Running head: NOISE IN PRAGMATIC REASONING

1

The Strategic Use of Noise in Pragmatic Reasoning
Leon Bergena,1 and Noah D. Goodmanba Department of Brain and Cognitive Sciences, MITb Department of Psychology, Stanford University

1 Corresponding author. Address: 43 Vassar St, Room 3037, Cambridge, MA 02139. Phone: 732-266-3200. Email:bergen@mit.edu

NOISE IN PRAGMATIC REASONING

2

Abstract
We combine two recent probabilistic approaches to natural language understanding, exploring
the formal pragmatics of communication on a noisy channel. We first extend a model of rational
communication between a speaker and listener, to allow for the possibility that messages arecorrupted by noise. In this model, common knowledge of a noisy channel leads to the use andcorrect understanding of sentence fragments. A further extension of the model, which allows the
speaker to intentionally reduce the noise rate on a word, is used to model prosodic emphasis. We
show that the model derives several well-known changes in meaning associated with prosodic
emphasis. Our results show that nominal amounts of actual noise can be leveraged forcommunicative purposes.
Keywords: Pragmatics; Communication; Prosody; Ellipsis; Bayesian modeling;
Noisy-channel

NOISE IN PRAGMATIC REASONING

3

The Strategic Use of Noise in Pragmatic Reasoning
Recent work in cognitive science has investigated the inferential aspects of humancommunication: how people use general-purpose reasoning to go beyond the literal content of
their language input. This work has generally taken one of two perspectives. Researchers haveused noisy channel models (Levy, Bicknell, Slattery, & Rayner, 2009; Gibson, Bergen, &Piantadosi, 2013) to investigate how people infer meanings from language input that may havebeen corrupted by noise. Work on probabilistic pragmatics models (Frank & Goodman, 2012;
Franke, 2009) has looked at how people use strategic reasoning to enrich the meanings that they
communicate with their conversational partners. The current work will combine theseperspectives, providing a unified explanation of two distinct phenomena: the use of sentence
fragments to communicate full propositions, and the use of prosody to shift pragmaticinterpretations.
People’s communication channels are limited in various ways, and successfulcommunication often requires complex inferences in order to overcome these limitations. When aspeaker in a conversation tries to communicate something to the listener, their intended signal
may be corrupted by speech errors (e.g., if they are excited or intoxicated), environmental noise(if they are in a loud setting, e.g. at a cocktail party), or perceptual noise (if the listener is notpaying attention). For the listener to successfully understand the speaker’s intended meaning, theymust take into account these possible sources of noise, and infer whether they heard what the
speaker actually intended. A growing body of experimental evidence suggests that people accountfor the possibility of noise when interpreting language; this has been successfully modeled asprobabilistic inference of the original message given the received message. A different body ofcomputational research has emphasized the implications for communication of probabilistic
models of social inference. This work views the agents in a conversation as rational andcooperative, and as having mutual knowledge of these properties. These agents are able to draw
sophisticated inferences about each others’ intentions by using this knowledge, explaining
pragmatic enrichment of sentence meanings.

NOISE IN PRAGMATIC REASONING

4

Here we ask whether novel pragmatic inferences can be driven by the physical properties of
the communication channel. Consider that agents in a conversation are typically mutually awareof the physical limitations of their communication system, e.g. that their utterances may becorrupted by noise. These agents can therefore take actions that exploit these limitations, andtheir conversational partners’ knowledge of these limitations. Although noise is typically viewed
as a problem for communication, we will argue that the possibility of noise is in many
circumstances a resource, without which certain types of communication would not be possible.
Using this perspective, we will explore two sets of phenomena that on the surface look only
loosely related (see Benz (2012) for the application of similar ideas to a different set of pragmatic
phenomena). The first of these phenomena is the effective use sentence fragments tocommunicate full propositions (Merchant, 2004). Consider, for example, the typical manner inwhich wh-questions are answered:(1)

A: Who went to the movies?
B: Bob

The utterance Bob is used to express the proposition that Bob went to the movies. This
proposition could have been expressed with the full sentence Bob went to the movies, but thiswould be redundant given the context; it is common knowledge between the two speakers that the
response will describe who went to the movies. Thus, given an appropriate context, the use of
sentence fragments allows people to remove extraneous words from their utterances. However,fragmentary sentences present a difficulty for probabilistic pragmatics models, which generally
require a complete truth-functional literal meaning to initiate pragmatic reasoning.
The second phenomenon is the use of prosodic stress—a change in how a sentence ispronounced—to change the interpretation of the sentence. In particular, speakers can use prosody
to place focus on particular parts of an utterance. Consider the following example:(2)

A: Who went to the movies?
B: BOB went to the movies.

