Author Full Name(s) Full title indicative conditionals, Rational-Speech-Act
model, computational pragmatics, causal Bayes nets, inferentialism

# Probabilistic modeling of rational communication with conditionals ††thanks:
We would like to thank three anonymous reviewers for their highly appreciated
and very valuable feedback which helped to improve the manuscript
substantially.

Britta Grusdt
Institut für Kognitionswissenschaften
Universität OsnabrückSchool of Philosophy, Psychology and Language Sciences
The University of EdinburghAllg. Sprachwissenschaft & Pragmatik
Eberhard Karls Universität Tübingen Daniel Lassiter
Michael Franke
School of Philosophy, Psychology and Language Sciences
The University of EdinburghAllg. Sprachwissenschaft & Pragmatik
Eberhard Karls Universität Tübingen B<EMAIL_ADDRESS>D
<EMAIL_ADDRESS>M<EMAIL_ADDRESS>
###### Abstract

While a large body of work has scrutinized the meaning of conditional
sentences, considerably less attention has been paid to formal models of their
pragmatic use and interpretation. Here, we take a probabilistic approach to
pragmatic reasoning about indicative conditionals which flexibly integrates
gradient beliefs about richly structured world states. We model listeners’
update of their prior beliefs about the causal structure of the world and the
joint probabilities of the consequent and antecedent based on assumptions
about the speaker’s utterance production protocol. We show that, when supplied
with natural contextual assumptions, our model uniformly explains a number of
inferences attested in the literature, including epistemic inferences,
conditional perfection and the dependency between antecedent and consequent of
a conditional. We argue that this approach also helps explain three puzzles
introduced by Douven2012b about updating with conditionals: depending on the
utterance context, the listener’s belief in the antecedent may increase,
decrease or remain unchanged.

###### keywords:

indicative conditionals, Rational-Speech-Act model, computational pragmatics,
causal Bayes nets, inferentialism

##  1 Introduction

Despite the long history of research on conditionals, no consensus has been
reached on crucial questions concerning their semantics and pragmatics, let
alone on fundamental questions such as whether conditionals have truth
conditions at all. What is inarguably true is the diversity of how
conditionals are interpreted, yet the circumstances that lead to different
interpretations of conditionals depending, among others, on the utterance
context are not well understood. An utterance of the conditional in (1) might
be disillusioning when given by a professional epidemiologist, as one would
naturally infer that the speaker is ignorant about the truth of the antecedent
and the truth of the consequent. But how does this inference about the
speaker’s uncertainty come about?

  1. (1)

If the pandemic is over next year, the conference will take place offline
again.

The conditional in (1) further suggests that the pandemic, mentioned in the
antecedent, is related to the form of the conference, mentioned in the
consequent. Does this inferred relationship originate from pragmatic reasoning
or do we draw this inference because it is inherently part of the semantics of
conditionals? The latter is advanced by prominent recent accounts of the
meaning of conditionals, so-called _inferentialism_ (e.g., Douven2017;
Douven2018).

One long-standing problem that has particularly gained attention in relation
with inferentialism is the problem of missing-link conditionals. Missing-link
conditionals are pragmatically infelicitous conditionals with no discernible
causal or conditional relationship between antecedent and consequent, as in
example (2).

  1. (2)

If Jo doesn’t pass the exam, it will rain tomorrow.

Contrary to inferentialist accounts, we argue that it is possible to explain
the observation that missing-link conditionals are infelicitous solely on the
basis of pragmatics, making use of a notion of _listener surprise_ about the
speaker’s utterance choice.

In this paper, we explore a formalization of interlocutors’ pragmatic
reasoning about the use and interpretation of conditionals within the
Rational-Speech-Act (RSA) modeling framework (Frank2012a; Franke2016;
Goodman2016).111When we write ‘conditional’, we refer to indicative
conditionals whose antecedent and consequent are simple propositions, to which
we restrict the considered kind of conditionals in this paper.  We aim to
contribute to the understanding of how the inferences commonly associated with
conditionals arise and how different utterance contexts influence the
interpretation of conditionals. In a nutshell, a probabilistic pragmatics
approach enables us to combine two aspects, which we here argue are key to
obtaining a more realistic picture of the pragmatics of conditionals:
pragmatic reasoning in the vein of Grice1975 and richly structured,
probabilistic world knowledge, particularly about _a priori_ plausible causal
relationships between antecedent and consequent.

To illustrate this, consider example (1) again. We argue that the inferences
outlined above can be explained by pragmatic reasoning about the speaker’s
utterance choice: a cooperative speaker who is in the position to assert the
antecedent and/or the consequent alone, should make this stronger claim in
order to be as informative as required (quine65; Grice1975; griceIC). Given
that the speaker did not claim the truth of the antecedent nor the consequent,
a listener would plausibly infer uncertainty on the side of the speaker about
whether the antecedent and the consequent are true. It is in this sense that
we expect Gricean pragmatic reasoning about the speaker’s protocol of choosing
utterances (e.g., governed by _Maxims of Conversation_ or similar) to matter
for the interpretation of conditionals.

As to the effects of prior world knowledge, the conditional in (1) also
illustrates that the listener’s prior beliefs may influence the strength of
the beliefs the listener holds after receiving information in form of a
conditional. When an assertion of (1) is accepted, the listener’s degree of
belief in the conference taking place offline conditional on the pandemic
being over will be high, simply as a result of the conditional’s semantic
meaning. But the degree to which the listener believes in the conference _not_
taking place offline conditional on the pandemic _not_ being over (arguably
one possible measure of a Condtional Perfection reading, see below) crucially
depends on the listener’s further beliefs about the world. The important,
though perhaps uncontroversial, point to notice in connection with this
example is that prior world knowledge is likely to impact the beliefs the
listener holds after accepting and interpreting an utterance of a conditional
sentence.

At large, it is challenging to account for how people update beliefs in light
of new conditional information whereas there is a widely adopted general
account of how people update beliefs in light of new factual information,
_probabilistic conditioning_. In this approach, factual information is
represented by a probability distribution over a set of possible worlds. When
a new piece of information $E$ is received, an updated belief state is formed
by assuming that the information is true and adjusting the distribution
accordingly. The consequence is that $P(A)$, the prior belief distribution on
proposition $A$, is updated to a posterior distribution $P(A\mid E)$ so that:

$P(A\mid E)=\frac{P(A\wedge E)}{P(E)}\,.$

This idea lies at the heart of Bayesian models in epistemology and decision
theory (e.g., earman92) as well as computational cognitive science
(tenenbaumetal11). Conditional sentences present a challenge to the generality
of the Bayesian approach to learning, because there is no consensus about how
the update should work for conditionals, or whether conditioning is even
applicable (e.g., Grove1997; Douven2011a). Many authors have argued that
conditionals cannot be regarded as having truth conditions at all (for a
recent review see Rothschild2015), but in its standard form, Bayesian update
cannot be applied to sentences without truth-conditions. Even if conditionals
do have truth-conditions, it remains unclear how to implement a plausible
update operation for conditional information.

In an influential paper, Douven2012b argues that there is no uniform way to
capture the update effect of conditionals within Bayesian theory, regardless
of the assumed semantics. The argument hinges on the relationship between the
listener’s prior and posterior beliefs about the antecedent. Douven shows that
depending on the context in which a conditional is uttered the inferred
probability of the antecedent may either decrease, increase or remain
unchanged compared to the listener’s beliefs prior to the uptake of the
conditional. Surveying a variety of possible semantics for conditionals, he
shows that all of them predict that Bayesian update should have a uniform
effect on the probability of the antecedent. We argue that it is possible to
explain the non-uniform update effect of conditionals, that we will refer to
as _Douven’s puzzle_ , by standard Bayesian belief revision if we also take
into account pragmatic reasoning and world knowledge. Specifically, prior
knowledge about how the relevant propositions are likely to relate to each
other influence assumptions about what a rational speaker would say under
various circumstances. Assumptions about a speaker’s production protocol in
turn influence the listener’s interpretation of an utterance of a conditional.
From this perspective, Douven’s examples make a strong case for the importance
of pragmatic and contextual information in the interpretation of conditionals,
and provide a useful test suite for our model which attempts to formalize
these factors.222Like-minded proposals to account for Douven’s puzzle have
been advanced recently (e.g., Gunther2018; Eva2019; Vandenburgh2020), making
use of the different intuitive causal structures of Douven’s scenarios.
However, none of these proposals considers the crucial role of pragmatic
reasoning.

We argue that prior knowledge about the likely causal relationship between
antecedent and consequent is not only important to account for Douven’s
puzzle, but generally mediates a listener’s interpretation of a conditional.
For instance, in (1) our background knowledge supports the intuitive inference
that there is a causal and probabilistic relationship between the form of a
conference and the state of the pandemic. However, no such relationship seems
reasonable in missing-link conditionals like (2). But why does background
knowledge affect our interpretation in this way, leading to a sense of
infelicity when a dependency between consequent and antecedent is implausible?
We argue that assumptions about the causal relationship between antecedent and
consequent have an (implicit) influence on the speaker’s utterance choice and
thereby also on the listener’s interpretation, assuming a Gricean pragmatic
interpretation of conditionals uttered by a cooperative speaker. As we will
show, Gricean pragmatic reasoning formalized in the Rational-Speech-Act model
and supplemented by structured representations of causal world knowledge is
better able to account for missing-link conditionals than inferentialists have
claimed. Overall, the key to the success of our model stems from leveraging
the natural stochastic association between (i) whether antecedent and
consequent are causally related and (ii) whether “If _A_ , _C_ ” (henceforth
$A\rightarrow C$) is the most informative true utterance the speaker can make.

The paper is structured as follows. Section 2 introduces the vanilla RSA model
and details our implementation of the model for reasoning about the use of
conditionals. Section 3 and 4 describe how this model of pragmatic reasoning
helps explain key phenomena of interest, on the basis of unspecific as well as
contextualized, concrete examples, including Douven’s puzzles, conditional
perfection readings, and general inferences about the causal relationship
between antecedent and consequent. Section 5 discusses how the presented
approach positions itself with respect to missing-link and biscuit
conditionals. Section 6 reflects critically and concludes.

##  2 A Rational-Speech-Act model for communication with conditionals

To provide a principled and uniform formal treatment of pragmatic reasoning
which explains the observations raised in the introduction, we turn to
probabilistic pragmatic modeling. The Rational-Speech-Act (RSA) model is a
prominent instance of a formalization of Gricean pragmatic reasoning using
tools from probability calculus, decision and game theory (see Franke2016;
Goodman2016, for an overview). RSA models are probabilistic, data-driven and
based on the assumption that linguistic behavior is goal-oriented and in this
regard (approximately) optimal. A noteworthy benefit of a probabilistic and
data-driven approach is that the predictions by the models can often be
directly compared to quantitative aspects of experimental data, thereby
allowing the statistical comparison of theoretically relevant models based on
empirical data (e.g. Qing2015; Degen2020; Franke2020). Moreover, as
probabilistic modeling is prominent in other areas in the cognitive sciences,
it becomes relatively easy to integrate insights from these other areas as
well, such as belief formation (Goodman2013; Herbstritt2019), sequential
adaptation (Schuster), or learning biases (Brochhagen2018). Since it was first
introduced by Frank2012a in the context of a language reference game, the
number of RSA models and modeled phenomena has been growing; scalar
implicatures (Goodman2013), hyperboles (Kao2014a) as well as more complex
phenomena such as the interpretation of vague or polite language
(lassitergoodman17; Yoon2016), projective content (Qing2016), metaphors
(Kao2014) or social meaning (Burnett2019) have been investigated by means of
RSA.333A hands-on introduction to RSA modeling, using the probabilistic
programming language WebPPL (Goodman2014a), that we also use to implement our
model, is provided by Scontras2017.

There are several reasons why RSA also seems promising for modeling the
interpretation of conditionals. First of all, we expect an influence of the
availability of non-conditional utterances that the speaker might have chosen
instead of the conditional. In this way, RSA formalizes a Gricean account of
listeners’ interpretations and the speakers’ utterance choices. Since RSA is
able to flexibly integrate contextual knowledge, we can model quite different
utterance contexts, including situations in which richly structured world
knowledge is relevant. This will be crucial in our account of the effects of
causal knowledge on the interpretation of conditionals.

In the following, Section 2.1 gives a general introduction to the vanilla
version of the RSA model. Subsequent sections then elaborate on the specific
adaptations necessary in order to capture reasoning about the use of
conditionals against the background of richly structured causal world
knowledge. We use Section 2.3 to demonstrate the main underlying ideas of the
model by means of a toy example before we lay out the assumed priors on the
generalized set of states in the sections thereafter.

###  2.1 The vanilla Rational-Speech-Act model

At the heart of the vanilla RSA model lies the formalization of a cooperative
Gricean speaker who, when trying to communicate a state $s$, probabilistically
selects an utterance $u$ by preferably choosing utterances that are not only
true, but also maximize the amount of relevant information conveyed to a
literal listener. The pragmatic listener is modeled as a rational interpreter
who combines prior beliefs with the speaker’s protocol of choosing utterances
by using Bayes’ rule.

In order to capture the pragmatic speaker’s behavior, in particular, in order
to ground out a notion of truth and informativity, the RSA model considers a
literal listener first, whose interpretation behavior is defined as a
conditional probability distribution $P_{\text{lit}}(s\mid u)$ obtained by
updating any prior beliefs $P_{\text{prior}}(s)$ about likely world states $s$
with the set of states $[\\![u]\\!]$ where utterance $u$ is permissible.444The
formula in Equation [1], and in Equations [3] and [4] for other model
components, gives the probability up to proportionality ($\propto$), leaving
the normalizing constant of the probability distribution implicit. If
$F(x)\geq 0$ is a non-normalized score for any $x\in X$ with $X$ a finite set,
the notation $P(x)\propto F(x)$ is shorthand for
$P(x)=\frac{F(x)}{\sum_{x^{\prime}}F(x^{\prime})}$. Moreover,
$\delta_{s\in\sv{u}}$ is the Kronecker delta function, which returns 1 when
its Boolean argument the denotation function of utterance $u$ applied to state
$s$ is true, otherwise 0.

