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Psycholinguistic evidence for restricted quantification

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Abstract

Quantificational determiners are often said to be devices for expressing relations. For example, the meaning of every is standardly described as the inclusion relation, with a sentence like every frog is green meaning roughly that the green things include the frogs. Here, we consider an older, non-relational alternative: determiners are tools for creating restricted quantifiers. On this view, determiners specify how many elements of a restricted domain (e.g., the frogs) satisfy a given condition (e.g., being green). One important difference concerns how the determiner treats its two grammatical arguments. On the relational view, the arguments are on a logical par as independent terms that specify the two relata. But on the restricted view, the arguments play distinct logical roles: specifying the limited domain versus supplying an additional condition on domain entities. We present psycholinguistic evidence suggesting that the restricted view better describes what speakers know when they know the meaning of a determiner. In particular, we find that when asked to evaluate sentences of the form every F is G, participants mentally group the Fs but not the Gs. Moreover, participants forego representing the group defined by the intersection of F and G. This tells against the idea that speakers understand every F is G as implying that the Fs bear relation (e.g., inclusion) to a second group.

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Notes

  1. The expression ‘ιX:Frogs(X)’ in (5a), glossed as the frogs, is shorthand for ‘ιX[∀x[X(x) ≡ Frog(x)]]’: the thingsX such that for each thingx, itx is one of themX iff itx is a frog. In using the iota operator here, we do not commit ourselves to the view that sentences like every frog is green presuppose the existence of some contextually relevant frogs. The expression ‘ιX[∀x[X(x) ≡ Frog(x)]]’ indicates the frogs; for present purposes, we remain agnostic about the nature of this plurality and any presuppositional commitments. In contrast, expressions like ‘Frog(x)’ and ‘Green(x)’ do not, on their own, imply the existence of a particular group (the frogs or the green things).

  2. Unless one thinks that most means something like “significantly more than half” for some pragmatically determined threshold value; see Denić and Szymanik (2022) for discussion of this possibility. In which case, one might prefer to replace “with at least one remainder” with “with a significant remainder, given some pragmatically determined threshold”.

  3. Given a manifestly collective predicate, like all students gathered in the hall, it might be less surprising for participants to group the students. After all, an individual student cannot gather. But at least as it is used in this experimental context, be blue is a distributive predicate in the sense that all the circles are blue if and only if each individual circle is blue. Even so, the studies cited above find that participants psychologically group the big circles upon encountering every big circle or all big circles to a greater extent than when encountering each big circle in the very same experimental context.

  4. A reviewer rightly notes that these results do not rule out the following possibility: maybe the things named by the external argument are mentally grouped at some stage of processing, but that group is quickly discarded before the subsequent cardinality question. While logically possible, this strikes us as unlikely, given what is known about the workings of the Approximate Number System (especially the result noted above that observers can represent the cardinality of three groups simultaneously with no apparent cost over and above representing just one; Halberda et al. 2006; Zosh et al. 2011). In any case, for the present argument, all that is needed is a difference in how both of the quantifier’s arguments are treated, as the relational view predicts them to be treated on a par and the restricted view predicts them to be treated asymmetrically. Even if the present results merely reflect a difference in likelihood of the extension of either argument being retained in memory, the finding would still fit better with the restricted view.

  5. Post-verification cardinality questions probing each feature were compared against baseline questions probing that same feature, controlling for the possibility that one of the features is more visually salient than the other. To further control for this possibility, Knowlton et al. (2021b) conducted a “swapped argument” version of Experiment 1, in which participants were asked to evaluate sentences like every blue circle is big instead of every big circle is blue.

  6. This is the hypothesis indicated by the representation in (6c), namely ‘{x: x is a big circle} = {x: x is a big circle} ∩ {x: x is blue}’. A reviewer rightly points out that the expectations about the behavioral repercussions of this representation differ based on how one interprets ‘∩’. For simplicity, we assumed ‘{x: x is a big circle} ∩ {x: x is blue}’ and ‘{x: x is a big circle that is blue}’ are identical. But one might instead suppose that ‘{x: x is a big circle} ∩ {x: x is blue}’ calls for representation both of the big circles and of the blue circles and then arrives at representation of the big blue circles by means of a further computation: intersecting those two sets. In this case, one might expect (6c) to lead to representation of three sets: the big circles, the blue circles, and the big blue circles. In any case, the data do not bear out this prediction.

  7. The question arises whether the difference in logical role itself is responsible for this difference in which groups are represented. Alternatively, this latter difference might be explained by a difference in whether the terms are themselves group-denoting as opposed to being an open formula including a singular variable. In this context, distinctions between each, every, and all might be relevant. The proposed meaning in (5) is for every. Knowlton (2021) argues that both each and every have restricted meanings, and both distributively apply the predicate supplied by the external argument to the members of the restricted domain, but only every calls for grouping that restricted domain; each calls for treating it as a series of independent individuals. On the other hand, all is restricted and calls for grouping the restricted domain (like every), but might not call for distributively applying the predicate supplied by the external argument.

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Acknowledgements

We thank Amy Rose Deal and two anonymous Natural Language Semantics reviewers for their thoughtful comments on previous versions of this manuscript. For helpful discussion about this project, we thank Alexis Wellwood, Valentine Hacquard, Norbert Hornstein, and audiences at ELM 1, CUNY/HSP 33, and SALT 30. We also thank Simon Chervenak for help collecting the data. This work was supported by funding from the National Science Foundation (#BCS-2017525 and #NRT-1449815).

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Authors and Affiliations

  1. MindCORE, University of Pennsylvania, Philadelphia, PA, USA

    Tyler Knowlton

  2. Department of Philosophy, Rutgers University, New Brunswick, NJ, USA

    Paul Pietroski

  3. Department of Linguistics and Department of Philosophy, University of Maryland, College Park, MD, USA

    Alexander Williams

  4. Department of Psychological & Brain Sciences, Johns Hopkins University, Baltimore, MD, USA

    Justin Halberda

  5. Department of Linguistics, University of Maryland, College Park, MD, USA

    Jeffrey Lidz

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  1. Tyler Knowlton
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Knowlton, T., Pietroski, P., Williams, A. et al. Psycholinguistic evidence for restricted quantification. Nat Lang Semantics 31, 219–251 (2023). https://doi.org/10.1007/s11050-023-09209-w

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