In This Article Formal Epistemology

  • Introduction
  • General Overviews and Textbooks
  • Anthologies
  • Journal Articles
  • Reference Works
  • Probability
  • From Logical Probability to Probabilistic Logic
  • Difficulties Concerning Subjective Probability
  • Probability and Issues of Descriptive Adequacy
  • Nonclassical Probabilities and Other Representations of Uncertainty
  • Bayesian Confirmation Theory
  • Probabilism and Updating Probabilities
  • Belief Revision
  • Decision Theory
  • Utilities
  • Causal Decision Theory
  • Decision Making under Uncertainty
  • Decision Theory and Issues of Descriptive Adequacy
  • Interactive Epistemology
  • Formal Learning Theory

Philosophy Formal Epistemology
by
Jeffrey Helzner, Vincent F. Hendricks
  • LAST MODIFIED: 26 August 2011
  • DOI: 10.1093/obo/9780195396577-0140

Introduction

Formal epistemology is a fairly recent field of study in philosophy dating back to the end of the 20th century. This is not to say that formal epistemological studies have not been conducted prior to the late 1990s, but rather that the term introduced to cover the philosophical enterprise was coined around this time. Formal epistemology denotes the formal study of crucial concepts in general or mainstream epistemology, including knowledge, belief and belief-change, certainty, rationality, reasoning, decision, justification, learning, agent interaction, and information processing. The formal tools may be drawn from a wide variety of areas, including logic, probability theory, game theory, decision theory, formal learning theory, and distributed computing, and is thus not simply a purely philosophical province. Its practitioners include philosophers, computer scientists, social scientists, cognitive psychologists, theoretical economists, mathematicians, and theoretical linguists.

General Overviews and Textbooks

Since formal epistemology is a newcomer, the number of general overviews and textbooks is limited. Epistemology is but one field of philosophy where formal methods may be of use, so some of the general overviews and textbooks focus on the formalization and the use of formal methods for various philosophical issues and problems rather than on simpler epistemological ones. Glymour 1992 is an introductory treatment of formal methods in philosophy and includes a few chapters on formal epistemology with particular emphasis on knowledge, reliability, and computability studies. Knowledge and reliability are furthermore taken for a systematic formal treatment in Hendricks 2006, whereas Williamson 2010 looks at knowledge from the point of view of mainstream epistemology coupled with epistemic logic.

  • Glymour, C. Thinking Things Through. Cambridge, MA: MIT Press, 1992.

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    An introduction to the use of formal methods in philosophy with some chapters dedicated to epistemological issues such as knowledge, reliability, and computability.

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    • Hendricks, V. F. Mainstream and Formal Epistemology. New York: Cambridge University Press, 2006.

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      A monograph describing and analyzing the common denominators of selected mainstream and formal epistemological programs.

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      • Williamson, T. Knowledge and Its Limits. Oxford: Oxford University Press, 2010.

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        Addresses a host of important epistemological issues among them sensitivity, skepticism, evidence, probability and knowability using assorted formal machinery.

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        Anthologies

        Anthologies of articles on formal philosophy exist in greater numbers than textbooks, but those with formal epistemology as the centerpiece have only recently begun to surface. Arlo-Costa, et al. 2011 includes all the canonical texts of formal epistemology. Hendricks 2006 is a collection of seminal papers reflecting on formal epistemology from epistemic logic to decision theory and agency. Hendricks and Symons 2005 includes reflections on formal methods in philosophy and epistemology by some of the most influential scholars in the field, whereas Hendricks and Pritchard 2008 does the same with particular emphasis on the intersection between mainstream and formal epistemology.

        • Arlo-Costa, H., J. V. van Benthem, and V. F. Hendricks, eds. Readings in Formal Epistemology. Cambridge, UK: Cambridge University Press, 2011.

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          A collection of classical sources in formal epistemology divided into different formal epistemological programs.

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          • Hendricks, V. F., ed. Special Issue: Eight Bridges Between Mainstream and Formal Epistemology. Philosophical Studies 128.1 (2006): 1–227.

            DOI: 10.1007/s11098-005-4068-5E-mail Citation »

            A collection of papers merging mainstream and formal epistemological concerns.

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            • Hendricks, V. F., and D. Pritchard, eds. Epistemology: 5 Questions. New York: Automatic, 2008.

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              A volume of interviews presenting five pertinent questions to formal and mainstream epistemologists alike.

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              • Hendricks, V. F., and J. Symons, eds. Formal Philosophy. New York: Automatic Press, 2005.

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                A volume of interviews presenting five pertinent questions to formal philosophers with particular emphasis on epistemological issues.

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                Journal Articles

                Purely methodological reflections on the use of formalization and formal methods in philosophy are not too common in journal articles, but a few have surfaced in recent years, in particular Bicchieri 2006 and Hansson 2000.

                • Bicchieri, C. “Philosophy: What Is to Be Done?” Topoi 25 (2006): 21–23.

                  DOI: 10.1007/s11245-006-0025-yE-mail Citation »

                  A plea for the interdisciplinary nature of philosophy discussing the interplay between conceptual analysis and scientific analysis.

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                  • Hansson, S. O. “Formalization in Philosophy.” Bulletin of Symbolic Logic 6.2 (2000): 162–175.

                    DOI: 10.2307/421204E-mail Citation »

                    A general introduction to the use of formal methods in philosophy with particular emphasis on the advantages and disadvantages of formalization.

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                    Reference Works

                    Reference works in philosophy have been in vogue in the late 20th and early 21st centuries, but reference works in formal philosophy are harder to come by, although recently the Handbook of Modal Logic (Blackburn, et al. 2007) has come out, along with Philosophy of Information (Adriaans, et al. 2008). The reference works all touch on various topics in formal epistemology, but there is no systematic reference work in formal epistemology.

                    • Adriaans, P., J. van Benthem, and D. Gabbay, eds. Philosophy of Information. Handbook of the Philosophy of Science 8. Amsterdam: North-Holland, 2008.

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                      Much of formal philosophy may be characterized as tools for information processing, whether in metaphysics, epistemology, philosophy of science and social science, etc. This handbook brings together most of the significant perspectives on the concept of information.

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                      • Blackburn, P., J. van Benthem, and F. Wolter. Handbook of Modal Logic. Amsterdam: Elsevier, 2007.

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                        Although formal philosophy and epistemology use other methods than the ones to be found in modal logic, the handbook gives a comprehensive overview of all the tools and applications of modal logic to formal philosophy.

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                        Epistemic Logic

                        Epistemic logic started with the study of proper axiomatizations for predicates and operators that represent concepts of knowledge, belief, certainty, and other epistemic attitudes. Hintikka 1962 inaugurated the field by focusing on axiomatizing knowledge and belief in mainly mono-agent systems, although this work does include the first pointers to multi-agent systems as well. Today the field is a rich blend of studies ranging from mono-agent and multi-agent axiomatizations of knowledge, belief, certainty, uncertainty, doubt, ignorance, and a host of other epistemic attitudes; models of the interplay between knowledge and games, preferences, actions, and decisions; active agency, learning, belief revision, models of agent interaction in multi-agent systems; combined multi-agent and multi-modal systems, in which for instance the development of knowledge over time may be scrutinized, relations between knowledge and deontic commitments investigated, divisions of cognitive labor modeled, and so forth. There is a good number of general overviews and textbooks that, suitably combined—say between Hintikka 1962; Lenzen 1978; Fagin, et al. 1995; and van Ditmarsch, et al. 2008 but also Meyer and van der Hoek 1995, Sowa 2000, and Halpern 2003—will also account for the development of epistemic logic from the early days to contemporary venues of research. Gabbay and Guenthner 1984 includes important chapters on modal logic, temporal logic, and other logics of interest to epistemic logic and formal epistemology. Hendricks and Roy 2010 includes a collection of interviews by some of the most prominent practitioners in the field and points to new venues of interdisciplinary research in epistemic logic.

                        • Fagin, R., J. Halpern, Y. Moses, and M. Vardi. Reasoning about Knowledge. Cambridge, MA: MIT Press, 1995.

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                          The seminal book on the modern epistemic logic, including some of the first multi-agent and multi-modal systems.

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                          • Gabbay, D., and F. Guenthner. Handbook of Philosophical Logic. Vol. 2, Extensions of Classical Logic. Dordrecht, The Netherlands: Reidel, 1984.

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                            A collection of both introductory and advanced papers on modal logic, nonmonotonic inference, defeasible reasoning, etc.

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                            • Halpern, J. Y. Reasoning about Uncertainty. Cambridge, MA: MIT Press, 2003.

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                              Provides a thorough analysis of uncertainty with formal systems for representing this notion, including probability, possibility, and plausibility measures.

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                              • Hendricks, V. F, and O. Roy, eds. Epistemic Logic: 5 Questions. New York: Automatic Press, 2010.

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                                Includes a fine spread of the finest scholars in epistemic logic discussing the aim, scope, and future direction of the field.