[1] $\displaystyle P_{\text{lit}}(s\mid u)\propto\delta_{s\in\sv{u}}\cdot
P_{\text{prior}}(s)$

The pragmatic speaker is then defined in terms of a notion of _utterance
utility_ $U(u;s)$, reflecting how informative $u$ is for communicating
$s$.555A common modification of the basic definition of utterance utility in
terms of informativity further accounts for differences among utterances by
using utterance cost regarding, for instance, their complexity, saliency or
social compatibility (e.g., see Qing2016; Gates2018). We do not include
utterance costs in our model, but they could easily be integrated.

[2] $\displaystyle U(u;s)=\text{log }P_{\text{lit}}(s\mid u)$

The informativity of an utterance $u$ as description of a state $s$ is defined
as the log-likelihood (negative surprisal) of the literal listener’s beliefs
for state $s$ after hearing utterance $u$. The probability that a speaker in
state $s$ will choose utterance $u$ is then defined as a soft-max operation on
utility scores:

[3] $\displaystyle P_{\text{S}}(u\mid s)\propto\text{exp}(\alpha\cdot U(u;s))$

$\alpha$ is a free model parameter governing how closely the speaker
approximates utility maximization: the higher $\alpha$, the more likely the
speaker is to choose the utility-maximizing utterance. At the extremes, a
hyperrational speaker ($\alpha\rightarrow\infty$) would only choose utterances
that maximize utility, while a randomizing speaker ($\alpha=0$) would choose
randomly among true utterances.

Finally, the pragmatic listener’s interpretation is captured by a conditional
probability distribution, $P_{\text{PL}}(s\mid u)$, which represents the
listener’s _a posteriori_ beliefs (after having heard utterance _u_) about the
probability of state $s$, taking the priors over states and a Gricean
speaker’s utterance-choice behavior into account.

[4] $\displaystyle P_{\text{PL}}(s\mid u)\propto P_{S}(u\mid s)\cdot
P_{\text{prior}}(s)$

###  2.2 World states, utterances & assertability

#### World states.

Conditionals like $A\rightarrow C$ are often associated with the speaker’s
uncertainty about whether $A$ and/or $C$ are true. Therefore, to model
pragmatic reasoning about conditionals, we include potentially uncertain
speakers into our modeling. Concretely, we look at the partition of possible
worlds into the four types of worlds which agree on the truth values of $A$
and $C$: $W=\\{w_{\emptyset},w_{A},w_{C},w_{AC}\\}$. The concrete set of
states used are probability distributions over worlds $w\in W$.

There are at least two prominent possibilities of how to interpret probability
distributions: as precise _objective_ chances or as _subjective_ beliefs.
Consequently, world states in our model can be interpreted in different ways,
too. For one, we can think of $s\in S$ as the true beliefs of a maximally
competent speaker about the objective chance of each type of world. In this
case, the conversational goal, implicitly defined in the vanilla RSA model, is
to communicate the true (objective, but intrinsically stochastic) world state
known to the speaker. For another, we can conceptualize world states as a
representation of an uncertain speaker’s subjective beliefs $s\in S$ about the
true (non-probabilistic) state of the world $w\in W$. Under this
interpretation, the vanilla RSA model implicitly treats the conversational
goal as that of communicating the speaker’s belief state (see
Aloni_2007:FC_BiOT; Franke2011:Quantity-Implic). The interpretation of world
states, as either objective or subjective, may matter for the interpretation
of the assertability conditions presented below. If not stated otherwise, we
refer to the subjective interpretation of model states.

#### Utterance alternatives.

Predictions of Gricean pragmatic reasoning strongly depend on the assumed set
of alternative utterances. There has been much discussion of alternative sets
for scalar items like _some_ , _warm_ and _or_ (e.g. Matsumoto1995;
Katzir2007), but much less for pragmatic reasoning about conditionals (for
some discussion see VanDerAuwera1997; VonFintel2001). The selection of
alternative utterances considered in our model is largely governed by the
desire to present a balanced set of alternative utterances sufficient to
describe the most salient differences in the set of world states. Utterances
are compositionally built from literals, possibly negated. They may be
combined with _and_ to form a conjunction, with _if_ to form a conditional or
with the word _likely_. Table 1 lays out all alternative utterances together
with the rule used to compute the update effects of each its “assertability
condition”, to be introduced next.

utterance type | assertability in state $s$ | example:
---|---|---
utterance $u$ |  assertability $u$ in state $s$
conjunction | $P^{(s)}(\phi,\psi)\geqslant\theta$ | $A\wedge\neg C$ | $P^{(s)}(A=a,C=\neg c)\geqslant\theta$
literal | $P^{(s)}(\phi)\geqslant\theta$ | $A$ | $P^{(s)}(A=a)\geqslant\theta$
conditional | $P^{(s)}(\psi\mid\phi)\geqslant\theta$ | $A\rightarrow\neg C$ | $P^{(s)}(C=\neg c\mid A=a)\geqslant\theta$
likely + literal | $P^{(s)}(\phi)>0.5$ | $likely\;\neg C$ | $P^{(s)}(C=\neg c)>0.5$
Table 1: Types of utterances with corresponding assertability conditions and
an example, ordered from most informative utterance on top to least
informative at the bottom. For conditionals and conjunctions, $\phi\neq\psi$.

#### Assertability conditions.

Given a world state $s\in S$ and an utterance $u\in U$, the definition of the
semantics $[\\![u]\\!]\subseteq S$ serves as the anchoring of pragmatic
reasoning in literal interpretation, defined in Equation [1]. As especially
the semantic meaning of conditionals is highly controversial, we would like to
stay as uncommitted and encompassing as possible. This is possible, to a
certain extent, if we focus not on the nature of the denotation function
$[\\![u]\\!]$ but rather at the functional role it plays in the architecture
of the pragmatic reasoning model. In particular, since the utility function in
Equation [2] and the speaker rule in Equation [3] entail that whenever
$s\not\in[\\![u]\\!]$, the speaker will not choose $u$ when in state $s$, the
main effect of $[\\![u]\\!]$ is to give _assertability conditions_ and as a
side-effect information about how informative each utterance is.666The
informativity of the modeled utterances generally follows the order shown in
Table 1. As we were pointed at by an anonymous reviewer, depending on the
chosen prior over states, it is, however, possible for a conditional (e.g.,
$A\rightarrow C$) to be _literally_ more informative than a literal (e.g.,
_C_) since the assertability of a literal does not per se entail the
assertability of a conditional, e.g. $P^{(s)}(c)\geqslant\theta\not\Rightarrow
P^{(s)}(c\mid a)\geqslant\theta$. This is an interesting observation, in
particular in the context of conditionals whose consequent is independent of
the antecedent as for instance in concessive conditionals (e.g., “ _Even if …_
”) which is certainly worth looking at in future work, but beyond the scope of
what we can cover in this paper. Consequently, our model lays out a general
method of computing update effects of conditionals at the level of a literal
interpreter with the ulterior goal of defining reasonable speaker behavior,
while avoiding as much as possible concrete commitment to a specific semantic
interpretation.

We treat utterances of literals like “$A$” as conveying that the target state
$s$ makes the probability that $A$ is true (i.e., $A=a$) high enough for
conversational purposes; the corresponding probability, e.g., $P^{(s)}(A=a)$
for literal “A”, must exceed a certain threshold $\theta$ for the respective
utterance to be assertable.777We write $P^{(s)}$ to refer to probabilities
_within_ states to distinguish them from probabilities _across_ states;
$P^{(s)}(X)$ denotes the probability assigned to any event $X$ that may be
inferred from state $s$, e.g., $P^{(s)}(A=a)$ is the probability of $A$ to be
true in $s$. In a model of objective chance, as we assume on an objective
interpretation of the probability distributions building up the set of world
states, determinate truth corresponds to probability 1. That is, a speaker
will treat $A$ as true just in case $A$ is determinately true in $s$ when
$P^{(s)}(A=a)=1$. The assumption that speakers sometimes assert things that
are not certain, but very likely true, yet justifies a threshold below 1 as
assertability condition. We also make use of this assumption on a subjective
interpretation of world states; a factual sentence $A$ is thus assertable as
long as the speaker’s subjective belief in $A$ is sufficiently large
($P^{(s)}(A=a)\geqslant\theta$, respectively $P^{(s)}(A=\neg
a)\geqslant\theta$ when $u=\neg A$).

Similarly, “ _likely A_ ” directly conveys that the subjective probability of
$A$ in $s$ is greater than $0.5$ as we assume that “ _likely A_ ” is
assertable in $s$ iff $P^{(s)}(A=a)>0.5$. This aspect of our account is
reminiscent of expressivist accounts of probability language (yalcin12a;
moss15; swanson15). On the objective interpretation, this means that _likely_
expresses high objective chance, consistent with the empirical findings of
fox15 and lassiter17. Note, however, that these authors show that _likely_ can
also express subjective uncertainty.

In parallel fashion, we render the assertability conditions of an indicative
conditional $A\rightarrow C$ as $P^{(s)}(C=c\mid A=a)\geqslant\theta$,
corresponding either to a high objective chance of $C$ to be true when $A$ is
true or to a strong belief in this conditional probability (on part of the
speaker), assuming an objective and subjective interpretation respectively.
Since we do not want or need to commit to a specific semantic theory of
conditionals here, we settle for motivating this condition as a plausible
minimal bound on assertability that should be acceptable from a wide range of
theoretical perspectives. For theories that are able to support the equation
$P(A\rightarrow C)=P(C\mid A)$ while avoiding triviality results (e.g.,
lewis76; hajek89), the derivation is strictly parallel to the factual case
above. This includes non-propositional theories (edgington95; bennett03),
trivalent theories (definetti36; milne97; lassiter19) and various others
(vanfraassen76; stalnakerjeffrey94; kaufmann04; khoo16b, among others). Our
assertability condition is also particularly natural for theories that render
the truth-condition of $A\rightarrow C$ as $P(C\mid A)=1$ (e.g., moss15).

The status of our assertability condition for conditionals is somewhat murkier
from the perspective of other prominent theories such as stalnaker68 and
kratzer91, as well as strict conditional theories. Because of the complexity
of the way that they assign truth-values to epistemically possible worlds
where the antecedent $A$ is false, these theories can make $P(A\rightarrow C)$
high even while $P(C\mid A)$ is low. As a result, our assertability condition
is stronger than these accounts would predict. However, we note that there is
by now an enormous body of empirical evidence supporting the equation between
the probability of a conditional and the corresponding conditional probability
(hadj; evansover04; Douven2010, among many others). This evidence
problematizes a key prediction made by the latter group of theories: that a
conditional can be judged highly probable simply because of the likely
falsehood of its antecedent. Instead, situations where the antecedent is false
are generally judged irrelevant to the probability of a conditional, in a
probabilistic analogue of the classic paradoxes of the material conditional
(edgington95). We do not doubt that the theories under consideration have
theoretical resources available that may allow them to avoid this problem—for
example, by using pragmatic reasoning to explain why false-antecedent cases
are not considered relevant in assertion (lewis76; griceIC). But doing so
would be tantamount to adopting our assertability condition or something quite
close to it. Therefore, we believe that our results should be relevant to
theorists with a wide variety of semantic commitments, including those for
whom probabilistic reasoning has not previously played a major theoretical
role.

###  2.3 Toy example

Let us consider a toy example to illustrate model predictions, and to motivate
further generalizations to be introduced hereafter. Concretely, we will
consider a case with just three world states, all equally likely and just
constructed for the sake of illustration. The topic of conversation in this
example is whether Alex and Chris are likely to go to a party. The states we
look at differ in the probability they assign to all four logical
possibilities of Alex and/or Chris going to the party.

  1. ($s_{1}$)

Alex and Chris both really like to go out. Both are seen at most parties, but
whether either comes is unrelated to whether the other comes.

  2. ($s_{2}$)

Alex and Chris go slightly more often than not, but usually not without each
other (e.g., they are a couple, best friends, etc.).

  3. ($s_{3}$)

Alex and Chris each go out more often than not, but have no connection with
each other.

These three scenarios are translated into the probability distributions
$P(A,C)$ shown at the top of Table 2 where $A$ ($C$) denotes whether Alex
(Chris) comes to the party.

With this context in mind, imagine the following utterance Bigi directs at
Wobo:

  1. (3)

Bigi: _Chris will come to the party_. [“ _C_.”]

Given that Bigi only says what she believes to be true and $s_{1},s_{2}$ and
$s_{3}$ describe all possible scenarios, Wobo should infer that Bigi describes
the situation modeled in $s_{1}$ since this utterance (_C_) is neither
assertable in $s_{2}$ nor $s_{3}$.

Now, imagine Bigi to utter the following conditional instead:

  1. (4)

Bigi: If Alex comes to the party, Chris comes too. [“If $A$, $C$.”]

