Forcing in Łukasiewicz predicate logic
type: Item
instance of: scholarly article
title: Forcing in Łukasiewicz predicate logic
zbMATH Open document ID: 1170.03013
author: Antonio Di Nola
author: George Georgescu
author: Luca Spada
published in: Studia Logica
publication date: 2008-07-31T00:00:00Z
review text: This paper studies the notion of forcing for Łukasiewicz predicate logic along the lines of Robinson's forcing in classical model theory. First, the basic notions and results on the syntax and semantics of Łukasiewicz propositional logic (Ł\(_\infty\)) and Łukasiewicz predicate logic (Ł\(\forall\)) are given. Finite forcing for Ł\(\forall\) is introduced as a translation of Keisler's forcing. Finitely generic structures are defined and a many-valued version of the Generic Model Theorem is proved. Infinite forcing values of sentences in Ł\(\forall\) are introduced as a semantical concept arising from \textit{A. Robinson}'s infinite forcing in classical model theory [``Infinite forcing in model theory'', in: Proc. 2nd Scandinav. Logic Sympos. 1970, Stud. Log. Found. Math. 63, 317--340 (1971; Zbl 0222.02057)]. A number of results describe the behavior of this new semantics w.r.t. the logical operations of Ł\(\forall\). Finally, infinitely generic structures are introduced, and it is shown that any Ł\(\forall\) structure can be embedded in a generic one. This provides the existence of generic structures, as well as the existence of existentially complete structures. A global characterization of the class of generic models is given. Previous knowledge of classical forcing is not required to understand this article. Definitions are built from scratch and detailed proofs are given.
reviewed by: Albert Hoogewijs
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B22
Mathematics Subject Classification ID: 03C25
zbMATH DE Number: 5306430
zbMATH Keywords: first-order many-valued logic
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: Robinson forcing
zbMATH Keywords: generic structures
MaRDI profile type: Publication
full work available at URL: s11225-008-9120-y
OpenAlex ID: W2140360098
cites work: Completing theories by forcing
cites work: Q4680302
cites work: Simplicial structures in MV-algebras and logic
cites work: Algebraizable logics
cites work: Algebraic Analysis of Many Valued Logics
cites work: A New Proof of the Completeness of the Lukasiewicz Axioms
cites work: The Axiom of Comprehension in Infinite Valued Logic.
cites work: Omitting types of prenex formulas
cites work: Algebraic foundations of many-valued reasoning
cites work: On theories and models in fuzzy predicate logics
cites work: On the Consistency of the Axiom of Comprehension in the Lukasiewicz Infinite Valued Logic.
cites work: Metamathematics of fuzzy logic
cites work: On arithmetic in the Cantor-Łukasiewicz fuzzy set theory
cites work: Q4450450
cites work: Rational Pavelka predicate logic is a conservative extension of Łukasiewicz predicate logic
cites work: Forcing, arithmetic, division rings
cites work: Q4180327
cites work: A theorem about infinite-valued sentential logic
cites work: Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus
cites work: Omitting types in fuzzy logic with evaluated syntax
cites work: Q3663260
cites work: Q5626647
cites work: Fragments of Many-Valued Statement Calculi
cites work: Die nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz
DOI: 10.1007/S11225-008-9120-Y
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description: scientific article; zbMATH DE number 5306430