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Averaging the truth-value in Łukasiewicz logic. (English) Zbl 0836.03016
Summary: Chang’s MV algebras are the algebras of the infinite-valued sentential calculus of Łukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of ‘average degree of truth’ of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AF \(C^*\)-algebras stand to commutative AF \(C^*\)-algebras, states are naturally related to noncommutative \(C^*\)-algebraic measures.

MSC:
03B50 Many-valued logic
03G25 Other algebras related to logic
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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