On Some Problems in Defining Sets for Q-Analysis

At the basis of Q -analysis are the notions of well-defined sets, and of hierarchies constructed out of the power sets of such sets. It is here argued that such set-theoretic hierarchies are in general inadequate in themselves for representing the structure of empirical phenomena, because sets defined as collections of individual objects or of other sets cannot, on the whole, be made to correspond directly to specific empirical entities. This is because empirical concepts correspond to more or less complex structures , rather than collections, of other concepts, whereas a set is, by definition, an unstructured collection of elements. This point is illustrated through several examples of sets of ‘things’ that purport to represent certain higher-order ‘things’, but in fact do not. There follows an analysis of the problem that draws from mathematical philosophy and logic on the one hand, and from the theory of Boolean algebras on the other. Both these perspectives point to the same conclusion, namely, that Q -analysis is a specialized methodology applicable to a rather restricted, though potentially very interesting class of problems. The paper ends with an evaluation of the significance of such a shift in our understanding of Q -analysis, and some tentative ideas about possible areas of fruitful application.