Teaching Theory of Knowledge

Probability and Induction

CONTRIBUTORS: Brad Armendt, Martin Curd.

      Some philosophers have affirmed that induction and probable reasoning lead to knowledge. Others have been skeptical. Some philosophers have thought that probability was an ingredient in the traditional analysis of knowledge. Others have thought that a theory of probability should replace traditional epistemology. It is clear, however, that an understanding of induction, probability, and confirmation are essential for a resolution of these issues as well as others pertaining to epistemology. It is the intention of this module to provide the materials for study that would yield understanding of the relevant issues.

Divisions in the module:

  1. Introduction to probability and induction
  2. Hume's problem
    1. Historical treatment of Hume's argument.
    2. Responses to the problem(s) Hume has been perceived as raising.
  3. Goodman's new riddle of induction
  4. The origins of the modern concept and theory of probability
  5. Philosophical interpretations of the concepts of probability
    1. Logical
    2. Relative frequency
    3. Subjective
    4. Objective chance
    5. Epistemic probability
  6. The Bayesian/subjectivist program
    1. Decision-theoretic roots: coherence, Dutch books, rational preference, probabilism vs. acceptance.
    2. Belief revision/learning from experience.
  7. Theory of confirmation

      Most sections include a brief discussion of the issues as well as a list of references for that topic. Starred members [*] of the list are more central to the topic, or are good introductions to it; unstarred members are usually more specialized and often more difficult than the starred entries. We have made an effort to provide suggestions that represent a variety of approaches, but the recommendations that follow are unavoidably biased in accordance with our interests and knowledge.

      Anthologies and collections of papers. The following books are cited by references in the sections that follow. Some are collections of papers that are of general interest. and some are more specialized.

Carnap, R. and Jeffrey, R. (eds.). Studies in Inductive Logic and Probability. Vol. I, University of California Press, 1971.

Foster, M. and Martin, M. (eds.). Probability, Confirmation, and Simplicity. Odyssey Press, 1966.

Harper, W. and Hooker, C. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science. Vol. 3.. Dordrecht: Reidel. 1976.

Hintikka, J. and Suppes, P. Aspects of Inductive Logic. NorthHolland, 1966.

Jeffrey, R. Studies in Inductive Logic and Probability. Vol. II, University of California Press, 1980.

Kyburg, H. and Nagel, E. Induction: Sorne Current Issues. Wesleyan University Press, 1963.

Kyburg, H. and Smokler, H. (eds.). Studies in Subjective Probability. Wiley, 1964.

Lakatos, I. The Probkm of Inductive Logic. North-Holland, 1968

Luckenbach, S. (ed.). Probabilities, Problerns, and Paradoxes. Dickenson, 1972.

Schilpp, P. A. The Philosophy of Rudolf Carnap. Open Court, 1963.

Swain, M. (ed.). Induction, Acceptance, and Rational Belief. Narth-Rniland. 1970.

      Mathematical books: These are all fairly demanding; vol. 1 of Feller is the most accessible.

Billingsley, P. Probability and Measure. Wiley, 1979.

Feller, W. An Introduction to Probability Theory and its Applications. Wiley, Vol. I (3rd ed.) 1968; Vol. II (2nd ed.) 1971.

Fine, T. Theories of Probability. Academic Press, 1973.

Kolmogorov, A. N. Foundations of the Theory of Probability. Chelsea 1950. (2nd ed.) 1956.

1. Introduction to Probability and Induction. Three books stand out as philosophical introductions to induction and probability:

  1. Skyrms, B. Choice and Chance. (3rd ed.). Wadsworth, 1986. This book contains a general introduction to induction and inductive logic, a chapter that introduces the elementary probability calculus (ch. 5), and treatments of topics 2B, 3, 5, 6, and 7. The treatments of topics 2B, 3, and 6 are particularly thorough; topic 7 is represented by a chapter on Mill's methods; topic 5 is covered, though not in as much detail as in Salmon (1966). Some sections are more difficult, but most of the book is quite accessible to students with little background in philosophy. Substantial parts of the book do presuppose some acquaintance with propositional logic.

  2. Salmon, W. The Foundations of Scientific Inference. University of Pittsburgh Press, 1966. This book complements Choice and Chance. It has a more detailed comparison of the various interpretations of probability (topic 5), and it has more on confirmation.

  3. Kyburg, H. Probability and Inductive Logic. Macmillan, 1970. This book is written at a level less accessible to most students new to the area. It contains extensive discussion of topics 5 and 7, and some discussion of topics 2B and 3.

2. Hume's Problem of Induction. David Hume poses his problem concerning induction in several places, including the Treatise, Book I, Part III, section 6, and the Inquiry, Section IV, Part II. Exactly what argument Hume is giving is a controversial matter; the references in part A are particularly concerned with this issue. Hume has usually been interpreted as offering a very general skeptical argument directed against induction, and many philosophers have replied to Hume's argument so interpreted. A small but influential part of this literature is listed in part B.

2A. Hume's argument and historical treatments of it.

* Hume, David. A Treatise of Human Nature.

* Hume, David. An Inquiry Concerning Human Understanding.

Beauchamp, T. and Rosenberg, A. Hume and the Problem of Causation. Oxford University Press, 1981.

Smith, N. K. The Philosophy of David Hume. Macmillan, 1949.

Stove, D. C. Probability and Hume's Inductive Skepticism. Oxford University Press, 1973.

