A catalog of Boolean concepts

doi:10.1016/S0022-2496(02)00025-1

Journal of Mathematical Psychology

Volume 47, Issue 1, February 2003, Pages 75-89

https://doi.org/10.1016/S0022-2496(02)00025-1 Get rights and content

Abstract

Boolean concepts are concepts whose membership is determined by a Boolean function, such as that expressed by a formula of propositional logic. Certain Boolean concepts have been much studied in the psychological literature, in particular with regard to their ease of learning. But research attention has been somewhat uneven, with a great deal of attention paid to certain concepts and little to others, in part because of the unavailability of a comprehensive catalog. This paper gives a complete classification of Boolean concepts up to congruence (isomorphism of logical form). Tables give complete details of all concepts determined by up to four Boolean variables. For each concept type, the tables give a canonic logical expression, an approximately minimal logical expression, the Boolean complexity (length of the minimal expression), the number of distinct Boolean concepts of that type, and a pictorial depiction of the concept as a set of vertices in Boolean D-space. Some psychological properties of Boolean concepts are also discussed.

Section snippets

Boolean concepts

A Boolean variable is a variable that can take one of two distinct values, e.g., 0 or 1, often thought of as “truth values” with 1 meaning “true” and 0 meaning “false” (Boole, 1854/1958). Boolean D-space is the space created by crossing D Boolean variables. Such a space can be conveniently thought of as a D-dimensional cube or hypercube, with each vertex corresponding to one possible combination of truth values for each of the D variables, that is, a D-dimensional Boolean object.

A Boolean

A complete classification of Boolean concepts

As mentioned above, Shepard et al. (1961) used an explicit typology for D=3,P=4. A complete typology for all D⩽4 and all P was given in Aiken et al.'s remarkable 1951 monograph, but in somewhat antiquated and difficult notation (directed at early designers of vacuum-tube switching circuits). The main purpose of this paper is to present the typology in more modern notation, organized by values of D and P, and giving certain additional information about each concept, most importantly (i)

Psychological properties

The remainder of my comments focus on what is known about psychological properties of various concept types.

Several studies have employed one or another type in studies of generalization. Such studies have typically trained subjects on examples from a single fixed concept; interest then focuses on the degree to which novel objects are judged as members of the target concept, as well as the confidence with which the trained objects are correctly classified. Thus such studies have usually not

Acknowledgments

I am very grateful to Lyle Bourne for bringing Robert Haygood's doctoral dissertation to my attention, and for sending me a copy; and to Howard Barnum, for bringing Michael Harrison's (1965) book to my attention; and to Chester Stromswold, for bringing to my attention the beautiful and pioneering treatise produced in 1951 by Howard Aiken and his colleagues at the Computation Laboratory of Harvard University. Preparation of this manuscript was supported in part by NSF SBR 9875175.

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Theoretical noteA catalog of Boolean concepts

Abstract

Section snippets

Boolean concepts

A complete classification of Boolean concepts

Psychological properties

Acknowledgments

Cognition

Cognition

Cognitive Psychology

Information and Control

An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities

Knowing and using concepts

Psychological Review

Concept learning as a function of the conceptual rule and the availability of positive and negative instances

Journal of Experimental Psychology

A study of thinking

On the length of programs for computing finite binary sequences

Journal of the Association for Computing Machinery

A cross-cultural study of conjunctive and disjunctive concept learning

Child Development

Minimization of Boolean complexity in human concept learning

Nature

Computers and intractability: A guide to the theory of NP-completeness

Introduction to switching circuit theory

Introduction to switching and automata theory

Attribute- and rule-learning aspects of conceptual behavior

Psychological Review

Theoretical note
A catalog of Boolean concepts