Probabilistic Semantics and Pragmatics:
Uncertainty in Language and Thought
Noah D. Goodman and Daniel Lassiter
Stanford University{ngoodman,danlassiter}@stanford.edu

Language is used to communicate ideas. Ideas are mental tools for coping witha complex and uncertain world. Thus human conceptual structures should be
key to language meaning, and probability—the mathematics of uncertainty—
should be indispensable for describing both language and thought. Indeed,probabilistic models are enormously useful in modeling human cognition (Tenenbaum et al., 2011) and aspects of natural language (Bod et al., 2003; Chater
et al., 2006). With a few early exceptions (e.g. Adams, 1975; Cohen, 1999b),
probabilistic tools have only recently been used in natural language semantics
and pragmatics. In this chapter we synthesize several of these modeling advances, exploring a formal model of interpretation grounded, via lexical semantics and pragmatic inference, in conceptual structure.
Flexible human cognition is derived in large part from our ability to imagine possibilities (or possible worlds). A rich set of concepts, intuitive theories,
and other mental representations support imagining and reasoning about possible worlds—together we will call these the conceptual lexicon. We posit that
this collection of concepts also forms the set of primitive elements available forlexical semantics: word meanings can be built from the pieces of conceptual
structure. Larger semantic structures are then built from word meanings bycomposition, ultimately resulting in a sentence meaning which is a phrase in
the “language of thought” provided by the conceptual lexicon. This expression is truth-functional in that it takes on a Boolean value for each imagined
world, and it can thus be used as the basis for belief updating. However,the connection between cognition, semantics, and belief is not direct: because
language must flexibly adapt to the context of communication, the connection between lexical representation and interpreted meaning is mediated bypragmatic inference.

A draft chapter for the Wiley-Blackwell Handbook of Contemporary Semantics —second edition, edited by Shalom Lappin and Chris Fox. This draft formatted on
25th June 2014.

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2Noah D. Goodman and Daniel Lassiter

There are a number of challenges to formalizing this view of language:
How can we formalize the conceptual lexicon to describe generation of possible
worlds? How can we appropriately connect lexical meaning to this conceptual
lexicon? How, within this system, do sentence meanings act as constraints on
possible worlds? How does composition within language relate to composition
within world knowledge? How does context affect meanings? How is pragmaticinterpretation related to literal meaning?
In this chapter we sketch an answer to these questions, illustrating theuse of probabilistic techniques in natural language pragmatics and semantics
with a concrete formal model. This model is not meant to exhaust the spaceof possible probabilistic models—indeed, many extensions are immediately
apparent—but rather to show that a probabilistic framework for natural language is possible and productive. Our approach is similar in spirit to cognitive semantics (Jackendoff, 1983; Lakoff, 1987; Cruse, 2000; Taylor, 2003), inthat we attempt to ground semantics in mental representation. However, we
draw on the highly successful tools of Bayesian cognitive science to formalize these ideas. Similarly, our approach draws heavily on the progress madein formal model-theoretic semantics (Lewis, 1970; Montague, 1973; Gamut,
1991; Heim & Kratzer, 1998; Steedman, 2001), borrowing insights about how
syntax drives semantic composition, but we compose elements of stochastic
logics rather than deterministic ones. Finally, like game-theoretic approaches(Benz et al., 2005; Franke, 2009), we place an emphasis on the the refinementof meaning through interactional, pragmatic reasoning.
In section 1 we provide background on probabilistic modeling and stochastic
λ-calculus, and introduce a running example scenario: the game of tug-of-war.
In section 2 we provide a model of literal interpretation of natural language
utterances and describe a formal fragment of English suitable for our runningscenario. Using this fragment we illustrate the emergence of non-monotonic
effects in interpretation and the interaction of ambiguity with backgroundknowledge. In section 3 we describe pragmatic interpretation of meaning asprobabilistic reasoning about an informative speaker, who reasons about aliteral listener. This extended notion of interpretation predicts a variety of
implicatures and connects to recent quantitative experimental results. In section 4 we discuss the role of semantic indices in this framework and show thatbinding these indices at the pragmatic level allows us to deal with several
issues in context-sensitivity of meaning, such as the interpretation of scalaradjectives. We conclude with general comments about the role of uncertainty
in pragmatics and semantics.

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1 Probabilistic models of commonsense reasoning
Uncertainty is a key property of the world we live in. Thus we should expect
reasoning with uncertainty to be a key operation of our cognition. At the sametime our world is built from a complex web of causal and other structures,
so we expect structure within our representations of uncertainty. Structured
knowledge of an uncertain world can be naturally captured by generativemodels, which make it possible to flexibly imagine (simulate) possible worlds
in proportion to their likelihood. In this section, we first introduce the basic
operations for dealing with uncertainty—degrees of belief and probabilistic
conditioning. We then introduce formal tools for adding compositional structure to these models—the stochastic λ-calculus—and demonstrate how thesetools let us build generative models of the world and capture commonsense
reasoning. In later sections, we demonstrate how these tools can be used toprovide new insights into issues in natural language semantics and pragmatics.
Probability is fundamentally a system for manipulating degrees of belief.
The probability1 of a proposition is simply a real number between 0 and
1 describing an agent’s degree of belief in that proposition. More generally,a probability distribution over a random variable A is an assignment of aprobability P (A=a) toPeach of a set of exhaustive and mutually exclusive
outcomes a, such that a P (A=a) = 1. The joint probability P (A=a, B=b),of two random variable values is the degree of belief we assign to the proposition that both A=a and B=b. From a joint probability distribution,
P (A=a, B=b),
P we can recover the marginal probability distribution on A:
P (A=a) = b P (A=a, B=b).
The fundamental operation for incorporating new information, or assumptions, into prior beliefs is probabilistic conditioning. This operation takes us
from the prior probability of A, P (A), to the posterior probability of A given
proposition B, written P (A|B). Conditional probability can be defined, following Kolmogorov (1933), by:
P (A|B) =P (A, B)
P (B)

(1)

This unassuming definition is the basis for much recent progress in modelinghuman reasoning (e.g. Oaksford & Chater, 2007; Griffiths et al., 2008; Chater& Oaksford, 2008; Tenenbaum et al., 2011). By modeling uncertain beliefs inprobabilistic terms, we can understand reasoning as probabilistic conditioning.
In particular, imagine a person who is trying to establish which hypothesisH ∈ {h1 , . . . , hm } best explains a situation, and does so on the basis of a1

In describing the mathematics of probabilities we will presume that we are dealing
with probabilities over discrete domains. Almost everything we say applies equallywell to probability densities, and more generally probability measures, but themathematics becomes more subtle in ways that would distract from our main
objectives.

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4Noah D. Goodman and Daniel Lassiterseries of observations {oi }N
i=1 . We can describe this inference as the conditional
probability:
P (H)P (o1 , . . . , oN |H).
(2)
P (H|o1 , . . . , oN ) =P (o1 , . . . , oN )This useful equality is called Bayes’ rule; it follows immediately from the definition in equation 1. If we additionally assume that the observations provide no
information about each other beyond what they provide about the hypothesis,that is they are conditionally independent, then P (oi |oj , H) = P (oi |H) for all
i 6= j. It follows that:
P (H|o1 , . . . , oN ) =P (H)P (o1 |H)···P (oN |H)
P (o1 )···P (oN |o1 ,...,oN −1 )(H)P (oP1 |H)···P (oN0|H) 0= P 0 P (o1 |H 0 )PP(H.
0 )···H
H 0 P (oN |H )P (H |o1 ,...,oN −1 )

(3)
(4)

From this it is a simple calculation to verify that we can perform the conditioning operation sequentially rather than all at once: the a posteriori degreeof belief given observations o1 , . . . , oi becomes the a priori degree of belief
for incorporating observation oi+1 . Thus, when we are justified in making thisconditional independence assumption, understanding the impact of a sequenceof observations reduces to understanding the impact of each one separately.
Later we will make use of this idea to reduce the meaning of a stream ofutterances to the meanings of the individual utterances.
1.1 Stochastic λ-Calculus and Church
Probability as described so far provides a notation for manipulating degrees
of belief, but requires that the underlying probability distributions be specified separately. Frequently we wish to describe complex knowledge involving
relations among many non-independent propositions or variables, and thisrequires describing complex joint distributions. We could write down a probability for each combination of variables directly, but this quickly becomes
unmanageable—for instance, a model with n binary variables requires 2n − 1
probabilities. The situation is parallel to deductive reasoning in classical logic
via truth tables (extensional models ascribing possibility to entire worlds),
which requires a table with 2n rows for a model with n atomic propositions;this is sound, but opaque and inefficient. Propositional logic provides structured means to construct and reason about knowledge, but is still too coarse
to capture many patterns of interest. First- and higher-order logics, such as
λ-calculus, provide a fine-grained language for describing and reasoning about(deterministic) knowledge. The stochastic λ-calculus (SLC) provides a formal,
compositional language for describing probabilities about complex sets of interrelated beliefs.
At its core SLC simply extends the (deterministic) λ-calculus (Barendregt,
1985; Hindley & Seldin, 1986) with an expression type (L⊕R), indicating random choice between the sub-expressions L and R, and an additional reduction
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rule that reduces such a choice expression to its left or right sub-expression
with equal probability. A sequence of standard and random-choice reductions
results in a new expression and some such expressions are in normal form
(i.e. irreducible in the same sense as in λ-calculus); unlike λ-calculus, the normal form is not unique. The reduction process can be viewed as a distribution
over reduction sequences, and the subset which terminate in a normal-form
expression induces a (sub-)distribution over normal-form expressions: SLC expressions denote (sub-)distributions over completely reduced SLC expressions.
It can be shown that this system can represent any computable distribution(see for example Ramsey & Pfeffer, 2002; Freer & Roy, 2012).
The SLC thus provides a fine-grained compositional system for specifying
probability distributions. We will use it as the core representational system forconceptual structure, for natural language meanings, and (at a meta-level) for
specifying the architecture of language understanding. However, while SLC issimple and universal, it can be cumbersome to work with directly. Goodman
et al. (2008a) introduce Church, an enriched SLC that can be realized as aprobabilistic programming language—parallel to the way that the programming language LISP is an enriched λ-calculus. In later sections we will use
Church to actually specify our models of language and thought. Church starts
with the pure subset of Scheme (which is itself essentially λ-calculus enrichedwith primitive data types, operators, and useful syntax) and extends it withelementary random primitives (ERPs), the inference function query, and thememoization function mem. We must take some time to describe these key, butsomewhat technical, pieces of Church before turning back to model construction. Further details and examples of using Church for cognitive modeling canbe found at http://probmods.org. In what follows we will assume passing
familiarity with the Polish notation used in LISP-family languages (fully parenthesized and operator initial), and will occasionally build on ideas from programming languages—Abelson & Sussman (1983) is an excellent backgroundon these ideas.
Rather than restricting to the ⊕ operation of uniform random choice(which is sufficient, but results in extremely cumbersome representations),
Church includes an interface for adding elementary random primitives (ERPs).
These are procedures that return random values; a sequence of evaluations ofsuch an ERP procedure is assumed to result in independent identically distributed (i.i.d.) values. Common ERPs include flip (i.e. Bernoulli), uniform,
and gaussian. While the ERPs themselves yield i.i.d. sequences, it is straightforward to construct Church procedures using ERPs that do not. For instance((λ (bias) (λ () (flip bias))) (uniform 0 1)) creates a function that “flips a coin”
of a specific but unknown bias. Multiple calls to the function will result in asequence of values which are not i.i.d., because they jointly depend on theunknown bias. This illustrates how more complex distributions can be built
by combining simple ones.
To represent conditional probabilities in SLC and Church we introduce
the query function. Unlike simpler representations (such as Bayes nets) where

