A note on strong axiomatization of G\"odel Justification Logic

TL;DR

This paper extends G"odel justification logic, a fuzzy variant of classical justification logic, by developing a strong completeness framework using fuzzy semantics and exploring its foundational properties.

Contribution

It introduces a fuzzy semantics for G"odel justification logic and proves strong completeness theorems for various extensions, advancing the understanding of fuzzy epistemic reasoning.

Findings

Established strong completeness theorems for fuzzy G"odel justification logic

Extended canonical semantics to fuzzy logic with minimum t-norm

Bridged justification logic with fuzzy inference models

Abstract

Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators "t:", indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We replace this classical boolean base with G\"odel logic, one of the three most prominent fuzzy logics, i.e. special instances of many-valued logics, taking values in the unit interval [0,1], which are intended to model inference under vagueness. We extend the canonical possible-world semantics for justification logic to this fuzzy realm by considering fuzzy accessibility- and evaluation-functions evaluated over the minimum t-norm and establish strong completeness theorems for various fuzzy analogies of prominent extensions for basic justification logic.