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Dimensional Characterization in Finite Quasi-Analysis
Vol. 54, No. 1, Festschrift in Honour of Wilhelm K. Essler on the Occasion of His Sixtieth Birthday (2001), pp. 121-131
Published by: Springer
Stable URL: http://www.jstor.org/stable/20013039
Page Count: 11
You can always find the topics here!Topics: Similarity theorem, Mathematical lattices, Equivalence relation, Topological dimensions, Cardinality, Mathematical functions, Euclidean space, Mathematical problems, Topology, Mathematical sets
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The method of Quasi-Analysis used by Carnap in his program of the constitution of concepts from finite observations has the following two goals: (1) Given unsharp observations in terms of similarity relations the true properties of the observed objects shall be obtained by a suitable logical construction. (2) From a single relation on a finite domain, different dimensions of qualities shall be reconstructed and identified. In this article I show that with a slight modification Quasi-Analysis is capable of fulfilling the first goal for a single observable dimension. We obtain a partition of the so-called Quality Classes representing the pairwise disjoint and exhaustive extensions associated to the "values" of the observable. On the other hand, an example demonstrates that the method fails, as Goodman has pointed out, for a relation expressing similarity with regard to at least one out of many properties. Since it seems to be impossible in general to reconstruct more-dimensional qualities from a single similarity relation, the constitution of at least as many similarity relations as there are qualities have to be presumed. Then it is possible to state adequate sufficient conditions for the dimension of the observable space, even if some of the similarity relations might depend on others. The concept of topological dimension cannot be used for this purpose on finite sets of observations. We replace it by a set-algebraical condition on the Quality Classes.
Erkenntnis (1975-) © 2001 Springer