The Information Flow Framework: A Descriptive Category Metatheory

The Information Flow Framework (IFF) is a descriptive category metatheory currently under development that provides an important practical application of category theory. It is an experiment in foundations, which follows a bottom-up approach to logical description. The IFF forms the structural aspect of the IEEE P1600.1 Standard Upper Ontology (SUO) project. The categorical approach of the IFF provides a principled framework for the modular design of object-level ontologies. The IFF represents metalogic, and as such operates at the structural level of ontologies. In the IFF, there is a precise boundary between the metalevel and the object level. The modular architecture of the IFF consists of metalevels, namespaces and meta-ontologies. There are four metalevels, Lower, Upper, Top and Ur, corresponding to the set-theoretic distinctions between the “small”, the “large”, the “very large” and the “generic”, respectively. Each metalevel services the levels below by providing a metalanguage used to declare and axiomatize those levels. Corresponding to the four metalevels are the four nested metalanguages, meta-lower ⊒ meta-upper ⊒ meta-top ⊒ meta-ur, where each metalanguage axiomatization includes specialization of the one immediately above. Within each metalevel, the terminology is partitioned into namespaces, and various namespaces are collected together into meaningful composites called meta-ontologies. The IFF contains meta-ontologies representing category theory, information flow, formal concept analysis, institutions, multitudes, equational logic, first order logic, the common logic standard, etc. All of the various meta-ontologies in the IFF are anchored to the IFF metastack Set ⊆ Cls ⊆ Col ⊆ Ur. The IFF development is largely driven by the principles of conceptual warrant, categorical design and institutional logic. The main application of the IFF is institutional: the notion of institutions and their morphisms are being axiomatized in the upper metalevels of the IFF, and the lower metalevel of the IFF has axiomatized various institutions in which semantic integration has a natural expression. “Philosophy cannot become scientifically healthy without an immense technical vocabulary. We can hardly imagine our great-grandsons turning over the leaves of this dictionary without amusement over the paucity of words with which their grandsires attempted to handle metaphysics and logic. Long before that day, it will have become indispensably requisite, too, that each of these terms should be confined to a single meaning which, however broad, must be free from all vagueness. This will involve a revolution in terminology; for in its present condition a philosophical thought of any precision can seldom be expressed without lengthy explanations.” — Charles Sanders Peirce, Collected Papers 8:169 2000 Mathematics Subject Classification: 18A15.