Aleksandr Yu. Konovalov
Skip Abstract Section
Abstract
Let V be a set of number-theoretical functions. We define a notion of absolute V-realizability for predicate formulas and sequents in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove that Basic Predicate Calculus is sound with respect to the semantics of absolute V-realizability if V satisfies some natural conditions.
- Mohammad Ardeshir. 1999. A translation of intuitionistic predicate logic into basic predicate logic. Stud. Logic. 62 (1999), 341–352. https://doi.org/10.1023/A:1005196310070Google Scholar
Cross Ref
- Zlatan Damnjanovic. 1994. Strictly primitive recursive realizability. J. Symbol. Log. 59, 4 (Dec. 1994), 1210–1227. https://doi.org/10.2307/2275700 Google ScholarDigital Library
- Zlatan Damnjanovic. 1995. Minimal realizability of intuitionistic arithmetic and elementary analysis. J. Symbol. Log. 60, 4 (Dec. 1995), 1208–1241. https://doi.org/10.2307/2275884Google Scholar
- Stephen C. Kleene. 1945. On the interpretation of intuitionistic number theory. J. Symbol. Log. 10, 4 (Dec. 1945), 109–124. https://doi.org/10.2307/2269016Google Scholar
- Aleksandr Yu. Konovalov. 2016. Arithmetical realizability and basic logic. Moscow Univ. Math. Bull. 71, 1 (2016), 35–38. https://doi.org/10.3103/S0027132216010071Google ScholarCross Ref
- Aleksandr Yu. Konovalov. 2016. Arithmetical realizability and primitive recursive realizability. Moscow Univ. Math. Bull. 71, 4 (2016), 166–169. https://doi.org/10.3103/S0027132216040069Google ScholarCross Ref
- Aleksandr Yu. Konovalov. 2018. The intuitionistic logic is not sound with respect to L-realizability. Intell. Syst. Theor. Applic. 22, 3 (2018), 41–44. Retrieved from http://mi.mathnet.ru/ista148Google Scholar
- Aleksandr Yu. Konovalov. 2018. The V-realizability for L-formulas coincides with the classical semantics iff V contains all L-definable functions. Intell. Syst. Theor. Applic. 22, 3 (2018), 127–130. Retrieved from http://mi.mathnet.ru/eng/ista153Google Scholar
- Aleksandr Yu. Konovalov. 2019. Absolute L-realizability and intuitionistic logic. Moscow Univ. Math. Bull. 74, 2 (2019), 79–82. https://doi.org/10.3103/S0027132219020086Google ScholarCross Ref
- Aleksandr Yu. Konovalov. 2019. Generalized realizability for extensions of the language of arithmetic. Moscow Univ. Math. Bull. 74, 4 (2019), 167–170. https://doi.org/10.3103/S0027132219040065Google ScholarCross Ref
- Aleksandr Yu. Konovalov. 2019. Markov’s principle is uniformly V-realizable in any V-enumerable domain. Intell. Syst. Theor. Applic. 23, 1 (2019), 99–103. Retrieved from http://mi.mathnet.ru/eng/ista219Google Scholar
- Aleksandr Yu. Konovalov. 2020. General recursive realizability and basic logic. Algeb. Log. 59 (2020), 367–384. https://doi.org/10.1007/s10469-020-09610-yGoogle ScholarCross Ref
- Aleksandr Yu. Konovalov. 2020. Generalized realizability and Markov’s principle. Moscow Univ. Math. Bull. 75, 1 (2020), 38–41. https://doi.org/10.3103/S0027132220010064Google ScholarCross Ref
- Aleksandr Yu. Konovalov and Valery E. Plisko. 2015. On hyperarithmetical realizability. Math. Notes 98, 5 (2015), 778–797. https://doi.org/10.1134%2FS0001434615110073Google ScholarCross Ref
- Ben H. Park. 2003. Subrecursive Realizability and Predicate Logic. Ph.D. Dissertation. Lomonosov Moscow State University, Moscow, Russian Federation.Google Scholar
- Valery E. Plisko. 1974. Recursive realizability and constructive predicate logic. Doklady Akad Nauk SSSR 214, 3 (1974), 520–523. http://mi.mathnet.ru/eng/dan38065.Google Scholar
- Valery E. Plisko. 1977. The nonarithmeticity of the class of realizable predicate formulas. Math. USSR - Izvestiya 11, 3 (1977), 453–471. https://doi.org/10.1070/IM1977v011n03ABEH001731Google ScholarCross Ref
- Valery E. Plisko. 1984. Absolute realizability of predicate formulas. Math. USSR - Izvestiya 22, 2 (1984), 291–308. https://doi.org/10.1070/IM1984v022n02ABEH001444Google ScholarCross Ref
- Hartley Rogers. 1987. Theory of Recursive Functions and Effective Computability. The MIT Press, Cambridge, MA. Google ScholarDigital Library
- Wim Ruitenburg. 1993. Basic logic and Fregean set theory. Dirk Van Dalen Festschrift, Quaestiones Infinitae 5 (1993), 121–142. Retrieved from https://www.mscsnet.mu.edu/ wim/publica/120519_baslog.pdfGoogle Scholar
- Wim Ruitenburg. 1998. Basic predicate calculus. Notre Dame J. Form. Log. 39, 1 (1998), 18–46. https://doi.org/10.1305/ndjfl/1039293019Google ScholarCross Ref
- Saeed Salehi. 2003. Provably total functions of basic arithmetic. Math. Log. Quart. 49, 3 (May 2003), 316–322. https://doi.org/10.1002/malq.200310032Google ScholarCross Ref
- Albert Visser. 1981. A propositional logic with explicit fixed points. Stud. Log. 40 (1981), 155–175. https://doi.org/10.1007/BF01874706Google ScholarCross Ref
- Dmitry A. Viter. 2002. Primitive Recursive Realizability and Constructive Theory of Models. Ph.D. Dissertation. Lomonosov Moscow State University, Moscow, Russian Federation.Google Scholar
Index Terms
- Generalized Realizability and Basic Logic
Recommendations
A Generalized Realizability and Intuitionistic Logic
Let V be a set of number-theoretical functions. We define a notion of V-realizability for predicate formulas in such a way that the indices of functions in V are used for interpreting the implication and the universal quantifier. In this article, we prove ...
Hintikka's Independence-Friendly Logic Meets Nelson's Realizability
Inspired by Hintikka's ideas on constructivism, we are going to `effectivize' the game-theoretic semantics (abbreviated GTS) for independence-friendly first-order logic (IF-FOL), but in a somewhat different way than he did in the monograph `The ...
Realizability for Monotone and Clausular (Co)inductive Definitions
We develop an extension of second order logic (AF2) with monotone, and not only positive, (co)inductive definitions and a clausular feature which simplifies considerably the defining mechanism. A sound realizability interpretation, where the extracted ...
Comments