V.A. Uspensky and V.E. Plisko "Intuitionistic Logic", In S.M. Nikol'ski, ed. A.N. Kolmogorov, Collected works. Mathematics and Mechanics, pp. 394-404, 1985 (in Russian).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....made Brouwer s definition of intuitionistic truth explicit, though informal, by introducing what is now known as Brouwer Heyting Kolmogorov (BHK) semantics. BHK semantics is widely recognized as the intended semantics for intuitionistic logic ( 18] 19] 20] 24] 37] 47] 50] 72] 73] 74] [75], 76] BHK semantics gives an informal explanation of the truth of intuitionistic connectives. A statement is true if it has a proof, and a proof of a logically compound statement is given in terms of the proofs of its components. The description uses the unexplained primitive notions of ....

....semantics for the latter. On the other hand, classical realizabilities (Kleene realizability [32] function realizability [33] modified realizability [38] Medvedev s calculus of finite problems [50] and its variants) give conditions necessary but not sufficient for Int(cf. 18] 71] 74] [75]) It turned out that the natural deduction proofs for Int can be transliterated by the CurryHoward isomorphism into the language of typed terms (see, for example, 24] 20] 72] The inductive definition of the Curry Howard isomorphism goes along the lines of BHK clauses, where terms play the ....

V.A. Uspensky and V.E. Plisko "Intuitionistic Logic", In S.M. Nikol'ski, ed. A.N. Kolmogorov, Collected works. Mathematics and Mechanics, pp. 394-404, 1985 (in Russian).

....within the classical mathematics. Technical Report CFIS 98 06, Cornell University y 627 Rhodes Hall, Cornell University, Ithaca NY, 14853 U.S.A. email:artemov hybrid.cornell.edu and Moscow University, Russia. 1 Kleene realizability [22] Medvedev finite problems [33] and its variants ( 50] [51]) are regarded (cf. 48] 13] 50] 51] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [25] turned out to be based ....

....Report CFIS 98 06, Cornell University y 627 Rhodes Hall, Cornell University, Ithaca NY, 14853 U.S.A. email:artemov hybrid.cornell.edu and Moscow University, Russia. 1 Kleene realizability [22] Medvedev finite problems [33] and its variants ( 50] 51] are regarded (cf. 48] 13] 50] [51]) as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [25] turned out to be based on an inconsistent theory (cf. 52] ....

V.A. Uspensky and V.E. Plisko "Intuitionistic Logic", In S.M. Nikol'ski, ed. A.N. Kolmogorov, Collected works. Matematics and Mechanics, pp. 394-404, 1985 (in Russian).

....of problems. Introduction In 1932 Kolmogorov ( 16] gave an informal description of the calculus of problems in classical mathematics and conjectured that it coincides with intuitioinistic propositional logic Int. Kleene realizability [15] Medvedev finite problems [23] and its variants ( 36] [37]) are regarded (cf. 34] 10] 36] 37] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. In 1933 Godel ( 12] defined Int on the basis of the notion of proof in a classical ....

....( 16] gave an informal description of the calculus of problems in classical mathematics and conjectured that it coincides with intuitioinistic propositional logic Int. Kleene realizability [15] Medvedev finite problems [23] and its variants ( 36] 37] are regarded (cf. 34] 10] 36] [37]) as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. In 1933 Godel ( 12] defined Int on the basis of the notion of proof in a classical mathematical system, where proof may be ....

V.A. Uspensky and V.E. Plisko "Intuitionistic Logic", In S.M. Nikol'ski, ed. A.N. Kolmogorov, Collected works. Matematics and Mechanics, pp. 394-404, 1985 (in Russian).

....of interpretation of Int classical BHK semantics. Classical realizabilities: Kleene realizability [19] function realizability [20] modified realizability [24] Medvedev s calculus of finite problems [31] and its variants, give conditions necessary but not sufficient for Int(cf. 12] 45] 46] [47]) Each of them realizes some formulas Department of Mathematics, Cornell University, email:artemov math.cornell.edu and Moscow University, Russia. not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [23] turned out to be based on an inconsistent theory (cf. ....

V.A. Uspensky and V.E. Plisko "Intuitionistic Logic", In S.M. Nikol'ski, ed. A.N. Kolmogorov, Collected works. Mathematics and Mechanics, pp. 394-404, 1985 (in Russian).

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