Knowledge and implicature: Modeling language understanding as social cognition
Noah D. Goodman (ngoodman@stanford.edu),
Department of Psychology, Stanford University

Andreas Stuhlmüller (ast@mit.edu),
Department of Brain and Cognitive Sciences, MIT
Abstract
Is language understanding a special case of social cognition?
To help evaluate this view, we can formalize it as the rational speech-act theory: listeners assume that speakers choose
their utterances approximately optimally, and listeners interpret an utterance by using Bayesian inference to “invert” thismodel of the speaker. We apply this framework to model scalar
implicature (“some” implies “not all”, and “N” implies “notmore than N”). This model predicts an interaction between the
speaker’s knowledge state and the listener’s interpretation. We
test these predictions in two experiments, and find good fit between model predictions and human judgements. Keywords:
Language; Bayesian model; Scalar implicature

To what extent does language understanding rely on extralinguistic knowledge and processes? One view of language processing suggests that it consists of largely separate,
special-purpose faculties; another view, that it depends critically on domain general inference mechanisms, and evenon intuitive theories that are not specific to language. Indeed, many thinkers have viewed speech as an action with
communicative goals, such as informing a listener (Grice,
1975; Levinson, 2000; Clark, 1996). A listener making thisassumption can make stronger inferences than an utterance
would allow from its literal meaning—pragmatic effects canstrengthen, or change, the interpreted meaning.
Recent work has aimed to formalize the social inference
view of pragmatics using tools of Bayesian statistics and information theory (Frank & Goodman, under review); we referto this formal framework as the rational speech-act theory oflanguage understanding. It views pragmatic competence asfollowing naturally from an intuitive theory of speech production, which in turn is a special case of intuitive theory of mind.
More precisely, listeners have an internal model which describes speakers as choosing their utterances approximately
optimally on the basis of certain social goals, such as conveying information to the listener; listeners then interpret anutterance by using Bayesian inference to “invert” this modelof the speaker, drawing conclusions about the world state andspeaker’s intention from the utterance and any other relevant
world knowledge. These two rationality assumptions, for listener and speaker, have a role similar to those made in ideal
observer models of perception (Geisler, 2003): they provide astarting point for a quantitative understanding of the complex
interactions involved in language understanding. Indeed, thisview has provided good quantitative models for pragmatic inference in a number of simple settings (Frank, Goodman, Lai,
& Tenenbaum, 2009; Frank & Goodman, under review).
Because rational decision making predicts that action selection is related to the expected utility—a quantity that de-

SOME OF THE
APPLES ARE RED.

SOME

ALL

Figure 1: How will the listener interpret the speaker’s utterance? How will this change if she knows that he can see only
two of the objects?

pends on the actor’s belief distribution—a listener who views
the speaker as rational should be sensitive to the speaker’s
belief state. The rational speech-act theory thus predicts aninteraction between (shared) knowledge about a speaker’s
knowledge state and a listener’s interpretation of his utterance. This is a very general prediction of the framework,
which could easily prove to be false—if pragmatic inferenceis a highly modularized computation, for instance, we would
not expect such general knowledge to affect it. Deriving andtesting precise predictions about this interaction thus provides
an important test of the rational speech-act theory.
If you hear “some of the apples are red,” you will infer that
not all of the apples are. Pragmatic effects of this sort are
called scalar implicatures (Horn, 2004), and they provide awindow on the interactions between the language faculty andgeneral cognition. Consider Fig. 1: If the speaker has seen all
the apples, his utterance would be interpreted as “some, but
not all, of the apples are red.” However, if the speaker had
only looked at two of the apples, the listener might draw adifferent conclusion. Indeed, we show below that the implicature “not all” can be canceled by facts about the speaker’s
perceptual access. This interaction between language understanding and general knowledge is not predicted by strongly
modular theories of scalar implicature, and even moderately
modular theories (Chierchia, Fox, & Spector, 2011) predictthat such influences can only turn off scalar implicatures. We
show that the interaction of knowledge and implicature ismore fine grained: the details of a speaker’s belief distribu-

