States on polyadic MV-algebras
type: Item
instance of: scholarly article
title: States on polyadic MV-algebras
zbMATH Open document ID: 1219.03078
author: George Georgescu
published in: Studia Logica
publication date: 2010-04-26T00:00:00Z
review text: The paper generalises Gaifman's approach to probabilistic models of first-order classical logic to the case of first-order Łukasiewicz infinite-valued logic. The paper is structured as follows: Section 2 recalls fundamental notions of the theory of MV-algebras, which constitutes the algebraic semantics of propositional Łukasiewicz logic, while Section 3 recalls the definition and basic properties of polyadic MV-algebras, which constitute the algebraic semantics of first-order Łukasiewicz logic, in the sense that Lindenbaum algebras of first-order Łukasiewicz logic are polyadic MV-algebras. New results are given in Sections 4 and 5. In particular, in Section 4, the author introduces the notion of polyadic MV-state and proves that for any MV-subalgebra \(D\) of a polyadic MV-algebra \(A\), and for any state \(\mu\) (the MV analog of a probability) there is a polyadic MV-state \(m\) extending \(\mu\) in the sense that the restriction of \(m\) to \(D\) coincides with \(\mu\). In Section 5, this result is carried over from algebra to the first-order Łukasiewicz logic \(\L\forall\) as follows: Given a set of new constant symbols \(U\), an MV-state \(m\) over the language of \(\L\forall\) extended with \(U\) is called a Gaifman state if for any formula \(\varphi(x)\) it holds that \[ m(\exists x\varphi(x))= \sup\{m(\bigvee_{i=1}^n \varphi(c_i)) \colon n \in \omega,\, c_1,\ldots,c_n \in U\}\,. \] Then a Gaifman MV-model of a state \(\mu\) over an MV-algebra \(D\) is a pair \((U,m)\) for \(U\) as before and \(m\) a Gaifman state over \(\L\forall\) extended with \(U\) such that the restriction of \(m\) to \(D\) coincides with \(\mu\). The final theorem then proves that any MV-state \(\mu\) has a Gaifman MV-model.
reviewed by: Stefano Aguzzoli
Mathematics Subject Classification ID: 03G20
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 28E15
zbMATH DE Number: 5701676
zbMATH Keywords: polyadic MV-algebra
zbMATH Keywords: polyadic MV-state
zbMATH Keywords: Łukasiewicz first-order logic
zbMATH Keywords: Gaifman model
MaRDI profile type: Publication
full work available at URL: s11225-010-9233-y
OpenAlex ID: W2088869122
cites work: Q4888332
cites work: Algebraic Analysis of Many Valued Logics
cites work: Algebraic foundations of many-valued reasoning
cites work: Algebraic logic for rational Pavelka predicate calculus
cites work: Q5607972
cites work: Concerning measures in first order calculi
cites work: Some model theory for probability structures
cites work: Bosbach states on fuzzy structures
cites work: Metamathematics of fuzzy logic
cites work: Q3282928
cites work: Representation and extension of states on MV-algebras
cites work: Averaging the truth-value in Łukasiewicz logic
cites work: Q4788820
cites work: Polyadic MV-Algebras
cites work: Q5599800
DOI: 10.1007/S11225-010-9233-Y
description: scientific article; zbMATH DE number 5701676