Stroud, B. Hume. Routledge & Kegan Paul, 1977.

2B. Responses to skepticism about induction. Most of the references below are not very recent. However, they do represent three prominent kinds of reponse to skepticism about induction:

  1. attempts to inductively justify inductive methods (Black),
  2. attempts to give pragmatic justifications or vindications of induction (Reichenbach, Salmon), and
  3. attempts to dissolve the problem of justifying induction (Edwards, Strawson).
Work on the topic has not died out; recent discussions of Hume's problem tend to be fragments of more general works, some of which are listed in sections 5E and 6B. The references below provide a good beginning for the topic.

Black, M. "The Justification of Induction." In Language and Philosophy, Cornell University Press, 1949.

Black, M. "Inductive Support of Inductive Rules." In Problems of Analysis, Cornell University Press, 1954.

Edwards P. "Russell's Doubts about Induction." Mind 58 (1949), 141-163.

Feigl, H. "De principiis non disputandum...?" In Max Black (ed.), Philosophical Analysis, a Collection of Essays, PrenticeHall. 1963. Also in Luckenbach (1972).

Jeffrey, R. "Probabilism and Induction." Topoi 5 (1986),51SR

Reichenbach, H. "A Frequentist Theory of Probability and Induction." 1949. In Luckenbach (1972); excerpts from The Theory of Probability.

Reichenbach, H. "On the Justification of Induction." In Feigel, H. and Sellars, W. (eds.), Readings in Philosophical Analysis, Appleton-Century-Crofts, 1949.

Salmon, W. "On Vindicating Induction." In Kyburg and Nagel (1963). Also in Luckenbach (1972).

* Skyrms, B. Choice and Chance. (3rd ad.). Wadsworth, 1986. The chapter on Hume's problem is good.

Strawson. P. Introduction to Logical Theory. 1952, 248-263.

Will, F. "Will the Future Be Like the Past?" Mind 56 (1947), 332-347.

3. Goodman's New Riddle of Induction. The obvious starting point is Goodman's Fact, Fiction, and Forecast (1955). In chapter 3 Goodman poses the problem; in chapter 4 he offers his own entrenchment theory of projectibility as a solution. There is a good, simple account of this material in Scheffler (1963). An important modification of Goodman's theory of entrenchment (Goodman, Schwartz, and Scheffler 1970) and replies to his critics are reprinted in Goodman (1972). More recent criticisms by Zabludowski (1974, 1975, 1972, 1982) and replies by Ullian and Goodman (1975, 1976, 1978, 1980, 1982) are listed in the bibliography. (Note: the Zabludowski saga is complicated and does not lend itself to an elementary presentation).

      In reviewing the literature on this topic, it is helpful to bear certain points in mind. (1) As Hesse (1969) points out, Goodman's problem is new, and it is about induction. Thus, the project is not to justify inductive inferences per se (which is the old, Humean problem of induction) but rather to adequately characterize the inductive inferences we do in fact make. In the context of Goodman's paradigm "grue" example, this requires finding a relevant asymmetry between "All emeralds are green" and "All emeralds are grue" that makes the former hypothesis projectible and the latter not. (Or, more realistically, the task is to explain why "bent" hypotheses have much lower degrees of projectibility than "straight" ones). Moreover, since the topic is inductive inference, the asymmetry must be relevant to our expectation that future emeralds will be green and not grue. Goodman's own position is that the only relevant asymmetries between projectible and unprojectible hypotheses stem from pragmatic and historical differences in the frequency with which their constituent predicates (actually, for Goodman, coextensive classes of predicates) have been (or could have been) projected.

      (2) Definitions of "grue" and other bent predicates differ in the literature. These differences can be significant. The most common variations are those which refer to "time-slices" of objects (e.g., "x is grue at time t") rather than to objects throughout their entire history ("x is grue" simpliciter), and those which omit the notion of "being examined" which was a feature of Goodman's original definition.

      Responses to Goodman's problem can be crudely divided into the following categories. Needless to say, these groupings are neither exclusive nor exhaustive.

  1. Goodman's theory of entrenchment is unnecessary since there is relevant epistemological asymmetry between "grue" and "green". Often this point is expressed by claiming that unlike "green", "grue" is essentially positional. See Barker and Achinstein (1960), Goodman (1960), Hesse (1969), Swinburne (1973), Shoemaker (1980), Shirley (1981).
  2. Goodman's paradox arises only because we mistakenly accept Nicod's criterion as sufficient for inductive support. In fact, it is false that generalizations are always confirmed by their positive, Nicodian, instances. See Cohen (1970), Rosenkrantz (1982).
  3. Goodman's theory of entrenchment fails because of his commitment to radical extensionalism. In particular, radical extensionalism undermines Goodman's conceptions of inherited entrenchment and the role of overhypotheses which play a crucial role in his theory of projectibility. See Bertolet (1976), Teller (1969), Zabludowski (1974).
  4. The solution to Goodman's problem lies in Bayesian confirmation theory, where degrees of projectibility are determined by the prior probabilities of rival hypotheses on background information. See Fain (1967), Teller (1969), Rosenkrantz (1982), Horwich (1982).

Barker, S. F. and Achinstein, P. "On the New Riddle of Induction." Philosopical Review 69 (1960), 511-22.

Bertolet, R. J. "On the Merits of Entrenchment." Analysis 37 (1976), 29-31.

Cohen, L. J. The Implications of Induction. Methuen: London, 1970.