NOISE IN PRAGMATIC REASONING

5

Here capital letters are used to indicate that the speaker has placed prosodic stress on the word
“Bob." A stressed utterance will be interpreted exhaustively (Groenendijk & Stokhof, 1984;
Von Stechow, 1991): the speaker intends to communicate that Bob was the only person who went
the movies. Without stress, this utterance is compatible with other people having gone to themovies in addition to Bob. One explanation of this relationship between prosody and meaning isthat prosody simply carries conventional meaning which composes with lexical meaning. An
alternative, which we will explore here, is that the meaning of prosodic emphasis comes as apragmatic enrichment given the effect that prosody has on the physical acoustic channel ofspeech.
We will demonstrate how both the use of sentence fragments and the meaning effects ofprosody emerge from pragmatic reasoning given the possibility of noise. Our results will betheoretical, showing that the machinery of noisy-channel and probabilistic pragmatics models canbe combined and exploring the predictions of this combination. Introspective data from examples
will be strong enough to motivate our explorations, but our theoretical results also suggest
experimental questions for future research.
BackgroundNoisy-channel models provide an account how people understand language despite thepossibility of noise corrupting their language inputs. Under noisy-channel accounts, thecomprehender has a model of the noise process that generated their observed utterance, and they
determine the speaker’s intended utterance by integrating this knowledge with their priorexpectations about what the speaker is likely to say. This is formalized as a Bayesian decoding
procedure. The listener has a prior distribution over utterances P(ui ), which gives the probability
that the speaker intends to use utterance ui . The listener’s model of the noise process isrepresented by the distribution PN (u p |ui ), which gives the probability that the listener will
perceive utterance u p given that the speaker intended ui . The posterior probability that the speaker

NOISE IN PRAGMATIC REASONING

6

intended ui given that the listener perceived u p is given by:L(ui |u p ) ∝ P(ui )PN (u p |ui )(1)

Probabilistic pragmatics models give an account of a different aspect of people’s
linguistic knowledge: how the participants in a conversation draw inferences that go beyond theliteral meanings of the utterances that they hear. In several contemporary models (Frank &Goodman, 2012; Franke, 2009; Jager, 2010; Goodman & Stuhlmüller, 2013), this is formalizedwith a hierarchy of agents of increasing strategic sophistication. Here we will present a slightly
modified version of the Rational Speech Acts model of (Frank & Goodman, 2012). The speakerhas a meaning m ∈ M that they want to communicate to the listener, where M is the set of possible
meanings, or worlds. The set of utterances U specifies the grammatical utterances available to the
speaker. The literal meaning JuK of an utterance u is the set of possible worlds in which theutterance is true: m ∈ JuK if and only if the utterance is true of the meaning.
The least sophisticated agent in the model, the listener L0 , does not reason about why the
speaker would have chosen a particular utterance. Rather, they interpret an utterance according to
its literal meaning by Bayesian inference:
L0 (m|u) ∝ P(m)
 0

if m ∈ JuK(2)otherwise.

The term P(m) is the prior distribution on meanings that the speaker may want to communicate.
The speaker Sn chooses utterances by considering which would most effectively
communicate their intended meaning to the listener Ln−1 . They want to minimize the informationthat the listener would still lack to recover meaning m after perceiving utterance u p . This can beformalized as the surprisal: − log Ln−1 (m|u p ) (Cover & Thomas, 1991). The speaker wants tominimize surprisal, thus has utility function:U(u|m) = log Ln−1 (m|u p ) − c(u),(3)

where c(u) is the cost of utterance u. The distribution over utterances for the speaker Sn is definedusing the Luce-choice rule, which describes the choice behavior of approximately rational agents

NOISE IN PRAGMATIC REASONING

7(Sutton & Barto, 1998):
Sn (u|m) ∝ eλUn (u|m)(4)

The term λ > 0 controls the speaker’s degree of rationality.
Finally, listener Ln (n ≥ 1) interprets an utterance u by considering which meanings would
make the speaker Sn likely to choose u. This reasoning is captured with Bayes’ rule:Ln (m|u) ∝ P(m)Sn (u|m)(5)

