Morphisms in Logic , Topology , and Formal Concept Analysis

The general topic of this thesis is the investigation of various notions of morphisms between logical deductive systems, motivated by the intuition that additional (categorical) structure is needed to model the interrelations of formal specifications. This general task necessarily involves considerations in various mathematical disciplines, some of which might be interesting in their own right and which can be read independently. To find suitable morphisms, we review the relationships of formal logic, algebra, topology, domain theory, and formal concept analysis (FCA). This leads to a rather complete exposition of the representation theory of algebraic lattices, including some novel interpretations in terms of FCA and an explicit proof of the cartestian closedness of the emerging category. It also introduces the main concepts of “domain theory in logical form” for a particularly simple example. In order to incorporate morphisms from FCA, we embark on the study of various context morphisms and their relationships. The discovered connections are summarized in a hierarchy of context morphisms, which includes dual bonds, scale measures, and infomorphisms. Finally, we employ the well-known means of Stone duality to unify the topological and the FCA-based approach. A notion of logical consequence relation with a suggestive proof theoretical reading is introduced as a morphism between deductive systems, and special instances of these relations are identified with morphisms from topology, FCA, and lattice theory. Especially, scale measures are recognized as topologically continuous mappings, and infomorphisms are identified both with coherent maps and with Lindenbaum algebra homomorphisms.