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Noah D. Goodman and Daniel Lassiterconditioning is an operation that happens to a model from the outside, query
can be defined within the SLC itself as an ordinary function. One way to
do this is via rejection sampling. Imagine we have a distribution represented
by the function with no arguments thunk, and a predicate on return values
condition. We can represent the conditional distribution of return values fromthunk that satisfy condition by:( define conditional(λ ()( define val ( thunk ) )( if ( condition val ) val ( conditional ) ) ) )where we have used a stochastic recursion (conveniently specified by thenamed define) to build a conditional. Conceptually this recursion samples
from thunk until a value is returned that satisfies condition; it is straightforward
to show that the distribution over return values from this procedure is exactly
the ratio used to define conditional probability in equation 1 (when both aredefined). That is, the conditional procedure samples from the conditional distribution that could be notated P ((thunk)=val|(condition val)=True). For parsimony,
Church uses a special syntax, query, to specify such conditionals:( query... def inition s...
qexpr
condition )where ...definitions... is a list of definitions, qexpr is the expression of interest
whose value we want, and condition is a condition expression that must return
true. This syntax is internally transformed into a thunk and predicate thatcan be used in the rejection sampling procedure:( define thunk (λ () ... defin itions. .. ( list condition qexpr ) ) )( define predicate (λ ( val ) ( equal ? true ( first val ) ) ) )Rejection sampling can be taken as the definition of the query interface, but it
is very important to note that other implementations that approximate thesame distribution can be used and will often be more efficient. For instance, see
Wingate et al. (2011) for alternative implementations of query. In this chapter
we are concerned with the computational (or competence) level of description
and so need not worry about the implementation of query in any detail.
Memoization is a higher-order function that upgrades a stochastic function to have persistent randomness—a memoized function is evaluated fully
the first time it is called with given arguments, but thereafter returns this“stored” value. For instance (equal? (flip) (flip)) will be true with probability
0.5, but if we define a memoized flip, (define memflip (mem flip)), then (equal?
(memflip) (memflip)) will always be true. This property is convenient for representing probabilistic dependencies between beliefs that rely on common properties, for instance the strengths and genders of people in a game (as illustrated
below). For instance, memoizing a function gender which maps individuals totheir gender will ensure that gender is a stable property, even if it is not known

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in advance what a given individual’s gender is (or, in effect, which possible
world is actual).2
In Church, as in most LISP-like languages, source code is a first-class data
type: it is represented by lists. The quote operator tells the evaluation process
to treat a list as a literal list of symbols, rather than evaluating it: (flip) results
in a random value true or false, while '(flip) results in the list (flip) as a value.
For us this will be important because we can “reverse” the process by callingthe eval function on a piece of reified code. For instance, (eval '(flip)) results ina random value true or false again. Usefully for us, evaluation triggered by evalhappens in the local context with any bound variables in scope. For instance:( define expression '( flip bias ) )( define foo ((λ ( bias ) (λ ( e ) ( eval e ) ) ) ( uniform 0 1) ) )( foo expression )In this snippet the variable bias is not in scope at the top level where expression
is defined, but it is in scope where expression is evaluated, inside the function
bound to foo. For the natural language architecture described below this allows
utterances to be evaluated in the local context of comprehension. For powerfulapplications of these ideas in natural language semantics see Shan (2010).
Church is a dynamically typed language: values have types, but expressions don’t have fixed types that can be determined a priori. One consequenceof dynamic typing for a probabilistic language is that expressions may take ona distribution of different types. For instance, the expression (if (flip) 1 true)
will be an integer half the time and Boolean the other half. This has interesting implications for natural language, where we require consistent dynamic
types but have no particular reason to require deterministically assigned static
types. For simplicity (and utility below) we assume that when an operator isapplied to values outside of its domain, for instance (+ 1 'a), it returns a special value error which is itself outside the domain of all operators, except theequality operator eq?. By allowing eq? to test for error we permit very simple
error handling, and allow query (which relies on a simple equality test to decide
whether to “keep going”) to filter out mis-typed sub-computations.
1.2 Commonsense knowledge
In this chapter we use sets of stochastic functions in Church to specify theintuitive knowledge—or theory—that a person has about the world. To illustrate this idea we now describe an example, the tug-of-war game, which we will
use later in the chapter as the non-linguistic conceptual basis of a semantics
2

A technical, but important, subtlety concerns the “location” where a memoized
random choice is created: should it be at the first use, the second, ...? In order toavoid an artificial symmetry breaking (and for technical reasons), the semantics
of memoization is defined so that all random values that may be returned by amemoized function are created when the memoized function is created, not where
it is called.

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8Noah D. Goodman and Daniel Lassiterand pragmatics for a small fragment of English. Tug-of-war is a simple gamein which two teams pull on either side of a rope; the team that pulls hardest
will win. Our intuitive knowledge of this domain (and indeed most similar
team games) rests on a set of interrelated concepts: players, teams, strength,
matches, winners, etc. We now sketch a simple realization of these concepts
in Church. To start, each player has some traits, strength and gender, that
may influence each other and his or her contribution to the game.
( define gender ( mem (λ ( p ) ( if ( flip ) ' male ' female ) ) ) )( define g e n d e r - m e a n - s t r e n g t h ( mem (λ ( g ) ( gaussian 0 2) ) ) )( define strength( mem (λ ( p ) ( gaussian ( g e n d e r - m e a n - s t r e n g t h ( gender p ) ) 1) ) ) )We have defined the strength of a person as a mixture model : strength depends
on a latent class, gender, through the (a priori unknown) gender means. Notethat we are able to describe the properties of people (strength, gender) without
needing to specify the people—instead we assume that each person is represented by a unique symbol, using memoized functions from these symbols toproperties to create the properties of a person only when needed (but then holdthose properties persistently). In particular, the person argument, p, is neverused in the function gender, but it matters because the function is memoized—agender will be persistently associated to each person even though the distribution of genders doesn’t depend on the person. We will exploit this patternoften below. We are now already in a position to make useful inferences. We
could, for instance observe the strengths and genders of several players, andthen Pat’s strength but not gender, and ask for the latter:( query( define gender ( mem (λ ( p ) ( if ( flip ) ' male ' female ) ) ) )( define g e n d e r - m e a n - s t r e n g t h ( mem (λ ( g ) ( gaussian 0 2) ) ) )( define strength( mem (λ ( p ) ( gaussian ( g e n d e r - m e a n - s t r e n g t h ( gender p ) ) 1) ) ) )( gender ' Pat )( and ( equal ? ( gender ' Bob ) ' male ) (= ( strength ' Bob ) -1.1)( equal ? ( gender ' Jane ) ' female ) (= ( strength ' Jane ) 0.5)
( equal ? ( gender ' Jim ) ' male ) (= ( strength ' Jim ) -0.3)(= ( strength ' Pat ) 0.7) ) )The result of this query is that Pat is more likely to be female than male(probability .63). This is because the observed males are weaker than Jane,the observed female, and so a strong player such as Pat is likely to be female
as well.
In the game of tug-of-war players are on teams:( define players '( Bob Jim Mary Sue Bill Evan Sally Tim Pat Jane Dan Kate ) )( define teams '( team1 team2 ... team10 ) )( define team-size ( uniform-draw '(1 2 3 4 5 6) ) )( define pl ay er s -o n- te a m ( mem (λ ( team ) ( draw-n team-size players ) ) ) )Here the draw-n ERP draws uniformly but without replacement from a list.
(For simplicity we draw players on each team independently, allowing players
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to potentially be on multiple teams.) In addition to players and teams, wehave matches: events that have two teams and a winner. The winner depends
on how hard each team is pulling, which depends on how hard each teammember is pulling.
( define tea ms-in-ma tch ( mem (λ ( match ) ( draw-n 2 teams ) ) ) )( define p l a y e r s - i n - m a t c h (λ ( match ) ( apply append ( map pl ay er s -o n- te a m( te ams-in- match match ) ) ) ) )( define pulling ( mem (λ ( player match )(+ ( strength player ) ( gaussian 0 0.5) ) ) ) )( define team-pulling ( mem (λ ( team match )( sum ( map (λ ( p ) ( pulling p match ) ) ( p la ye rs - on -t e am team ) ) ) ) ) )( define ( winner match )( define teamA ( first ( teams- in-matc h match ) ) )( define teamB ( second ( team s-in-mat ch match ) ) )( if ( > ( team-pulling teamA ) ( team-pulling teamB ) ) teamA teamB ) )Notice that the team pulling is simply the sum of how hard each member ispulling; each player pulls with their intrinsic strength, plus or minus a randomamount that indicates their effort on this match.

( define( define( define( define

players '( Bob Jim Mary Sue Bill Evan Sally Tim Pat Jane Dan Kate ) )teams '( team1 team2 ... team10 ) )matches '( match1 match2 match3 match4 ) )individuals ( append players teams matches ) )( define gender ( mem (λ ( p ) ( if ( flip ) ' male ' female ) ) ) )( define g e n d e r - m e a n - s t r e n g t h ( mem (λ ( g ) ( gaussian 0 2) ) ) )( define strength ( mem (λ ( p ) ( gaussian ( g e n d e r - m e a n - s t r e n g t h ( gender p ) )1) ) ) )( define team-size ( uniform-draw '(1 2 3 4 5 6) ) )( define pl ay er s -o n- te a m ( mem (λ ( team ) ( draw-n team-size players ) ) ) )( define tea ms-in-ma tch ( mem (λ ( match ) ( draw-n 2 teams ) ) ) )( define p l a y e r s - i n - m a t c h (λ ( match ) ( apply append ( map pl ay er s -o n- te a m( te ams-in- match match ) ) ) ) )( define pulling ( mem (λ ( player match ) (+ ( strength player ) ( gaussian 00.5) ) ) ) )( define team-pulling ( mem (λ ( team match )( sum ( map (λ ( p ) ( pulling p match ) ) ( p la ye rs - on -t e am team ) ) ) ) ) )( define ( winner match )( let ([ teamA ( first ( teams-in -match match ) ) ][ teamB ( second ( team s-in-ma tch match ) ) ])( if ( > ( team-pulling teamA match ) ( team-pulling teamB match ) )teamA
teamB ) ) )Figure 1. The collected Church definitions forming our simple intuitive theory (orconceptual lexicon) for the tug-of-war domain.

The intuitive theory, or conceptual lexicon of functions, for the tug-of-war
domain is given altogether in Figure 1. A conceptual lexicon like this one
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Noah D. Goodman and Daniel Lassiter

to be. We can use this knowledge to reason from observations to predictions
or latent states by conditioning (i.e. query). Let us illustrate how a generativemodel is used to capture key patterns of reasoning. Imagine that Jane isplaying Bob in match 1; we can infer Jane’s strength before observing theoutcome of this match:( query... ToW theory...
( strength ' Jane ) ; ; variable of interest( and ; ; conditioning expression( equal ? ( pl ay e rs -o n -t ea m ' team1 ) '( Jane ) )( equal ? ( pl ay e rs -o n -t ea m ' team2 ) '( Bob ) )( equal ? ( te ams-in- match ' match1 ) '( team1 team2 ) ) ) )In this and all that follows ...ToW theory... is an abbreviation for the definitionsin Figure 1. The result of this inference is simply the prior belief about Jane’s
strength: a distribution with mean 0 (Figure 2). Now imagine that Jane wins
this match:( query... ToW theory...
( strength ' Jane ) ; ; variable of interest( and ; ; conditioning expression( equal ? ( pl ay e rs -o n -t ea m ' team1 ) '( Jane ) )( equal ? ( pl ay e rs -o n -t ea m ' team2 ) '( Bob ) )( equal ? ( te ams-in- match ' match1 ) '( team1 team2 ) )( equal ? ( winner ' match1 ) ' team1 ) ) )If we evaluate this query we find that Jane is inferred to be relatively strong:
her mean strength after observing this match is around 0.7, higher than hera priori mean strength of 0.0.

