## 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

- 1.
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). - 2.
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).

- 3.
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.

- 4.
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).

- 5.
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.

- 6.
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. - 7.
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.

- 8.
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. - 9.
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).

- 10.
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.

- 11.
Note that \(P_{L_{0/1}}(V)\) is an improper prior if the range of

*V*extends to infinity in either direction. - 12.
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.

- 13.
“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.

- 14.
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. - 15.
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.

- 16.
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?

- 17.
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.

- 18.
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. - 19.
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.

## References

Adams, E. W. (1966). Probability and the logic of conditionals.

*Studies in Logic and the Foundations of Mathematics*,*43*, 265–316.Adams, E. W. (1975).

*The logic of conditionals: An application of probability to deductive logic*. Dordvecht: Springer.Bale, A. C. (2011). Scales and comparison classes.

*Natural Language Semantics*,*19*, 169–190.Barker, C. (2002). The dynamics of vagueness.

*Linguistics and Philosophy*,*25*(1), 1–36.Bartsch, R., & Vennemann, T. (1973).

*Semantic structures: A study in the relation between semantics and syntax*. New York: Athenäum.Benz, A., Jäger, G., & van Rooij, R. (2005).

*Game theory and pragmatics*. London: Palgrave Macmillan.Bergen, L., Goodman, N. D., & Levy, R. (2012). That’s what she (could have) said: How alternative utterances affect language use. In N. Miyake, D. Peebles, & R. P. Cooper (Eds.),

*Thirty-fourth annual meeting of the cognitive science society*(pp. 120–125). Berlin: Cognitive Science Society.Black, M. (1937). Vagueness. An exercise in logical analysis.

*Philosophy of Science*,*4*(4), 427–455.Borel, É. (1907). Sur un paradoxe économique: Le sophisme du tas de blé et les vérités statistiques.

*Revue du Mois*,*4*, 688–699.Büring, D. (2007b). More or less. In

*Proceedings from the Annual Meeting of the Chicago Linguistic Society*(pp. 3–17). Chicago Linguistic Society.Büring, D. (2007a). Cross-polar nomalies. In T. Friedman & M. Gibson (Eds.),

*Semantics and linguistic theory (SALT) 17*. Ithaca: CLC Publications.Clark, H. H. (1996).

*Using language*. Cambridge: Cambridge University Press.Douven, I., & Decock, L. (2015). What verities may be.

*Mind*(in press).Edgington, D. (1992). Validity, uncertainty and vagueness.

*Analysis*,*2*, 193–204.Edgington, D. (1995). On conditionals.

*Mind*,*104*(414), 235. doi:10.1093/mind/104.414.235.Edgington, D. (1997). Vagueness by degrees. In R. Keefe & P. Smith (Eds.),

*Vagueness: A reader*(pp. 294–316). Cambridge: MIT Press.Égré, P. to appear. Vagueness: Why do we believe in tolerance?.

*Journal of Philosophical Logic*.Égré, P. (2011). Perceptual ambiguity and the sorites. In R. Nouwen, R. van Rooij, U. Sauerland, & H.-C. Schmitz (Eds.),

*Vagueness in communication*(pp. 64–90). Bordeaux: Springer.Égré, P., & Barberousse, A. (2014). Borel on the heap.

*Erkenntnis*,*79*(5), 1043–1079.Fara, D. G. (2000). Shifting sands: An interest-relative theory of vagueness.

*Philosophical Topics*,*20*, 45–81.Fox, D., & Katzir, R. (2011). On the characterization of alternatives.

*Natural Language Semantics*,*19*(1), 87–107.Frank, M. C., & Goodman, N. D. (2012). Predicting pragmatic reasoning in language games.

*Science*,*336*(6084), 998.Franke, M. (2009).

*Signal to act: game theory in pragmatics*. (University of Amsterdam dissertation).Franke, M. (2011). Quantity implicatures, exhaustive interpretation, and rational conversation.

