A Four-Valued Dynamic Epistemic Logic
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A Four-Valued Dynamic Epistemic Logic Yuri David Santos1
© The Author(s) 2020
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. Keywords Many-valued logics · Epistemic logic · Paraconsistent logics · Public announcements · Multi-agent systems · Evidence
1 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. Despite being very useful, this simple model leaves out of the discussion an important factor in the formation of beliefs: evidence. Belnap (1977) gave an interpretation to first degree entailment (FDE) (Dunn 1976; Priest 2008), a four-valued logic, centered on the idea of evidence. In that logic, a
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Yuri David Santos [email protected] Faculty of Philosophy and Institute of Artificial Intelligence, University of Groningen, Groningen, The Netherlands
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Y. D. Santos
Fig. 1 An epistemic model (left) and a four-valued epistemic model (right)
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. For example, if both is the value assigned to p, then 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. Later, modal extensions of FDE have been developed, such as K FDE (Priest 2008) and BK (Odintsov and Wansing 2010). As remarked by Fitting in the conclusion of Fitting (1991), very little has been said about intuitions underlying many-valued modal logics, a situation which seems to persist in the current literature. One of our main objectives in this paper is to extend the Belnapian epistemic interpretation of FDE to a modal setting. By doing this, we sim
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