establishing the
following equivalences. (Different notations will be explained immediately afterwards.)
Theorem 4.4.5 (Completeness theorem). The following conditions are equivalent:
1. ?LPP ?;
2. ?&? ?;
3. ?L&? ?;LPP+
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4. ?L ?.LPP
First, we show that (1) entails (2) (viz., ? is a theorem of LPP only if ? is valid in the
class of PTS) by proving soundness. Second, let LLPP be the Lindenbaum-Tarski algebra
of the language of LPP; we show that (4) entails (1) by proving that LPP is complete with
respect to LLPP . Third, let us define another Lindenbaum-Tarski algebra LLPP+ using the
positive fragment of LPP that does not contain ?; let L&?LPP+ be the PTS generated from the
Lindenbaum-Tarski algebra L &?LPP+. Since LLPP+ is a special PTS, (2) entails (3) immedi-
ately. Last, we show that L&?LPP+ and LLPP are indeed isomorphic, thereby establishing the
equivalence between (3) and (4). I shall also remind the readers that in (2)-(4), validity is
defined in term of the inquisitive valuation. For any atomic formula p, the valuation will
assign p and (p? ?)? ? to the same element.
Lemma 4.4.6 (Soundness lemma). If ?LPP ?, then ?&? ?.
Proof. We can directly show that all the axiom schemas from above are valid with respect
to PTS. The proofs for AS1?8 from the positive fragment are straightforward. Take AS1 as
an example. Let v be an arbitrary valuation on an arbitrary PTS, and put v(?) = ?x,x??
and v(?) = ?y, y??. Then, it follows that v(? ? (? ? ?)) = ?x,x?? ??R (?y, y?? ??R ?x,x??) =
?x ? (y ? x), (x ? (y ? x))??. Since x and y are elements from a Heyting algebra,
x ? (y ? x) = ?. Therefore, v(? ? (? ? ?)) = ??,?? = 1. Likewise for AS2?7, since
no negation is involved, we can show that the corresponding formulas hold in the Heyting
algebra.
The validity of AS9 immediately follows from the definition of ? in PTS.
To prove the validity of ex falso as stated by AS10, again let v(?) = ?x,x?? and v(?) =
?y, y??. It follows that v(?? ? (? ? ?)) = ??x,x?? ?? (?x,x?? ?? ?y, y?R R ?) = ?x?, x? ??R ?x ?
y, (x ? y)?? = ?x? ? (x ? y), (x? ? (x ? y))??. Now, given that, x? ? (x ? y) = (x? ? x) ?
y = ?? y = ?, v(??? (?? ?)) = ??,?? = 1. Hence, ?&? ??? (?? ?).
The validity of AS15 immediately follows from the fact that valuations on PTS map every
atomic proposition p to an element of H&??? = {(?x,x???? 0)??R R 0 ??x,x?? ? H&?}.
We have already proven the validity of AS16 in the last subsection.
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Lastly, to prove that modus ponens holds, again let v(?) = ?x,x?? and v(?) = ?y, y??.
Assume v(?) = 1 and v(?? ?) = 1, that is ?x,x?? = ??,?? and ?x? y, (x? y)?? = ??,??. We
want to show that v(?) = 1, that is, ?y, y?? = ??,??. This is straightforward because for any
two elements x and y in a Heyting algebra, if x = ? and x? y = ?, it must follow that y = ?.
And given that y ? y? = ?, it must follow that y? = ?. Thus, ?y, y?? = ??.??.
Definition 4.4.7 (Equivalence relation ?). Define ? as follows: ? ? ? iff ?LPP ? ? ? and
?LPP ??? ??.
Proposition 4.4.8. The relation ? is a congruence on the propositional formula algebra of
LPP: F = ?F ,?,?,?,?,??.
Definition 4.4.9 (Lindenbaum-Tarksi algebra of LPP). Let F /? be the set of congruence
classes ??? induced by ? on the set of formulas F . The Lindenbaum-Tarski algebra of LPP
is defined as follows:
LLPP = ?F /?,?,?,?,?,?,??
where ??? ? ??? ?= ?? ? ??, ??? ? ??? ?= ?? ? ??, ??? ? ??? ?= ?? ? ??, ???? ?= ????,
? ?= ??? ??, and ? ?= ???.
