Structural completeness in many-valued logics with rational constants

Abstract

The logics $\mathbf{R} \mathbf{\L}, \mathbf{R} \mathbf{P}$, and $\mathbf{R G}$ have been obtained by expanding $\{L}$ukasiewicz logic $\mathbf{L}$, product logic $\mathbf{P}$, and Gödel-Dummett logic $\mathbf{G}$ 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 $\mathbf{} \mathbf{,} \mathbf{P}$, and $\mathbf{G}$. Namely, $\mathbf{R} \mathbf{L}$ is hereditarily structurally complete. $\mathbf{R} \mathbf{P}$ 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 RP, show their decidability, and characterize passive structural completeness for extensions of $\mathbf{R P}$. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of $\mathbf{R P}$, and this is also the case for extensions of RG, where in turn passive structural completeness is characterized by the equivalent algebraic semantics having the joint embedding property. For nontrivial axiomatic extensions of $\mathbf{R G}$ we provide a base of admissible rules. We leave the problem open whether the variety of rational Gödel algebras is $\mathcal{Q}$-universal.