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                                • Hintikka, J. Knowledge and Belief: An Introduction to the Logic of the Two Notions. Ithaca, NY: Cornell University Press, 1962.

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                                  Re-compiled by Hendricks and Symons and reissued by King’s College in 2005. The seminal book on the logic of knowledge and belief and their first axiomatizations in mono-agent systems.

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                                  • Lenzen, W. “Recent Work in Epistemic Logic.” Acta Philosophica Fennica 30 (1978): 1–219.

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                                    Provides a thorough philosophical discussion of the plausibility of various axioms governing knowledge and belief in mono-agent systems.

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                                    • Meyer, J. J., and W. van der Hoek. Epistemic Logic for AI and Computer Science. Cambridge Tracts in Theoretical Computer Science 41. Cambridge, UK: Cambridge University Press, 1995.

                                      DOI: 10.1017/CBO9780511569852E-mail Citation »

                                      A thorough introduction to epistemic logic with particular emphasis on its applications in theoretical computer science and artificial intelligence.

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                                      • Sowa, John F. Knowledge Representation: Logical, Philosophical, and Computational Foundations. Pacific Grove, CA: Brooks/Cole, 2000.

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                                        This monograph puts together logic, philosophy, linguistics, and computer science into the study of knowledge and its various models and implementations.

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                                        • van Ditmarsch, H., W. van der Hoek, and B. Kooi. Dynamic Epistemic Logic. Dordrecht, The Netherlands: Springer, 2008.

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                                          A monograph devoted to the latest developments in epistemic logic, including models of multi-agent systems, multi-modal systems, and applications in game theory, Decision Theory, and some Interactive Epistemology.

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                                          Recent Developments in Epistemic Logic

                                          There are now numerous papers on various aspects and developments within epistemic logic. Papers of immediate interest to epistemology include Arlo-Costa and Pacuit 2006, in which neighborhood semantics are introduced in order to come to terms with logical omniscience among other items of epistemological interest. Artemov and Nogina 2005 introduces justification in epistemic logic; the interplay between epistemic logic and games is scrutinized in van Benthem 2001; the seminal paper on epistemic logic and public announcement is Baltag, et al. 1998; agency and epistemic logic is dealt with in Hendricks 2003; the epistemological plausibility of the common axioms of epistemic logic is discussed by Stalnaker 2006; and the combination of epistemic logic and belief revision is the topic of Baltag and Smets 2008. For a general overview of the interface between epistemology and epistemic logic, refer to Hendricks and Symons 2005.

                                          • Arlo-Costa, H., and E. Pacuit. “First Order Classical Modal Logic.” Studia Logica 84.2 (2006): 171–210.

                                            DOI: 10.1007/s11225-006-9010-0E-mail Citation »

                                            Introduces neighborhood semantics, as opposed to classical Kripke semantics, for use in epistemic logic and other modal logics.

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                                            • Artemov, S., and E. Nogina. “On Epistemic Logic with Justification.” In Theoretical Aspects of Rationality and Knowledge: Proceedings of the Tenth Conference: June 10–12, 2005, National University of Singapore. Edited by R. van der Meyden, 279–294. New York: Association for Computing Machinery, 2005.

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                                              Introduces a justification term in epistemic logic and deals with a host of issues related to the justification clause in mainstream epistemology, including the Gettier problems.

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                                              • Baltag, A., L. S. Moss, and S. Solecki. “The Logic of Public Announcements, Common Knowledge, and Private Suspicion.” In Theoretical Aspects of Rationality and Knowledge: Proceedings of the Seventh Conference (TARK 1998), July 22–24, 1998, Evanston, Illinois. Edited by Itzhak Gilboa, 43–56. San Francisco: Morgan Kaufmann, 1998.

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                                                Formulates a multi-modal system in which common knowledge, public announcement operations, and belief may be modeled and studied.

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                                                • Baltag, A., and Smets, S. “A Qualitative Theory of Dynamic Interactive Belief Revision.” Paper presented at the 7th Conference on Logic and the Foundations of Game and Decision Theory, Liverpool, July 2006. In Logic and the Foundations of Game and Decision Theory (LOFT7). Edited by Giacomo Bonanno, Wiebe van der Hoek, and Michael Wooldridge, 11–58. Texts in Logic and Games 3. Amsterdam: Amsterdam University Press, 2008.

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                                                  Introduces belief revision mechanics into dynamic epistemic logic.

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                                                  • Hendricks, V. F. “Active Agents.” Journal of Logic, Language and Information 12 (2003): 469–495.

                                                    DOI: 10.1023/A:1025059002654E-mail Citation »

                                                    Considers active agency at the crossroads between epistemic logic and formal learning theory.

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                                                    • Hendricks, V. F., and J. Symons. “Epistemic Logic.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2005.

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                                                      A general introduction to the nuts and bolts of epistemic logic with pointers to applications in formal epistemology.

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                                                      • Stalnaker, R. “On Logics of Knowledge and Belief.” Philosophical Studies 128.1 (2006): 169–199.

                                                        DOI: 10.1007/s11098-005-4062-yE-mail Citation »

                                                        Discusses epistemic logic and its broader relations to formal and mainstream epistemology. Particular attention devoted to the epistemological plausibility of the axioms of epistemic and doxastic logic.

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                                                        • van Benthem, J. “Games in Dynamic Epistemic Logic.” Bulletin of Economic Research 53.4 (2001): 219–224.

                                                          DOI: 10.1111/1467-8586.00133E-mail Citation »

                                                          The paper considers the interplay between games and knowledge with particular emphasis on the logical languages that games give rise to.

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                                                          Probability

                                                          The standard axioms for the concept of a finite probability measure admit a wide variety of interpretations, but only some of these appear to have anything to do with our informal conceptions of probability. The classical interpretation is restricted by its requirement that the most basic possibilities be equally probable. The limiting frequency interpretation, as suggested by von Mises 1957, is perhaps the one that is most familiar to working scientists, although propensity interpretations such as Popper 1959 might be closer to what they have in mind. Moreover, concerned with long-term behavior, the limiting frequency interpretation is not immediately relevant to particular events (e.g., the outcome of the next toss of a given coin). The logical interpretation of probability, advocated by Keynes 1921 and then Carnap (see From Logical Probability to Probabilistic Logic), attempts to locate probability through an extension of the relation of logical consequence, but there are many such extensions, and the success of the logicist program seems to require nearly self-evident principles that would suffice to privilege one of these extensions over the others. F. P. Ramsey expressed serious doubts about Keynes’s logical interpretation in his 1931 “Truth and Probability” and then presented his subjectivist interpretation of probability, according to which the axioms of a finitely additive probability measure serve as a standard of consistency on the agent’s degrees of belief; see Kyburg and Smokler 1964 for various classic papers concerning the subjective interpretation. Yet the standard of consistency provided by the probability axioms is so lax that it can be met in many intuitively unreasonable ways. This would seem to present a challenge for the subjectivist interpretation if it is to be taken seriously as a basis for scientific inference. Objective Bayesians such as Jeffreys 1948 attempt to address this challenge, but doubts about such attempts remain, as seen in Seidenfeld 1979. Useful surveys concerning interpretations of probability include Howson 1995 and relevant chapters of Suppes 2002.

                                                          • Jeffreys, H. Theory of Probability. Oxford: Clarendon, 1948.

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                                                            A classic reference for objective Bayesianism, an interpretation that shares some characteristics with various logical interpretations.

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                                                            • Howson, C. “Theories of Probabilities.” British Journal for the Philosophy of Science 46.1 (1995): 1–32.

                                                              DOI: 10.1093/bjps/46.1.1E-mail Citation »

                                                              A survey article that recalls Terry Fine’s classic Theories of Probability (New York and London: Academic Press, 1973). Regrettably, Fine’s excellent book is not widely available.

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                                                              • Keynes, J. M. A Treatise on Probability. London: Macmillan, 1921.

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                                                                An early statement of the logical interpretation of probability that was criticized famously by Ramsey. Though superseded by later developments in the logicist tradition, Keynes’s book contains worthwhile insights: for example, it anticipates some of the discussion concerning the distinction between risk and uncertainty that was revived by Ellsberg about forty years later.

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                                                                • Kyburg, H. E., Jr., and H. E. Smokler, eds. Studies in Subjective Probability. New York: Wiley, 1964.

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                                                                  An important collection that includes seminal papers on subjective probability, such as F. P. Ramsey’s 1931 “Truth and Probability” and a translation of B. de Finetti’s “La prevision: ses lois logiques, ses sources subjectives” (1937).

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                                                                  • Popper, K. R. “The Propensity Interpretation of Probability.” British Journal for the Philosophy of Science 10 (1959): 26–42.

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                                                                    The second of two very influential articles on propensities by one of its major supporters.

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                                                                    • Seidenfeld, T. “Why I’m Not an Objective Bayesian.” Theory and Decision 11 (1979): 413–440.

                                                                      DOI: 10.1007/BF00139451E-mail Citation »

                                                                      A penetrating and general critique of objective Bayesianism.