Lemma 4.4.10 (Completeness of LPP w.r.t. LLPP ). If ?L ?, then ?LPP LPP ? .
Proof. We prove its contrapositive. Suppose ?LPP ?. Then ??? ? ?. Let us define a canonical
valuation v0 ? F ? LLPP from the set of formulas to the carrier of LLPP as follows: for
every atomic proposition p, v0(p) = ?p?. Note that given the presence of AS15, we have
?(p ? ?) ? ?? = ?p? for every atomic proposition p, which in turn ensures that v0 still
qualifies as an inquisitive valuation. Now, it follows that for all formulas ?, v0(?) = ???.
Hence, v0(?) = ???. And since ??? ? ?, v0(?) ? ?. Thus, ?L ?.LPP
Definition 4.4.11 (Equivalence relation ?). Define ? as follows: ? ? ? iff ?LPP ?? ?.
Proposition 4.4.12. The relation ? is a congruence on the propositional formula algebra
of the negation-free fragment of LPP: F+ = ?F +,?,?,?,??.
Definition 4.4.13 (Lindenbaum-Tarski algebra L +LPP+). Let F /? be the set of congruence
classes [?] induced by ? on F +. The Lindenbaum-Tarski algebra LLPP+ of the negation-free
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fragment of LPP is defined as follows:
L +LPP+ = ?F /?,?,?,?,?,??
where [?]?[?] ?= [???], [?]?[?] ?= [???], [?]? [?] ?= [?? ?], ? ?= [?? ?], and ? ?= [?].
Definition 4.4.14 (PTS generated from the Lindenbaum-Tarski algebra LLPP+). We define
the PTS of LLPP+ as follows:
L&?LPP+ = ?L&? ?LPP+,?R,??R,??R,?,1,0?
where L&?LPP+ = {?[?], [?]??? [?] ? F +/?} ? {?[?]?, [?]?? [?] ? F +/?}, and
?[?], [??]? ??R ?[?], [??]? ?= ?[?] ? [?], ([?] ? [?])??
?[?], [??]? ??R ?[?], [??]? ?= ?[?] ? [?], ([?] ? [?])??
?[?], [??]???R ?[?], [??]? ?= ?[?]? [?], ([?]? [?])??
??[?], [??]? ?= ?[??], [?]?
1 ?= ??,??
0 ?= ??,??
Next, we want to show that LLPP and L&?LPP+ are indeed isomorphic. To prove this, we
first define two maps between LLPP and L&?LPP+. We then show that they are homomorphisms
that are also inverses of each other.
Definition 4.4.15. We define a map h ? L &?LPP ? LLPP+ from the carrier of LLPP to the
carrier of L&?LPP+:
h(?p?) = ?[p], [p]??
h(?? ? ??) = h(???) ??R h(???)
h(?? ? ??) = h(???) ??R h(???)
h(??? ??) = h(???)??R h(???)
h(????) = ?h(???)
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h(?) = 1
h(?) = 0
Proposition 4.4.16. h is a homomorphism.
Proof. This is obvious given how h is defined.
Definition 4.4.17. We define a converse map h? ? L&?LPP+ ? LLPP :
???
?(?[ ] [ ]?) = ?
???? if [?] = [?]?
h ? , ? ?????????? if [?] = [?]
?
Example 4.4.18. To illustrate, let us consider three examples:
1. h?(?[?A], [?]?) = ??A?,22 provided that the second coordinate [?] is the pseudo-complement
of the first coordinate [?A]?that is, [?] = [?A? ?]?but not vice versa.
2. h?(?[?], [?A]?) = ???A?, provided that the first coordinate [?] is the pseudo-complement
of the second coordinate [?A], but not vice versa.
3. h?(?[A ? A], [?]?) = ?A ? A? = ????, provided that [A ? A] and [?] are pseudo-
complement of each other. In other words, we have that h?(1) = ? = ??.
Proposition 4.4.19. For any ?[?], [?]?, if [?] = [?]? and [?] = [?]?, then h?(?[?], [?]? =
??? = ????.
Proof. Recall again that [?]? = [?? ?]. Since [?] = [?]? and [?] = [?]?, we have ?LPP ??