Fain, H. "The Very Thought of Grue." Philosophical Review 76 (1967), 61-73.

* Goodman, N. Fact, Fiction, and Forecast. (4th ed.). (1955). Harvard University Press, 1983.

* Goodman, N. "Positionality and Pictures." Philosophlcal Review 69 (1960), 523-25. Reprinted in Goodman (1972).

Goodman, N. Problems and Projects. Indianapolis: BobbsMerrill, 1972. See especially chapter 8.

Hesse, M. "Ramifications of 'Grue'." British Journal of Philosophy of Science 20, (1969) 13-25.

Horwich,P. Probability and Evidence. Cambridge:Cambridge University Press, 1982.

Rosenkrantz, R. D. "Does the Philosophy of Induction Rest on a Mistake?" Journal of Philosophy 79 (1982), 78-97.

Scheffler, I. The Anatomy of Inquiry. New York: Knopf, 1963.

Shirley, E. S. "An Unnoticed Flaw in Barker and Achinstein's Solution to Goodman's New Riddle of Induction." Philosophy of Science 48 (1981), 611-17.

Shoemaker, S. "Properties and Inductive Projectibility." In Cohen, L. J. and Hesse, M. (eds.), Applications of Inductive Logic. Oxford: Clarendon Press, 1980, 291-312.

Slots, M. A. "Entrenchment and Validity." Analysis 34 (1974), 204-7.

* Symposium on Goodman's New Riddle of Induction. Journal of Philosophy 63 (1966). See especially papers by R. C Jeffrey, J. J. Thomson, and J. R. Wallace.

Swinburne, R. G. An Introduction to Confirmation Theory. London: Methuen, 1973.

* Teller, P. "Goodman's Theory of Projection." British Journal of the Philosophy of Science 20 (1969), 219-38.

Ullian, J. "Wanton Embedding Revised and Secured." Journal of Philosophy 77 (1980), 487-95.

Ullian, J. "The Ninth Inning." Journal of Philosophy 79 (1982), 332-34.

Ullian, J. and Goodman, N. "Bad Company: A Reply to Mr. Zabludowski and Others." Journal of Philosophy 72 (1975), 142-45.

Ullian, J. and Goodman, N. "Projectibility Unscathed." Journal of Philosophy 73 (1976), 527-31.

Ullian, J. and Goodman, N. "The Short of It." Journal of Philosophy 75 (1978), 263-64.

Zabludowski, A. "Concerning a Fiction About How Facts Are Forecast." Journal of Philosophy 71 (1974), 97-112.

Zabludowski, A. "Good or Bad, But Deserved: A Reply to Ullian and Goodman." Journal of Philosophy 72 (1975), 779

Zabludowski, A. "Quod Periit, Periit." Journal of Philosophy 74 (1977), 541-52.

Zabludowski, A. "Revised But Not Secured: A Reply to Ullian." Journal of Philosophy 79 (1982), 329-32.

4. The Origins of Probability. The best way to begin this topic is with Hacking (1975) and the contrasting views of Garber and Zabell (1979) and Jeffrey (1984). The only really suitable course text is Hacking; most of the other members on the list are primary sources. David (1962) is quite readable and interesting. Todhunter (1865) is a classic work that concentrates on the development of the mathematics.

Arnauld, A. Logic, or, The Art of Thinking (The Port-Royal Logic) 1662. Dickoff, J. and James, P. (tr.). Bobbs-Merrill,

Bayes, T. "An Essay Towards Solving a Problem in the Doctrine of Chances." 1763. In Two Papers by Bayes, 1940, 1963, and in Pearson and Kendall, 1970.

Bernoulli, D. "Specimen theorie novae de mensura sortis." 1738. ("Exposition of a new theory of the measurement of risk"). (tr.). In Econometrica 22, 23-36, 1954.

Bernoulli, J. Ars Conjectandi (The Art of Conjecturing). 1713.

* David, F. N. Games, Gods, and Gambling. Charles Griffin, 1962.

De Moivre, A. The Doctrine of Chances. 1717, 1738, 1756.

Fermat, Pierre de. "Correspondence with Pascal." See David 1962, 229-253.

* Garber, D. and Zabell, S. "On the Emergence of Probability." Archive for History of Exact Sciences, 21 (1979), 33-53.

* Hacking, I. The Emergence of Probability. Cambridge University Press, 1975.

Huygens, C. De Ratiociniis in Aleae Ludo (On Calculating in Games of Luck). 1657.

* Jeffrey, R. "Probability and the Art of Judgment." In Achinstein, P. and Hannaway, O. (ads.), Experiment and Observation in Modern Science. Bradford/MIT Press, 1984.

Laplace, P. S. Theorie Analytique des Probabilities. 1812, 1814, 1820.

Pascal, B. Pensees. 1670. Krailsheimer, A. J. (tr.). Penguin, 1966, also Trotter, F. W. Dutton, 1958.

Pearson, E. S. and Kendall, M. G. Studies in the History of Statistics and Probability, 1970.

* Todhunter, I. A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace. 1865. Chelsea, 1949.

