Revisiting the Adequacy Theorem for Fragments of Łukasiewicz Logic

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.