Truth-Tracking by Belief Revision

Abstract

Abstract.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.

On the Solvability of Inductive Problems: A Study in Epistemic Topology

“…This special case is a topological translation of one of our previous results[3,4]. However, the result about problemsolving universality is not only new and much more general, but also much harder to prove, involving new topological notions and results 2.…”

“…However, there does exist a line of research that combines belief revision with learning-theoretic notions, line pursued by Kelly [21,26], Kelly, Schulte and Hendricks [19], Martin and Osherson [28] and ourselves [13,3,4,14]. In this paper we continue this research program, using topological characterizations and methods.…”

Tracking probabilistic truths: a logic for statistical learning

“…Instead, it may eventually allow her to come to believe the true probability (at least, with a high degree of accuracy). This belief may even stabilize, to such a degree that it approaches the 'softer', defeasible notion of 'knowledge', which is the main focus in Epistemology (Lehrer 1990;Stalnaker 1996;Rott 2004) and (inductive) Learning Theory (Gold 1967;Baltag et al 2019a). This convergence in belief and the resulting acquisition of statistical knowledge is what we aim to capture in this paper.…”

“…Connections to belief revision theory Grove sphere models (in non-probabilistic form, consisting of possible worlds instead of distributions) form the standard semantic framework in Belief Revision Theory (Grove 1988). Plausibility models (again, in their non-probabilistic version) are well-known equivalent relational descriptions of sphere models, that are preferred in Dynamic Epistemic Logic (Baltag and Smets 2008a, b;Baltag et al 2019a;van Benthem 2007van Benthem , 2011, as well as in the "dynamic interactive epistemology" approach developed by game-theorists (Board 2004). These are in fact adaptations to doxastic modeling of the older setting of Lewis spheres, with its equivalent description in terms of a comparative similarity relation (Lewis 2000).…”

Learning to act: qualitative learning of deterministic action models

“…This is not the first application of learning theoretic tools to dynamic epistemic logic (see [13][14][15][16]) or to the logical theories of belief revision (see, e.g. [4,5,19]). The present work is however pioneering in studying the learning of the internal structure of actions in dynamic epistemic logic.…”