Table 2: Model predictions for the scenario from Section 2.3 where $\alpha=1,\theta=0.9$. (a) States $s_{1},s_{2},s_{3}$. (b) Assertability of utterances given states with respectively most informative utterances in bold. (c) Literal interpretation. (d) Speaker production likelihoods. (e) Pragmatic interpretation. |  | $s_{1}$ | $c$ | $\neg c$ |  | $s_{2}$ | $c$ | $\neg c$ |  | $s_{3}$ | $c$ | $\neg c$
---|---|---|---|---|---|---|---|---|---|---|---|---
(a) | $P(A,C)$ | $a$ | 0.81 | 0.09 |  | $a$ | 0.6 | 0.05 |  | $a$ | 0.36 | 0.24
$\neg a$ | 0.09 | 0.01 |  | $\neg a$ | 0.05 | 0.3 |  | $\neg a$ | 0.24 | 0.16
(b) | $u=\text{likely}\;C$ | 1 |  | 1 |  | $\bm{1}$
$u=A\rightarrow C$ | 1 |  | $\bm{1}$ |  | 0
$u=C$ | $\bm{1}$ |  | 0 |  | 0
$u=A\wedge C$ | 0 |  | 0 |  | 0
(c) | $P_{\text{lit}}(s_{i}\mid u=likely\;C)$ | $\nicefrac{{1}}{{3}}$ |  | $\nicefrac{{1}}{{3}}$ |  | $\nicefrac{{1}}{{3}}$
$P_{\text{lit}}(s_{i}\mid u=A\rightarrow C)$ | $\nicefrac{{1}}{{2}}$ |  | $\nicefrac{{1}}{{2}}$ |  | 0
$P_{\text{lit}}(s_{i}\mid u=C)$ | 1 |  | 0 |  | 0
$\sum_{u\prime}P_{\text{lit}}(s_{i}\mid u\prime)$ | $\nicefrac{{11}}{{6}}$ |  | $\nicefrac{{5}}{{6}}$ |  | $\nicefrac{{1}}{{3}}$
(d) | $P_{\text{S}}(u=likely\;C\mid s_{i})$ | $\nicefrac{{\frac{1}{3}}}{{\frac{11}{6}}}=\nicefrac{{2}}{{11}}$ |  | $\nicefrac{{\frac{1}{3}}}{{\frac{5}{6}}}=\nicefrac{{2}}{{5}}$ |  | $\nicefrac{{\frac{1}{3}}}{{\frac{1}{3}}}=1$
$P_{\text{S}}(u=A\rightarrow C\mid s_{i})$ | $\nicefrac{{\frac{1}{2}}}{{\frac{11}{6}}}=\nicefrac{{3}}{{11}}$ |  | $\nicefrac{{\frac{1}{2}}}{{\frac{5}{6}}}=\nicefrac{{3}}{{5}}$ |  | 0
$P_{\text{S}}(u=C\mid s_{i})$ | $\nicefrac{{1}}{{\frac{11}{6}}}=\nicefrac{{6}}{{11}}$ |  | 0 |  | 0
(e) | $P_{\text{PL}}(s_{i}\mid u=likely\;C)$ | $\nicefrac{{\frac{2}{11}}}{{(\frac{2}{11}+\frac{2}{5}+1)}}=\nicefrac{{10}}{{87}}$ |  | $\nicefrac{{\frac{2}{5}}}{{(\frac{2}{11}+\frac{2}{5}+1)}}=\nicefrac{{22}}{{87}}$ |  | $\nicefrac{{1}}{{(\frac{2}{11}+\frac{2}{5}+1)}}=\nicefrac{{55}}{{87}}$
$P_{\text{PL}}(s_{i}\mid u=A\rightarrow C)$ | $\nicefrac{{\frac{3}{11}}}{{(\frac{3}{11}+\frac{3}{5})}}=\nicefrac{{5}}{{16}}$ |  | $\nicefrac{{\frac{3}{5}}}{{(\frac{3}{11}+\frac{3}{5})}}=\nicefrac{{11}}{{16}}$ |  | 0
$P_{\text{PL}}(s_{i}\mid u=C)$ | $\nicefrac{{\frac{6}{11}}}{{\frac{6}{11}}}=1$ |  | 0 |  | 0

As noted in Table 2, $A\rightarrow C$ is assertable in states $s_{1}$ and
$s_{2}$. Therefore, under a literal interpretation of Bigi’s utterance,
$s_{1}$ and $s_{2}$ are considered equally likely ($P_{\text{lit}}(s_{1}\mid
u=A\rightarrow C)=P_{\text{lit}}(s_{2}\mid u=A\rightarrow C)=0.5$). However,
under a pragmatic interpretation of $A\rightarrow C$, $s_{2}$ is judged as
more likely than $s_{1}$ ($\nicefrac{{11}}{{16}}$ vs. $\nicefrac{{5}}{{16}}$).
This is because Bigi could have chosen a more informative utterance to
communicate $s_{1}$.

###  2.4 Inferring latent causal relations

The example in Table 2 shows how the model introduced so far yields the
inference that $A\rightarrow C$ is most likely associated with $s_{2}$, by
rather straight-forward Gricean reasoning. It also demonstrates how, on top of
inferring a state $s$, Wobo might draw inferences about the likely causal
relation between $A$ and $C$ which may have led to Bigi’s beliefs as captured
in $s_{1}$, $s_{2}$ or $s_{3}$. By motivation of the example, the coming of
Alex and Chris was assumed to be _independent_ in states $s_{1}$ and $s_{3}$,
while the very reason for writing a table like in $s_{2}$ was because we
assumed that there was a stochastic relationship between Alex’s and Chris’
coming to the party. Suppose that there are two equally likely relations
$r\in\\{\text{independent},\text{dependent}\\}$, meaning that $A$ and $C$ are
either independent or dependent. Intuitively, $P(S=s_{2}\mid
r=\text{independent})$ is much smaller than $P(S=s_{1}\mid
r=\text{independent})$ or $P(S=s_{3}\mid r=\text{independent})$, and also
$P(S=s_{2}\mid r=\text{dependent})$ is much higher than $P(S=s_{1}\mid
r=\text{dependent})$ or $P(S=s_{3}\mid r=\text{dependent})$. In this way, by
Bayesian inference, we can obtain an indirect inference of a likely
causal/stochastic relation between $A$ and $C$ just from probabilistic
pragmatic reasoning _and_ natural assumptions about the differential
likelihood between different causal/stochastic relations $r$ and states $s$.
Note the importance of the pragmatic reasoning for drawing an inference about
the likely relation in this example: under a literal interpretation of the
conditional $A\rightarrow C$, Wobo would not show a preference between
$s_{1}$, where $A$ and $C$ are likely dependent, and $s_{2}$, where they are
likely independent.

$R$$S$$\mathcal{U}$ Figure 1: Through world knowledge, beliefs about the
causal relation $r\in R$ inform probabilities $s\in S$ which are, in turn,
considered as known by the speaker. Probabilities in $s$ then directly
influence the speaker’s utterance choice $u\in\mathcal{U}$.

We can think of this model as a sequence of inferences: beliefs about $r$
stochastically inform $s$ via world knowledge or intuitions about
dependence/independence and $s$, in turn, stochastically informs the speaker’s
utterance choice $u$ due to pragmatic constraints on what counts as a good
utterance. Schematically: $P(r)\Rightarrow P(s\mid r)\Rightarrow P(u\mid s)$
which is depicted in Figure 1. We can then derive $P(r\mid u)$ via Bayes’
rule. In the following we specify and motivate a concrete prior structure, in
particular for the $P(s\mid r)$ part, so as to be able to derive general
predictions from this model for what we may consider a _default context_ ,
where no specific world knowledge is assumed to be available regarding
antecedent and consequent.

###  2.5 Prior over world states in default context

causal relation ($R$) |  instance causal relation ($r$) | interpretation
---|---|---
$A\rightsquigarrow C$ |  $A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$ | Truth of $A$ increases probability for truth of $C$
$A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$ | Falsity of $A$ increases probability for truth of $C$
$C\rightsquigarrow A$ |  $C\overset{\scriptscriptstyle{++}}{\rightsquigarrow}A$ | Truth of $C$ increases probability for truth of $A$
$C\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}A$ | Falsity of $C$ increases probability for truth of $A$
Table 3: Notation for dependent causal relations (types and instances). The
instance of the causal relation provides information about the associated
joint probability tables, spelled out in column ‘interpretation’.

A state $s$ where variables $A$ and $C$ are assumed to be independent
($r=A\rotatebox[origin={c}]{90.0}{$\models$}C$) is represented by
probabilistically independent distributions where
$P^{(s)}(A,C)=P^{(s)}(A)\cdot P^{(s)}(C)$. Since in the default context, we do
not make any specific assumptions, $A$ and $C$ are both assigned a uniform
prior probability over the interval $[0,1]$:

[5] $P^{(s)}(A=a),\;P^{(s)}(C=c)\sim\text{Uniform}(0,1)$

Together with the assumption of independence, this yields the following
probability distribution over partitions of possible worlds, representing a
single state $s$ when $r=A\rotatebox[origin={c}]{90.0}{$\models$}C$:

$\displaystyle P(w_{AC})=P^{(s)}(A=a)\cdot P^{(s)}(C=c),$ $\displaystyle
P(w_{A})=P^{(s)}(A=a)-P(w_{AC})$ $\displaystyle
P(w_{C})=P^{(s)}(C=c)-P(w_{AC}),$ $\displaystyle
P(w_{\emptyset})=1-(P(w_{AC})+P(w_{A})+P(w_{C}))$

To derive the probability distributions over partitions of possible worlds
that represent states where variables $A$ and $C$ are assumed to be dependent,
we distinguish between two possible _types_ of causal relation: either $A$ has
causal power to provoke $C$ ($R=A\rightsquigarrow C$) or vice versa
($R=C\rightsquigarrow A$).888 The choice to model only a single relevant cause
is corroborated by many empirical studies that have shown people’s tendency to
neglect alternative causes (e.g., see Fernbach2010; Fernbach2011;
Fernbach2013; Hagmayer; Krynski2007); also, in a theoretical model from
Icard2015, the loss of information resulting from neglecting alternative
causes was on average too small as to justify their consideration.  While the
causal relation, $R$, merely provides information about the causal direction,
the concrete _instances_ of a causal relation, denoted as $r$, are
distinguished based on how exactly they affect the conditional probabilities,
as shown in Table 3. $R=A\rightsquigarrow C$, for instance, tells us that the
outcome of variable $A$ has direct influence on the outcome of variable $C$,
and the instance of the causal relation, $r$, further tells us _how_ the
outcome of $A$ influences the outcome of $C$. For example, we write
$r=$$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$ to pick out the
class of probability distributions where the truth of $A$ increases the
probability that $C$ is true, whereas
$r=$$A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$ picks out
probability distributions where the _falsity_ of $A$ increases the probability
that $C$ is true. Putting this differently, when
$r=$$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$, $w_{AC}$ will be
considerably more likely than $w_{A}$ ($P(w_{AC}\gg w_{A}$), and when
$r=$$A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$, $P(w_{C})\gg
P(w_{\emptyset})$.

$C$$A$$B$$\tau$$\beta$

$\displaystyle P(C=c\mid
A=a)=1-(1-\tau)\cdot(1-\beta)=\tau+\beta-\tau\cdot\beta$ $\displaystyle
P(C=c\mid A=\neg a)=\beta$

Figure 2: Graphical representation of a leaky noisy-or model (left) with a
single explicitly modeled cause $A$ and the corresponding conditional
probabilities of $C$ when $A$ is true or false (right). Variable $B$
summarizes all potential other causes of $C$; $\tau$ and $\beta$ denote the
causal power of $A$, respectively $B$, to induce the truth of $C$.

Formally, the probability distributions of the dependent states can be
described as leaky-noisy or model with binary variables allowing for positive
as well as negative causes. Noisy-or models (pearl88) describe the
relationship between an effect variable and its cause variables where each
cause is capable of producing the effect independently of all other causes;
corresponding to a logical OR-function where the effect is only true if at
least one of its causes is true. Leaky noisy-or models
(diezParameterAdjustmentBayes1993a) comprise “background noise” represented by
an additional cause variable that is always present and summarizes all
potential causes of the effect that are not explicitly modeled. Figure 2 shows
a graphical representation of the leaky noisy-or model underlying the
dependent states in our model with the corresponding conditional probabilities
of the effect $C$ to be true when $A$ is true or false respectively, assuming
non-deterministic relations where variables $A$ and $B$ have causal power
$\tau$, respectively $\beta$, to provoke $C$. Since we want to stick to the
simplest set of induced probability distributions, but still need to cover all
possible stochastic relations between the two explicitly modeled variables $A$
and $C$ to have a balanced set of states that preserves the informativity of
utterances, we use a generalization of the classical noisy-or model which
further allows negative causes: not only the truth, but also the falsity, of a
cause can have an influence on the truth of the effect (e.g., see
hyttinenNoisyORModelsLatent2011).999If we only used positive causes, states
where $P(w_{AC})$ and $P(w_{\emptyset})$ tend to be low would be very rare and
thus, conditionals like $A\rightarrow\neg C$ would only be assertable in very
few states which would, in turn, render this utterance very informative. As a
result, this particular conditional would become more informative than for
instance literals, which does not seem reasonable. Using states with positive
and negative causes results in a balanced set of states such that any
conjunction is more informative than any literal and any literal is, in turn,
more informative than any conditional.  Therefore, $C$ is likely true when $A$
is false and a negative cause
($r=$$A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$) or when $A$ is
true and a positive cause
($r=$$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$); in both cases
the truth of $C$ may independently be due to background noise $B$.

$R:\;A\rotatebox[origin={c}]{90.0}{$\models$}C$$\begin{aligned}
&P(A=a)\sim\text{Uniform}(0,1)\\\ &P(C=c)\sim\text{Uniform}(0,1)\end{aligned}$
$\nicefrac{{1}}{{2}}$$A,C$ dependent$R:\;A\rightarrow C$$r:\;$
$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$$\begin{aligned}
&\tau\sim\text{Beta}(10,1),\\\ &\beta\sim\text{Beta}(1,10),\\\
&\upsilon_{p}\sim\text{Uniform}(0,1)\end{aligned}$
$\nicefrac{{1}}{{2}}$$A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$…$\nicefrac{{1}}{{2}}$$\nicefrac{{1}}{{2}}$$C\rightarrow
A$……$\nicefrac{{1}}{{2}}$$\nicefrac{{1}}{{2}}$ Figure 3: Graphical
representation of the procedure for sampling a state $s$ from the prior in the
default context.

To instantiate the set of probability distributions for the respective
dependent causal relations, we specify the prior distributions over the
respective causal power ($\tau$), noise ($\beta$) and prior probability of the
parent variable ($\upsilon_{p}$) as shown in Figure 3.101010 Note that even
though the notion of _causal_ relations is eventually not important as the
modeling hinges on certain probabilistic dependencies between events
(depending on the relation), it is important for the choice of our prior
distributions which is motivated by relations that are causal by nature.  The
values of the hyperparameters for the beta distributions of the causal power
and noise are chosen such that the mean of the distribution of the causal
power $\tau$ exceeds the assertability threshold, set to 0.9 in all
simulations below; for the causal power of the background noise, the
parameters are simply reversed such that the prior distribution $P(\beta)$ is
skewed towards 0. As we consider a default context here, where no further
information is available, the prior probability of the parent variable
($\upsilon_{p}$) is sampled from a uniform distribution over the interval
$[0,1]$. The joint probability distributions over $A,C$ are then derived based
on $\tau,\beta$ and $\upsilon_{p}$. Depending on the relation $R$ and the
cause variable being a positive or a negative cause, $\beta$, $\upsilon_{p}$
and $\upsilon_{c}$ correspond to different (conditional) probabilities, as
listed in Table 4 where $\upsilon_{c}$ denotes the conditional probability of
the effect to be true when the cause variable is true:
$\upsilon_{c}=\tau+\beta-\tau\cdot\beta$ (see Figure 2).

