# Exhaustivity and anti-exhaustivity in the RSA framework:
Testing the effect of prior beliefs††thanks: We would like to thank Leon
Bergen, Lucas Champollion, Judith Degen, Danny Fox, Michael Franke, Roni
Katzir, MH Tessler for precious discussion, as well as the audiences of the
LPCS research seminar at the University of Latvia, the System analysis group
at Vilnius University, the NYU semantics group, the ESSLLI workshop
“Approaches to implicature: Rational choice and/or exhaustification”, the
LINGUAE seminar at ENS Paris, and the semantics research seminar at UCL. We
gratefully acknowledge funding from the Agence Nationale de la Recherche
(FrontCog Project ANR-17-EURE-0017 and ProbaSem project ANR-19-CE28-0004-01),
the Mind Brain Behavior Interfaculty Initiative at Harvard, and the Research
Council of Lithuania and European Social Fund under Measure 09.3.3-LMT-K-712.

Alexandre Cremers Vilnius University, Filologijos Fakultetas  Ethan G. Wilcox
Harvard University  Benjamin Spector Institut Jean Nicod, Département d’études
cognitives, ENS, EHESS, PSL University, CNRS

###### Abstract

During communication, the interpretation of utterances is sensitive to a
listener’s probabilistic prior beliefs, something which is captured by one
currently influential model of pragmatics, the Rational Speech Act (RSA)
framework. In this paper we focus on cases when this sensitivity to priors
leads to counterintuitive predictions of the framework. Our domain of interest
is exhaustivity effects, whereby a sentence such as Mary came is understood to
mean that _only_ Mary came. We show that in the baseline RSA model, under
certain conditions, _anti-exhaustive_ readings are predicted (e.g., Mary came
would be used to convey that both Mary and Peter came). The specific question
we ask is the following: should exhaustive interpretations be derived as
purely _pragmatic_ inferences (as in the classical Gricean view, endorsed in
the baseline RSA model), or should they rather be generated by an encapsulated
_semantic_ mechanism (as argued in some of the recent formal literature)? To
answer this question, we provide a detailed theoretical analysis of different
RSA models—some of which include such an encapsulated semantic mechanism—, and
evaluate these models against data obtained in a new study which tested the
effects of prior beliefs on both production and comprehension, improving on
previous empirical work. We found no anti-exhaustivity effects, but observed
that message choice is sensitive to priors, as predicted by the RSA framework
overall. The best models turn out to be those which include an encapsulated
exhaustivity mechanism (in line with other studies which reached the same
conclusion on the basis of very different data). We conclude that, on the one
hand, in the division of labor between semantics and pragmatics, semantics
plays a larger role than is often thought, but, on the other hand, the
tradeoff between informativity and cost which characterizes all RSA models
does play a central role for genuine pragmatic effects.

##  1 Introduction

The effect of linguistic utterances in context depends in part on the prior
beliefs of speakers and hearers. Suppose that Mary and Peter are known to be a
couple, who usually go out together, so that, upon learning that Mary came to
a certain party to which Peter was also invited, one would assign a high
probability to the possibility that Peter came as well. If someone is asked
Did Mary come to the party? and answers with she did, the questioner would be
warranted to infer that Peter probably came too. Furthermore, if the question
is instead Did Peter come to the party?, an answer such as I am not sure, but
Mary did would again suggest that it is quite likely that Peter came to the
party. Typically, though, pragmatic inferences go well beyond what can be
predicted with such a simple model of linguistic interpretation. They also
involve, for instance, reasoning about other sentences that the speaker could
have uttered given some assumptions about their communicative goals (Grice
1975). For instance, in response to the question Among Gloria, Adam, Fred and
Sue, who attended the show today?, an answer such as Adam did tends to trigger
the inference that the others did not, even if there is no expectation that
what any of them does depends on what the others do. Such exhaustivity effects
can be accounted for in terms of Grice’s maxim of quantity: if in fact both
Adam and Gloria had attended the show, a knowledgeable speaker would say that
rather than just talking about Adam. Alternatively, several arguments have
been put forward in the literature according to which such exhaustivity
effects are linguistically encoded and result from the presence, in the
logical form of the sentence, of an exhaustivity operator (see, e.g.,
Chierchia . 2012), whose meaning is, in first approximation, akin to that of
_only_ (_Only Mary did_ entails that nobody else attended the show). In both
types of accounts (Gricean and operator-based), a crucial ingredient is the
idea that the sentence being interpreted (Mary did) is enriched with the
negation of some alternative sentences (e.g, Peter and Mary did).

Now, in some situations, these two types of effects (effect of prior beliefs,
exhaustivity effects) are pitted against each other (for a recent discussion
of this, see Wilcox Spector, 2019; Schreiber Onea, 2021). Consider again the
case where Peter and Mary are known to typically go out together, and someone
asks Among Peter, Mary, Fred and Sue, who attended the show today?. What
should we expect an answer such as _Mary did_ to convey? On the one hand,
given prior probabilistic knowledge, one should be led to conclude that Peter
probably attended the show. On the other hand, the answer might trigger an
exhaustive interpretation, whereby one would conclude the opposite, namely
that only Mary attended the show.

Our goal in this paper is to investigate the relationship between prior
beliefs and exhaustivity effects, working within a pragmatic modeling
framework, the Rational Speech Act framework (Frank Goodman, 2012; Goodman
Stuhlmüller, 2013). By modeling pragmatics as joint probabilistic reasoning
between speaker/listeners layers, the RSA makes it possible to incorporate
prior probabilistic knowledge into both message interpretation and message
choice. We start at the theoretical level, presenting a detailed comparison of
nine different variants of the RSA framework and an analytic description of
the conditions under which each will produce exhaustivity effects. We
demonstrate that, in addition to exhaustivity, many of the model variants
tested are predicted to produce anti-exhaustivity effects; a theoretical
prediction distinctive to the RSA framework in which under certain conditions
a sentence such as _Mary came_ ends up conveying, for instance, that both
Peter and Mary came.111Degen . (2015), which we will discuss in detail in
section 4.1, as well as Wilcox Spector (2019), Schreiber Onea (2021), and Fox
Katzir (2021), have pointed out before us that the predictions of the RSA
model (and related frameworks in the case of of Fox Katzir 2021) are extremely
sensitive to priors, in a way that is sometimes problematic. To evaluate the
model variants, we deploy an experimental paradigm that assesses the effect of
prior beliefs on the interpretation and production of sentences, allowing us
to test our RSA models as joint models of comprehension and production. We
find strong evidence for exhaustivity effects, as well as that these effects
can be influenced by prior knowledge; however, we find no evidence for anti-
exhaustivity effects in human participants. Our joint data reveal that, for
this pragmatic phenomenon, neither production nor comprehension can be both
rational and well-informed, meaning that the best-fitting models are either
rational but do not share some information between speakers/listeners, or
models which depart from the rationality assumption for one of the agents. In
addition, the lack of anti-exhaustivity effects in the data mean that the
best-fitting RSA models are those which break the exhaustivity/anti-
exhaustivity symmetry in the semantics via an encapsulated, non-pragmatic
mechanism such as grammatical exhaustification.

In order to set up a crucial dimension along which different RSA variants can
differ from each other, we start in Section 2 with a discussion the so-called
_symmetry problem_ for exhaustivity effects. We show that In contrast to
‘logic-based’ approaches to exhaustivity, the baseline RSA model can predict
not only exhaustive readings, but also, and quite surprisingly, anti-
exhaustive readings, whereby sentences are enriched with (the positive form
of) some of their alternatives.

##  2 The symmetry problem: exhaustivity and anti-exhaustivity

Consider an utterance of a certain sentence A, in a context where the truth-
values of two sentences A and B are relevant. Suppose the speaker is fully
informed about A and B. If she believes that A is true and B is false, then
the message A and not B is the most informative sentence she can use
truthfully. If she believes that both A and B are true, then A and B is the
most informative sentence she can use truthfully. So, if the speaker’s choice
of a message were only determined by informativity considerations, A would
never be used by a fully informed speaker, and should therefore give rise to
the inference that the speaker does not know B’s truth-value. Since, in fact,
an utterance of A typically gives rise to an exhaustivity effect, one must
include a mechanism to break the symmetry between the two more informative
messages A and B and A and not B.

In the neo-Gricean tradition, the solution is based on the notion of
_alternatives_. It is assumed that A and B is an alternative of A, but that A
and not B is not. The exhaustive reading of a sentence is obtained by negating
the alternatives that entail it (or, depending on the details, the
alternatives that are not entailed by it). So in the case of A, we add the
negation of A and B, and we get $A\wedge\neg B$. This approach is shared both
by pragmatic accounts in the spirit of Grice (e.g., Sauerland, 2004) and by
the so-called ‘grammatical approach’, based on exhaustivity operators.

In recent years, several probabilistic, decision-theoretic models of
pragmatics have proposed, among other things, a new formalization of
exhaustivity effects (see, a.o, Parikh 2001; Benz . 2005; Jäger 2012; Franke
2011; Rothschild 2013; Frank Goodman 2012; Goodman Stuhlmüller 2013; Bergen .
2016). We specifically focus here on the Rational Speech Act model (Frank
Goodman 2012; Goodman Stuhlmüller 2013; Bergen . 2016), which has become
extremely influential. While we will present the details of the framework in
Section 3, what is sufficient for the current discussion is that in this
model, pragmatic reasoning is treated as probabilistic recursive reasoning
between different ‘speaker’ and ‘listener’ layers. The speaker’s choice of a
message is governed by a tradeoff between informativity and cost: that is, the
speaker tries to pick a message that will be maximally informative, in an
information-theoretic sense, while minimizing the cost of the message. While
in some specific applications, the set of messages that are considered in a
given model is defined in terms of the classical neo-Gricean alternatives,
neo-Gricean alternatives can in fact be entirely dispensed with, because cost
can essentially serve the same purpose.222Of course, for any model to be
tractable, it will include a finite number of messages, so in a sense every
model makes certain commitments regarding alternative messages. In specific
applications, typically, only a narrow range of messages is considered. This
is sometimes done on the basis of the classical neo-Gricean notion of
alternative, in which case there is no need to include a cost for messages
(see Franke, 2009, §2.3.1 for an early discussion of whether to use cost or
neo-Gricean alternatives to break symmetry in a probabilistic model of
implicatures). But the underlying assumption of an RSA model that includes a
notion of cost is that the availability of messages to the speaker depends on
their cost and, in particular, that cost is responsible for breaking the
symmetry between alternative messages. A model with cost of course includes a
finite set of messages, but without relying on, say, Horn scales to solve the
symmetry problem. The cost of a message might be viewed as determining the
prior probability that a certain message will be used, independently of the
speaker’s communicative intent (the higher the cost, the less likely it is to
be used), so that not including a message amounts to assigning it an infinite
cost. In the conclusion, we come back to the implications of relying on the
neo-Gricean notion of alternatives rather than on cost within the RSA
framework.  Things work as follows. A and not B is more costly than A and B.
For this reason, a speaker will typically be more likely to choose the less
informative message A when A is true and B is false than when both A and B are
true. In both cases, the more informative message, which is either A and not B
or A and B, is more costly than A itself, but the cost disadvantage is more
pronounced in the case of A and not B. As a result, the speaker has a stronger
incentive to use the cheap but underinformative message A when A is true and B
is false than when both A and B are true. Upon hearing A, then, a Bayesian
listener will infer that the former situation is more likely than the latter.

However, while symmetry _can_ be broken by cost, it can also be broken by the
other term of the tradeoff that governs message-choice, namely informativity,
which is defined in probabilistic terms. Consider a context where the prior
probability of B being true given that A is true is very high. In such a
context, if the speaker believes that A is true and B is false, then the
message A, in terms of its literal meaning, is very bad at communicating the
speaker’s belief: upon learning A, a literal listener will assign a very high
probability to B being true. Even if it does well on the cost side of the
tradeoff, it does not do well at all on the informativity side. On the other
hand, if the speaker believes that both A and B are true, then A is a very
good message to communicate her belief: by simply saying A, the speaker is
sure that the listener will assign a very high probability to B being true, so
the message A is almost as good as the message A and B in terms of
informativity, and it is otherwise better in terms of cost.333In comparison,
even when the conditional probability that, say, all $X$s $Y$ed given that
some $X$s $Y$ed is very high, a some-message can never be better than an all-
message to convey that all $X$s $Y$ed, because all is no more costly than
some, but is more informative. While Degen . (2015) showed that the baseline
RSA model predicts too strong an influence of priors in such case, a model
where the speaker is fully rational would not have this problem (some would
never be used by a fully rational speaker to communicate all). In contrast
with this, even a fully rational speaker may prefer to use A to convey that A
and B is true instead of A and B, because A and B is more costly than A.

In other words, the speaker might now be _more likely_ to say A and not B when
A and not B is true than she is to say A and B when A and B is true. This
means that she would be comparatively more likely to use A when A and B is
true than when A and not B is. The symmetry between the two alternative
messages A and B and A and not B is now broken by informativity
considerations, in favor of A and not B. For this reason, the baseline RSA
model predicts cases of _anti-exhaustivity_ : when the priors are sufficiently
biased, i.e. when $P(B$ true$|A$ true$)$ is high enough, the pragmatic
listener, after processing the message A, will assign a higher probability to
B being true than what the priors by themselves warrant on the basis of the
literal meaning of A: $P(B\text{ true}|\text{ speaker said`}A\text{'})$ (the
posterior probability assigned to B being true after processing the message A)
will be higher than $P(B\text{ true}|A\text{ true})$ (the conditional prior
belief).

Now, as has been previously noted (Degen ., 2015; Javangula, 2019; Wilcox
Spector, 2019; Fox Katzir, 2021; Schreiber Onea, 2021), this prediction seems
to be undesirable. Consider a context where Mary is known to be nearly always
accompanied by Peter anywhere she goes. In the context of a question such as
Did Mary come to the party?, the answer Yes she did could plausibly license
the inference that Peter probably was there too. In such a case, though, it is
not part of the speaker’s communicative intention to convey that Peter was
there, given the underlying question, which is only about Mary. In contrast,
if the question is Who came among Peter, Mary and Sue?, the answer Mary did is
very unlikely to be used, it seems to us, by a speaker who wants to convey
that Mary and Peter came but Sue didn’t, and would not be so interpreted (see
section 5.1 for previous experimental results confirming this point).