These models are each intended to capture distinct phenomena, and as currently presented
the models are not compatible with each other. The probabilistic pragmatics models assume that
there is no noise in the transmission of the utterance from the speaker to the listener; if the
speaker intends for the listener to receive utterance u, then this is what the listener will receive.
On the other hand, the noisy-channel models do not say anything about how a comprehender can
extract non-literal meanings from a speaker’s utterance; from the perspective of these models, thespeaker is a black box, not a rational agent with sophisticated intentions.
Neither type of model can, on its own, explain the phenomena illustrated above. The
pragmatics models only deal with grammatical utterances whose literal meanings are specified inadvance, ruling out the use of non-grammatical sentence fragments within these models (unless asemantics for these utterances is given as an input). In addition, without stipulating that prosody
shifts the literal meaning of an utterance, these models cannot explain why prosody influences
pragmatic interpretation (see Bergen, Goodman, and Levy (2012) for a discussion of related
issues). Noisy-channel accounts, which do not provide an endogenous model of how the speakerchooses utterances, do not explain why the speaker would choose ungrammatical sentence
fragments as utterances, or place prosodic stress on their utterances.
Ellipsis
As example 1 illustrates, people can use sentence fragments to communicate full
propositions; indeed, this is often the most natural way to communicate. A basic question, then, is

NOISE IN PRAGMATIC REASONING

8

what linguistic knowledge allows for the successful interpretation of sentence fragments. One
possibility is that this knowledge is conventionalized as part of a language’s grammar or
semantics, i.e., the language provides a procedure for interpreting fragments that are used in
response to questions, or in other contexts. The language might contain a rule stipulating that thename of an agent is a grammatical response to a “who"-question, and that the answer to such aquestion can be obtained by substituting this name into the question.
It is unlikely, however, that such a grammatical rule would account for the productivity ofsentence fragment usage. Consider the following examples (see Stainton (1998) for related cases):(3)

A: I think I’ve discovered the culprit.
B: The butler!

(4)

A: I got laughed out of the seminar again today.
B: Your pragmatic theories.
A: Yeah, everyone thinks they’re crazy.

In neither case is there an apparent grammatical marker which licenses the fragment’s use as anutterance. Rather, in both examples, the first speaker’s utterance raises a topic for discussion, andthe second speaker offers a fragment that can be understood given this topic. These examples
suggest that the interpretation of sentence fragments is not fixed by the grammar, but rather isdriven by general inferential mechanisms.
We will be pursuing the hypothesis that people interpret sentence fragments throughpragmatic inference, not grammatical rules. The speaker can successfully communicate using an
ungrammatical, fragmentary utterance by exploiting the possibility—known to the listener—that
noise may have corrupted the utterance. When a strategically naive listener hears a sentence
fragment, they will believe that noise corrupted it, because sentence fragments are notgrammatical parts of the language. They will therefore infer what sentence the speaker actuallyintended to use. A strategically sophisticated speaker, reasoning about the naive listener, will
therefore choose a sentence fragment to express their intended meaning if it is parsimonious andNOISE IN PRAGMATIC REASONING

9likely to be interpreted correctly by the listener. A more sophisticated listener will expect suchfragments, and so on. Neither speaker nor listener assign a literal meaning to the sentence
fragment itself.
ModelThis reasoning can be formalized by combining the Rational Speech Acts model of (Frank& Goodman, 2012) with a simple Bayesian model for interpreting noisy language input (Levy,
2008; Gibson et al., 2013). The model starts with the literal listener L0 who interprets utterances
according to their literal meanings. There is an important difference between the definition of thisliteral listener and the one in Equation 2. Because noise will sometimes corrupt the speaker’s
utterances, the listener L0 needs to infer which utterance the speaker intended before assigning it aliteral interpretation. The distribution PN (u p |ui ) defines the noise process: the probability that thelistener will perceive utterance u p given that the speaker intends to use utterance ui . The literal
listener assigns a probability to meaning m by integrating the prior probability of m with theprobability that m is literally consistent with the speaker’s intended utterance:
L0 (m|u p ) ∝ P(m)∑P(ui )PN (u p |ui ).