The joint probability distributions over partitions of possible worlds are
then computed as follows when $R=A\rightsquigarrow C$ (similarly for
$R=C\rightsquigarrow A$):

$\displaystyle P(w_{AC})=P(c\mid a)\cdot P(a),$ $\displaystyle P(w_{A})=P(\neg
c\mid a)\cdot(P(a)$ $\displaystyle P(w_{C})=P(c\mid\neg a)\cdot P(\neg a),$
$\displaystyle P(w_{\emptyset})=P(\neg c\mid\neg a)\cdot P(\neg a)$

instance causal relation ($r$) | $\upsilon_{p}$ | $\upsilon_{c}$ | $\beta$
---|---|---|---
$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$ | $P^{(s)}(A=a)$ | $P^{(s)}(C=c\mid A=a)$ | $P^{(s)}(C=c\mid A=\neg a)$
$A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$ | $P^{(s)}(A=\neg a)$ | $P^{(s)}(C=c\mid A=\neg a)$ | $P^{(s)}(C=c\mid A=a)$
$C\overset{\scriptscriptstyle{++}}{\rightsquigarrow}A$ | $P^{(s)}(C=c)$ | $P^{(s)}(A=a\mid C=c)$ | $P^{(s)}(A=a\mid C=\neg c)$
$C\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}A$ | $P^{(s)}(C=\neg c)$ | $P^{(s)}(A=a\mid C=\neg c)$ | $P^{(s)}(A=a\mid C=c)$
Table 4: Probabilities ($\upsilon_{p},\upsilon_{c},\beta$) that define the
joint probability distribution of a state $s$, $P^{(s)}(A,C)$, for each
instance of a dependent causal relation. $\upsilon_{p}$ is the prior
probability of the cause, $\upsilon_{c}$ is the conditional probability of the
effect to be true when the cause is true and $\beta$ is the power of the
unmodeled variables to provoke the effect corresponding to the conditional
probability of the effect to be true when the explicitly modeled cause is
false.

Figure 4: Histograms of the probabilities of the four possible worlds,
$w_{AC},w_{A},w_{C}$ and $w_{\emptyset}$ of all sampled probability tables
$P^{(s)}(A,C)$ with $r=$
$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$ (left), $r=$
$C\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}A$ (middle) and
$r=A\rotatebox[origin={c}]{90.0}{$\models$}C$ (right). Numbers in the upper
right corners are the expected values for the respective worlds.

In total, we sample 10,000 probability distributions that comprise the set of
world states used by our model. More concretely, we first sample a causal
relation $r$ from its prior distribution given in Equation [6]. Our choice to
put a prior on the causal relation, which determines the shape of the
associated sampled probability distributions, is primarily based on work from
tenenbaum_theory-based_2003; griffiths_structure_2005; they were the first who
put priors on the causal structure itself to predict human causal judgments
instead of focusing on learning the causal strength of a, possibly non-
existent, causal link (e.g., see Cheng1997).

[6] $P(r)=\begin{cases}\nicefrac{{1}}{{2}}&\text{if
}r=A\rotatebox[origin={c}]{90.0}{$\models$}C\\\ \nicefrac{{1}}{{8}}&\text{if
}r\in\\{A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C,A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C\\}\\\
\nicefrac{{1}}{{8}}&\text{if
}r\in\\{C\overset{\scriptscriptstyle{++}}{\rightsquigarrow}A,C\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}A\\}\end{cases}$

Based on the causal relation $r$, we then sample probability distributions
according to the procedure described above. A visualization of the sampled
states is given in Figure 4 which shows histograms of the probabilities for
each of the four possible worlds across all sampled states for three selected
causal relations.111111The source code and all modeling results are publicly
available: https://osf.io/6bshq/?view_only=1703a3417a1343a5a66b78ac8ce206c2.

###  2.6 Communicating causal information implicitly via conditionals

So far, we have specified how the joint probability distributions are derived
under the assumption of particular causal relations. That is, strictly
speaking, our world states have two components, a probability distribution $s$
and a causal relation $r$. Independently of the causal relation from which $s$
originates, the choice probabilities of a speaker who aims to communicate her
beliefs about the world can then be written as:

[7] $\displaystyle P_{\text{S}}(u\mid r,s)=P_{\text{S}}(u\mid
s)\propto\text{exp}(\alpha\cdot(\text{log P}_{\text{lit}}(s\mid u)))$
$\displaystyle\text{where }P_{\text{lit}}(s\mid
u)=\sum_{r^{\prime}}P_{\text{lit}}(r^{\prime},s\mid u)$

The speaker’s goal of communication, when using conditional sentences, that we
assume in this paper, is first and foremost to convey their beliefs about the
antecedent and the consequent. In other words, we start by exploring a
probabilistic model of communication with conditionals from the most austere
assumption, namely that not only the assertability of a conditional does _not_
hinge on any putative causal relation necessarily, but that also the purpose
of communication itself is _not_ to directly communicate information about the
causal relation.121212 There are circumstances where the goal of the
communication reasonably includes the causal relation which we leave for
future work here.

Before we discuss the results of our simulations, we would like to add a final
note on the interpretation of the world states in our model. Instead of
considering them as simple pairs consisting of a probability distribution and
a causal relation, we can also think of them as _causal Bayesian networks_
(pearl88; Pearl2009; Pearl2014), henceforth abbreviated as ‘Bayes nets’. They
have, similar to the closely related formalism of Structural Causal Models, a
rich tradition of supporting semantic theories of conditionals already, (e.g.
hiddleston05; Pearl2009; Pearl2013; Rips2010; briggs12; kaufmann13; Lucas2015;
santorio16; Lassiter2017). Bayes nets represent sets of variables (here binary
variables) and the dependencies among them; they consist of a directed,
acyclic graph that defines the relations among variables, and a set of
conditional probabilities for each variable given all possible instantiations
of all its _direct_ parent nodes, which simplify to unconditional
probabilities for variables without parent nodes. This set of (conditional)
probabilities is sufficient to define the joint probability distribution over
all variables represented in the graph; see Figure 5 for a Bayes net that
represents the joint probability distribution $s_{2}$ from Table 2.

In sum, the model we explore here can be seen as capturing the implicit
communication of causal information by (i) treating world states as causal
Bayes nets, (ii) identifying the purpose of utterance (the _question under
discussion_ (Roberts2012:Information-Str) or the relevance projection
(Kao2014a)) to be the precise communication of the probability table $s$ (the
speaker’s beliefs about joint truth of $A$ and $C$) and (iii) a pragmatic
process of utterance generation favoring true and informative utterances.

$A$$C$

$\displaystyle P(A=a)=0.65$ $\displaystyle P(C=c\mid
A=a)=\nicefrac{{12}}{{13}}$ $\displaystyle P(C=c\mid A=\neg
a)=\nicefrac{{1}}{{7}}$

Figure 5: Bayes net representing $s_{2}$ from Table 2 consisting of a
graphical representation (left) and a set of associated (conditional)
probabilities (right) that define $P^{(s_{2})}(A,C)$.

##  3 Pragmatic inferences from conditionals in default contexts

In this section, we explore model predictions in default contexts. We can
think of these as the model’s predictions generalized over a wide range of
more specific contexts, or, relatedly, as the model’s predictions for the
interpretation of utterances of conditionals in unbiased, out-of-the-blue
contexts. We will particularly look at the listener’s inferences from an
utterance of a conditional about the speaker’s uncertainty about $A$ and $C$,
the speaker’s beliefs about any systematic relation between $A$ and $C$, and
the strength of a conditional perfection reading. Doing so, we show how
informativity-driven pragmatic choice of utterances leads to _de facto_
assertability conditions as postulated by inferentialist accounts without
having to stipulate these directly.131313All results reported below were
obtained with the literal meaning threshold $\theta=0.9$, and the rationality
parameter $\alpha=3$. Qualitatively identical results were obtained for
parameters from a grid with $\theta\in[0.9,0.95,0.975],\alpha\in[1,3,5,10]$.

#### Setting the scene.

Let us reconsider example (4), repeated here from above, now uttered in an
out-of-the-blue context.

  1. (4)

Bigi: If Alex comes to the party, Chris comes too. [“If $A$, $C$.”]
Wobo: Who are these guys? What party are you talking about?

Even without any strong prior convictions, Wobo is likely to draw pragmatic
inferences from Bigi’s utterance. Intuitively, Wobo would infer that Bigi is
uncertain about whether Alex comes to the party, similarly for Chris, and that
Chris’ coming to the party is not entirely unrelated to Alex’s coming.
Moreover, the implicit dependency between antecedent and consequent may be
interpreted to be so strong that the conditional is understood as
biconditional; it does not seem unnatural for Wobo to infer that, according to
Bigi, Chris comes to the party _only if_ Alex comes.

To analyse the listener’s _a posteriori_ beliefs and the speaker’s utterance
choices we make use of the following definitions concerning the modeled
states. The set of states in which the speaker is _uncertain about event $X$_
contains all and only states $s$ such that the probability of $X$ in $s$,
$P^{(s)}(X)$, is neither too low nor too high:

[8] $\text{Uncertain}(X)=\\{s\mid 1-\theta\leqslant
P^{(s)}(X)\leqslant\theta\\}$

A similar construction captures the speaker’s _certainty about whether $X$ is
true_:

[9] $\text{Certain}(X)=\\{s\mid P^{(s)}(X)>\theta\\}\cup\\{s\mid
P^{(s)}(X)<1-\theta\\}$

The uncertainty / certainty about $X$ given $Y$ is captured analogously.

#### Hyperrational utterance choices in default contexts.

According to the model presented here, the listener’s inferences about the
speaker’s epistemic state can be put down, at least in part, to a
Q-implicature (Atlas1981a; Horn1984): from the fact that the speaker decided
_not_ to utter a more specific and thus more informative utterance than
$A\rightarrow C$, the pragmatic listener should deem it unlikely that the
speaker refers to a state $s$ in which a more informative utterance would also
apply. To see whether such an alternatives-based explanation is endorsed by
the present modeling setup, we first look at a speaker who always chooses the
utterance with the highest utility (hyperrational speaker where rationality
parameter $\alpha\rightarrow\infty$). Figure 6 shows model predictions from
the speaker’s point of view for different probabilistic beliefs with respect
to propositions $A$ and $C$ given the causal relation between antecedent and
consequent. It clearly shows that a (hyperrational) speaker chooses utterances
in dependence of her belief state: when the speaker is uncertain about both
propositions, the best utterance is either a conditional or “ _likely_ $\Phi$”
which are the two least informative utterance types.141414For simplicity, we
discuss all results assuming an epistemic interpretation of the probabilities
in our world states, but an interpretation based on objective chance is
equally applicable. On the other hand, the speaker’s best utterance is either
a conjunction or a literal when she is certain about both propositions, that
is, when she is certain that $A$ is true (or false) and certain that $C$ is
true (or false), but she is not necessarily certain that a conjunctive event
occurs (e.g. $A=a\wedge C=\neg c$). This is also the reason, why conjunctions
do not seem to be preferred over literals when $A$ and $C$ are independent and
the speaker is certain about $A$ and certain about $C$ (Figure 6 (i)). In this
case, when the speaker’s best utterance is a literal, there simply is no
conjunction that truthfully describes the given state (e.g.,
$P^{(s)}(a,c)=0.82,P^{(s)}(a,\neg c)=0.08,P^{(s)}(\neg a,c)=0.08,P^{(s)}(\neg
a,\neg c)=0.02$). When the speaker is only certain about one proposition, her
best utterance is a literal (Figure 6 (iii)).

(i) A certain, C certain(ii) A uncertain, C uncertain(iii) A xor C
certain0.000.250.500.751.000.000.250.500.751.000.000.250.500.751.00conditionalconjunctionlikely
+ literalliteralproportioncausal relation $R$$C\rightsquigarrow
A$$A\rightsquigarrow C$$A\rotatebox[origin={c}]{90.0}{$\models$}C$

Figure 6: Relative frequency of how often each utterance type is the speaker’s
_best_ choice for a set $S$ of 10,000 states sampled from the prior (default
context), given that the speaker is (i) certain or (ii) uncertain about $A$
and about $C$, i.e., $\forall s\in S:s\in\text{Certain}(A)\wedge
s\in\text{Certain}(C)$, respectively $\forall s\in
S:s\in\text{Uncertain}(A)\wedge s\in\text{Uncertain}(C)$, or (iii) the speaker
is uncertain about the truth of one proposition but certain about the truth of
the other, e.g., $\forall s\in S:s\in\text{Uncertain}(A)\wedge
s\in\text{Certain}(C)$.

Figure 6 (ii) further reveals that, when the speaker is uncertain about both
$A$ and $C$, her utterance choice is strongly influenced by their causal
relation: for these states the speaker’s best utterance will always be “
_likely_ $\Phi$” when $A$ and $C$ are independent, whereas when $A$ and $C$
are dependent, it will almost certainly be a conditional.

#### Inferences about causal dependency.

Since pragmatic interpretation is here modeled as backwards-inference based on
the speaker’s utterance choice protocol, we can already anticipate from the
above results of hyperrational speakers that pragmatic interpreters may draw
rather specific inferences about the causal relation between $A$ and $C$ from
an utterance of $A\rightarrow C$. In what follows, we look at inferences about
the causal relationship in more detail, assuming “normal speakers”
($\alpha=3$).