But while the baseline model is subject to potentially unwanted anti-
exhaustivity effects, it is not the only one model within the RSA framework,
and different RSA models will make different predictions. For instance, Degen
. (2015) address a closely related problem and offer a solution based on a
refined RSA model, where the listener is, in a sense, uncertain about the
priors that the speaker is assuming (we will discuss their proposal in details
in section 4.1). Let us also note that the baseline RSA model, despite making
_prima facie_ undesirable anti-exhaustivity inferences, makes at the same time
a plausible prediction about the influence of priors on message choice.
Namely, it predicts that if $A\wedge\neg B$ is antecedently much less likely
to be true than $A\wedge B$, a speaker who wants to convey $A\wedge\neg B$
will have an extra incentive to use the very explicit message A and not B
rather than simply say A. For instance, in the above situation, the speaker
might be more likely to say Mary, but not Peter than in a situation where
there is no expectation that Peter and Mary would typically come to the party
together (see Chemla Romoli, 2016, §3 for a similar observation regarding
some/some but not all).

The rest of this paper proceeds as follows: In section 3 we present the
baseline model, and provide an analytic description of when it produces both
exhaustivity and anti-exhaustivity effects. In section 4 we present
alternative variants of the baseline RSA model, such as the ‘wonky worlds’
model (Degen ., 2015) as well as models in which sentences are potentially
ambiguous, which makes it possible to treat exhaustive interpretations as
being derived in the semantics: the Lexical Uncertainty model (Bergen .,
2016), Degen . (2015), the Lexical Intentions model (Franke Bergen, 2020), as
well as another variant of the RSA framework that we call the
‘supervaluationist model’ (based on Spector, 2017). In each case, we discuss
whether these models can in principle predict anti-exhaustive readings and
whether they predict an influence of priors on message choice. Finally, in
Section 5, we will report the results of an experiment probing both
interpretation and production and compare how well different models are able
to account for the results.

##  3 The baseline RSA model and anti-exhaustivity effects

###  3.1 The base-line RSA model

The RSA framework is a game-theoretic and decision-theoretic model of
pragmatics. Speakers are modeled as approximately rational speakers whose
choice of a message is governed by a tradeoff between the _informativity_
(measured in information-theoretic terms) and the _cost_ of messages.
Listeners are viewed as Bayesian agents who update their beliefs about the
world and the communicative intentions of the speaker based on the message
that the speaker used. The model defines a sequence of listeners and speakers
of increasing sophistication. Starting with a literal listener who simply
conditionalizes her belief state (viewed as a probability distribution) on the
meaning of the message, the model defines a first-level pragmatic speaker who
believes she is talking to the literal listener, and chooses her message
accordingly. It then defines a first-level pragmatic listener who uses Bayes’
rule to update her beliefs, on the assumption that the received message was
produced by the pragmatic speaker. An even more sophisticated (second-level)
pragmatic speaker can then be defined as one who believes that she is speaking
to the first-level pragmatic listener, and so on.

The baseline RSA model includes: a set of possible world states (worlds for
short); a probability distribution $P$ over world states which can be thought
of as representing the listener’s information state, known to the speaker,
before she heard any message; a set of messages, each of which have a fixed
meaning (a function from worlds to truth values). Each message $m$ has a
_cost_ $c(m)$. Finally, the model includes a ‘rationality parameter’ $\lambda$
(a positive real number) that regulates how rational the speaker is when
choosing a message (the higher $\lambda$ is, the more likely it is that the
speaker will choose the best message).

The literal listener, $L_{0}$ interprets messages by conditionalizing their
distribution over world states, $P$, with the proposition expressed by the
literal meaning of the message. If $u$ is a message, we identify its meaning
with the set of worlds in which it is true, and notate this set $\llbracket
u\rrbracket$, and we notate $\llbracket u\rrbracket(w)$ the truth-value of $u$
in $w$ . $L_{0}$ is defined by the probability it assigns to a world $w$ after
having processed a message $u$:

. $L_{0}(w|u)=P(w|\llbracket
u\rrbracket)=\left\\{\begin{array}[]{cl}0&\text{if $w\not\in\llbracket
u\rrbracket$,}\\\ \frac{P(w)}{P(\llbracket u\rrbracket)}&\text{if
$w\in\llbracket u\rrbracket$}\end{array}\right.$
Equivalently (using the proportionality notation): $L_{0}(w|u)\propto
P(w)\llbracket u\rrbracket(w)$

The 1st-level pragmatic speaker is characterized by a utility function which
assigns a utility to a message-world pair, based on how the literal listener
interprets $u$ and the cost of the message, notated $c(u)$:

. $U_{1}(u,w)=\log(L_{0}(w|u))-c(u)$

Importantly, this function expresses a tradeoff between informativity and
cost. The first term is an increasing function of the probability assigned by
$L_{0}$ to $w$ after having processed $u$, and so represents how effective $u$
at communicating $w$ to $L_{0}$.

The level-1 speaker is characterized by a function $S_{1}$ which determines
how likely the speaker is to choose a message $u$ when she wants to
communicate $w$, and is defined by the following equation:

. $S_{1}(u|w)=\dfrac{\exp(\lambda
U_{1}(u,w))}{\sum\limits_{u^{\prime}}\exp\left(\lambda
U_{1}(u^{\prime},w)\right)}$

When $\lambda$ approaches infinity, this speaker approaches a fully rational
speaker who, in world $w$ will choose a message which maximizes $U_{1}(u|w)$
with probability 1. When $\lambda$ is finite, the speaker is only
approximately rational, and messages that yield greater utility have a grater
probability of being used than less useful messages, but they are not certain
to be used.

Then we can define a pragmatic listener $L_{1}$ who believes that she is
receiving a message from $S_{1}$ and uses Bayes’ rule to update a probability
distribution over worlds when processing a message $u$:

.
$L_{1}(w|u)=\dfrac{P(w)S_{1}(u|w)}{\sum\limits_{w^{\prime}}P(w^{\prime})S_{1}(u|w^{\prime})}$

On this basis, higher-level speakers and listeners are defined recursively as
follows:

. . $U_{n+1}(u|w)=\log(L_{n}(w|u))-c(u)$ .̱ $S_{n+1}(u|w)=\dfrac{\exp(\lambda
U_{n+1}(u,w))}{\sum\limits_{u^{\prime}}\exp(\lambda U_{n+1}(u^{\prime},w))}$
.̧
$L_{n}(w|u)=\dfrac{P(w)S_{n}(u|w)}{\sum\limits_{w^{\prime}}P(w^{\prime})S_{n}(u|w^{\prime})}$

###  3.2 The symmetry problem and the tradeoff between informativity and cost:
exhaustivity and anti-exhaustivity

Coming back to exhaustivity, let us consider a very simple RSA model, with
just two worlds, $w_{a}$ (in which A is true but B is false) and $w_{ab}$ (in
which both A and B are true), and three messages: A, $A\wedge B$ (true if both
A and B are true) and $A\wedge\neg B$ (true if A is true and B is false). We
set the cost of A to 0, and we notate the costs of the two other messages
$c_{A\wedge B}$ and $c_{A\wedge\neg B}$. In such a model, A is uninformative,
as it is true in both worlds. Suppose the speaker is fully informed about the
world. In order to communicate $w_{a}$, the speaker will choose between A and
$A\wedge\neg B$ (the message $A\wedge B$ is false in $w_{A}$, and so its
utility will be infinitely negative). In order to communicate $w_{ab}$, the
speaker will choose between A and $A\wedge B$. The statements in describe the
precise conditions under which anti-exhaustivity effects are expected (see
Appendix A.1 for proofs):

. Anti-exhaustivity

$\begin{array}[]{llcl}\text{a.}&S_{1}(A|w_{ab})>S_{1}(A\wedge
B|w_{ab})&\text{iff}&-\log(P(w_{ab}))~{}<~{}c_{A\wedge B}\\\
\text{{b}.}&L_{1}(w_{ab}|A)>P(w_{ab})&\text{iff}&\log(P(w_{ab}))-\log(P(w_{a}))>c_{A\wedge\neg
B}-c_{A\wedge B}\end{array}$

[a] expresses the condition for observing an anti-exhaustivity effect on the
speaker side. It means that in $w_{ab}$, the level-1 speaker will prefer to
use the underinformative message A rather than $A\wedge B$ if the information
gain that would be brought by $A\wedge B$ (measured by $-\log(P(w_{ab}))$) is
not worth its cost: this will happen when $w_{ab}$ has a very high prior
probability, so that the speaker has no incentive to use a costly message to
convey $w_{ab}$, given that the listener already assigns a high probability to
$w_{ab}$. [b] expresses the condition for observing anti-exhaustivity on the
listener side. This condition is the mathematically precise version of the
fact that symmetry between alternative messages can be broken by
informativity: when the bias in favor of $w_{ab}$ compared to $w_{a}$
(measured by the log-odds $\log(P(w_{ab}))-\log(P(w_{a}))$) exceeds the cost
disadvantage of $A\wedge\neg B$ compared to $A\wedge B$, A is more likely to
be used in $w_{ab}$ than in $w_{a}$, and as a result the listener tends to
interpret A as conveying $w_{ab}$.

Figure 1: Illustration of the anti-exhaustivity effect with high priors in
baseline RSA ($\lambda=3$, $c_{A\wedge B}=.5$, $c_{A\wedge\neg B}=1$).

To illustrate, Figure 1 represents the predicted interpretation of A as a
function of the prior probability of $w_{ab}$, for a certain choice of values
for the parameters of the model (costs, rationality parameter). While anti-
exhaustivity is a problematic prediction of the baseline RSA model, [a] has a
pendant which appears to be a good prediction:

. $S_{1}(A\wedge\neg B|w_{a})>S_{1}(A|w_{a})$ iff
$-\log(P(w_{a}))>c_{A\wedge\neg B}$

In words: in $w_{a}$, the speaker will prefer the explicit and fully
informative message $A\wedge\neg B$ over the less informative message A if the
prior probability of $w_{a}$ is low enough, so that the informativity gain
brought by the fully informative message (measured by $-\log(P(w_{a}))$)
exceeds its cost.

To sum up, the baseline RSA makes a number of interesting predictions when
priors are such that $P(w_{ab})$ is much higher than $P(w_{a})$:

  * •

A speaker who believes $w_{a}$ is much more likely to say $A\wedge\neg B$ than
in the case of non-biased priors.

  * •

A speaker who believes $w_{ab}$ will tend use the message A rather than
$A\wedge B$, even when the truth-value of both A and B are relevant (anti-
exhaustivity on the speaker side).

  * •

The message A will tend to be interpreted as conveying $w_{ab}$ (anti-
exhaustivity on the listener side).

As has been established by previous studies (Javangula, 2019; Wilcox Spector,
2019; Schreiber Onea, 2021), and as we will find in our own experiments
discussed in Section 5, the last two predictions are problematic insofar as
human subjects do not produce anti-exhaustivity effects. The first one,
however, is at least qualitatively correct. In the next section, we turn to
more sophisticated RSA models and ask whether they can solve the symmetry
problem without giving rise to anti-exhaustivity, but without losing the
predicted effect of priors on the choice between A and A$\wedge\neg$B in the
$w_{a}$ situation.444A more complex model that also includes the messages B
and not B makes the same qualitative predictions. In such a model, we need to
include worlds in which $A$ is false but $B$ is true, since otherwise B is
simply equivalent to A and B. Under uniform priors, in such a model, the
speaker is even more likely to use A in $w_{a}$ than in $w_{ab}$, because in
$w_{ab}$ the message B, which is no more costly that A, could be used, while
it could not be used in $w_{a}$ (while not B, which is true in $w_{a}$, is
more costly than $A$). Under biased priors (i.e. when
$P(w_{ab}|\\{w_{a},w_{ab}\\})$ is very high), A can still be more likely to be
used in $w_{ab}$ than in $w_{a}$, because A would be a very bad message to use
in $w_{a}$, while it is a quite good message to communicate $w_{ab}$.
Furthermore, A might be the best message to convey $w_{ab}$, because A and B
is more costly and only slightly more informative, while B, while no more
costly, is not necessarily more informative (since it is compatible with a
world where only $B$ is true, whose prior probability could be high).

##  4 More sophisticated RSA models

As mentioned earlier, the baseline RSA model can be enriched in various ways.
For instance, it can incorporate, as variables that speakers and listeners
reason about, uncertainty about priors, uncertainty about what information is
contextually relevant, or the Question Under Discussion (as in Kao . 2014),
message ambiguity (as in the Lexical Uncertain model of Bergen ., 2016, the
Lexical Intention model of Franke Bergen, 2020, or Spector’s 2017
‘supervaluationist’ model described in section 4.2).555In all these cases, a
fixed parameter of the baseline RSA becomes a probabilistic variable which is
subject to Bayesian inference – often then called a ‘lifted variable’ – or
enters into the speaker’s utility function (as in Spector 2017). In
particular, the RSA framework is by itself compatible with the ‘grammatical
approach’ to exhaustivity, whereby sentences are systematically ambiguous
between a ‘plain’ and an ‘exhaustified’ meaning. In this section, we describe
and discuss the various models we will compare, namely Degen . (2015), Spector
(2017), the Lexical Uncertainty model (Bergen ., 2016) and the Lexical
Intention model (Franke Bergen, 2020), focusing in particular on whether these
models predict antiexhaustivity effects, in the sense discussed in Section
3.2.

###  4.1 The “wonky world” RSA model

Degen . (2015), to our knowledge, is the only proposal within the RSA
framework which is intended to address the problem of anti-exhaustivity,
focusing on the case of some vs. all. Suppose that someone throws marbles into
a lake. Typically, one expects that if one them sinks, then all of them will.
That is, the conditional probability that all marbles sank given that some did
is very high. For this reason, in the baseline RSA model, an utterrance of
Some of the marbles sank (some for short) ends up conveying that all of the
marbles sank. This is not intuitively a good result. Degen . discuss the
results of an experiment in which they tested the influence of priors on
interpretation, by varying the predicates used in the frame _Some NPs VP_.
Schematically, they collected priors on the number of $NP$s expected to have
the property denoted by the $VP$ before any message is presented, and then
posteriors after a some-message is presented. While they found an influence of
priors on interpretation, the best-fitting baseline RSA model does not fit the
data well because it assigns too high a probability to the possibility that
all the marbles sank after the some-message is processed. This was so even
though in their implementation, Degen . did not include the message some but
not all, so that the baseline RSA model could not, properly speaking, give
rise to genuine anti-exhaustivity (where the listener would increase the
probability that all the marbles sank after hearing some beyond what the
priors would warrant). If we add to the set of messages the some but not all
message and assign it a cost higher than that of other messages but not
infinite, the predictions are even worse.