5. Interpretations of Probability. Three major schools of thought about the nature of the inductive probabilities have been influential. On one interpretation, probabilities are (inductive) logical relations between propositions, or between some substitutes for propositions. They are knowable a priori, at least in principle in the way logical entailment relations are knowable. Another interpretation, or class of interpretations, takes probabilities to be relative frequencies. These theories have various forms: probabilities may be interpreted as actual relative frequencies of occurrent events, or as limiting relative frequencies of infinite sequences of occurrent and imagined events, or as modal relative frequencies of possible events. The third kind of interpretation regards probabilities as subjective; they are degrees of partial belief held by a rational agent. The degrees of belief are thought of as betting quotients -- the odds at which an agent in idealized betting situations would be equally inclined to take either side of a bet on the truth of the proposition in question.

      Each of these interpretations has been accused by critics of being incoherent, or of being artificial and uninteresting. In addition, each has been criticized as an inadequate account of probability as it is used in everyday discourse, or in science, or in decision making, or in philosophical (e.g. epistemic) theory, and so on. Some who favor only one of the interpretabons try to show how the favored interpretation can account for the intuitions which underlie the other interpretations. Others admit the legitimacy in distinct domains, of more than one of the interpretations, and they discuss the connections between the kinds of probability.

      Besides the three interpretations above, two other kinds of probability are often discussed. One is probability thought of as objective chance or propensity. Chances are taken to be out in the world, independent of beliefs about them; they are properties of events or "chance setups". The probabilities of quantum theory are standard examples. They might be subsumed under a relative frequency interpretation (typically not actual relative frequencies), but they are often taken to be something else, perhaps basic, especially by those who find limiting or modal relative frequency interpretabons unenlightening. Those who do this strive to relate chance to their favored interpretation(s) of probability.

      The other kind of probability is epistemic probability. This is probability thought of as a measure of degree of justification or warrant. Like subjective probability it is a concept of probability tied to belief systems of a rational agent, but it is not typically developed in terms of variably strong dispositions to act or choose. The normative standards imposed on the rational believer's corpus of beliefs have their source in intuitions about justification, and are generally more restrictive than the subjectivist's.

      The three books listed in section 1, Skyrms' Choice and Chance, Salmon's Foundations of Scientific Inference, and Kyburg's Probability and Inductive Logic, are the best introductions to the interpretations of probability. They should be kept in mind as references for all of the following subsections.

5A. Logical Interpretation of Probability. The central figure is Rudolf Carnap. The three starred papers provide a good introduction to his views; read the "Two Concepts" paper first and the other two in chronological order. Carnap's late work, posthumously published, is not listed below. It appears in Carnap and Jeffrey (1971) and Jeffrey (1980); see the list of anthologies.

* Carnap, R. "On Inductive Logic." Philosophy of Science 12 (1945), 72-97. Also in Luckenbach (1972) and Foster and Martin (1966).

* Carnap, R. "The Two Concepts of Probability." Philosophy and Phenomenological Research, 1945. Also in Readings in Philosophical Analysis, Feigl and Sellars, (eds.), Appleton-Century-Crofts 1949.

Carnap, R. Logical Foundations of Probability. 1950. University of Chicago Press, 2nd ed. 1962.

Carnap, R. The Continuum of Inductive Methods. University of Chicago Press, 1960.

* Carnap, R. "The Aim of Inductive Logic." In Nagel, Suppes, Tarski (eds.), Logic, Methodology, and the Philosophy of Science. Proceedings of the 1960 International Congress. Stanford, Calif., 1962. Also in Luckenbach (1972).

Carnap, R. "Inductive Logic and Inductive Intuition." In Lakatos, 1968. With comments by and replies to Bunge, Watkins, Bar-Hillel, Popper, and Hintikka.

Carnap, R. "Inductive Logic and Rational Decisions." Expanded version of "The Aim of Inductive Logic" (1960), in Carnap and Jeffrey (1971).

Carnap, R. "Notes on Probability and Induction." Synthese 25 (1973), 269-298; notes from Carnap's course at UCLA, Fall 1955.

* Hintikka, J. "A Two-Dimensional Continuum of Inductive Methods." 1966. In Hintikka and Suppes (1968).

Hintikka, J. "Carnap and Essler versus Inductive Generalization." Erkenntnis 9 (1975), 235-244; reply to Essler's comments in same issue on Hintikka's system in Hintikka (1966).

Jeffrey, R. "Carnap's Inductive Logic," Synthese 25 (1973), 299-306.

Keynes, J. M. A Treatise on Probability. Macmillan & Co, 1921.

5B. Relative Frequency Interpretations of Probability. The central figure here is Hans Reichenbach. Begin with Reichenbach (1933) and Salmon (1966). These selections represent the limiting relative frequency interpretation. For a modal relative frequency interpretation, see van Fraassen (1980).

Jeffrey, R. "Mises Redux." In Butts, R. and Hintikka, J. (eds.), Basic Problems in Methodology and Linguistics, Part III. Reidel, 1977.

Kyburg, H. The Logical Foundations of Statistical Inference. Dordrecht Reidel, 1974.

* Reichenbach, H. "The Logical Foundations of the Theory of Probability." In Feigl and Sellars (eds.), Readings in Philosophical Analysis, Appleton-Century-Crofts, Inc. 1949; English trans. of 1933 Erkenntnis article. Also in Feigl and Brodbeck (eds.), Readings in the Philosophy of Science, Appleton-Century-Crofts, Inc. 1953.

* Reichenbach, H. The Theory of Probability. 1935. University of California Press 1949, repr. 1971.

Reichenbach, H. "Philosophical Foundations of Probability." In Proceedings of the Berkeley Symposium on Probability and Statistics, University of California Press, 1943.

