Kolmogorov A. (1932) Zur Deutung der intuitionistischen Logik. Mathematische Zeitschrift, 35, 58--65. English translation in P. Mancosu, Ed., From Brouwer to Hilbert : the debate on the foundations of mathematics in the 1920s, Oxford University Press, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....logic where frequently occurring constructions in intuitionistic mathematics have a logical counterpart. One of these constructions is the proof of an implication. Heyting [24] describes the proof of an implication a ) b as: Deriving a solution for the problem b from the problem a. Kolmogorov [28] is even more explicit, and describes a proof of a ) b as the construction of a method that transforms each proof of a into a proof of b. This means that a proof of a ) b can be seen as a (constructive) function from the proofs of a to the proofs of b. In other words, the proofs of the proposition ....

Kolmogorov, A.N., Zur Deutung der Intuitionistischen Logik, Mathematisches Zeitschrift, Volme 35, Pages 58-65, 1932.

....A is not provable in VPL, while it is provable in IPL. Visser treated VPL as a preliminary for a study of FPL. VPL, however, was also motivated by a revision of the Brouwer Heyting Kolmogorov (BHK) proof interpretation introduced in Ruitenburg [Rui91] and Ruitenburg [Rui92] see also Kolmogorov [Kol32] and Heyting [Hey56] Ruitenburg s interpretation for A B is a construction that uses the assumption A to produce a proof of B while the standard BHK interpretation looks like B is a construction that converts proofs of A into proofs of B. Ruitenburg argued that using assumption A, ....

A. N. Kolmogorov, Zur Deutung der intuitionistischen Logik, Mathematische Zeitschrift, 35, 1932, pp. 58--65.

....logic incorrect. Church [8] and Curry [11] introduced the simply typed calculus (STLC) to provide logic while avoiding Russell s paradox in a manner similar to RTT. Unfortunately, like RTT, the STLC is too restrictive. The areas, Logics, Types and Rewriting converge. Heyting [20] Kolmogorov [32], Curry and Feys [11] improved by Howard [22] and de Bruijn [38] all observed the propositions as types or proofs as terms (PAT) correspondence. In PAT, logical operators are embedded in the types of terms rather than in the propositions and terms are viewed as proofs of the ....

A. N. Kolmogorov. Zur Deutung der Intuitionistischen Logik. Mathematisches Zeitschrift, 35:58-65, 1932.

....second gives an overview of the work reported in the body of the this thesis. 1.1 Background This idea that a proof is fundamentally a kind of program has a long history. It is more or less explicit in the very origins of constructive logic and mathematics. An early expression is in Kolmogoro# s [55] explanations of the constructive interpretation of first order logic, according to which its statements express problems, and to assert a statement is to claim to know an e#ective solution to the problem expressed by the statement. If putting a solution into e#ect is thought of as running a ....

A. N. Kolmogorov. Zur deutung der intuitionistischen logik. Matematische Zeitschrift, 35:58--65, 1932.

....to Per Martin Lof, Bengt Nordstrom and Kent Petersson for many discussions. 2 Specifications as sets The idea of of viewing a specification of computer programs as a set in Martin Lof s type theory has its origin both in understanding propositions as sets and in Kolmogorov s explanation in [9] of propositions as problems. Kolmogorov explains the sentential constants in the following way: A B is the problem of solving both of the problems A and the problem B . A B is the problem of solving at least one of the problems A and B . A oe B is the problem of solving the problem B ....

A. N. Kolmogorov. Zur Deutung der intuitionistischen Logik. Matematische Zeitschrift, 35:58 --65, 1932.

....by explaining what is to be regarded as a proof of c(A 1 ;A 2 ; A n ) assuming one knows what counts as a proof of each of A 1 , A 2 , A n . This was made explicit, #rst by Heyting (1898 1966) in 1930 [22] and more fully in 1934 [23] and independently by Kolmogorov (1903 1987) in 1932 [32]. Actually, Heyting and Kolmogorov regarded their respective formulations as distinct. To convey the idea, let us consider the explanations of # (disjunction, or ) # (implication, implies ) and (negation, not ) in Heyting s formulation: A.S. Troelstra Theoretical Computer Science 211 ....

A.N. Kolmogorov, Zur Deutung der intuitionistischen Logik, Math. Z. 35, (1932) 58--65.

....to non posetal fibrations, and thus closer to toposes accomodating categorical proof theory. 1 Introduction The basic ideas of constructive logic cristallised very slowly. The Brouwer Heyting Kolmogorov interpretation of proofs as constructions had evolved through the first half of this century [4, 16, 23]. It motivated the well known debate about foundations and, indirectly, Imperial College, London, England; e mail: D.Pavlovic doc.ic.ac.uk the creation of realizability in the fourties. Most of the time, however, this conception of strong constructivism was overshadowed by simpler ideas, ....

A.N. Kolmogorov, Zur Deutung der Intuitionistischen Logik, Math. Z. 35(1932) 58--65

....to non posetal fibrations, and thus closer to toposes accommodating categorical proof theory. 1 Introduction The basic ideas of constructive logic cristallised very slowly. The Brouwer HeytingKolmogorov interpretation of proofs as constructions had evolved through the first half of this century [4, 16, 23]. It motivated the well known debate about foundations and, indirectly, the creation of realizability in the fourties. Most of the time, however, this conception of strong constructivism was overshadowed by simpler ideas, boiling down to the rejection of Excluded Middle. The search for a ....

