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Abstrakt

In the present paper, two kinds of constructive logics are distinguished. Contents-constructive logic is an instrument of building constructive systems in science; the formally constructive logic is an instrument of constructive building of science. Some problems of the former kind of constructive logic are dealt with in the present study. By means of the analysis of the concept of construction and recoursive functions, the author arrives at a conclusion that though the concept of construction must be basic it need not be either unique or freely chosen. It must be relative, viz. relative from the viewpoint of the position of mathematics in the whole science and from the viewpoint of the conception of the verificative-demonstrational function of science. This enables the rise of various kinds of constructivism and logical systems which are adequate to them. It is shown in the present paper how various nonconstructive elements came into mathematics, viz. in a logical way by means of indirect proof, and in a mathematical way by a broad definition of function and by a broad understanding of the concept of set. Constructive logics are then compared with nonconstructive ones and the latter are shown necessarily to have an ontological background. An opinion is spread widely enough that the 2-valued propositional calculus is eminently finitistic and hence contents-constructive. By means of the analysis of the matrix method of the propositional calculus, the conclusions are drawn that there are nonconstructive elements even in the propositional calculus. The first metamathematical Tarski’s axiom U =< Ν0 where U stands for the set of all propostions, serves as a starting point. The consequences of the application of ultraintuitionistic arithmetic are studied according to which U < Ν would hold good, then the consequences are studied of both potential and actual understanding of the infinite, i. e., the consequences of U =< Ν0. In the end, attention is devoted, from the viewpoint of the infinite, to the axiomatical method of propositional calculus using the rule of substitution.