Degen . (2015) propose a modified RSA model which they argue is much better at
predicting the interpretation of some-sentences when priors are biased. The
core idea is that the listener is uncertain about which priors the speaker is
assuming, and she jointly reasons about these priors as well as the state of
the world. In a nutshell, when the priors are very biased in favor of all, the
message some has a relatively low probability of use. Upon receiving some, the
listener then reasons that the speaker must have been assuming a less biased
prior distribution, which prevents her from deriving anti-exhaustivity.

In Degen .’s model, the listener and the speaker are communicating about some
dimension of interests (for instance whether none, some but not all, or all of
the marbles sank), but the listener is uncertain about the ‘background
assumptions’ that the speaker takes to be common knowledge. For instance, the
‘default’ background assumption might be that all marbles are made out of the
same material, so that either all will sink or none will. The ‘wonky’
background assumption would be one where the marbles are assumed not to be
made of the same material, so that there is a very high probability that some
but not all of them will sink. There are two values for the background
assumption, call them $b_{1}$ and $b_{2}$, and the prior probability of a
given world $w$ relative to a certain background assumption $b$ is $P(w|b)$.
The listener herself has a prior over $b_{1}$ and $b_{2}$ (that is, her prior
distribution is a joint distribution over $(w,b)$). The situation described by
the model is one where the speaker is (i) fully informed both about $b$ and
$w$, and (ii) believes (possibly wrongly) that the value of $b$ is common
knowledge. At the same time, the listener—while ignorant about $b$—knows that
the speaker knows the value of $b$ and that she believes (wrongly) that the
listener knows it. $P(w|b)$ thus represents both ‘the probability of $w$ given
$b$’ and ‘the probability that $w$ is the case given that the speaker believes
that $b$ is common knowledge’ (since the speaker believes that $b$ is common
knowledge if and only if $b$ is in fact the case). The model is based on the
following equations:

  1. 1.

$L_{0}(w|u,b)\propto P(w|b)\times\llbracket u\rrbracket^{w}$

  2. 2.

$U_{1}(u|w,b)=log(L_{0}(w|u,b))-c(u)$

  3. 3.

$S_{1}(u|w,b)\propto e^{\lambda U_{1}(u|w,b)}$

  4. 4.

$L_{1}(w,b|u)\propto S_{1}(u|w,b)\times P(w|b)\times P(b)$

The equation for $L_{0}$ describes the behavior of a listener who believes
that the background assumption $b$ is true. The equations for $U_{1}$ and
$S_{1}$ make sense for a speaker who believes that the listener is assuming
the background assumption $b$. The level-1 listener computes the joint
posterior distribution on $(w,b)$ using Bayes’ rule. This equation captures
the following kind of reasoning: given that the speaker used message $u$, I
should give higher credence to pairs $(w,b)$ such that the speaker was
particularly likely to use $u$ if $b$ and $w$ were in fact the case and the
speaker assumed that I know $b$.

In their experiments, Degen . collected priors from participants for different
lexical choices, so as to vary the priors, as well as posterior beliefs about,
e.g., how many marbles sank given that the message some of the marbles sank
was used. They evaluated their model by fitting various parameters to the
observed data. Among the parameters that are fitted is the prior on background
assumptions $b_{1}$ and $b_{2}$, where $b_{1}$ is an assumption that
corresponds to a uniform prior (the ‘wonky world’) over $w$, and $b_{2}$
corresponds to the measured prior over $w$ ($w$ being the number of marbles
that sank). They obtained the best fit with a nearly equiprobable prior on
$b_{1}$ and $b_{2}$, and the model is much better at predicting the observed
interpretation that the baseline RSA model.

However, on the assumption that the measured distribution over worlds truly
represents prior probabilities for the relevant cognitive computation, the
implementation of the model proposed in Degen . is, in a fundamental sense,
non-Bayesian. The reason is that this implementation uses the measured
distribution as the value for the “non-wonky” prior $P(w|b=b_{2})$, yet the
prior over $w$ that the model actually assigns to the pragmatic listener is
the _unconditional_ prior $P(w)=P(b_{1})\times P(w|b_{1})+P(b_{2})\times
P(w|b_{2})$. Because, in the best-fitting model, $b_{1}$ (the ‘wonky-world’
prior) reaches 0.5, this means that the model attributes to the listener a
prior over $w$ which is quite different from the one that is measured
experimentally. Importantly, what makes this implementation non-Bayesian is
the way the data and the model are linked, not the model itself and its
equations.

This non-Bayesian aspect is not by itself problematic. The idea would be that,
in most situations, including most experimental settings, listeners are simply
oblivious to some possibilities which, upon reflection, they might realize
have a significant probability. For instance, they would completely ignore the
possibility that the marbles do not have all the same density, might be made
of unusual material, etc. In other words, listeners might be unable to access
their true prior, and report a prior restricted to a sub-part of the logical
universe where things are ‘normal’ enough. However, this does not strike us as
very plausible if in fact the ‘unnatural’ background assumption (i.e. the
abnormal part of the logical universe) has a prior probability as high as 0.5,
which is needed for the model to deliver good predictions—while at the same
time listeners would entirely ignore this possibility when reporting their
priors. Alternatively, the parameter being fitted can be thought of as a
backoff parameter, or the degree to which prior knowledge should be abandoned
in favor of uniform belief. Indeed, with $P(b_{1})=0$, Degen .’s 2015 model
just becomes the baseline model based on the measured prior, while with
$P(b_{1})=1$, what we get is the baseline model with complete ignorance of the
measured prior, and uniform priors instead. In a sense, the proposed model
could therefore be viewed as gradually incorporating the possibility of ‘base-
rate neglect’, whereby listeners would not take into account their prior
(Kahneman Tversky, 1973).

One possible way of modifying Degen .’s model is to drop the assumption that
the speaker’s belief about ‘background’ assumptions is viewed by the listener
as necessarily true. In particular, such a model would no longer equate
$P(w|$‘the speaker believes $b$’$)$ with $P(w|b)$. Rather, the model now
assumes that the listener uses her own prior over $w$ (which corresponds to
the non-‘wonky’ priors, $b_{2}$) to update her beliefs, and has otherwise a
prior over the various background assumptions that the speaker might be
entertaining (possibly wrongly). The modified model is described by the
following equations:

  1. 1.

$L_{0}(s|u,b)\propto P(s|b)\times\llbracket u\rrbracket^{s}$

  2. 2.

$U_{1}(u|s,b)=log(L_{0}(s|u,b))-c(u)$

  3. 3.

$S_{1}(u|s,b)\propto e^{\lambda U_{1}(u|s,b)}$

  4. 4.

$L_{1}(s,b|u)\propto P(s|b_{2})S_{1}(u|s,b).P(b)$

  5. 5.

$L_{1}(s|u)\propto P(s|b_{2})\sum\limits_{b}S_{1}(u|s,b)P(b)$

The important change is at $L_{1}$, who now uses the measured prior to
determine the prior on $w$ ($P(w|b=b_{2})$) but does not assume that the
speaker shares her prior, and considers all possibilities that correspond to
different speaker beliefs about the prior (under this model, $P(b)$ does not
denote the prior probability that the listener assigns to the background
assumption $b$, but the prior probability assigned by the pragmatic listener
to the speaker’s beliefs about the literal listener’s prior). Although
arguably more complex, such a model is theoretically advantageous insofar as
it allows us to use the measured prior to model the data while still viewing
the listener as a fully Bayesian agent. It is bound to predict a stronger
influence of the ‘normal’ prior on interpretation than the original model, and
as shown in Appendix A.2, it may require higher values of $P(b_{1})$ in order
to get rid of anti-exhaustivity.

Finally, we point out that besides the prior $P(b_{1})$, the wonky/backoff
prior $P(w|b_{1})$ itself is not uniquely defined and should be seen as a
hidden parameter, the value of which substantially affects the output in both
versions of the model. Indeed, worlds can be arbitrarily fine-grained, making
it impossible to properly define a uniform distribution over $w$. The wonky
prior thus has to be a uniform distribution over the cells of some Question
Under Discussion (QUD), i.e. sets of worlds which only differ along dimensions
which are irrelevant to the current discussion (see next section for a formal
introduction of the concept).666To model their data, Degen . (2015) implicitly
assume that the QUD is “How many marbles sank?”, which is the question
participants had to answer, but not necessarily the question they thought the
speaker of target utterances was addressing. Under this how-many QUD, the
probability assigned by wonky prior to the ‘all’-situation given that some is
true is $\frac{1}{15}$. Yet this probability could rise as high as
$\frac{1}{2}$ if the QUD were “Did all of the marbles sink?” or fall as low as
$2^{-15}$ with “Which marbles sank?”. Our results suggest that a choice of QUD
which increases the probability $P(w_{all}|b_{1})$ requires increasing the
wonkiness prior $P(b_{1})$ accordingly to prevent the model from predicting
anti-exhaustivity. Overall, the degrees of freedom corresponding to the prior
on wonkiness and the implicit QUD encoded in the wonky prior raise the
question whether the wRSA model offers more than a baseline RSA model with
priors that would not be those that are actually measured, but would be fitted
so as to get a maximally good correspondence with the data.

In the next sections, we introduce three different RSA models in which
sentences can be ambiguous in terms of their literal meaning. This will allow
us to compare models which do or do not allow for the presence of exhaustivity
operators in the logical forms of sentences (i.e. where a sentence could be
ambiguous between its ‘plain’ meaning and its exhaustive interpretation).

###  4.2 Supervaluationist RSA

Spector (2017) proposed a model with a completely different purpose in mind,
but which can be applied to the case at hand. The model was designed to
capture the interpretation of plural definite descriptions, and assumed that
their meaning is underdetermined between an existential meaning and a
universal meaning. To apply this model to our case, we will treat sentences as
ambiguous between their ‘plain’ meaning and an ‘exhaustified’ meaning. In
Spector’s 2017 model, the first-level pragmatic speaker takes into account the
fact that sentences may have mutliple meanings:777This is different from what
happens in the Lexical Uncertainty model presented in section 4.3, where
semantic underspecification is taken into account by the first-level pragmatic
listener, but not by the first-level pragmatic speaker. the speaker is
uncertain about which meaning the literal listener will assume, and so chooses
her message so as to maximize the _expected utility_ of her utterance, i.e.
the weighed average of the utilities that would be achieved for each possible
interpretation, given some prior distribution over interpretations. On top of
that, the model incorporates uncertainty about which components of the world
state are relevant to the local communicative goal, as formalized with the
notion of Questions Under Discussion (QUDs, cf. Roberts 2012). The model views
the speaker as wanting to communicate both the answer to a QUD, and the
identity of the QUD she is addressing.
The model is described by the following equations:

  1. 1.

Ingredients: a set of messages, each of which has a cost, a set of QUDs (which
are denoted with variable $Q$), a prior probability distributions over QUDs,
interpretation functions ($i$) and world states ($w$), where each of these
variables are probalistically independent. $\llbracket u\rrbracket^{i}(w)$
denotes the truth value of $u$ in $w$ under interpretation $i$.

  2. 2.

A QUD $Q$ is viewed as an equivalence relation over worlds, notated
$\sim_{Q}$, and $Q(w)$ denotes the set of worlds equivalent to $w$ relative to
$Q$ (i.e. worlds which do not differ from $w$ along any dimension relevant to
the question denoted by $Q$)

  3. 3.

$L_{0}(w,Q|u,i)\propto P(w)P(Q)\llbracket u\rrbracket^{i}(w)$

  4. 4.

$U_{1}(u|w,Q)=\sum\limits_{i}P(i)\log(L_{0}(Q(w),Q)|u,i))=\sum\limits_{i}P(i)\log(\sum\limits_{v\sim_{Q}w}L_{0}(v,Q|u,i))-c(u)$

  5. 5.

$S_{1}(u|w,Q)\propto\exp(\lambda U_{1}(u|w,Q))$

  6. 6.

For $n\geq 1$, $L_{n}(w,Q|u)\propto P(w)P(Q)S_{n}(u|w,Q)-c(u)$

  7. 7.

For $n\geq 1$,
$U_{n+1}(w,Q|u)=\log(L_{n}(Q(w),Q|u))-c(u)=\log(\sum\limits_{v\sim_{Q}w}\log(L_{n}(v,Q|u))-c(u)$

  8. 8.

For $n\geq 1$, $S_{n+1}(w,Q|u)\propto\exp(\lambda U_{n+1}(u|w,Q))$

This model can be adapted to our case in the following manner. We now view
exhaustification as a _semantic_ phenomenon that involves an operator $exh$
(along the lines of Chierchia ., 2012). The message A is viewed as ambiguous
between $exh(A)$ (= $A\wedge\neg B$) and its literal interpretation A. The
model includes two worlds, $w_{a}$, $w_{ab}$, and three messages, A,
$A\wedge\neg B$, $A\wedge B$, where the last two messages have a positive
cost. There are two interpretation functions, $i_{lit}$ and $i_{exh}$, which
only differ regarding the meaning of A: under $i_{lit}$, A is true in both
$w_{a}$ and $w_{ab}$, but under $i_{exh}$, A is interpreted as $exh(A)$ and
true only in $w_{a}$. Finally, given the two worlds we consider, there are
only two possible QUDs: the total question $Q_{total}$ where the speaker wants
to communicate the actual world (i.e. for either world $w_{a}$ and $w_{ab}$,
$Q_{total}(w)=\\{w\\}$), and the partial QUD $Q_{A}$ where the speaker only
cares about communicating the truth-value of A. In this scenario, given that A
is true in both worlds, this amounts to a question whose answer is already
known (i.e. for either world, $Q_{A}(w)=\\{w_{a},w_{ab}\\}$).