Figure 7 shows the causal inferences that listeners draw about $A$ and $C$
when the speaker utters the conditional $A\rightarrow C$. We observe that _a
posteriori_ the listener (literal and pragmatic) assigns very low probability
to states where $r=$ $A\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}C$ or
$r=$ $C\overset{\scriptscriptstyle{-+}}{\rightsquigarrow}A$, as these are very
unlikely to give rise to a probability table in which the conditional
$A\rightarrow C$ is assertable.

Pragmatic listenerLiteral listenerPrior0.000.250.500.751.00Degree of
beliefrelation $R$$A\rightsquigarrow C$$C\rightsquigarrow
A$$A\rotatebox[origin={c}]{90.0}{$\models$}C$instance $r$A+C+A-C+C-A+C+A+A,C
independent

Figure 7: Degree of belief in each causal relation (A+C+ is shorthand for
$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$, analogous for other
relations) at all three levels of interpretation; prior to uptake of the
conditional $A\rightarrow C$ and _a posteriori_ given a literal / pragmatic
listener.

Interestingly, the listener is not committed to a single underlying causal
relation, but instead merely infers that there is a positive dependency
between antecedent and consequent: $A$ tends to be true (false) when $C$ is
true (false) and vice versa. The pragmatic listener assigns almost the entire
probability mass to the corresponding causal relations
($A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$,
$C\overset{\scriptscriptstyle{++}}{\rightsquigarrow}A$), the literal listener
approximately 75%.

This result suggests that also under a pragmatic interpretation, the listener
needs further knowledge to disambiguate the underlying causal structure since,
in this most general context, it is not possible to infer that the antecedent
is a cause of the consequent or that, as under a diagnostic reading of the
conditional $A\rightarrow C$, the consequent is a cause of the antecedent.

Another interesting result concerns the states where $A$ and $C$ are
independent: even though the literal listener largely diminishes her beliefs
in the independence of $A$ and $C$ as compared to her beliefs prior to the
speaker’s utterance of the conditional $A\rightarrow C$, the pragmatic
listener assigns a still smaller probability to states where antecedent and
consequent are causally independent. This is an important and interesting
result that bears emphasis. Since the pragmatic listener combines a rich
representation of the speaker’s beliefs about truth of propositions and their
causal relation with Gricean pragmatic reasoning, the pragmatic listener
concludes more about the (speaker’s beliefs about the) causal structure of the
world than is entailed by the semantics of a conditional. This additional
causal-pragmatic inference essentially rides piggyback on standard Gricean
Quantity reasoning.

To see this, let us consider the perspective of the hyperrational speaker
again (Figure 6). Those states where $A$ and $C$ are independent are
exclusively states for which an informative Gricean speaker will either prefer
to utter a bare proposition (conjunction or literal) which is more informative
than a conditional or “ _likely_ $\Phi$”. As a result, the pragmatic listener
infers a causal relation from entirely unbiased assertability conditions for
conditionals and standard Gricean Quantity reasoning. Causal inference comes
up as a pragmatic inference without having to stipulate an additional
pragmatic constraint concerning a (causal) relation between propositions, let
alone hard-coding such a requirement in the semantic meaning.

Pragmatic listenerLiteral listenerPrior0.00.20.40.60.8Degree of beliefvalue
$\mathbb{E}[P^{(s)}(a\mid c)]$$\mathbb{E}[P^{(s)}(\neg c\mid\neg a)]$

Figure 8: Degree of belief in the two CP-related conditional probabilities
$P^{(s)}(\neg c\mid\neg a)$ and $P^{(s)}(a\mid c)$; prior to the uptake of the
conditional $A\rightarrow C$ and _a posteriori_ , given a literal or pragmatic
listener.

#### The strength of conditional perfection readings.

In the example given in the beginning of this section, it does not seem
astounding to interpret the “if” in “ _If Alex comes to the party, Chris comes
too_ ” as “ _if and only if_ ”. It might even seem to be a quite acceptable,
if not natural, inference although we did not specify any further context and
although, from a logical point of view, this inference is _not_ valid.

The phenomenon that conditionals are sometimes interpreted as biconditionals,
known as _conditional perfection_ (CP), has caught much attention in the
literature, especially since Geis1971. CP remains a topic of ongoing debate;
no consensus has for instance been found concerning its prevalence or the
circumstances that trigger a CP reading (e.g. see newstead_conditional_1997;
VonFintel2001; oberauer_meanings_2003;
moldovanDenyingAntecedentConditional2013).

Neither exists a standard measure that quantifies the degree to which a
conditional receives a CP reading. Here we will refer to two inferences that
are prominently considered in the literature on conditional reasoning, ‘
_Denying the antecedent_ ’ (DA) and ‘ _Affirming the consequent_ ’ (AC), shown
below. The endorsement of DA or AC inferences suggests that participants
interpret the conditional as biconditional since only then these are logically
valid inferences (e.g., newstead1997conditional;
evansThinkingConditionalsStudy2007).

  * DA:

$A\rightarrow C.\ \ \neg A.\ \ \therefore\ \ \neg C$.

  * AC:

$A\rightarrow C.\ \ C.\ \ \therefore\ \ A$.

To learn how strongly $A\rightarrow C$ is interpreted as biconditional, we
will therefore look at the listener’s expected beliefs (about the speaker’s
beliefs) $P^{(s)}(\neg c\mid\neg a)$ and $P^{(s)}(a\mid c)$. Only when “ _if_
” is interpreted as “ _iff_ ” the two considered quantities should be large.

As can be seen in Figure 8, without any further contextual assumptions our
model predicts a quite strong CP-reading for the interpretation of
conditionals in the default context. The speaker’s utterance of $A\rightarrow
C$ elicits an increase in the listener’s beliefs (about the speaker’s beliefs)
in the conditional probabilities $P^{(s)}(\neg c\mid\neg a)$ and
$P^{(s)}(a\mid c)$ as compared to her prior beliefs. This is true for the
literal and the pragmatic interpretation, yet the pattern is more pronounced
in the latter. This result is to a great extent due to the representation of
dependent world states as noisy-or models (see Section 2.5). In this way, this
interpretation is explained as something akin to an I-implicature (Atlas1981a;
Levinson2000), as suggested by Horn2000; CP-readings are supported in large
part by what is arguably a cognitively economic, perhaps stereotypical
representation format of causal dependency between events. However, we also
see that pragmatic reasoning about alternatives further strengthens the CP-
reading quantitatively, similar to the accounts of VanDerAuwera1997 and
VonFintel2001 which share with our model the feature that CP is a result of a
listener reasoning about the speaker’s production protocol given VonFintel2001
account, including, crucially, the possibility that the speaker could simply
have asserted the consequent.

#### Deriving inferentialist assertability conditions.

So far, we showed that the current modeling setup predicts that speakers will
use conditional sentences predominantly in cases where there is a
(causal/inferential) relationship between antecedent and consequent and that
listeners, therefore, infer such a relationship from an utterance of a
conditional.151515 Our model predicts the dependency relation to be a
defeasible inference, but it is not predicted to arise in any context (e.g.,
see the discussion on missing-link and biscuit conditionals in Section 5).
Therefore, it is also compatible with lassiter_decomposing_nodate account of
when the dependency between antecedent and consequent will (or will not) arise
based on discourse coherence. These predictions particularly challenge the
idea advanced by advocates of inferentialism that a (causal/inferential)
relation between antecedent and consequent is part, in whatever form, of the
core semantics of conditionals. We here argue that an austere assertability
condition for conditionals in combination with rich representations of
contextual (causal) world knowledge and pragmatic reasoning is sufficient to
derive the kind of assertability conditions postulated by inferentialist
accounts. In particular, we will consider the assertability condition
formulated in Equation [10] which was proposed by VanRooij2019 improving on
like-minded work of Douven2008.

[10] $A\rightarrow C\text{ is acceptable/assertable only if
}\Delta^{*}P=\frac{P(c\mid a)-P(c\mid\neg a)}{1-P(c\mid\neg
a)}\geqslant\theta\text{.}$

The general idea behind assertability criteria of this kind is that an
utterance of a conditional $A\rightarrow C$ is acceptable only if $P(c\mid a)$
is high (like we assume as well here) and, in addition, $P(c\mid\neg a)$ is
low, that is, $C$ being true is only likely when $A$ is true.

Figure 9: Distribution of $\Delta^{*}P$ values for 10,000 states randomly
sampled from (i) the prior (bottom), (ii) from the prior given $A\rightarrow
C$ is assertable (middle), and (iii) for states from (ii) where additionally
$A\rightarrow C$ is the pragmatic speaker’s best choice (top). Causal
relations are abbreviated, e.g., A+C+ is shorthand for
$A\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$.

We do not argue that Equation [10] necessarily holds, empirical research is
needed to find out whether the acceptance/assertability of conditionals is
related to such criteria. 161616While some of the empirical studies that have
been conducted to date suggest that the relationship between antecedent and
consequent has an influence on the assertability of conditionals (e.g.,see
douven_indicatives_2012; Skovgaard-Olsen2016), others found no such effect
(e.g., see singmann_new_2014; oberauer_what_2007). Instead, we aim to
investigate how the model that we propose here relates to accounts that
propose this kind of assertability conditions for conditionals, possibly
bringing Inferentialists’ ideas and pragmatic reasoning closer together.

We find that our pragmatic model for the use of conditionals derives the
criterion from Equation [10], in the sense that whenever the model predicts a
(hyperrational) speaker to use a conditional in some state $s$, the value
$\Delta^{*}P$ calculated from the probability table entailed by $s$ is indeed
very high. To see this, Figure 9 shows the distribution of $\Delta^{*}P$
measures for three sets of states: (i) sampled from the prior (default
contexts), (ii) sampled from the prior conditioned on $A\rightarrow C$ being
assertable (literal speaker condition), and (iii) the subset of states sampled
in the literal speaker condition, in which a hyperrational speaker would utter
$A\rightarrow C$ (pragmatic speaker condition). Figure 9 reveals,
reassuringly, that $\Delta^{*}P$ is not always high for any state sampled from
the prior. It also shows that just from our austere assertability condition
alone, namely $P^{(s)}(c\mid a)\geqslant\theta$, the average associated
$\Delta^{*}P$ increases, even if there are still quite a number of cases where
what we may call a “literal speaker” might use $A\rightarrow C$ while the
measure $\Delta^{*}P$ is quite low. This clearly shows that, despite biases
for “simple situations” introduced by _noisy-or_ parameterization of state
priors, the assertability condition that $P^{(s)}(c\mid a)\geqslant\theta$
alone does not guarantee that $\Delta^{*}P$ is high. But for the pragmatic
speaker, we see that the $\Delta^{*}P$ measure is exclusively very high, thus
lending support to the idea that a straightforward model of Gricean pragmatic
use of conditionals explains an otherwise stipulative assertability condition
for conditionals.

0.000.250.500.751.00literalconjunction(other)
conditional$\Delta^{*}P^{(s)}$causal relation
$R$$A\rotatebox[origin={c}]{90.0}{$\models$}C$$A\rightsquigarrow
C$$C\rightsquigarrow A$

Figure 10: $\Delta^{*}P^{(s)}$ values, zoomed into the range where
$0\leqslant\Delta^{*}P^{(s)}\leqslant 1$ for $s$ from a set of 10,000 states
sampled from the prior given $A\rightarrow C$ is assertable (literal speaker
condition in Figure 9) where $A\rightarrow C$ is _not_ the best choice of a
hyperrational speaker ($\alpha=\infty$).

We saw that when $A\rightarrow C$ is asserted by our pragmatic speaker,
$\Delta^{*}P$ is large, but is it also the case that a large value of
$\Delta^{*}P$ is sufficient for a hyperrational pragmatic speaker to utter the
conditional $A\rightarrow C$? If it was a sufficient condition, our pragmatic
speaker should prefer this conditional whenever $\Delta^{*}P$ is large. Figure
10 shows boxplots in the range $0\leqslant\Delta^{*}P\leqslant 1$ of the
$\Delta^{*}P$-values of states where $A\rightarrow C$ is assertable, but _not_
the most likely utterance for a hyperrational speaker.171717For states where
$R=C\rightsquigarrow A$, $\Delta^{*}P$ has a minimum value of -212 when the
hyperrational speaker’s best utterance is a literal, and a minimum value of
-69.3 when it is a conjunction. When
$R=A\rotatebox[origin={c}]{90.0}{$\models$}C$, $\Delta^{*}P^{(s)}$ clusters
closely around 0. The utterance type of the hyperrational speaker’s best
choice for the respective states a literal, a conjunction or a conditional
other than $A\rightarrow C$ is shown on the x-axis; color codes represent the
causal relation of the states. Clearly, there are states where
$\Delta^{*}P^{(s)}$ is large and the speaker chooses a different utterance
than $A\rightarrow C$. Particularly interesting for us are those states where
the speaker does not choose a different conditional but an utterance that is
more informative than a conditional, that is, a conjunction or a literal.
These are situations where the predictions of our model diverge from the
predictions of accounts arguing that a conditional is a assertable/acceptable
when $\Delta^{*}P$ is large: although $A\rightarrow C$ is (literally)
assertable, the speaker may have good reasons to choose a different utterance
(to be maximally informative) and so, the conditional $A\rightarrow C$ might
still be rejected. This explains why a criterion like the one defined in
Equation [10] might be a necessary, but not a sufficient condition for the
assertability of a conditional.

Overall the predictions from our model corroborate the hypothesis that the
dependency relation does not need to be incorporated into the semantics of
conditionals. The model shows that pragmatic reasoning about the speaker’s
conditional utterance $A\rightarrow C$ is a possible way to derive the
dependency relation given an appropriate representation of the modeled
variables, in particular their underlying causal structure.

##  4 Interpretation in concrete contexts: Douven’s puzzle

So far we showed that an RSA model combined with unbiased default priors is
able to explain general pragmatic inferences associated with utterances of
conditionals. In the following we will turn towards predictions for utterances
of conditionals in rather specific and explicitly given contexts of use. In
particular, we investigate how the present setup helps explain a puzzle put
forward by Douven2012b. Douven2012b contrasts three cases of conditionals
uttered in concrete contexts, that we will fully cite below. In these
contexts, learning a conditional $A\rightarrow C$ either leads to an increase
(the Skiing Example given in (5), discussed in Section 4.1) or a decrease (the
Garden Party Example given in (6), discussed in Section 4.2) in the listener’s
degree of belief in the truth of $A$ or the listener’s degree of belief in the
antecedent $A$ does not change at all (the Sundowners Example given in (7),
see Section 4.3).

