Abstract
It is well known that the usual versions of probability kinematics have serious limitations. According to the classical notion of conditioning when one learns a piece of information A its probability raises to its maximum (one). Moreover no further instance of learning will be capable of defeating A. Once a piece of information is learned one should be maximally confident about it and this confidence should remain unaltered forever. It is clear that there are many instances of learning that cannot be accommodated in this Procrustean bed. There are various ways of amending this limited picture by enriching the Bayesian machinery. For example, one can appeal to a notion of primitive conditional probability capable of making sense of conditioning on zero measure events. But the detailed consideration of this alternative leads to similar limitations: the picture of learning that thus arises continues to be cumulative. There are many ways of overcoming these important limitations. Williamson considers one possible way of doing so in his essay reprinted in the section on Bayesian epistemology. One of the lessons that have been learned in recent years is that there is no apparent way of circumventing this rigidity of Bayesianism without introducing in some way a qualitative doxastic or epistemic notion as a primitive alongside probability. Here are two examples: Williamson proposes a model where knowledge is a primitive, while Levi appeals to a primitive notion of full belief.
Horacio Arló-Costa was deceased at the time of publication.
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Suggested Further Reading
There are a number of recent surveys and books that complement the reprinted papers appearing here. Regarding surveys the two most recent surveys are: Logic of Belief Revision, in Stanford Encyclopedia of Philosophy, 2006, by Sven Ove Hansson; and: Belief Revision in The Continuum Companion to Philosophical Logic, (eds.) L. Hornsten and R. Pettigrew, by Horacio Arlo-Costa and Paul Pedersen. These surveys contain references to previous surveys in the field. A classic book in this area that continues to be useful is Peter Gärdenfors’s monograph: Knowledge in Flux: Modeling the Dynamic of Epistemic States, College Publications (June 2, 2008). A very useful textbook presentation of some of the main results in the theory of belief change is: A Textbook of Belief Dynamics: Theory Change and Database Updating, Springer 2010, by Sven Ove Hansson. The book focuses mainly on syntactic presentations of belief change and it contains a very detailed presentation of belief base updating. Some more recent topics like iterated belief change are not treated in detail though.
Decision theoretic foundations for belief change are provided in various books by Hans Rott and Isaac Levi (independently). A book-length argument articulating Rott’s account (and extending the content of the article reprinted here) appears in: Change, Choice and Inference: A Study of Belief Revision and Non-monotonic Reasoning, Oxford Logic Guides, 2001. Some challenges to this type of foundational strategy are considered by Arlo-Costa and Pedersen in: “Social Norms, Rational Choice and Belief Change,” in Belief Revision Meets Philosophy of Science, (eds.) E.J. Olsson and S. Enqvist, Springer, 2011. Isaac Levi has also published various essays where he presents decision theoretic foundations for belief change (but his account is rather different than Rott’s). The most recent book presenting Levi’s current views about belief change is: Mild Contraction: Evaluating Loss of Information Due to Loss of Belief, Oxford, 2004. Further references to his work can be found in this book.
The previous accounts tried to justify principles of belief change in the broader context of Bayesian or neo-Bayesian theory. An almost orthogonal view consists in deriving principles of belief change by taking some form of formal learning theory as an epistemological primitive. While all the previous accounts focused on justifying the next step of inquiry (or a finite and proximate sequence of steps) this second strategy focuses on selecting belief change methods capable of learning the truth in the long run. One important paper in this tradition is Kevin Kelly’s: Iterated Belief Revision, Reliability, and Inductive Amnesia, Erkenntnis, 50, 1998 pp. 11–58. Daniel Osherson and Eric Martin present a similarly motivated account that nevertheless is formally quite different from Kelly’s theory in: Elements of Scientific Inquiry, MIT, 1998.
There are various attempts to extend the theory of belief revision to the multi-agent case and to present a theory of belief change as some form of dynamic epistemic logic. The idea in this case is to use traditional formal tools in epistemic logic to represent the process of belief change. Hans van Ditmarsch, Wiebe van der Hoek, and Barteld Kooi have recently published a textbook with some basic results in this area: Dynamic Epistemic Logic, Springer, 2011. Krister Segerberg has developed his own brand of dynamic doxastic logic in a series of articles since at least the mid 1990’s. One recent paper including rather comprehensive results in this area is: “Some Completeness Theorems in the Dynamic Doxastic Logic of Iterated Belief Revision,” Review of Symbolic Logic, 3(2):228–246, 2010.
The notion of relevance is quite central for a representation of belief and belief change. In a Bayesian setting there are standard ways of articulating relevance. But there is recent work that has used proof theoretic techniques to deal with relevance rather than probability theory. Rohit Parikh initiated this area of research with an article published in 1999: Beliefs, belief revision, and splitting languages, Logic, language, and computation (Stanford, California) (Lawrence Moss, Jonathan Ginzburg, and Maarten de Rijke, editors), vol. 2, CSLI Publications, pp. 266–278. Recently David Makinson has contributed as well an important article in collaboration with George Kourousias,: Parallel interpolation, splitting, and relevance in belief change, Journal of Symbolic Logic 72 September 2007 994-1002. This article contains a detailed bibliography of recent work in this area.
One recent paper including rather comprehensive results in this area is: “Some completeness theorems in the dynamic doxastic logic of iterated belief revision,” Review of Symbolic Logic, 3, 02, 2010. For more on iterated belief revision please refer to: Darwiche and Pearl (Darwiche, A., & Pearl, J. (1996). On the logic of iterated belief revision. Artificial Intelligence, 89, 1–29) appears in: Change, choice and inference: A study of belief revision and non-monotonic reasoning, Oxford Logic Guides, 2001.
And there is also more to be found in Pagnucco and Rott (Pagnucco, M., & Rott, H. (1999). Severe withdrawal – and recovery. Journal of Philosophical Logic, 28, 501–547. See publisher’s “Erratum” (2000), Journal of Philosophical Logic, 29, 121) and Lindström (Lindström, S. (1991). A semantic approach to nonmonotonic reasoning: Inference operations and choice. Uppsala Prints and Preprints in Philosophy, no. 6/1991, University of Uppsala).
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Arló-Costa, H., Hendricks, V.F., van Benthem, J. (2016). Introduction. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_12
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