Consider now a first-level pragmatic speaker who wants to answer the question
$Q_{total}$ and believes $w_{ab}$. Under the basic assumption that
$P(i_{exh})>0$, the message A is not usable at all for such a speaker. This is
because under $i_{exh}$, A is false in $w_{ab}$, and for this reason the
utility achieved by A under $i_{exh}$ is infinitely negative, hence the
expected utility of A is itself infinitely negative.888One of the terms in the
sum that defines $U_{1}(A,|w_{a},Q_{total})$, namely
$P(i_{exh})\log(L_{0}(Q_{total}(w_{ab}),Q_{total}|A,i_{exh}))$, is infinitely
negative because $L_{0}(w_{ab},Q_{total}|A,i_{exh})=0$. Therefore a speaker
who _wants_ to convey $w_{ab}$ has to say $A\wedge B$, and so there can’t be
any anti-exhaustive use of A.999This means that in this model, relative to a
maximally fine-grained QUD, a message cannot be used unless it is true on all
its possible meanings, and will therefore be interpreted as entailing the
conjunction of all its possible meanings. For this reason we call this model
the _supervaluationist_ RSA model. In philosophical logic, supervaluationism
(Van Fraassen 1966; Fine 1975) refers to a trivalent semantics for
semantically underspecified sentences (e.g., vague sentences) where sentences
are said to be _supertrue_ if they are true under all their possible
construals, _superfalse_ if they are false under all their possible
construals, and undefined otherwise. Now, if the speaker wants to address
$Q_{A}$, then A is a good message in $w_{ab}$, even under $i_{exh}$: even
though the exhaustive reading of A is false, it still singles out the true
answer to the question. As a consequence, from the first-level pragmatic
listener’s ($L_{1}$) perspective, the message A will still be compatible with
the world $w_{ab}$, since $L_{1}$ cannot categorically exclude that the
intended question is $Q_{A}$, and so some influence of the prior probabilities
on worlds is expected. What is important, however, is that there can not be
any anti-exhaustivity effect on the interpretation side: under $Q_{total}$, A
singles out $w_{a}$, and under $Q_{A}$, it does not distinguish $w_{ab}$ from
$w_{a}$, so $L_{1}$, who is uncertain about the QUD, is bound to increase the
probability of $w_{a}$ compared to that of $w_{ab}$ (because of the
possibility that the QUD is $Q_{total}$). The extent to which the
interpretation will be exhaustive, i.e. to which the posterior probability of
$w_{a}$ will exceed its prior probability, will depend on the priors. If the
priors are such that that $w_{a}$ is antecedently extremely unlikely, then
$L_{1}$ will think, upon hearing A, that probably the speaker meant to address
$Q_{A}$, and so the exhaustivity effect will be small, but still non-zero. For
a formal proof, see Appendix A.3.

Finally, will a speaker who wants to convey $w_{a}$ choose A or $A\wedge\neg
B$? Suppose the level-1 speaker $S_{1}$ wants to answer $Q_{total}$ and to
communicate $w_{a}$. Both A and $A\wedge\neg B$ are usable. In terms of
informativity, $A\wedge\neg B$ is better than A: $A\wedge\neg B$ singles out
$w_{a}$ under both $i_{lit}$ and $i_{exh}$, while A does so only under
$i_{exh}$. But A is of course better in terms of cost. The outcome of the
tradeoff between informativity and cost will, as in the baseline RSA model,
depend on the prior distribution over worlds. If the priors are extremely
biased in favor of $w_{ab}$, the informativity disadvantage of A will be more
pronounced than in a less biased situation (because under $i_{lit}$, the
listener would assign a high probability to $w_{ab}$ after interpreting A). So
$S_{1}$’s probability of choosing $A\wedge\neg B$ will be an increasing
function of the prior probability of $w_{ab}$.

While we have discussed what happens at $S_{1}$ and $L_{1}$, we should note
that things do not substantially change at $S_{2}$. $S_{2}$ still chooses her
message depending on the QUD, and, when she wants to convey $w_{ab}$ to answer
$Q_{total}$, she will have to use the message A and B. This is because $S_{2}$
thinks she talks to $L_{1}$, and upon receiving A, $L_{1}$ infers that either
the QUD is $Q_{A}$, or the QUD is $Q_{total}$ and the world is $w_{a}$. So the
message A cannot be used to communicate the pair $(w_{ab},Q_{total})$. The
fact that the speaker cares about communicating the QUD plays a crucial role
here. Furthermore, when she wants to communicate $w_{a}$ (in the context of
$Q_{total}$), the speaker’s choice between A and $A\wedge\neg B$ will still be
sensitive to priors. One difference between $S_{2}$ and $S_{1}$ is that
$S_{2}$ is talking to a listener who is able to draw inferences about the
intended QUD. Since $S_{2}$ cares about communicating the QUD (and not just
the answer to the QUD), she will take into account $L_{1}$’s inferences about
QUDs. In particular, when the priors are extremely biased in favor of
$w_{ab}$, the message A prompts $L_{1}$ to increase the probability she
assigns to $Q_{A}$ (because under $Q_{total}$, A was unlikely to be used,
since it would communicate the unlikely world $w_{a}$). This will create an
extra incentive for $S_{2}$ to use $A\wedge\neg B$ when she wants to convey
the pair $(w_{a},Q_{total})$.101010However, in cases where the prior is, on
the contrary, extremely biased in favor of $w_{a}$, something quite
counterintuitive could happen regarding the interpretation of $A\wedge B$: if
$w_{ab}$ is now extremely unlikely, $L_{1}$ will reason that the intended QUD
is probably $Q_{A}$ (since under $Q_{total}$, the message $A\wedge B$ would
probably be false), and then interpret $A\wedge B$ as compatible with $w_{a}$.
Furthermore, using $A\wedge B$ could be then an excellent way for $S_{2}$ to
signal that the intended QUD is $Q_{A}$, and so $A\wedge B$ might become the
preferred way of communicating that the intended QUD is $Q_{A}$, irrespective
of whether B is true. Note however that for $S_{1}$, A and B is never the best
message to convey $(w_{a},Q_{A})$, so this puzzling behavior disappears as the
rationality parameter increases.

To conclude, this model categorically excludes the possibility of an anti-
exhaustive use or interpretation of A, but it still predicts an effect of
priors on message choice and also on interpretation, in the sense that the
exhaustivity inference triggered by A will be modulated by priors, as will be
the speaker’s choice between the two messages A and $A\wedge\neg B$. This
model essentially makes the prediction that A triggers the inference that
either the speaker considers B to be irrelevant, or the speaker believes B to
be false. It is thus in line with some recent proposals which argue that
exhaustivity inferences are obligatory when relevant, such as Buccola Haida
(2019).

###  4.3 The Lexical Uncertainty Model

The Lexical Uncertain Model (LU for short, see Bergen ., 2016) is another
model in which messages can be ambiguous. It includes all the ingredients of
the baseline RSA model, and adds to it a set of interpretation functions. Just
like the model presented in the previous section, the literal listener $L_{0}$
is relativized to a specific interpretation function (hence there are as many
literal listeners as there are interpretation functions). The first-level
pragmatic speaker $S_{1}$ is also relativized to a specific interpretation
function, insofar as she chooses messages for a literal listener who has a
particular interpretation function (this is unlike the supervaluationist model
discussed in the previous section, where $S_{1}$ is uncertain about which
literal listener she is talking to). It is only at the level of the first
pragmatic listener $L_{1}$ that uncertainty about meaning is factored in.
$L_{1}$ has a prior distribution over worlds and interpretation functions,
which are assumed to be probabilistically independent, and uses Bayes’ rule to
infer a joint posterior distribution over worlds and interpretation functions.
Her posterior distribution over worlds is obtained by marginalizing on
interpretation functions. Then higher-level listeners/speakers are defined
just as in the baseline model (since $L_{1}$’s behavior is no longer dependent
on a specific interpretation function). The model is defined by the following
equations:

  1. 1.

$L_{0}(w|u,i)\propto P(w)\llbracket u\rrbracket^{i}(w)$

  2. 2.

$U_{1}(u|w,i)=\log(L_{0}(w|u,i))-c(u)$

  3. 3.

$S_{1}(u|w,i)\propto\exp(\lambda U_{1}(u|w,i))$

  4. 4.

$L_{1}(w|u)\propto P(w)\sum\limits_{i}P(i)S_{1}(u|w,i)$

  5. 5.

For $n\geq 1$, $U_{n+1}(u|w)=\log(L_{n}(w|u))-c(u)$

  6. 6.

For $n\geq 1$, $S_{n+1}(u|w)\propto\exp(\lambda U_{n+1}(u|w))$

  7. 7.

For $n\geq 2$, $L_{n}(w|u)\propto\log(S_{n}(w|u))$

We can then consider two versions of this model, depending on the set of
interpretation functions we include in the model. In one version, which we
call the EXH-LU model, the possible interpretation functions are, like in the
previous model, $i_{lit}$ and $i_{exh}$. This is the version of LU used in
Franke Bergen (2020). In the second version, which we call the FREE-LU model,
we add to these two interpretation functions a third one, $i_{\textit{anti-
exh}}$, where A means $A\wedge B$. This version corresponds to the
unrestricted LU model used in Bergen . (2016); Potts . (2016), in which the
possible meanings of a message consist of all the possible ways of
strengthening their literal meaning. In the EXH-LU model, symmetry between A
and B and A and not B is broken in the grammar, while the FREE-LU, just like
the baseline model, can only break symmetry with cost and priors.

In the EXH-LU model, if the two interpretation functions are equiprobable and
as long as the cost of $A\wedge B$ does not exceed that of $A\wedge\neg B$, it
can be shown that there cannot be any anti-exhaustivity effect at $L_{1}$.
However, anti-exhaustivity effects can arise at $S_{2}$ (in the sense that
$S_{2}$ may use A to convey $w_{ab}$) and at higher-level listeners. This is
essentially because at higher levels of recursion, the model is
indistinguishable from the baseline model. If the priors are sufficiently
biased in favor of $w_{ab}$, $L_{1}(w_{ab}|A)$ will itself be close to 1
(though not higher than $P(w_{ab})$), and so $S_{2}$ might be more likely to
use A in $w_{ab}$ than in $w_{a}$ (because A and not B would be a much better
message that A in $w_{a}$), following exactly the same logic as in our
discussion of the baseline RSA model.

As to the FREE-LU model, unsurprisingly, it does not exclude anti-exhaustive
inferences even at $L_{1}$, because one of its interpretation functions makes
A equivalent to $A\wedge B$.

###  4.4 The Lexical Intentions Model

Finally, we will consider another model in which messages can be ambiguous—the
Lexical Intentions Model, recently proposed by Franke Bergen
(2020).111111Franke Bergen (2020) also discuss a Global Intentions model,
which is equivalent to the LI model for the cases we consider in this
manuscript. In this model each message can again have multiple meanings. The
crucial difference with the LU model is that the first-level speaker jointly
chooses a message and an interpretation function for this message, so as to
maximize the conditional probability of the intended world given the meaning
that is chosen. This corresponds to a speaker who chooses a message with a
certain interpretation in mind, and based on a literal listener who will
always interpret the message under the interpretation that the speaker has in
mind. If we again consider two possible interpretations functions, $i_{lit}$
and $i_{exh}$,121212This amounts to counting unambiguous messages twice each,
since they can be paired to both $i_{lit}$ and $i_{exh}$, with no change in
meaning. This increases the probability of unambiguous messages to be used,
all else being equal, and is in line with Franke Bergen’s 2020 implementation
(Michael Franke, p.c.). Other choices are possible, with no important
qualitative change. then the best message-interpretation pair for a speaker
who wants to convey $w_{a}$ is the pair $(A,i_{exh})$, which is true only in
$w_{a}$ and is less costly than any pair $(A\wedge\neg B,i)$.

The first level listener, $L_{1}$, performs a joint inference both on worlds
and intended interpretation. The prior distribution over $i$ is uniform (and
for this reason can be entirely ignored when stating proportionality
statements). The model is characterized by the following equations:

  1. 1.

$L_{0}(w|u,i)\propto P(w)\llbracket u\rrbracket^{i}(w)$

  2. 2.

$U_{1}(u,i|w)=\log(L_{0}(w|u,i))-c(u)$

  3. 3.

$S_{1}(u,i|w)\propto\exp(\lambda U_{1}(u,i|w))$

  4. 4.

$L_{1}(w,i|u)\propto P(w)S_{1}(u,i|w)$ priors on $i$ are implicitly uniform

Predictions about the speaker’s choice of a message and the listener’s
inferences about the world are obtained by marginalizing over meanings:

  1. 5.

$S_{1}(u|w)=\sum\limits_{i}S_{1}(u,i|w)$

  2. 6.

$L_{1}(w|u)=\sum\limits_{i}L_{1}(w,i|u)\propto
P(w)\sum\limits_{i}S_{1}(u,i|w)$

We will consider a model with the same set of worlds and messages as before,
and where the message A is ambiguous between the plain and the exhaustified
meaning, while the two other messages are unambiguous. It can be proved that
as long as the cost of $A\wedge B$ is no greater than that of $A\wedge\neg B$,
A can never be the preferred message to convey $w_{ab}$, and A is never
interpreted anti-exhaustively. The reason is that $S_{1}$ will never be more
likely to use A in $w_{ab}$ than in $w_{a}$. In $w_{a}$, A is always the best
message to choose (under the interpretation $i_{exh}$). In $w_{ab}$, A might
be the best message to use if the priors are extremely biased in favor of
$w_{ab}$, but even in such a case it will be no more likely to be used in
$w_{ab}$ than in $w_{a}$ because (i) it competes against A$\wedge$B which is
less costly that A$\wedge\neg$B, and (ii) it must be interpreted according to
$i_{lit}$, and face a small but positive informativity penalty compared to the
interpretation $i_{exh}$ in $w_{a}$. Now, while Franke Bergen (2020) do not
define higher-level speakers and listeners for the Lexical Intention model,
the most natural way to extend their proposal to higher-level speakers and
listeners would be to define them as in the baseline RSA model, as shown
below, in which case anti-exhaustivity effects could reappear at higher levels
of recursion, as discussed in the previous section in relation with the LU
model.131313Note that since $L_{1}$ infers the interpretation $i$ intended by
$S_{1}$ based on the surface form of the message only, there is no way for
$S_{2}$ to influence the interpretation inferred by $L_{1}$ for a given
message. Therefore, the intention mechanism cannot be recursively extended
beyond $S_{1}$.

  1. 7.

For $n\geq 1$, $U_{n+1}(u|w)=\log(L_{n}(w|u))-c(u)$

  2. 8.

For $n\geq 1$, $S_{n+1}(u|w)\propto\exp(\lambda U_{n+1}(u|w))$

  3. 9.

For $n\geq 2$, $L_{n}(w|u)\propto P(w)S_{n}(u|w)$

##  5 Experiment

The experiment was pre-registered on OSF (https://osf.io/7xts3/). All data and
scripts are available at
https://github.com/Alex-Cremers/RSA-Exh-Priors.

