Invariant measures in Lukasiewicz logic
Giovanni Panti · arXiv (Cornell University) · 2005
Abstract. We prove that on the finitely generated free MV-algebras the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure. 1. Preliminaries To fix notation, we recall that an MV-algebra is an algebra (A, ⊕, ¬, 0) such that (A, ⊕, 0) is a commutative monoid and the identities ¬¬f = f, f ⊕ ¬0 = ¬0, and ¬(¬f ⊕g) ⊕g = ¬(¬g ⊕f) ⊕f are satisfied; as usual, we define f ⊙g = ¬(¬f ⊕ ¬g) and 1 = ¬0. MV-algebras stand to ̷Lukasiewicz infinite-valued propositional logic as Boolean algebras stand to classical two-valued logic; we assume familiarity with the basics of the theory, see [5], [4], [1], [3]. In [7], Mundici defined a state on the MV-algebra A as a function m: A → [0, 1] such that m(0) = 0, m(1) = 1, and m(f ⊕ g) = m(f) + m(g) provided that f ⊙g = 0. If A is viewed as the Lindenbaum algebra of some theory in ̷Lukasiewicz logic, then m is a function assigning an “average truth-value ” to the elements of A, i.e., to the propositional formulas modulo the theory. The set of all states of A is a