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                                                                      • Suppes, P. Representation and Invariance of Scientific Structures. Stanford, CA: CSLI, 2002.

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                                                                        A masterful book in the general philosophy of science, the fifth chapter is an excellent and extended discussion of the various interpretations of probability.

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                                                                        • von Mises, R. Probability, Statistics and Truth. 2d ed. New York: Macmillan, 1957.

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                                                                          A classic statement of frequentism.

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                                                                          From Logical Probability to Probabilistic Logic

                                                                          There are various possibilities for interactions between probability and logic. For example, one may get a general sense of the notion of logical implication from classical logic, the model theory of classical logic, or the language of classical logic. The first of these options is perhaps the one that is most associated with the logicist tradition of Keynes and Carnap concerning the foundations of probability. Yet, Carnap’s seminal works (Carnap 1945 and Carnap 1950) also inspired generalizations of the second sort—for example, Gaifman 1964, Scott and Krauss 1966, and Gaifman and Snir 1982—that are not directed toward foundations. Likewise, efforts at generalizations of the third sort, as in Halpern 2003, are not foundational in nature. Rather, by enriching the language of classical logic to allow for the expression of statements involving probability and related notions, one aims for logical frameworks that can be used for reasoning about systems in which probability and related notions are salient. Without falling squarely into any of the three sorts mentioned, Kyburg 1974 does maintain the logical status of probability statements, but Kyburg’s program of “epistemological probability” differs in various ways from Carnap’s logicist program; for example, Kyburg makes important use of interval-valued probabilities and does not require deductive closure for the evidential states on which such probabilities are grounded. Finally, the Progic Conference Series has emerged as an important forum for introducing new research concerning the intersection of probability and logic.

                                                                          • Carnap, R. “On Inductive Logic.” Philosophy of Science 12.2 (1945): 72–97.

                                                                            DOI: 10.1086/286851E-mail Citation »

                                                                            Classic article-length statement of Carnap’s views on probability.

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                                                                            • Carnap, R. The Logical Foundations of Probability. Chicago: University of Chicago, 1950.

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                                                                              Probably the most influential work on the logical interpretation of probability.

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                                                                              • Gaifman, H. “Concerning Measures in First-Order Calculi.” Israel Journal of Mathematics 2 (1964): 1–18.

                                                                                DOI: 10.1007/BF02759729E-mail Citation »

                                                                                A student of Carnap, Gaifman leaves behind some of Carnap’s foundational concerns and opens up a new line of research by introducing probabilistic generalizations of the concepts of model and theory from first-order logic.

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                                                                                • Gaifman, H., and M. Snir. “Probabilities of Rich Languages, Testing and Randomness.” Journal of Symbolic Logic 47 (1982): 495–548.

                                                                                  DOI: 10.2307/2273587E-mail Citation »

                                                                                  Further advances in the tradition of Gaifman 1964 and Scott and Krauss 1966.

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                                                                                  • Halpern, J. Y. Reasoning about Uncertainty. Cambridge, MA: MIT Press, 2003.

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                                                                                    Includes extensions of classical logic that allow for reasoning about probability and related notions.

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                                                                                    • Kyburg, H. E., Jr. The Logical Foundations of Statistical Inference. Dordrecht, The Netherlands: Reidel, 1974.

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                                                                                      An important contribution that did not receive the hearing it deserved. It seems that the style of the presentation was sufficiently off-putting that some reviewers never really engaged the ideas in the book. For an informed review, see Teddy Seidenfeld’s review from Journal of Philosophy 74.1.

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                                                                                      • Progic Conference Series.

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                                                                                        A conference series dedicated to various interactions between probability and logic.

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                                                                                        • Scott, D., and P. Krauss. “Assigning Probabilities to Logical Formulas.” In Aspects of Inductive Logic. Edited by J. Hintikka and P. Suppes. Amsterdam: North-Holland, 1966.

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                                                                                          Also extends the approach taken in Gaifman 1964.

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                                                                                          Difficulties Concerning Subjective Probability

                                                                                          The subjective interpretation of probability is an attractive option for those who are concerned to represent the credal states of rational agents. Nonetheless, there are several well-known difficulties associated with the subjective interpretation. Some of these difficulties concern subjective probability in the context of confirmation or decision. We restrict our attention here to a few examples that are relatively autonomous. First, viewed as consistency requirement, the subjective interpretation demands credal states that can be represented by a finitely additive probability function. These demands are relatively weak in the sense that subjective probabilities need not be informed by objective chances. Principles of direct inference may be invoked to require certain relations between credal probability and chance (see Kyburg 1974, Levi 1977, and Lewis 1980). Second, measuring subjective probabilities might be more difficult than is sometimes assumed. Of course, one could take subjective probability as a theoretical concept, perhaps partially interpreted by various reduction sentences but never reaching a fully operational characterization. This does not appear to be the attitude that Ramsey takes in “Truth and Probability.” Important difficulties concerning Ramsey’s betting-rate approach to the measurement of subjective probabilities are considered in Seidenfeld, et al. 1990. An alternative approach to the measurement of subjective probabilities is discussed in Krantz, et al. 1971 within the wider context of the representation theory of measurement. Third, as a consistency standard for credal states, the axioms for a finitely additive probability function entail significant computational demands. One might wonder if the advantages of satisfying such a consistency standard are offset by such computational costs. Issues of this sort are considered in Hacking 1967 and Gaifman 2004. Finally, Hajek 2005 raises concerns about the Dutch-book arguments that seem to demand that rational credence must satisfy the axioms for a finitely additive probability measure.

                                                                                          • Gaifman, H. “Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements.” Synthese 140 (2004): 97–119.

                                                                                            DOI: 10.1023/B:SYNT.0000029944.99888.a7E-mail Citation »

                                                                                            Written as part of special issue dedicated to the work of Isaac Levi. Gaifman’s paper addresses some of the same issues that are considered in Hacking 1967. Also see Levi’s response to Gaifman in that same issue of Synthese.

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                                                                                            • Hacking, I. “Slightly More Realistic Personal Probability.” Philosophy of Science 34 (1967): 311–325.

                                                                                              DOI: 10.1086/288169E-mail Citation »

                                                                                              Hacking offers a modified account that is meant to address concerns about the computational demands that are imposed by taking the probability axioms as a standard for rational credence. Hacking is in part responding to a challenge raised by L. J. Savage in an article from the same issue of Philosophy of Science.

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                                                                                              • Hajek, A. “Scotching Dutch Books?” Epistemology: Philosophical Perspectives 19 (2005): 139–151.

                                                                                                DOI: 10.1111/j.1520-8583.2005.00057.xE-mail Citation »

                                                                                                Questions the standard interpretation of the Dutch-book arguments that are often used to motivate introduction of subjective probabilities.

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                                                                                                • Krantz, D., D. Luce, P. Suppes, and A. Tversky. The Foundations of Measurement. Vol. 1, Additive and Polynomial Representations. New York: Academic Press, 1971.

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                                                                                                  The fifth chapter of this classic is dedicated to issues concerning the measurement of probability.

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                                                                                                  • Kyburg, H. E., Jr. The Logical Foundations of Statistical Inference. Dordrecht, The Netherlands: Reidel, 1974.

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                                                                                                    Includes a detailed account of direct inference.

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                                                                                                    • Levi, I. “Direct Inference.” Journal of Philosophy 74 (1977): 5–29.

                                                                                                      DOI: 10.2307/2025732E-mail Citation »

                                                                                                      An important critique of Kyburg’s account of direct inference (Kyburg 1974).

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                                                                                                      • Lewis, D. “A Subjectivist’s Guide to Objective Chance.” In Studies in Inductive Logic. Vol. 2. Edited by R. Jeffrey, 199–221. Berkeley: University of California Press, 1980.

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                                                                                                        A basic reference for many philosophers who are concerned with the relationship between credence and chance.

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                                                                                                        • Seidenfeld, T., M. J. Schervish, and J. B. Kadane. “When Fair Betting Odds are Not Degrees of Belief.” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1 (1990): 517–524.

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                                                                                                          Ramsey, like Percy Bridgman, seems to have admired the operationalism made famous by Einstein in his analysis of simultaneity. Seidenfeld, Schervish, and Kadane show that there are serious problems with the fair-betting-odds analysis of credence that was advocated by Ramsey, and later by de Finetti and Savage.

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                                                                                                          Probability and Issues of Descriptive Adequacy

                                                                                                          While concerns in formal epistemology tend to be normative, an absolute division between the normative and the descriptive may be untenable; minimally, normative theories ought to be descriptive in some limited range of circumstances—for example, of sophisticated agents in situations where computational demands are modest. Currently, much of the influential work on the psychology of probability grows out of the “heuristic and biases” tradition of D. Kahneman and A. Tversky (see Kahneman, et al. 1982 and Kahneman and Tversky 1996). A second tradition, which predates the former at least in its roots going back to the seminal work of H.A. Simon (see Decision Theory and Issues of Descriptive Adequacy), has informed notable critique—as in Gigerenzer 1996—of the heuristic and biases tradition. Samuels, et al. 2002 examines the philosophical relevance of the relationship between the two traditions that have been mentioned.

