The mental representation of universal quantifiers

Knowlton, Tyler; Pietroski, Paul M.; Halberda, Justin; Lidz, Jeffrey

doi:10.1007/s10988-021-09337-8

Cited by 9 publications

(39 citation statements)

References 58 publications

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“…Researchers are starting to look at the interface between generalized universal quantifiers and nonsymbolic quantities. Knowlton et al (2022) present experimental results suggesting that verifying sentences with the quantifiers "all" and "every" against visual displays triggers a representation of the cardinality of a set, whereas "each" does not.…”

Section: Open Question 5 Do the Proportional Quantifiers Extract Rati...mentioning

confidence: 79%

“…Knowlton et al. (2022) present experimental results suggesting that verifying sentences with the quantifiers “all” and “every” against visual displays triggers a representation of the cardinality of a set, whereas “each” does not.…”

Section: Shared Processing Mechanismsmentioning

confidence: 99%

“…For instance, while “each” tends to refer to individuals and their properties, “all” and “every” usually refer to sets of objects. Recently acquired preliminary evidence suggests that this difference translates into variability in mental representations of the universal quantifiers (Knowlton, Pietroski, Halberda, & Lidz, 2022). Here, we do not take a position on whether generalized universal quantifiers should recruit nonsymbolic quantities but only highlight it.…”

Section: Preliminariesmentioning

confidence: 99%

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Questions About Quantifiers: Symbolic and Nonsymbolic Quantity Processing by the Brain

Szymanik,

Kochari,

Bremnes

2023

Cognitive Science

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One approach to understanding how the human cognitive system stores and operates with quantifiers such as “some,” “many,” and “all” is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the relationship between processing exact number symbols (such as “3” or “three”) and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing.

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Section: Open Question 5 Do the Proportional Quantifiers Extract Rati...mentioning

confidence: 79%

Section: Shared Processing Mechanismsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Questions About Quantifiers: Symbolic and Nonsymbolic Quantity Processing by the Brain

Szymanik,

Kochari,

Bremnes

2023

Cognitive Science

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“…Queries involving 'more' can also invoke the ANS, though the method people use appears distinct from that for 'most' [72]. In further work, Knowlton et al [73] show that different English expressions for universal quantification in fact elicit different representations altogether: while 'all' and 'every' prompt representations of ensembles and their cardinalities, 'each' seems to elicit an individual-level procedural strategy more like semantic automata.…”

Section: Dynamic Modalitiesmentioning

confidence: 99%

Interleaving Logic and Counting

VAN BENTHEM,

ICARD

2023

Bull. symb. log

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Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus 'grassroots mathematics'.We begin with a brief review of FO(#), first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, MFO(#). We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced.Next, we investigate a series of strengthenings of MFO(#), again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system ML(#) that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of ML(#), and provide a new kind of bisimulation matching the expressive power of the language.As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of FO(#) by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of 'mass' or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite 'semantic automata'. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.

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“…Szabo is right that in CM, I didn't show how to extend the idea to adverbial quantifiers; that's hard, and I'm not at all confident about the details. But regarding quantificational determiners,Knowlton et al (2022) describe some experimental findings that point in the direction I suggested.…”

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confidence: 99%