We argue here that an explicit representation of contextually-grounded world
knowledge that does not only comprise plausible prior beliefs about the
probability of the represented variables, but also caters for their causal
structure, is sufficient to explain how pragmatic listeners adjust their
beliefs about the antecedent in one way or another after receiving information
in form of a conditional. The key to the explanation we propose here is that,
in each case, the listener learns a piece of causal information, that is, the
listener learns that propositions are causally related where this was
previously not expected or deemed rather unlikely.181818A perhaps more
realistic picture would be to model a listener as completely _unaware_ of the
causal relation in question; that agent’s stock of explicitly entertained
alternatives would just not represent that contingency. Since adding agent’s
unawareness to a model of pragmatic reasoning is possible but technically
rather involved (e.g. HeifetzMeier2006:Interactive-Una; FrankeJagerde-
Jager2010:Now-that-you-me; Franke2012:Pragmatic-Reaso), this paper makes the
simplifying assumption that the listener is aware of the possible causal
connection but deems it very unlikely to begin with.

###  4.1 The skiing case

The Skiing Example is a case where, intuitively, the listener’s degree of
belief in the antecedent increases.

#### The Skiing Example (Douven2012b).

Harry sees his friend Sue buying a skiing outfit. This surprises him a bit,
because he did not know of any plans of hers to go on a skiing trip. He knows
that she recently had an exam and thinks it unlikely that she passed. Then he
meets Tom, another friend of Sue’s, who is just on his way to Sue to hear
whether she passed the exam, and who tells him:

  1. (5)

If Sue passed the exam, her father will take her on a skiing vacation.
$\leadsto$ listener belief in antecedent _increases_

In this example, there are three relevant propositions: $E$ (Sue passed the
exam), $S$ (Sue goes skiing) and $C$ (Sue buys skiing clothes). The listener
Harry has observed $C$, so his beliefs in $C$ are high, possibly 1. The
speaker Tom utters the conditional $E\rightarrow S$. We want to explain how
this can lead to an increase in Harry’s probabilistic beliefs about $E$.

Our explanation hinges on assuming that Harry has certain plausible beliefs
about the propositions involved and their causal relationship. It proceeds in
three steps:

  1. 1.

From pragmatic reasoning, the listener infers from the utterance of
$E\rightarrow S$ that the speaker likely believes in a causal relation
$E\overset{\scriptscriptstyle{++}}{\rightsquigarrow}S$: Sue passing the exam
increases her chance to go on a skiing trip.

  2. 2.

The listener takes the speaker to be an authority on the matter and, at least
to a certain extent, also increases degrees of beliefs in the causal relation
$E\overset{\scriptscriptstyle{++}}{\rightsquigarrow}S$.

  3. 3.

Since the listener also has a high degree of belief in $C$, and given that it
is plausible to assume that in general a relation of the kind
$S\overset{\scriptscriptstyle{++}}{\rightsquigarrow}C$ holds, the listener
ends up with a higher degree of belief in $E$ after processing the utterance.

$E$$P(e)=0.2$$S$$\begin{aligned} &P(s\mid e)=1\\\ &P(s\mid\neg
e)=0\end{aligned}$ (a) $s_{dep}$: $E,S$ dependent

$E$$P(e)=0.2$$S$$P(s)=1$ (b) $s_{ind}$: $E,S$ independent

$\mathbb{E}[P^{(s)}(e)]$0.00.10.2Pragmatic interpretationLiteral
interpretationPrior beliefstate$s_{dep}$$s_{ind}$

(c) Listener’s degree of belief in the antecedent

Figure 11: Bayes nets and results for the pragmatic reasoning part in the
Skiing Example with $\mathcal{U}=\\{S,\text{likely }S,E\rightarrow S\\}$,
$\alpha=1$, $E$: pass exam, $S$: go skiing. Both states, $s_{dep}$ and
$s_{ind}$, are assigned equal prior probability.

To illustrate this reasoning schema, we offer one concrete context model for
the Skiing Example in Figure 11.191919A more realistic choice than using a
single independent state with $P^{(s_{ind})}(s)>\theta$, would be to include
several independent states, e.g. with $P^{(s)}(s)\in[0,0.5,1]$ which would,
however, require a larger set of utterances such that for every state there is
at least one utterance assertable. For the sake of a simpler discussion, here
and in the following examples, we give just one set of concrete numbers of the
relevant (conditional) probabilities.  There are two Bayes nets that are in
line with the speaker’s utterance $E\rightarrow S$, one in which passing the
exam stands in a direct causal relation to going on a skiing trip (Figure
11(a)), and one in which it does not (Figure 11(b)).202020Even though it seems
less probable, there is the possibility that the causal relation between $E$
and $S$ is reversed (i.e., $S\rightarrow E$ instead of $E\rightarrow S$); Sue
may for instance study extra hard _because_ her father invites her to go on a
skiing trip. Note that in this case, the listener’s observation of Sue buying
skiing clothes would, however, not increase the listener’s degree of belief in
Sue passing the exam.

Their entailed joint probability distributions are spelled out in Table 5. The
situation model in Figure 11 gives the listener’s (Harry’s) beliefs about what
Tom might believe about the relation between $E$ and $S$. Given the context-
story, there is no indication that Harry believes that Tom believes that the
dependent Bayes net is any more or less likely than the independent one.
Therefore, we assign equal prior probability to both Bayes nets, even though
this is not crucial for the case at hand.

| $s$ | $\neg s$
---|---|---
$e$ | $P(e)\cdot P(s\mid e)=0.2$ | $P(e)\cdot P(\neg s\mid e)=0$
$\neg e$ | $P(\neg e)\cdot P(s\mid\neg e)=0$ | $P(\neg e)\cdot P(\neg s\mid\neg e)=0.8$
(a) $P^{(s_{dep})}(E,S)$

| $s$ | $\neg s$
---|---|---
$e$ | $P(e)\cdot P(s)=0.182$ | $P(e)\cdot P(\neg s)=0.018$
$\neg e$ | $P(\neg e)\cdot P(s)=0.728$ | $P(\neg e)\cdot P(\neg s)=0.072$
(b) $P^{(s_{ind})}(E,S)$

Table 5: Entailed joint probability distributions of variables $E,S$ of the
two Bayes net $s_{ind},s_{dep}$ shown in Figure 11; $E:$‘pass exam’, $S$: ‘go
on skiing trip’.

With the set of alternative utterances $\mathcal{U}=\\{S,\text{likely
}S,E\rightarrow S\\}$, $\alpha=1$ and the probabilities and causal relations
as shown in Figure 11, we get the following probability distribution for the
pragmatic listener when $u=E\rightarrow S$:
$P_{\text{PL}}(s=s_{\text{dep}}\mid u=E\rightarrow S)=\nicefrac{{5}}{{6}}$ and
$P_{\text{PL}}(s=s_{\text{ind}}\mid u=E\rightarrow S)=\nicefrac{{1}}{{6}}$;
see Table 6 for the respective values of other model components. The dependent
Bayes net becomes more likely under a pragmatic interpretation only: by taking
into account the fact that the speaker could have chosen a more informative
utterance (e.g. “Sue goes on a skiing trip” (S)) to refer to the independent
Bayes net (see Table 6), the listener learns that, most likely, the speaker
believes in a connection between Sue going on a skiing trip and her passing
the exam. Contrary to that, under a literal interpretation both Bayes nets
remain equally likely as the chosen utterance, $E\rightarrow S$, is literally
true in both states.

The predicted interpretation of Tom’s utterance, $E\rightarrow S$, with
respect to the probability of the antecedent is shown in Figure 11(c): the
listener’s degree of belief about the speaker’s beliefs in the antecedent
remains at an expected value of 0.2, only the degree of belief related to the
causal relation between $E$ and $S$ is influenced by the speaker’s utterance
of the conditional. Crucially, since we so far only modelled the listener’s
pragmatic inferences about the speaker’s beliefs, the listener’s observation
($C=c$) is not considered yet.

How should a listener change their beliefs in the light of a (pragmatically
derived) belief about the speaker’s beliefs? That depends on the more general
assumptions the listener makes about the speaker: is she trustworthy and well-
informed on the subject matter at hand? Since nothing in the scenario
described in Example (5) gives us reason to expect otherwise, we may follow
the usual assumptions in Gricean belief-based reasoning that the speaker is
cooperative and knowledgeable (e.g. Geurts2010:Quantity-Implic). Even if the
precise effect of integrating beliefs of a cooperative and knowledgeable agent
into one’s own beliefs are elusive, the effect in a scenario like the one at
hand is most likely that the listener increases their own beliefs in the
relation $E\overset{\scriptscriptstyle{++}}{\rightsquigarrow}S$. To keep
matters simple for a fully fleshed out numerical example, we just assume that
the listener adopts exactly the same probabilistic beliefs as the inferred
speaker beliefs.

Table 6: Distributions for the Skiing Example with two Bayes nets $s_{\text{dep}}$, $s_{\text{ind}}$, utterances $\mathcal{U}=\\{E\rightarrow S,S,\text{likely }S\\}$ and $\alpha=1$.  |  Literal meaning $\delta_{s\in\sv{u}}$ |  |  |  Speaker $P_{\text{S}}(u\mid s)$
---|---|---|---|---
$u$ | $E\rightarrow S$ | $S$ | $\text{likely }S$ |  | $u$ | $E\rightarrow S$ | $S$ | $\text{likely }S$
$s_{\text{dep}}$ | 1 | 0 | 0 |  | $s_{\text{dep}}$ | 1 | 0 | 0
$s_{\text{ind}}$ | 1 | 1 | 1 |  | $s_{\text{ind}}$ | $\nicefrac{{1}}{{5}}$ | $\nicefrac{{2}}{{5}}$ | $\nicefrac{{2}}{{5}}$
|  Literal listener $P_{\text{LL}}(s\mid u)$ |  |  |  Pragmatic listener $P_{\text{PL}}(s\mid u)$
$u$ | $E\rightarrow S$ | $S$ | $\text{likely }S$ |  | $u$ | $E\rightarrow S$ | $S$ | $\text{likely }S$
$s_{\text{dep}}$ | $\nicefrac{{1}}{{2}}$ | 0 | 0 |  | $s_{\text{dep}}$ | $\nicefrac{{5}}{{6}}$ | 0 | 0
$s_{\text{ind}}$ | $\nicefrac{{1}}{{2}}$ | 1 | 1 |  | $s_{\text{ind}}$ | $\nicefrac{{1}}{{6}}$ | 1 | 1

This takes us to the last step of the explanation, where we look at the effect
of the listener’s private knowledge that $C$ is the case. Put differently, we
look at the listener’s inference of the speaker’s beliefs about Sue passing
the exam under the assumption that the speaker makes the same observation as
the listener. Similar to Douven2012b, we draw on world knowledge about the
relation between $C$ and $S$, where the former is highly unlikely when one
does not go on a skiing trip.212121We argue that the relation between $C$ and
$S$ is essential for the desired interpretation of the conditional. Imagine
that Harry, the listener, had no idea what skiing is. In this case, both
states should be modeled without the link between $S$ and $C$ making Harry’s
observation irrelevant with respect to his belief about Sue passing the exam
($E$). Therefore, his belief in the antecedent would remain unchanged. See
Gunther2018 who also makes use of causal Bayes nets but argues that the
intuitive interpretation of the Skiing example is independent of whether or
not the relation between variables $S$ and $C$ is modeled. This is reflected
in the model by setting $P(c\mid\neg s)$ to 0 and $P(c\mid s)$ to 0.5 (see
Figure 12(c)) as the context story does not provide any information concerning
Sue’s shopping behavior.

$S$$P(s)$$C$$\begin{aligned} &P(c\mid s)=0.5\\\ &P(c\mid\neg
s)=0\end{aligned}$ (c) $s_{wk}$: $S,C$ dependent

$\mathbb{E}[P^{(s)}(e\mid c)]$0.00.10.20.30.40.50.60.70.80.9Pragmatic
interpretationwith listener’s observationPragmatic interpretationw/o
listener’s observationstate$s_{dep}$$s_{ind}$

(d) Expected value of the listener’s degree of belief in the antecedent

Figure 12: Bayes net for the assumed world knowledge in the Skiing Example
(left) and results for the the listener’s inference about the antecedent
(right); $E$: pass exam, $S$: go skiing, $C$: buy skiing clothes.

Based on this (world) knowledge about $S$ and $C$ (Bayes net $s_{wk}$), we can
compute the listener’s updated belief in the probability that Sue goes on a
skiing trip given the listener’s observation of Sue buying skiing clothes:

[11] $P^{s_{wk}}(s\mid c)=\frac{P^{s_{wk}}(c\mid s)\cdot
P^{s_{wk}}(s)}{P^{s_{wk}}(c)}=\frac{0.5\cdot P^{s_{wk}}(s)}{0.5\cdot
P^{s_{wk}}(s)+0\cdot P^{s_{wk}}(\neg s)}=1$

Assuming the updated probability of Sue going on a skiing trip,
$P^{s_{wk}}(s\mid c)$ given in Equation [11], the updated probability of Sue
passing the exam ($E=e$) for the two modeled states $s_{dep}$ and $s_{ind}$ is
given in Equations [12] and [13]:

[12] $\begin{split}P^{(s_{dep})}(e\mid c)&=P^{(s_{dep})}(s,e\mid
c)+P^{(s_{dep})}(\neg s,e\mid c)\\\ &=P^{(s_{dep})}(e\mid s)\cdot
P^{(s_{wk})}(s\mid c)+P^{(s_{dep})}(e\mid\neg s)\cdot P^{(s_{wk})}(\neg s\mid
c)\\\ &=1\cdot 1+0\cdot 0=1\end{split}$ [13] $\begin{split}P^{(s_{ind})}(e\mid
c)&=P^{(s_{ind})}(s,e\mid c)+P^{(s_{ind})}(\neg s,e\mid c)\\\
&=P^{(s_{ind})}(e)\cdot P^{(s_{wk})}(s\mid c)+P^{(s_{ind})}(e)\cdot
P^{(s_{wk})}(\neg s\mid c)\\\ &=0.2\cdot 1+0.2\cdot 0=0.2\end{split}$

These values emphasize the importance of taking into account the causal
relations among variables: only for $s_{dep}$ the degree of belief in $E$ is
influenced by the listener’s observation of $c$. Given $s_{ind}$, the
probability of Sue going on a skiing trip remains the same as without the
listener’s observation. The listener’s updated belief in the antecedent, given
the listener’s independent observation of Sue buying skiing clothes, is then
equal to the expected value of $P^{(s)}(e\mid c)$ ($s\sim P_{\text{PL}}(s\mid
u=E\rightarrow S),s\in[s_{dep},s_{ind}]$) which is approximately 0.87, as
spelled out in Equation [14]. Remember that $P_{\text{PL}(s\mid u=E\rightarrow
S)}$ describes the listener’s beliefs after the uptake of the conditional,
that is, the listener is approximately $83\%$ confident that the speaker’s
beliefs correspond to $s_{dep}$ and approximately $17\%$ confident that they
correspond to $s_{ind}$. Figure 12(d) shows the predictions for the listener’s
posterior beliefs in the antecedent, with and without consideration of the
listener’s observation.

