Логика на рубеже тысячелетий.

Abstract

Development of logic at the end of 20th century generated such questions as «What is logic?» or «What is logical system?». The aim of this paper is to find out and analyze the phenomena that led to such unexpected questions. Firstly, that is a continual variety of logical systems; secondly, an extension of logical system as a result of its restriction; thirdly, an embedding (translation) of richer logical systems into weaker systems; fourthly, a tendency to the investigation of classes of logical systems in the fashion of certain constructions; fifthly, a completion of algebraization of logic; sixtly, the demands of computer revolution, etc. The result of logical concretization of two mathematical constructions - a closure operator introduced at the beginning of this century and a concept of category appeared at the middle of the century - was the beginning of investigations of different arrays of logical systems in a form of some lattice or category construction. Logic apparently loses the quality of science concerned with correctness of reasoning (and this is the phenomenon which expresses a crisis of modeling of truly human logic) and becomes the science about constructions that have an extremely abstract logical nature. In this sense logic is transformed just in metalogic with its new role. And the main point is that Logic comes out of limits of what is properly logical.

Content of the paper: 1) Introduction; 2) Closure operator and deductive systems; 3) Logical comprehension of continuum; 4) An extension of classical logic as a consequence of its restriction (translations and embeddings); 5) Lattices of theories and logics; 6) Lattices of calculi and other constructions; 7) Constructions called as «category» and «topos»/ 8) Closure operator and category together; 9) Algebraization of logic; 10) In a search of logical system.