Abstract
We derive a probabilistic account of the vagueness and context-sensitivity of scalar adjectives from a Bayesian approach to communication and interpretation. We describe an iterated-reasoning architecture for pragmatic interpretation and illustrate it with a simple scalar implicature example. We then show how to enrich the apparatus to handle pragmatic reasoning about the values of free variables, explore its predictions about the interpretation of scalar adjectives, and show how this model implements Edgington’s (Analysis 2:193–204,1992, Keefe and Smith (eds.) Vagueness: a reader, 1997) account of the sorites paradox, with variations. The Bayesian approach has a number of explanatory virtues: in particular, it does not require any special-purpose machinery for handling vagueness, and it is integrated with a promising new approach to pragmatics and other areas of cognitive science.
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Notes
More precisely, these are characteristics which are robust especially (perhaps only) for the relative adjectives on which we focus here. Absolute adjectives such as full, empty, wet, dry, open, closed, safe, dangerous are somewhat different: see Kennedy and McNally (2005), Kennedy (2007), Lassiter (2015), Morzycki (to appear).
The ratio definition of conditional probability is convenient and simple, but not at all crucial for us. We would also be content to take conditional probability as basic, as many philosophers of probability have recommended (e.g., Hájek 2003).
We will not address speaker modeling here, but our model does make predictions: to the extent that listeners do reason to level 1, a reflective speaker should choose utterances by reference to an \(L_1\) model who is reasoning about an \(S_1\) who is reasoning about an \(L_0\). See also Qing and Franke (2014) for a model of vague interpretation which builds on the one described here, but places greater emphasis on speaker modeling.
On the equivalence between random variables and question denotations (on the Groenendijk and Stokhof (1984) interpretation), see van Rooij (2003). Note in particular that an answer A is just a set of possible worlds, i.e., a proposition which is a cell in the partition which the question denotes.
A notable simplification in our model is the assumption that the speaker knows the true answer with certainty. This assumption could be relaxed either by allowing that the speaker samples a possible world from his personal probability distribution and then proceeds with the calculations described here, or by using expected informativity instead of simple informativity in the utility function. For relevant discussion see Goodman and Lassiter (2015).
Plausibly, there is a domain restriction variable in the quantifier’s meaning which must be inferred (Stanley and Szabó 2000). In the case at hand, the domain restriction does not vary across the alternatives under consideration, and so we can safely ignore this variable. However, there are many cases in which this variable is not clearly given, and its inference will likely interact with other aspects of the pragmatic reasoning considered here.
Note that the precise numerical values derived here depend on the Luce choice parameter \(\alpha \), here set to 4. The qualitative preference for ALL does not depend on this parameter, though. For example, with \(\alpha = 10\), the dispreference for ALL over some when \(n=6\) would not be on the order of .001 / .999, but a much stronger \([1.7 \times 10^{-8}] / [1 - 1.7 \times 10^{-8}]\)—almost indistinguishable from a deterministic preference for the maximum utility choice. With \(\alpha = 1\), the dispreference for SOME has the more moderate value .16 / .84. Ultimately we can only fit this model parameter to data, or attempt to manipulate it qualitatively in an experimental setting.
This is a \(\langle d,\langle e,t \rangle \rangle \) semantics for adjectives in the style of von Stechow (1984). For current purposes, it does not matter whether we use this analysis or an \(\langle e,d \rangle \) treatment as recommended by Bartsch and Vennemann (1973), Kennedy (1997), Kennedy (2007). The only modification needed would be in the definition of the POS morpheme/type-shifter.
In the context of our pragmatic model, assuming two unrelated thresholds for tall and short is enough to derive reasonable interpretations, as we will see below; it even derives the fact that tall and short are contraries without stipulation. We could, if we wanted, add a lexical stipulation to this effect (\(\theta _{tall} \ge \theta _{short}\)), but we do not know of any compelling reason to do so. It would also be possible to assume that these expressions are contradictories (\(\theta _{tall} = \theta _{short}\), so that short \(\equiv \) not tall). Whether the latter assumption is reasonable is a matter of debate: see Horn (1989), Heim (2006), Heim (2008), Büring (2007a), Büring (2007b) among many others for arguments pro and con.
We do not, of course, want to claim that listeners making such inferences in real time actually consider an infinite set of \(\langle A, V \rangle \) pairs as hypotheses. The problem is, however, a very general one about how humans make inferences with a very large hypothesis space—something that we are surprisingly good at. The model presented in this paper is a high-level computational theory for which a variety of techniques are available that can make inference tractable: this includes lazy computation (reason only about variables that are actually present in the utterance and need to be inferred) and Markov Chain Monte Carlo techniques such as the one that we use below to simulate Bayesian posteriors. For further discussion see, for example, Griffiths et al. (2012), Vul et al. (2014), Goodman and Tenenbaum (2015).
