Discovering Quantification and Number in a Role-Filler Model.

Discovering Quantification and Number in a Role-Filler Model Aaron J. Hamer (ahamer@hawaii.edu) University of Hawaii at Manoa, Linguistics 1890 East-West Road, Honolulu, HI 96822 USA Leonidas A. A. Doumas (leonidas@hawaii.edu) University of Hawaii at Manoa, Psychology 2530 Dole Street, Honolulu, HI 96822 USA Abstract Behavioral Data Quantification plays a central role in human reasoning and models thereof, but the discovery and development of quantification remains an open question. We present a theory of how such concepts are learned from experience in the DORA model, a neurally-plausible computational model of relational learning and reasoning (Doumas et al., 2008). The same theory accounts for how concepts of number are acquired in this class of model. We are unaware of any prior model that accounts for the development of both quantification and number from unstructured (e.g., perceptual) input. Quantification Facts Behavioral evidence suggests that there are three broad areas of difficulty with the acquisition of quantification: quantifier spreading, mapping issues, and superlative quantifiers. Keywords: number; quantification; relational discovery; computational modeling. Introduction Quantification and number are key representational constructs in human cognition. These concepts are foundational in science, mathematics, music, and many other domains of human achievement. Many models of cognition rely on these representational primitives (e.g., any symbolic model that relies on first-order predicate calculus, many Bayesian models such as Piantadosi, Tenenbaum, & Goodman’s (2013) model of quantifier discovery, etc.), but as Carey (2009, p. 456) notes, “There is no proposal I know for a learning mechanism available to non-linguistic creatures that can create representations of objects, number, agency, or causality from perceptual primitives.” These concepts share significant semantic overlap ranging from their function as predicates over sets of objects (Barwise & Cooper, 1981) to innate, scalar ordering (e.g., one, two, three & some, many, most; Horn, 1972). Both sets of concepts can be derived from a small set of axioms via set theory (i.e., set membership, identity; Van Heijenoort, 1977). It does not seem unreasonable to consider the problem of their acquisition jointly. While there have been attempts to explain their acquisition in terms of a developmental trajectory from number to quantifiers or vice versa (e.g., Gelman & Gallistel, 1978; Carey, 2004), we are unaware of any existing model that accounts for the development of representations of both quantification and number from unstructured (i.e., perceptual) input. Quantifier Spreading Philip and his colleagues (1991a, 1991b) popularized the term quantifier spreading to describe a phenomenon first reported by Inhelder & Piaget (1964). Children aged six to seven were unable to restrict universal quantifiers to a subset of items present in an array based on a shared feature. When presented with three purple triangles and a purple circle and asked “Are all the triangles purple?” the children would respond in the negative. When asked for an explanation, a typical response was “The circle is purple, too.” Mapping Issues Brooks & Braine (1996) demonstrated that children have more rigid mappings for the quantifiers all and each than adults. Children preferred a grouped interpretation of all in scenarios such as “All of the roses are in a vase” and a distributed, one-to-one interpretation of each in scenarios such as “Each of the roses is in a vase”. They interpreted scenes where roses were distributed over more than one vase as false for the all quantifier and scenes where there was not a one-to-one mapping of roses-to-vases (e.g., more roses than vases, more vases than roses) as false for the each quantifier. Children achieve adult-like performance reasoning about all at around age five but do not reach adult-like performance reasoning about each until age nine. Superlative Quantifiers Scalar quantifiers can be divided into two types: superlative quantifiers that include their endpoints (e.g., at most three, three or more) and comparative quantifiers that exclude their endpoints (e.g., less than four, more than two). Musolino (2004) showed that five-year-old children performed worse on tasks relying on superlative quantifiers versus comparative quantifiers. Geurts et al. (2010) investigated this phenomenon further and showed that the difficulty of acquiring superlative quantifiers extended to 11-year-old children. Geurts et al. also showed that superlative quantifiers were more difficult for adults to process (as shown by higher RTs). Hurewitz et al. (2006) found that three-year-olds interpret some as