of the International Congress of Mathematicians Vancouver , 1974 On Superintuitionistic Logics * A . V . Kuznetsov

Since Brouwer has proclaimed in 1908 the untrustworthiness of classical logic by rejecting the law of the excluded middle, intuitionistic logic, managing without this law, began little by little to develop. As a calculus, it has been presented in a wellknown paper of Heyting (1930), preceded by the interesting papers of A. N. Kolmogorov (1925) and V. I. Glivenko (1929). Soon after that, Kolmogorov (1932) proposed an interpretation of the intuitionistic logic as the logic of problems, which showed that it is valuable not only for intuitionists. This became quite clear after the appearance of the theory of algorithms and the constructive tendency in mathematics. Just the connection of the truth of the mathematical proposition with the problem of its demonstration, its falsity with the problem of its refutation, and the law of the excluded middle with the problem of construction of the algorithm allowing to prove or refute any proposition has generated ineradicable doubt in the validity of this law. The papers of Tarski, Rasiowa, Curry and other mathematicians give the precise algebraic and topological interpretations of intuitionistic logic and its easy immersion into a modal logic £4 detected by Godei. However in 1932 Godei [3] proved that it is impossible to give exactly the intuitionistic logic by any finite truth matrix; though, as Jaskowski [22] showed later on, it may be approximated by a sequence of such matrices. The attempts to give the exact pithy-semantical (meaningful) construction of the intuitionistic logic by means of, for example, precisely stating Kolmogorov's logic of problems have unexpectedly led to logics, slightly different from the intuitionistic—to the logic of recursive readability of S. Kleene and G. Rose (see [5], [14]) and to the logic of finite problems of Ju. T. Medvedev [10] (as V. E. Plisko [11] has shown recently, these two logics are incomparable). The