##
###
Open Problems in Logical Dynamics
[chapter]

Johan van Benthem

2006
*
Mathematical Problems from Applied Logic I
*

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. 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
## more »

... tter-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. 2 Next, there is also a longer term to information flow beyond individual update or revision steps. For, these do not occur in isolation. There is a history of past interactions, determining our trust in our interlocutor, as well as a future of things yet to be said. Eventually, this calls for a merge of epistemic and a temporal logic which allows for statements of regularities over time, often in the form of 'protocols' (Section 7). Another longer-term perspective concerns the purpose of a question. Behind every ordinary question there is a "Why" question: what is the point, and what are people trying to achieve? This leads to current game logics for describing strategic behaviour in games, of which there exist quite a few by now. These are just some of the current bridges between between logic and game theory, which deserve a separate treatment. Section 8 provides a very brief introduction with pointers. Finally, Section 9 summarizes a few general issues playing across all of the above phenomena. A word of clarification. 'Open problems' in new areas like this may be of several different sorts. Some are clear-cut mathematical questions, where all notions have crystallized out, and what remains to be done is some hard-nosed theorem proving. But other significant questions concern mathematical modeling, finding perspicuous formal mechanisms for information update, or temporal evolution, or say, the logical behaviour of different types of agent. Often, questions like this are triggered by the challenge of describing some communicative practice, or some given game. Other questions at this conceptual level have to do with relating different logical paradigms trying to describe the same phenomena. Third, and finally, there are interesting and highly non-trivial questions of computational implementation and cognitive realism in studying the fit between all these logics and actual behaviour of men and machines. In this survey, the main emphasis will be on the first type of question, but there are a few of the second kind as well. We only mention issues of the third kind in passing. Finally, a caveat. This paper is not a self-contained introduction to logics for epistemic update and games. It is rather intended for readers with at least some background in the area, who can then see a coherent panorama of directions to be pursued. Standard Epistemic Logic The basis for all our further topics is standard epistemic logic, created originally by Hintikka for analyzing philosophical issues in epistemology -but linked more closely with computer science, and even economics, for a long time now. Excellent introductions to epistemic logic with a computational slant are Fagin, Halpern, Moses

doi:10.1007/0-387-31072-x_3
fatcat:f5il7g4k6zcdndjrl3ymqaetl4