A temporal logic of epistemic and normative justifications, with an application to the Protagoras paradox

TL;DR

This paper introduces a combined temporal, epistemic, and deontic logic with justification assertions, providing semantics and axioms, and applies it to analyze the Protagoras paradox and related normative reasoning.

Contribution

It develops a novel logic integrating temporal, epistemic, and normative justifications with semantics and axioms, and applies it to formalize and analyze a classic paradox.

Findings

Established soundness and completeness of the logic.

Provided formal semantics based on Fitting and neighborhood models.

Analyzed the Protagoras paradox within the new logical framework.

Abstract

We combine linear temporal logic (with both past and future modalities) with a deontic version of justification logic to provide a framework for reasoning about time and epistemic and normative reasons. In addition to temporal modalities, the resulting logic contains two kinds of justification assertions: epistemic justification assertions and deontic justification assertions. The former presents justification for the agent's knowledge and the latter gives reasons for why a proposition is obligatory. We present two kinds of semantics for the logic: one based on Fitting models and the other based on neighborhood models. The use of neighborhood semantics enables us to define the dual of deontic justification assertions properly, which corresponds to the notion of permission in deontic logic. We then establish the soundness and completeness of an axiom system of the logic with respect to these semantics. Further, we formalize the Protagoras versus Euathlus paradox in this logic and present a precise analysis of the paradox, and also briefly discuss Leibniz's solution.