SLIDE 4 Overview Hempel, Carnap & Popper Modern Bayesianism The “Fallacy” References
As Carnap [2] showed, c(H1 & H2, E) > c(H1, E) is possible. Moreover, it seems to be (intuitively) true in the Linda case. As Tversky & Kahneman themselves [18] say: “feminist bank teller is a better hypothesis about Linda than bank teller”. Two (independent) robust2 confirmation-theoretic results:
Theorem 1. For all3 Bayesian relevance measures c, if
(i) c(H2, E | H1) > 0 and (ii) c(H1, E) ≤ 0,
then c(H1 & H2, E) > c(H1, E). Theorem 2. For all Bayesian relevance measures c, if
(i) Pr(E | H1 & ∼H2) < Pr(E | H1 & H2) and (ii′) Pr(E | H1 & ∼H2) ≤ Pr(E | ∼H1 & H2), (ii′′) Pr(E | H1 & ∼H2) ≤ Pr(E | ∼H1 & ∼H2),
then c(H1 & H2, E) > c(H1, E).
2Some authors [8, 15] have offered non-robust confirmational analyses.
But, these analyses rest on invidious (and non-normative) choices of c [6].
3By “all”, I mean all relevance measures satisfying Joyce’s (WLL) [9], which
seems to include all relevance measures that have appeared in the literature.
Branden Fitelson Probability, Confirmation, and the “Conjunction Fallacy” fitelson.org Overview Hempel, Carnap & Popper Modern Bayesianism The “Fallacy” References
Inequalities equivalent to our (i) [Pr(E | H1 & H2) > Pr(E | H1)] have been empirically vindicated in traditional CF cases [14]. Our (ii)’s have not been explicitly tested. But, we suspect these will obtain (empirically) in the the traditional CF cases. We are performing experiments to test these (i)/(ii) and (i)/(ii′)/(ii′′) accounts of the traditional CF cases [4]. Interestingly, there are other (non-traditional) sorts of CF cases in which the (ii)’s seem much less intuitive. E.g. [16]
E = John is Scandanavian.4 H1 = John has blue eyes. H2 = John has blond hair.
This example also seems to exhibit a conjunction fallacy pattern (this time, symmetric with respect to H1 and H2). Even in this broader class of CFs, however, we think that some confirmation-theoretic conditions will be useful for predicting and explaining observed patterns of response.
4Scandinavia has the greatest percentage of people with blond hair and
blue eyes (though every possible combination of hair and eye color occurs there). E = John is sampled at random from the Scandinavian population.
Branden Fitelson Probability, Confirmation, and the “Conjunction Fallacy” fitelson.org Overview Hempel, Carnap & Popper Modern Bayesianism The “Fallacy” References
In the John case, (i) seems intuitively plausible, but (ii) does
- not. What about (ii′) and (ii′′)? While (ii′) is not patently
false (the = case, anyway), (ii′′) seems pretty clearly false. Moreover, (i) alone cannot undergird a robust account. It
- nly suffices for some c’s, which lack normative force [6].
Descriptively, we suspect confirmation-theoretic relations between H1 and H2 themselves may be involved in this CF. Specifically, the terms c(Hi, Hj) seem to be salient. We bet they are explanatorily relevant. There is some preliminary evidence in the literature for this (but no general model). Psychologically, we think there are two important sets of confirmation-theoretic factors involved in CF cases:
c(H1, E), c(H2, E), c(H1, E | H2), c(H2, E | H1). [Traditional CF] c(H1, H2), c(H2, H1), c(H1, H2 | E), c(H2, H1 | E). [NT CF]
We are working on more general confirmaiton-theoretic models which we hope will account for (i.e., predict and explain, if not rationalize) all known conjunction fallacies.
Branden Fitelson Probability, Confirmation, and the “Conjunction Fallacy” fitelson.org Overview Hempel, Carnap & Popper Modern Bayesianism The “Fallacy” References [1]
- N. Bonini, K. Tentori and D. Osherson, (2004), “A different conjunction fallacy,” Mind & Language 19.
[2]
- R. Carnap, (1950), Logical foundations of probability, 1st ed., U. Chicago Press.
[3]
- R. Carnap, (1962), Logical foundations of probability, 2nd ed., U. Chicago Press.
[4]
- V. Crupi, B. Fitelson and K. Tentori, (2006), “Comparative Probability, Comparative Confirmation, and
the Conjunction Fallacy,” manuscript. [5]
- B. Fitelson, (2001), Studies in Bayesian confirmation theory, PhD. thesis, University of Wisconsin,
URL: http://fitelson.org/thesis.pdf. [6] , forthcoming, “Likelihoodism, Bayesianism, and comparative confirmation,” Synthese, URL: http://fitelson.org/synthese.pdf. [7]
- C. Hempel, (1945), “Studies in the logic of confirmation,” Mind 54.
[8] Hertwig, R. and Chase, V. M. (1998), “Many reasons or just one: How response mode affects reasoning in the conjunction problem,” Thinking and Reasoning 4. [9]
- J. Joyce, (2003), “Bayes’ Theorem,” Stanford Encyclopedia of Philosophy.
[10]
- J. Keynes, A treatise on probability, Macmillan, London, 1921.
[11]
- I. Levi, (1985), “Illusions about Uncertainty”, The British Journal for the Philosophy of Science 36.
[12]
- K. Popper, (1954), “Degree of confirmation,” British Journal for the Philosophy of Science 5.
[13]
- W. Salmon, (1975), “Confirmation and relevance,” in Induction, Probability, and Confirmation,
Maxwell and Anderson eds., University of Minnesota Press. [14]
- E. Shafir, E. Smith and D. Osherson, “Typicality and reasoning fallacies,” Memory and Cognition 18.
[15]
- A. Sides, D. Osherson, N. Bonini, and R. Viale, (2002), “On the reality of the conjunction fallacy,”
Memory and Cognition 30. [16]
- K. Tentori, N. Bonini, and D. Osherson, (2004), “The conjunction fallacy: a misunderstanding about
conjunction?,” Cognitive Science 28. [17]
- K. Tentori, V. Crupi, N. Bonini and D. Osherson, (2005), “Comparison of confirmation measures,”
Cognition, in press. [18]
- A. Tversky and D. Kahneman, (1983), “Extensional vs. intuitive reasoning: the conjunction fallacy in
probability judgment,” Psychogical Review 90. Branden Fitelson Probability, Confirmation, and the “Conjunction Fallacy” fitelson.org
