Revisiting the Adequacy Theorem for Fragments of Łukasiewicz Logic

Miguel Pérez-Gaspar; Juan Manuel Ramírez-Contreras; Juan Sebastián Slagter

doi:10.18778/0138-0680.2026.08

Authors

Miguel Pérez-Gaspar Universidad Nacional Autónoma de México (UNAM), Facultad de Ingeniería, Ciudad de México, México Author https://orcid.org/0000-0002-6761-4571
Juan Manuel Ramírez-Contreras Universidad Digital del Estado de México (UDEMEX), Informática Administrativa, Toluca, Méxicov Author
Juan Sebastián Slagter Universidad Nacional del Sur, Departamento de Matemática, Bahía Blanca, Argentina Author https://orcid.org/0000-0002-3440-9264

DOI:

https://doi.org/10.18778/0138-0680.2026.08

Keywords:

implicational fragment of Łukasiewicz logic, 3-valued Łukasiewicz logic, Δ operator, first-order logics

Abstract

A. V. Figallo introduced the 3-valued Super Łukasiewicz logic expanded with the Δ operator, denoted as C₃^↣,Δ, in 1990. This operator is used in the definition of 3-valued Łukasiewicz algebras, and it is not possible to recover Δ through implication and top in Super Łukasiewicz logic. On the other hand, Baaz introduced the Δ operator in Gödel logic, both in its propositional and quantified versions. Subsequently, this operator was extensively studied in the field of fuzzy logic.
In this paper, we prove a strong version of the Adequacy Theorem for C₃^↣,Δ3. As a consequence, we demonstrate that the Deduction Theorem does not hold in this calculus. Furthermore, we introduce the first-order version of C₃^↣,Δ3 and establish soundness and completeness results by adapting a recently developed algebraic technique. In this context, our presentation differs from others in the literature because we need to construct a special homomorphism, brought from the algebraic study of C₃^↣,Δ3, in the syntactic setting. This homomorphism is also necessary to determine the generating algebras. While we can ascertain that the logical system is algebraizable by a (quasi-)variety of algebras, we cannot know a priori which are the subdirectly irreducible algebras.

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