                                                                                                          • Gigerenzer, G. “On Narrow Norms and Vague Heuristics: A Reply to Kahneman and Tversky.” Psychological Review 103 (1996): 592–596.

                                                                                                            DOI: 10.1037/0033-295X.103.3.592E-mail Citation »

                                                                                                            A well-known part of the “heuristic and biases” program of Kahneman and Tversky.

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                                                                                                            • Kahneman, D., P. Slovic, and A. Tversky. Judgment Under Uncertainty: Heuristics and Biases. Cambridge, UK: Cambridge University Press, 1982.

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                                                                                                              A classic collection of papers in the heuristics and biases tradition.

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                                                                                                              • Kahneman, D., and A. Tversky. “On the Reality of Cognitive Illusions: A Reply to Gigerenzer’s Critique.” Psychological Review 103 (1996): 582–591.

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                                                                                                                Kahneman and Tversky respond to Gigerenzer 1996.

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                                                                                                                • Samuels, R, and S. Stich, and M. Bishop. “Ending the Rationality Wars: How to Make Disputes about Human Rationality Disappear.” In Common Sense, Reasoning and Rationality. Edited by R. Elio, 236–268. New York: Oxford University Press, 2002.

                                                                                                                  DOI: 10.1093/0195147669.001.0001E-mail Citation »

                                                                                                                  Philosophers weigh in on the significance of what they characterize as a dispute between evolutionary psychologists (e.g., Gigerenzer) and the heuristic and biases tradition of Kahneman and Tversky.

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                                                                                                                  Nonclassical Probabilities and Other Representations of Uncertainty

                                                                                                                  While probability measures are the most well-known representation of uncertainty, they are hardly the only option. Various considerations, from confirmation to decision making, have been taken as motivation to relax some of the requirements imposed by probability representations. For example, Levi 1974 argues that the credal states of rational agents ought to be represented by (convex) sets of probability measures. Walley 1991, recalling the work of B. de Finetti, develops an approach to imprecise probabilities based on upper and lower previsions. Shafer 1976 advocates the use of belief functions, a concept that was introduced by A. Dempster in the late 1960s. Roughly, and from a purely formal point of view, Dempster-Shafer belief functions can be viewed as generalized probability measures in which the so-called inclusion-exclusion principle is relaxed to an inequality. Halpern 2003 covers each of the representations mentioned above, as well as other important representations such as possibility measures (see Dubois and Prade 1990) and ranking functions (as in Spohn 2009). Finally, the Society for Imprecise Probabilities: Theories and Applications has emerged as an important forum for presenting new research on these various alternatives to traditional probability measures.

                                                                                                                  • Dubois, D., and H. Prade. “An Introduction to Possibilistic and Fuzzy Logics.” In Readings in Uncertain Reasoning. Edited by G. Shafer and J. Pearl, 742–761. San Mateo: Morgan Kaufman, 1990.

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                                                                                                                    Nice coverage of a topic that was introduced by Zadeh in the 1970s and developed by others, including Dubois and Prade.

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                                                                                                                    • Halpern, J. Y. Reasoning about Uncertainty. Cambridge, MA: MIT Press, 2003.

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                                                                                                                      A comprehensive textbook that surveys all of the main options regarding the representation of uncertainty.

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                                                                                                                      • Levi, I. “On Indeterminate Probabilities.” Journal of Philosophy 71 (1974): 391–418.

                                                                                                                        DOI: 10.2307/2025161E-mail Citation »

                                                                                                                        Argues for the use of sets of probability measures in representing the credal states of rational agents.

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                                                                                                                        • Shafer, G. A Mathematical Theory of Evidence. Princeton, NJ: Princeton University Press, 1976.

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                                                                                                                          An influential work that builds on the work of A. Dempster.

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                                                                                                                          • Society for Imprecise Probabilities: Theories and Applications.

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                                                                                                                            SIPTA has been running its main conference in odd years since 1999. Electronic proceedings from each of those meetings can be found through SIPTA’s main page.

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                                                                                                                            • Spohn, W. “A Survey of Ranking Theory.” In Degrees of Belief. Edited by F. Huber and C. Schmidt-Petri. Dordrecht, The Netherlands: Springer, 2009.

                                                                                                                              DOI: 10.1007/978-1-4020-9198-8E-mail Citation »

                                                                                                                              A recent survey article written by the person who introduced ranking functions in the 1980s, although an important interpretation of ranking functions can be traced back to the late 1960s in the work of G.L.S. Shackle. It is worth noting that the entire Degrees of Belief collection is likely to interest many philosophers. Arlo-Costa’s NPDR review of Degrees of Belief can be found online.

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                                                                                                                              • Walley, P. Statistical Reasoning with Imprecise Probabilities. New York: Chapman and Hall, 1991.

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                                                                                                                                Highly regarded and the central reference for many in the Society for Imprecise Probabilities: Theories and Applications (SIPTA).

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                                                                                                                                Bayesian Confirmation Theory

                                                                                                                                Conditional probability, which can be taken as a primitive notion or as a derived notion, may serve as the basis for a theory of confirmation; roughly, evidence E confirms (i.e., supports) hypothesis H just in case the conditional probability of H given E is greater than the unconditional probability of H. Bayes’s theorem shows how this relationship between the conditional probability of H given E and the unconditional probability of H is influenced by what H says about E—in other words, by how likely E is according to H (and according to alternatives to H). Confirmation theory has a long and storied tradition within the philosophy of science, starting at least as far back as the classic works of Hempel and Carnap. The following works are primarily restricted to the Bayesian approach to confirmation. Popper 1968 and Hajek 2003 should prove helpful to philosophers who require a minimal background on conditional probability and related topics. Salmon 1975 should be useful to those who are interested in contrasting the Bayesian approach to earlier work on confirmation. Maher 1993 and Fitelson 2001 provide comprehensive accounts of the Bayesian position. Glymour 1980, Garber 1983, and Kelly and Glymour 2004 focus on well-known challenges to Bayesian confirmation theory.

                                                                                                                                • Fitelson, B. Studies in Bayesian Confirmation Theory. PhD diss., University of Wisconsin-Madison, 2001.

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                                                                                                                                  Comprehensive treatment by a leading figure of contemporary work on confirmation theory.

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                                                                                                                                  • Garber, D. “Old Evidence and Logical Omniscience in Bayesian Confirmation Theory.” In Testing Scientific Theories. Edited by J. Earman, 99–131. Midwest Studies in the Philosophy of Science 10. Minneapolis: University of Minnesota Press, 1983.

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                                                                                                                                    Revisits Glymour’s problem of old evidence.

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                                                                                                                                    • Glymour, C. Theory and Evidence. Princeton, NJ: Princeton University Press, 1980.

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                                                                                                                                      Includes the “problem of old evidence” for Bayesian confirmation theory.

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                                                                                                                                      • Hajek, A. “What Condition Probability Could Not Be.” Synthese 137 (2003): 273–323.

                                                                                                                                        DOI: 10.1023/B:SYNT.0000004904.91112.16E-mail Citation »

                                                                                                                                        Criticizes Kolmogorov’s definition of conditional probability in terms of unconditional probability.

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                                                                                                                                        • Kelly, K., and C. Glymour. “Why Probability Does Not Capture the Logic of Scientific Justification.” In Contemporary Debates in the Philosophy of Science. Edited by Christopher Hitchcock, 89–123. London: Blackwell, 2004.

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                                                                                                                                          Argues against the Bayesian approach to confirmation.

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                                                                                                                                          • Maher, P. Betting on Theories. Cambridge, UK: Cambridge University Press, 1993.

                                                                                                                                            DOI: 10.1017/CBO9780511527326E-mail Citation »

                                                                                                                                            Develops a qualified Bayesian position.

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                                                                                                                                            • Popper, K. The Logic of Scientific Discovery. 3d ed. London: Hutchinson, 1968.

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                                                                                                                                              The appendices include a presentation of “Popper functions” as a way of taking conditional probability as primitive notion.

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                                                                                                                                              • Salmon, W. “Confirmation and Relevance.” In Induction, Probability, and Confirmation. Edited by G. Maxwell and R. Anderson, 3–36. Minnesota Studies in the Philosophy of Science 6. Minneapolis: University of Minnesota Press, 1975.

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                                                                                                                                                A classic paper that should also be helpful to readers who are interested in tracing the development of confirmation theory from the work of Hempel and others to more contemporary views.

