Stalnaker (Philosophical Studies, 128(1), 169–199 2006) introduced a combined epistemic-doxastic logic that can formally express a strong concept of belief, a concept of belief as ‘subjective certainty’. In this paper, we provide a topological semantics for belief, in particular, for Stalnaker’s notion of belief defined as ‘epistemic possibility of knowledge’, in terms of theclosure of the interior operatoron extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge (as the interior operator) with a new topological semantics for belief. We prove that the belief logicKD45is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. We also study (static) belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based onhereditarily extremally disconnected spaces, and axiomatize the belief logic of hereditarily extremally disconnected spaces. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic.

We propose a number of powerful dynamic-epistemic logics for multi-agent information sharing and acts of publicly or privately accessing other agents’ information databases. The static base of our logics is obtained by adding to standard epistemic logic comparative epistemic assertions for groups or individuals, as well as a common distributed knowledge operator (that combines features of both common knowledge and distributed knowledge). On the dynamic side, we introduce actions by which epistemic superiority can be acquired: “sharing all one knows” (by e.g. giving access to one’s information database to all or some of the other agents), as well as more complex informational events, such as hacking. We completely axiomatize several such logics and prove their decidability.

We study the learning power of iterated belief revision methods. Successful learning is understood as convergence to correct, i.e., true, beliefs. We focus on the issue of universality: whether or not a particular belief revision method is able to learn everything that in principle is learnable. We provide a general framework for interpreting belief revision policies as learning methods. We focus on three popular cases: conditioning, lexicographic revision, and minimal revision. Our main result is that conditioning and lexicographic revision can drive a universal learning mechanism, provided that the observations include only and all true data, and provided that a non-standard, i.e., non-well-founded prior plausibility relation is allowed. We show that a standard, i.e., well-founded belief revision setting is in general too narrow to guarantee universality of any learning method based on belief revision. We also show that minimal revision is not universal. Finally, we consider situations in which observational errors (false observations) may occur. Given a fairness condition, which says that only finitely many errors occur, and that every error is eventually corrected, we show that lexicographic revision is still universal in this setting, while the other two methods are not.

We investigate the issues of inductive problem-solving and learning by doxastic agents. We provide topological characterizations of solvability and learnability, and we use them to prove that AGM-style belief revision is "universal", i.e., that every solvable problem is solvable by AGM conditioning.Comment: In Proceedings TARK 2015, arXiv:1606.0729

In this paper, we propose a unified logical framework for representing and analyzing various forms of correlated information change. Our main thesis is that “logical dynamics,” in the sense of van Benthem (Exploring logical dynamics. CSLI Publications, Stanford, 1996; Logical dynamics of information and interaction. Cambridge University Press, Cambridge, 2011), and in particular dynamic epistemic notions of conditional, as developed in Baltag and Smets (Electron Notes Theor Comput Sci 165:5–21, 2006a; Stud Log 89:185–209, 2008a; Texts in logic and games. Amsterdam University Press, Amsterdam, pp 9–58, 2008b), play a central role in understanding and modeling a wide range of apparently very different information-gathering phenomena which do have one specific feature in common, namely the very act of learning new information may directly change the reality that is being learned. On the one hand, we focus on the way in which an introspective agent changes her beliefs when learning new higher-order information, i.e., information that may refer to her own beliefs. On the other hand, we analyze situations in which an observer learns about a phenomenon by performing observations that may perturb the very phenomenon under study, as in the case of quantum measurements, or observations in social sciences, psychology and medicine. Our formal techniques are based on ideas from dynamic logic and on the modeling of “dynamic conditionals.” We offer a semantics based on “test frames,” i.e., Kripke frames labeled by propositional formulae which yields a unified setting for the two types of correlated information change under study. We show how this framework can be used to analyze the ontic and epistemic–informational aspects of quantum measurements and to compare them with other types of observation, testing, belief revision, counterfactual conditionals, etc.