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AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC $\mathbf {MMTL}\boldsymbol {\forall }$

Part of: Distributive lattices Algebraic logic Boolean algebras (Boolean rings)

Published online by Cambridge University Press:  18 October 2023

JUNTAO WANG ,
HONGWEI WU ,
PENGFEI HE  and
YANHONG SHE
JUNTAO WANG
Affiliation:
SCHOOL OF SCIENCE XI’AN SHIYOU UNIVERSITY XI’AN, SHAANXI 710065 CHINA E-mail: wjt@xsyu.edu.cn
HONGWEI WU
Affiliation:
SCHOOL OF SCIENCE XI’AN SHIYOU UNIVERSITY XI’AN, SHAANXI 710065 CHINA E-mail: wuhw@snnu.edu.cn
PENGFEI HE
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS SHAANXI NORMAL UNIVERSITY XI’AN, SHAANXI 710119 CHINA E-mail: hepengf1986@126.com
YANHONG SHE*
Affiliation:
SCHOOL OF SCIENCE XI’AN SHIYOU UNIVERSITY XI’AN, SHAANXI 710065 CHINA
*
E-mail: yanhongshe@xsyu.edu.cn
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Abstract

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$-valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

Keywords

monadic fuzzy predicate logicS5-like fuzzy modal logicmonadic MTL-algebrafunctional representationsubdirectly irreduciblecompleteness

MSC classification

Primary: 06D35: MV-algebras
Secondary: 03G05: Boolean algebras
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 27
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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