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                                                                                                                                                Probabilism and Updating Probabilities

                                                                                                                                                In addition to its role in theories of confirmation, Bayes’s rule can also be interpreted dynamically as a rule for updating credences (see Joyce 2008). If one accepts the view that degrees of belief are the only basic states of epistemological interest and that such states can be represented by probabilities, then it would seem that questions concerning rational changes to such epistemic states can be understood in terms of rules for updating probabilities. Rejecting such a view invites consideration of the topics considered in the section on Belief Revision. Various rules for updating subjective probabilities have been proposed, with Bayes’s rule being the most well known. Diaconis and Zabell 1982 surveys various rules for updating subjective probabilities. The view that all other epistemic notions (e.g., knowledge or full belief) can be reduced to subjective probabilities is sometimes called “radical probabilism” and is most famously associated with the work of Richard Jeffrey (Jeffrey 1992). Van Fraassen 1995 proposes a different variety of radical probabilism. Arlo-Costa 2001 extends and develops these views. Kaplan 1996 is an introductory book that includes a very accessible discussion of probabilism. Joyce 1998 concerns what may be regarded as the foundations of probabilism, while Maher 2002 is a response to Joyce 2008.

                                                                                                                                                • Arlo-Costa, H. “Bayesian Epistemology and Epistemic Conditionals: On the Status of the Export-Import Laws.” Journal of Philosophy 98.11 (2001): 555–593.

                                                                                                                                                  DOI: 10.2307/3649472E-mail Citation »

                                                                                                                                                  Builds on some of the ideas in van Fraassen 1995.

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                                                                                                                                                  • Diaconis, P., and S. Zabell. “Updating Subjective Probability.” Journal of the American Statistical Society 77.380 (1982): 822–830.

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                                                                                                                                                    Examines various proposals for updating subjective probabilities.

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                                                                                                                                                    • Jeffrey, R. C. Probability and the Art of Judgment. New York: Cambridge University Press, 1992.

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                                                                                                                                                      A classic statement of Jeffrey’s radical probabilism.

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                                                                                                                                                      • Joyce, J. “A Nonpragmatic Vindication of Probabilism.” Philosophy of Science 65.4 (1998): 575–603.

                                                                                                                                                        DOI: 10.1086/392661E-mail Citation »

                                                                                                                                                        An important paper on the foundations of probabilism. The idea that one ought to have his or her credal judgments satisfy the conditions for a finitely additive probability measure is often motivated through Dutch-book arguments. Joyce argues that such arguments are unsatisfying from an epistemological point of view and that such norms can be motivated by considering the extent to which the agent’s credences are an accurate representation of the world.

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                                                                                                                                                        • Joyce, J. “Bayes’ Theorem.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008.

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                                                                                                                                                          A nice discussion that covers various forms of Bayes’ theorem as well as its relevance to certain accounts of evidence and learning.

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                                                                                                                                                          • Kaplan, M. Decision Theory as Philosophy. Cambridge, UK: Cambridge University Press, 1996.

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                                                                                                                                                            Contains an introductory-level discussion of probabilism.

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                                                                                                                                                            • Maher, P. “Joyce’s Argument for Probabilism.” Philosophy of Science 69.1 (2002): 73–81.

                                                                                                                                                              DOI: 10.1086/338941E-mail Citation »

                                                                                                                                                              Maher responds to Joyce 2008.

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                                                                                                                                                              • van Fraassen, B. “Fine-Grained Opinion, Probability, and the Logic of Full Belief.” Journal of Philosophical Logic 24.4 (1995): 349–377.

                                                                                                                                                                DOI: 10.1007/BF01048352E-mail Citation »

                                                                                                                                                                Presents a version of radical probabilism that differs from the one proposed by Jeffrey.

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                                                                                                                                                                Belief Revision

                                                                                                                                                                The study of formal models of belief revision is a relatively new development, the origins of which can be found in fields such as computer science, especially work in database updating, as well as in work from the late 1970s and 1980s done by philosophers such as Peter Gärdenfors, William Harper, and Isaac Levi. As this is a young field addressing a fundamental notion, it is perhaps not surprising that several theoretical options remain in play. Gärdenfors 1992 and Hansson 2009 are both good places to start for someone looking to get a sense of the field. Hansson 1999 would be helpful to those seeking a textbook presentation. Likewise, Halpern 2009, while not dedicated to belief revision, is a textbook that covers updating operations for various representations of (uncertain) beliefs (see Nonclassical Probabilities and Other Representations of Uncertainty) and includes a chapter on belief revision that emphasizes a perspective that is rather different from the other references mentioned below. The theory presented in Alchourrón, et al. 1985 is almost surely the most well-known formal account of belief revision. Another central account is offered by Isaac Levi as part of his highly original and thought-provoking approach to epistemology. Levi 1980 is perhaps the most well-known presentation of Levi’s views on the subject, although he has developed and refined his theory in subsequent articles and books. Levi’s account is very much informed by his work on decision theory, and Arlo-Costa and Levi 2006 serves to make some of the decision-theoretic underpinnings formally explicit. Finally, Rott 1993 also serves to connect work in belief revision with frameworks that are most typically associated with practical, as opposed to theoretical, choice—Rott accomplishes this by locating the AGM account within the economists’ study of rational choice functions.

                                                                                                                                                                • Alchourrón, C., P. Gärdenfors, and D. Makinson. “On the Logic of Theory Change: Partial Meet Functions for Contraction and Revision.” Journal of Symbolic Logic 50 (1985): 510–530.

                                                                                                                                                                  DOI: 10.2307/2274239E-mail Citation »

                                                                                                                                                                  The classic reference for what is now known as the AGM theory of belief change.

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                                                                                                                                                                  • Arlo-Costa, H, and I. Levi “Contraction: On the Decision-Theoretical Origins of Minimal Change and Entrenchment.” Synthese 152.1 (2006): 129–154.

                                                                                                                                                                    DOI: 10.1007/s11229-005-0351-4E-mail Citation »

                                                                                                                                                                    A detailed formal analysis based on some of Levi’s earlier proposals.

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                                                                                                                                                                    • Gärdenfors, P., ed. Belief Revision. Cambridge, UK: Cambridge University Press, 1992.

                                                                                                                                                                      DOI: 10.1017/CBO9780511526664E-mail Citation »

                                                                                                                                                                      A collection of papers on belief revision that includes contributions from many of the central figures in the field. Also includes a substantial introductory essay by Peter Gärdenfors, the “G” of the AGM theory of belief revision.

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                                                                                                                                                                      • Halpern, J. Y. Reasoning about Uncertainty. Cambridge, MA: MIT Press, 2009.

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                                                                                                                                                                        Considers updating rules for several different representations of belief.

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                                                                                                                                                                        • Hansson, S. O. A Textbook of Belief Dynamics: Theory Change and Database Updating. Dordrecht, The Netherlands: Kluwer Academic, 1999.

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                                                                                                                                                                          A textbook treatment written by a leading figure in the field.

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                                                                                                                                                                          • Hansson, S. O. “Logic of Belief Revision.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2009.

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                                                                                                                                                                            Detailed and highly readable survey article. Includes discussions of the history and motivation of the subject, basic conceptual issues, and technical matters concerning the representation of belief states and revision operators.

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                                                                                                                                                                            • Levi, I. The Enterprise of Knowledge. Cambridge, MA: MIT Press, 1980.

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                                                                                                                                                                              The classic statement of Levi’s far-reaching epistemological program in which justification of changes to the agent’s belief state, rather than justification of the agent’s belief state, are made central.

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                                                                                                                                                                              • Rott, H. “Belief Contraction in the Context of the General Theory of Rational Choice.” Journal of Symbolic Logic 58.4 (1993): 1426–1450.

                                                                                                                                                                                DOI: 10.2307/2275152E-mail Citation »

                                                                                                                                                                                Rott identifies important connections between the theory of rational choice functions, as studied by economists, and the theory of belief contraction presented in (Alchourrón, et al. 1985).

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                                                                                                                                                                                Decision Theory

                                                                                                                                                                                Typically, decision theories attempt to integrate two factors: beliefs and desires (or preferences). The former are often represented by a probability measure, although the accounts mentioned in (Nonclassical Probabilities and Other Representations of Uncertainty) provide alternatives. The latter are often represented by utilities, which are discussed in (Utilities). The most well-known decision theories are the varieties of expected utility maximization that were initially proposed as an improvement over the earlier theory of expected monetary value, which had been shown to have serious limitations such as those illustrated by Bernoulli’s “St. Petersburg” example from the early 18th century. Savage 1972 is probably the single most important book in subjective expected utility theory. Jeffrey 1965 offers an alternative account that has been popular among philosophers. Luce and Raiffa 1957 provides a highly readable survey. Fishburn 1981 and Kreps 1988, though a bit more demanding mathematically, also provide excellent surveys of the field. Kaplan 1996 and Peterson 2009 might serve as good introductions, especially for philosophers who are new to the area or lack the mathematical background for something like Kreps 1988. Gärdenfors and Sahlin 1988 is collection of important articles, most of which are concerned with various challenges to the received view such as those considered in Causal Decision Theory and Decision Making under Uncertainty.

                                                                                                                                                                                • Fishburn, P. C. “Subjective Expected Utility Theory: A Review of Normative Theories.” Theory and Decision 13 (1981): 139–199.

                                                                                                                                                                                  DOI: 10.1007/BF00134215E-mail Citation »

                                                                                                                                                                                  A thorough review article written by one of the foremost contributors to the mathematics of decision theory.