Reichenbach, H. "A Frequentist Theory of Probability and Induction." 1949. In Luckenbach 1972; excerpts from The Theory of Probability.

Salmon, W. "What Happens in the Long Run? Philosophical Review 74, 1965, 373-78.

* Salmon, W. The Foundations of Scientific Inference. University of Pittsburgh Press, 1966.

van Fraassen, B. The Scientific Image. Clarendon Press, 1980. See chapter 6.

von Mises, R. Probability, Statistics, and Truth. Macmillan, loss

5C. Subjective Interpretation of Probability. The chapters on subjective probability in Skyrms' Choice and Chance are especially strong parts of the book. The best single reading on the subject is the very rich paper in which it was invented: Ramsey (1926). The interpretation was developed independently in de Finetti (1937). Savage (1954) and Jeffrey (1965, 1983) are both important, influential books. Savage is quite technical, but also has interesting sections of philosophical discussion. Some chapters of Jeffrey's book are intricate, but they do not require advanced mathematics; the philosophical parts of the book are accessible treatments of diverse topics. The 1983 edition has very useful comments and references at the end of each chapter. There is considerable overlap between the works listed here and those listed in section 6B on Bayesian theory: the list below contains the works that obvious, central references on subjective probability, together with a few others that are somewhat arbitrarily picked. For more references, see section 6.

* de Finetti, B. "Foresight: Its Logical Laws, Its Subjective Sources." 1937. In Kyburg and Smokler, 1964.

de Finetti, B. Theory of Probability. 2 vols. Wiley, 1970.

de Finetti B. Probability, Seduction, and Statistics. 1972.

Hacking, I. "Slightly More Realistic Personal Probability." Philosophy of Science, 34 (1967), 311-325.

* Jeffrey, R. The Logic of Decision. McGraw Hill, 1965; (2nd ed.). University of Chicago Press, 1983.

Kyburg, H. "Subjective Probability: Criticisms, Reflections, and Problems." Journal of Philosophical Logic 7 (1978), 157-180.

Kyburg, H. and Smoker, H. (eds.). Studies in Subjective Probability. Wiley, 1964.

* Ramsey, F. P. "Truth and Probability." 1926. In The Foundations of Mathematics, Routledge & Kegan Paul, 1931. Also in Kyburg and Smokler (1964).

* Savage, L. J. The Foundations of Statistics. Wiley, 1954; also 1972, Dover.

Savage, L. J. "Difficulties in the Theory of Personal Probability." Philosophy of Science 34 (1967). 305-310.

5D. Chance.

Braithwaite, R. B. "On Unknown Probabilities." In S. Korner (ed.), Observation and Interpretation. Butterworth's Scientific Publications, 1957.

Fetzer, J. "Dispositional Probabilities." Boston Studies in the Philosophy of Science 8 (1971), 473-482. Dordrecht Reidel.

Giere, R. "A Laplacean Formal Semantics for Single-Case Propensities." Journal of Philosophical Logic 5 (1976), 321-353.

Hacking, I. Logic of Statistical Inference. Cambridge University Press, 1965.

Jeffrey, R. "Judgmental Probability and Objective Chance." Erkenntnis 24 (1986), 5-16.

Kyburg, H. "Chance." Journal of Philosophlcal Logic 5 (1976), 355-393.

Kyburg, H. "Propensities and Probabilities." British Journal for the Philosophy of Science 25 (1974), 358-375.

Levi, I. Gambling with Truth. Knopf, 1967.

Levi, I. "On Indeterminate Probabilities." Journal of Philosophy 71 (1974), 391-418.

Lewis, D. "A Subjectivist's Guide to Objective Chance." In Jeffrey (1980).

* Mellor, D. H. The Matter of Chance. Cambridge University Press, 1971.

Popper, K. "The Propensitv Interpretation of Probability." British Journal for the Philosophy of Science 10 (1959), 2542.

Skyrms, B. "Resiliency, Propensity, and Causal Necessity." Journal of Philosophy 74 (1977), 704-713.

Suppes, P. "New Foundations for Objective Probability: Axioms for Propensities." In Suppes, et. al. (eds.), Logic, Methodology, and Philosophy of Science, vol. 5. North Holland. 1973.

van Fraassen, B. "A Temporal Framework for Conditionals and Chance." Philosophical Review 89 (1980), 91-108.

< a name="5e">5E. Epistemic Probability.

Bogdan, R. (ed.). Profile of Kyberg and Levi. Dordrecht: Reidel, 1981.

Kyburg, H. "Epistemological Probability." Synthese 23 (1971), 309-326.

Kyburg, H. Logical Foundations of Statistical Inference. Dordecht: Reidel, 1974.

Pollock, J. Knowledge and Justification. Princeton University Press. 1974.

Pollock, J. "Epistemology and Probability." Synthese 55 (1983), 231-252.

6A. Bayesian Decision Making, Coherence, Preference, Acceptance. Several themes appear in this section. The decision-theoretic roots of Bayesianism are more fully represented here than in section 5C. There are two strands in the literature on this subject: First, the presentation and evaluation of the well-known Dutch book arguments that rational degrees of belief are coherent, i.e., satisfy the probability calculus. The idea of the argument is mentioned in Ramsey (1926) without much elaboration; it appears repeatedly in the important Bayesian works, and a number of papers have appeared, some fairly recently, that are specifically devoted to the argument. Dutch book arguments have also appeared in discussions of diachronic rules for changing beliefs, references containing them appear in section 6B. The literature mentioned below deals with arguments for coherent synchronic belief systems.

de Finetti, B. "Foresight: Its Logical Laws, Its Subjective Sources." 1937. In Kyburg and Smokler (1964).

