First-order Logic Based on Set Approximation: a Partial Three-valued Approach

After presenting a very general framework of set approximation the author shows that it can be the set-theoretical base of the semantics of a partial three-valued first-order logic. Approximative functors can appear in object language, and so the properties of set approximation can be given as logical laws. Permitting semantic partiality gives a possibility to make correct difference between the following different cases: a predicate is true, false, uncertain or undefined for an object. Some important laws are proved in order to show the characteristic behavior of introduced logical system. They open doors before the investigation of different consequence relations in order to show how one can make a valid inference relying on represented and not total knowledge. Partiality appears in many information systems, and so the theoretical results can be applied in practice in the future.