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Journal of Logic and Computation Advance Access published online on August 17, 2009
Journal of Logic and Computation, doi:10.1093/logcom/exp045
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© The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original Papers

On two Attempts of Describing Propositional Realizability Logic

Valery Plisko

Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119991, Russia.
E-mail: veplisko{at}yandex.ru

Received 6 February 2009.


  Abstract

The study of propositional realizability logic was initiated in the 50th of the last century. Unfortunately, no description of the class of realizable propositional formulas is found up to now. Nevertheless, some attempts of such a description were made. In 1974, the author proved that every known realizable propositional formula has the property that every one of its closed arithmetical instances is deducible in the system obtained by adding Extended Church's Thesis and Markov Principle as axiom schemes to Intuitionistic Arithmetic. Visser calls this system Markov's Arithmetic. In 1990, another attempt of describing the class of realizable propositional formulas was made by Varpakhovskii who proposed a calculus in an extended propositional language and proved that all known realizable propositional formulas are deducible in this calculus. In this article we prove that every propositional formula deducible in Varpakhovskii's calculus has the property that each of its closed arithmetical instances is deducible in Markov's Arithmetic.

Keywords: Propositional logic; recursive realizability; Markov's Arithmetic


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