Justification Logics in a Fuzzy Setting
Meghdad Ghari
TL;DR
This work introduces fuzzy justification logics by integrating Artemov-style justification with several t-norm based fuzzy logics, replacing classical propositional bases with ${ m BL}$, ${ m Ł}$, ${ m G}$, ${ m igtriangleup}$, and ${ m RPL}$ and developing corresponding fuzzy models with crisp accessibility. It establishes soundness for all resulting systems and proves a graded-style completeness theorem for the Rational Pavelka extension, ${ m RPLJ}$, including canonical-model constructions and Truth Lemmas. The paper also analyzes truth-constants extensions and discusses the inclusion of principles like jT and jD in the fuzzy setting, mapping out a program that broadens justification logic to reasoning under vagueness and uncertain evidence. Overall, it provides a rigorous semantic and proof-theoretic foundation for representing evidential degrees in justification-based epistemic reasoning and outlines directions to relate these systems to fuzzy modal logic ${ m K}$ and soft accessibility frameworks.
Abstract
Justification Logics provide a framework for reasoning about justifications and evidences. Most of the accounts of justification logics are crisp in the sense that agent's justifications for a statement is convincing or is not. In this paper, we study fuzzy variants of justification logics, in which an agent can have a justification for a statement with a certainty degree between 0 and 1. We replaced the classical base of the justification logics with some known fuzzy logics: Hajek's basic logic, Łukasiewicz logic, Gödel logic, product logic, and rational Pavelka logic. In all of the resulting systems we introduced fuzzy models (fuzzy possible world semantics with crisp accessibility relation) for our systems, and established the soundness theorems. In the extension of rational Pavelka logic we also proved a graded-style completeness theorem.