By Jaakko Hintikka
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Extra resources for Aspects of Inductive Logic
Such an ideographic space is not too unwieldly when the number of possible strings is small, but as has already been remarked, when this is the case the whole apparatus of a property space and the imposition of structure on this space is scarcely necessary. We may just as well use a straightforward stimulussampling or Bayesian model. A more natural space of properties, which would generalize to longer strings, is the following. Dimension 1 characterizes the first position. The value 1 on the first dimension indicates that symbol 1 must occur in the first position of a string, and the value 2 that symbol 0 must occur in this position.
The point of the present formulation is to impose such a structure. The space of properties is conceived as a multi-dimensional space with each dimension corresponding to a property. (Admittedly in many applications the space will consist of a finite set of points and thus will not satisfy the usual mathematical definition of a multi-dimensional space, but that is not a matter of serious concern here. , a set with a structure), that it is possible to talk about motion in the space. The concept of motion in an arbitrary set with no postulated structure is not well defined.
The idea is to express any possible concepts for solving the problem of classification as a point in the space of properties associated with the stimulus material of the problems. A new concept is formed by moving to a new point in the property space. In these terms the theory of concept formation relevant to solving a given set of problems consists of two parts: first, characterizing the appropriate space of properties, and, secondly, characterizing the laws of motion in the space. In terms of the kind of formulation of stimulus-sampling theory considered in earlier sections, it may be thought that the phrase, "laws of motion", is too grandiose, and that what is described are simply the assumptions for sampling properties or stimuli.