Figure 2. An example of explaining away. Lines show the distribution on Jane’s
inferred strength after (a) no observations; (b) observing that Jane beat Bob, whosestrength is unknown; (c) learning that Bob is very weak, with strength -8. (d)learning that Jane and Bob are different genders

However, imagine that we then learned that Bob is a weak player:( query

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11... ToW theory...
( strength ' Jane ) ; ; variable of interest( and ; ; conditioning expression( equal ? ( pl ay e rs -o n -t ea m ' team1 ) '( Jane ) )( equal ? ( pl ay e rs -o n -t ea m ' team2 ) '( Bob ) )( equal ? ( te ams-in- match ' match1 ) '( team1 team2 ) )( equal ? ( winner ' match1 ) ' team1 )(= ( strength ' Bob ) -8.0) ) )This additional evidence has a complex effect: we know that Bob is weak, andthis provides evidence that the mean strength of his gender is low; if Jane is
the same gender, she is also likely weak, though stronger than Bob, who shebeat; if Jane is of the other gender, then we gain little information about her.
The distribution over Jane’s strength is bimodal because of the uncertainty
about whether she has the same gender as Bob. If we knew that Jane and
Bob were of different genders then information about the strength of Bob’s
gender would not affect our estimate about Jane:( query... ToW theory...
( strength ' Jane ) ; ; variable of interest( and ; ; conditioning expression( equal ? ( pl ay e rs -o n -t ea m ' team1 ) '( Jane ) )( equal ? ( pl ay e rs -o n -t ea m ' team2 ) '( Bob ) )( equal ? ( te ams-in- match ' match1 ) '( team1 team2 ) )( equal ? ( winner ' match1 ) ' team1 )(= ( strength ' Bob ) -8.0)( equal ? ( gender ' Bob ) ' male )( equal ? ( gender ' Jane ) ' female ) ) )Now we have very little evidence about Jane’s strength: the inferred mean
strength from this query goes back to (almost) 0, because we gain no information via gender mean strengths, and Jane beating Bob provides little
information given that Bob is very weak. This is an example of explaining
away (Pearl, 1988): the assumption that Bob is weak has explained the observation that Jane beat Bob, which otherwise would have provided evidence
that Jane is strong. Explaining away is characterized by a priori independent variables (such as Jane and Bob’s strengths) becoming coupled together
by an observation (such as the outcome of match 1). Another way of saying
this is that our knowledge of the world, the generative model, can have a significant amount of modularity; our inferences after making observations will
generally not be modular in this way. Instead, complex patterns of influencecan couple together disparate pieces of the model. In the above example wealso have an example of screening off : the observation that Bob and Janeare of different genders renders information about Bob’s (gender’s) strength
uninformative about Jane’s. Screening off describes the situation when two
variables that were a priori dependent become independent after an observation (in some sense the opposite of explaining away). Notice that in thisexample we have gone through a non-monotonic reasoning sequence: Our degree of belief that Jane is strong went up from the first piece of evidence,
down below the prior from the second, and then back up from the third.

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Noah D. Goodman and Daniel Lassiter

Such complex, non-monotonic patterns of reasoning are extremely common
in probabilistic inference over structured models.
There are a number of other patterns of reasoning that are common results of probabilistic inference over structured models, including Occam’s razor(complexity of hypotheses is automatically penalized), transfer learning (aninductive bias learned from one domain constrains interpretation of evidence
in a new domain), and the blessing of abstraction (abstract knowledge can belearned faster than concrete knowledge). These will be less important in what
follows, but we note that they are potentially important for the question of language learning—when we view learning as an inference, the dynamics of probabilistic inference come to bear on the learning problem. For detailed examplesof these patterns, using Church representation, see http://probmods.org.
1.3 Possible worlds
We have illustrated how a collection of Church functions—an intuitive theory—
describes knowledge about the world. In fact, an intuitive theory can be interpreted as describing a probability distribution over possible worlds. To see
this, first assume that all the (stochastic) functions of the intuitive theory
are memoized.3 Then the value of any expression is determined by the values of those functions called (on corresponding inputs) while evaluating theexpression; any expression is assigned a value if we have the values of all thefunctions on all possible inputs. A possible world then, can be represented by
a complete assignment of values to function-argument pairs, and a distribution over worlds is defined by the return-value probabilities of the functions,as specified by the intuitive theory.
We do not need to actually compute the values of all function-argument
pairs in order to evaluate a specific expression, though. Most evaluations will
involve just a fraction of the potentially infinite number of assignments needed
to make a complete world. Instead, Church evaluation constructs only a partial
representation of a possible world containing the minimal information needed
to evaluate a given expression: the values of function applications that are
actually reached during evaluation. Such a “partial world” can be interpreted
as a set of possible worlds, and its probability is the sum of the probabilitiesof the worlds in this set. Fortunately this intractable sum is equal to the
product of the probabilities of the choices made to determine the partial world:
the partial world is independent of any function values not reached during
evaluation, hence marginalizing these values is the same as ignoring them.
In this way, we can represent a distribution over all possible worlds implicitly, while explicitly constructing only partial worlds large enough to be
relevant to a given query, ignoring irrelevant random values. The fact that3

If not all stochastic functions are memoized, very similar reasoning goes through:now each function is associated with an infinite number of return values, individuated by call order or position.

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infinite sets of possible worlds are involved in a possible worlds semantics
has sometimes been considered a barrier to the psychological plausibility ofthis approach. Implementing a possible worlds semantics via a probabilistic
programming language may help defuse this concern: a small, finite subsetof random choices will be constructed to reason about most queries; the remaining infinitude, while mathematically present, can be ignored because thequery is statistically independent of them.

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2 Meaning as condition
Following a productive tradition in semantics (Stalnaker, 1978; Lewis, 1979;
Heim, 1982, etc.), we view the basic function of language understanding as
belief update: moving from a prior belief distribution over worlds (or situations) to a posterior belief distribution given the literal meaning of a sentence. Probabilistic conditioning (or query) is a very general way to describe
updating of degrees of belief. Any transition from distribution Pbefore to distribution Pafter can be written as multiplying by a non-negative, real-valued
function and then renormalizing, provided Pbefore is non-zero whenever Pafter
is.4 From this observation it is easy to show that any belief update which
preserves impossibility can be written as the result of conditioning on some(stochastic) predicate. Note that conditioning in this way is the natural analogue of the conception of belief update as intersection familiar from dynamic
semantics.
Assume for now that each sentence provides information which is logicallyindependent of other sentences given the state of the world (which may include
discourse properties). From this it follows, parallel to the discussion of multiple
observations as sequential conditioning above, that a sequence of sentences canbe treated as sequentially updating beliefs by conditioning—so we can focus
on the literal meaning of a single sentence. This independence assumption
can be seen as the most basic and important compositionality assumption,
which allows language understanding to proceed incrementally by utterance.
(When we add pragmatic inference, in section 3, this independence assumption
will be weakened, but it remains essential to the basic semantic function ofutterances.)
How does an utterance specify which belief update to perform? We formalize the literal listener as:( define ( l i t e r a l - l i s t e n e r utterance QUD )( query... theory...
( eval QUD )( eval ( meaning utterance ) ) ) )This function specifies the posterior distribution over answers to the Question Under Discussion (QUD) given that the literal meaning of the utterance istrue.5 Notice that the prior distribution for the literal listener is specified by aconceptual lexicon—the ...theory...—and the QUD will be evaluated in the local
environment where all functions defined by this theory are in scope. That is,4

For infinite spaces we would need a more general condition on the measurabilityof the belief update.
5
QUD theories have considerable motivation in semantics and pragmatics: seeGinzburg 1995; Van Kuppevelt 1995; Roberts 2012; Beaver & Clark 2008 among
many others. For us, the key feature of the QUD is that it denotes a partition of W
that is naturally interpreted as the random variable of immediate interest in theconversation.

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the question of interest is determined by the expression QUD while its answer isdetermined by the value of this expression in the local context of reasoning by
the literal listener: the value of (eval QUD). (For a description of the eval operator
see section 1.1 above.) Hence the semantic effect of an utterance is a function
from QUDs to posteriors, rather than directly a posterior over worlds. Using the
QUD in this way has two beneficial consequences. First, it limits the holism ofbelief update, triggering representation of only the information that is needed
to capture the information conveyed by a sentence about the question of current interest. Second, when we construct a speaker model the QUD will be used
to capture a pressure to be informative about the topic of current interest, as
opposed to global informativity about potentially irrelevant topics.
2.1 Composition
The meaning function is a stochastic mapping from strings (surface forms) to
Church expressions (logical forms, which may include functions defined in...theory...). Many theories of syntactic and semantic composition could be
used to provide this mapping. For concreteness, we consider a simple system
in which a string is recursively split into left and right portions, and themeanings of these portions are combined with a random combinator. The
first step is to check whether the utterance is syntactically atomic, and if solook it up in the lexicon:( define ( meaning utterance )( if ( lexical-item ? utterance )( lexicon utterance )( compose utterance ) ) )Here the predicate lexical-item? determines if the (remaining) utterance is asingle lexical item (entry in the lexicon), if so it is looked up with the lexicon
function. This provides the base case for the recursion in the compose function,
which randomly splits non-atomic strings, computes their meanings, and combines them into a list:( define ( compose utterance )( define subs ( random-split utterance ) )( list ( meaning ( first subs ) ) ( meaning ( second subs ) ) ) )The function random-split takes a string and returns the list of two substrings
that result from splitting at a random position in the length of the string.6
Overall, the meaning function is a stochastic mapping from strings to Church
expressions. In literal-listener we eval the representation constructed by meaning
6

While it is beyond the scope of this chapter, a sufficient syntactic system would
require language-specific biases that favor certain splits or compositions on nonsemantic grounds. For instance, lexical items and type shifters could be augmented with word-order restrictions, and conditioning on sentence meaning could be
extended to enforce syntactic well-formedness as well (along the lines of Steedman
2001). Here we will assume that such a system is in place and proceed to compute
sample derivations.

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Noah D. Goodman and Daniel Lassiterin the same environment as the QUD. Because we have formed a list of the submeanings, evaluation will result in forward application of the left sub-meaning
to the right. Many different meanings can get constructed and evaluated in thisway, and many of them will be mis-typed. Critically, if type errors are interpreted as the non-true value error (as described in section 1.1), then mis-typed
compositions will not satisfy the condition of the query in the literal-listener
function—though many ill-typed compositions can be generated by meaning,they will be eliminated from the posterior, leaving only well-typed interpretations.
To understand what the literal-listener does overall, consider rejection
sampling: we evaluate both the QUD and meaning expressions, constructing
whatever intermediate expressions are required; if the meaning expression has
value true, then we return the value of QUD, otherwise we try again. Random
choices made to construct and evaluate the meaning will be reasoned about
jointly with world states while interpreting the utterance; the complexity ofinterpretation is thus an interaction between the domain theory, the meaning
function, and the lexicon.
2.2 Random type shifting
The above definition for meaning always results in composition by forward application. This is too limited to generate potential meanings for many sentences. For instance “Bob runs” requires a backward application to apply
the meaning of “runs” to that of “Bob”. We extend the possible composition
methods by allowing the insertion of type-shifting operators.
( define ( meaning utterance )( if ( lexical-item ? utterance )( lexicon utterance )( shift ( compose utterance ) ) ) )( define ( shift m )( if ( flip )m( list ( uniform-draw type-shifters ) ( shift m ) ) ) )( define type-shifters '( L G AR1 AR2 ...) )Each intermediate meaning will be shifted zero or more times by a randomly
chosen type-shifter; because the number of shifts is determined by a stochastic
recursion, fewer shifts are a priori more likely. Each lexical item thus has thepotential to be interpreted in any of an infinite number of (static) types,
but the probability of associating an item with an interpretation in some typedeclines exponentially with the the number of type-raising operations required
to construct this interpretation. The use of a stochastic recursion to generate
type ambiguities thus automatically enforces the preference for interpretation
in lower types, a feature which is often stipulated in discussions of type-shifting(Partee & Rooth, 1983; Partee, 1987).
We choose a small set of type shifters which is sufficient for the examplesof this chapter:
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•
•
•