*Semantics and Pragmatics*,*4*, 1–82.Franke, M. (2012a). On scales, salience and referential language use. In M. Aloni, F. Roelofsen, & Kn Schulz (Eds.),

*Amsterdam colloquium 2011*. Amsterdam: Springer.Franke, M. (2012b). Scales, Salience and referential safety: The benefit of communicating the extreme. In Thomas C. Scott-Phillips, Mónica Tamariz, Erica A. Cartmill, & James R. Hurford (Eds.),

*The evolution of language: Proceedings of the 9th international conference (evolang 9)*(pp. 118–125).Frazee, J, & Beaver, D. (2010). Vagueness is rational under uncertainty. In Maria, A, Harald, B, Tikitu, de J, & Katrin, S. (Eds.),

*Logic, language and meaning: 17th Amsterdam colloquium, Amsterdam, The Netherlands, December 16–18, 2009, revised selected papers, pp. 153–162*. Springer.Gallin, D. (1975).

*Intensional and higher-order modal logic: With applications to Montague semantics*. New York: Elsevier.Gärdenfors, P. (2000).

*Conceptual spaces: The geometry of thought*. Cambridge: MIT press.Ginzburg, J. (1995a). Resolving questions, I.

*Linguistics and Philosophy*,*18*(5), 459–527.Ginzburg, J. (1995b). Resolving questions, II.

*Linguistics and Philosophy*,*18*(6), 567–609.Golland, D, Liang, P, & Klein, D. (2010). A game-theoretic approach to generating spatial descriptions. In

*Empirical methods in natural language processing (EMNLP)*(pp. 410–419).Goodman, N.D., & Tenenbaum, JB. (2015). Electronic. Probabilistic models of cognition. Retrieved February 5, 2015 from http://probmods.org.

Goodman, N. D., & Lassiter, D. (2015). Probabilistic semantics and pragmatics: Uncertainty in language and thought. In S. Lappin & C. Fox (Eds.),

*Handbook of contemporary semantic theory*(2nd ed.). Malden: Wiley-Blackwell.Goodman, N. D., & Stuhlmüller, A. (2012). Knowledge and implicature: Modeling language understanding as social cognition.

*Topics in Cognitive Science*,*5*(1), 173–184.Grice, H. P. (1957). Meaning.

*The Philosophical Review*,*66*(3), 377–388.Grice, H. P. (1975). Logic and conversation. In P. Cole & J. Morgan (Eds.),

*Syntax and semantics 9: Pragmatics*(pp. 41–58). New York: Academic Press.Grice, H. P. (1989).

*Studies in the way of words*. Cambridge: Harvard University Press.Griffiths, T. L., Kemp, C., & Tenenbaum, J. B. (2008). Bayesian models of cognition. In R. Sun (Ed.),

*Cambridge handbook of computational psychology*(pp. 59–100). Cambridge: Cambridge University Press.Griffiths, T. L., Vul, E., & Sanborn, A. N. (2012). Bridging levels of analysis for probabilistic models of cognition.

*Current Directions in Psychological Science*,*21*(4), 263–268.Groenendijk, J., & Stokhof, M. (1984).

*Studies in the semantics of questions and the pragmatics of answers*. Ph.D. thesis, University of Amsterdam.Hájek, A. (2003). What conditional probability could not be.

*Synthese*,*137*(3), 273–323.Heim, I. (1982).

*The semantics of definite and indefinite noun phrases*. Ph.D. thesis, University of Massachusetts at Amherst.Heim, I. (2008). Decomposing antonyms. In

*Proceedings of Sinn und Bedeutung*, (Vol. 12, pp. 212–225).Heim, I. (2006).

*Remarks on comparative clauses as generalized quantifiers*. Ms., MIT.Horn, L. (1989).

*A natural history of negation*. Chicago: University of Chicago Press.Jacobson, P. (1999). Towards a variable-free semantics.

*Linguistics and Philosophy*,*22*(2), 117–185.Jäger, G., & Ebert, C. (2009). Pragmatic rationalizability. In

*Proceedings of Sinn und Bedeutung*(Vol. 13, pp. 1–15).Jäger, G. (2007). Game dynamics connects semantics and pragmatics. In A.-V. Pietarinen (Ed.),

*Game Theory and Linguistic Meaning*(pp. 103–118). Elsevier.Kamp, H. (1975). Two theories about adjectives. In E. Keenan (Ed.),

*Formal semantics of natural language*(pp. 123–155). Cambridge: Cambridge University Press.Kamp, H. (1981). The paradox of the heap. In U. Mönnich (Ed.),

*Aspects of philosophical logic*(pp. 225–277). New York: Springer.Kao, J. T., Wu, J. Y., Bergen, L., & Goodman, N. D. (2014). Nonliteral understanding of number words.

