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Forcing in Łukasiewicz predicate logic. (English) Zbl 1170.03013
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 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.

##### MSC:
 03B50 Many-valued logic 03B22 Abstract deductive systems 03C25 Model-theoretic forcing
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