17

L: (λ (x) (λ (y) (y x)))G: (λ (x) (λ (y) (λ (z) (x (y z)))))
AR1: (λ (f) (λ (x) (λ (y) (x (λ (z) ((f z) y))))))
AR2: (λ (f) (λ (x) (λ (y) (y (λ (z) ((f x) z))))))

Among other ways they can be used, the shifter L enables backward application and G enables forward composition. For instance, Bob runs has anadditional possible meaning ((L 'Bob) runs) which applies the meanings of runs
to that of Bob, as required.
Type shifters AR1 and AR2 allow flexible quantifier scope as described in
Hendriks (1993); Barker (2005). (The specific formulation here follows Barker,
2005, pp.453ff.) We explore the ramifications of the different possible scopes
in section 2.5. This treatment of quantifier scope is convenient, but others
could be implemented by complicating the syntactic or semantic mechanisms
in various ways: see e.g. May (1977); Steedman (2012).
2.3 Interpreting English in Church: the Lexicon
Natural language utterances are interpreted as Church expressions by themeaning function. The stochastic λ-calculus (implemented in Church) thus functions as our intermediate language, just as the ordinary, simply-typed λcalculus functions as an intermediate translation language in the fragmentof English given by Montague (1973). A key difference, however, is that theintermediate level is not merely a convenience as in Montague’s approach.
Conceptual representations and world knowledge are also represented in thislanguage as Church function definitions. The use of a common language torepresent linguistic and non-linguistic information allows lexical semantics tobe grounded in conceptual structure, leading to intricate interactions between
these two types of knowledge. In this section we continue our running tugof-war example, now specifying a lexicon mapping english words to Church
expressions for communicating about this domain.
We abbreviate the denotations of expressions (meaning α) as [[α]]. The
simplest case is the interpretation of a name as a Church symbol, which serves
as the unique mental token for some object or individual (the name-bearer).
••••[[Bob]]: 'Bob[[Team 1 ]]: 'team1[[Match 1 ]]: 'match1...

Interpreted in this way names are directly referential since they are interpretedusing the same symbol in every situation, regardless of inferences made during
interpretation.
A one-place predicate such as player or man is interpreted as a function
from individuals to truth-values. Note that these denotations are groundedin aspects of the non-linguistic conceptual model, such as players, matches, andgender.
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Noah D. Goodman and Daniel Lassiter

•
•
•
•
•

[[player ]]: (λ (x) (element? x players))[[team]]: (λ (x) (element? x teams))[[match]]: (λ (x) (element? x matches))[[man]]: (λ (x) (equal? (gender x) 'male))[[woman]]: (λ (x) (equal? (gender x) 'female))

Similarly, transitive verbs such as won denote two-place predicates. (We simplify throughout by ignoring tense.)••[[won]]: (λ (match) (λ (x) (equal? x (winner match))))[[played in]]: (λ (match) (λ (x) (or (element? x (teams-in-match match)) (element? x (players-in-match
match)))))•[[is on]]: (λ (team) (λ (x) (element? x (players-on-team team))))Intensionality is implicit in these definitions because the denotations ofEnglish expressions can refer to stochastic functions in the intuitive theory.
Thus predicates pick out functions from individuals to truth-values in any
world, but the specific function that they pick out in a world can depend on
random choices (e.g., values of flip) that are made in the process of constructing the world. For instance, player is true of the same individuals in every
world, because players is a fixed list (see Figure 1) and element? is the deterministic membership function. On the other hand, man denotes a predicate which
will be a priori true of a given individual (say, 'Bob) in 50% of worlds—because
the memoized stochastic function gender returns 'male 50% of the time when it
is called with a new argument.
For simplicity, in the few places in our examples where plurals are required,
we treat them as denoting lists of individuals. In particular, in a phrase like
Team 1 and Team 2, the conjunction of NPs forms a list:•[[and ]] = (λ (x) (λ (y) (list x y)))Compare this to the set-based account of plurals described in Scha & Winter2014 (this volume). To allow distributive properties (those which require
atomic individuals as arguments) to apply to such collections we include a
type-shifting operator (in type-shifters, see section 2.2) that universally quantifies the property over the list:•

DIST: (λ (V) (λ (s) (all (map V s))))

For instance, Bob and Jim played in Match 1 can be interpreted by shifting
the property [[played in Match 1 ]] to a predicate on lists (though the order of
elements in the list will not matter).
We can generally adopt standard meanings for functional vocabulary, suchas quantifiers.
••••[[every]]: (λ (P) (λ (Q) (= (size P) (size (intersect P Q)))))[[some]]: (λ (P) (λ (Q) (< 0 (size (intersect P Q)))))[[no]]: (λ (P) (λ (Q) (= 0 (size (intersect P Q)))))[[most]]: (λ (P) (λ (Q) (< (size P) (* 2 (size (intersect P Q))))))

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For simplicity we have written the quantifiers in terms of set size; the sizefunction can be defined in terms of the domain of individuals as (λ (S) (length
7(filter S individuals))).
We treat gradable adjectives as denoting functions from individuals todegrees (Bartsch & Vennemann, 1973; Kennedy, 1997, 2007). Antonym pairs
such as weak/strong are related by scale reversal.
••[[strong]]: (λ (x) (strength x))[[weak ]]: (λ (x) (- 0 (strength x)))This denotation will require an operator to bind the degree in any sentenceinterpretation. In the case of the relative and superlative forms this operator
will be indicated by the corresponding morpheme. For instance, the superlative morpheme -est is defined so that strongest player will denote a property
that is true of an individual when that individual’s strength is equal to the
maximum strength of all players:8•[[-est]]: (λ (A) (λ (N) (λ (x) (= (A x) (max-prop A N)))))

For positive form sentences, such as Bob is strong, we will employ a type
shifting operator which introduces a degree threshold to bind the degree—seesection 4.
2.4 Example interpretations
To illustrate how a (literal) listener interprets a sequence of utterances, weconsider a variant of our explaining-away example from the previous section.
For each of the following utterances we give one expression that could be
returned from meaning (usually the simplest well-typed one); we also show eachmeaning after simplifying the compositions.
•••Utterance 1: Jane is on Team 1.
meaning: ((L 'Jane) (λ (team) (λ (x) (element? x (players-on-team team))) 'team1))
simplified: (element? 'Jane (players-on-team 'team1))
Utterance 2: Bob is on Team 2.
meaning: ((L 'Bob) (λ (team) (λ (x) (element? x (players-on-team team))) 'team2))
simplified: (element? 'Bob (players-on-team 'team2))
Utterance 3: Team 1 and Team 2 played in Match 1.
meaning: ((L ((L 'team 1) ((λ (x) (λ (y) (list x y))) 'team2))) (DIST ((λ (match) (λ(x) (element? x (teams-in-match match)))) 'match1)))

simplified: (all (map (λ (x) (element? x (teams-in-match 'match1)))) '(team1 team2))
7

In the examples below, we assume for simplicity that many function words, forexample is and the, are semantically vacuous, i.e., that they denote identity functions.
8
The set operator max-prop implicitly quantifies over the domain of discourse, similarly to size. It can be defined as (lambda (A N) (max (map A (filter N individuals)))).

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Noah D. Goodman and Daniel Lassiter

Utterance 4: Team 1 won Match 1.
meaning: ((L 'team1) ((λ (match) (λ (x) (equal? x (winner match)))) 'match1))
simplified: (equal? 'team1 (winner 'match1))

The literal listener conditions on each of these meanings in turn, updating
her posterior belief distribution. In the absence of pragmatic reasoning (seebelow), this is equivalent to conditioning on the conjunction of the meaningsof each utterance—essentially as in dynamic semantics (Heim, 1992; Veltman,
1996). Jane’s inferred strength (i.e. the posterior on (strength 'Jane)) increases
substantially relative to the uninformed prior (see Figure 3).
Suppose, however, the speaker continues with the utterance:•

Utterance 5: Bob is the weakest player.
meaning: ((L 'Bob) (((L (λ (x) (- (strength x)))) (λ (A) (λ (N) (λ (x) (= (A x) (max-prop
A N)))))) (λ (x) (element? x players))))simplified: (= (- (strength 'Bob)) (max (λ (x) (- (strength x))) (λ (x) (element?
x players))))This expression will be true if and only if Bob’s strength is the smallest of any
player. Conditioning on this proposition about Bob, we find that the inferred
distribution of Jane’s strength decreases toward the prior (see Figure 3)—
Jane’s performance is explained away. Note, however, that this non-monotonic
effect comes about not by directly observing a low value for the strength of
Bob and information about his gender, as in our earlier example, but by conditioning on the truth of an utterance which does not entail any precise valueof Bob’s strength. That is, because there is uncertainty about the strengthsof all players, in principle Bob could be the weakest player even if he is quite
strong, as long as all the other players are strong as well. However, the otherplayers are most likely to be about average strength, and hence Bob is particularly weak; conditioning on Utterance 5 thus lowers Bob’s expected strengthand adjusts Jane’s strength accordingly.
2.5 Ambiguity
The meaning function is stochastic, and will often associate utterances withseveral well-typed meanings. Ambiguities can arise due to any of the following:••

Syntactic: random-split can generate different syntactic structures for an utterance. If more than one of these structures is interpretable (using thetype-shifting operators available), the literal listener will entertain interpretations with different syntactic structures.
Compositional: Holding the syntactic structure fixed, insertion of different(and different numbers of) type-shifting operators by shift may lead towell-typed outputs. This can lead, for example, to ambiguities of quantifierscope and in whether a pronoun is bound or free.

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Figure 3. A linguistic example of explaining away, demonstrating that the literal
listener makes non-monotonic inferences about the answer to the QUD “How strong
is Jane?” given the utterances described in the main text. Lines show the probability
density of answers to this QUD after (a) utterances 1-3; (b) utterances 1-4; (c)utterances 1-5.

•

Lexical: the lexicon function may be stochastic, returning different options
for a single item, or words may have intrinsically stochastic meanings. (Theformer can always be converted to the latter.)In the literal interpretation model we have given above, literal-listener, thesesources of linguistic ambiguity will interact with the interpreter’s beliefs about
the world. That is, the query implies a joint inference of sentence meaning andworld, given that the meaning is true of the world. When a sentence is ambiguous in any of the above ways, the listener will favor plausible interpretations
over implausible ones, because the interpreter’s model of the world is more
likely to generate scenarios which make the sentence true.
For example, consider the utterance “Most players played in some match”.
Two (simplest, well-typed) interpretations are possible. We give an intuitive
paraphrase and the meanings for each (leaving the leaving lexical items inplace to expose the compositional structure):••

Subject wide scope:“For most players x, there was a match y such that x played in y.”((L ([[Most]] [[players]])) ((AR2 (AR1 [[played in]])) ([[some]] [[match]])))
Object wide scope:“For some match y, most players played in y.”((L ([[Most]] [[players]])) ((AR1 (AR2 [[played in]])) ([[some]] [[match]])))

Both readings equally a priori probable, since the meaning function draws typeshifters uniformly at random. However, if one reading is more likely to be true,
given background knowledge, it will be preferred. This means that we can
influence the meaning used, and the degree to which each meaning influences
the listener’s posterior beliefs, by manipulating relevant world knowledge.
To illustrate the effect of background knowledge on choice of meaning,
imagine varying the number of matches played in our tug-of-war example.
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Noah D. Goodman and Daniel Lassiter

Recall (see Figure 1) that all teams are of size team-size, which varies acrossworlds and can be anywhere from 1 to 6 players, with equal probability. If thenumber of matches is large (say we (define matches '(match1 ... match10))), then thesubject-wide scope reading can be true even if team-size is small: it could easily
happen that most players played in one or another of ten matches even if eachteam has only one or two players. In contrast, the object-wide scope reading,which requires most players on a single match, can be true only if teams arelarge enough (i.e. team-size is ≥ 4, so that more than half of the players arein each match). The literal-listener jointly infers team-size and the reading of
the utterance, assuming the utterance is true; because of the asymmetry inwhen the two readings will be true, there will be a preference for the subjectwide reading if the number of matches is large—it is more often true. If thenumber of matches is small, however, the asymmetry between readings will
be decreased. Suppose that only one match was played (i.e. (define matches'(match1))), then both readings can be true only if the team size is large. The
listener will thus infer that team-size≥ 4 and the two readings of the utterance
are equally probable. Figure 4, left panel, shows the strength of each reading
as the number of matches varies from 1 to 10, with the number of teams fixed
to 10. The right panel shows the mean inferred team size as the number ofmatches varies, for each reading and for the marginal. Our model of language
understanding as joint inference thus predicts that the resolution of quantifierscope ambiguities will be highly sensitive to background information.

