We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. It is shown that, although these sentences are known to have a diagrammatic expression (in the form of the Gergonne circles) which constitute a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. (...) It is hypothesized that the quantified sentences have an abstract underlying representation common to the formulas and their associated sets of diagrams (models). Nine predictions are derived (three semantic, two pragmatic, and four mixed) regarding people's assessment of how well each of the five diagrams expresses the meaning of each of the quantified sentences. The results from three experiments, using Gergonne's circles or an adaptation of Leibniz lines as external representations, are reported and shown to support the predictions. (shrink)

We present a set-theoretic model of the mental representation of classically quantified sentences (All P are Q, Some P are Q, Some P are not Q, and No P are Q). We take inclusion, exclusion, and their negations to be primitive concepts. We show that although these sentences are known to have a diagrammatic expres- sion (in the form of the Gergonne circles) that constitutes a semantic representation, these concepts can also be expressed syntactically in the form of algebraic formulas. (...) We hypothesized that the quantified sen- tences have an abstract underlying representation common to the formulas and their associated sets of dia- grams (models). We derived 9 predictions (3 semantic, 2 pragmatic, and 4 mixed) regarding people’s as- sessment of how well each of the 5 diagrams expresses the meaning of each of the quantified sentences. We report the results from 3 experiments using Gergonne’s (1817) circles or an adaptation of Leibniz (1903/ 1988) lines as external representations and show them to support the predictions. (shrink)

Children's comprehension of the universal quantifiers all and each was explored in a series of experiments using a picture selection task. The first experiment examined children's ability to restrict a quantifier to the noun phrase it modifies. The second and third experiments examined children's ability to associate collective, distributive, and exhaustive representations with sentences containing universal quantifiers. The collective representation corresponds to the "group" meaning (for All the flowers are in a vase all of the flowers are in the same (...) vase). The distributive representation implies a pairing (e.g., each flower paired with a vase for Each flower is in a vase). The exhaustive representation exhausts both sets (e.g., for The flowers are in the vases all the flowers are in vases and all the vases have flowers in them). Four- to 10-year-olds children had little difficulty restricting the quantifier all to the noun it modified in a task which required them to attend to the group feature of all. In contrast, only 9- and 10-year-olds were able to solve the task when the quantifier was each and the pictures showed entities in partial one-to-one correspondence. Children showed a preference for associating collective pictures with sentences containing all and distributive pictures with sentences containing each. The results suggest that between the ages of 5 and 10 years, children's semantic representations undergo less radical changes than others have proposed. Instead, developmental change may occur gradually as children acquire linguistic cues which map onto existing semantic representations. (shrink)