Cognition, Semantics and Philosophy pp 47-73 | Cite as

# Changing Beliefs Rationally: Some Puzzles

Chapter

## Abstract

The puzzles I shall discuss belong to the subject names (by Richard Jeffrey’) “Probability Kinematics,” that is, the question of how probability judgments should change in the light of new information. But I start with a bit of the pre-history of this subject as I see it.

## Keywords

Conditional Probability Belief Revision Inductive Logic Labour Party Probability Judgment
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## Notes

- 1.There is less agreement aey,
*The Logic of Decision*, (McGraw-Hill, 1965) chapter 11.Google Scholar - 2.I have in mind the work of R. Carnap, especially
*The Logical Foundations of Probability.*Google Scholar - 3.B. van Fraassen, “A Demonstration of the Jeffrey Conditionalization Rule,”
*Erkenntnis*, 24, 1986,pp. 17–24.Google Scholar - 4.B. van Fraassen,
*op. cit.*, pp. 22–23.Google Scholar - 5.Peter Williams, “Bayesian conditionalisation and the Principle of Minimum Information,”
*British Journal for the Philosophy of Science*, 1980.Google Scholar - 6.Bas C. van Fraassen, “A Problem for Relative Information Minimizers in Probability Kinematics,”
*British Journal for the Philosophy of Science*, 1981.Google Scholar - 7.
*Op. cit.*Google Scholar - 8.R.I.G. Hughes and B.C. van Fraassen, “Symmetry Arguments in Probability Kinematics” in P. Kitcher and P. Asquith (eds),
*PSA*1984,vol 2 (East Lansing Michigan); and Bas C. van Fraassen, R.I.G. Hughes,and Gilbert Harman A promble for Relative Information Minimizers,Continued (unpublished).Google Scholar - 9.There is less agreement about how to explain this fact. David Lewis gives a Gricean explanation in “Probabilities of Conditionals and Conditional Probabilities,”
*Philosophical Review*85,1976,pp. 297–315,reprinted in his*Philosophical Papers*Volume II (Oxford, 1986). He later abandoned this in favour of Frank Jackson’s explanation in “On Assertion and Indicative Conditionals,”*Philosophical Review*88 (1979), pp. 565–589, and “Conditionals and Possibilia,”*Proceedings of the Aristotelian Society*81 (1981) pp. 125–137. See also Ernest Adams,*The Logic of Conditionals*, (Reidel, 1975); and my “Do Conditionals Have Truth Conditions?,”*Critica*, 1986, pp. 3–39, forthcoming in Frank Jackson (ed.)*Conditionals*, (Oxford: Oxford Readings in Philosophy, 1990).Google Scholar - 10.This puzzle is a (depoliticised) version of one put by John Skorupski to Crispin Wright as a problem for an intuitionist-style conditional; and also, I heard, by David Lewis to Frank Jackson as a problem for Jackson’s claim that conditionals are assertible only if they are “fit for modus ponens.”
*See*Frank Jackson,*op. cit..*I have since discovered by solution to it is to be found, in essence, in Robert Stalnaker,*Inquiry*, (MIT Press, 1984),p. 105. See also David Lewis,*Philosophical Papers*void’,*op. cit*, pp. 154–156.Google Scholar - 11.I believe, following Ernest Adams.
*op. cit.*, that conditional judgments express high conditional probabilities. But you do not need to agree with me about that. Here, the puzzle is about high conditional probabilities, and conditionalisation.Google Scholar - 12.My treatment of this example shows that it is not a counterexample to Jackson’s claim: It is just that one can never get into the position to use
*modus ponens*(one would if one could!), for one can never get the information that the antecedent is true without getting further relevant information. Whether the intuitionist-style conditional can be similarly saved is part of the general question of how to adapt logical constants designed for mathematics to empirical discourse. It is not my problem. See Crispin Wright,*Realism*,*Meaning and Truth*,(Blackwell, 1988), Appendix.Google Scholar - 13.I am grateful to my ex-student Ruth Weintraub both for introducing me to this puzzle, and for being the first to suggest that A’s probability of being hanged should not change in response to this information.Google Scholar
- 14.According to this theorem of the probability calculusGoogle Scholar
- 15.Alex Edgington, my son, suggests the following (which is somewhat above my head). (It is framed in terms of a second question-and-answer equivalent to my 2’. “Given that we’re in R, how likely is it that we’re in R
_{2}7” “0”): “another way of showing that the amount of information we get depends on how we ask the question is as follows: suppose Judy Benjamin assigns some initial probability density to the speaker’s degrees of belief (which are themselves probabilities) in the four possibilities. From this density function, which is over four real variables, we can derive density functions corresponding to variables defined in terms of those four—in this case we are interested in ”R_{2}“ and ”R_{2}/(R,+R_{2}).“ When we evaluate these derived densities at the answer received (0 in both cases), we get a measure of how plausible she would have thought that answer, and hence (inversely) of how much information it gives her. The point is that value is different for different variables (”R_{2}“ and ”R_{2}/R_{1}+R_{2})“), even if the answers tell her the same fact about R_{2}. For example, suppose Judy assigns the following distribution to the speaker’s beliefs: R2 is distributed uniformly on [0,1] and R, uniformly on [0,1-R_{2}]. It is then easy to show that the p.d.f.of ”R_{2}“ evaluated at 0 is 1, whereas the p.d.f. of ”R_{2}/(R,+R_{2})“ evaluated at 0 is infinite (so she gets more information in the first case).”Google Scholar - P(A&B) = P(A)P(B/A) = P(B)P(A/B). So P(B/A) = P(B) iff P(A/B) =P(A).Google Scholar
- 17.“Probabilities of Conditionals and Conditional Probabilities,”
*op. cit.*Google Scholar - 18.See my “Do Conditionals Have Truth Values?,”
*op. cit.*Google Scholar - 19.See Simon Blackburn, in Charles Travis (ed.)
*Meaning and Truth*, (Blackwell, 1986).Google Scholar - 20a.Bas van Fraassen, “Probabilities of Conditionals” in W.L. Harper and C.A. Hooker (eds)
*Foundations of Probability Theory*,*StatisticalInference*,*and Statistical Theories of Science*, Volume 1 (Reidel, 1976)Google Scholar - 20b.Brian Ellis,
*Rational Belief Systems*(Blackwell, 1979), pp. 74–80;Google Scholar - 20c.Brian Ellis “Two Theories of Indicative Conditionals,”
*Australasian Journal of Philosophy*, 1984.Google Scholar

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