(6)

ui :m∈Jui K

The speaker Sn (for n ≥ 1) chooses utterances by maximizing the probability that thelistener Ln−1 will recover their intended meaning, while minimizing effort. The speaker needs toconsider that their utterance may be corrupted by noise; whether the listener successfully recovers
the speaker’s intended meaning will depend on what utterance the listener perceives. Thus, thespeaker’s utility function Un generalizes the one defined in Equation 3, and now takes the
expectation over the surprisal of their intended meaning:
Un (ui |m) = ∑ PN (u p |ui ) log (Ln−1 (m|u p )) − c(ui ).

(7)

up

The speaker’s distribution over utterances is defined using the Luce-choice rule, as in Equation 4.
The listener Ln (for n ≥ 1) has a model of the speaker Sn , and, to a first approximation,interprets an utterance by reasoning about what intended meanings would have made this speaker

NOISE IN PRAGMATIC REASONING

10

likely to use the utterance. However, the listener accounts for the possibility that noise corrupted
the utterance she received; combining Equations 5 and 1:
Ln (m|u p ) ∝ P(m) ∑ Sn (ui |m)PN (u p |ui ).

(8)

ui

Results
We will use example 1 to illustrate how the model assigns interpretations to sentence
fragments. Suppose that it is common knowledge, following the question, that if anyone went to
the movies, then it was either Alice or Bob who went. Thus the speaker either wants tocommunicate that Alice went to the movies, that Bob went, or that nobody went. For simplicity,
we will assume that the utterances available to the speaker are of the form “X went to themovies," where X is “Alice," “Bob," or “Nobody"; it is straightforward to generalize thereasoning here to larger sets of alternative utterances. Finally, we will suppose that the noise
process consists of three types of noise (Levy, 2008; Gibson et al., 2013): insertions of words into
the sentence, deletions of words, and replacements of words, which are assumed to occur
independently. Only deletions will be relevant for the current example.
Figure 1 shows model results: robust interpretation of “Bob" as meaning that Bob went to
the movies, almost independent of the actual amount of noise. This can be explained as follows.
The naive listener L0 will interpret the two grammatical utterances according to their literalmeanings. If the listener hears the sentence fragments “Alice" or “Bob," then they will believethat this was not the speaker’s intent, but rather that noise corrupted the intended utterance. Afterhearing “Bob," the most probable inference is that the speaker intended to say “Bob went to themovies": this requires positing the smallest number of string edits, i.e. the deletion of the last fourwords. If the first word of the utterance is deleted, so that the listener hears “went to the movies"(or any subset of these words), then the speaker’s intent will be completely lost, because either
grammatical utterance will have been equally likely to produce the perceived utterance.
The speaker S1 reasons about the listener L0 , and knows that this listener will usuallyinterpret the utterance “Bob" as meaning that Bob went to the movies, and similarly for “Alice."

NOISE IN PRAGMATIC REASONING

11

If this speaker wants to communicate that Bob went to the movies, then choosing the fragment“Bob" is a reasonable strategy: it will usually be intepreted correctly, and requires less effort than
“Bob went to the movies." This explains why sentence fragments will be pragmatically licensed
in many contexts to communicate propositional meanings.

Probability of correct interpretation

1.0
0.8
0.6
0.4
0.2
0.0
10

-5

10

-4

10

-3

Noise

10

-2

10

-1

Use of Sentence Fragments
Probability of using sentence fragment

Interpretation of Sentence Fragments
1.0
0.8
0.6
0.4
0.2
0.0

10-5

10-4

10-3

Noise

10-2

10-1

Figure 1. The left panel shows the probability the listener will assign the correct meaning to thesentence fragment “Bob." The right panel shows the probability the speaker will use a sentence
fragment, as a function of the true noise rate.

It is important to note that this reasoning will work even if the noise rate is arbitrarily close
to 0, so long as it is positive. This is illustrated in Figure 1. When the listener L0 hears an
ungrammatical sentence fragment, they know that they perceived something that the speaker did
not intend. Conditional on having heard this ungrammatical utterance, they know that noise musthave corrupted the speaker’s intended utterance, however a priori unlikely that is. As a result, thespeaker S1 may utter sentence fragments much more frequently than would be expected under thenoise model. The frequency with which the speaker S1 chooses a sentence fragment is determinedby how successfully it communicates their intended meaning, and its cost relative to agrammatical sentence.

