The Logic of Partial Information

This book presents the foundations of reasoning with partial information and a theory of common sense reasoning based on monotonic logic and partial structures. This theory was designed specifically for the needs of practicing computer scientists and provides easily implementable algorithms. Starting from first principles, following the logic of discovery of Karl Popper and Imre Lakatos, and the semantics of programming languages, the book develops a system of reasoning with partial information, and applies it to a comprehensive study of the problem examples from the literature of common sense reasoning. Proof-theoretic and model-theoretic views are considered in the applications, as well as logical problems of theoretical physics, such as issues related to Heisenberg's uncertainty principle. The book points out that customary expositions of common-sense reasoning are based on a flawed non-monotonic reasoning paradigm and that the resulting solutions proposed for major problems, such as the frame problem, are either ad hoc or inadequate. It is shown that non-monotonicity results from hiding information that should not be hidden. The essential research in common-sense reasoning has been developed in isolation from the disciplines of theoretical computer science and classical logic. This work breaks the isolation and establishes deep links. The book will be of interest to computer scientists, mathematicians, logicians, and philosophers interested in the foundations and applications of reasoning with partial information.

One must be able to say at all times - in- stead of points, straight lines, and planes - tables, chairs and beer mugs. (David Hilbert) One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled "discarded nonsense. " (Eric T. Bell) This book discusses reasoning with partial information. We investigate the proof theory, the model theory and some applications of reasoning with par- tial information. We have as a goal a general theory for combining, in a principled way, logic formulae expressing partial information, and a logical tool for choosing among them for application and implementation purposes. We also would like to have a model theory for reasoning with partial infor- mation that is a simple generalization of the usual Tarskian semantics for classical logic. We show the need to go beyond the view of logic as a geometry of static truths, and to see logic, both at the proof-theoretic and at the model-theoretic level, as a dynamics of processes. We see the dynamics of logic processes bear with classical logic, the same relation as the one existing between classical mechanics and Euclidean geometry.

This monograph presents the foundations of reasoning with partial information and a theory of common-sense reasoning based on monotonic logic and partial structures. It points out that customary expositions of common-sense reasoning in Artificial Intelligence are based on a flawed non-monotonic reasoning paradigm.