The term “existence” is important in philosophy, mathematics and logic. Various approaches to logic research of existential expressions are possible. The question of relation of terms to the objects designated by them is important. These problems were considered by Aristotle in the names theory developed by him. Scholastic … of suppositions regard this problem as well. From the middle of the 19th century these problems were studied by J. Mill, W. Jevons, F. Brentano, A. Meinong, E. Husserl. Mill made a significant statement: names only have a task of naming something but to express that this something exists the predicate “to exist” is needed. Nevertheless F. Frege, B. Russell and R. Carnap believed that the expression of the language should be considered as a name only when it designates a real existing object, then the predicate of existence is unnecessary. We believe that everything that we can think about or see or feel exists, whatever it is, though it exists in different meanings and in different degrees. The kinds of existence need to be differentiated. There is a round square even as an idea. But there exists a bridge of ideas forming a concept of a round square, for example, as a dynamic structure transforming with the time in topology or as a set of figures, some of them being similar to a square while others - to a circle. There exists x: x > 0, x < 0 also as an idea, as a set of two maximum consistent subsystems of the system. There is a generalized existence, for example according to Chebyshev’s idea (when all the predicates of the concept are diminished) and in the meaning of committee constructions when all consistent subsystems of predicates are considered. There is an infinite point - a nonintrinsic element in the plane.