*Proceedings of the National Academy of Sciences*,*111*(33), 12002–12007.Kehler, A., & Rohde, H. (2013). A probabilistic reconciliation of coherence-driven and centering-driven theories of pronoun interpretation.

*Theoretical Linguistics*,*39*, 1–37.Kennedy, C. (1997).

*Projecting the adjective: The syntax and semantics of gradability and comparison*: U.C., Santa Cruz Dissertation.Kennedy, C. (2007). Vagueness and grammar: The semantics of relative and absolute gradable adjectives.

*Linguistics and Philosophy*,*30*(1), 1–45.Kennedy, C., & McNally, L. (2005). Scale structure, degree modification, and the semantics of gradable predicates.

*Language*,*81*(2), 345–381.Klein, E. (1980). A semantics for positive and comparative adjectives.

*Linguistics and Philosophy*,*4*(1), 1–45.Klein, E. (1991). Comparatives. In A. von Stechow & D. Wunderlich (Eds.),

*Semantik: Ein internationales Handbuch der zeitgenössischen Forschung*. New York: Walter de Gruyter.Kratzer, A. (1998). Scope or pseudo-scope? Are there wide-scope indefinites? In S. Rothstein (Ed.),

*Events and grammar*. Dordrecht: Kluwer.Kyburg, A., & Schubert, L. (1993). Reconciling sharp true/false boundaries with scalar vagueness. In

*First Conference of the Pacific Association for Computational Linguistics (PACLING 1993)*(pp. 53–62). Simon Fraser University.Kyburg, A. (2000). When vague sentences inform: A model of assertability.

*Synthese*,*124*(2), 175–191.Lassiter, D. (2011). Vagueness as probabilistic linguistic knowledge. In R. Nouwen, R. van Rooij, U. Sauerland, & H.-C. Schmitz (Eds.),

*Vagueness in communication*(pp. 127–150). Berlin: Springer.Lassiter, D. (2012). Presuppositions, provisos, and probability.

*Semantics & Pragmatics*,*5*, 1–37.Lassiter, D. (2015). Adjectival modification and gradation. In S. Lappin & C. Fox (Eds.),

*Handbook of contemporary semantic theory*(2nd ed.). New York: Wiley-Blackwell.Lassiter, D., & Goodman, N. D. (2013). Context, scale structure, and statistics in the interpretation of positive-form adjectives. In T. Snider (Ed.),

*Semantics & linguistic theory (SALT) 23*(pp. 587–610). Ithaca: CLC Publications.Lawry, J. (2008). Appropriateness measures: An uncertainty model for vague concepts.

*Synthese*,*161*(2), 255–269.Leitgeb, H. (2014). The stability theory of belief.

*Philosophical Review*,*123*(2), 131–171.Lewis, D. (1969).

*Convention: A philosophical study*. Cambridge: Harvard University Press.Lewis, D. (1970). General semantics.

*Synthese*,*22*(1), 18–67.Lin, H., & Kelly, K. T. (2012). Propositional reasoning that tracks probabilistic reasoning.

*Journal of Philosophical Logic*,*41*(6), 957–981.Luce, R. D. (1959).

*Individual choice behavior: A theoretical analysis*. New York: John Wiley.MacKay, D. J. C. (2003).

*Information theory, inference, and learning algorithms*. Cambridge: Cambridge University Press.Montague, R. (1973). The proper treatment of quantification in ordinary English. In J. Hintikka, J. Moravcsik, & P. Suppes (Eds.),

*Approaches to natural language*(Vol. 49, pp. 221–242). Dordrecht: Reidel.Morzycki, M. To appear.

*Modification*. Cambridge University Press.Neal, RM. (1993).

*Probabilistic inference using markov chain monte carlo methods*. Department of Computer Science, University of Toronto (Technical Report CRG-TR-93-1).Oaksford, M., & Chater, N. (2007).