A.N. Kolmogorov, Zur Deutung der Intuitionistischen Logik, Math. Z. 35(1932) 58--65

....rule. Such reduction rules preserve the well known functional interpretation of intuitionistic connectives and proofs which Brouwer, Heyting, Kolmogorov and others proposed in order to allow a better understanding of the constructive features of intuitionistic logic (BHK interpretation) [6] [4] Since irreducible proofs explicitly represent mathematical constructions, reduction rules for intuitionistic logic turn out to have a computational meaning. The BHK interpretation was also helpful to the development of typed functional languages and of computer science in general, for ....

A.N. Kolmogorov, "Zur Deutung der Intuitionistischen Logik", Mathematische Zeit., 35, 1932.

....Abstract We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator. 1 Introduction It is an old idea that proofs in formal logics are certain functions and objects. The Brower Heyting Kolmogorov (BHK) interpretation [15,51,40], in the form stated by Heyting [40] states that a proof of an implication P Q is a construction which transforms any proof of P into a proof of Q. This idea was formalized independently by Kleene s realizability interpretation [46,47] in which proofs of intuitionistic number theory are ....

A. Kolmogorov. Zur Deutung der intuitionistischen Logik. Mathematische Zeitschrift, 35:58--65, 1932.

....the rules of grammar are obeyed but also very close to the way mathematicians have always been writing . The appearance of types in Automath finds its roots in de Bruijn s contacts with Heyting, who made de Bruijn familiar with the intuitionistic interpretation of the logical connectives (see [24], 18] The interpretation of the proof of an implication A B as an algorithm to transform any proof of A in a proof of B, so in fact a function from proofs of A to proofs of B, gave rise to interpret a proposition as a class (a type) of proofs. De Bruijn, who was not influenced by developments ....

A.N. Kolmogorov. Zur Deutung der Intuitionistischen Logik. Mathematisches Zeitschrift, 35:58--65, 1932.

....be regarded as a proof of c(A 1 ; A 2 ; A n ) assuming one knows what counts as a proof of each of A 1 , A 2 , A n . This was made explicit, first by A. Heyting (1898 1966) in 1930 ( Hey31] and more fully in 1934 ( Hey34] and independently by A. Kolmogorov (1903 1987) in 1932 ([Kol32]) Actually, Heyting and Kolmogorov regarded their respective formulations as distinct. To convey the idea, let us consider the explanations of (disjunction, or ) implication, implies ) and : negation, not ) in Heyting s formulation: ffl A proof of A B is given by exhibiting either a ....

A. N. Kolmogorov. Zur Deutung der intuitionistischen Logik. Mathematische Zeitschrift, 35:58--65, 1932.

....22] a proposition is interpreted as a set whose elements represent the proofs of the proposition. 2 B. Nordstr#m, K. Petersson and J. M. Smith It is also possible to view a set as a problem description in a way similar to Kolmogorov s explanation of the intuitionistic propositional calculus [25]. In particular, a set can be seen as a speci cation of a programming problem; the elements of the set are then the programs that satisfy the speci cation. An advantage of using type theory for program construction is that it is possible to express both speci cations and programs within the same ....

A. N. Kolmogorov. Zur Deutung der intuitionistischen Logik. Matematische Zeitschrift, 35:5865, 1932.

....it can be defined as we note below. Once we have combined propositions P (x) with the set constructors, an idea also used in Morse s set theory [55] we can also form P Theta Q; P Q and x : A P (x) It is remarkable that these operations make sense in terms of a logic of problems. [47]. We have already seen P Q in the definition of Q P . To say P Q is to say that there is an effective function that converts a proof of P to a proof of Q. This function f reduces Q to P . The operation P Theta Q corresponds to the combined problem; to solve P Theta Q we must solve P and ....

....define the classical notion of disjunction, P classical or Q, symbolized as P flQ. It is defined as P fl Q iff : P :Q) The concept of negation used here is that :P holds when P ) F alse, i.e. F alse P . 5. 2 axiomatizing a logic of problems It is a remarkable fact discovered by Kolmogorov [47] and Heyting [37] that the operators Decidable(P; Q) Q P; 9; 8 used to make distinctions about decidability, reducibility, construction and uniform solution obey the ordinary laws of logic of the corresponding classical operators with only one exception. Namely, to show Decidable(P; Q) we must ....

A. N. Kolmogorov. Zur deutung der intuitionistischen logik. Mathematische Zeitschrift, 35:58-- 65, 1932.

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Kolmogorov A. (1932) Zur Deutung der intuitionistischen Logik. Mathematische Zeitschrift, 35, 58--65. English translation in P. Mancosu, Ed., From Brouwer to Hilbert : the debate on the foundations of mathematics in the 1920s, Oxford University Press, 1998.

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A. N. Kolmogorov. Zur deutung der intuitionistischen logik. Matematische Zeitschrift, 35:58--65, 1932.

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