tion affect the details of an implicature.
This paper provides a formal model of the pragmatic inference that leads to scalar implicatures, building on the rational
speech-act framework. We directly model the possibility thatthe speaker may have incomplete knowledge and the effects
this has on the listener’s interpretation. We derive predictionsof this model for interpretation of the quantifier “some” andthe number words (“one”, “two”, etc.). The model both explains the standard implicatures for these words, and predicts
that these implicatures can be canceled, completely or partially, when the speaker has incomplete knowledge. We test
these predictions in two experiments and find good fits to human judgements, both qualitatively and quantitatively.

A rational speech-act model
We view language comprehension as a rational inference
based on an intuitive theory of language production. Our setting is illustrated in Fig. 1. The listener infers the world state,s, given the speaker’s utterance, w, and shared information
about the speaker’s (possibly incomplete) information access,a. By Bayes’ rule:
Plistener (s|w, a) ∝ Pspeaker (w|s, a)P(s).

(1)

Where P(s) captures the listener’s prior beliefs about theworld state, and Pspeaker (w|s, a) describes the listener’s intuitive theory of how the speaker chooses words.
The speaker chooses an utterance in accord with Bayesian
decision theory (Berger, 1985): she acts to approximately optimize expected utility. Imagine a speaker who makes observation o about the true state of the world. (For instance, inFig. 1, the speaker has perceptual access to two of three apples, and observes that those two are red). The speaker selects
an utterance w to convey information about the world state toa listener, and does so by soft-max optimizing expected utility:
Pspeaker (w|o, a) ∝ exp α EP(s|o,a) [U(w; s)](2)
The speaker’s utility function, U(w; s), captures the value ofsaying w if the world is actually s. The expectation is takenover the speaker’s belief state, P(s|o, a), because the speaker
may still be uncertain about the state of the world. The parameter α controls the deviation from optimality.
So far, nothing in the model is unique to language—indeed,similar models have been used to model social cognition
more generally (Baker, Saxe, & Tenenbaum, 2009; Goodman, Baker, & Tenenbaum, 2009; Ullman et al., 2009). To
capture a motivation to be informative, utility must be related
to the information conveyed in the utterance.1 More specifically, utility is related to the amount of information that aliteral listener would not yet know about state s after hearing it described by utterance w—this is the negative surprisal(Cover & Thomas, 1991):
U(w; s) = ln(Plex (s|w)),(3)

1 Other communicative motivations could be added to this utility,
such as a complexity term influencing the manner of expression.

where the literal interpretation probability Plex (s|w) is determined by the lexicon—here we will assume that each utterance has a truth function, Fw : s 7→ {0, 1}, and the distribution
is otherwise uninformative: Plex (s|w) ∝ δFw (s) .
We assume that the speaker’s access a is common knowledge of speaker and listener, but the listener still does notknow what observation the speaker made, hence:
Pspeaker (w|s, a) = ∑ Pspeaker (w|o, a)P(o|a, s)(4)oIn the experiments below (as in Fig. 1), the state of the worldis always a set of objects that may have a given property
and observations consist of looking at a subset of the objects. Thus the observation probability P(o|a, s) is a hypergeometric distribution (i.e. the probability of drawing N ballsof a given color, without replacement, from an urn containing
a given set of colored balls). In this setting, it is also reasonable to assume that the prior probability, P(s), is a binomial
distribution (i.e. draws with replacement); we will initially assume so, and will later measure the distribution empirically.
Overall, the above equations describe the inferences thata rational listener will make to comprehend a speaker that
she believes to be approximately rational and have a goal
to be informative. Importantly, these inferences depend on
shared knowledge about the aspects of the world to which the
speaker has access. Thus the rational speech-act theory predicts that the speaker’s access affects utterance interpretation;
we test this prediction below. To derive more specific predictions we must describe the set of alternative utterances.