NOISE IN PRAGMATIC REASONING

12

ProsodyIn the previous section, we showed how participants in a conversation can use thepossibility of noise to enrich the meanings that their utterances communicate. We now consider
the possibilities for communication that are opened when the speaker can affect the noise rate of
their utterances.
As example 2 illustrates, the interpreted meaning of an utterance is changed by prosodic
stress. In that example, the speaker uses stress to signal that they have exhaustive knowledge
about the question under discussion, namely who went to the mall. Several related uses of
prosodic stress have been noted in the literature. Consider the following example:(5)

A: Who did John introduce at the party?
B: John introduced MARY to Bill.

By placing stress on “Mary," the speaker is able to indicate that Mary was the only person thatJohn introduced to Bill; this leaves open the possibility that Mary was introduced to other people
by John. This contrasts with the meaning that results from placing stress on “Bill":(6)

B: John introduced Mary to BILL.

In this case, the speaker intends to communicate that Bill was the only person that Mary was
introduced to; it is still possible, however, that other people were introduced to Bill. Thus,
prosodic stress in these cases is used to indicate the dimension along which the speaker has
exhaustive knowledge.
Prosody can also be used to shift the interpretation of scalar items. Consider the following
contrast:(7)

I passed the test.

(8)

I PASSED the test.

The first example generates a weak scalar implicature: the speaker at least passed the test, andthey do not know if they aced it. In contrast, the second example generates a strong implicature:

NOISE IN PRAGMATIC REASONING

13

the stress on the scalar item “passed" indicates that the speaker knows that they did not ace thetest. More generally, prosodic stress communicates that the speaker knows that the stronger scalaralternatives to their utterance are false.
Why does prosodic stress have these effects on the pragmatic interpretation of utterances?
We propose that these effects result from the acoustic signature of prosody. There are three main
acoustic changes associated with prosodic stress: increased loudness, duration, and fundamental
frequency (Breen, Fedorenko, Wagner, & Gibson, 2010).2 The increases in loudness and duration
are most clearly relevant for our purposes, as they will make the utterance more robust to noise.
An utterance that is louder and longer is less likely to get swamped by sounds in the environment,
and more likely to be the focus of the listener’s attention.3 Thus, by placing stress on part of anutterance, the speaker is choosing to decrease the noise rate for that part of the utterance. The
speaker’s choice to reduce the noise rate for a particular part of the utterance is an intentional
action, and as a result can receive a pragmatic interpretation by the listener.
Our work is closely related to the approach of Schmitz (2008), who also proposes a
pragmatic derivation of the interpretive effects of prosody. The current work differs from thisprevious work in many details, because it provides an explicit formal account of the speaker’s andlistener’s reasoning. Future work will consider whether these differences result in distinct
empirical predictions.
Model
As with the previous model, this version begins with a listener L0 who interprets utterances
literally. Before defining this listener, we will consider several changes to the speaker model.
Previously, we assumed that the speaker was fully knowledgeable, so that the speaker’s intended
meaning could be modeled by a point distribution on meanings. The current model will relax this2 See Hirst and Di Cristo (1998) for an overview of prosodic systems across a number of languages.
3 The changes in pitch associated with prosodic stress may also increase the robustness of the utterance to noise, ei-

ther by focusing the listener’s attention or through other acoustic mechanisms. Further experimental work is necessary
to determine the effects of pitch changes on the noise rate.

NOISE IN PRAGMATIC REASONING

14

assumption, so that the speaker’s knowledge state is modeled by a non-trivial distribution overmeanings (Goodman & Stuhlmüller, 2013); the listener will try to infer this knowledge state. In
particular, assume that the speaker made an observation o, resulting in a posterior distribution
over meanings: P(m|o) ∝ P(o|m)P(m), where P(o|m) is the probability that the speaker madeobservation o given that the true world state is m.
The noise rate of the speaker’s communication channel is a function of the prosody that the
speaker chooses. This is captured by the term PN (·|ui , s), which is the distribution over utterancesthat will be perceived by the listener, given that the speaker intends utterance ui and uses prosodys. The probability that the listener accurately perceives ui will be higher if the speaker chooses toapply prosodic stress to the utterance.
The speaker Sn wants to choose an utterance and prosody that will communicate theirknowledge state most effectively, while simultaneously minimizing their effort. Because thespeaker’s knowledge state is no longer represented by a single meaning, but rather by a
distribution over meanings, the informativeness of the speaker’s utterance is no longer measured
by surprisal, as it was in Equation 7. Instead, we will use KL-divergence to measure the distance
between the speaker’s belief distribution P(·|o) and the listener’s posterior distribution after
hearing the speaker’s chosen utterance and prosody:Un (ui , s|o) = − ∑ PN (u p |ui , s)KL(P(·|o)||Ln−1 (·, o|u p , s)) − c(ui )up