* Eells, E. Rational Decision and Causality. Cambridge University Press. 1982.

Heilig, K. "Carnap and de Finetti on Bets and the Probability of Singular Events: the Dutch Book Argument Reconsidered." British Journal for the Philosophy of Science 29 (1978), 325-346.

Jackson, F. and Pargetter, R. "A Modified Dutch Book Argument." Philosophical Studies 29 (1976), 403-407.

Jeffrey, R. The Logic of Decision. McGraw Hill, 1965; (2nd ed.), University of Chicago Press, 1983.

Kemeny, J. G. "Fair Bets and Inductive Probabilities." Journal of Symbolic Logic 20 (1955) 263-273.

Kennedy, R. and Chihara, C. "The Dutch Book Argument: Its Logical Flaws, Its Subjective Sources." Philosophical Studies 36 (1979), 19-33.

Lehmann, R. S. "On Confirmation and Rational Betting." Journal of Symbolic Logic 20 (1955), 251-262.

Ramsey, F. P. "Truth and Probability." In The Foundations of Mathematics. Routledge & Kegan Paul, 1931. Also in Kyburg and Smokler (1964).

Savage, L. J. The Foundations of Statistics. Wiley, 1954; also 1972, Dover.

Shimony, A. "Coherence and the Axioms of Confirmation." Journal of Symbolic Logic 20 (1955), 1-28.

* Skyrms, B. Choice and Chance. (3rd ed.). Wadsworth, 1986.

      The second strand in the literature on decision-theoretic roots of Bayesianism consists of treatments of subjective probability (rational degrees of belief) in terms of rational preference systems. In the foundations of Bayesian decision theory the idea is that axiomatically described rational systems of preferences can be shown to be represented (and to an appropriate extent, uniquely represented) by utility assignments and probability functions that satisfy an appropriate Bayesian decision rule. Such foundations figure prominently in the classics: Ramsey (1926), Savage (1954), and Jeffrey (1965, 1983). Fishburn (1981) surveys the large number of theories put forward by the time he wrote. An account of the influential approach to the foundations of decision theory developed by Suppes, Luce, and others can be found in Krantz, et. al. (1971). Eells (1982) contains a less technical discussion of the Bayesian foundations. In many ways Jeffrey's theory (which makes use of formal results of Boker -- see Jeffrey for references), is philosophically the most satisfactory foundation for subjective probability, but Jeffrey's decision theory has been attacked recently as inappropriate in certain decision situations, and several critics have advanced versions of "causal decision theory" as a substitute. The remaining entries on the list deal with this issue.

Armendt, B. "A Foundation for Causal Decision Theory." Topoi 5 (1986), 3-19.

* Eells, E. Rational Decision and Causality. Cambridge University Press, 1982.

Fishburn, P. "Subjective Expected Utility: A Review of Normative Theories." Theory and Decision 13 (1981), 139-199. This paper contains a good bibliography of the technical work in this area.

Cibbard, A. and Harper, E. "Counterfactuals and Two Kinds of Expected Utility." In If s, Harper, Stalnaker, and Pearce (eds.), Dordrecht: Reidel, 1976.

* Jeffrey, R. The Logic of Decision, McGraw Hill, 1965; (2nd ed.). University of Chicago Press, 1983.

Jeffrey, R. "The Logic of Decision Defended." Synthese, 48 (1981), 473-492.

Krantz, D., Luce, R. D., Suppes, P., and Tversky, A. Foundations of Measurement. Vol. I, Academic Press, 1971.

* Ramsey, F. P. "Truth and Probability." In The Foundations of Mathematics, Routledge & Kegan Paul, 1931. Also in Kyburg and Smokler (1964).

* Savage, L. J. The Foundations of Statistics. Wiley, 1954; also 1972, Dover.

Skyrms, B. Causal Necessity. Yale University Press, 1979.

Skyrms, B. Pragmatics and Empiricism. Yale University Press, 1984, chapters 2, 4.

The last topic of this section is the question of how the Bayesian notion of partial belief and the normative principles Bayesians endorse are connected to the notion of (rational) acceptance. Many Bayesians have reasons for thinking that it is rarely or never rational to give full belief (degree of belief 1) to empirical propositions; many others have reasons for thinking that any epistemology for humans must permit full belief, i.e., acceptance, of empirical propositions without condemning it as irrational. Several views are possible here: 1) the Bayesian's normative standards are incompatible with acceptance, and so much the worse for the Bayesian's standards; 2) they are incompatible, and so much the worse for the notion of rational acceptance; 3) there is some way of resolving the apparent conflict without completely rejecting either notion. The place to begin this topic is with Levi (1970) and Jeffrey's (1970) reply.

* Jeffrey, R. "Dracula Meets Wolfman: Acceptance vs. Partial Belief." In Swain 1970.

Kaplan, M. "A Bayesian Theory of Rational Acceptance." Journal of Philosophy 78 (1980), 305-330.

* Levi, I. "Probability and Evidence." In Swain, 1970.

Levi, I. The Enterprise of Knowledge. MIT Press, 1980.