[14] $\begin{split}E[P^{(s)}(e\mid
c)]&=\sum_{s_{i}\in\\{s_{\text{dep}},s_{sind}\\}}P^{(s_{i})}(e|c)\cdot
P_{\text{PL}}(s_{i}\mid u=E\rightarrow S)=\\\ &P^{(s_{\text{dep}})}(e\mid
c)\cdot P_{\text{PL}}(s_{\text{dep}}\mid u=E\rightarrow S)\;+\\\
&P^{(s_{\text{ind}})}(e\mid c)\cdot P_{\text{PL}}(s_{\text{ind}}\mid
u=E\rightarrow S)=1\cdot\nicefrac{{5}}{{6}}+0.2\cdot\nicefrac{{1}}{{6}}\approx
0.87\end{split}$

Conceptually, the expected value may be interpreted as integration of the
listener’s own observation with the information received from the speaker
about the speaker’s beliefs about the world that the listener takes over.

###  4.2 The Garden Party Example

Despite its structural similarity to the Skiing Example, for completeness, let
us also briefly consider the Garden Party Example where, intuitively, the
listener’s degree of belief in the antecedent decreases.

#### The Garden Party Example (Douven2012b).

Betty knows that Kevin, the son of her neighbors, was to take his driving test
yesterday. She has no idea whether or not Kevin is a good driver; she deems it
about as likely as not that Kevin passed the test. Betty notices that her
neighbors have started to spade their garden. Then her mother, who is friends
with Kevin’s parents, calls her and tells her the following:

  1. (6)

If Kevin passed the driving test, his parents will throw a garden party.
$\leadsto$ listener belief in antecedent _decreases_

$G$$\begin{aligned} &P(g\mid d)=1\\\ &P(g\mid\neg
d)=0.5\end{aligned}$$D$$P(d)=0.5$ (a) $s_{dep}$: $D,G$ dependent

$G$$P(g)=0.95$$D$$P(d)=0.5$ (b) $s_{ind}$: $D,G$ independent

$\mathbb{E}[P^{(s)}(d)]$0.00.10.20.30.40.5Pragmatic interpretationLiteral
interpretationPrior beliefstate$s_{dep}$$s_{ind}$

(c) Expected value of the listener’s degree of belief in the antecedent

Figure 13: Bayes nets and results for Garden Party Example with
$\mathcal{U}=\\{G,D\rightarrow G,\text{likely }G\\}$ and $\alpha=3$, $D$: pass
driving exam, $G$: throw garden party. Both states, $s_{dep}$ and $s_{ind}$,
are assigned equal prior probability.

The difference between the Bayes nets shown in Figure 13 to the Bayes nets
used in the Skiing example only lies in the intuitive instantiation of the
prior probabilities. The results for the interpretation of the conditional in
the Garden Party Example (Figure 13(c)) also falls in with the results in the
Skiing Example: only under a pragmatic interpretation, the listener increases
her belief in the Bayes net where both variables are connected ($s_{dep}$),
but the listener’s belief in the probability of the antecedent remains as
prior to the speaker’s utterance.

Again, to consider the listener’s own beliefs, we assume that the listener
simply takes over the speaker’s beliefs communicated by the utterance of the
conditional. The world knowledge that we draw on in this example, shown in
Figure 14(a), concerns the fact that throwing a garden party is incompatible
with spading the garden ($S=s$). Therefore, $P(s\mid g)$ is set to 0 and, due
to the lack of more concrete information, $P(s\mid\neg g)$ is set to 0.5.
Contrary to the Skiing Example, the listener’s observation in this example is
not evidence, but counterevidence for the antecedent: Betty observes her
neighbors spading the garden, thus, the probability for a garden party
decreases, $P^{(s_{wk})}(g\mid s)=0$. While in $s_{ind}$, the degree of belief
in $D$ is not influenced by the truth or falsity of $G$,
$P^{(s_{dep})}(d\mid\neg g)=0$ (as opposed to $P^{(s_{dep})}(d\mid
g)=\nicefrac{{2}}{{3}}$). The combination of world knowledge about the
connection between $S$ and $G$, the speaker’s utterance ($D\rightarrow G$),
communicating a likely connection between $D$ and $G$, and the listener’s
observation related to $G$ therefore make Betty decrease her belief in the
antecedent (Kevin passing the driving test), shown in Figure 14(b).

$G$$P(g)$$S$$\begin{aligned} &P(s\mid g)=0\\\ &P(s\mid\neg
g)=0.5\end{aligned}$ (a) $s_{wk}$: $S,G$ dependent

$\mathbb{E}[P^{(s)}(d\mid s)]$0.00.10.20.30.40.5Pragmatic interpretationwith
listener’s observationPragmatic interpretationw/o listener’s
observationstate$s_{dep}$$s_{ind}$

(b) Expected value of the listener’s degree of belief in the antecedent

Figure 14: Bayes net for the assumed world knowledge in the Garden Party
Example (left) and results for the the listener’s inference about the
antecedent (right); $D$: pass driving exam, $G$: throw garden party, $S$:
spade garden.

###  4.3 The Sundowners Example

The Sundowners Example is a case where, intuitively, the listener’s degree of
belief in the antecedent does not change much, if at all.

#### The Sundowners Example (Douven2011a).

Sarah and Marian have arranged to go for sundowners at the Westcliff hotel
tomorrow. Sarah feels there is some chance that it will rain, but thinks they
can always enjoy the view from inside. To make sure, Marian consults the staff
at the Westcliff hotel and finds out that in the event of rain, the inside
area will be occupied by a wedding party. So she tells Sarah:

  1. (7)

If it rains tomorrow, we cannot have sundowners at the Westcliff.
$\leadsto$ listener belief in antecedent remains _unchanged_

$S$$\begin{aligned} &P(s\mid r)=0\\\ &P(s\mid\neg
r)=1\end{aligned}$$R$$P(r)=0.5$ (a) $s_{dep}$: $R,S$ dependent ($R\rightarrow
S$)

$R$$P(r)=0.5$$S$$P(s)\in[0.05,0.95]$ (b) $R,S$ independent
($R\rotatebox[origin={c}]{90.0}{$\models$}S$) with
$P^{(s_{ind\\_low})}(s)=0.05$ and $P^{(s_{ind\\_high})}(s)=0.95$

$R$$P(r)=0.5$$W$$\begin{aligned} &P(w\mid r)=1\\\ &P(w\mid\neg
r)=0\end{aligned}$$S$$\begin{aligned} &P(s\mid w)=0\\\ &P(s\mid\neg
w)=1\end{aligned}$ (c) Examplary Bayes net where $R$ and $S$ are related via a
mediating variable ($W$)

$\mathbb{E}[P^{(s)}(R)]$$\mathbb{E}[P^{(s)}(r)]$$\mathbb{E}[P^{(s)}(r,s)]$0.00.20.50.81.00.00.20.50.81.00.00.20.50.81.0Pragmatic
interpretationLiteral interpretationPrior beliefcausal relation
$R$$R\rightarrow S$$R\rotatebox[origin={c}]{90.0}{$\models$}S$

(d) Expected value of the listener’s degree of belief in the causal relation
between $R$ and $S$ (left), the probability of the antecedent (middle) and the
joint event of antecedent and consequent (right), given the Bayes nets from
Figure 15(a) and Figure 15(b).

Figure 15: Bayes nets and results for Sundowners Example with
$\mathcal{U}=\\{R\rightarrow\neg S,\text{likely}S,\text{likely}\neg S,S,\neg
S\\}$, $\alpha=3$ and a prior probability of 0.85 for Bayes net
$s_{ind\\_high}$, which is _a priori_ most likely according to the context
story; $s_{ind\\_low}$ and $s_{dep}$ are both assigned a prior probability of
0.075.

The intuition behind the formal treatment of the Sundowners Example, given
below, is as follows. Even though the speaker’s utterance of the conditional
does not seem infelicitous, it seems less natural than the conditionals in the
previous two examples. This oddness is reflected in a response from the
listener that is not far to seek: why should rain prevent them from having
sundowners? Put differently, we expect the listener, who has a strong prior
belief in the independence of $R$ and $S$, to be somewhat surprised by the
speaker’s utterance. The most rational explanation for the speaker’s utterance
choice of the conditional $R\rightarrow\neg S$ is to give up the assumption of
independence at least if the integrity of the speaker is taken for granted.
The listener’s surprise may be resolved by accommodating a third, latent
variable which is neither observed nor part of the speaker’s utterance and
acts as mediator between ‘rain’ and ‘having sundowners’. This would justify
the speaker’s choice to utter the conditional $R\rightarrow\neg S$ which
strongly suggests that (the speaker knows that) there is a (causal) connection
between ‘rain’ and ‘having sundowners’. In other words, we propose that, as
opposed to the previous examples, the speaker’s utterance in the Sundowners
Example forces the listener to accommodate a mitigating variable, which forms
a “causal bridge” between antecedent and consequent, so as to rationalize the
speaker’s utterance. As nothing further is known about this newly introduced
“causal bridge” or any other relevant variable, the result is that the
listener’s beliefs about the variable ‘rain’ remains largely the same.

A formally explicit treatment of these ideas makes use of the Bayes nets shown
in Figure 15(a)–15(c). As for the Skiing and the Garden Party Example, the
speaker’s utterance of the conditional provokes a shift from the Bayes net
where $R$ and $S$ are independent (Figure 15(b)) towards the Bayes net where
they are causally related (Figure 15(a)). Yet, as can be seen in the leftmost
panel of Figure 15(d), in this example, the listener has a strong prior belief
in the former which explains the listener’s surprise resulting from the
speaker’s utterance choice. Assuming independence of $R$ and $S$, the listener
would rather expect the speaker to choose a more informative utterance
assertable than the conditional (e.g. ,$\neg S$).

The RSA model of pragmatic language production entails a notion of _surprising
utterances_. Given prior beliefs about states $P_{\text{prior}}(s)$ and the
speaker’s assumed production probabilities $P_{\text{S}}(u\mid s)$, the
pragmatic listener expects utterance $u$ with probability:

$\displaystyle P_{\text{PL}}(u)=\sum_{s}P_{\text{prior}}(s)\
P_{\text{S}}(u\mid s)$

If the pragmatic listener only considers the three states represented in
Figures 15(a) and 15(b), with a prior probability of 0.85 for $s_{ind\\_high}$
and a prior probability of respectively 0.075 for $s_{ind\\_low}$ and
$s_{dep}$, the conditional $R\rightarrow\neg S$ is highly surprising in the
sense that its expected occurrence probability is very low ($\approx 0.08$
when $\alpha=3$).222222$P_{\text{S}}(u=R\rightarrow\neg S\mid
s_{ind\\_low})\approx 0.06,P_{\text{S}}(u=R\rightarrow\neg S\mid s_{dep})=1$
This notion of listener surprise in the light of an unexpected utterances
helps explain the intuition that an utterance of $R\rightarrow\neg S$ may
trigger the desire to go look for an additional explanation which may
rationalize the observed utterance.

One possibility of how the listener may rationalize a surprising utterance
through a mitigating variable is shown in Figure 15(c) where $W$ denotes the
event of ‘a wedding taking place inside the hotel’ that represents any event
which may possibly prevent the interlocutors from having sundowners at the
hotel. Rather than denying the speaker’s integrity due to the speaker’s
somewhat puzzling utterance, pragmatic reasoning, eminently causal-pragmatic
reasoning, therefore allows the listener to infer a previously unexpected
relation between $R$ and $S$, that is able to explain the speaker’s utterance.
Nevertheless, the conditional still seems to be incomplete as one could expect
the speaker to be more informative about the reasons why the event of rain may
prevent them from having sundowners.

Contrary to the Skiing and the Garden Party Example, in the Sundowners
Example, the listener does not intuitively update her belief in the
probability of the antecedent. Figure 15(d) (middle) shows that the listener’s
predicted degree of belief in the antecedent remains at its prior value of 0.5
after processing the speaker’s utterance, assuming a literal or a pragmatic
interpretation. The crucial difference here is that the listener does not make
any further observations as in the previous examples. Yet, the speaker’s
utterance of the conditional has a strong effect on the the listener’s beliefs
concerning the joint event of the antecedent and the consequent, shown in the
right panel of Figure 15(d). Contrary to her prior belief that $P(r,s)$ is
almost a matter of chance, the listener judges it almost impossible that the
two events both hold at the same time upon receiving the conditional
information $R\rightarrow\neg S$: in $s_{dep}$, $R$ and $S$ are mutually
exclusive, thus $P(r,s)=0$ and in $s_{ind\\_low}$ where $R\rightarrow\neg S$
is assertable, $P(r,s)$ is close to 0.

In sum, the model is therefore able to account by a single mechanism for
different interpretations with respect to the listener’s degree of belief in
the antecedent, namely by the interplay of pragmatic reasoning and an
adequate, explicit representation of the interlocutors’ prior probabilistic
and in particular, _causal_ beliefs.