The speaker’s distribution Sn (ui , s|o) over utterances is again defined using the Luce-choice rule
(Equation 4).
The literal listener L0 interprets an utterance u p with prosodic stress s in a manner thatgeneralizes Equation 6:
L0 (m, o|u p , s) ∝ P(m)P(o|m)∑

P(ui )PN (u p |ui , s)(9)ui :m∈Jui K

Similarly, after perceiving utterance u p and prosody s, the listener Ln infers the world state andspeaker’s observations by:Ln (m, o|u p , s) ∝ P(m)P(o|m) ∑ Sn (ui , s|o)PN (u p |ui , s)
ui

(10)

NOISE IN PRAGMATIC REASONING

15

Results
We will first consider how the model derives the interpretation of sentences like example 2.
Assume that following the question in that example, there are three possible meanings that the
speaker may want to communicate: Alice went to the movies alone, Bob went alone, or Alice andBob both went. Assume for simplicity that at least one of them went. The speaker may, however,be ignorant about whether both Alice and Bob went. The utterances available to the speaker will
be of the form “X went to the movies," where X can be “Alice," “Bob," or “Alice and Bob." The
speaker can adopt one of two prosodic strategies: they can either use no prosodic stress, or placeprosodic stress on the names of the agents in the sentence. The base rate of noise is assumed to be
1% and the use of prosodic stress is assumed to decrease the noise rate by a factor of 2 (though
the model’s predictions are invariant to a range of base noise rates and noise reduction factors).
Figure 2 shows that at higher recursion depths—i.e., for more sophisticated speakers andlisteners—the model interprets prosodic stress on “Bob" as indicating that the speaker knows that
Alice did not go to the movies. This can be explained as follows. The listener L0 is strategicallyunsophisticated, and interprets utterances literally. This listener will not gain information fromprosodic stress, as prosodic stress is semantically inconsequential under the current account.
The pragmatic effects of prosody first emerge with the speaker S1 . If this speaker knowsthat only Bob went to the movies, then they have a very strong preference for the listener to hear
“Bob went to the movies": if the listener accidentally hears “Alice went to the movies" she willinfer that Alice went to the movies, when in fact the speaker knows that this is false. This speaker
will therefore be relatively likely to place prosodic stress on the word “Bob," in order to decrease
the noise rate and ensure that the listener perceives this accurately. In contrast, if the speakerknows that Bob went to the movies, but is uncertain about whether Alice also went, then they willbe less likely to place stress on the word “Bob." If the listener accidentally hears “Alice went to
the movies" in this case, it is still a problem for the speaker, as it will imply something that the
speaker does not know. However, in contrast to the previous case, Alice going to the movies is atleast compatible with this speaker’s knowledge. Prosody is thus marginally less valuable for the

NOISE IN PRAGMATIC REASONING

16

Use of Prosody

0.8

1.0
Prosody
Prosodic Stress
No Prosodic Stress

Probability of prosodic stress

Probability of exhaustive interpretation

Interpretation of Prosody
1.0

0.6
0.4
0.2
0.0

0.8

Knowledge
Knowledgeable
Ignorant

0.6
0.4
0.2
0.0

2

4

Depth

6

8

10

2

4

Depth

6

8

10

Figure 2. The left panel shows the probability that the listener will assign an exhaustive reading to“BOB went to the movies," as a function of recursion depth. The right panel shows theprobability that the speaker will use prosodic stress when they have full or partial knowledge
about the situation.
ignorant speaker; this effect is amplified by further levels of pragmatic recursion.
Next consider the interpretation of prosody in example 5, where selective prosody
exhaustifies only part of the interpretation. The modeling assumptions for this casestraightforwardly extend those for the previous case. The pragmatic effects of prosody again ariseat the level of the speaker S1 . If the speaker S1 knows that Mary was the only person who wasintroduced to Bill, but is uncertain about whether Mary was introduced to anyone else, then they
will be more likely to place prosodic stress on “Mary" than “Bill." This can be seen byconsidering the effect of the listener mishearing either “Mary" or “Bill." If “Mary" is misheard,then the listener’s beliefs will be incompatible with the speaker’s knowledge: the speaker knowsthat Mary, and nobody else, was introduced to Bill. In contrast, if “Bill" is misheard, then thelistener’s beliefs may still be compatible with the speaker’s knowledge: by assumption, thespeaker does not know if Mary was introduced to anyone besides Bill.
A symmetric argument shows that if the speaker S1 knows that Mary was only introduced toBill, but does not know if anyone else was introduced to Bill, then they will be more likely toNOISE IN PRAGMATIC REASONING