6B. Bayesian Learning/Belief Revision. A central part of the Bayesian program is the treatment of belief revision. All the papers listed below deal with the topic; this has been a very active area in recent years. The best place to begin is chapter 11 of Jeffrey (1965, 1983) for mention of the rule of simple conditionalization (applicable when learning is the acquisition of evidence having probability 1), and an account of Jeffrey's generalization of that rule for uncertain evidence. Probability kinematics is a term that sometimes refers to a general picture of how learning may occur, but is more often used to refer specifically to Jeffrey's rule. Another idea is that rational belief revisions are minimal belief revisions given the new evidence, the rule of maximum entropy inference is based on this idea. Some of the papers below explore the connections between conditionalization and maximum entropy inference. Jeffrey's notes at the end of chapter 11 of his book mention most of the papers below. Two important papers that appear after those notes were written are van Fraassen (1984) and Skyrms (1986).

Armendt, B. "Is There a Dutch Book Argument for Probability Kinematics?" Philosophy of Science 47 (1980), 583-588.

Diaconis, P. and Zabell, S. "Updating Subjective Probability." Journal of the Arnerican Statistical Association 77 (1982) 822-830.

Domotor, Z. "Probability Kinematics, Conditionals, and Entropy Principles." Synthese 63 (1985), 75-114.

Field, H. "A Note on Jeffrey Conditionalization." Philosophy of Science 45 (1978), 361-367.

Freedman, D. and Purves, R. "Bayes' Method for Bookies." Annals of Mathematical Statistics 40 (1969), 1177-1186.

Garber, D. "Discussion: Field and Jeffrey Conditionalization " Philosophy of Science 47 (1980), 142-145.

Harper, W. "Rational Belief Change, Popper Functions, and Counterfactuals." Synthese 30 (1975), 221-262. Also in Harper and Hooker, 1976.

Harper, W. "Ramsey Test Conditionals and Iterated Belief Change." 1976. In Harper and Hooker, 1976.

Harper, W. "Bayesian Learning Models with Revision of Evidence." Philosophia 7 (1978), 357-367.

* Jeffrey, R. The Logic of Decision. McGraw Hill, 1965; (2nd ed.), University of Chicago Press, 1983. See especially chapter 11

Jeffrey. R. "Probable Knowledge." In Lakatos (1968).

Lehrer, K. "Truth, Evidence, and Inference." American Philosophy Ouarterly (1974), 79-92.

Lehrer K. "Induction, Rational Acceptance, and Minimally Inconsistent Sets." In Maxwell, G., and Anderson, R. M., Jr., (eds ), (1975) Minnesota Studies in the Philosophy of Science. Vol 4, University of Minnesota Press, 1975, 295-323.

Levi, I. "Probability Kinematics." Philosophy of Science 18 (1967), 197-209.

Skyrms, B. Pragmatics and Empiricism. Yale University Press. 1984. See especially chapter 3.

Skyrms, B. "Maximum Entropy Inference as a Special Case of Conditionalizaion." Synthese 63 (1985), 55-74.

Skyrms, B. "Dynamic Coherence and Probability Kinematics. 1986. Circulated manuscript.

Teller, P. "Conditionalization and Observation." Synthese 26 (1973), 218-258. Also in Harper and Hooker (1976) vol. I.

van Fraassen, B. "Rational Belief and Probability Kinematics." Philosophy of Science 47 (1980), 165-178.

van Fraassen, B. "A Problem for Relative Information Minimizers in Probability Kinematics." British Journal for the Philosophy of Science 32 (1981), 367-379.

van Fraassen, B. "Belief and the Will." Journal of Philosophy 81 (1984), 235-256.

Williams, P. M. "Bayesian Conditionalization and the Principle of Minimum Information." British Journal for the Philosophy of Science 31 (1980), 131-144.

7. Theory of confirmation. The seminal work in this area is Hempel (1945). There are systematic discussions of Hempel's arguments, with useful bibliographies in Scheffler (1963) and Swinburne (1971, 1973). In teaching this material it is vital to distinguish clearly between two different things that get called "paradoxes of confirmation" in the literature:

  1. Hempel's reductio ad absurdum of Nicod's Criterion based on the logical inconsistency between Nicod's Criterion (regarded as necessary for confirmation), the Scientific Laws Condition, and the Equivalence Condition (if e confirms h then e confirms any j that is logically equivalent to h). The "Scientific Laws Condition" is Swinburne's term for the claim that H, H3, and H4 are logically equivalent where H = "All ravens are black," H3 = "Everything which is either a raven or a nonraven is either black or a nonraven," and H4 = "All nonblack things are nonravens." Nicod's Criterion is the assertion that hypotheses of the form "All Xs are Ys" are confirmed by observation reports e if and only if e is of the form "Xa & Ya."
  2. The derivation from Hempels own Satisfaction Criterion of the results that H = "All ravens are black" (which Hempel writes as (x)(Rx --> Bx), where "-- >" is material implication) is confirmed not only by observation reports of black ravens (b1 = "Ra & Ba"), but also by reports of black nonravens (b3 = "~Ra & Ba"), and by reports of nonblack nonravens (b4 = "~Ra & ~Ba"). Hempel himself regards the b3 and b4 results as correct (and hence nonparadoxical) arguing that our intuitive belief to the contrary is a psychological illusion.

      One important class of attempts to vindicate Hempcl's judgment that observations of nonravens confirm H comes from adopting what Swinburne calls the "quantitative argument." (Rody (1978) refers to this as "the class-size approach"). The best accounts of this approach are Alexander (1958), Mackie (1963), and Swinburne (1971, 1973). Much of this material can be presented very simply by using Venn diagrams in which areas are proportional to probabilities.

