On the multi dimensional modal logic of substitutions

TL;DR

This paper develops a comprehensive algebraic framework for multi-dimensional modal logics with structured states, establishing key logical properties such as completeness, interpolation, and decidability.

Contribution

It introduces an algebraic approach using complex algebras of Kripke semantics for multi-dimensional modal logics with structured states, extending existing algebraic logic methods.

Findings

Proved completeness, interpolation, and decidability for the logic.

Established an omitting types theorem in this context.

Connected the algebraic structures to cylindric and polyadic algebras.

Abstract

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing (varieties generated by) complex algebras of Kripke semantics for such logic. Those algebras, whose elements are sets of states are common reducts of cylindric and polyadic algebras