Work by prof. A. and . Popova is the first Soviet book that gives a general outline of mathematical logic. The book gives a brief historical overview of the emergence of mathematical logic, The main directions of modern mathematical logic are popularly described , special attention is paid to the question of the relationship between mathematical and classical (formal) logic , The place of mathematical logic in the system of scientific knowledge is considered .. The paper identifies practical applications of mathematical logic as the basis for the design and operation of various “smart” and “thinking” machines. The book is intended for researchers, graduate students, students and teachers of secondary school, as well as for a wide range of readers interested in philosophy, logic and mathematics.