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Vague judgment: a probabilistic account

  • S.I.: Vagueness and Probability
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Abstract

This paper explores the idea that vague predicates like “tall”, “loud” or “expensive” are applied based on a process of analog magnitude representation, whereby magnitudes are represented with noise. I present a probabilistic account of vague judgment, inspired by early remarks from E. Borel on vagueness, and use it to model judgments about borderline cases. The model involves two main components: probabilistic magnitude representation on the one hand, and a notion of subjective criterion. The framework is used to represent judgments of the form “x is clearly tall” versus “x is tall”, as involving a shift of one’s criterion, and then to derive observed patterns of acceptance for borderline contradictions, namely sentences of the form “x is tall and not tall”, relative to the acceptance of their conjuncts.

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Notes

  1. A more common characterization of the analog-digital distinction is in terms of a continuous versus a discrete system of representations. The model that follows, however, is not committed to the assumption that analog representation is necessarily a mapping to a continuum. In that, the present definition agrees with Maley (2011), according to whom “analog representation is representation in which the represented quantity covaries with the representational medium, regardless of whether the representational medium is continuous or discrete.” Unlike Maley, I do not assume the covariation is necessarily given by a linear function.

  2. In this example and in the examples that follow, I am making the assumption that the random variable \(X_{i}\) is the same for each i, i.e. does not depend on the position of the increment. That assumption is not essential, in particular it can be relaxed in the application of the Central Limit Theorem, provided adequate convergence conditions are satisfied. For present purposes, however, I confine myself to the assumption of an identical distribution.

  3. See Wright (1995) on the idea of an essential connection between vagueness and faultless variability. See Kennedy (2013) for a recent discussion.

  4. What I call the standard here may therefore equally be an ideal value, an actual value or an average value, used as a specific comparison point. See Egré and Cova (2015) and Bear and Knobe (2016) for more on the distinction between statistical versus normative comparison points. There can exist multiple standards or comparison points along a single dimension that are active for the same predicate. What I here call the criterion is the personal threshold value determined as a function of those (possibly multiple) comparison points. In Egré and Cova (2015), we use the term “standard” for what I here call “criterion”.

  5. See Stevens (1966) for a similar argument concerning the perceived value of money.

  6. Further support for the idea that even digital numbers are analogically represented can be found in the distance effect evidenced by Moyer and Landauer (1967), who found that the time required to compare Arabic digits as well as the accuracy in comparison are inversely proportional to the numerical distance between the digits. This effect, widely replicated including with number words (see Dehaene 1997; Feigenson et al. 2004), indicates that precise quantities too are transduced with noise. In Moyer and Landauer’s original experiment, the mapping operates without even the need for unit specification. It is very likely that it remains operative when digital numbers are communicated in relation to specific quantities (such as prices, heights, etc).

  7. For the green curves, we computed the values at each multiple of a dm, and interpolated the other points.

  8. The claim may seem counterintuitive, but by this I mean that the coarser the granularity, the more reliable our judgments will be that a magnitude should be assigned to this or to that number on the corresponding scale. A subject asked to estimate the height of a building with a margin of error of 10 m is less likely to make errors than one asked to evaluate the height of that building with a margin of 1 m.

  9. Stevens (1957) calls “prothetic” those continua on which “discrimination is mediated by an additive or prothetic process at the physiological level”. The model I propose basically handles vague predicates as prothetic in that general sense. Whether the sort of process here envisaged has any psychological reality is certainly not obvious. In particular, the model shares a number of assumptions with what is known as the Rasch model of categorization (see in particular Verheyen et al. 2010). The Rasch model, as far as I can see, does not make specific assumptions about underlying processes. A comparison between both models lies beyond the scope of this paper.

  10. See also Sassoon (2010) for a discussion of the typology of gradable adjectives in relation to measurement scales. Sassoon argues that several adjectives, among them negative adjectives such as “short”, do not appear to come with a ratio scale, but only come with an interval scale instead. However, her discussion of the typology of ratio scales mostly concerns the lexicalization of ratio phrases, and is not about psychophysics proper. While Sassoon expresses skepticism toward Stevens’ hypothesis of the universality of ratio scales for prothetic predicates, she is nevertheless cautious not to reject it (see the discussion in 3.6 of her paper). For example, she admits that an adjective such as “happy”, however multidimensional the predicate might be, could be used with a ratio scale in mind.

