Theoretical noteA catalog of Boolean concepts
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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