Figure 4. The probability of the listener interpreting the utterance Most players
played in some match according to the two possible quantifier scope configurations
depends in intricate ways on the interpreter’s beliefs and observations about the
number of matches and the number of players on each team (left). This, in turn, influences the total information conveyed by the utterance (right). For this simulation
there were 10 teams.

More generally, an ambiguous utterance may be resolved differently, andlead to rather different belief update effects, depending on the plausibility ofthe various interpretations given background knowledge. Psycholinguistic re-Page: 22

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search suggests that background information has exactly this kind of graded
effect on ambiguity resolution (see, for example, Crain & Steedman, 1985; Altmann & Steedman, 1988; Spivey et al., 2002). In a probabilistic framework,
preferences over alternative interpretations vary continuously between the extremes of assigning equal probability to multiple interpretations and assigningprobability 1 to a single interpretation. This is true whether the ambiguity is
syntactic, compositional, or lexical in origin.
2.6 Compositionality
It should be clear that compositionality has played a key role in our model oflanguage interpretation thus far. It has in fact played several key roles: Church
expressions are built from simpler expressions, sequences of utterances areinterpreted by sequential conditioning, the meaning function composes Church
expressions to form sentence meanings. There are thus several, interlocking“directions” of compositionality at work, and they result in interactions that
could appear non-compositional if only one direction was considered. Let us
focus on two: compositionality of world knowledge and compositionality of
linguistic meaning.
Compositionality of world knowledge refers to the way that we use SLC
to build distributions over possible worlds, not by directly assigning probabilities to all possible expressions, but by an evaluation process that recursively
samples values for sub-expressions. That is, we have a compositional language
for specifying generative models of the world. Compositionality of linguistic
meaning refers to the way that conditions on worlds are built up from simpler pieces (via the meaning function and evaluation of the meaning). This is
the standard approach to meaning composition in truth-conditional semantics.
Interpreted meaning—the posterior distribution arrived at by literal-listener—
is not immediately compositional along either world knowledge or linguistic
structure. Instead it arises from the interaction of these two factors. The glue
between these two structures is the intuitive theory; it defines the conceptual
language for imagining particular situations, and the primitive vocabulary forsemantic meaning.
An alternative approach to compositional probabilistic semantics would
be to let each linguistic expression denote a distribution or probability directly, and build the linguistic interpretation by composing them. This appears
attractive: it is more direct and simpler (and does not rely on complex generative knowledge of the world). How would we compose these distributions?
For instance take “Jack is strong and Bob is strong”. If “Jack is strong” hasprobability 0.2 and “Bob is strong” has probability 0.3, what is the probabilityof the whole sentence? A natural approach would be to multiply the two probabilities. However this implies that their strengths are independent—which is
intuitively unlikely: for instance, if Jack and Bob are both men, then learningthat Jack is strong suggests than men are strong, which suggests that Bill isstrong. A more productive strategy is the one we have taken: world knowledge
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Noah D. Goodman and Daniel Lassiterspecifies a joint distribution on the strength of Bob and Jack (by first sampling
the prototypical strength of men, then sampling the strength of each), and thesentence imposes a constraint on this distribution (that each man’s strength
exceeds a threshold). The sentence denotes not a world probability simpliciter,
but a constraint on worlds which is built compositionally.
2.7 Extensions and related work
The central elements of probabilistic language understanding as describedabove are: grounding lexical meaning into a probabilistic generative model of
the world, taking sentence meanings as conditions on worlds (built by composing lexical meanings), and treating interpretation as joint probabilistic inference of the world state and the sentence meaning conditioned on the truthof the sentence. It should be clear that this leaves open many extensions andalternative formulations. For instance, varying the method of linguistic composition, adding static types that influence interpretation, and including othersources of uncertainty such as a noisy acoustic channel are all straightforward
avenues to explore.
There are several related approaches that have been discussed in previous
work. Much previous work in probabilistic semantics has a strong focus onvagueness and degree semantics: see e.g. Edgington 1997; Frazee & Beaver
2010; Lassiter 2011, discussed further in section 4 below and in Lassiter 2014
(this volume). There are also well-known probabilistic semantic theories ofisolated phenomena such as conditionals (Adams, 1975; Edgington, 1995, andmany more) and generics (Cohen, 1999a,b). We have taken inspiration fromthese approaches, but we take the strong view that probabilities belong at thefoundation of an architecture for language understanding, rather than treating
it as a special-purpose tool for the analysis of specific phenomena.
In Fuzzy Semantics (Zadeh, 1971; Lakoff, 1973; Hersh & Caramazza, 1976,
etc.) propositions are mapped to real values that represent degrees of truth,
similar to probabilities. Classical fuzzy semantics relies on strong independence assumptions to enable direct composition of fuzzy truth values. This
amounts to a separation of uncertainty from language and non-linguistic
sources. In contrast, we have emphasized the interplay of linguistic interpretation and world knowledge: the probability of a sentence is not defined
separate from the joint-inference interpretation, removing the need to definecomposition directly on probabilities.
A somewhat different approach, based on type theory with records, is described by Cooper et al. (2014). Cooper et al.’s project revises numerous basic
assumptions of model-theoretic semantics, with the goals of better explaining
semantic learning and “pervasive gradience of semantic properties.” The work
described here takes a more conservative approach, by enriching the standard framework while preserving most basic principles. As we have shown, thisgives rise to gradience; we have not addressed learning, but there is an extensive literature on probabilistic learning of structured representations similar to
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those required by our architecture: see e.g. Goodman et al. 2008b; Piantadosi
et al. 2008, 2012; Tenenbaum et al. 2011. It may be, however, that stronger
types than we have employed will be necessary to capture subtleties of syntax and facilitate learning. Future work will hopefully clarify the relationshipbetween the two approaches, revealing which differences are notational andwhich are empirically and theoretically significant.

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3 Pragmatic interpretation
The literal-listener described above treats utterances as true information about
the world, updating her beliefs accordingly. In real language understanding,however, utterances are taken as speech acts that inform the listener indirectly
by conveying a speaker’s intention. In this section we describe a version of
the Rational Speech Acts model (Goodman & Stuhlmüller, 2013; Frank &Goodman, 2012), in which a sophisticated listener reasons about the intentionof an informative speaker.
First, imagine a speaker who wishes to convey that the question underdiscussion (QUD) has a particular answer (i.e. value). This can be viewed as aninference: what utterance is most likely to lead the (literal) listener to thecorrect interpretation?
( define ( speaker val QUD )( query( define utterance ( langua ge-prio r ) )utterance( equal ? val ( l i t e r a l - l i s t e n e r utterance QUD ) ) ) )The language-prior forms the a priori (non-contextual and non-semantic) distribution over linguistic forms, which may be modeled with a probabilistic
context free grammar or similar model. This prior inserts a cost for each utterance: using a less likely utterance will be dispreferred a priori. Notice that
this speaker conditions on a single sample from literal-listener having the correct val for the QUD—that is, he conditions on the literal-listener “guessing”
the right value. Since the listener may sometimes accidentally guess the rightvalue, even when the utterance is not the most informative one, the speaker
will sometimes choose sub-optimal utterances. We can moderate this behavior
by adjusting the tendency of the listener to guess the most likely value:( define ( speaker val QUD )( query( define utterance ( langua ge-prio r ) )utterance( equal ? val (( power l i t e r a l - l i s t e n e r alpha ) utterance QUD ) ) ) )Here we have used a higher-order function power that raises the return distribution of the input function to a power (and renormalizes). When the poweralpha is large the resulting distribution will mostly sample the maximum of
the underlying distribution—in our case the listener that speaker imagines will
mostly sample the most likely val.
Writing the distribution implied by the speaker function explicitly can beclarifying:
P (ut|val, QUD) ∝ P (ut)Plistener (val|ut, QUD)α∝e

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Thus, the speaker function describes a speaker who chooses utterances usinga soft-max rule P (utt) ∝ eαU (utt) (Luce, 1959; Sutton & Barto, 1998). Herethe utility U (utt) is given by the sum of••

the informativity of utt about the QUD, formalized as negative surprisal of
the intended value: ln(Plistener (val|ut, QUD)),a cost term ln(P (utt)), which depends on the language prior.

Utterance cost plausibly depends on factors such as length, frequency, andarticulatory effort, but the formulation here is noncommittal about precisely
which linguistic and non-linguistic factors are relevant.
A more sophisticated, pragmatic, listener can now be modeled as a
Bayesian agent updating her belief about the value of the question underdiscussion given the observation that the speaker has bothered to make aparticular speech act:
( define ( listener utterance QUD )( query... theory...
( define val ( eval QUD ) )val( equal ? utterance ( speaker val QUD ) ) ) )Notice that the prior over val comes from evaluating the QUD expression given
the theory, and the posterior comes from updating this prior given that the
speaker has chosen utterance to convey val.
The force of this model comes from the ability to call the query function
within itself (Stuhlmueller & Goodman, 2013)—each query models the inference made by one (imagined) communicator, and together they capture
sophisticated pragmatic reasoning. Several observations are worth making:First, alternative utterances will enter into the computation in sampling (ordetermining the probability of) the actual utterance from speaker. Similarly,
alternative values are considered in the listener functions. Second, the notionof informativity captured in the speaker model is not simply information transmitted by utterance, but is new information conveyed to the listener about the
QUD. Information which is not new to the listener or which is not relevant to
the QUD will not contribute to the speaker’s utility.
3.1 Quantity implicatures
We illustrate by considering quantity implicatures: take as an example thesentence “Jane played in some match”. This entails that Jane did not play inzero matches. In many contexts, it would also be taken to suggest that Jane
did not play in all of the matches. However, there are many good reasons forthinking that the latter inference is not part of the basic, literal meaning of
the sentence (Grice, 1989; Geurts, 2010). Why then does it arise? Quantity
implicatures follow in our model due to the pragmatic listener’s use of “counterfactual” reasoning to help reconstruct the speaker’s intended message from

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Noah D. Goodman and Daniel Lassiterhis observed utterance choice. Suppose that the QUD is “How many matches
did Jane play in?” (interpreted as [[the number of matches Jane played in]]).
The listener considers different answers to this question by simulating partialworlds that vary in how many matches Jane played in and considering what
the speaker would have said for each case. If Jane played in every match, then“Jane played in every match” would be used by the speaker more often than“Jane played in some match”. This is because the speaker model favors more
informative utterances, and the former is more informative: a literal speaker
will guess the correct answer more often after hearing “Jane played in every
match”. Since the speaker in fact chose the less informative utterance in thiscase, the listener infers that some precondition for the stronger utterance’s
use—e.g., its truth—is probably not fulfilled.
For example, suppose that it is common knowledge that teams have four
players, and that three matches were played. The speaker knows exactly whoplayed and how many times, and utters “Jane played in some match”. How
many matches did she play in? The speaker distribution is shown in Figure 5.
If Jane played in zero matches, the probability that the speaker will use either
utterance is zero (instead the speaker will utter “Jane played in no match”).
If she played in one or two matches, the probability that the speaker will utter“Jane played in some match” is non-zero, but the probability that the speaker
will utter “Jane played in every match” is still zero. However, the situation
changes dramatically if Jane in fact played in all the matches: now the speakerprefers the more informative utterance “Jane played in every match”.