###  5.1 Background and motivation

Several experimental studies have investigated the influence of priors on
exhaustive interpretations, starting with Degen . (2015). In the analogous
case of some/all, they found that the prior probability for the all-situation
only had a small effect on the interpretation of some, which was
overwhelmingly interpreted as some but not all, and this was the motivation
for the wRSA model. Javangula (2019) investigated effects of priors and QUD
manipulations on comprehension, and found that priors modulated the strength
of exhaustivity inferences, but did not find evidence for anti-exhaustivity.
Schreiber Onea (2021) looked directly at the type of exhaustivity inferences
discussed in the introduction, and found that exhaustive interpretations are
essentially constant across domain sizes and priors. Finally, Wilcox Spector
(2019) looked at the effect of priors on both production and comprehension.
They only found a small effect on exhaustive interpretations, and did not
observe the predicted effect on production, but the results were too noisy to
draw firm conclusions.

The goal of our experiment was to improve on previous work in several
respects. First, we wanted to obtain cleaner results on production, where the
predictions of the baseline RSA are intuitively correct, at least
qualitatively. Second, none of the cited studies measured priors from the same
participants who took part in the comprehension/production task. We decided to
gather prior data from the same participants we tested for
comprehension/production in order to maximize our chances to find correlations
between priors and linguistic behavior.141414Using a noisy predictor leads to
systematic underestimation of the coefficient in a linear regression (a
phenomenon known as _regression dilution_), so reducing the noise on the prior
measure should increase the measured correlation with
comprehension/production.

###  5.2 Methods and materials

####  5.2.1 Experimental design

Participants were recruited on Amazon’s Mechanical Turk, and the survey was
hosted directly there. We first introduced a context in which a character,
Alex, is choosing what she will have for lunch. An example is given in Fig. 2.
Among the options are a sandwich and a soup, which can be bought together for
a discount. The survey focuses on whether Alex will buy only the sandwich (our
$w_{a}$ situation), or both the sandwich and the soup (a $w_{ab}$ situation).
We manipulated the price of the lunch deal from $12 (the soup comes for free
with the sandwich) to $19 (only $1 discount), in order to generate various
expectations about Alex’s preference between the lunch deal and the sandwich
alone.

We first asked participants to calculate the value of the discount (i.e. to
subtract the last two prices in the context story). This was an attention
check allowing us to filter out participants who did not pay enough attention,
but it also drew attention to our manipulation in order to maximize its
effect. We then measured participants’ prior belief in Alex ordering the soup
conditional on her ordering the sandwich (i.e.
$P(w_{ab}|\\{w_{a},w_{ab}\\})$), using a slider. After answering the question
prior, participants could click a “next question” button which brought them to
either a production question or a comprehension question, and froze the slider
for the prior question (so they could still see their response but could not
edit it).

Figure 2: Example survey, before submitting answer to the prior question.

In the production survey, the last question was , and participants had to type
in their answer (up to 50 characters). The rationale for the free response was
to make it more effortful to provide longer messages, thereby increasing the
likelihood that cost would factor in participants’ choice of a message (in
contrast with a multiple choice question where choosing the most informative
message has no disadvantage). Free response also allowed us to record the
length of the different messages, which we used as a proxy for cost. We
manipulated the information given about Alex’s order so as to place
participants in either a $w_{a}$ situation or a $w_{ab}$ situation.

.  In fact, Alex ordered { only the sandwich / both the sandwich and the
soup}. What do you think she would answer to someone who asks what she
ordered? …text response field … p

In the production survey, the next question was , which participants could
answer using a slider identical to the one used for measuring their prior.
This question presupposed that Alex did order the sandwich, and measured
participants’ posterior belief $P(w_{ab}|u)$, where the utterance was either A
(“the sandwich”) or A&B (“the sandwich and the soup”). Crucially, since their
answer to the prior question was still visible, it was very easy for
participants to intentionally place the slider lower or higher for the
posterior (i.e. to indicate an exhaustive or anti-exhaustive interpretation,
respectively).

. You ask Alex what she ordered, and she answers {“The sandwich” / “The
sandwich and the soup”}.
Given her answer, how likely is it that she also ordered the soup?

Production and comprehension data were collected separately to avoid
confusion. In all, there were $8\times(2+2)$ versions of the survey: eight
possible prices for the lunch deal, two possible worlds for the production
question and two possible utterances for the comprehension question.

####  5.2.2 Participants

Mechanical Turk workers were recruited using the following filters: US
Location, HIT approval of at least 98% (we initially tried 99% but too few
Turkers met the criterion), number of approved HITs greater than 1000.
Participants were paid $0.47 for their participation (the survey took about
3min to complete).

For each of the 8 prices, data from 20 participants were collected for
utterance A (comprehension) and world $w_{a}$ (production), and from 10
participants for utterance A&B (comprehension) and world $w_{ab}$
(production).151515A pilot study had shown very little variation in
participants’ behavior in the latter two conditions. In total, 239
participants were recruited for the comprehension survey and 237 for the
production survey (goal: 240 for each: $8\times(20+10)$). Among them, 222 and
224 passed the comprehension check respectively, amounting to an error rate of
about 6%.

####  5.2.3 Data analysis

The production data were coded into the following 6 categories (examples
provided): A (“the sandwich”), A&B (“the sandwich and the soup”, “the lunch
deal”), A&$\neg$B (“just the sandwich”), B (“soup”), $\neg$B (“I didn’t want
the soup”), and Other/NA (“I wasn’t hungry today”). The number of characters
used to convey the message was recorded. NA responses were excluded from the
analyses. Prior measures were compressed from $[0,1]$ to $[.005,.995]$.

We first ran theory-neutral analyses to test the association between the
priors and posteriors in comprehension data with linear regression, and the
association between priors and production with binomial (logistic) regressions
(excluding literally false messages, e.g. A&B in the $w_{a}$ world). We then
fitted the parameters of each of the models of interest to our data set. We
only observe 1 B response and 12 $\neg$B responses, so we decided to merge
them with A&B and A&$\neg$B respectively for model fits. For production,
strictly-speaking, no linking hypothesis is needed since each model
immediately generates a prediction for the likelihood to observe any given
message. However, models often predict probabilities of $0$, so following
Franke Bergen (2020), we added an error parameter to our model of production
data: if $S(u)$ is the theoretical predicted probability for observing message
$u$, we used $\frac{S(u)+\epsilon}{1+n\epsilon}$ as our predictor, where $n$
is the number of candidate messages. The parameter $\epsilon$ was fitted
together with model parameters. For comprehension, the models return a point
estimate for the posterior probability. We assumed that data is normally
distributed around the prediction, with censoring at 0 and 1 (the boundaries
of the slider), as in tobit regression. This is a direct improvement on the
usual least-square fit, which amounts to assuming an uncensored normal
distribution (including impossible values beyond 0 and 1). We assumed
different noise parameters $\sigma_{a}$ and $\sigma_{ab}$ for the A and A&B
condition.

Parameter estimation was done by maximizing the likelihood of the whole
dataset (production and comprehension). Models were evaluated by their AIC.

###  5.3 Results

###  5.4 Preliminary analyses

(a) Effect of the price manipulation on conditional prior beliefs
$P(\text{soup}|\text{sandwich})$, with linear correlation line.

(b) Length of elicited responses in number of characters, by expressed
message, with lower (1) and upper (50) limits.

(c) Effect of prior beliefs on production, after exclusion of literally false
messages. For plotting (and only for plotting), priors are binned into
$\left\\{[0,24],[25,49],[50,74],[75,99],\\{100\\}\right\\}$, with the last bin
represented by a large dot. Shade indicates sample size in each bin. The
colored lines correspond to logistic regressions.

(d) Effect of prior beliefs on comprehension for each utterance. The bisecting
dashed line indicates the “posterior=conditional prior line”, which represents
a literal interpretation of utterance A. Points below this line indicate
exhaustivity, while points above the line indicate anti-exhaustivity (for the
A message). The colored lines correspond to linear regressions.

Figure 3: Descriptive results for prior elicitation, production and
comprehension questions.

##### Priors:

The reported priors covered the whole slider range, although they were biased
towards higher values ($M=.70,SD=.27$), as intended. The price manipulation
had a clear effect on the reported prior ($F(1,409)=33,p<.001$). In all the
following, our predictor will be the measured prior for each participant, and
we ignore the price variable.

##### Production:

We observed all categories of responses, but B appeared only once as “SOUP”.
The otherwise shortest message on average was A ($M=22.8,SD=8.9$), followed by
$\neg$B ($M=30.5,SD=12$), then A&$\neg$B ($M=35.7,SD=9.7$, often expressed as
“just the sandwich”) and finally A&B ($M=35.8,SD=11$, expressed either as a
conjunction or “the lunch deal”). We observed a small effect of priors on the
production of an exhaustive form to convey $w_{a}$
($\chi^{2}(1)=4.95,p=.026$): when the prior for the soup is higher,
participants are more likely to explicitly say that they didn’t order it. In
contrast, A was never used to convey $w_{ab}$ (A&B was always used), and so
there was no influence of priors regarding the choice of a message in
$w_{ab}$.

##### Comprehension:

We observed an effect of prior on the interpretation of A ($\beta=.49$,
$F(1,145)=28.2,p<.001$), but no evidence for the kind of anti-exhaustivity
predicted by the RSA: 15 out of 147 responses to the target comprehension
survey indicated a posterior value higher than the prior, but while the base
RSA predicts anti-exhaustivity to occur when the prior is high, we found that
the probability to observe such a response was inversely correlated with the
prior (logistic regression: $\beta=-.91,z=-3.5,p<.001$). This is exactly what
we would expect if these responses were the result of random errors, which are
more likely to fall above the prior when the prior is lower. By contrast,
there was no effect of priors on the interpretation of A&B ($\beta=-.050$,
$F(1,73)=1.2,p=.28$).

##### Connecting production and comprehension:

We can already see that participants who took the comprehension survey do not
behave as Bayesian listeners with respect to participants’ behavior in the
production survey. Indeed, no participant produced A in the $w_{ab}$ condition
(0/72), so a Bayesian listener making interpretations based on a speaker model
that is similar to our measured production data should conclude that
$P(w_{ab}|\textit{A})\approx 0$, irrespective of the prior. Yet comprehension
responses to the A condition show a clear effect of prior on comprehension,
with posterior values high above 0. This does not mean that listener
participants are not Bayesian, but if they are, their model of the speaker
does not match what participants do in the production survey.

Conversely, we can fit the costs and rationality parameters of an RSA-speaker
addressing a listener who behaves like our participants in the comprehension
task.161616The empirical listener was modeled as a linear regression on
arcsine-transformed priors and posteriors. We tested polynomial fits up to
degree 5, but a linear fit was BIC-optimal. The resulting RSA speaker yielded
a good fit of the data, but with implausible parameters
$(\lambda=1.6,c_{\text{A\&B}}=-8.9,c_{\text{A\&$\neg$B}}=.34)$. In short, the
model avoids predicting uses of A to convey $w_{ab}$ by assigning a large
negative cost to the message A&B. Forcing the model to assign a positive costs
to the messages A&B and A&$\neg$B immediately worsens the predictions (log-
likelihood falls from $-131$ to $-137$ and the model predicts A to be used to
convey $w_{ab}$ with probability up to $.19$).

In short, participants in the comprehension task are not Bayesian with respect
to the behavior of participants in the production task. Likewise, participants
in the production task—in the sense of the RSA—do not seem to be rational with
respect to participants in the comprehension task either.

##### Preliminary discussion:

We didn’t detect any anti-exhaustivity effect except for a few data points
that would be best explained as random errors. In comprehension, the
probability assigned to $w_{ab}$ after receiving the message A was influenced
by prior probabilities but was never higher on average than its prior
probability, except for very low prior values where a few outliers are
sufficient to bring the posterior value over the prior. In production, A was
never used to convey $w_{ab}$. These results thus provide _prima facie_
evidence in favor of models that cannot predict anti-exhaustivity and against
those that can. We now turn to evaluating alternatives to the base RSA model.

###  5.5 Models evaluation

####  5.5.1 Tested implementations

We tested how well variants of the baseline RSA model fit our production and
comprehension data. Besides the baseline RSA model, we evaluated the following
models:

  * •

The “wonky world” RSA model (wRSA) of Degen . (2015), as well as the Bayesian
variant described in §4.1 (BwRSA). Both versions come with an extra parameter,
which corresponds to the “wonkiness” prior.

  * •

Spector’s supervaluationist model (svRSA), as described in §4.2. Our
experiment did not measure participants’ assumptions about the QUD, so we fit
the prior on the total QUD as a parameter of the model. We test two versions
of the model which differ in their definition of the speaker: svRSA1 models
production as a mixture of the speakers trying to convey the partial QUD and
the total QUD (in proportion of the prior on QUD), while svRSA2 simply models
production as a speaker who tries to convey the total QUD. We use a uniform
prior on interpretations (exhaustive or literal).

  * •

The lexical uncertainty model, in its free version (FREE-LU, which allows both
exhaustive and anti-exhaustive strengthening in line with Bergen ., 2016) and
in its grammatical version (EXH-LU, which only considers strengthening derived
from a grammatical theory of exhaustification, as in Franke Bergen, 2020). For
both models, we used a uniform prior on interpretations.

  * •

The lexical intentions model with grammatical exhaustification. Here again we
test two versions of the model, which differ in terms of production: RSA-LI1
uses the marginal $S_{1}$ speaker, while RSA-LI2 uses $S_{2}$. The crucial
difference is that $S_{1}$ picks simultaneously a message and a meaning in
order to communicate a world, while $S_{2}$ simply picks a message and talks
to a listener who has already factored in ambiguity.

All models use $L_{1}$ to model comprehension, and all models except RSA-LI1
use the $S_{2}$ speaker to model production. This is because in the wRSA and
the LU models, $S_{1}$ is a theoretical construct that is not readily
interpretable, as it is relativized to a lifted variable. Furthermore, the
choice of $L_{1}$ rather than $L_{2}$ was motivated by the observation that
the comprehension participants cannot be Bayesian with respect to the
empirical speaker. The prediction of all models were computed under the
assumption that the possible utterances are A, A&B and A&$\neg$B and the
worlds to convey are $w_{a}$ and $w_{ab}$. Models are parametrized by the
rationality parameter $\lambda$, the costs of A&B and A&$\neg$B relative to A
($\Delta_{ab}$ and $\Delta_{a\neg b}$), and an extra prior parameter for the
wonky RSA and the two supervaluationist models. The first three parameters are
constrained to be positive, and the prior parameters are constrained to
$[0,1]$. Appendix B gives the exact implementation for each model. For
additional details, see the R script in Supplementary Materials.