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                                                                                                                                                                                  • Gärdenfors, P., and N. Sahlin, eds. Decision, Probability and Utility. Cambridge, UK: Cambridge University Press, 1988.

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                                                                                                                                                                                    A collection of classic papers on decision theory. Accessible and informative essays, written by Gärdenfors and Sahlin, introduce each section.

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                                                                                                                                                                                    • Jeffrey, R. C. The Logic of Decision. New York: McGraw-Hill, 1965.

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                                                                                                                                                                                      Within certain philosophical traditions, Jeffrey’s book has been even more influential than Savage’s 1972 book.

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                                                                                                                                                                                      • Kaplan, M. Decision Theory as Philosophy. Cambridge, UK: Cambridge University Press, 1996.

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                                                                                                                                                                                        Introductory treatment of decision theory and its relevance to philosophy.

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                                                                                                                                                                                        • Kreps, D. Notes on the Theory of Choice. Boulder, CO: Westview, 1988.

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                                                                                                                                                                                          Terse, but provides excellent coverage of classic results in modern decision theory.

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                                                                                                                                                                                          • Luce, R. D. and H. Raiffa. Games and Decisions: Introduction and Critical Survey. New York: Wiley, 1957.

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                                                                                                                                                                                            A highly readable and influential reference covering classic work on decision theory and game theory.

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                                                                                                                                                                                            • Peterson, M. An Introduction to Decision Theory. Cambridge, UK: Cambridge University Press, 2009.

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                                                                                                                                                                                              A recent textbook on decision theory.

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                                                                                                                                                                                              • Savage, L. J. The Foundations of Statistics. New York: Dover, 1972.

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                                                                                                                                                                                                Savage’s classic analysis of subjective expected utility theory. First published in 1954 (New York: Wiley).

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                                                                                                                                                                                                Utilities

                                                                                                                                                                                                While the beliefs side of decision theory has surely received the bulk of attention in formal epistemology, an increasing amount of attention is being given to the values side. Of course, the representation of desires, preferences, and values through the general concept of utility has a long tradition in economics and related areas that goes back at least as far as Daniel Bernoulli (b. 1700–d. 1782) who credits Gabriel Cramer (b. 1704–d. 1752) with the introduction of the concept (see Savage 1972). The relevant modern concept of utility, which was placed on a solid foundation by Neumann and Morgenstern 1947, must be distinguished from the earlier, riskless notions of utility that were discredited with the rise of the so-called ordinalist revolution in economics. Subsequent works, such as Debreu 1960 and the theory of conjoint measurement, which is discussed at length in Krantz, et al. 1972, offer a different foundation for utility theory, one that avoids the notion of risk that is central to the account given by von Neumann and Morgenstern. Rather than introducing risk, such accounts exploit tradeoffs between the various components, or attributes, of the relevant alternatives. The study of multi-attribute utility is a large field, and much of the literature is application driven. Keeney and Raiffa 1976 and Weirich 2001 should be helpful in giving the interested reader a good a sense of some of the relevant conceptual issues. Finally, Hansson 2001 draws upon the “logic of preference” tradition that was developed by philosophers in parallel to the aforementioned developments within economics.

                                                                                                                                                                                                • Debreu, G. “Topological Methods in Cardinal Utilility.” In Mathematical Methods in the Social Sciences. Edited by K. J. Arrow, S. Karlin, and P. Suppes, 16–26. Stanford, CA: Stanford University Press, 1960.

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                                                                                                                                                                                                  Presents an analysis of utility based on tradeoffs between the components of the alternatives. This approach stands in contrast to the standard approach of von Neumann and Morgenstern that is based on introduction of risk in the form of lotteries.

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                                                                                                                                                                                                  • Hansson, S. O. The Structure of Values and Norms. Cambridge, UK: Cambridge University Press, 2001.

                                                                                                                                                                                                    DOI: 10.1017/CBO9780511498466E-mail Citation »

                                                                                                                                                                                                    A nice study that should be helpful in trying to bridge the work that grew out of the statistics and economics communities with the preference logic literature developed by philosophers.

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                                                                                                                                                                                                    • Keeney, R., and H. Raiffa. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: Wiley, 1976.

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                                                                                                                                                                                                      A classic text on multi-attribute utility theory.

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                                                                                                                                                                                                      • Krantz, D., D. Luce, P. Suppes, and A. Tversky. The Foundations of Measurement. Vol. 1. New York: Academic, 1972.

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                                                                                                                                                                                                        Utility theory may be studied within the more general framework of measurement theory. Chapter 6 of this volume is especially relevant, as it is dedicated to additive conjoint measurement, which has wide-ranging significance for mathematical psychology and has connections with Debreu’s topological approach to utility.

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                                                                                                                                                                                                        • Luce, R. D., and H. Raiffa. Games and Decisions: Introduction and Critical Survey. New York: Wiley, 1957.

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                                                                                                                                                                                                          The second chapter is dedicated to utility theory.

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                                                                                                                                                                                                          • Neumann, J. von, and O. Morgenstern. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press, 1947.

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                                                                                                                                                                                                            Although it is best known as a classic on game theory, it also revived interest in (cardinal) utility by placing the concept on a solid foundation.

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                                                                                                                                                                                                            • Savage, L. J. The Foundations of Statistics. New York: Dover, 1972.

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                                                                                                                                                                                                              The fifth chapter, which is dedicated to utility, ends with a very nice historical and critical discussion.

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                                                                                                                                                                                                              • Weirich, P. Decision Space: Multidimensional Utility Analysis. Cambridge, UK: Cambridge University Press, 2001.

                                                                                                                                                                                                                DOI: 10.1017/CBO9780511498602E-mail Citation »

                                                                                                                                                                                                                Some overlap with Keeney and Raiffa 1976 but perhaps a bit more clearly philosophical.

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                                                                                                                                                                                                                Causal Decision Theory

                                                                                                                                                                                                                According to some philosophers, problems of the sort discussed in Nozick 1969 pose serious problems for accounts that recommend maximizing conditional expected utility. Though some of these problems might seem unrealistic enough to be ignored, Weirich 2008 reminds us that sufficiently realistic versions of these are available. Moreover, a problem of the relevant sort, now known as a “Newcomb Problem” can be related to other important problems in rational choice such as the Prisoner’s Dilemma (see Lewis 1979). Causal decision theory, which may be regarded as a rival to so-called evidential decision theories, depends crucially on assigning probabilities to conditionals, which is not to be confused with traditional concept of conditional probability. The origins of this influential proposal can be traced back to a suggestion by Robert Stalnaker, who had previously done seminal work on the analysis of subjunctive conditionals. Harper, et al. 1981 includes several relevant papers on conditionals, including the aforementioned paper by Stalnaker. An early but important philosophical discussion of causal decision theory can be found in Lewis 1981. Armendt 1986 and Joyce 1999 are concerned with the foundations of causal decision theory. Meek and Glymour 1994 offers a different take on the relationship between causal and evidential decision theories.

                                                                                                                                                                                                                • Armendt, B. “A Foundation for Causal Decision Theory.” Topoi 5 (1986): 3–19.

                                                                                                                                                                                                                  DOI: 10.1007/BF00137825E-mail Citation »

                                                                                                                                                                                                                  Presents a representation theorem for causal decision theory.

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                                                                                                                                                                                                                  • Harper, W., R. Stalnaker, and G. Pearce, eds. Ifs: Conditionals, Belief, Decision, Chance, and Time. Dordrecht, The Netherlands: Reidel, 1981.

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                                                                                                                                                                                                                    An important collection that includes reprints of classics such as Stalnaker’s “A Theory of Conditionals” (1972), Gibbard and Harper’s “Counterfactuals and Two Kinds of Expected Utility” (1978), and Stalnaker’s famous 1972 letter to Lewis that suggests using probabilities of conditionals rather than conditional probabilities in expected utility calculations.

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                                                                                                                                                                                                                    • Joyce, J. The Foundations of Causal Decision Theory. Cambridge, UK: Cambridge University Press, 1999.

                                                                                                                                                                                                                      DOI: 10.1017/CBO9780511498497E-mail Citation »

                                                                                                                                                                                                                      Very influential book on foundations. This is especially notable for providing a representation theorem that may be seen as the basis for a unified account of causal and evidential decision theory.

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                                                                                                                                                                                                                      • Lewis, D. “Prisoner’s Dilemma is a Newcomb Problem.” Philosophy and Public Affairs 8 (1979): 235–240.

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                                                                                                                                                                                                                        Argues that the Prisoner’s Dilemma, an important topic in game theory, is an instance of the Newcomb Problem that continues to be a major concern for many who work on the philosophy of decision theory. If correct, the Prisoner’s Dilemma would seem to provide a more common illustration of the points that are thought to be illuminated in the traditional formulations of the Newcomb Problem that some disregarded on grounds that they are improbable.

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                                                                                                                                                                                                                        • Lewis, D. “Causal Decision Theory.” Australasian Journal of Philosophy 59 (1981): 5–30.