Figure 5. Normalized probability that the speaker will utter “Jane played in no/some/every match” in each situation, generated by reasoning about which utterance
will most effectively bring the literal listener to select the correct answer to the QUD“How many matches did Jane play in?”. (The parameter alpha is set to 5.)

The pragmatic listener still does not know how many matches Jane playedin but can reason about the speaker’s utterance choice. If the correct answer
were 3 the speaker would probably not have chosen “some”, because the literal
listener is much less likely to choose the answer 3 if the utterance is “some”
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Figure 6. Interpretation of “Jane played in some match” by the literal and pragmatic listeners, assuming that the only relevant alternatives are “Jane played in
no/every match”. While the literal listener (left pane) assigns a moderate probability to the “all” situation given this utterance, the pragmatic listener (right pane)
assigns this situation a very low probability. The difference is due to the fact thatthe pragmatic listener reasons about the utterance choices of the speaker (Figure
5 above), taking into account that the speaker is more likely to say “every” than“some” if “every” is true.

as opposed to “every”. The listener can thus conclude that the correct answer probably is not 3. Figure 6 shows the predictions for both the literaland pragmatic listener; notice that the interpretation of “some” differs only
minimally from the prior for the literal listener, but is strengthened for the
pragmatic listener. Thus, our model yields a broadly Gricean explanation ofquantity implicature. Instead of stipulating rules of conversation, the content
of Grice’s Maxim of Quantity falls out of the recursive pragmatic reasoning
process whenever it is reasonable to assume that the speakers is making aneffort to be informative. (For related formal reconstructions of Gricean reasoning about quantity implicature, see Franke 2009; Vogel et al. 2013.)
3.2 Extensions and related work
The simple Rational Speech Acts (RSA) framework sketched above has been
fruitfully extended and applied to a number of phenomena in pragmatic understanding; many other extensions suggest themselves, but have not yet been
explored. In Frank & Goodman 2012 the RSA model was applied to explain
the results of simple reference games in which a speaker attempted to communicate one of a set of objects to a listener by using a simple property todescribe it (e.g. blue or square). Here the intuitive theory can be seen as simply
a prior distribution, (define ref (ref-prior objects)) over which object is the referent in the current trial, the QUD is simply ref, and the properties have theirstandard extensions. By measuring the ref-prior empirically Frank & Goodman (2012) were able to predict the speaker and listener judgements withhigh quantitative accuracy (correlation around 0.99).
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In Goodman & Stuhlmüller 2013 the RSA framework was extended to take
into account the speaker’s belief state. In this case the speaker should choose
an utterance based on its expected informativity under the speaker’s belief
distribution. (Or, equivalently, the speaker’s utility is the negative KullbackLeibler divergence of the listener’s posterior beliefs from the speaker’s.) Thisextended model makes the interesting prediction that listeners should not
draw strong quantity implicatures from utterances by speakers who are notknown to be informed about the question of interest (cf. Sauerland, 2004;
Russell, 2006). The experiments in Goodman & Stuhlmüller (2013) show that
this is the case, and the quantitative predictions of the model are borne out.
As a final example of extensions to the RSA framework, the QUD itself canbe an object of inference. If the pragmatic listener is unsure what topic the
speaker is addressing, as must often be the case, then she should jointly infer
the QUD and its val under the assumption that the speaker chose an utterance
to be informative about the topic (whatever that happens to be). This simple
extension can lead to striking predictions. In Kao et al. (2014); Kao et al.
such QUD inference was shown to give rise to non-literal interpretations: hyperbolic and metaphoric usage. While the literal listener will draw an incorrect
inference about the state of the world from an utterance such as “I waited
a million hours”, the speaker only cares if this results in correct information
about the QUD; the pragmatic listener knows this, and hence interprets the utterance as only conveying information about the QUD. If the QUD is inferred to bea non-standard aspect of the world, such as whether the speaker is irritated,then the utterance will convey only information about this aspect and notthe (false) literal meaning of the utterance: the speaker waited longer than
expected and is irritated about it.
The RSA approach shares elements with a number of other formal approaches to pragmatics. It is most similar to game theoretic approaches topragmatics. In particular to approaches that treat pragmatic inference as iterated reasoning, such as the Iterated Best Response (IBR) model (Franke,
2009; Benz et al., 2005). The IBR model represents speakers and listeners
recursively reasoning about each other, as in the RSA model. The two maindifferences are that IBR specifies unbounded recursion between speaker andlistener, while RSA as presented here specifies one level, and the IBR specifies that optimal actions are chosen, rather than soft-max decisions. Neitherof these differences is critical to either framework. We view it as an empirical question whether speakers maximize or soft-maximize and what level ofrecursive reasoning people actually display in language understanding.

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4 Semantic indices
In formal semantics sentence meanings are often treated as intensions: functions from semantic indices to truth functions (Lewis, 1970, 1980; Montague,1973). The semantic theory has little or nothing to say about how these indices are set, except that they matter and usually depend in some way oncontext. We have already seen that a probabilistic theory of pragmatic interpretation can be used to describe and predict certain effects of context andbackground knowledge on interpretation. Can we similarly use probabilistic
tools to describe the ways that semantic indices are set based on context?
We must first decide how semantic indices should enter into the probabilistic
framework presented above (where we have so far treated meanings simply as
truth functions). The simplest assumption is that they are random variables
that occur (unbound) in the meaning expression and are reasoned about by
the literal listener:( define ( l i t e r a l - l i s t e n e r utterance QUD )( query... theory...
( define index ( index-prior ) )( define val ( eval QUD ) )val( eval ( meaning utterance ) ) ) )Here we assume that the meaning may contain an unbound occurrence of index
which is then bound during interpretation by the (define index ...) definition.
Because there is now a joint inference over val and index, the index will tend
to be set such that the utterance is most likely to be true.
Consider the case of gradable adjectives like strong. In section 2.3 we havedefined [[strong]] = (λ (x) (strength x)); to form a property from the adjective ina positive form sentence like Bob is strong, we must bind the degree returned
from strength in some way. A simple way to do this is to add a type-shifterthat introduces a free threshold variable θ—see, for example, Kennedy 2007
and Lassiter 2014 (this volume). We extend the set of type shifters that canbe inserted by shift (see section 2.2) with:•

POS: (λ (A) (λ (x) (>= (A x) θ)))In this denotation the variable θ is a free index that will be bound during interpretation as above. Now consider possible denotations that can be generated
by meaning.
•••[[Bob is strong]]=('Bob (λ (x) (strength x)))[[Bob is strong]]=((L 'Bob) (λ (x) (strength x)))[[Bob is strong]]=((L 'Bob) (POS (λ (x) (strength x))))

The first of these returns error because 'Bob is not a function; the second
applies strength to 'Bob and returns a degree. Both of these meanings will be removed in the query of literal-listener because their values will never equal true.
The third meaning tests whether Bob is stronger than a threshold variable andPage: 31

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Noah D. Goodman and Daniel Lassiter

returns a Boolean—it is the simplest well-typed meaning. With this meaning
the utterance “Bob is strong” (with QUD “How strong is Bob?”) would be interpreted by the literal listener (after simplification, and assuming for simplicity
a domain of -100 to 100 for the threshold) via:
( query... theory...
( define θ ( uniform -100 100) )( define val ( strength ' Bob ) )val( >= ( strength ' Bob ) θ) )Figure 7 shows the prior (marginal) distributions over θ and Bob’s strength,
and the corresponding posterior distributions after hearing “Bob is strong”.
The free threshold variable has been influenced by the utterance: it changes
from a uniform prior to a posterior that is maximum at the bottom of its
domain and gradually falls form there—this makes the utterance likely to betrue. However, this gives the wrong interpretation of Bob is strong. Intuitively,
the listener ought to adjust her estimate of Bob’s strength to a fairly highvalue, relative to the prior. Because the threshold is likely very low, the listenerinstead learns very little about the variable of interest from the utterance: theposterior distribution on Bob’s strength is almost the same as the prior.

Figure 7. The literal listener’s interpretation of an utterance containing a freethreshold variable θ, assuming an uninformative prior on this variable. This listener’s
exclusive preference for true interpretations leads to a tendency to select extremely
low values of θ (“degree posterior”). As a result the utterance conveys little information about the variable of interest: the strength posterior is barely different fromthe prior.

What is missing is the pressure to adjust θ so that the sentence is not
only true, but also informative. Simply including the informative speaker andpragmatic listener models as defined above is not enough: without additionalchanges the index variables will be fixed by the literal listener with no pragmatic pressures. Instead, we lift the index variables to the pragmatic level.
Imagine a pragmatic listener who believes that the index variable has a value
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that she happens not to know, but which is otherwise common knowledge (i.e.
known by the speaker, who assumes it is known by the listener):( define ( listener utterance QUD )( query... theory...
( define index ( index-prior ) )( define val ( eval QUD ) )val( equal ? utterance ( speaker val QUD index ) ) ) )( define ( speaker val QUD index )( query( define utterance ( langua ge-prio r ) )utterance( equal ? val ( l i t e r a l - l i s t e n e r utterance QUD index ) ) ) )( define ( l i t e r a l - l i s t e n e r utterance QUD index )( query... theory...
( define val ( eval QUD ) )val( eval ( meaning utterance ) ) ) )In most ways this is a very small change to the model, but it has importantconsequences. At a high level, index variables will now be set in such a waythat they both make the utterance likely to be true and likely to be pragmatically useful (informative, relevant, etc); the tradeoff between these two
factors results in significant contextual flexibility of the interpreted meaning.

Figure 8. The pragmatic listener’s interpretation of an utterance such as “Bob isstrong,” containing a free threshold variable θ that has been lifted to the pragmaticlevel. Joint inference of the degree and the threshold leads to a “significantly greaterthan expected” meaning. (We assume that the possible utterances are to say nothing(cost 0) and “Bob is strong/weak” (cost 6), and alpha= 5, as before.)

