First-Order Typed Fuzzy Logics and their Categorical Semantics: Linear Completeness and Baaz Translation via Lawvere Hyperdoctrine Theory

Date

2020

Authors

Maruyama, Yoshihiro

Journal Title

Journal ISSN

Volume Title

Publisher

IEEE

Abstract

It 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.

Description

Keywords

first-order typed fuzzy logic, categorical semantics, completeness, Baaz translation, Lawvere hyperdoctrine

Citation

Source

2020 IEEE International Conference on Fuzzy Systems, FUZZ 2020

Type

Conference paper

Book Title

Entity type

Access Statement

License Rights

Restricted until

2099-12-31