(? ? ?), and ?LPP ? ? (? ? ?). Next, since both ? and ? appearing in [?] and [?] are
formulas from the negation-free fragment of LPP, they do not contain any ?. By Proposition
4.4.2 where we establish that for any non-negated formula ?, ?LPP ?? ? (? ? ?), it then
follows that ?LPP ? ? ??, and ?LPP ? ? ??. And given that ?LPP ??? ? ?, we have
?LPP ??? ? ??. Since we have obtained both ?LPP ? ? ?? and ?LPP ??? ? ??, by the
definition of the congruence relation on LPP, ? and ?? belong to the same congruence class,
that is, ??? = ????.
22?A abbreviates A ? (A? ?).
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Proposition 4.4.20. h? is a homomorphism.
Proof. We reason by cases.
1. For negation, we have that h?(??[?], [?]?) = h?(?[?], [?]?) =
???????? = ????? = ????? = ?h?(?[?], [?]?) if [?] = [?]?;??
???????? = ???? = ?h
?(?[?], [?]?) if [?] = [?]?.
Hence, regardless of whether the first coordinate is the pseudo-complement of the
second or vice versa, h?(??[?], [?]?) = ?h?(?[?], [?]?).
2. For conjunction, we establish the following equivalences first: h?(?[?], [??]? ??R
?[?], [??]?) = h?(?[?] ? [?], ([?] ? [?])??) = h?(?[? ? ?], [? ? ?]??) = ?? ? ??, given
that [? ? ?]? is the pseudo-complement of [? ? ?].
Next, we show that the following equations hold: ?? ? ?? =
???????? ? ?? if [?
?] = [?]? and [??] = [?]?;
?
????????? ? ???? if [??] = [?]? and [?] = [??]?;
??
???? ????? ? ?? if [?] = [?
?]? and [??] = [?]?;
???
??
?
?????
? ? ???? if [?] = [??]? and [?] = [??]?.
The first case is obvious. It then follows that ?? ? ?? = ??? ? ??? = h?(?[?], [??]?) ?
h?(?[?], [??]?). Therefore, we have that when [??] = [?]? and [??] = [?]?,
h?(?[?], [??]? ??R ?[?], [??]?) = h?(?[?], [??]?) ? h?(?[?], [??]?).
Consider the second case. We want to show that when [??] = [?]? and [?] = [??]?,
that is, when [??] = [? ? ?] and [?] = [?? ? ?], both ? ?LPP (? ? ?) ? (? ? ?? ) and
? ? ?LPP ?(???)? ?(???? ) obtain. First, note that since both ??? and ???? are non-
negated formulas, by the restricted contraposition established in Corollary 4.4.1.1, if we
can prove that ?LPP (???)? (?????), we can prove that ? ?LPP ?(???)? ?(???? ).
Now, consider the proof of ? ?LPP (? ? ?) ? (? ? ?? ). Given that [?] = [?? ? ?], we
have ?LPP ?? (?? ? ?). We can then easily show that ?LPP (???)? (??(?? ? ?)).
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Since ?? is a formula from the negation-free fragment of LPP, it does not contain ?.
Thus, by Proposition 4.4.3 which establishes the equivalence between toggle negation
and intuitionistic negation when they are embedded below a binary connective, it
follows that ?LPP (? ? ?) ? (? ? ???). And since both ? ? ? and ? ? ??? are non-
negated formulas, by contraposition, we have ?LPP ?(? ? ?)? ?(? ? ???).
Because we have established both ?LPP (? ? ?) ? (? ? ???) and ?LPP ?(? ? ?) ?
?(? ? ???), it follows that ?? ? ?? = ?? ? ????. Continuing with the proof that h? is
a homomorphism, we have ?? ? ???? = ??? ? ????? = h?(?[?], [??]?) ? h?(?[?], [??]?),
provided that [??] = [?]? and [?] = [??]?.
We can apply similar reasoning to the last two cases and show that for all four cases,
h?(?[?], [??]? ?? ?[?], [??R ]?) = h?(?[?], [??]?) ? h?(?[?], [??]?).
3. For disjunction, by the same method, we can show that h?(?[?], [??]? ?? ?[?], [??R ]?) =
h?(?[?], [??]?) ? h?(?[?], [??]?).
4. Likewise for implication, we can show h?(?[?], [??]? ??R ?[?], [??]?) = h?(?[?], [??]?) ?
h?(?[?], [??]?).
