Structural Completeness in Many-Valued Logics with Rational Constants

Abstract

The logics $\mathbf{R}\mathbf{\text{Ł}}$, $\mathbf{RP}$, and $\mathbf{RG}$ have been obtained by expanding Łukasiewicz logic $\mathbf{\text{Ł}}$, product logic P, and Gödel–Dummett logic G with rational constants. We study the lattices of extensions and structural completeness of these three expansions, obtaining results that stand in contrast to the known situation in $\mathbf{\text{Ł}}$, P, and G. Namely, $\mathbf{R}\mathbf{\text{Ł}}$ is hereditarily structurally complete. $\mathbf{RP}$ is algebraized by the variety of rational product algebras that we show to be $\mathcal{Q}$-universal. We provide a base of admissible rules in $\mathbf{RP}$, show their decidability, and characterize passive structural completeness for extensions of $\mathbf{RP}$. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of $\mathbf{RP}$, and this is also the case for extensions of $\mathbf{RG}$, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of $\mathbf{RG}$, we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is $\mathcal{Q}$-universal.