Probability measures in the logic of nilpotent minimum
type: Item
instance of: scholarly article
title: Probability measures in the logic of nilpotent minimum
zbMATH Open document ID: 1209.03054
author: Stefano Aguzzoli
author: Brunella Gerla
published in: Studia Logica
publication date: 2010-04-26T00:00:00Z
review text: 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)].
reviewed by: Daniele Mundici
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 28A60
Mathematics Subject Classification ID: 60A05
zbMATH DE Number: 5701671
zbMATH Keywords: MV-algebra
zbMATH Keywords: Kroupa-Panti theorem
zbMATH Keywords: MV-algebraic state
zbMATH Keywords: nilpotent minimum logic
zbMATH Keywords: probability measure
zbMATH Keywords: Gödel logic
zbMATH Keywords: NM-algebra
MaRDI profile type: Publication
full work available at URL: s11225-010-9228-8
OpenAlex ID: W2067787770
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DOI: 10.1007/S11225-010-9228-8
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description: scientific article; zbMATH DE number 5701671