####  5.5.2 Results

Figure 4 shows the model fits against the data, and Table 1 gives the
parameter estimates. The best models are the wonky world RSA model (wRSA) and
the first version of supervaluationist model (svRSA1), which fit the data
equally well (wRSA is better at capturing comprehension and svRSA1 better at
capturing production). The fit of comprehension data illustrates our earlier
point about how the wRSA model is, in some sense, non-Bayesian, since its
predictions are not constrained to 0 and 1 when the prior is 0 and 1,
respectively. Note as well that RSA-LI2 and EXH-LU are strictly equivalent
when the priors on interpretation functions are uniform and the costs are
null, which happens to be where the maximum likelihood is reached.

Interestingly, all models but the supervaluationist model (svRSA) end up
assigning a cost of 0 to A&B—presumably to minimize anti-exhaustivity—, while
the svRSA model assigns it a very high cost (presumably so that A&B is never
used to convey the partial QUD, which would allow its use in the $w_{a}$
world). Since A&B and A&$\neg$B had roughly the same length in participants’
responses (35.8 vs. 35.7 characters on average), we refitted the models with
an added constraint $\Delta_{ab}=\Delta_{a\neg b}$ as a post-hoc analysis.
From a theoretical point of view, this analysis allows us to evaluate how much
each model relies on costs to break symmetry.

(a) Model fits on the production data. We only present the data from the
$w_{a}$ world as the production for the $w_{ab}$ world consisted only of A&B
(i.e., the orange line should be as high as possible). Note that unlike Fig
3(c), this plot includes literally false messages (captured by the noise
parameter in the model fit but discarded in the preliminary analysis).

(b) Model fits on the comprehension data. We represent the median and quartile
predictions (given the censored gaussian model for noise, the theoretical
predictions correspond to the median, not the mean).

Figure 4: Model fits on production and comprehension. Likelihood was maximized on the combined data. Model | $\lambda$ | $\Delta_{ab}$ | $\Delta_{a\neg b}$ | $\xi$ | $\sigma_{a}$ | $\sigma_{ab}$ | $\epsilon$ | AIC
---|---|---|---|---|---|---|---|---
wRSA | 3.9 | 0 | .37 | .86 | .33 | .22 | .022 | 492
svRSA1 | .68 | 44 | .20 | .89 | .38 | .22 | .003 | 492
svRSA2 | .54 | 131 | 0 | .89 | .38 | .22 | .022 | 509
RSA-LI1 | 8.6 | 0 | .054 |  | .42 | .22 | .022 | 551
BwRSA | 5.8 | 0 | .36 | 1 | .37 | .22 | .052 | 561
EXH-LU | 1.0e+3 | 0 | 0 |  | .48 | .22 | .037 | 593
RSA-LI2 | 1.0e+3 | 0 | 0 |  | .48 | .22 | .037 | 593
FREE-LU | .57 | 0 | 1.9 |  | .48 | .22 | .028 | 653
Base-RSA | .41 | 0 | 2.3 |  | .48 | .22 | .027 | 659
Table 1: Parameter fit for each model. $\lambda$ is the rationality parameter,
$\Delta_{ab}$ the cost difference between A&B and A, $\Delta_{a\neg b}$ the
cost difference between A&$\neg$B and A, and $\xi$ the extra prior used by
some models (wonkiness prior in the wonky RSA model, prior on the total QUD in
the supervaluationist RSA model). $\sigma_{a}$ and $\sigma_{ab}$ are the noise
parameters for the comprehension data, $\epsilon$ the noise parameter for the
production data. Models are sorted by AIC.

####  5.5.3 Posthoc analysis: equal costs for A&B and A&$\neg$B

Imposing equal costs for A&B and A&$\neg$B immediately penalizes the four
models which cannot break symmetry within the semantics (wRSA, BwRSA, FREE-LU
and base RSA), which fall to the bottom of Table 2. Undesirable behavior is
evident on both the speaker and listener side, shown in Figure 5: On the
production side, the probability of the A&B message (i.e. the orange line) is
now clearly below 1, meaning that these models predict the use of A in the
$w_{ab}$ world. On the comprehension side, three of them now predict the
listener to just guess (i.e. the posterior is equal to the prior along the
dashed line). By contrast, the EXH-LU, RSA-LI1, and RSA-LI2 already assigned
these two messages the same null costs, so they are unaffected by this
constraint. The two versions of the supervaluationist model (svRSA1 and
svRSA2) remain better than the rest, even though they don’t perform as well as
with the extra cost on A&B. In particular, the prediction discussed in
footnote 10 surfaces: when the prior on $w_{ab}$ falls sufficiently lower than
the prior on the partial QUD, A&B can become a good message to convey the
partial QUD even in $w_{a}$. This is visible in the production curve of svRSA1
for message A in $w_{a}$—which is not monotone as A competes with A&B on the
left and A&$\neg$B on the right—, but not in that of svRSA2, as the latter
assumes the total QUD for production. While some A&B messages do show up in
the $w_{a}$ condition of the production survey, they do not appear to come
from particularly low prior values (means: $0.70$ vs. $0.72$ for other
messages, $t(9.6)=-.26,p=.80$), and are more likely due to errors. The
problematic prediction is most visible in its effect on comprehension, as A&B
is predicted to be compatible with $w_{a}$, although it never reduces the
posterior below the prior (i.e., A&B is never predicted to _convey_ $w_{a}$).
Quantitatively, this problematic prediction does not penalize the fit of the
model much, since there are very few comprehension data points for A&B at low
prior values.

Model | $\lambda$ | $\Delta$ | $\xi$ | $\sigma_{a}$ | $\sigma_{ab}$ | $\epsilon$ | AIC
---|---|---|---|---|---|---|---
svRSA1 | .85 | .41 | .91 | .38 | .28 | .003 | 514
svRSA2 | .68 | .18 | .91 | .39 | .28 | .022 | 529
RSA-LI1 | 10 | .013 |  | .43 | .22 | .023 | 555
RSA-LI2 | 1.0e+3 | 0 |  | .48 | .22 | .037 | 591
EXH-LU | 1.0e+3 | 0 |  | .48 | .22 | .037 | 591
wRSA | 1.4 | 6.3e-4 | 1 | .39 | .22 | .026 | 591
BwRSA | .26 | 0 | 1 | .51 | .22 | .026 | 670
FREE-LU | .21 | 0 |  | .51 | .22 | .026 | 671
RSA | .17 | 0 |  | .52 | .22 | .026 | 674
Table 2: Parameter fit for each model when imposing equal costs for A&B and
A&$\neg$B. $\lambda$ is the rationality parameter, $\Delta$ the cost
difference between A&B/A&$\neg$B and A, and $\xi$ the extra prior used by some
models (wonkiness prior in the wonky RSA, prior on the total QUD in the
supervaluationist model). $\sigma_{a}$ and $\sigma_{ab}$ are the noise
parameters for the comprehension data, $\epsilon$ the noise parameter for the
production data. Models are sorted by AIC.

(a) Model fits on the production data. We only present the data from the
$w_{a}$ world as the production for the $w_{ab}$ world consisted only of A&B
(i.e., the orange line should be as high as possible).

(b) Model fits on the comprehension data. We represent the median and quartile
predictions (given the censored gaussian model for noise, the theoretical
predictions correspond to the median, not the mean).

Figure 5: Model fits on production and comprehension after imposing equal
costs for A&B and A&$\neg$B.

##  6 Discussion

By measuring priors in the same experiment as target behavior, our design
improved on previous studies and allowed us to detect the effect of priors on
production, which had been predicted but not established. We also found an
effect of priors on comprehension, with an effect size almost 8x larger than
Degen . (2015) when it comes to the correlation between priors and posteriors
($\beta=.49$ vs $\beta=.06$), yet we still do not find any trace of anti-
exhaustivity.171717Javangula (2019) did find an even stronger correlation than
we did: $\beta=.79$, albeit in a more restricted range of priors, from 0.30 to
0.80 prior conditional probability for the $w_{ab}$ world, where we would not
expect anti-exhaustivity anyway.

At the highest level, our results pose a challenge to simple information-
theoretic models of pragmatics, like the base RSA model, insofar as they are
not compatible with either production and comprehension being at the same time
rational and well-informed. If our production data is taken as the basis for
RSA-style comprehension, the participants are not rational, in a Bayesian
sense. In this respect, our conclusions align with Sikos . (2021), who also
found that listeners don’t align with speakers in a recent replication of
Frank Goodman’s 2012 experiment.181818RSA models do not generally predict that
the listener and the speaker will be rational with respect to each other: if
we use $L_{1}$ and $S_{1}$ as models for the listener and speaker, then the
speaker is assumed to be (approximately) rational with respect to $L_{0}$, not
$L_{1}$. If instead we take $L_{0}$ to be the model for the listener and
$S_{1}$ for the speaker, then we do not model the listener as being rational
with respect to the speaker. Yet RSA models do assume that at least one of the
two participants is rational with respect to the other: with $L_{1}$ and
$S_{1}$, the listener is modeled as rational with respect to the speaker (but
not vice-versa). With $L_{0}$ and $S_{1}$, or $L_{1}$ and $S_{2}$, the speaker
is modeled as (approximately) rational with respect to the listener, but not
vice-versa.

Likewise, if our comprehension data is taken as the basis for utterance
production, then anti-exhaustivity effects can only be avoided with
implausible costs. Our results, however, do not exclude models in which
speakers and listeners are rational but do not share the same information
(e.g., knowledge of extra parameters which can be introduced in the RSA
framework), or models which depart from the rationality assumption for one of
the agents (e.g., by allowing the listener to revise her priors, as in some
implementations of “wonky world” RSA). Lastly, we found that models which can
break symmetry in the semantics by implementing grammatical exhaustification
have a clear advantage over models which must rely on informativity and costs
alone (see Champollion ., 2019 and Franke Bergen, 2020 for similar conclusions
from different phenomena).

The EXH-LU and LI2 models converged to parameters that turn them into a
categorical model: null costs and extremely high rationality mean that the
speaker always selects the most informative message. In turn the listener can
retrieve the world with near absolute certainty, so priors play no role on the
interpretation side.

The supervaluationist model (svRSA) emerged as the best candidate to model our
data set. It appears to be particularly good at capturing speaker/listener
asymmetries thanks to its QUD parameter, and its approach to ambiguity
resolution allows it to take full advantage of exhaustification. Unlike the LI
model, in which ambiguity offers the speakers more opportunities to use a
given message, the supervaluationist model restricts the use of ambiguous
messages to situations where they are true under all possible interpretations.
This constraint blocks anti-exhaustivity in both production and comprehension,
independently of costs parameters, and can only be relaxed when a partial QUD
makes exhaustivity irrelevant. The model does however make a problematic
prediction regarding the use of A&B to convey the pair ($w_{a}$, partial QUD)
when priors become too extreme and the cost of A&B too low.

Like previous studies, our experiment used written stimuli in English. Since
English usually marks focus prosodically, this leaves the focus structure of
our message A underspecified. Many of the models we tested assumed ambiguity,
but focus and covert exhaustification are usually assumed to correlate, so
prosody could contribute to disambiguation, and this could take many forms.
First, we could assume that prosody fully disambiguates the intended meaning,
so that unfocused A is obligatorily literal and focused A obligatorily
exhaustive. Second, we could assume that the exhaustification operator
requires focus, so that unfocused A is unambiguously literal, but focused A
would remain ambiguous. Conversely, we could assume that focus requires
exhaustification, so that focused A is obligatorily exhaustive but unfocused A
is now ambiguous. The problem becomes even more complex if we consider these
connections to be probabilistic rather than deterministic. Alternatively, the
connection between focus and exhaustification could be indirect, for instance
if focus conveys information about the QUD.

One reason why previous investigations may have underestimated the issue of
anti-exhaustivity in RSA models is that model predictions are often derived
without taking into account explicit exhaustive messages such as A and not B,
relying instead on a predefined set of alternatives defined along the lines of
standard neo-Gricean approach (as discussed in footnote 2). This is the case
in both Degen . (2015) and Potts . (2016), which tested wRSA and RSA-LU
respectively. Doing so amounts to artificially breaking the symmetry by
assigning an infinite cost to this message, which prevents anti-exhaustivity
even in the base RSA model. Yet, our production data show that such messages
can easily be expressed with only or just and are produced as often as the
weak message A, so they shouldn’t be ignored in modeling. Furthermore, we did
detect the expected influence of priors on message choice when the the
intended meaning is $w_{a}$. As predicted by virtually all RSA models,
participants are more likely to use an explicit message whose meaning is
$A\wedge\neg B$ when conditional probability of $A\&B$ given $A$ is very high.
The trade-off between cost and informativity in message choice, which is a key
feature of RSA models with cost, is a virtue (see also Chemla Romoli, 2016 and
Enguehard Spector, 2021, which include clear introspective judgments
suggesting that the choice between _some_ and _not all_ is sensitive to priors
in the expected way), but it appears that RSA models, to be able to predict
such effects without predicting at the same time anti-exhaustivity effects,
need to include exhaustification in their semantic component.

Fox Katzir (2021) have previously discussed the fact that probabilistic models
of scalar implicatures and exhaustivity effects (such as models couched within
the RSA framework) let us expect a strong influence of prior probabilities on
such phenomena – cases where different priors should lead to drastically
different interpretations for the very same sentence. They argue that, if in
fact such an influence is not found in these domains, such a finding should
shed doubt on pragmatic theories of exhaustivity that rely on such models.
They point out that even if some relatively complex probabilistic models
manage to eliminate the undesirable effect of priors, their success would then
not be due to what is _distinctive_ about probabilistic approaches. Our
results are consistent with this view, since they provide new evidence that
exhaustive interpretations should be derived within the compositional
_semantic_ system rather than being viewed as a purely _pragmatic_ phenomenon
(in line with some other recent studies in the RSA literature, e.g., Franke
Bergen 2020). The relevant semantic mechanism is indeed completely
encapsulated from probabilistic reasoning. Now, precisely because this
mechanism is not a pragmatic mechanism but rather belongs to compositional
semantics, this conclusion is by itself fully compatible with the view that
pragmatic reasoning proper is probabilistic in nature and, specifically, that
the notion of informativity relevant to pragmatics is the information-
theoretic notion of _surprisal_ used in the RSA framework. Our data,
specifically on the production side, suggest that this is in fact the case. In
a semantic theory that includes an exhaustivity mechanism, sentences are
typically ambiguous between an exhaustive and a non-exhaustive reading, and so
an account of the use and interpretation of potentially ambiguous sentences
needs to include an explicit formalization of how speakers and listeners deal
with this ambiguity. The RSA framework provides the means to articulate a
semantic view of exhaustivity with such a formally explicit pragmatic model,
in which the role of prior probabilistic expectations can be captured.