The alternatives
We have assumed that the interpretation of an utterance ismade with respect to a set of alternative utterances. These
alternatives could be all possible utterances, or could be alimited set generated by replacing key words in the actual
utterance with related words. The alternatives may, for instance, be generated by a grammatical mechanism as in Fox
and Katzir (2011). For our results the details of this process
are unimportant; what is important is that there exists a set ofalternative expressions and a (truth-functional) literal meaning for each. We make standard assumptions in both respects.
Consider the case of the quantifier words “some” and “all”.
Under the standard semantics, “all the balls are red” is true exactly when N of the N balls are red, while “some of the balls
are red” is true when M ≥ 1 of the N balls are red. In particular, the literal meaning of “some” allows the state where all
N balls are red. We use a lexicon that consists of the standard
meanings for “none”,“some”, and “all”. Model predictions
are shown in Fig. 2a for the listener’s interpretation of “someof the balls are red” when there are 3 objects, under varying
conditions of speaker access.
When the speaker has perceptual access to 3 of the 3 objects (hence complete knowledge) and says “some”, there is alower probability on 3 than 2—this is the standard “some but
not all” implicature. To understand why the model predicts
this implicature, we can first simplify the speaker model: inExperiment

(a)

(c)

"Some", access 2"Some", access 3

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Figure 2: (a,b) Model prediction for probability of each world state (number of objects with property), varying the word the
speaker used and the speaker’s perceptual access. The prior is assumed to be binomial with base rate 0.62, and the speaker
optimality parameter is set to α = 3.4. (c,d) Mean participant bet on each world state, varying the word the speaker used andthe speaker’s perceptual access. Data have been filtered to include only trials where the participant’s bet that the speaker had
complete knowledge was greater than 70 in the expected direction. Error bars are standard error of the mean.
a complete knowledge situation the speaker’s belief distribution is concentrated on the true state, so Eqs. 2, 3, and 4 become:Pspeaker (w|s, a) ∝ exp (α ln(Plex (s|w)))∝ (Plex (s|w))α .

(5)

Using the standard meanings described above, we see that
if the state were 3 the speaker would be more likely to say“all” (Plex (s|w)=1) than “some” (Plex (s|w)= 31 ); conversely,
if the state were 2, the speaker would say “some”, since theprobability for “all” is now Plex (s|w)=0. Now consider Eq. 1
and imagine for the moment a uniform prior over states. In
this case the listener will infer each state s in proportion to
how likely the speaker was to say “some” given this state.
Overall, this leads to the implicature that “some” is unlikely
to be interpreted as 3—“some but not all”.

In contrast, when the speaker has only partial access thecalculation is more complex, involving the inferred belief distribution of the speaker. Comparing across the three panelsof Fig. 2a, we see the probability of 3 is much higher whenaccess is 1 or 2 than when it is 3. When access is 1, no implicature is predicted (the probability of 3 is approximately
the same as the probability of 2); when access is 2, only avery slight implicature. Overall, we predict that incomplete
speaker knowledge can cancel the standard “some but not all”
implicature.
The case of numerals (“one”, “two”, ...) is similar but more
subtle. It has been argued that number words have a lowerbound meaning (Horn, 1972) (e.g. “two balls are red” means
M ≥ 2 of the balls are red), and the intuitive, exact, meanings
arise as a pragmatic implicature—“one but not two, three,etc.”. In Fig. 2b we show model predictions based on thelower-bound semantics for number words, varying speaker’s
access. We see that exact meanings do arise as an inference
when the speaker has complete access, but there is an interaction: number words do not receive an exact interpretation
when the speaker has incomplete knowledge. Of particularinterest is the case where the speaker has seen 2 out of 3 objects and says “one”: here a partial implicature is predicted,
with the probability of 3 low, but 1 and 2 high.