      The class-size approach has two main ingredients: (1) the introduction of background knowledge, k, about the ratio of ravens to nonravens (x: 1-x), and the ratio of black things to nonblack things (y: 1-y) in the universe; and, (2) a probabilistic theory of confirmation based on what Mackie (1963) calls "the Inverse Principle." (Following Hooker and Stove (1967), Mackie (1969) later refers to this principle as "the Relevance Criterion"). The Inverse Principle states that h is confirmed by an observation report b relative to background knowledge k if and only if Pr(b/k&h)< Pr(b/k), and that b confirms h the better the more the adding to h to k raises the probability of b.

      The clearest motivation for the Inverse Principle comes from adopting what Salmon (1973, 1975), following Carnap, calls the "relevance" or "incremental" concept of confirmation. On this conception, b confirms h on k just in case Pr(h/b&k)<Pr(h/k). The Inverse Principle then follows from Bayes' Theorem.

      The class-size approach vindicates Hempel's judgment about the positive confirming power of b4 instances but it also contradicts his verdict about b3 instances. On the one hand, b4 instances always confirm H, but when x >>(1-y) their confirming power is very small and quite negligible compared with that provided by b1 reports. This would plausibly explain our psychological conviction that b4 reports are confirmationally neutral since we normally take for granted that the number of nonblack things in the universe vastly exceeds the number of ravens. On the other hand, b3 reports always disconfirm H whenever y<1 and x>0. Hooker and Stove (1967) and Rody (1978) regard this as a decisive objection to the class-size approach.

      As Salmon (1975), Rody (1978), and Rosenkrantz (1982) have noted, the Hempelian concept of confirmation is fundamentally at odds with the Bayesian concept underlying the class-size approach. For example the inverse condition (sometimes called "Huyghens' Principle") that h is always confirmed by its logical consequences is rejected as false by Hempel, but it is an immediate consequence of Bayes' Theorem.

      Various proposals have been made within the Bayesian tradition for resolving the b3 problem and for providing a more convincing account of confirmation in general (Salmon, Good, Horwich, Rosenkrantz, Rody). Rody (1978), for example, defends the Relevance Criterion by adopting a theory of selective confirmation in which b confirms h on k with respect to a competing hypothesis h', if and only if Pr(b/h&k)>Pr(b/h'&k). The general features of theories of selective confirmation and their implications for the Equivalence Condition are discussed in Grandy (1967). (The basic point is that logically equivalent hypotheses do not entail the same contraries and so the Equivalence Condition cannot be retained in its original formulation).

      Other attempts to solve the paradoxes of confirmation include Popperian approaches (Watkins 1964) which stress the importance of the method by which observation reports are obtained, and nonformal approaches influenced by Quine's views on natural kinds (Fisch 1984).


Alexander, H. G. "The Paradoxes of Confinnation." British Journal of the Philosophy of Science 9 (1958), 227-33.

Fisch, M. "Hempel's Ravens, The Natural Classification of Hypotheses and the Growth of Knowledge." Erkenntnis 21 (1984), 45-62.

Good, I. J. "The Paradox of Confirmation.'' British Journal of the Philosophy of Science 11 (1960). 145-49.

Good, I. J. "The Paradox of Confirmation (II)." British Journal of the Philosophy of Science 12 (1961), 63-64.

Grandy, R. E. "Some Comments on Confirmation and Selective Confirmation." Philosophical Studies 18 (1967), 19-24.

* Hempel, C. G. "Studies in the Logic of Confirmation.'' Mind 54 (1945), 1-26, 97-121. Reprinted with a postscript in Hempel, C. G., Aspects of Scientific Explanation. New York: The Free Press, 1965, 1-51.

Hooker, C. A. and Stove, D. "Relevance and the Ravens." British Journal of the Philosophy of Science 18 (1968), 305-15.

Horwich, P. Probability and Evidence. Cambridge: Cambridge University Press, 1982.

Mackie, J. L. "The Relevance Criterion of Confirmation.'' British Journal of the Philosophy of Science 20 (1963), 27-40.

Rody, P. J. "(C) Instances, The Relevance Criterion, and the Paradoxes of Confirmation." Philosophical Studies 45 (1978), 289-302.

Rosenkrantz, R. D. "Does the Philosophy of Induction Rest on a Mistake?" Journal of Philosophy 79 (1982), 78-97.

Salmon, W. C. "Confirmation." Scientific American 228 (1973), 75-83. (May issue)

Salmon, W. C. ''Confirmation and Relevance." In G. Maxwell and R. M. Anderson. Jr., (eds.), Induction, Probability, and Confirmation: Minnesota Studies in the Philosophy of Science, vol. 6. Minneapolis: University of Minnesota Press, 1975, 3-36.

Scheffler, I. The Anatomy of Inquiry. New York: Knopf, 1963.

Smokler, H. "The Equivalence Condition." American Philosophical Quarterly 4 (1967),300-7.

Swinburne, R. G. "The Paradoxes of Confirmation -- A Survey." American Philosophical Quarterly 8 (1971), 318-30.

Swinburne, R. G. An Introduction to Confirmation Theory. London: Methuen, 1973.

Watkins, J. W. N. ''Confirmation, Paradox and Positivism." In M. Bunge (ed.), The Critical Approach to Science and Philosophy. New York: The Free Press, 1964.