Boolean algebras with an automorphism group: a framework for Łukasiewicz logic

Authors:	Thomas Vetterlein
Title:	Boolean algebras with an automorphism group: a framework for Łukasiewicz logic
Journal:	Journal of Multiple-Valued Logic and Soft Computing
Volume	14
Number
Pages:	51 - 67
Year:	2008

Abstract

We introduce a framework within which reasoning according to Lukasiewicz logic can be represented. We consider a separable Boolean algebra $\mathcal B$ endowed with a (certain type of) group $G$ of automorphisms; the pair $({\mathcal B},G)$ will be called a Boolean ambiguity algebra. $\mathcal B$ is meant to model a system of crisp properties; $G$ is meant to express uncertainty about these properties.

We define fuzzy propositions as subsets of $\mathcal B$ which are, most importantly, closed under the action of $G$. By defining a conjunction and implication for pairs of fuzzy propositions in an appropriate manner, we are led to the algebraic structure characteristic for Lukasiewicz logic.