On the Equipollence of the Calculi Int and KM

TL;DR

This paper proves the equivalence of certain propositional calculi, extends the result to modal logics based on intuitionistic logic, and explores algebraic interpretations, contributing to the understanding of logical systems and their relationships.

Contribution

It restores Kuznetsov's syntactic proof of equipollence between intuitionistic calculus and KM, and generalizes this property to a broad class of modal logics on an intuitionistic basis.

Findings

Proves assertoric equipollence of Int and KM calculi

Extends equipollence to modalized Heyting calculus mHC

Provides algebraic interpretation of equipollence for KM subsystems

Abstract

Following A. Kuznetsov's outline, we restore Kuznetsov's syntactic proof of the assertoric equipollence of the intuitionistic propositional calculus and the proof-intuitionistic calculus KM (Kuznetsov's Theorem). Then, we show that this property is true for a broad class of modal logics on an intuitionistic basis, which includes, e.g., the modalized Heyting calculus mHC. The last fact is one of two key properties necessary for the commutativity of a diagram involving the lattices of normal extensions of four well-known logics. Also, we give an algebraic interpretation of the assertoric equipollence for subsystems of KM.