Logical analysis of data and knowledge without using an axiomatic approach

Logical analysis of data and knowledge is required not only in artificial intelligence systems, but also in mathematical proofs. This work shows the difficulties of logical analysis caused by the indisputable application of the formal axiomatic approach adopted at the turn of the XIX and XX centuries as the only theoretical basis of modern logic and the foundations of mathematics. In this approach, the main role is played by the axioms and methods of syntactic analysis, which underlie predicate calculus. Instead of set theory, we propose to use more simple algebra of sets as a theoretical basis of such a logic, for which the substantiation of the basic laws does not require axioms. A new approach to logical analysis based on the laws of set algebra is considered. Some advantages of this approach are shown, and a list of unresolved problems is provided.