In the case of the adjective strong, Figure 8, the listener’s posterior estimate of strength is shifted significantly upward from the prior, with mean
at roughly one standard deviation above the prior mean (though the exact
distribution depends on parameter choices). Hence strong is interpreted as

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Noah D. Goodman and Daniel Lassitermeaning “significantly stronger than average”, but does not require maximal
strength (most informative) or permit any strength (most often true). Thismodel of gradable adjective interpretation (which was introduced in Lassiter& Goodman 2013) has a number of appealing properties. For instance, theprecise interpretation is sensitive to the prior probability distribution on answers to the QUD. We thus predict that gradable adjective interpretation
should display considerable sensitivity to background knowledge. This is indeed the case, as for example in the different interpretations of “strong boy”,“strong football player”, “strong wall”, and so forth. Prior expectations about
the degree to which objects in a reference class have some property frequently
plays a considerable role in determining the interpretation of adjectives. Thisaccount also predicts that vagueness should be a pervasive feature of adjective
interpretation, as discussed below. See Lassiter & Goodman 2013 for detailed
discussion of these features.
We can motivate from this example a general treatment of semantic indices: lift each index into the pragmatic inference of listener, passing themdown to speaker and on to literal-listener, allowing them to bind free variablesin the literal meaning. As above all indices will be reasoned over jointly withworld states. Any index that occurs in a potential meaning of an alternativeutterance must be lifted in this way, to be available to the literal-listener. If wewish to avoid listing each index individually, we can modify the above treatment with an additional indirection: For instance by introducing a memoizedfunction index that maps variable names to (random) values appropriate for
their types.
4.1 Vagueness and indeterminate boundaries
Probabilistic models of the type described here make it possible to maintainthe attractive formal precision of model-theoretic semantics while also making room for vagueness and indeterminate boundaries in both word meanings
and psychological categories. There is considerable evidence from both psychological (e.g. Rosch, 1978; Murphy, 2002; Hampton, 2007) and linguistic(Taylor, 2003) research that a lack of sharp boundaries is a pervasive featuresof concept and word usage. Linguistic indeterminacy and vagueness can beunderstood as uncertainty about the precise interpretation of expressions incontext. As discussed in section 2.5, uncertainty can enter from a number ofsources in constructing meaning from an utterance; to those we can now add
uncertainty that comes from a free index variable in the meaning, which isresolved at either the literal or pragmatic listener levels. Each source of uncertainty about the meaning leads to an opportunity for context-sensitivity
in interpretation. These sources of context-sensitivity predict a number ofimportant features of vagueness. We illustrate this by discussing how key features of vagueness in adjective interpretation are predicted by our treatment ofgradable adjectives, above. For more discussion of vagueness and an overview
of theories see Lassiter 2014 (this volume).
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Borderline cases. While the underlying semantics of Bill is strong yields
a definite boundary, introduced to the meaning by POS, there is posterior
uncertainty over the value of this threshold. Hence, an individual whose degreeof strength falls in the middle of the posterior distribution (see Figure 8)
will be a borderline case of strong. In the example above, an individual withstrength 3 will have a roughly equal chance of counting as strong and as notstrong.
Tolerance principles. Suppose Bill has strength 4.5 and Mary has
strength 4.4. It would be odd for someone to confidently agree to the claimthat Bill is strong, but to deny confidently that Mary is strong. Our model
explains this intuition: when two individuals’ strength are separated by asmall gap, the posterior probability that the threshold falls in this gap is verysmall—hence it is very rarely the case that one counts as strong and the other
does not. Indeed, this could happen only if the posterior distribution overstrength had a sharp discontinuity, which in turn would imply that the prior
had an abrupt boundary (Lassiter & Goodman, 2013).
The sorites paradox. The following is an instance of a famous puzzle:•••

Bill is strong.
A person who is slightly less strong than a strong person is also strong.
Therefore, everyone is strong, no matter how weak.

People generally find the premises plausible, but the conclusion (which follows
logically by induction) not at all plausible. Evidently something is wrong withthe second premise, but what?
Our probabilistic approach, built as it is upon a bivalent logic, requires thatthe conclusion is true in a given world if the premises are true. However, if thesecond premise is interpreted as universally quantified it will rarely be true: if
there are enough individuals, there will be two separated by a small amount,but on either side of the threshold. Yet this answer—that the second premise
is in fact false in most relevant situations—does not explain the psychologicalaspect of the puzzle (Graff, 2000): people express high confidence in the second
premise.
Lassiter & Goodman (2013) argue that the second premise is not interpreted in a simple universally quantified way, but is evaluated probabilisticallyas a conditional: given that person x (of a priori unknown strength) is strong,form the posterior distribution over θ as above; under this distribution what isthe probability that a person with strength slightly smaller is strong, i.e. theprobability that (- (strength x) )> θ.9 This probability depends on the priordistribution, but for reasonably gradual priors and fairly small gaps  it will
be quite high. Figure 4.1 shows the probability of the inductive premise as
a function of the gap for the setup used before. This account builds on previous probabilistic approaches to the vagueness and the sorites (Borel, 1907;
9An extension to the linguistic fragment described above would be necessary toderive this interpretation formally. One approach would be to treat the relative
clause an embedded query.

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Noah D. Goodman and Daniel LassiterBlack, 1937; Edgington, 1997; Lawry, 2008; Frazee & Beaver, 2010; Égré,
2011; Lassiter, 2011; Sutton, 2013), but is the first to offer a specific accountof why vague adjectives should have context-sensitive probabilistic interpretations, and of how the distribution is determined in a particular context ofutterance.

Figure 9. With prior distributions and parameters as above, the probability of thesecond premise of the sorites paradox is close to 1 when the inductive gap is small,but decreases as the size of the gap increases.

4.2 Extensions and related work
Another interpretation of the above modeling approach (indeed, the original
interpretation, introduced in Bergen et al. (2012)) is as the result of lexical uncertainty: each index represents a lingering uncertainty about word meaning
in context which the listener must incorporate in the interpretation process.10
This interpretation is appealing in that it connects naturally to language acquisition and change (Smith et al., 2013). For instance, upon hearing a newword a learner would initially treat its meaning as underdetermined—in effect,
as an index variable ranging over all expressions of the appropriate type—andinfer its meaning on each usage from contextual cues. Over time the prior overthis ‘index’ would tighten until only the correct meaning remained, and nocontextual flexibility was left. A difficulty with the lexical uncertainty interpretation is explaining why certain aspects of a word’s meaning are so much
more flexible than others and why this appears to be regular across words of agiven type. The free-index interpretation accounts for this naturally because
the dimensions of flexibility are explicitly represented as unbound variablesin lexical entries or in type shifters used in the compositional construction of
meaning. A more structured (e.g. hierarchical) notion of lexical uncertainty
may be able to reconcile these interpretations, which are essentially equivalent.
10

Note that lexical uncertainty is a form of lexical ambiguity, but is the special form
in which the choice of ambiguous form is lifted to the pragmatic listener.

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The use of lifted semantic indices, or lexical uncertainty, can account for anumber of puzzling facts about language use beyond those considered above.
The original motivation for introducing these ideas (Bergen et al., 2012) was
to explain the Division of Pragmatic Labor (Horn, 1984) : why are (un)marked
meanings assigned to (un)marked utterances, even when the utterances havethe same literal semantics? The basic RSA framework cannot explain thisphenomena. If however we assume that the meanings can each be refined tomore precise meanings, the correct alignment between utterances and interpretations is achieved.
An important question is raised by this section: which, if any, ambiguities
or under-specifications in meaning are resolved at the literal listener level, andwhich are lifted to the pragmatic listener? This choice has subtle but important consequences for interpretation, as illustrated above for scalar adjectives,
but it is empirical question that must be examined for many more cases before
we are in a position to generalize.

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5 Conclusion
In this chapter we have illustrated the use of probabilistic modeling to study
natural language semantics and pragmatics. We have described how stochastic
λ-calculus, as implemented in Church, provides compositional tools for probabilistic modeling. These tools helped us to explicate the relationship between
linguistic meaning, background knowledge, and interpretation.
On the one hand we have argued that uncertainty, formalized via probability, is a key organizing principle throughout language and cognition. On
the other hand we have argued, by example, that we must still build detailed models of natural language architecture and structure. The system wehave described here provides important new formalizations of how context
and background knowledge affect language interpretation—an area in which
formal semantics has been largely silent. Yet the enterprise of formal semantics
has been tremendously successful, providing insightful analyses of many phenomena of sentence meaning. Because compositional semantics plays approximately its traditional role within our architecture, many of the theoreticalstructures and specific analyses will be maintained. Indeed, seen one way, ourprobabilistic approach merely augments traditional formalizations with a theory of interpretation in context—one that makes good on many promissory
notes from the traditional approaches.
There are several types of uncertainty and several roles for uncertainty inthe architecture we have described. While the fundamental mechanisms forrepresenting and updating beliefs are the same for discrete variables (such as
those that lead to scope ambiguity for quantifiers) and continuous variables(such as the threshold variable we used to interpret scalar adjectives in thepositive form), there are likely to be phenomenological differences as well as
similarities. For instance, continuous variables lend themselves to borderline
cases in a way that discrete variables don’t, while both support graded judgements. Similarly, the point at which a random variable is resolved—within
the literal listener, in the pragmatic listener, or both—can have profoundeffects on its role in language understanding. Variables restricted to the literal listener show plausibility but not informativity effects; variables in the
pragmatic listener that are not indices show informativity but limited context
sensitivity; etc. Overall then, uniform mechanisms of uncertainty can lead toheterogeneous phenomenology of language understanding, depending on the
structure of the language understanding model.
In the architecture we have described, uncertainty is pervasive throughall aspects of language understanding. Pervasive uncertainty leads to complex interactions that can be described by joint inference of the many random
choices involved in understanding. Joint inference in turn leads to a great
deal of flexibility, from non-monotonic effects such as explaining away (section 2), through ambiguous compositional structure (section 2.5) and pragmatic strengthening (section 3), to vagueness and context-specificity of indices(section 4). It is particularly important to note that even when the archi-

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tecture specification is relatively modular, for instance separate specificationof world knowledge (the ...theory...) and meaning interpretation (the meaning
function), the inferential effects in sentence interpretation will have complex,
bi-directional interactions (as in the interaction of background knowledge andquantifier scope ambiguity in section 2). That is, language understanding isanalyzable but not modular.

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6 Acknowledgements
We thank Erin Bennett for assistance preparing this chapter, including simulations and editting. We thank Henk Zeevat, Shalom Lappin, Scott Martin,
and Adrian Brasoveanu for helpful comments on early versions of this chapter
or related presentations.
This work was supported in part by a John S. McDonnell Foundation
Scholar Award (NDG), and Office of Naval Research grants N000141310788
and N000141310287 (NDG).

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References
Abelson, Harold & Gerald Jay Sussman (1983), Structure and interpretation ofcomputer programs .
Adams, Ernest W. (1975), The logic of conditionals: An application of probability to
deductive logic, Springer.
Altmann, Gerry & Mark Steedman (1988), Interaction with context during humansentence processing, Cognition 30(3):191–238.
Barendregt, Hendrik Pieter (1985), The lambda calculus: Its syntax and semantics,
volume 103, North Holland.
Barker, Chris (2005), Remark on jacobson 1999: Crossover as a local constraint,
Linguistics and Philosophy 28(4):447–472.
Bartsch, Renate & Theo Vennemann (1973), Semantic structures: a study in therelation between semantics and syntax, Athenäum.
Beaver, David & Brady Clark (2008), Sense and sensitivity: How focus determines
meaning, Wiley-Blackwell, ISBN 1405112646.
Benz, Anton, Gerhard Jäger, & Robert van Rooij (2005), Game theory and Pragmatics, Palgrave Macmillan.
Bergen, L., N.D. Goodman, & R. Levy (2012), That’s what she (could have) said:
How alternative utterances affect language use, in Proceedings of the 34th Annual
Meeting of the Cognitive Science Society.
Black, Max (1937), Vagueness. an exercise in logical analysis, Philosophy of science
4(4):427–455.
Bod, R., J. Hay, & S. Jannedy (2003), Probabilistic linguistics, The MIT Press.
Borel, Émile (1907), Sur un paradoxe économique: Le sophisme du tas de blé et lesvérités statistiques, Revue du Mois 4:688–699.
Chater, Nick, Christopher D Manning, et al. (2006), Probabilistic models of language
processing and acquisition, Trends in cognitive sciences 10(7):335–344.
Chater, Nick & Mike Oaksford (2008), The probabilistic mind: Prospects forBayesian cognitive science, Oxford University Press.
Cohen, Ariel (1999a), Generics, frequency adverbs and probability, Linguistics andPhilosophy 22:221–253.
Cohen, Ariel (1999b), Think generic! The Meaning and Use of Generic Sentences,
CSLI.
Cooper, Robin, Simon Dobnik, Shalom Lappin, & Staffan Larsson (2014), A probabilistic rich type theory for semantic interpretation, in Proceedings of the EACL
2014 Workshop on Type Theory and Natural Language Semantics (TTNLS),(72–79).
Crain, Stephen & Mark Steedman (1985), On not being led up the garden path:
The use of context by the psychological parser :320–358.
Cruse, D Alan (2000), Meaning in language, volume 2, Oxford University Press
Oxford.
Edgington, Dorothy (1995), On conditionals, Mind 104(414):235, doi:10.1093/mind/
104.414.235.
Edgington, Dorothy (1997), Vagueness by degrees, in R. Keefe & P. Smith (eds.),
Vagueness: A Reader, MIT Press, (294–316).
Égré, Paul (2011), Perceptual ambiguity and the sorites, in Vagueness in Communication, Springer, (64–90).