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##  Appendix A Formal conditions leading to Exhaustivity and Anti-Exhaustivity

###  A.1 Baseline RSA model

Consider the baseline RSA model with two worlds, $w_{a}$ and $w_{ab}$ and
three messages, A, $A\wedge B$ and $A\wedge\neg B$. Let us set the cost of A
at $0$, and notate $c_{A\wedge B}$ and $c_{A\wedge\neg B}$ the costs of the
two other messages.As usual, $\lambda$ is a positive real number, and $P$ is
the prior probabilty distribution over the two worlds.

We will first prove the following resuts:

. . $S_{1}(A|w_{ab})>S_{1}(A|w_{a}))$ if and only if
$\log(P(w_{ab}))-\log(P(w_{a}))>c_{A\wedge\neg B}-c_{A\wedge B}$ .̱
$L_{1}(w_{ab}|A)>P(w_{ab}|\\{w_{a},w_{ab}\\})$ if and only if
$\log(P(w_{ab}))-\log(P(w_{a}))>c_{A\wedge\neg B}-c_{A\wedge B}$

Let us note that in this simple model, since there are only two worlds,
$P(w_{ab}|\\{w_{a},w_{ab}\\})=P(w_{ab})$, and the message A is in fact the
tautology. For this reason, for either world $w$ in $\\{w_{a},w_{ab}\\}$,
$L_{0}(w|A)=P(w)$.

Given the equations of the RSA model, and the observations just made, we have:

. . $U_{1}(A|w_{a})=\log(P(w_{A}))$ .̱ $U_{1}(A\wedge\neg
B|w_{A})=\log(\underbrace{L_{0}(w_{a}|A\wedge\neg B)}_{1})-c_{A\wedge\neg
B}=-c_{A\wedge\neg B}$ .̧ $U_{1}(A|w_{ab})=\log(P(w_{ab}))$ .̣ $U_{1}(A\wedge
B|w_{ab})=\log(\underbrace{L_{0}(w_{ab}|A\wedge B)}_{1})-c_{A\wedge
B}=-c_{A\wedge B}$

Given this, applying the definitions, we have:

.
$S_{1}(A|w_{a})=\dfrac{e^{\lambda\log(P(w_{a}))}}{e^{\lambda\log(P(w_{a}))}+e^{-\lambda
c_{A\wedge\neg B}}}=\dfrac{1}{1+e^{-\lambda(c_{A\wedge\neg
B}+\log(P(w_{a}))}}$

Likewise:

.
$S_{1}(A|w_{ab})=\dfrac{e^{\lambda\log(P(w_{ab}))}}{e^{\lambda\log(P(w_{ab}))}+e^{-\lambda
c_{A\wedge B}}}=\dfrac{1}{1+e^{-\lambda(c_{A\wedge B}+\log(P(w_{ab}))}}$

Since the logistic function $\frac{1}{1+e^{-\lambda x}}$ is increasing,

$S_{1}(A|w_{ab})~{}>~{}S_{1}(A|w_{a})\textrm{~{}iff~{}}c_{A\wedge~{}B}+\log(P(w_{ab}))~{}>~{}c_{A\wedge\neg~{}B}+\log(P(w_{a}))$

This is in turn equivalent to:

$\log(P(w_{ab}))-\log(P(w_{a}))>c_{A\wedge\neg B}-c_{A\wedge B}$

Let us now prove that:

. $L_{1}(w_{ab}|A)>P(w_{ab})$ iff $S_{1}(A|w_{ab})>S_{1}(A|w_{a})$

We have:

$L_{1}(w_{ab}|A)=\dfrac{P(w_{ab})\times S_{1}(A|w_{ab})}{P(w_{ab})\times
S_{1}(A|w_{ab})+P(w_{a})\times S_{1}(A|w_{a})}$

With $p=P(w_{ab})\in(0,1)$, we have:

$\displaystyle L_{1}(w_{ab}|A)>P(w_{ab})$ $\displaystyle\equiv$
$\displaystyle\dfrac{p\times S_{1}(A|w_{ab})}{p\times
S_{1}(A|w_{ab})+(1-p)\times S_{1}(A|w_{a})}>p$ $\displaystyle\equiv$
$\displaystyle\dfrac{S_{1}(A|w_{ab})}{p\times S_{1}(A|w_{ab})+(1-p)\times
S_{1}(A|w_{a})}>1$ $\displaystyle\equiv$ $\displaystyle
S_{1}(A|w_{ab})>p\times S_{1}(A|w_{ab})+(1-p)\times S_{1}(A|w_{a})$
$\displaystyle\equiv$ $\displaystyle(1-p)S_{1}(A|w_{ab})>(1-p)S_{1}(A|w_{a})$
$\displaystyle\equiv$ $\displaystyle S_{1}(A|w_{ab})>S_{1}(A|w_{a})$

###  A.2 Wonky RSA models

In this model, the $S_{1}$ speaker is conditionalized on the wonkiness
parameter, and therefore not interpretable as an actual speaker. We therefore
consider anti-exhaustivity at the $L_{1}$ listener level only. The full
expression of $L_{1}$ in the standard wRSA and our Bayesian implementation
BwRSA are given in [a] and [b] respectively, where $p=P(w_{ab})$ is the non-
wonky prior on the world which makes A and B true, $\omega$ is the wonkiness
prior, $\lambda$ the rationality parameter, and $c_{A\wedge B}$ and
$c_{A\wedge\neg B}$ the costs of the A and B and A and not B messages,
respectively. $f_{\lambda}$ represents the logistic function with rate
$\lambda$.

. . $\displaystyle L_{1}(w_{ab}|A)=\frac{p(1-\omega)f_{\lambda}(\log
p+c_{A\wedge B})+\frac{1}{2}\omega f_{\lambda}(c_{A\wedge B}-\log
2)}{p(1-\omega)f_{\lambda}(\log p+c_{A\wedge B})+\frac{1}{2}\omega
f_{\lambda}(c_{A\wedge B}-\log
2)+(1-p)(1-\omega)f_{\lambda}(\log(1-p)+c_{A\wedge\neg B})+\frac{1}{2}\omega
f_{\lambda}(c_{A\wedge\neg B}-\log 2)}$ .̱ $\displaystyle
L_{1}(w_{ab}|A)=p\frac{(1-\omega)f_{\lambda}(\log p+c_{A\wedge B})+\omega
f_{\lambda}(c_{A\wedge B}-\log 2)}{p[(1-\omega)f_{\lambda}(\log p+c_{A\wedge
B})+\omega f_{\lambda}(c_{A\wedge B}-\log
2)]+(1-p)[(1-\omega)f_{\lambda}(\log(1-p)+c_{A\wedge\neg B})+\omega
f_{\lambda}(c_{A\wedge\neg B}-\log 2)]}$

In both cases, we define listener-side anti-exhaustivity as any situation
where the posterior exceeds the measured prior, i.e.
$L_{1}(w_{ab}|A)>P(w_{ab}|b_{1})$.

For the wRSA, an analytical solution would be difficult to express, but we can
already make two observation: if $\omega>0$, the model always predicts anti-
exhaustivity when $p\to 0$, and always predicts exhaustivity when $p\to 1$.
This is just a consequence of it being non-Bayesian, as the posterior value
never converges to exactly 0 and 1, respectively. The anti-exhaustivity when
$p\to 0$ is unrelated to anti-exhaustivity in the baseline RSA, which happens
when $p$ is high, and is arguably unproblematic. The main question is whether
anti-exhaustivity can arise in the wRSA when $p$ is high but not too close to
1. With numerical simulations, we can answer positively. For instance, if we
set the parameters to $\omega=0.1,c_{A\wedge B}=1,c_{A\wedge\neg
B}=1.2,\lambda=3$, we find anti-exhaustivity for $p<.04$ but also for
$p\in[0.57,0.95]$. The problematic anti-exhaustivity inherited from the
baseline RSA fades away with sufficient increase in either $\omega$ or
$c_{A\wedge\neg B}$.

For the Bayesian version BwRSA, we can show that anti-exhaustivity happens if
and only if:

$(1-\omega)\left[f_{\lambda}(\log p+c_{A\wedge
B})-f_{\lambda}(\log(1-p)+c_{A\wedge\neg
B})\right]+\omega\left[f_{\lambda}(c_{A\wedge B}-\log
2)-f_{\lambda}(c_{A\wedge\neg B}-\log 2)\right]>0$

Since this is a monotone increasing function of $p$, we can simply look at the
situation when $p\to 1$, and it follows that a sufficient and necessary
condition on the presence of anti-exhaustivity in this model is:

$\omega<\frac{f_{\lambda}(c_{A\wedge B})}{f_{\lambda}(c_{A\wedge
B})-f_{\lambda}(c_{A\wedge B}-\log 2)+f_{\lambda}(c_{A\wedge\neg B}-\log 2)}$

In practice, this means that anti-exhaustivity is very difficult to avoid in
this model. Numerically, $\omega$ would have to be over 0.9 with the parameter
values used in the wRSA case.

###  A.3 Supervaluationist model

At the level of $S_{1}$, the model predicts no anti-exhaustivity in the sense
that for any QUD $Q$, no matter the priors, $S_{1}(A|w_{ab},Q)\leq
S_{1}(A|w_{a},Q)$. Indeed, if $Q$ distinguishes between $w_{a}$ and $w_{ab}$,
the fact that A can be interpreted as $exh(A)$ entails $S_{1}(A|w_{ab},Q)=0$,
while if $Q$ does not distinguish between these worlds,
$S_{1}(A|w_{ab},Q)=S_{1}(A|w_{a},Q)$.

Turning to the pragmatic listener $L_{1}$, we can prove that the posterior on
$w_{ab}$ after hearing A never exceeds the prior (with equality only if the
prior is exactly 0 or 1).

$\begin{array}[]{rcl}L_{1}(w_{ab}|A)&=&L_{1}(w_{ab}|A,Q_{A})+L_{1}(w_{ab}|A,Q_{\text{total}})\\\
L_{1}(w_{ab}|A)&=&\displaystyle\frac{P(w_{ab})P(Q_{A})S_{1}(A|w_{ab},Q_{A})+P(w_{ab})P(Q_{\text{total}})\overbrace{S_{1}(A|w_{ab},Q_{\text{total}})}^{=0}}{\sum_{w,Q}P(w)P(Q)S_{1}(A|w,Q)}\\\
&&\rule[-12.91663pt]{0.0pt}{25.83325pt}\text{Noting that
}S_{1}(A|w_{a},Q_{A})=S_{1}(A|w_{ab},Q_{A})=S_{1}(A|Q_{A})\text{ we can
factorize the denominator and simplify:}\\\
L_{1}(w_{ab}|A)&=&\displaystyle\frac{P(w_{ab})P(Q_{A})S_{1}(A|Q_{A})}{(P(w_{a})+P(w_{ab}))P(Q_{A})S_{1}(A|Q_{A})+P(w_{a})P(Q_{\text{total}})S_{1}(A|w_{a},Q_{\text{total}})}\\\
L_{1}(w_{ab}|A)&=&\displaystyle\frac{P(w_{ab})}{\displaystyle
1+P(w_{a})\underbrace{\frac{P(Q_{\text{total}})S_{1}(A|w_{a},Q_{\text{total}})}{P(Q_{A})S_{1}(A|Q_{A})}}_{>0}}\leq
P(w_{ab})\\\ \end{array}$

##  Appendix B Implementation of the models

Here we list the formulas used to implement the different models. See the file
“RSAExh-ModelFunctions.R” in supplementary materials for details on how we
translated these formulas to R. In all models, $\lambda$ represents the
rationality parameter, $\Delta_{ab}$ and $\Delta_{a\neg b}$ the costs of A&B
and A&$\neg$B relative to A, and $p$ the conditional prior on $w_{ab}$ given
$\\{w_{a},w_{ab}\\}$. Since we consider only two worlds, we only give marginal
listener functions for $w_{ab}$. The listener function for $w_{a}$ is simply
the complementary.

###  B.1 Base RSA

$L_{1}$ listener:

$\begin{array}[]{rcl}L_{1}(w_{ab}|A)&=&\frac{1}{1+\frac{(1-p)\left(1+\mathrm{e}^{-\lambda(\log(p)+\Delta_{ab})}\right)}{p\left(1+\mathrm{e}^{-\lambda(\log(1-p)+\Delta_{a\neg
b})}\right)}}\\\ L_{1}(w_{ab}|A\&B)&=&1\\\ L_{1}(w_{ab}|A\&\neg
B)&=&0\end{array}$

$\log S_{2}$ speaker:

$\begin{array}[]{rcl}\log
S_{2}(A|w_{a})&=&-\log\left(1+\mathrm{e}^{-\lambda(\log(1-L_{1}(w_{ab}|A))+\Delta_{a\neg
b})}\right)\\\ \log S_{2}(A\&\neg
B|w_{a})&=&-\log\left(1+\mathrm{e}^{\lambda(\log(1-L_{1}(w_{ab}|A))+\Delta_{a\neg
b})}\right)\\\ \log S_{2}(A|w_{ab})&=&-\log\left(1+\mathrm{e}^{-\lambda(\log
L_{1}(w_{ab}|A)+\Delta_{ab})}\right)\\\ \log
S_{2}(A\&B|w_{ab})&=&-\log\left(1+\mathrm{e}^{\lambda(\log
L_{1}(w_{ab}|A)+\Delta_{ab})}\right)\end{array}$

###  B.2 wRSA and BwRSA

The $S_{1}$ is parametrized by the wonkiness parameter $b$ which can take
values $b_{1}$ (wonky) or $b_{2}$ (usual), and a parameter $\omega=P(b=b_{1})$
which is the prior on wonkiness. The “ususal” $S_{1}$ speaker is the same as
the base RSA speaker. The “wonky” $S_{1}$ speaker is given by:

$\begin{array}[]{rcl}S_{1}(A|w_{a},b_{1})&=&\frac{1}{1+\mathrm{e}^{\lambda(\log
2-\Delta_{a\neg b})}}\\\ S_{1}(A\&\neg
B|w_{a},b_{1})&=&\frac{1}{1+\mathrm{e}^{-\lambda(\log 2-\Delta_{a\neg b})}}\\\
S_{1}(A|w_{ab},b_{1})&=&\frac{1}{1+\mathrm{e}^{\lambda(\log
2-\Delta_{ab})}}\\\
S_{1}(A\&B|w_{ab},b_{1})&=&\frac{1}{1+\mathrm{e}^{-\lambda(\log
2-\Delta_{ab})}}\end{array}$

For wRSA, the $L_{1}$ speaker updates her prior to reflect her assumptions
about the speaker. Here we consider the marginal posterior on the world.

