Mathematical Support for Empirical Theory Building

Abstract

Abstract Empirical theory building as common in physics is successfully represented in mathematical models, for instance in euclidean vector spaces. Those formal models support the respective disciplinary theory with expressive structure which particularily stabilizes the theory at longer term. In social sciences such support is very seldom. In representational measurement theory, mathematical models for social science theories are specified as relational structures, but the range of their applications is quite restricted because they are designed for being represented by numerical structures. What is needed for formal representations of qualitative theories, are non-numerical mathematical models which make the theories more transparent and communicable. Those models can be derived by methods of Formal Concept Analysis which offers a mathematical theory for developing and analysing conceptual structures. Using this approach, the basic models for representing qualitative theories are formal contexts (also called conceptual scales) together with their concept lattices. Implicationally simple conceptual scales form a useful type of models for formalizing those parts of qualitative theories which are expressible by implications between attributes (properties) with one-element premise and by inconsistency conditions for sets of attributes. The concept lattices of those scales are (up to isomorphism) the finite distributive lattices truncated by identifying all elements of an order ideal with 0. Similarily, other types of conceptual scales can by decribed for logical assumptions concerning attributes. For representing a qualitative theory as a whole, the conceptual scales describing parts of the theory can be combined by the semiproduct. The resulting formal context yields a model which can be used for testing the theory empirically. Such testing can be graphically supported by the program system TOSCANA which allows to inspect the empirical data with respect to the given theory by suitably combining the concept lattices of the involved conceptual scales. In this way, formal and material argumentations can be activated to justify conceptual scales as proper representations of parts of the theory or to modify conceptual scales and even parts of the theory. TOSCANA may even support an iterative process of empirically grounded theory building where its history of development is kept. Examples are given to demonstrate those processes of theory building.