  11. See also Krantz et al. (1971, p. 141), who write (admittedly with some caution) “it may be possible to obtain orderings of intervals from a properly designed rating scale.”

  12. I am indebted to James Hampton for bringing this issue of the interaction between clarity judgments and steepness to my attention a few years ago.

  13. Thanks to a reviewer and to E. Chemla on that point.

  14. For more on the pragmatic explanation of the acceptance of borderline contradictions, I refer to Alxatib et al. (2013), Cobreros et al. (2014) and Egré and Zehr (2016).

  15. In the color experiment, the response curves for “x is blue” and “x is not blue” do intersect at a probability of about 0.5, and are roughly symmetrical. It is therefore natural to refer both sentences to the same “baseline” criterion. If on average participants were using distinct criteria for “blue” and “not blue”, the curves should intersect at a distinct position, as shown in Fig. 8. Hence, we do not need to assume that “blue” and “not blue”, when presented in isolation, are evaluated relative to separate strict criteria. This is an interesting difference with the setting of Alxatib and Pelletier (2011)’s experiment, where some subjects who check True to “x is tall and not tall” check False to “x is tall” and to “x is not tall”. To explain the behavior of those subjects, it can’t be the case that the separate sentences “x is tall” and “x is not tall” are referred to a common criterion. Instead, we need to assume that the conjuncts, when uttered in isolation, are assessed relative to distinct criteria. The difference may be attributable to the fact that, in the color categorization experiment, the conjunctive sentence and the conjuncts were never presented together on the same screen, but always in separate blocks, whereas in the setting of Alxatib and Pelletier’s experiment, they are presented simultaneously. A test of the present model on Alxatib and Pelletier’s data based on those assumption confirms that the observed proportions of True answers to the “and” and “neither” descriptions fall, in the case of the man of middling height, in the interval predicted from the observed proportions for the conjuncts. However, the number of items is too limited and the predicted intervals too narrow for that test to be reliable. Also, the present model does not concern itself with finer differences in acceptance between “x is tall and not tall” versus “x is neither tall nor not tall”. I refer to Egré and Zehr (2016) for a study of that question.

  16. My use of “algebraic” versus “probabilistic” is inspired from Luce (1959), who draws a similar opposition in relation to his account of imperfect discrimination.

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Acknowledgments

Special thanks to Denis Bonnay, Emmanuel Chemla, Vincent de Gardelle, Joshua Knobe, David Ripley, Jérôme Sackur, Benjamin Spector, Steven Verheyen, and to Richard Dietz and four anonymous reviewers for very helpful comments and criticisms. I am particularly grateful to a reviewer for pointing out a mistaken independence assumption in the first version of the paper, and to E. Chemla for suggesting the proper way to fix it. I also thank Galit Agmon, Nick Asher, Stefan Buijsman, Heather Burnett, Hartry Field, Yossi Grodzinsky, James Hampton, Roni Katzir, Chris Kennedy, Uriah Kriegel, Joëlle Proust, Diana Raffman, François Récanati, Jack Spencer, Jason Stanley, for further comments or conversations, as well as audiences at the Institut Jean-Nicod in Paris, at the Language, Logic and Cognition Center in Jerusalem, at Tel Aviv University, at MIT, at the University of Chicago, and at the Swedish Collegium for Advanced Study in Uppsala. Thanks to the ANR Program TrilLogMean ANR-14-CE30-0010-01 for support, as well as to grants ANR-10-LABX-0087 IEC et ANR-10-IDEX-0001-02 PSL* for research carried out at the Department of Cognitive Studies of ENS. The first version of this paper was written and submitted during the summer of 2014, and the latest version was revised while on a research fellowship at the Swedish Collegium for Advanced Study.

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  1. Institut Jean-Nicod (CNRS, EHESS, ENS), Paris, France

    Paul Égré

  2. Department of Cognitive Studies & Department of Philosophy, Ecole Normale Supérieure, Paris Sciences et Lettres (PSL), 29 rue d’Ulm, 75005, Paris, France

    Paul Égré

  3. Swedish Collegium for Advanced Study, Uppsala, Sweden

    Paul Égré

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Égré, P. Vague judgment: a probabilistic account. Synthese 194, 3837–3865 (2017). https://doi.org/10.1007/s11229-016-1092-2

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