$\begin{array}[]{rcl}L_{1}(w_{ab}|A)&=&\frac{\frac{1}{2}\omega
S_{1}(A|w_{a},b_{1})+p(1-\omega)S_{1}(A|w_{a},b_{2})}{\frac{1}{2}\omega\left(S_{1}(A|w_{a},b_{1})+S_{1}(A|w_{ab},b_{1})\right)+(1-\omega)(pS_{1}(A|w_{a},b_{2})+(1-p)S_{1}(A|w_{ab},b_{2}))}\end{array}$

In the BwRSA, the $L_{1}$ speaker makes inferences about the speaker’s priors,
but sticks to her own priors when inferring $w$. Again, we consider the
marginal posterior on the world:

$\begin{array}[]{rcl}L_{1}(w_{ab}|A)&=&p\frac{\omega
S_{1}(A|w_{a},b_{1})+(1-\omega)S_{1}(A|w_{a},b_{2})}{\omega\left(pS_{1}(A|w_{a},b_{1})+(1-p)S_{1}(A|w_{ab},b_{1})\right)+(1-\omega)(pS_{1}(A|w_{a},b_{2})+(1-p)S_{1}(A|w_{ab},b_{2}))}\end{array}$

In both cases, the strong messages are interpreted literally:
$L_{1}(w_{ab}|A\&B)=1$ and $L_{1}(w_{ab}|A\&\neg B)=0$. At the level of
$S_{2}$, the parameter $b$ does not appear, and the speaker only tries to
convey the world, so the formulas are identical to base RSA.

###  B.3 svRSA1 and svRSA2

Here we consider two possible QUDs: $Q_{A}$ is the partial QUD and corresponds
to the partition $\\{\\{w_{a},w_{ab}\\}\\}$ (which is trivial when considering
only two worlds) and $Q_{total}$ is the total QUD and corresponds to the
partition $\\{\\{w_{a}\\},\\{w_{ab}\\}\\}$. We write $q=P(Q=Q_{total})$ the
prior on QUDs. We write $\chi=P(i_{exh})$ the prior on exhaustification, which
will be fixed to $\frac{1}{2}$ in the model fit.

For $Q_{A}$, the world does not affect the choice of a message since
$Q_{A}(w_{a})=Q_{A}(w_{ab})$, and only costs distinguish the three messages.

$\begin{array}[]{rcl}S_{1}(A|w,Q_{A})&=&\frac{1}{1+\mathrm{e}^{-\lambda\Delta_{a\neg
b}}+\mathrm{e}^{-\lambda\Delta_{ab}}}\\\ S_{1}(A\&\neg
B|w,Q_{A})&=&\frac{1}{1+\mathrm{e}^{\lambda\Delta_{a\neg
b}}+\mathrm{e}^{\lambda(\Delta_{a\neg b}-\Delta_{ab})}}\\\
S_{1}(A\&B|w,Q_{A})&=&\frac{1}{1+\mathrm{e}^{\lambda\Delta_{ab}}+\mathrm{e}^{\lambda(\Delta_{ab}-\Delta_{a\neg
b})}}\\\ \end{array}$

For $Q_{total}$, the world and therefore the meaning of the messages matter:

$\begin{array}[]{rcl}S_{1}(A|w_{a},Q_{total})&=&\frac{1}{1+\mathrm{e}^{-\lambda((1-\chi)\log(1-p)+\Delta_{a\neg
b})}}\\\ S_{1}(A\&\neg
B|w_{a},Q_{total})&=&\frac{1}{1+\mathrm{e}^{\lambda((1-\chi)\log(1-p)+\Delta_{a\neg
b})}}\\\ S_{1}(A|w_{ab},Q_{total})&=&0\\\
S_{1}(A\&B|w_{ab},Q_{total})&=&1\end{array}$

Note that under the total QUD, A can only be used to convey $w_{a}$, so only
A&B can convey $w_{ab}$.

$L_{1}$ infers both the QUD and the world. We give the joint posterior
distribution here (which is needed to derive $S_{2}$), but the fit of the
comprehension data was based on the marginal world posterior.

$\begin{array}[]{rcl}L_{1}(w_{a},Q_{A}|A)&=&\frac{(1-p)(1-q)S_{1}(A|w,Q_{A})}{(1-q)S_{1}(A|w,Q_{A})+q(1-p)S_{1}(A|w_{a},Q_{total})}\\\
L_{1}(w_{a},Q_{total}|A)&=&\frac{(1-p)qS_{1}(A|w_{a},Q_{total})}{(1-q)S_{1}(A|w,Q_{A})+q(1-p)S_{1}(A|w_{a},Q_{total})}\\\
L_{1}(w_{ab},Q_{A}|A)&=&\frac{p(1-q)S_{1}(A|w,Q_{A})}{(1-q)S_{1}(A|w,Q_{A})+q(1-p)S_{1}(A|w_{a},Q_{total})}\\\
L_{1}(w_{a},Q_{A}|A\&B)&=&\frac{(1-p)(1-q)S_{1}(A\&B|w,Q_{A})}{(1-q)S_{1}(A\&B|w,Q_{A})+qp}\\\
L_{1}(w_{ab},Q_{A}|A\&B)&=&\frac{p(1-q)S_{1}(A\&B|w,Q_{A})}{(1-q)S_{1}(A\&B|w,Q_{A})+qp}\\\
L_{1}(w_{ab},Q_{total}|A\&B)&=&\frac{qp}{(1-q)S_{1}(A\&B|w,Q_{A})+qp}\\\
L_{1}(w_{a},Q_{A}|A\&\neg B)&=&\frac{(1-p)(1-q)S_{1}(A\&\neg
B|w,Q_{A})}{(1-q)S_{1}(A\&\neg B|w,Q_{A})+q(1-p)S_{1}(A\&\neg
B|w_{a},Q_{total})}\\\ L_{1}(w_{ab},Q_{A}|A\&\neg
B)&=&\frac{p(1-q)S_{1}(A\&\neg B|w,Q_{A})}{(1-q)S_{1}(A\&\neg
B|w,Q_{A})+q(1-p)S_{1}(A\&\neg B|w_{a},Q_{total})}\\\
L_{1}(w_{a},Q_{total}|A\&\neg B)&=&\frac{(1-p)qS_{1}(A\&\neg
B|w_{a},Q_{total})}{(1-q)S_{1}(A\&\neg B|w,Q_{A})+q(1-p)S_{1}(A\&\neg
B|w_{a},Q_{total})}\\\ \end{array}$

All other combinations involve $Q_{total}$ and world that makes the message
literally false, and therefore receive a posterior probability of 0.

We can now give the $\log S_{2}$ functions. The functions for $Q_{A}$ only
depend on the $L_{1}$ marginal $Q_{A}$ posterior, since a speaker who wants to
convey $Q_{A}$ does not care about $w$.

$\begin{array}[]{rcl}\log
S_{2}(A|w,Q_{A})&=&-\log\left(1+\mathrm{e}^{-\lambda\left(\log\frac{L_{1}(Q_{A}|A)}{L_{1}(Q_{A}|A\&\neg
B)}+\Delta_{a\neg
b}\right)}+\mathrm{e}^{-\lambda\left(\log\frac{L_{1}(Q_{A}|A)}{L_{1}(Q_{A}|A\&B)}+\Delta_{ab}\right)}\right)\\\
\log S_{2}(A\&\neg
B|w,Q_{A})&=&-\log\left(1+\mathrm{e}^{\lambda\left(\log\frac{L_{1}(Q_{A}|A)}{L_{1}(Q_{A}|A\&\neg
B)}+\Delta_{a\neg
b}\right)}+\mathrm{e}^{-\lambda\left(\log\frac{L_{1}(Q_{A}|A\&\neg
B)}{L_{1}(Q_{A}|A\&B)}+\Delta_{ab}-\Delta_{a\neg b}\right)}\right)\\\ \log
S_{2}(A\&B|w,Q_{A})&=&-\log\left(1+\mathrm{e}^{-\lambda\left(\log\frac{L_{1}(Q_{A}|A\&B)}{L_{1}(Q_{A}|A\&\neg
B)}+\Delta_{a\neg
b}-\Delta_{ab}\right)}+\mathrm{e}^{\lambda\left(\log\frac{L_{1}(Q_{A}|A)}{L_{1}(Q_{A}|A\&B)}+\Delta_{ab}\right)}\right)\\\
\log
S_{2}(A|w_{a},Q_{total})&=&-\log\left(1+\mathrm{e}^{-\lambda\left(\log\frac{L_{1}(w_{a},Q_{total}|A)}{L_{1}(w_{a},Q_{total}|A\&\neg
B)}+\Delta_{a\neg b}\right)}\right)\\\ \log S_{2}(A\&\neg
B|w_{a},Q_{total})&=&-\log\left(1+\mathrm{e}^{\lambda\left(\log\frac{L_{1}(w_{a},Q_{total}|A)}{L_{1}(w_{a},Q_{total}|A\&\neg
B)}+\Delta_{a\neg b}\right)}\right)\\\ \log
S_{2}(A\&B|w_{ab},Q_{total})&=&0\end{array}$

In practice, we tested two implementations of the supervaluationist model for
production. In svRSA1, the production was modelled as
$(1-q)S_{2}(u|w,Q_{A})+qS_{2}(u|w,Q_{total})$, representing uncertainty on the
QUD the speaker should convey. In svRSA2 it was simply modelled as
$S_{2}(u|w,Q_{total})$, which meant that the speaker had no uncertainty on
which QUD to convey, but the listener still had to infer the QUD and the
speaker was aware of the listener’s uncertainty.

###  B.4 FREE-LU and EXH-LU

We first introduced a full unrestricted LU model with all possible
strengthening and free priors over these, from which we will define our two
implementations. We consider three possible interpretations
$i_{lit},i_{exh},i_{\textit{anti-exh}}$ such that:

$\begin{array}[]{lccl}\llbracket
A\rrbracket^{i_{lit}}&=&\\{w_{a},w_{ab}\\}&\quad\text{literal}\\\ \llbracket
A\rrbracket^{i_{exh}}&=&\\{w_{a}\\}&\quad\text{exhaustive}\\\ \llbracket
A\rrbracket^{i_{\textit{anti-exh}}}&=&\\{w_{ab}\\}&\quad\text{anti-
exhaustive}\end{array}$

For messages A&B and A&$\neg$B, all $i$’s return the literal interpretations,
as they cannot be further strengthened.

$\begin{array}[]{rcl}S_{1}(A|w_{a},i_{lit})&=&\frac{1}{1+\mathrm{e}^{-\lambda(\log(1-p)+\Delta_{a\neg
b})}}\\\ S_{1}(A|w_{ab},i_{lit})&=&\frac{1}{1+\mathrm{e}^{-\lambda(\log
p+\Delta_{ab})}}\\\
S_{1}(A|w_{a},i_{exh})&=&\frac{1}{1+\mathrm{e}^{-\lambda\Delta_{a\neg b}}}\\\
S_{1}(A|w_{ab},i_{\textit{anti-
exh}})&=&\frac{1}{1+\mathrm{e}^{-\lambda\Delta_{ab}}}\end{array}$

Moving to $L_{1}$, we need to consider the priors on these three
interpretations $\rho_{lit},\rho_{exh},\rho_{\textit{anti-exh}}$. The
posteriors for $w_{ab}$ after A&B and A&$\neg$B remain 1 and 0, respectively.

$L_{1}(w_{ab}|A)=p\frac{\rho_{lit}S_{1}(A|w_{ab},i_{lit})+\rho_{\textit{anti-
exh}}S_{1}(A|w_{ab},i_{\textit{anti-
exh}})}{(1-p)\rho_{lit}S_{1}(A|w_{a},i_{lit})+p\rho_{lit}S_{1}(A|w_{ab},i_{lit})+(1-p)\rho_{exh}S_{1}(A|w_{a},i_{exh})+p\rho_{\textit{anti-
exh}}S_{1}(A|w_{ab},i_{\textit{anti-exh}})}$

The $S_{2}$ speaker is defined as in baseline RSA. We derive the FREE-LU model
by fixing $\rho_{lit}=\rho_{exh}=\rho_{\textit{anti-exh}}=\frac{1}{3}$ and the
EXH-LU model by fixing $\rho_{lit}=\rho_{exh}=\frac{1}{2}$ and
$\rho_{\textit{anti-exh}}=0$.

###  B.5 RSA-LI

For the RSA-LI model, we keep interpretations $i_{lit}$ and $i_{exh}$ from the
LU model, but the speaker now tries to convey which interpretation they meant.
There is no prior on $i$ anymore (in other words, the listener has a uniform
prior on $i$). We can look directly at the marginal $S_{1}$ speaker.

$\begin{array}[]{rcl}S_{1}(A|w_{a})&=&\frac{1+\mathrm{e}^{\lambda\log(1-p)}}{1+\mathrm{e}^{\lambda\log(1-p)}+2\mathrm{e}^{-\lambda\Delta_{a\neg
b}}}\\\ S_{1}(A|w_{ab})&=&\frac{1}{1+2\mathrm{e}^{-\lambda(\log
p+\Delta_{ab})}}\\\ S_{1}(A\&\neg
B|w_{a})&=&\frac{2}{2+\mathrm{e}^{\lambda\left(\log(1-p)+\Delta_{a\neg
b}\right)}+\mathrm{e}^{\lambda\Delta_{a\neg b}}}\\\
S_{1}(A\&B|w_{ab})&=&\frac{2}{2+\mathrm{e}^{\lambda(\log
p+\Delta_{ab})}}\par\end{array}$

The marginal $L_{1}$ listener is then described by:

$\begin{array}[]{rcl}L_{1}(w_{ab}|A)&=&\frac{1}{1+\frac{(1-p)\left(1+2\mathrm{e}^{-\lambda(\log
p+\Delta_{ab})}\right)\left(1+\mathrm{e}^{\lambda\log(1-p)}\right)}{p\left(1+\mathrm{e}^{\lambda\log(1-p)}+2\mathrm{e}^{-\lambda\Delta_{a\neg
b}}\right)}}\end{array}$

The $S_{2}$ speaker is defined as the usual RSA speaker (she cannot select a
specific meaning, as $L_{1}$—unlike $L_{0}$—is assumed to be blind to the
intended meaning). Because of this crucial difference between $S_{1}$ and
$S_{2}$, we tested both speakers as models of production ($S_{1}$ in RSA-LI1
and $S_{2}$ in RSA-LI2).