17

Probability of exhaustive interpretation of "Mary"

Interpretation of Complex Prosody
1.0

0.8

Prosody
Prosodic Stress on "Mary"
Prosodic Stress on "Bill"No Prosodic Stress

0.6

0.4

0.2

0.0
2

4

Depth

6

8

10

Figure 3. The probability that “John introduced Mary to Bill" will communicate that the speakerknows that only Mary was introduced to Bill, given different prosodic actions.
place prosodic stress on “Bill" than on “Mary." As a consequence, the speaker’s choice of whereto place prosody is informative about the dimension along which they have exhaustive knowledge
as shown in Figure 3.
Finally, we consider the model’s derivation of strong implicatures from prosodic stress, as
in example 8. Suppose that the two scalar alternatives available to the speaker are “passed" and“aced" (where “aced" entails “passed"). If the speaker S1 knows that they passed the test, but did
not ace it, then it is important for the listener to correctly hear their utterance. If the listener
accidentally hears “aced" instead of “passed," then they will believe something that the speakerknows is false. This speaker will therefore be relatively likely to stress “pass."
In contrast, if the speaker knows that they at least passed, but does not know if they aced thetest, then it is not as important for the listener to correctly perceive their utterance. If the listener
accidentally hears “aced" in this case, then their beliefs will be consistent with what the speakerknows. As a result, the speaker with partial knowledge will be less likely to place prosody on thescalar term. Figure 4 shows how prosody affects the interpretation and usage of scalar items.

NOISE IN PRAGMATIC REASONING

18

0.8

Use of Prosody for Scalar Items
1.0

Probability of prosodic stress

Probability of exhaustive interpretation

Interpretation of Prosody
Prosody
Prosodic Stress
No Prosodic Stress

0.6
0.4
0.2

0.8

Knowledge
Knowledgeable
Ignorant

0.6
0.4
0.2
0.0

2

4

Depth

6

8

10

2

4

Depth

6

8

10

Figure 4. The left panel shows the probability that the listener will assign a strong(knowledgeability) implicature to “I PASSED the test," as a function of recursion depth. The rightpanel shows the probability that the speaker will use prosodic stress when they have full or partial
knowledge.
Conclusion
We have provided a unified pragmatic account of two distinct linguistic phenomena: the useof sentence fragments to communicate full propositions, and the use of prosody to shift theinterpretation of utterances. We have argued that both of these phenomena result from thestrategic exploitation of noise, which emerges naturally when noisy channel and probabilistic
pragmatics models are integrated.
The first of these phenomena, the communicative use of sentence fragments, can be derived
from the assumption that noise is possible in the discourse. When the listener encounters asentence fragment, they will infer that noise corrupted the speaker’s intended message, and thatthe speaker actually intended a grammatical sentence; the speaker exploits this fact tocommunicate full propositions using sentence fragments. The second phenomenon, the use of
prosodic stress to strengthen an utterance’s meaning, can be derived from the assumption thatprosodic stress allows the speaker to decrease the noise rate of their utterance. The speaker will
use prosodic stress to decrease the noise rate on the parts of their utterance which are most

NOISE IN PRAGMATIC REASONING

19

important for communicating their intended meaning; the listener can then use the placement ofprosodic stress to infer which parts of the utterance were most important for the speaker.
We have thus shown that the knowledge of these phenomena does not need to be
encapsulated in the language’s grammar. The grammar does not need to specify a set of rules forinterpreting sentence fragments under different conditions, nor does it need to use focus to shift
the semantic content of prosodically marked utterances. Domain-general pragmatic reasoning issufficient.
There are a number of possible directions for extending the models in this paper.
Throughout, we have assumed that the noise process takes an especially simple form: words in an
utterance are independently inserted, deleted, or replaced. Though this was sufficient to explain
the phenomena in this paper, there are related phenomena where it is unclear whether this is true.
In particular, in order to account for more complex types of ellipsis, it may be necessary for thenoise model to be sensitive to the phrase structure of utterances (Frazier & Clifton, 2001).
Prosodic stress is often used to communicate aspects of meaning beyond those discussed inthis paper. It can provide information about the topic/question under discussion; indicate contrast
with previous information in the discourse; and shift the interpretation of focusing adverbs suchas only and even (Rooth, 1992; Büring, 1999). As defined in this paper, our model of prosodic
stress does not provide an explanation of these phenomena. Future work will consider whether
these phenomena can be explained by integrating more information about the discourse structure
into the model, in particular, information about the speaker’s and listener’s communicative goals(Kao, Wu, Bergen, & Goodman, 2014).
Acknowledgments
We are grateful to Michael Franke, Chris Potts, and audience members at CogSci 2014 forcomments on this work. This work was supported by an NSF Graduate Research Fellowship andan NSF Dissertation grant (LB), and by a James S. McDonnell Foundation Scholar Award, ONR
grant N00014-13-1-0788, and ONR grant N00014-13-1-0287 (NDG).

