A Four-Valued Dynamic Epistemic Logic

Yuri David Santos

doi:10.1007/s10849-020-09313-8

Abstract

Epistemic logic is usually employed to model two aspects of a situation: the factual and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with these situations, where agents have only knowledge about the available information (or evidence), which can be incomplete or conflicting, but not explicitly about facts. This layer of available information or evidence, which is the object of the agents’ knowledge, can be seen as a database. By adopting this sceptical posture in our semantics, we prepare the ground for logics where the notion of knowledge—or more appropriately, belief—is entirely based on evidence. The technical results include a set of reduction axioms for public announcements, correspondence proofs, and a complete tableau system. In summary, our contributions are twofold: on the one hand we present an intuition and possible application for many-valued modal logics, and on the other hand we develop a logic that models the dynamics of evidence in a simple and intuitively clear fashion.

Highlights

Epistemic logic usually features a set of propositions about the world, and models a group of agents and their knowledge about these propositions
We simultaneously achieve two goals: (i) we design a four-valued modal logic suited to model situations where there is a publicly available body of potentially conflicting or incomplete evidence, and a group of agents that might be uncertain about what evidence is there and about what others know about this evidence; and (ii) we provide an epistemic intuition to many-valued logics, contributing to their practical applicability
We find that V ( i φ, s) = ∅ means that agent i considers it possible that there is no evidence for φ and she knows that there is no evidence against it

Summary

Introduction

Epistemic logic usually features a set of propositions about the world, and models a group of agents and their knowledge (or beliefs) about these propositions. Santos proposition p can be, besides true or false, both (true and false) or neither (true nor false). He interpreted these truth-values as the status of information possibly coming from several sources. If both is the value assigned to p, this means that some source supports the truth and another the falsity of p. The value none means that no information is available about p In this way, the valuation already has an epistemic (not ontic) character

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Conclusion