Tripos and Topos Theory for Fuzzy Logic: Baaz Δ Translation as Lawvere-Tierney Cotopology and Higher-Order Hyperdoctrinal Completion
Yoshihiro Maruyama, Satoshi Nakata, Taichi Uemura, Takeo Uramoto, Arisa Yasuda · 2025
Categorical semantics of first-order and higher-order fuzzy logic has recently been developed in terms of Lawvere hyperdoctrine and Hyland-Johnstone-Pitts tripos. Building upon them, we provide a novel categorical formulation of Baaz delta translation in terms of Lawvere-Tierney (co)topology, which is enabled via the internalisation of truth-value objects as in topos theory. As a by-product of this, we obtain a general notion of topos for fuzzy logic. At the same time, we prove a higher-order completion theorem which allows us to construct higher-order fuzzy logical hyperdoctrines (models of higher-order fuzzy logic) from first-order fuzzy logical hyperdoctrines (models of first-order fuzzy logic). The higher-order completion method is even new for intuitionistic logic as well as fuzzy logic.