Page: 41

job: Goodman-HCS-final

macro: handbook.cls

date/time: 25-Jun-2014/8:41

42

Noah D. Goodman and Daniel LassiterFrank, M.C. & N.D. Goodman (2012), Predicting pragmatic reasoning in language
games, Science 336(6084):998–998.
Franke, M. (2009), Signal to act: Game theory in pragmatics, Ph.D. thesis, Institutefor Logic, Language and Computation, University of Amsterdam.
Frazee, Joey & David Beaver (2010), Vagueness is rational under uncertainty, Proceedings of the 17th Amsterdam Colloquium .
Freer, Cameron E & Daniel M Roy (2012), Computable de finetti measures, Annalsof Pure and Applied Logic 163(5):530–546.
Gamut, L.T.F. (1991), Logic, Language, and Meaning, volume 1: Introduction to
Logic, volume 1, University of Chicago Press.
Geurts, Bart (2010), Quantity implicatures, Cambridge University Press.
Ginzburg, J. (1995), Resolving questions, I, Linguistics and Philosophy 18(5):459–
527.
Goodman, Noah D., Vikash K. Mansinghka, Daniel Roy, Keith Bonawitz, &Joshua B. Tenenbaum (2008a), Church: A language for generative models, inUncertainty in Artificial Intelligence 2008.
Goodman, Noah D & Andreas Stuhlmüller (2013), Knowledge and implicature: Modeling language understanding as social cognition, Topics in cognitive science5(1):173–184.
Goodman, Noah D., Joshua B. Tenenbaum, Jacob Feldman, & Thomas L. Griffiths(2008b), A rational analysis of rule-based concept learning, Cognitive Science32(1):108–154.
Graff, Delia (2000), Shifting sands: An interest-relative theory of vagueness, Philosophical Topics 20:45–81.
Grice, H. Paul (1989), Studies in the Way of Words, Harvard University Press.
Griffiths, Thomas L., Charles Kemp, & Joshua B. Tenenbaum (2008), Bayesianmodels of cognition, in R. Sun (ed.), Cambridge Handbook of Computational
Psychology, Cambridge University Press, (59–100).
Hampton, J.A. (2007), Typicality, graded membership, and vagueness, Cognitive
Science 31(3):355–384.
Heim, Irene (1982), The semantics of definite and indefinite noun phrases, Ph.D.
thesis.
Heim, Irene (1992), Presupposition projection and the semantics of attitude verbs,
Journal of Semantics 9(3):183, ISSN 0167-5133.
Heim, Irene & Angelika Kratzer (1998), Semantics in Generative Grammar, Blackwell.
Hendriks, H.L.W. (1993), Studied flexibility: Categories and types in syntax andsemantics, Institute for Logic, Language and Computation.
Hersh, Harry M & Alfonso Caramazza (1976), A fuzzy set approach to modifiersand vagueness in natural language., Journal of Experimental Psychology: General105(3):254.
Hindley, James Roger & Jonathan Paul Seldin (1986), Introduction to Combinators and (Lambda) Calculus, volume 1, Cambridge [Cambridgeshire]; New York:
Cambridge University Press.
Horn, Laurence (1984), Toward a new taxonomy for pragmatic inference: Q-based
and r-based implicature, in Deborah Schiffrin (ed.), Meaning, Form, and Use inContext: Linguistic Applications, Georgetown University Press, (11–42).
Jackendoff, Ray (1983), Semantics and cognition, volume 8, The MIT Press.

Page: 42

job: Goodman-HCS-final

macro: handbook.cls

date/time: 25-Jun-2014/8:41

Probabilistic Semantics and Pragmatics

43

Kao, Justine T, Leon Bergen, & Noah D Goodman (2014), Formalizing the pragmatics of metaphor understanding, in Proceedings of the 36th Annual Meetingof the Cognitive Science Society.
Kao, Justine T, Jean Y Wu, Leon Bergen, & Noah D Goodman (????), Nonliterallanguage understanding for number words, under review.
Kennedy, C. (2007), Vagueness and grammar: The semantics of relative and absolute
gradable adjectives, Linguistics and Philosophy 30(1):1–45.
Kennedy, Chris (1997), Projecting the adjective: The syntax and semantics of gradability and comparison, Ph.D. thesis, U.C., Santa Cruz.
Kolmogorov, Andrey (1933), Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius
Springer.
Lakoff, George (1973), Hedges: A study in meaning criteria and the logic of fuzzy
concepts, Journal of philosophical logic 2(4):458–508.
Lakoff, George (1987), Women, fire, and dangerous things: What categories reveal
about the mind .
Lassiter, D. (2011), Vagueness as probabilistic linguistic knowledge, in R. Nouwen,
R. van Rooij, U. Sauerland, & H.-C. Schmitz (eds.), Vagueness in Communication, Springer, (127–150).
Lassiter, Daniel (2014), Adjectival modification and gradation, in Shalom Lappin& Chris Fox (eds.), Handbook of Contemporary Semantic Theory, 2nd edition,
Blackwell.
Lassiter, Daniel & Noah D. Goodman (2013), Context, scale structure, and statistics in the interpretation of positive-form adjectives, to appear in Semantics &Linguistic Theory (SALT) 23.
Lawry, Jonathan (2008), Appropriateness measures: an uncertainty model for vague
concepts, Synthese 161(2):255–269.
Lewis, David (1970), General semantics, Synthese 22(1):18–67.
Lewis, David (1979), Scorekeeping in a language game, Journal of Philosophical
Logic 8(1):339–359, ISSN 0022-3611, doi:10.1007/BF00258436.
Lewis, Davis (1980), Index, context, and content, in Stig Kanger & Sven Öhman(eds.), Philosophy and Grammar, Reidel, (79–100).
Luce, R.D. (1959), Individual choice behavior: A theoretical analysis, John Wiley.
May, Robert (1977), The grammar of quantification, Ph.D. thesis, Massachusetts
Institute of Technology.
Montague, Richard (1973), The proper treatment of quantification in ordinary English, in J. Hintikka, J. Moravcsik, & P. Suppes (eds.), Approaches to Natural
Language, Reidel, volume 49, (221–242).
Murphy, Gregory (2002), The Big Book of Concepts, MIT Press.
Oaksford, Mike & Nick Chater (2007), Bayesian rationality: The probabilistic approach to human reasoning, Oxford University Press.
Partee, Barbara (1987), Noun phrase interpretation and type-shifting principles,
Studies in discourse representation theory and the theory of generalized quantifiers 8:115–143.
Partee, Barbara & Mats Rooth (1983), Generalized conjunction and type ambiguity,
Formal Semantics: The Essential Readings :334–356.
Pearl, Judea (1988), Probabilistic Reasoning in Intelligent Systems: Networks ofPlausible Inference, Morgan Kaufmann, ISBN 1558604790.
Piantadosi, Steven T., Noah D. Goodman, Benjamin A. Ellis, & Joshua B. Tenenbaum (2008), A bayesian model of the acquisition of compositional semantics, inPage: 43

job: Goodman-HCS-final

macro: handbook.cls

date/time: 25-Jun-2014/8:41

44

Noah D. Goodman and Daniel Lassiter

Proceedings of the Thirtieth Annual Conference of the Cognitive Science Society,(1620–1625).
Piantadosi, Steven T, Joshua B Tenenbaum, & Noah D Goodman (2012), Bootstrapping in a language of thought: A formal model of numerical concept learning,
Cognition 123(2):199–217.
Ramsey, Norman & Avi Pfeffer (2002), Stochastic lambda calculus and monads ofprobability distributions, in ACM SIGPLAN Notices, ACM, volume 37, (154–165).
Roberts, Craige (2012), Information structure in discourse: Towards an integrated
formal theory of pragmatics, Semantics & Pragmatics 5:1–69.
Rosch, Eleanor (1978), Principles of categorization, in Eleanor Rosch & Barbara B.
Lloyd (eds.), Cognition and categorization, Lawrence Erlbaum, (27–48).
Russell, Benjamin (2006), Against grammatical computation of scalar implicatures,
Journal of semantics 23(4):361–382.
Sauerland, Uli (2004), Scalar implicatures in complex sentences, Linguistics andphilosophy 27(3):367–391.
Scha, Remko & Yoad Winter (2014), The formal semantics of plurality, in Shalom
Lappin & Chris Fox (eds.), Handbook of Contemporary Semantic Theory, 2ndedition, Blackwell.
Shan, Chung-chieh (2010), The character of quotation, Linguistics and Philosophy
33(5):417–443.
Smith, N. J., N. D. Goodman, & M. C. Frank (2013), Learning and using language
via recursive pragmatic reasoning about other agents, in NIPS 2013.
Spivey, Michael J, Michael K Tanenhaus, Kathleen M Eberhard, & Julie C Sedivy(2002), Eye movements and spoken language comprehension: Effects of visualcontext on syntactic ambiguity resolution, Cognitive Psychology 45(4):447–481.
Stalnaker, R. (1978), Assertion, in P. Cole (ed.), Syntax and Semantics 9: Pragmatics, Academic Press.
Steedman, Mark (2001), The Syntactic Process, MIT press.
Steedman, Mark (2012), Taking Scope: The Natural Semantics of Quantifiers, MIT
Press.
Stuhlmueller, A. & N. D. Goodman (2013), Reasoning about reasoning by nested
conditioning: Modeling theory of mind with probabilistic programs, Journal of
Cognitive Systems Research .
Sutton, Peter (2013), Vagueness, Communication and Semantic Information, Ph.D.
thesis.
Sutton, R.S. & A.G. Barto (1998), Reinforcement learning: An introduction, MIT
Press.
Taylor, John R (2003), Linguistic Categorization, Oxford University Press.
Tenenbaum, J.B., C. Kemp, T.L. Griffiths, & N.D. Goodman (2011), How to growa mind: Statistics, structure, and abstraction, Science 331(6022):1279.
Van Kuppevelt, Jan (1995), Discourse structure, topicality and questioning, Journalof linguistics 31(1):109–147.
Veltman, Frank (1996), Defaults in update semantics, Journal of Philosophical Logic
25(3):221–261, ISSN 0022-3611, doi:10.1007/BF00248150.
Vogel, Adam, Max Bodoia, Christopher Potts, & Dan Jurafsky (2013), Emergenceof Gricean maxims from multi-agent decision theory, in Human Language Technologies: The 2013 Annual Conference of the North American Chapter of the

Page: 44

job: Goodman-HCS-final

macro: handbook.cls

date/time: 25-Jun-2014/8:41

Probabilistic Semantics and Pragmatics

45

Association for Computational Linguistics, Association for Computational Linguistics, Atlanta, Georgia, (1072–1081).
Wingate, David, Andreas Stuhlmueller, & Noah D Goodman (2011), Lightweight
implementations of probabilistic programming languages via transformational
compilation, in Proceedings of the 14th international conference on Artificial
Intelligence and Statistics, (131).
Zadeh, Lotfi Asker (1971), Quantitative fuzzy semantics, Information sciences
3(2):159–176.

Page: 45

job: Goodman-HCS-final

macro: handbook.cls

date/time: 25-Jun-2014/8:41