5. For the two 0-ary operators, we immediately have h?(1) = ? and h?(0) = ?.
This completes our proof that h? is a homomorphism.
Lemma 4.4.21. L &?LPP and LLPP+ are isomorphic.
Proof. Since both h and h? are homomorphisms, we can show that for all x ? L ?LPP , h (h(x)) =
x, and for all X ? L&? ?LPP+, h(h (X)) =X.
1. We prove that for all x ? L , h?LPP (h(x)) = x via an induction.
Base case: for any atomic proposition p, h?(h(?p?)) = h?(?[p], [p]??) = ?p?.
Inductive step: Suppose h?(h(???) = ??? and h?(h(???) = ???. Then,
(a) h?(h(?? ? ??)) = h?(h(???) ??R h(???)) = h?(h(???)) ? h?(h(???)) = ??? ? ??? =
?? ? ??;
(b) By similar reasoning, h?(h(?? ? ??)) = ?? ? ??, and h?(h(??? ??)) = ??? ??;
(c) h?(h(????) = h?(?h(???) = ?h?(h(???) = ???? = ????.
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2. Given we have just established that for all x ? L ?LPP , h (h(x)) = x, we can show for all
X ? L&? ?LPP+, h(h (X)) = X via a reductio. Suppose, on the contrary, that h(h?(X)) ?
X, and let X = h(x). It follows that h(h?(h(x))) ? h(x). But since for all x ?
L , h?LPP (h(x)) = x, it also follows that h(h?(h(x))) = h(x). Therefore, by reductio,
h(h?(X)) =X.
Given that h and h? are inverses of each other, LLPP and L&?LPP+ are indeed isomorphic.
Corollary 4.4.21.1. ?L ? iff ?LPP L&? ?.LPP+
We have completed the proof of completeness.
4.5 Generalizations
As stated in the introduction, a second aim of this paper is to further popularize this method
of twist-construction. To this end, I will explore several generalizations of PTS in this last
section. To begin with, let me first make a conceptual remark regarding the use of twist-
construction. As notated at the beginning of this paper, to reject LEM, we could either adopt
a bivalent or a non-bivalent approach. And as we have seen, in both the case of N3 and the
case of LPP, we can transform a bivalent account to a non-bivalent one via twist-operation
which introduces a toggle negation that rejects LEM. Crucially, what twist-operation enables
is to convert a semantic theory that employs only one central notion to one that employs two
independent central notions. This is reflected in the fact that the carrier of a twist-structure
consists of pairs whose components are themselves elements of the carrier of the original
algebra from which the twist-structure is generated. As such, twist-structures are able to
enrich the semantic interpretation a sentence receives by encoding two separate bodies of
information embodied by it?e.g., its acceptance and rejection conditions.
To generalize, we can view PTS as one of the many structures that can be generated from
applying the twist-operation to a certain base algebra. In using this method, we need to
determine three things: first, we fix an algebra that serves as the base of the twist-operation,
then we define the carrier set of the twist-structure by filtering out a subset of elements from
the product algebra, and lastly we define the logical operators of the new algebra. This is not
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to say that these three parameters are fully independent of each other, since, for example,
selecting a base that does not contain a bottom element will also limit how the carrier can
be defined. In what follows, I shall examine one generalization along each dimension to
illustrate the utility of twist-construction.
4.5.1 The base
The base for the twist-operation is the algebra from which the product algebra is derived.
As we have seen, both PTS and BTS can be regarded as derived from Heyting algebras.
Thus, a natural generalization is to explore other algebras that can serve as the base for
twist-construction. One option is to explore algebras weaker than Heyting algebras. In the
existing literature, twist-structures have been defined using weaker bases such as implicative
lattices (Odintsov, 2004; Rivieccio, 2014) and subminimal algebras (Vakarelov, 2005).
For instance, using implicative lattices as the base gives rise to the so-called full twist-
structures (FTS). Note that since, different from a Heyting algebra, an implicative lattice
does not have to come with a unique bottom element, we cannot use the same carrier as
that of BTS; consequently, the carrier of FTS is simply identified with the carrier of the
whole product lattice. FTS have been used to provide an algebraic semantics for Nelson?s
paraconsistent logic N4, especially given that there is no requirement for the meet of the two
coordinates of any given pair in FTS to be the bottom element.
