This paper presents a simple decidable logic of functional dependence LFD, based on an extension of classical propositional logic with dependence atoms plus dependence quantifiers treated as modalities, within the setting of generalized assignment semantics for first order logic. The expressive strength, complete proof calculus and meta-properties of LFD are explored. Various language extensions are presented as well, up to undecidable modal-style logics for independence and dynamic logics of changing dependence models. Finally, more concrete settings for dependence are discussed: continuous dependence in topological models, linear dependence in vector spaces, and temporal dependence in dynamical systems and games.

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 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 process of truth-seeking by iterated belief revision with higher-level doxastic information. We elaborate further on the main results in Baltag and Smets (Proceedings of TARK, 2009a, Proceedings of WOLLIC’09 LNAI 5514, 2009b), applying them to the issue of convergence to truth. We study the conditions under which the belief revision induced by a series of truthful iterated upgrades eventually stabilizes on true beliefs. We give two different conditions ensuring that beliefs converge to “full” (complete) truth, as well as a condition ensuring only that they converge to true (but not necessarily complete) beliefs.