Open Problems in Logical Dynamics

Abstract

In recent years, a number of 'dynamic epistemic logics' have been developed for dealing with information, communication, and interaction. This paper is a survey of conceptual issues and open mathematical problems emanating from this development. 1 Logical Dynamics The traditional paradigm of logic is drawing a conclusion from some given premises. But derivation from data already at our disposal is just one way in which information can be obtained. We can also observe new facts, or just ask some better-informed person whom we trust. Concomitantly with all this information flow, our knowledge and beliefs change, and this adaptation process may even be triggered by further cues. Such cognitive actions are of logical interest per se, and their explicit study and its various repercussions has been described as a 'Dynamic Turn' in logic (van Benthem 1996). In particular, relevant actions in this broader setting need not be single-agent tasks such as drawing a conclusion or observing some fact. After all, perhaps the simplest logical scenario for getting or giving information is asking a question. But this essentially involves information flow between two agents, and their mutual epistemic and 'social' interactions as the question is asked and an answer is given. An excellent framework for multi-agent dynamic behaviour in communication is epistemic logic (introduced in Section 2), suitably 'dynamified' by using ideas from the dynamic logic of actions. Section 3 is about the best-explored system of this kind, viz. the dynamic logic of public announcements. Section 4 generalizes this to general dynamic-epistemic logic of actions or events whose observation conveys information. The resulting technical questions here blend into issues about more classical logical systems, which are discussed in Section 5 on first-order and fixed-point logics. But knowledge and ignorance are not the only attitudes of participants in a conversation. They also have beliefs about their current situation and expectations about the future. These are revised as observation and communication take place. Thus, epistemic dynamics runs into belief revision. Section 6 is devoted to links between dynamicepistemic logic and belief revision theory as developed in AI and related areas.