Boolean algebras with an automorphism group: a framework for Łukasiewicz logic. (English) Zbl 1236.03018

Summary: We introduce a framework within which reasoning according to Łukasiewicz 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 Łukasiewicz logic.


03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
03G05 Logical aspects of Boolean algebras
06D35 MV-algebras
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