Structural completeness in many-valued logics with rational constants

TL;DR

This paper investigates the structural completeness of three rational-constant-extended many-valued logics, revealing unique properties and algebraic characterizations that contrast with their base logics.

Contribution

It provides a detailed analysis of the lattices of extensions and structural completeness, including bases of admissible rules and algebraic properties for RL, RP, and RG.

Findings

RL is hereditarily structurally complete.

RP's algebraic semantics are Q-universal.

Structural completeness properties coincide for extensions of RP and RG.

Abstract

The logics RL, RP, and RG have been obtained by expanding Lukasiewicz logic L, product logic P, and G\"odel--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 L, P, and G. Namely, RL is hereditarily structurally complete. RP is algebraized by the variety of rational product algebras that we show to be Q-universal. We provide a base of admissible rules in RP, show their decidability, and characterize passive structural completeness for extensions of RP. Furthermore, structural completeness, hereditary structural completeness, and active structural completeness coincide for extensions of RP, 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 RG we provide a base of admissible rules. We leave the problem open whether the variety of rational G\"odel algebras is Q-universal.