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The mental representation of universal quantifiers

Abstract

A sentence like every circle is blue might be understood in terms of individuals and their properties (e.g., for each thing that is a circle, it is blue) or in terms of a relation between groups (e.g., the blue things include the circles). Relatedly, theorists can specify the contents of universally quantified sentences in first-order or second-order terms. We offer new evidence that this logical first-order vs. second-order distinction corresponds to a psychologically robust individual vs. group distinction that has behavioral repercussions. Participants were shown displays of dots and asked to evaluate sentences with each, every, or all combined with a predicate (e.g., big dot). We find that participants are better at estimating how many things the predicate applied to after evaluating sentences in which universal quantification is indicated with every or all, as opposed to each. We argue that every and all are understood in second-order terms that encourage group representation, while each is understood in first-order terms that encourage individual representation. Since the sentences that participants evaluate are truth-conditionally equivalent, our results also bear on questions concerning how meanings are related to truth-conditions.

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Notes

  1. For these purposes, extensions include functions in extension from possible worlds to sets of entities, from variables to assignable values, from functions to functions, etc. Church (1941) contrasted these abstracta—identifiable, given sufficient ontology, with sets of ordered pairs—with what he called functions in intension, or procedures that have extensions, explicitly allowing that distinct procedures can have the same extension; cp. Frege’s (1893) procedural notion of function, and Chomsky’s (1986) notion of I-language.

  2. Words like unicorn, centaur, and ghost present familiar difficulties for the extensional/externalist approach. These words are not true of anything, yet they have different meanings. Compare picture of a centaur watching a ghost ride a unicorn and picture of a ghost watching a unicorn ride a centaur; see, e.g., Goodman (1949, 1953), and Chomsky (1957). One response is to posit possible worlds at which unicorns exist (e.g., Lewis, 1986); though Kripke (1980) offered powerful reasons for not doing so. Alternatively, one can posit many “ways of presenting” the empty set; but this suggests that extensional semantic values are distinct from (even if determined by) lexical meanings.

  3. By contrast, assuming displacement of determiner phrases, we could adopt a syncategorematic treatment that posits sentence frames like [S [D every [N …]]i [S …ti…]]. Truth conditions for instances of this frame can be specified in a way that corresponds to (1): relative to any assignment A of values to indices, \( {\forall} \)x: ⟦[N …]⟧(x)[⟦[S …ti…]⟧A:x/i ]; i.e., for each individual mapped to truth by the semantic value of the noun (or noun phrase), truth is the semantic value of the embedded sentence relative to the assignment that is just like A except that the individual in question is assigned to the index of the relevant trace of displacement. This way of specifying an extensional semantic role for [S [D every [N …]]i [S …ti…]], which invites the further step of dispensing with appeals to truth values and functions as semantic values for predicates, is not without virtues; see Davidson (1967) and Higginbotham (1985). But our questions would remain, since the semantic role in question could also be specified in ways corresponding to (2–4).

  4. And in (17a), the matters; cp. All soldiers {*surrounded/admired} the fortress. Some collective predicates, like be numerous and be a good team, cannot be used with any of the universal quantifiers (Champollion, 2015; Dowty, 1987; Winter, 2002). But since these predicates also conflict with other quantifiers, including the manifestly second-order most, this is not a diagnostic for second-order quantification.

  5. Moreover, as Vendler notes, All of those dots are similar can be heard as true even if there is no single dimension on which each pair of dots is similar. But it’s much harder to find a sensible interpretation for Each of those dots is similar or Every dot is similar. There are also cases where sentences with each can give rise to distributive readings, but those with every cannot; see Sect. 5.

  6. Their claim is not that meanings are verification strategies or that people always use a certain strategy to evaluate a given sentence. In ordinary (uncontrolled) contexts, many considerations can be relevant to choosing a verification strategy. The claim is rather that the representational format of the meaning contributes a detectable influence over verification procedures. And if other considerations are controlled for, then variation in verification strategies for different expressions can reasonably be attributed to variation in the way the meanings of those expressions are mentally encoded.

  7. This model does not only apply to numerosities perceived visually. It has been applied to many other psychological dimensions as well (e.g., loudness, brightness, distance) across multiple modalities (e.g., vision, audition, touch) (Cantlon et al., 2009; Lu and Dosher, 2014; Odic et al., 2016; Stevens, 1964).

  8. The parameter β in (25–26) can either be thought of as representing the degree of compression/expansion of the signal or of the response code that the Gaussian distributions of Fig. 1 are mapped to (see Izard and Dehaene, 2008).

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Acknowledgements

This work was supported by the National Science Foundation (Grants #NRT-1449815 to T.K. and #BCS-2017525 to T.K. and J.L.) and by the James S. McDonnell Foundation (Grant on The Nature and Origins of the Human Capacity for Abstract Combinatorial Thought to P.P., J.H., and J.L.). We are grateful to two anonymous Linguistics and Philosophy reviewers for valuable feedback on previous drafts. For helpful discussion throughout this project, we thank: Alexander Williams, Zoe Ovans, Ellen Lau, Darko Odic, Nicolò Cesana-Arlotti, and Adam Liter, as well as audiences at MACSIM 2017, LSA 2019, and CUNY 2019.

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Knowlton, T., Pietroski, P., Halberda, J. et al. The mental representation of universal quantifiers. Linguist and Philos 45, 911–941 (2022). https://doi.org/10.1007/s10988-021-09337-8

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Keywords

  • Psychosemantics
  • Quantification
  • Semantics-cognition interface
  • Universal quantifiers