*Bayesian rationality: The probabilistic approach to human reasoning*. Oxford: Oxford University Press.Over, D. E. (2009). New paradigm psychology of reasoning.

*Thinking and Reasoning*,*15*(4), 431–438.Pearl, J. (2000).

*Causality: Models, reasoning and inference*. Cambridge: Cambridge University Press.Potts, C. (2008).

*Interpretive economy, Schelling points, and evolutionary stability*. Ms., University of Massachusetts at Amherst.Qing, C., & Franke, M. (2014). Gradable adjectives, vagueness, and optimal language use: A speaker-oriented model. In T. Snider (Ed.),

*Semantics & linguistic theory (SALT) 24*(pp. 23–41). Ithaca: CLC Publications.Roberts, C. (1996).

*Information structure in discourse: Towards an integrated formal theory of pragmatics*. Working Papers in Linguistics-Ohio State University Department of Linguistics, (pp. 91–136).Sassoon, G. W. (2013). A typology of multidimensional adjectives.

*Journal of Semantics*,*30*(3), 335–380.Shannon, C. E. (1948). A mathematical theory of communication.

*Bell System Technical Journal*,*27*, 379–423.Silverman, B. W. (1986).

*Density estimation for statistics and data analysis*. London: Chapman & Hall.Smith, N. J., Goodman, N., & Frank, M. (2013). Learning and using language via recursive pragmatic reasoning about other agents. In C. J. C. Burges, L. Bottou, M. Welling, Z. Ghahramani, & K. Q. Weinberger (Eds.),

*Advances in neural information processing systems, 26*(pp. 3039–3047). New York: Curran Associates Inc.Solt, S. (2011). Notes on the comparison class. In R. Nouwen, R. van Rooij, U. Sauerland, & H.-C. Schmitz (Eds.),

*Vagueness in communication*(pp. 189–206). Berlin: Springer.Sorensen, R. (2012). The sorites and the generic overgeneralization effect.

*Analysis*,*72*(3), 444–449.Stalnaker, R. (1978). Assertion. In P. Cole (Ed.),

*Syntax and semantics 9: Pragmatics*. Chicago: Academic Press.Stanley, J., & Szabó, Z. (2000). On quantifier domain restriction.

*Mind & Language*,*15*(2–3), 219–261.von Stechow, A. (1984). Comparing semantic theories of comparison.

*Journal of Semantics*,*3*(1), 1–77.Sutton, P. (2013).

*Vagueness, communication and semantic information*. Ph.D. thesis, King’s College, University of London.Sutton, R. S., & Barto, A. G. (1998).

*Reinforcement learning: An introduction*. Cambridge: MIT Press.Tenenbaum, J. B., Kemp, C., Griffiths, T. L., & Goodman, N. D. (2011). How to grow a mind: Statistics, structure, and abstraction.

*Science*,*331*(6022), 1279–1285.van Rooij, R. (2003). Questioning to resolve decision problems.

*Linguistics and Philosophy*,*26*(6), 727–763.Vogel, A., Bodoia, M., Potts, C., & Jurafsky, D. (2013a). Emergence of Gricean maxims from multi-agent decision theory. In

*Human language technologies: The 2013 annual conference of the North American chapter of the Association for Computational Linguistics*(pp. 1072–1081). Stroudsburg, PA: Association for Computational Linguistics.Vogel, A., Potts, C., & Jurafsky, D. (2013b). Implicatures and nested beliefs in approximate Decentralized-POMDPs. In

*Proceedings of the 2013 annual conference of the Association for Computational Linguistics*(pp. 74–80). Stroudsburg, PA: Association for Computational Linguistics.Vul, E., Goodman, N. D., Griffiths, T. L., & Tenenbaum, J. B. (2014). One and done? Optimal decisions from very few samples.

*Cognitive Science*,*38*(4), 599–637.Xu, F., & Tenenbaum, J. B. (2007). Word learning as Bayesian inference.

*Psychological Review*,*114*(2), 245–272.

## 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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### Keywords

- Vagueness
- Probability
- Cognitive science
- Sorites paradox