NOISE IN PRAGMATIC REASONING

20

References
Benz, A. (2012). Errors in pragmatics. Journal of Logic, Language and Information, 21(1),
97–116.
Bergen, L., Goodman, N. D., & Levy, R. (2012). ThatâĂŹs what she (could have) said: Howalternative utterances affect language use. In Proceedings of the thirty-fourth annual
conference of the cognitive science society.
Breen, M., Fedorenko, E., Wagner, M., & Gibson, E. (2010). Acoustic correlates of information
structure. Language and Cognitive Processes, 25(7-9), 1044–1098.
Büring, D. (1999). Topic. In P. Bosch & R. van der Sandt (Eds.), Focus — linguistic, cognitive,
and computational perspectives (pp. 142–165). Cambridge: Cambridge University Press.
Cover, T. M., & Thomas, J. A. (1991). Elements of information theory. John Wiley & Sons.
Frank, M. C., & Goodman, N. D. (2012). Predicting pragmatic reasoning in language games.
Science, 336(6084), 998–998.
Franke, M. (2009). Signal to act: Game theory in pragmatics.
Frazier, L., & Clifton, C. (2001). Parsing coordinates and ellipsis: Copy α. Syntax, 4(1), 1–22.
Gibson, E., Bergen, L., & Piantadosi, S. T. (2013). Rational integration of noisy evidence andprior semantic expectations in sentence interpretation. Proceedings of the National
Academy of Sciences, 110(20), 8051–8056.
Goodman, N. D., & Stuhlmüller, A. (2013). Knowledge and implicature: Modeling language
understanding as social cognition. Topics in cognitive science, 5(1), 173–184.
Groenendijk, J., & Stokhof, M. (1984). Studies in the semantics of questions and the pragmatics
of answers (Unpublished doctoral dissertation). University of Amsterdam.
Hirst, D., & Di Cristo, A. (1998). Intonation systems: a survey of twenty languages. Cambridge
University Press.
Jager, G. (2010). Game-theoretical pragmatics. Handbook of logic and language, 467.
Kao, J. T., Wu, J. Y., Bergen, L., & Goodman, N. D. (2014). Nonliteral understanding of number
words. Proceedings of the National Academy of Sciences, 111(33), 12002–12007.

NOISE IN PRAGMATIC REASONING

21

Levy, R. (2008). A noisy-channel model of rational human sentence comprehension under
uncertain input. In Proceedings of the conference on empirical methods in natural
language processing (pp. 234–243).
Levy, R., Bicknell, K., Slattery, T., & Rayner, K. (2009). Eye movement evidence that readers
maintain and act on uncertainty about past linguistic input. Proceedings of the National
Academy of Sciences, 106(50), 21086–21090.
Merchant, J. (2004). Fragments and ellipsis. Linguistics and Philosophy, 27(6), 661–738.
Rooth, M. (1992). A theory of focus interpretation. Natural language semantics, 1(1), 75–116.
Schmitz, H.-C. (2008). Accentuation and interpretation. Palgrave Macmillan.
Stainton, R. J. (1998). Quantifier phrases, meaningfulness “in isolation”, and ellipsis. Linguistics
and Philosophy, 21(3), 311–340.
Sutton, R. S., & Barto, A. G. (1998). Reinforcement learning: An introduction (Vol. 1) (No. 1).
Cambridge Univ Press.
Von Stechow, A. (1991). Focusing and backgrounding operators. Discourse Particles:
Pragmatics & Beyond, Amsterdam: John Benjamins, 37–84.