Mutually exclusive nuances of truth in Moisil logic

TL;DR

This paper refines the algebraic framework of Moisil logic by providing an alternative definition that preserves the principle of reasoning about vague information in subalgebras, and establishes a duality theory for these algebras.

Contribution

It introduces an equivalent definition of ulusiewicz-Moisil algebras that maintains the determination principle in subalgebras and develops a duality theory for these algebras.

Findings

New equivalent definition of ulusiewicz-Moisil algebras.

Duality result for ulusiewicz-Moisil algebras based on Boolean centers.

Extension of duality to MV-algebras.

Abstract

Moisil logic, having as algebraic counterpart \L ukasiewicz-Moisil algebras, provide an alternative way to reason about vague information based on the following principle: a many-valued event is characterized by a family of Boolean events. However, using the original definition of \L ukasiewicz-Moisil algebra, the principle does not apply for subalgebras. In this paper we identify an alternative and equivalent definition for the $n$-valued \L ukasiewicz-Moisil algebras, in which the determination principle is also saved for arbitrary subalgebras, which are characterized by a Boolean algebra and a family of Boolean ideals. As a consequence, we prove a duality result for the $n$-valued \L ukasiewicz-Moisil algebras, starting from the dual space of their Boolean center. This leads us to a duality for MV$_n$-algebras, since are equivalent to a subclass of $n$-valued \L ukasiewicz-Moisil algebras.