Experiment 1
Because a rational speaker chooses actions based on expected utility, the rational speech-act model predicts an effect of speaker’s knowledge on the listener’s interpretation of“some” statements. We tested the predictions of the model by
putting participants in the role of the listener and asking themto judge the state of the world in scenarios where perceptual
access (and hence knowledgeability) of the speaker varied.

Participants, Materials, and Methods
50 participants were recruited through Amazon’s Mechanical
Turk crowd-sourcing service and completed the experiment
for a small payment.
We constructed six scenarios in which a speaker had threeobjects that could have (or not) a given property. The speakerthen made a statement indicating the number of objects they
had looked at and a quantified (“some”) statement. We split
each scenario into setup and speech act phases. The setup
phase named the speaker and described the objects and therelevant property. For example:

may have gained knowledge by another route), we elicited
this directly:
Do you think Laura knows exactly how many of the 3
letters have checks inside?
Each response was given by a betting measure: participants
were instructed to divide “$100” among the options, betting
to indicate their confidence in each option. For the first twoquestions there were four options (0–3 of the objects have theproperty) and for the final question there were two options(the speaker does/doesn’t have complete knowledge). Before
the experiment began, participants were given a brief warmup, using unrelated questions, to familiarize them with thebetting measure.
Each scenario existed in forms varying speaker access (thenumber of objects the speakers had looked at) from 1 to 3.
Each participant saw each access condition once, in random
order, presented using randomly chosen scenarios (with no
duplicate scenarios). In terms of our predictions, we have
two partial-knowledge conditions (where we expect cancelation of the implicature) and one complete-knowledge “control” (where we expect the standard implicature).2

Qualitative Results

Now how many of the 3 letters do you think have checks
inside?

There was no effect of scenario, so we collapse across thisfactor in all analyses. As expected based related work
(Goodman et al., 2009), the speaker’s access statement(e.g. “I have looked at 2 of the 3 letters”) was not a perfectmanipulation of knowledgeability: in the partial access conditions some participants judged that the speaker was likely
to know exactly how many objects had the property. (The bet
that the speaker had complete knowledge when access=1 was
M = 27.1, SD = 4.9; when access=2, M = 34.8, SD = 5.7;
when access=3, M = 93.0, SD = 2.7.) Since we were interested in the effects of varying knowledgeability, we initially exclude trials in which the knowledge judgement wasless than 70 in the expected direction (we come back to thefull data set in the quantitative analysis below). Fig. 2c shows
the mean of bets on each option, as access varied. As predicted, there was an effect of speaker’s access on listener’sinterpretation (one-way ANOVA with bets on 3 as dependent
variable, F(2, 102) = 10.18, p<0.001).
We next performed our pre-planned comparisons to checkthat an implicature was drawn when the speaker had complete
knowledge, but not when the speaker had partial knowledge.
In the complete access condition bets on 3 were less than betson 2 (paired, directional t-test, t(43) = −10.2, p<0.001). In
the partial access conditions the implicature was canceled:bets on 2 did not exceed bets on 3 when speaker had access to
1 object (paired, directional t-test, t(31) = 0.77, p = 0.78) or
when access was 2 (paired, directional t-test, t(28) = −0.82,
p = 0.21). If we look at just bets on 3, we see significantly lower bets in the complete access condition than the

Finally, because the speech act might not be a perfect manipulation of speaker’s knowledgeability (for instance, the speaker

2 Expt. 1 may be viewed at http://goo.gl/3S5zz, Expt. 2 at
http://goo.gl/iSc6o.

Letters to Laura’s company almost always have checks
inside. Today Laura received 3 letters.
Because our model predicts greater effects when the a prioribase rate of the property is high (otherwise it is difficult totell an implicature from an a prior belief that it is unlikely for
all objects to have the property), we describe all properties as“almost always” holding. To make sure participants attended
to the setup, we asked them to report the a priori probability
that 0, 1, 2, or 3 objects had the property:
How many of the 3 letters do you think have checks inside?
The speech-act phase introduced a speech act in which the
speaker both declared how many of the objects they had observed and stated that some objects had the property:
Laura tells you on the phone: “I have looked at 2 of the
3 letters. Some of the letters have checks inside.”
We then again elicited judgements about how many objects
had the property:

