Boolean Algebras with an Automorphism Group: a Framework for Lukasiewicz Logic

We introduce a framework within which reasoning according to à Lukasiewicz logic can be represented. We consider a separable Boolean algebra B endowed with a (certain type of) group G of automorphisms; the pair (B, G) will be called a Boolean ambiguity algebra. 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 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 à Lukasiewicz logic.