Here, I wish to explore this generalization in the other direction, that is, to consider
taking algebra that is stronger than the Heyting algebra as the base for twist-construction.
In particular, we can consider taking the inquisitive algebra (Bezhanishvili et al., 2019) as
the base of our pseudo-complemented twist-operation.
Definition 4.5.1 (Inquisitive algebra). An inquisitive algebra I is a Heyting algebra H
that is regularly generated, is well-connected, and validates the Kreisel-Putnam axiom, viz.,
(??? (? ? ?))? ((??? ?) ? (??? ?)).
Definition 4.5.2 (Regularly generated). A Heyting algebra H is regularly generated if it is
generated by H?? = {??x?x ? H}.
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Definition 4.5.3 (Well-connected). A Heyting algebra H is well-connected if for every x
and y in H, if x ? y = 1, then either x = 1 or y = 1.23
Taking inquisitive algebras as the base of the pseudo-complemented twist-operation as
defined in ?4.4 produces a new class of twist-structures, call them Inq-PTS. Correspondingly,
we obtain an extension of LPP, call this logic Inq-LPP. In particular, since the base algebra
now satisfies the KP axiom, we add the following axiom to Inq-LPP:
AS17. ((?? ?)? (? ? ?))? ((?? ?)? ?) ? ((?? ?)? ?))
What we have above is a version of the KP schema where the relevant negation is intuition-
istic. To elucidate its validity, note that any pair from the carrier of Inq-PTS that comes to
interpret AS17 will be such that the pair?s first coordinate is necessarily identical to the top
element given that the base algebra from which the twist-structure is generated now satisfies
KP. Additionally, whenever ? is a non-negated formula, say ?, then given the conversion
between intuitionistic negation and toggle negation, we can derive from AS17 the following
theorem: ?Inq?LPP (?? ? (? ? ?)) ? ((?? ? ?) ? (?? ? ?)). Hence, we also obtain a
restricted version of KP schema for toggle negation.
4.5.2 The carrier
A second parameter we can vary is the twist-structure?s carrier set. In the case of FTS, there
is no restriction on the carrier set; hence every element in the product lattice is included in
the carrier of the resulting twist-structure. In the case of BTS, the restriction is that the
two components of each pair need to be such that their meet is the bottom element. And in
the case of PTS, we further restrict the carrier to pairs wherein either the first component
is the pseudo-complement of the other, or vice versa. In general, after we have fixed a base
algebra, we can explore different ways to determine the carrier set of the twist-structure.
23The two properties of being regularly generated and well-connected help us to further delimit the class
of Heyting algebras suitable for modeling inquisitive logic. The property of being regularly generated directly
follows from the fact that, since inquisitive valuations assign every atomic proposition to a member of H??,
all formulas are thus assigned to elements from the subalgebra generated by H??. On the other hand, well-
connectedness follows from the fact that inquisitive logic, just like intuitionistic logic, satisfies the disjunction
property, that is, if ??? is a theorem of the logic, then either ? must be a theorem or ? must be a theorem
of the logic.
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In this subsection, I shall explore one immediate generalization of PTS which shares
the same base as PTS but impose a further restriction on the carrier. As I briefly men-
tioned in ?4.4.1, we may further curtail the carrier set such that, for any given pair, their
two coordinates are pseudo-complements of each other. Call the resulting twist-structures
complemented twist-structures (CTS), which are generated in the following way.
Given a Heyting algebra H = ?H,?,?,?,?,??, we define a CTS, H&c = ?H&c,?c,?c,?c
,?,1,0?, as follows:
H&c = {?x,x??? x? = x? and x = x??}
?x,x?? ?c ?y, y?? ?= ?(x ? y)??, (x ? y)??
?x,x?? ?c ?y, y?? ?= ?(x ? y)??, (x ? y)??
?x,x???c ?y, y?? ?= ?(x? y)??, (x? y)??
??x,x?? ?= ?x?, x?
1 ?= ??,??
0 ?= ??,??
Complemented twist-structures are closed under the above operations, given that for any x
in a Heyting algebra, x?? and x? are necessarily pseudo-complements of each other.
Theorem 4.5.4. Every CTS is a Boolean algebra, and every Boolean algebra is isomorphic
to a suitable CTS.
