Mental models of Boolean concepts

Abstract

a b s t r a c tNegation, conjunction, and disjunction are major building blocks in the formation of concepts. This article presents a new model-based theory of these Boolean components. It predicts that individuals simplify the models of instances of concepts. Evidence corroborates the theory and challenges alternative accounts, such as those based on minimal descriptions, algebraic complexity, or structural invariance. A computer program implementing the theory yields more accurate predictions than these rival accounts. Two experiments showed that the numbers of models of a Boolean concept predict the difficulty of formulating a description of it. As mental models may also underlie deductive reasoning, the present theory integrates two hitherto separate areas of investigation.

Exploring the conceptual universe.

“…Although several authors have pointed out that rules can incorporate quantification and relations (Goodman et al, 2008;Goodwin & Johnson-Laird, 2011;Piantadosi, Goodman, & Tenenbaum, 2010), most of the rule-based models in the literature focus on rules formulated in propositional logic. The rule-plus-exception (RULEX) model proposes that humans learn concepts by constructing conjunctive rules and remembering exceptions to these rules .…”

“…The analyses most relevant to this article 10 KEMP are presented by Feldman (2006) and Goodwin and Johnson-Laird (2011), who both focus on Feldman's data set of 76 Boolean concepts. Feldman's algebraic complexity approach and the mental models approach both account well for the data, but the mental models approach achieves slightly higher quantitative fits.…”

A theory of reverse engineering and its application to Boolean systems

“…The standard conceptual learning task is to present individuals with REVERSE ENGINEERING 5 instances and noninstances of a concept, and their task is to learn to identify all and only the instances of the concept (e.g., Shepard, Hovland, & Jenkins, 1961). In a comparative study, Goodwin and Johnson-Laird (2011) showed that the difficulty of Boolean concept acquisition is likely to be dependent on the number of mental models underlying the concept. One of the model theory's predictions, which experimental results corroborate, is that it is easier to acquire a concept that is an exclusive disjunction*it has two models of its instances*than to acquire a concept that is an inclusive disjunction*it has three models of its instances.…”

“…In particular, exclusive disjunctions are not always more difficult to deal with than inclusive disjunctions. In deductive reasoning, exclusive disjunctions are reliably easier than inclusive disjunctions (see, e.g., Bauer & Johnson-Laird, 1993); the same difference occurs in the acquisition of concepts (see, e.g., Goodwin & Johnson-Laird, 2011). The reason is that exclusive disjunctions are consistent with fewer possibilities than inclusive disjunctions.…”

“…Given such assertions, individuals can also evaluate their truth or falsity, list the possibilities to which they refer, and find a possibility common to several assertions; conversely, given a set of possibilities, they can formulate a description of them (Goodwin & Johnson-Laird, 2011). Mental model theory, which explains these tasks, rests on the assumption that each possibility is represented in a separate mental model, and that the smaller the number of models of the premises, the easier the task should be.…”