access 1 condition (unpaired, directional t-test, t(47)= − 4.0,
p<0.001) or the access 2 condition (unpaired, directional ttest, t(43)= − 3.5, p<0.001). While there was no significant implicature in either partial-access condition, there isa slightly greater tendency toward implicature in the access
2 condition than the access 1 condition, as predicted by themodel (two-way ANOVA with bet as dependent variable, andaccess (1 or 2) and state (2 or 3) as independent variables,F(2, 294)=3.77, p<0.05).
Thus, the knowledge of the speaker affected listener’s interpretation of “some” in the way predicted by the rational
speech-act model. We examine the quantitative fit of themodel below.

Experiment 2
In Expt. 2 we tested the predictions of the rational speechact model for interpretation of numerals. We again expect tofind an effect of speaker’s knowledge, but in this case there
is a more detailed interaction: the implicature should be canceled when the speaker says “one” after seeing only one object and when the speaker says “two” after seeing two objects,
but it should only be partially canceled when the speaker says“one” after seeing two objects—this implies a fine-grainedinterplay between the speaker’s knowledge state and the interpretation of her utterance.

Participants, Materials, and Methods
50 participants were recruited through Amazon’s Mechanical
Turk crowd-sourcing service and completed the experiment
for a small payment.
We used the same stimuli as in Expt. 1, modifying the scenarios only in the speech act: the speaker now made a statement indicating the number of objects they had looked at andthe number that had the property. For instance:
Laura tells you on the phone: “I have looked at 2 of the
3 letters. 1 of the letters has checks inside.”
Each scenario existed in forms varying the speaker’s access, from 1 to 3, and the number word the speaker used, from
1 to 3; we limited to sensible situations where the word used
was no greater than the number of objects seen. Hence we had
six conditions, with access/word: 1/1, 2/1, 2/2, 3/1, 3/2, 3/3.
In terms of our predictions, we have three partial-knowledge
conditions (where we expect partial or complete cancelationof the implicature) and three complete-knowledge “controls”(where we expect the standard implicatures). The order ofscenarios and the order of conditions were randomized between participants.

Qualitative Results
There was again no effect of scenario, so we collapse acrossthis factor. As in Expt. 1, the speaker’s access statement
was not a perfect manipulation of knowledgeability (bets thatspeaker had complete knowledge in partial-access conditions:
M=42.0, SD=3.4, in complete access conditions: M=92.1,
SD=1.6). We once again limit to trials with the expectedjudgements of knowledgeability (with a threshold of 70). The
mean of participants’ bets are shown in Fig. 2d. To evaluate the overall effect of access, we performed an ANOVA
with access and word as independent measures and bet on
3 as dependent measure. We find a main effect of access(F(2, 205)=6.57, p<0.01), an interaction between word andaccess (F(1, 205)=34.7, p<0.001), and a main effect of word
(F(2, 205)=269.8, p<0.001).
We then explored the results in more detail using planned
comparisons to test whether implicatures were drawn (only)
when predicted. We found an implicature in the complete
access conditions: when the speaker said “two”, bets onstate 3 were less than on state 2 (paired, directional t-test,
t(43) = −10.2, p<0.001). When the speaker said “one”, bets
on state 1 were greater than on state 3 (paired, directional ttest, t(42)= − 13.1, p<0.001) or state 2 (paired, directionalt-test, t(42)= − 17.1, p<0.001). In contrast, there was no
implicature when access was 1 and the speaker says “one”—
bets on 1 were not greater than on 2 (paired, directional ttest, t(24)=1.9, p=0.96) or on 3 (paired, directional t-test,
t(24)=3.2, p=1.0)—and no implicature when access is 2 andthe speaker says “two”—bets on 2 were not greater than on 3
(paired, directional t-test, t(24)=1.1, p=0.87). When accessis 2 and the speaker says “one” we found the predicted partial
implicature: bets on state 1 were significantly greater than onstate 3 (paired, directional t-test, t(25)= − 3.9, p<0.001), but
not on state 2 (paired, directional t-test, t(25)=1.5, p=0.92).
These results again support the predictions of the rational
speech-act model, showing not merely an interaction between
the speaker’s knowledge and the listener’s tendency to draw
an implicature, but a fine-grained interaction that is unlikely
to result from a modular process of language understanding.
In addition, these result support the standard, but controversial (Huang, Snedeker, & Spelke, 2004; Barner & Bachrach,2010), view that number words have a “lower-bound” semantics which is only strengthened into an exact meaning bypragmatic inference.