Proof. we can prove that every CTS is a Boolean algebra by showing that elements of CTS
satisfy the equational definitions of Boolean algebras (cf. Rasiowa 1974). Take the equation
X ?c ?X = 1 as an example, where X is any element from the carrier of CTS. Suppose
X = ?x,x??. We want to show that if x? = x? and x = x??, then ?x,x?? ?c ?x?, x? = 1 = ??,??.
First, by definition, ?x,x?? ?c ?x?, x? = ?(x ? x?)??, (x ? x?)??. Next, since H is a Heyting
algebra, (x?x?)? = x? ?x??. And given that x and x? are pseudo-complements of each other,
x? ? x?? = x? ? x?? = ?. Thus, we have ?(x ? x?)??, (x ? x?)?? = ??,?? = 1.
Conversely, every Boolean algebra can also be regarded as coming from a CTS. Since
every Boolean algebra is a Heyting algebra, then given any Boolean algebra B, we can
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generate a CTS based on B and show that B is isomorphic to the resulting CTS. This can
be done by mapping every element x in B to the pair ?x,x?? in the resulting CTS.
4.5.3 The operator
After we have determined a base and a carrier, depending on how we choose to define the
logical operators, we may still derive different twist-structures. Here, I shall examine a
variation of PTS, call them left-pseudo-complemented twist-structures (LPTS), which share
the same base and carrier as PTS but with their operators defined differently.
We have remarked earlier that, under the inquisitive setting, there is an asymmetry
between support and rejection in terms of the amount of information required: support in
general demands more information than rejection. Since disjunction under inquisitive seman-
tics plays the role of introducing alternatives, to support a disjunction, an information state
needs to contain enough information to settle on at least one of the alternatives; by contrast,
a conjunction does not introduce any alternatives, and thus to reject a conjunction, it is not
mandatory for an information state to be able to reject one of the conjuncts. As a conse-
quence, in devising a suitable class of twist-structures for modeling inquisitiveness, we define
the support condition constructively but define the rejection condition non-constructively,
thereby taking binary connectives to produce pairs wherein the second-coordinate is always
the pseudo-complement of the first coordinate.
Now, as we go beyond the quest of modeling inquisitiveness, we may consider the asym-
metry in constructiveness between support and rejection as representing different global
standards for acceptance and rejection of information. Under PTS, we have a relatively
stringent standard on information acceptance, that is, we support fewer things, compared
to our standard on information rejection. In particular, to accept a disjunction, we must be
able to support one of its disjuncts.
As a natural generalization, we may entertain reversing the stringency associated with
information acceptance and rejection. We may adopt a relatively stringent standard on
information rejection, that is, we reject fewer things, compared to our standard on informa-
tion acceptance. In particular, to reject a conjunction, we must be able to reject one of its
conjuncts.
109
Formally, we may entertain the converse structures of PTS under which the three binary
connectives produce pairs wherein the first coordinate is always the pseudo-complement of
the second coordinate. Without delving too deep into how the rejection condition of a
conditional should be specified, let us assume that the second coordinates are defined in
accordance with the BTS as devised for the constructive logic N3. Hence, we supply the
following definition for left-pseudo-complemented twist-structures.
Given a Heyting algebra H = ?H,?,?,?,?,??, we define a LPTS, H&? &? ? ?L = ?H ,?L,?L,
??L,?,1,0?, as follows:
H&? = {?x,x??? either x? = x?,or x = x??}
?x,x?? ?? ?y, y?L ? ?= ?(x ? y)?, x ? y?
?x,x?? ?? ? ?L ?y, y ? ?= ?(x ? y) , x ? y?
?x,x???? ?y, y?? ?= ?(x ? y?)?L , x ? y??
??x,x?? ?= ?x?, x?
1 ?= ??,??
0 ?= ??,??
As an abnormality of LPTS, modus ponens is invalid. To illustrate, under LPTS, we
have ? (A ? ?A)? ? and ? ((A ? ?A)? ?)? ?(A ? ?A), but ? ?(A ? ?A). To see this, put
v(A) = ?a, a??. Then, v((A ? ?A) ? ?) = ??, a ? a?? ??L ??,?? = ??,?? = 1 and v(((A ? ?A) ?
?)? ?(A ? ?A)) = ??,????L ?a ? a?,?? = ??,?? = 1, but v(?(A ? ?A)) = ?a ? a?,?? ? 1.
