TY - GEN

T1 - First-order typed fuzzy logics and their categorical semantics

T2 - 2020 IEEE International Conference on Fuzzy Systems, FUZZ 2020

AU - Maruyama, Yoshihiro

N1 - Publisher Copyright:
© 2020 IEEE.

PY - 2020/7

Y1 - 2020/7

N2 - 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.

AB - 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.

KW - Baaz translation

KW - Categorical semantics

KW - Completeness

KW - First-order typed fuzzy logic

KW - Lawvere hyperdoctrine

UR - http://www.scopus.com/inward/record.url?scp=85090495566&partnerID=8YFLogxK

U2 - 10.1109/FUZZ48607.2020.9177695

DO - 10.1109/FUZZ48607.2020.9177695

M3 - Conference contribution

T3 - IEEE International Conference on Fuzzy Systems

BT - 2020 IEEE International Conference on Fuzzy Systems, FUZZ 2020 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 19 July 2020 through 24 July 2020

ER -