Quantitative model comparison
To evaluate the quantitative model predictions, we first compare model predictions with mean human ratings for the subset of trials in which participants gave knowledgeability ratings in the expected direction (greater than 70, as above). As
in the model description, we assume a binomial prior distribution. We fit the prior base rate parameter and the α parameter by minimizing the root mean squared error (RMSE) of
the model predictions and mean data for both experiments.
The resulting model fit is RMSE=9.01 and r=0.96. The
model predictions with the best-fitting parameters are shownin Fig. 2a,b and show striking correspondence with the humandata in Fig. 2c,d.
To consider all responses, including those previously removed due to unexpected knowledge judgements, we extend
the model by including an additional knowledge parameter:
the probability that the speaker did in fact have completeknowledge (regardless of perceptual access). Since we measured prior expectations and knowledgeability in each trial,
we compute model predictions, trial by trial, using these values. Because the betting interface encouraged participants
to round small bets to 0, but probability 0 is very different
than a non-zero small probability to the model, we changed 0
and 100 bets to 1 and 99. In addition, we assumed a simple
power-law relationship between subjective probability andbets (equivalent to the soft-max used to model the speaker,
Eq. 2). The parameter of this power law and the speaker optimality, α, were then fit by minimizing the RMSE of mean
model predictions to mean human judgements in both experiments. The resulting fit was again good: RMSE=8.36,
r=0.95. Looking at individual participants, we find mediancorrelation of r = 0.75 between a participant’s judgements
and the model predictions based on their prior and knowledge scores. These results suggest that the rational speech-act
model is able to capture the quantitative pattern of people’s
judgements both across the population and within individual
participants.

Conclusion
We have described a rational speech-act model of scalar implicatures and their interaction with speaker knowledge. Thismodel formalizes language understanding as social cognition,with language-specific goals and actions, using the tools ofBayesian statistics. In addition to predicting the standard implicatures (“some but not all”, etc) as an inference that depends on the alternative utterances, this model predicted thatthese implicatures could be canceled, completely or in part,when it was common knowledge that the speaker had incomplete knowledge. Expts. 1 and 2 verified these qualitative predictions, and showed tight quantitative fits with the model.
The predicted interaction between interpretation andspeaker’s knowledge was not a peculiarity of this set of words
or of scalar lexical items, instead it follows from the general fact that rational decision makers must choose actions
based on their expected utility. In contrast, a strongly modular of implicatures would predict no such interactions. One
could amend these theories (Chierchia et al., 2011) to allow a speaker’s ignorance to gate the implicature mechanism,
but this still would not predict the fine-grained interactions
demonstrated for number words: the details of a speaker’s
knowledge state influence a listener’s interpretation.
Our results support the rational speech-act framework formodeling pragmatics. More generally, they further boost the
momentum building for quantitative models of language asa branch of rational social cognition. In the words of Grice(1975): “One of [our] avowed aims is to see talking as a special case or variety of purposive, indeed rational, behavior...”Acknowledgments
We are grateful to Mike Frank, Josh Hartshorne, Danny Fox,and Irene Heim for comments on this work. This work was
supported by a John S. McDonnell Foundation Scholar Award
and ONR grant N00014-09-1-0124.

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