                                                                                                                                                                                                                          DOI: 10.1080/00048408112340011E-mail Citation »

                                                                                                                                                                                                                          Lewis compares his version of causal decision theory to other versions that have also been advanced to address the challenges illustrated by the Newcomb Problem.

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                                                                                                                                                                                                                          • Meek, C., and C. Glymour. “Conditioning and Intervening.” British Journal of Philosophy of Science 45 (1994): 1001–1021.

                                                                                                                                                                                                                            DOI: 10.1093/bjps/45.4.1001E-mail Citation »

                                                                                                                                                                                                                            Uses the framework developed in P. Spirtes, C. Glymour, and R. Scheines, Causation, Prediction and Search(New York: Springer-Verlag, 1993) to resolve some of the Newcomb-inspired rivalry between causal and evidential decision theory.

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                                                                                                                                                                                                                            • Nozick, R. “Newcomb’s Problem and Two Principles of Choice.” In Essays in Honor of Carl G. Hempel. Edited by N. Rescher, 114–146. Dordrecht, The Netherlands: Reidel, 1969.

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                                                                                                                                                                                                                              Early statement of a Newcomb Problem.

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                                                                                                                                                                                                                              • Weirich, P. “Causal Decision Theory.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008.

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                                                                                                                                                                                                                                Detailed and highly readable survey article.

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                                                                                                                                                                                                                                Decision Making under Uncertainty

                                                                                                                                                                                                                                The literature on decision making under uncertainty (i.e., decision making in contexts where no “objective” distribution is salient to the decision maker) has a long and distinguished history. Various decision rules have been recommended for these situations. Some of these rules were developed within the “statistical decision theory” tradition and are surveyed in Luce and Raiffa 1957. Although subjective expected utility theory is supposed to cover cases of decision making under uncertainty, since the rational agent is assumed to have a subjective probability even when no objective distribution is salient, there are serious doubts about the tenability of this position. Perhaps the most serious of these doubts grow out of well-known examples offered by Ellsberg 1961. Such examples have inspired works such as Levi 1974, Gärdenfors and Sahlin 1982, Gilboa and Schmeidler 1989, and Kadane, et al. 1999 to investigate various decision theories that make use of indeterminate or imprecise probabilities (see Nonclassical Probabilities and Other Representations of Uncertainty). In contrast to decision theories such as those proposed in Ellsberg 1961 and Gärdenfors and Sahlin 1982, which abandon a certain sort of independence assumption that is central to traditional theories of expected utility, the theory proposed in Levi 1974 relaxes another central assumption of expected utility, namely the assumption of a complete ordering over the space of alternatives. Seidenfeld 1988 evaluates these two sorts of theoretical alternatives. The general commitment to optimization against a complete ordering is also reconsidered in Sen 1997, which contrasts optimization with maximization, where the latter, unlike the former, is compatible with incomplete orderings. Kadane, et al. 1999 examines both the foundations and implications of dropping the requirement of a complete ordering.

                                                                                                                                                                                                                                • Ellsberg, D. “Risk, Ambiguity, and the Savage Axioms.” Quarterly Journal of Economics 75 (1961): 643–669.

                                                                                                                                                                                                                                  DOI: 10.2307/1884324E-mail Citation »

                                                                                                                                                                                                                                  A seminal paper that, according to Google Scholar, has been cited over 2,600 times. Ellsberg argues persuasively for a distinction between risk and uncertainty, which is a distinction not recognized by the subjective expected utility theory of L. J. Savage.

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                                                                                                                                                                                                                                  • Gärdenfors, P., and N. E. Sahlin. “Unreliable Probabilities, Risk Taking, and Decision Making.” Synthese 53 (1982): 361–386.

                                                                                                                                                                                                                                    DOI: 10.1007/BF00486156E-mail Citation »

                                                                                                                                                                                                                                    A thought-provoking article in which the authors argue in favor of -maximin style decision theory for imprecise probabilities.

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                                                                                                                                                                                                                                    • Gilboa, I., and D. Schmeidler. “Maximin Expected Utility with Non-Unique Prior.” Journal of Mathematical Economics 18 (1989): 141–153.

                                                                                                                                                                                                                                      DOI: 10.1016/0304-4068(89)90018-9E-mail Citation »

                                                                                                                                                                                                                                      Advances a theory similar to the one presented by Gärdenfors and Sahlin 1982, but Gilboa and Schmeidler’s paper is more demanding from a mathematic point of view.

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                                                                                                                                                                                                                                      • Kadane, J., M. Schervish, and T. Seidenfeld. Rethinking the Foundations of Statistics. Cambridge, UK: Cambridge University Press, 1999.

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                                                                                                                                                                                                                                        A deep and important collection of papers, most of which are written by the trio of Kadadane, Schervish, and Seidenfeld. Many of the papers in the collection are concerned with decision making under uncertainty and related topics.

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                                                                                                                                                                                                                                        • Levi, I. “On Indeterminate Probabilities.” Journal of Philosophy 71 (1974): 391–418.

                                                                                                                                                                                                                                          DOI: 10.2307/2025161E-mail Citation »

                                                                                                                                                                                                                                          Offers a novel decision theory based on indeterminate probabilities. Levi’s decision theory is particularly notable for the fact that it does not require an ordering on the set of alternatives.

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                                                                                                                                                                                                                                          • Luce, R. D., and H. Raiffa. Games and Decisions: Introduction and Critical Survey. New York: Wiley, 1957.

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                                                                                                                                                                                                                                            The thirteenth chapter includes an extensive discussion of various proposals for decision making under uncertainty.

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                                                                                                                                                                                                                                            • Seidenfeld, T. “Decision Theory without Independence or without Ordering: What is the Difference?” Economics and Philosophy 4 (1988): 267–290.

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                                                                                                                                                                                                                                              An important analysis that compares the merits of dropping the independence assumption to that of dropping the ordering assumption within the context of expected utility theory. Seidenfeld’s analysis favors the latter.

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                                                                                                                                                                                                                                              • Sen, A. “Maximization and the Act of Choice.” Econometrica 65.4 (1997): 745–779.

                                                                                                                                                                                                                                                DOI: 10.2307/2171939E-mail Citation »

                                                                                                                                                                                                                                                A general and insightful discussion contrasting maximization and optimization.

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                                                                                                                                                                                                                                                Decision Theory and Issues of Descriptive Adequacy

                                                                                                                                                                                                                                                Not surprisingly, many of the concerns mentioned in Probability and Issues of Descriptive Adequacy extend to decision theory, since choice is typically construed as being informed by probability judgments. Kahneman and Tversky 2000 extends the authors’ approach to the study of judgment to the study of choice. Simon 1955 and Simon 1956 are classic statements of Simon’s influential notion of bounded rationality. Rubinstein 1998 provides a formal framework that is inspired by Simon’s work.

                                                                                                                                                                                                                                                • Kahneman, D., and A. Tversky, eds. Choices, Values, and Frames. Cambridge, UK: Cambridge University Press, 2000.

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                                                                                                                                                                                                                                                  An influential collection of papers on the psychology of decision making from leaders of the “heuristics and biases” tradition.

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                                                                                                                                                                                                                                                  • Rubinstein, A. Modeling Bounded Rationality. Cambridge, MA: MIT Press, 1998.

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                                                                                                                                                                                                                                                    Offers formal models of bounded rationality. Also includes a final chapter dedicated to Simon’s criticisms of the approach covered in the earlier parts of the book.

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                                                                                                                                                                                                                                                    • Simon, H. “A Behavioral Model of Rational Choice.” Quarterly Journal of Economics 69.1 (1955): 99–118.

                                                                                                                                                                                                                                                      DOI: 10.2307/1884852E-mail Citation »

                                                                                                                                                                                                                                                      Classic statement of Simon’s views on bounded rationality, as directed toward economists.

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                                                                                                                                                                                                                                                      • Simon, H. “Rational Choice and the Structure of the Environment.” Psychological Review 63.2 (1956): 129–138.

                                                                                                                                                                                                                                                        DOI: 10.1037/h0042769E-mail Citation »

                                                                                                                                                                                                                                                        Classic statement of Simon’s views on bounded rationality, as directed toward psychologists.

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                                                                                                                                                                                                                                                        Interactive Epistemology

                                                                                                                                                                                                                                                        The term “interactive epistemology” was coined by Aumann 1999. It denotes an epistemic program studying shared knowledge and belief among two or more agents or players and was already an important epistemological component in the seminal paper Aumann 1976. This program has strong ties to game-theoretic reasoning and questions of common knowledge and belief (see Barwise 1988 and Stalnaker 1996), epistemic conditions for game equlibria (see Aumann and Brandenburger 1995), backward induction, equilibria and strategies in games, (im)perfect information games, (bounded) rationality, etc. (see Brandenburger 2007). Since its inauguration with Aumann’s text, the field has been dominated by scholars drawn from theoretical economics and computer science (e.g., Baltag and Moss 2004) rather than philosophy, but recently philosophers and philosophy-oriented logicians’ works such as van Benthem 2007 have begun to pay acute attention to this burgeoning program within formal epistemology.

