The higher dimensional propositional calculus

TL;DR

This paper introduces a sound and complete sequent calculus for n-dimensional Boolean algebra propositional logic, extending classical logic to multiple truth values and proving key properties.

Contribution

It develops the nLK calculus for nCL, generalizing propositional logic to multiple truth values with proofs of soundness and completeness.

Findings

nLK is sound and complete for nCL

The calculus enjoys the cut admissibility property

Completeness proofs include syntactic and semantic methods

Abstract

In recent research, some of the present authors introduced the concept of an n-dimensional Boolean algebra and its corresponding propositional logic nCL, generalising the Boolean propositional calculus to n>= 2 perfectly symmetric truth values. This paper presents a sound and complete sequent calculus for nCL, named nLK. We provide two proofs of completeness: one syntactic and one semantic. The former implies as a corollary that nLK enjoys the cut admissibility property. The latter relies on the generalisation to the n-ary case of the classical proof based on the Lindenbaum algebra of formulas and Boolean ultrafilters.