Primitive Recursive Realizability and Basic Propositional Logic

TL;DR: It is proved that every sequent deducible in Basic Propositional Calculus is strictly primitive recursively realizable.

Abstract: Two notions of primitive recursive realizability for arithmetic sentences are considered. The first one is strictly primitive recursive realizability introduced by Z. Damnjanovic in 1994. We prove that intuitionistic predicate logic is not sound with this kind of realizability. Namely there exists an arithmetic sentence which is deducible in the intuitionistic predicate calculus but is not strictly primitive recursively realizable. Another variant of primitive recursive realizability was introduced by S. Salehi in 2000. This kind of realizability is defined for the formulas of Basic Arithmetic introduced by W. Ruitenburg in 1998. We prove that these two notions of primitive recursive realizability are essentially di! erent. Namely there exists arithmetic sentence being also a sentence of Basic Arithmetic which is strictly primitive recursively realizable but is not realizable by Salehi. The negation of such a sentence is realizable by Salehi but is not strictly primitive recursively realizable. The relation between Basic Propositional Logic and strictly primitive recursive realizability is studied. We consider a sequent variant of Basic Propositional Calculus. Notions of strictly primitive recursive realizability for arithmetic and propositional sequents are defined. We prove that every sequent deducible in Basic Propositional Calculus is strictly primitive recursively realizable. An example of a sequent which is deducible in Intuitionistic Propositional Calculus but is not strictly primitive recursively realizable is proposed.