                                                                                                                                                                                                                                                        • Aumann, R. “Agreeing to Disagree.” Annals of Statistics 4.6 (1976): 1236–1239.

                                                                                                                                                                                                                                                          DOI: 10.1214/aos/1176343654E-mail Citation »

                                                                                                                                                                                                                                                          Provides a formal definition of common knowledge and uses it to prove “Aumann’s Agreement Theorem” describing conditions under which two “like minded” agents or players cannot “agree to disagree” in the sense that if the two players’ posteriors of some event are common knowledge then they must coincide.

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                                                                                                                                                                                                                                                          • Aumann, R. “Interactive Epistemology I.” International Journal of Game Theory 28.3 (1999): 263–300.

                                                                                                                                                                                                                                                            DOI: 10.1007/s001820050111E-mail Citation »

                                                                                                                                                                                                                                                            The foundational paper on interactive epistemology that provided the cornerstones for the program. Discusses two basic parallel approaches to the subject.

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                                                                                                                                                                                                                                                            • Aumann, R., and A. Brandenburger. “Epistemic Conditions for Nash Equilibrium.” Econometrica 63.5 (1995): 1161–1180.

                                                                                                                                                                                                                                                              DOI: 10.2307/2171725E-mail Citation »

                                                                                                                                                                                                                                                              The paper neatly ties together the sufficient conditions for a Nash-equilibrium in n-player games with the players’ knowledge and belief. Then the paper relates this to what the players know and believe about the game as well as the rationality of each other and their actions.

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                                                                                                                                                                                                                                                              • Baltag, A. and L. Moss. “Logics for Epistemic Programs.” Synthese 139 (2004): 165–224.

                                                                                                                                                                                                                                                                DOI: 10.1023/B:SYNT.0000024912.56773.5eE-mail Citation »

                                                                                                                                                                                                                                                                The paper formalizes epistemic changes in a multi-agent system such as “epistemic programs.” Epistemic actions and changes in the agents’ information, such as public or private announcements within a group, may be modeled as certain signatures that again give rise to various logical languages of information change and epistemic update.

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                                                                                                                                                                                                                                                                • Barwise, J. “Three Theories of Common Knowledge.” In Proceedings of the 2nd Conference on Theoretical Aspects of Reasoning about Knowledge: March 7–9, 1988, Pacific Grove, California. Edited by Moshe Y. Vardi, 365–379. Los Altos, CA: Morgan Kaufmann, 1988.

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                                                                                                                                                                                                                                                                  The paper investigates the relationship between three celebrated views of common knowledge and demonstrates that they are not equivalent contrary to common wisdom. It also discusses the pros and cons of the different proposals for both conceptual analysis and applications.

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                                                                                                                                                                                                                                                                  • Brandenburger, A. “The Power of Paradox: Some Recent Developments in Interactive Epistemology.” International Journal of Game Theory 35.4 (2007): 465–492.

                                                                                                                                                                                                                                                                    DOI: 10.1007/s00182-006-0061-2E-mail Citation »

                                                                                                                                                                                                                                                                    The paper scrutinizes two paradoxes in game theory that have been crucial to the development of interactive epistemology. There is also an instructive discussion of some fundamental notions in interactive epistemology, including rationality and common knowledge of rationality.

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                                                                                                                                                                                                                                                                    • Stalnaker, R. “Knowledge, Belief and Counterfactual Reasoning in Games.” Economics and Philosophy 12 (1996): 133–163.

                                                                                                                                                                                                                                                                      DOI: 10.1017/S0266267100004132E-mail Citation »

                                                                                                                                                                                                                                                                      The paper discusses reasoning and deliberation among game agents or players in light of counterfactual circumstances. It relates the adherent counterfactual reasoning processes to game theoretical considerations pertaining to motivation for setups and model building.

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                                                                                                                                                                                                                                                                      • van Benthem. “Logic Games, From Tools to Models of Interaction.” In Logic at the Crossroads. Edited by A. Gupta, R. Parikh, and J. van Benthem, 283–317. New Delhi: Allied, 2007.

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                                                                                                                                                                                                                                                                        The paper provides a crisp overview of “game logics” and incorporates notions from game theory in these logics including players’ preferences and imperfect information.

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                                                                                                                                                                                                                                                                        Formal Learning Theory

                                                                                                                                                                                                                                                                        Formal or computational learning theory begins with the problem of finding true or empirically adequate general theories from an ongoing stream of particular, empirical data. The basic idea is to seek epistemic justification in terms of truth-finding performance rather than in axiomatically regimented intuitive feelings. For example, one of the first publications in the area, Putnam 1963, involved a computational critique of the learning power of Carnap’s confirmation theory. After that auspicious beginning, the field was developed mainly by mathematicians and computer scientists interested in the foundations of machine learning (see, for instance, Jain, et al. 1999 for a compendium of results) until the late 1980s, when Glymour, Kelly, Schulte, Osherson, Weinstein, and others began again to apply it to more traditional, epistemological concerns. Such applications include explications of empirical under-determination and simplicity, critiques of Bayesianism, belief revision (see Kelly 1999), and internal realism, and the justification of inductive inference, Ockham’s razor, and causal discovery. With its point of departure in computability theory, some general overviews and textbooks dealing with this area lack a clear epistemological perspective. However, Kelly 1996, Martin and Osherson 1998, and Hendricks 2001 are important exceptions—in each of these works the epistemological perspective is unmistakably evident. Kelly 1996 is the most thorough introduction to the field, and Kelly 2000 together with Schulte 2008 provide solid introductions to the field. Along with the canonical tools of formal learning theory, it also provides philosophical motivations and results of pertinence to formal epistemology as in Kelly 2004.

                                                                                                                                                                                                                                                                        • Hendricks, V. F. The Convergence of Scientific Knowledge: A View from the Limit. Dordrecht, The Netherlands: Springer, 2001.

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                                                                                                                                                                                                                                                                          The book combines elements of formal learning theory with modal logic in order to study the properties of limiting convergent knowledge.

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                                                                                                                                                                                                                                                                          • Jain, S., D. Osherson, J. S. Royer, and A. Sharma. Systems That Learn: An Introduction to Learning Theory. 2d ed. Cambridge, MA: MIT Press, 1999.

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                                                                                                                                                                                                                                                                            The second edition of the first systematic monograph on formal learning theory from 1986, originally authored by Osherson, Stob, and Weinstein. Has close ties to the modeling of language acquisition in cognitive psychology.

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                                                                                                                                                                                                                                                                            • Kelly, K. T. The Logic of Reliable Inquiry. New York: Oxford University Press, 1996.

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                                                                                                                                                                                                                                                                              The seminal book on formal learning theory covering the philosophy of science, methodology, and formal epistemology.

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                                                                                                                                                                                                                                                                              • Kelly, K. T. “Iterated Belief Revision, Reliability, and Inductive Amnesia.” Erkenntnis 50 (1999): 11–58.

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                                                                                                                                                                                                                                                                                A paper systematically scrutinizing the learning theoretical powers of various belief revision paradigms.

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                                                                                                                                                                                                                                                                                • Kelly, K. T. “The Logic of Success.” British Journal for the Philosophy of Science 51.4 (2000): 639–660.

                                                                                                                                                                                                                                                                                  DOI: 10.1093/bjps/51.4.639E-mail Citation »

                                                                                                                                                                                                                                                                                  The paper portrays formal learning theory and places it squarely within philosophy of science covering topics from under-determination to convergence, realism, and naturalism in science.

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                                                                                                                                                                                                                                                                                  • Kelly, K. T. “Learning Theory and Epistemology.” In Handbook of Epistemology. Edited by I. Niiniluoto, M. Sintonen, and J. Smolenski. Dordrecht, The Netherlands: Kluwer, 2004.

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                                                                                                                                                                                                                                                                                    A paper that fleshes out the epistemological significance of the formal learning theoretical paradigm and is particularly pertinent to formal epistemologists of all stripes.

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                                                                                                                                                                                                                                                                                    • Martin, E., and D. Osherson. Elements of Scientific Inquiry. Cambridge, MA: MIT Press, 1998.

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                                                                                                                                                                                                                                                                                      The book ties formal learning theory to the theory of rational belief change and belief revision.

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                                                                                                                                                                                                                                                                                      • Putnam, H. “Degree of Confirmation and Inductive Logic.” In The Philosophy of Rudolf Carnap. Edited by P. A. Schilpp. La Salle, IL: Open Court, 1963.

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                                                                                                                                                                                                                                                                                        A seminal paper in which Putnam investigates Carnap’s inductive logic from a formal learning theoretical stance.

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                                                                                                                                                                                                                                                                                        • Schulte, O. “Formal Learning Theory.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008.

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                                                                                                                                                                                                                                                                                          A thorough yet very accessible introduction to the nuts and bolts of formal learning theory.

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