The failure of modus ponens is often a telling trait of certain non-bivalent paraconsistent
logics. For instance, both Priest?s (1979) Logic of Paradoxes and the paraconsistent variant
of N3 (Kapsner, 2015) fail to vindicate it. The reason why modus ponens fails in these
logics is that semantic consequence is defined not as preservation of truth or support but as
preservation of non-falsity or non-rejection. That is,
? ? ? iff in every model and at every w, if w ?0 ? for every ? in ?, then w ?0 ?.
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Consequently, if the implication ? is defined so that ?? ? can receive a non-0 value when ?
also receives a non-0 value even if ? receives 0, then modus ponens fails because the inference
does not preserve non-falsity.
However, for LPTS, the situation is rather different, as semantics consequence in LPTS
is still defined in terms of preservation of support. That is, ? ? ? iff for all v such that
v(?) = 1, v(?) = 1. As a result, LPTS is in fact not paraconsistent, since it does vindicate
ex contradictione quodlibet, viz., ?,?? ? ?.24 Nevertheless, since the rejection condition for
conjunction is constructive in the sense that to reject a conjunction we must be able to reject
either one of its conjuncts, the Law of Non-Contradiction (LNC), viz., ?(? ? ??) is in fact
invalid in LPTS. For instance, as we have just seen in the above counterexample to modus
ponens, ? ?(A ? ?A).
The contrast between the validity of ex contradictione and the invalidity of LNC has
an intuitive explanation. Given that our semantic consequence is defined as preservation of
acceptance. It means if one genuinely accepts both ? and ??, i.e., a contradiction, then one
is expected to accept everything. But this does not equate to saying that for any sentence
?, we are expected to either reject it or reject its negation. Since we might not have enough
information to countenance either option.
Moreover, since the failure of modus ponens is a direct consequence of how implication is
defined in LPTS, we can retain the constructiveness of the rejection condition for conjunction
while at the same time reinstate modus ponens by replacing ??L with a suitable implication.
One immediate candidate for this role is the previous operator ??R as employed in PTS, viz.,
?x,x?? ?? ?R ?y, y ? ?= ?x ? y, (x ? y)??. Modified in this way, the resulting twist-structure
is essentially identical to the PTS, given that the only difference between the two concerns
the definitions for conjunction and disjunction, and as it turns out, ?? ?L and ?L are already
definable in PTS as follows:
?x,x?? ??L ?y, y?? ?= ?(??x,x?? ??R ??y, y??)
24Here is a quick proof of this result. Given how semantic consequence is defined in LPTS, it is impossible
for there to be a valuation v such that both v(?) = ?x,x?? = ??,?? and v(??) = ?x?, x? = ??,??. Consequently,
anything follows.
111
?x,x?? ??L ?y, y?? ?= ?(??x,x?? ??R ??y, y??)
This means that PTS is already capable of modeling both constructive and non-constructive
acceptance for disjunction as well as both constructive and non-constructive rejection for
conjunction.
4.6 Conclusion
This paper highlighted two ways to reject the Law of Excluded Middle and explored a non-
bivalent approach to inquisitiveness via the construction of a suitable class of twist-structures.
Just as N3 extends the standard intuitionistic logic by introducing a notion of counterexam-
ples in addition to the existing notion of proofs, the logic of pseudo-complemented propo-
sitions (or to be more precise, the logic Inq-LPP) extends the standard inquisitive logic by
introducing a notion of rejection in addition to the existing notion of support. And just like
N3, LPP also incorporates a toggle negation which validates Double Negation Elimination
and functions as a switch that toggles between the support and rejection conditions of a
sentence. However, unlike N3, LPP embodies an asymmetry in constructiveness between
support and rejection: support is interpreted constructively but rejection is not. I have pre-
sented an algebraic semantics for LPP via pseudo-complemented twist-structures and proved
completeness. The general utility of twist-construction?that is, as a way to convert a biva-
lent semantics into a non-bivalent one?has been further highlighted via exploration of some
variations of PTS. By varying twist-construction along three dimensions?i.e., the base al-
gebra, the carrier of the twist-structure, and the definitions of the logical operators?we can
generate a wide range of twist-structures which could potentially serve diverse philosophical
purposes.
112
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