We are assuming that A and V are independent for the listener in the prior, and are related only via the interpretation process.
Note that we could continue to iterate to some higher \(L_n\) and pass the variable up. It is also possible to marginalize at non-maximal levels. We have found that, under many conditions, scalar adjectives receive implausibly weak interpretations if marginalization happens at \(L_0\) (Goodman and Lassiter 2015). However, many other possibilities remain to be explored, e.g., a level-3 pragmatic intepreter with level-1 or -2 marginalization.
Note that \(P_{L_{0/1}}(V)\) is an improper prior if the range of V extends to infinity in either direction.
The kernel density is a nonparametric estimate of the density of a continuous function from a finite number of samples, given certain assumptions about the smoothness of the function. See, for example, Silverman (1986).
Simulations took 5,000,000 samples from \(P_{L_1}(h, \theta _{tall} | \ u)\) with a burn-in of 5000 (i.e., the first 5000 samples were discarded because to avoid dependence on the starting point of the simulation). Unlike Lassiter and Goodman (2013) we do not rescale variables to fall within [0, 1], since we are not concerned with the difference between bounded and unbounded scales here.
“Approximately” because the normal distribution has support over the entire real line, but it does not make sense to talk about heights less than zero. However, for the priors that we will consider, only a negligible portion of the prior probability mass falls below zero.
We will not deal with higher-order vagueness in detail here, but two compatible lines of attack suggest themselves. First, we could treat definitely as a vague modal, following Lassiter (2011). Second, we could allow uncertainty about the relevant prior distribution, which would generate meta-uncertainty about the interpretation.
We call this inference “metalinguistic” because it requires us to reflect directly on linguistic interpretation and usage. Unfortunately for psycholinguists and field linguists, this is a difficult and fairly unusual task for most people: usually inferences about language serve as a means to communicate about the non-linguistic world, rather than as an end in themselves, and we have to figure out some more indirect way to get at them.
A cognitive theory of the interpretation of vague expressions does not answer the metaphysical question, but it does perhaps sharpen the subject matter of this question. If we had a satisfying probabilistic theory of how listeners interpret vague expressions and what information they extract from them, as well as an account of why and when speakers choose to use vague expressions, what phenomena would remain to be explained by an answer to the metaphysical question?
Our discussion follows Edgington’s seminal work most closely. For other related work connecting probability with philosophical and linguistic questions involving vagueness, see Borel (1907) (with translation and commentary in Égré and Barberousse (2014)), Black (1937), Kyburg and Schubert (1993), Kyburg (2000), Lawry (2008), Frazee and Beaver (2010), Lassiter (2011), Égré (2011), Sutton (2013); Égré to appear. A detailed comparative analysis of these accounts would take us too far afield here; see however Égré and Barberousse (2014); Égré to appear for some discussion along these lines.
We simplify here by assuming that the value of \(\theta _{tall}\) is fixed for all instances of “tall” in a single sentence whose arguments are drawn from a common prior. This entails that “y is tall and z is not” has probability 0 when y and z have a common prior and z is taller than y. The account of the sorites is not much different qualitatively if we relax this assumption, allowing that \(\theta _{tall}\) is drawn independently for the two instances of \(\theta _{tall}\).
Note also that, given the assumption of common prior and \(\theta _{tall}\), it doesn’t matter whether the utterance u referred to in Eq. 32 is “\(x_m\) is tall” or “\(x_{m-1}\) is tall”. On the Edgington-inspired account of the sorites that we are considering, the prior is what drives the interpretation, not the individuals’ actual height.
Considerations specific to the theory of conditionals are of great relevance here: for example, the inadequacies of the material conditional interpretation of English if are well-known, and we do not know whether the metalinguistic, suppositional account just sketched will ultimately be viable.
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Acknowledgments
Thanks to Michael Franke, Chris Potts, Chris Kennedy, Adrian Brasoveanu, Paul Égré, Alexis Wellwood, Lenhart Schubert, Richard Dietz, two Synthese reviewers, three SALT 23 reviewers, participants in our 2013 ESSLLI course “Probability in semantics and pragmatics”, participants in Lassiter’s 2014 NASSLLI course “Language understanding and Bayesian inference”, and audiences at SALT 23, Stanford, Northwestern, Brown, U. Chicago, and UT-Austin. This paper is modified and extended from Lassiter and Goodman (2013), which appeared in the proceedings of the conference Semantics & Linguistic Theory 23. This work was supported by a James S. McDonnell Foundation Scholar Award to NDG and by ONR Grant N00014-13-1-0788.
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Lassiter, D., Goodman, N.D. Adjectival vagueness in a Bayesian model of interpretation. Synthese 194, 3801–3836 (2017). https://doi.org/10.1007/s11229-015-0786-1
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DOI: https://doi.org/10.1007/s11229-015-0786-1