##  5 On missing links & biscuits

Previous sections showed how the model of communication with conditionals
presented here predicts that speakers use a conditional predominantly when
there is a causal/inferential relation between antecedent and consequent and
that therefore listeners will interpret conditionals accordingly. This brings
up the obvious question as to how the advocated approach positions itself with
regard to cases, prominently discussed in the literature, in which antecedent
and consequent are clearly _not_ causally or inferentially related. The
absence of a relation between antecedent and consequent can either result in
infelicity, as is the case in what we here call _missing-link conditionals_
(Douven2017), or trigger an altogether different kind of interpretation, as in
what we here address as _biscuit conditionals_ (e.g., Austin1956;
geisLycanNonconditional93). This section deals with each case in turn.

###  5.1 Missing-link conditionals

Missing-link conditionals have been used in support of the inferentialist
position that the requirement of a causal/inferential connection between
antecendent and consequent is a necessary condition for assertability of a
conditional, arguably situated in the semantics of conditionals because it is
unclear how else a pragmatic account could explain the _in_ felicity of
missing-link conditionals (e.g. Douven2008; Douven2017; Krzy2014; Skovgaard-
Olsen2016a). This position is concretely exemplified by the contrast pair in
(8), given by Douven2008.

  1. (8)

There will be at least one heads in the first 1,000,000 tosses of this fair
coin ($h_{10^{6}}$) _if_

    1. a.

there is a heads in the first ten tosses. ($h_{10}$)

    2. b.

Chelsea wins the Champions League. ($c$)

What is remarkable about the contrast between the two sentences in (8) is
this: on the one hand, both (8a) and (8b) arguably pass the minimal necessary
requirement for assertability, namely that the probability of the consequent
given the antecedent is high in fact the difference between the relevant
conditional probabilities is minute ($P(h_{10^{6}}\mid h_{10})=1$,
$P(h_{10^{6}}\mid c)\approx 1$); on the other hand, while (8a) is intuitively
acceptable, for instance in a context where the speaker wants to highlight the
entailment relation to a listener who might otherwise not attend to it
sufficiently, (8b) rather clearly is odd.

To account for the infelicity of (8b) and assertability of (8a),
inferentialism stipulates that a conceivable inferential relation between
antecedent and consequent is a necessary requirement for assertability,
howsoever the inferential link may exactly be defined. It is not limited to
deductive inferences, as modern inferentialism allows less strict inferential
relations such as induction or abduction (e.g. Krzy2014; Douven2017).232323
Several empirical studies (e.g., Krzyzanowska2013; Skovgaard-Olsen2016;
Skovgaard-Olsen2017) have shown that the link between antecedent and
consequent has an influence on whether conditionals are accepted. Participants
in a study from Douven2010 for instance interpreted conditionals differently
depending on the type of the link (deductive, inductive or abductive). Yet,
there is also empirical evidence supporting the view that the link between
antecedent and consequent is attributable to discourse pragmatics rather than
conventional semantics (e.g. see Cruz; lassiter_decomposing_nodate).
Inferentialists have criticized pragmatic explanations of the perceived
infelicity of missing-link conditionals for remaining too vague about how the
pragmatic processes may concretely look like (e.g., Douven2017).

We argue here that (8b) is similar in kind to the Sundowners Example, yet more
extreme, therefore leading to perceived infelicity. The Sundowners Example
from the previous section showcases that there are contexts in which the
utterance of a conditional is not questioned per se, but will neither be
accepted without further ado. As there is no _obviously_ conceivable
connection between ‘rain’ and ‘not having sundowners at the hotel’, it seems
quite natural for the listener to reply with a question asking for precisely
this connection. We consider infelicitous missing-link conditionals like (8b)
to be similar in kind but more extreme cases of the same variety: in (8b) the
speaker provides so little information that the listener does not have enough
cues to make sense of the conditional utterance from world knowledge alone. In
other words, we maintain that the infelicity of (8b) is a result of a failure
of the listener to see a connection between antecedent and consequent that
could rationalize the speaker’s utterance choice.

To see how this is predicted by our model, we take the perspective of a
listener, who tries to interpret a missing-link conditional and knows, from
common sense world knowledge, that the antecedent and the consequent are two
independent events. Given the assumption of independence, our hyperrational
speaker would never choose any of the conditional utterances (see Figure 6),
and even for a speaker with a less optimal rationality parameter ($\alpha=3$),
we observe that conditional utterances only have an expected utterance choice
probability $<5\%$ when considering the speaker’s predictions for states with
$r=A\rotatebox[origin={c}]{90.0}{$\models$}C$ from a set of 10,000 samples
from the default context prior, as shown in Figure 16.

0.00.20.40.6$A\rotatebox[origin={c}]{90.0}{$\models$}C$$A\rightsquigarrow
C$$C\rightsquigarrow A$causal relation $R$expected utterancechoice
probabilityutterance typelikely + literalconditionalliteralconjunction

Figure 16: Speaker’s expected utterance choice probabilities computed on
speaker’s predictions, where $\alpha=3$, for a set of 10,000 states sampled
from the default context prior, grouped by utterance type and causal relation.

The reason for this low probability of choosing a conditional when $A$ and $C$
are assumed to be independent, is that very likely there will be an
assertable, more informative utterance available; for the independent Bayes
net, the joint probability tables where $A\rightarrow C$ is assertable (i.e.,
$P^{(s)}(c\mid a)\geqslant\theta$) also satisfy the assertability condition
for a literal or a conjunction. Given the background knowledge of independence
of the antecedent and the consequent and given that in these cases the speaker
hardly ever chooses conditionals, the listener will naturally be highly
surprised by the speaker’s utterance of a conditional. It is this notion of
surprise on the part of the listener, based on the speaker’s expected
utterance choice probabilities, that allows us to infer the infelicity of
missing-link conditionals: due to the listener’s surprise, she will, arguably,
want to look for another way of rationalizing the assertion. If no such option
is forthcoming, the utterance feels infelicitous. Missing-link conditionals
are therefore _not_ accepted by the listener contrary to the conditional in
the Sundowners Example where the listener’s initial surprise can be repaired
at the content-level since the listener is, after all, able to accommodate a
connection between antecedent and consequent.

The infelicity of missing-link conditionals as we explain it here, may further
be couched within the resource-rational, sampling-based approach to explain
causal reasoning.242424Thanks to an anonymous reviewer for highlighting this
interpretation. For example, dasgupta_where_2017 show that cognitive biases
like _subadditivity_ or _superadditivity_ can be explained by a comparison of
people’s search for a plausible explanation of observed data with sampling
algorithms, in particular MCMC sampling. They argue that due to cognitive load
and time constraints, the number of samples that people can draw is
restricted, such that the optimal answer is not found, giving rise to common
cognitive biases. As regards to missing-link conditionals, the listener’s
search for plausible Bayes nets with sufficient explanatory value to
rationalize the speaker’s utterance, seems to fail (e.g., in Example 8b). Yet,
when it comes to the interpretation of Biscuit conditionals (see Section 5.2
below) or to conditionals like in the Sundowners example, it seems reasonable
for a listener to be able to quickly come up with more or less satisfying
candidates.

###  5.2 Biscuit conditionals

Similar to missing-link conditionals, biscuit conditionals (BCs) are commonly
considered a special kind of conditional as both lack the probably most
characteristic feature of conditionals, the relation between antecedent and
consequent. Unlike missing-link conditionals, biscuit conditionals are however
felicitous, indeed quite common; see (9) for the classical example from
Austin1956.

  1. (9)

If you’re hungry, there are biscuits on the sideboard.

While Inferentialists have mostly excluded BCs from their analysis of
conditionals altogether, we argue that the felicity of BCs can indeed be
explained as a pragmatic phenomenon.252525 Recently, VanRooij2020; Rooij2021
proposed a generalization of their account for the assertability of
conditionals which is able to account for BCs as well. The model presented
here helps explain why listeners, who will likely assume a priori that the
consequent and the antecedent in example (9) are independent, would be
surprised by an utterance of (9) if it were to be interpreted like a normal
conditional. It is plausible that for some conditionals, like BCs, the
listener will see a repair strategy which enables a rationalization of the
conditional after all. However, this rationalization does not take place at
the content-level by an attempt to find a relation between antecedent and
consequent as (unsuccessfully) in missing-link conditionals or (successfully)
in the Sundowners Example, but at a different level, for instance at the level
of speech-acts. Concretely, this idea could be integrated into our model by
making the speaker’s utterance choice additionally dependent on communicative
goals, for example, whether the speaker wants to perform a speech-act (e.g.,
offer the listener some cookies), or just wants to make a plain assertion
(e.g., inform the listener about the existence of biscuits on the sideboard).

Admittedly this sketch is just a first step towards a satisfying pragmatic
account of BCs, as there is certainly more needed to fully explain their use
and interpretation. We also do not, with emphasis, suggest that this kind of
“surprise-repair” interpretation is actively entertained during each reception
of a BC: BCs arguably provide sufficient secondary cues for the listener to
trigger a BC-like interpretation, including intonation or, in some languages,
word order (e.g., in German, where BCs can occur with verb-third (V3) verb
order and not with the usual verb-second (V2) as seen in simple indicative
conditionals).

###  5.3 Conditionals for the communication of _in_ dependence

We assumed that, first and foremost, the speaker’s aim is to communicate
(probabilistic) beliefs about certain events and saw that, based on this
assumption, the speaker should only choose to utter a conditional if no other,
more informative utterance is assertable. However, in some circumstances,
conditionals may be used in order to put emphasis on the fact that the
consequent is _not_ dependent on the antecedent. Consider the example given in
(10):

  1. (10)

If you study, you will pass and if you don’t study, you will pass, so don’t
worry, you will pass!

Along the lines that we have been arguing, a conditional alike should be
avoided since evidently the speaker could say the same thing with a more
informative, shorter alternative, e.g., “ _You will pass (no matter what)!_ ”.
Notwithstanding, the conditional in (10) seems a natural utterance to say, in
particular when the speaker wants to emphasize the relation, here the
_independence_ between studying and passing an exam. If we extended the set of
alternative utterances available to our speaker by including this kind of
combined conditionals ($A\rightarrow C\wedge\neg A\rightarrow C$), the
listener would infer that the consequent holds true independent of the
antecedent. We saw previously that our listener infers from the utterance of a
conditional, that the speaker is likely uncertain about the truth or falsity
of the antecedent. For simplicity, let us assume the speaker refers to a state
$s$ where $P^{(s)}(a)=P^{(s)}(\neg a)=0.5$. Since the speaker uttered
$A\rightarrow C$, $P^{(s)}(c\mid a)\geqslant\theta$ must hold, and
additionally, due to the speaker’s utterance of the second conditional, $\neg
A\rightarrow C$, $P^{(s)}(c\mid\neg a)\geqslant\theta$. Then,
$P^{(s)}(c)=0.5\cdot P^{(s)}(c\mid a)+0.5\cdot P^{(s)}(c\mid\neg
a)=0.5\cdot(P^{(s)}(c\mid a)+P^{(s)}(c\mid\neg a))\geqslant\theta$. Therefore,
in this state, it is also possible to assert the consequent straightaway. A
speaker whose aim it is to communicate her uncertain beliefs should therefore
prefer the simpler utterance, assuming higher cost for combined utterances
like $A\rightarrow C\wedge\neg A\rightarrow C$. However, if the speaker’s aim
was two-minded, including the communication of her uncertain beliefs as well
as highlighting the independence between the variables at hand, the combined
conditional should become a likely utterance choice for the speaker: for
almost all states in which both conditionals are assertable,
$R=A\rotatebox[origin={c}]{90.0}{$\models$}C$ whereas $C$ is also assertable
in many states where $R=$ $A\overset{\scriptscriptstyle{}}{\rightsquigarrow}C$
or $R=$ $C\overset{\scriptscriptstyle{}}{\rightsquigarrow}A$.

##  6 Conclusion

We have taken a probabilistic approach to pragmatic reasoning about
conditionals here. Our approach is a conservative one: we combined two
independently motivated, well-established formalizations, namely the Rational-
Speech-Act model from probabilistic pragmatics and causal Bayes nets, in order
to predict a listener’s posterior probabilistic beliefs about relevant events
and how they are causally related. Our contribution is a formal, computational
model of the use and interpretation of conditionals that may be a starting
point for other models in this vein to gain further insights on how rich prior
beliefs are updated in the light of conditionals.

We showed that our model vindicates a number of different pragmatic inferences
observed in communication with conditionals. These include inferences such as
the dependency between antecedent and consequent, the infelicity of missing-
link conditionals or the tendency to interpret conditionals as biconditionals
as in conditional perfection readings. When the model is supplied with
particular prior beliefs representing concrete utterance contexts, it also
makes predictions corresponding to what is considered their intuitive
interpretation; we showed that the model helps to explain the listener’s
varying inferences with respect to the probability of the antecedent in three
concrete utterance contexts from Douven2012b.

We explored the interpretation of conditionals under the assumption that
conditionals are used to communicate stochastic information about co-
occurrence, while information about the underlying causal structure is
communicated implicitly in our model. In some circumstances, the speaker’s
communicative goal might however go further than this and include the
communication about causal information as well, bringing into play explicit
causal language. An interesting extension of the model may investigate the
relationship between conditionals, causality and causal language further by
considering different alternative utterances, including explicit causal
expressions that we did not address here.

The flexibility of the model we presented moreover allows future extensions to
systematically investigate the factors influencing the interpretation of
special types of conditionals such as biscuit conditionals. Other salient
extensions of the present approach include applications to subjunctive
conditionals, nested conditionals or conditionals with other logically complex
antecedents or consequents. Further, the quantitative predictions of the model
offer the possibility to test and improve it by a comparison with empirical
data.

ritta Grusdt
Wachsbleiche 27
49090 Osnabrück, Germany
aniel Lassiter
Dugald Stewart Building
3 Charles Street, Edinburgh
EH8 9AD, United Kingdom
ichael Franke
Geschwister-Scholl-Platz
72074 Tübingen, Germany