Maruyama, Yoshihiro2024-01-19July 19-24978-172816932-3http://hdl.handle.net/1885/311638It is known that some fuzzy predicate logics, such as Łukasiewicz predicate logic, are not complete with respect to the standard real-valued semantics. In the present paper we focus upon a typed version of first-order MTL (Monoidal T-norm Logic), which gives a unified framework for different fuzzy logics including, inter alia, Hajek's basic logic, Łukasiewicz logic, and Gödel logic. And we show that any extension of first-order typed MTL, including Łukasiewicz predicate logic, is sound and complete with respect to the corresponding categorical semantics in the style of Lawvere's hyperdoctrine, and that the so-called Baaz delta translation can be given in the first-order setting in terms of Lawvere's hyperdoctrine. A hyperdoctrine may be seen as a fibred algebra, and the first-order completeness, then, is a fibred extension of the algebraic completeness of propositional logic. While the standard real-valued semantics for Łukasiewicz predicate logic is not complete, the hyperdoctrine, or fibred algebraic, semantics is complete because it encompasses a broader class of models that is sufficient to prove completeness; in this context, incompleteness may be understood as telling that completeness does not hold when the class of models is restricted to the standard class of real-valued hyperdoctrine models. We expect that this finally leads to a unified categorical understanding of Takeuti-Titani's fuzzy models of set theory.The author hereby acknowledges that the present work was financially supported by JST PRESTO (grant code: JPMJPR17G9) and JSPS KAKENHI (grant code: 17K14231).application/pdfen-AU© 2020 IEEEfirst-order typed fuzzy logiccategorical semanticscompletenessBaaz translationLawvere hyperdoctrine202010.1109/FUZZ48607.2020.91776952022-10-02