Topological Models of Propositional Logic of Problems and Propositions

Оноприенко, Анастасия Александровна

doi:10.3103/s0027132222050059

Cited by 6 publications

(1 citation statement)

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“…brings a version of the Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [4] (as well as the Lax Logic of Fairtlough and Walton [17] and the Russell-Prawitz modal logic of Aczel [1], see [65, p. 22] for further details). QHC also admits a topological semantics [66, section 4] as well as a Kripkestyle semantics described by Anastasia Onoprienko, which she used for establishing the completeness of QHC with respect to this semantics [73], [74].…”

Section: This Line Of Research Was Further Pursued In 1970s and 1980s...mentioning

confidence: 99%

A.N. Kolmogorov’s Calculus of Problems and Homotopy Type Theory

Rodin¹

2022

Вест. ПГУ. ФСП

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In 1932 A.N. Kolmogorov proposed an original version of mathematical intuitionism where the distinc-tion between problems and theorems plays a central role and which differs in its content from the versions of intuitionism developed by A. Heyting and other followers of L. Brouwer. In view of today’s historians and logicians, it remains a controversial point whether this distinction is to be treated as logical or Kol-mogorov’s «problems» should be regarded as propositions, provided the latter term is interpreted intui-tionistically. The popular BHK semantics of intuitionistic logic, so called after Brouwer, Heyting, and Kolmogorov and supposed to provide a synthesis of ideas of these authors, does not formally distinguish between problems and theorems and treats this distinction as contextual or purely linguistic. We argue that the distinction between problems and theorems is a crucial element of Kolmogorov’s approach and provide a logical interpretation of this distinction using homotopy type theory (HoTT). The notion of ho-motopy level of a given type in HoTT allows one to distinguish, after Kolmogorov, between problems re-duced to proving a sentence and problems that require realization of higher-order constructions. Thus, HoTT provides a support for Kolmogorov’s view on intuitionistic logic in his polemics with Heyting. At the same time, HoTT does not solve the problem of finding a constructive notion of negation applicable to general problems, which is raised by Kolmogorov in the same paper and still remains open.

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