Probability measures in the logic of nilpotent minimum
Nilpotent minimum logic (NML) was introduced in 2001 by Esteva and Godo as a formalization of the properties of nilpotent minimum t-norms. NML is obtained from MTL by adding the involutivity axiom for negation together with a special axiom taking care of the nilpotent minimum condition. NM-algebras are the algebras of NML. The present paper is devoted to the axiomatization of the notion of a ``state'' in the \(n\)-generator-free NM-algebra \(F(n)\) as a (de Finetti) coherent assessment on the events described by NML-formulas. It is well known that \(F(n)\) is an algebra of \((0,1)\)-valued functions defined on the unit real interval \([0,1]^n\). The following counterpart of the MV-algebraic Kroupa-Panti theorem is proved in the paper under review: The integral induces a one-one correspondence between states of \(F(n)\) and Soret probability measures on \([0,1]^n\). For background on the Kroupa-Panti theorem and de Finetti's coherent assessments of continuous-valued events see the present reviewer's monograph [Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic -- Studia Logica Library 35. Berlin: Springer (2011; Zbl 1235.03002)].
scientific article; zbMATH DE number 5701671
scientific article