Abstract
This paper is about the differences between probabilities and beliefs and why reasoning should not always conform to probability laws. Probability is defined in terms of urn models from which probability laws can be derived. This means that probabilities are expressed in rational numbers, they suppose the existence of veridical representations and, when viewed as parts of a probability model, they are determined by a restricted set of variables. Moreover, probabilities are subjective, in that they apply to classes of events that have been deemed (by someone) to be equivalent, rather than to unique events. Beliefs on the other hand are multifaceted, interconnected with all other beliefs, and inexpressible in their entirety. It will be argued that there are not sufficient rational numbers to characterise beliefs by probabilities and that the idea of a veridical set of beliefs is questionable. The concept of a complete probability model based on Fisher's notion of identifiable subsets is outlined. It is argued that to be complete a model must be known to be true. This can never be the case because whatever a person supposes to be true must be potentially modifiable in the light of new information. Thus to infer that an individual's probability estimate is biased it is necessary not only to show that the estimate differs from that given by a probability model, but also to assume that this model is complete, and completeness is not empirically verifiable. It follows that probability models and Bayes theorem are not necessarily appropriate standards for people's probability judgements. The quality of a probability model depends on how reasonable it is to treat some existing uncertainty as if it were equivalent to that in a particular urn model and this cannot be determined empirically. Bias can be demonstrated in estimates of proportions of finite populations such as in the false consensus effect. However the modification of beliefs by ad hoc methods like Tversky and Kahneman's heuristics can be justified, even though this results in biased judgements. This is because of pragmatic factors such as the cost of obtaining and taking account of additional information which are not included even in a complete probability model. Finally, an analogy is drawn between probability models and geometric figures. Both idealisations are useful but qualitatively inadequate characterisations of nature. A difference between the two is that the size of any error can be limited in the case of the geometric figure in a way that is not possible in a probability model.
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References
Abelson, R.P. (1986) Beliefs are like possessions,Journal of the Theory of Social Behaviour, 16, pp. 223–250.
Bachrach, A.J. (1965)Psychological research. An introduction.2 nd edition (New York, Random House).
Barlow, R.E., Bartholomew, D.J., Bremner, J.M. & Brunk, H.D. (1972)Statistical inference under order restrictions (New York, Wiley).
Baron, J. & Frisch D. (1994) Ambiguous Probabilities and the paradoxes of expected utility in G. Wright & P. Ayton (Eds.),Subjective probability (Chichester, Wiley).
Cohen, L.J. (1981) Can human irrationality be experimentally demonstrated?,Brain and Behavioral Sciences, 4, pp. 317–370.
Dennett, D.C. (1998) Can machines think? in D.C. Dennett (Ed.),Brainchildren (London, Penguin).
Dawes, R.M. (1989) Statistical criteria for establishing a truly false consensus effect,Journal of Experimental Social Psychology, 25, pp. 1–17.
Dawes, R.M. (1990) The potential nonfalsity of the false consensus effect, in R.M. Hogarth (Ed.),Insights into decision making, (Chicago, University of Chicago Press).
de Finetti, B. (1972)Probability, induction & statistics: The art of guessing (London, Wiley).
Dulany, D.E. & Hilton, D.J. (1991) Conversational implicature, conscious repreesentation, and the conjunction fallacy,Social Cognition, 9, pp 85–110.
Eco, U. (1999)Serendipities. Translated by Weaver W. (London, Weidenfeld & Nicolson).
Eagly, A.H. & Chaiken, S. (1998) Attitude structure and function, in D.T. Gilbert, S.T. Fiske, & G. Lindzey (Eds.),The handbook of social psychology, Vol. 2., 4th ed., (Boston, Mcgraw-Hill).
Edwards, W. (1962) Subjective probabilities inferred from decisions,Psychological Review, 69, pp. 109–135.
Feller, W. (1957)An introduction to probability theory and its applications, 2nd ed. (New York, Wiley).
Fisher, R.A. (1973)Statistical methods and scientific inference, 3rd ed. (Hafner, New York).
Fisher, R.A. (1957) The underworld of probability,Sankhya, 18, pp. 201–210.
Fernández-Armesto, F. (1997)Truth. A history and a guide for the perplexed (London, Bantam).
Gigerenzer, G. (1994) Why the distinction between single-event probabilities and frequencies is important for psychology (and vice versa), in G. Wright & P. Ayton (Eds.),Subjective probability (Chichester, Wiley).
Hacking, I. (1975)The emergence of probability (Cambridge, C.U.P.).
Hilton, D. J. (1995) The social context of reasoning—conversational inference and rational judgment,Psychological Bulletin, 118, pp. 248–271.
Howson, C. & Urbach, P (1993)Scientific reasoning: the Bayesian approach, 2nd ed. (Peru, Illinios: Open Court).
Jaynes, E. T. (1973) The well posed problem,Foundations of Physics, 3, pp. 477–493.
Keynes, J. M. (1921)A treatise on probability (London, Macmillan).
Kac, M. (1983) What is random?,American Scientist, 71, pp. 405–406.
Krech, D., Crutchfield, R.S. & Ballachey E.L. (1962)An individual in society (New York, McGraw-Hill).
Kingman, J.F.C. (1982) The thrown string,Journal of the Royal Statistical Society, Series B, 44, pp. 109–138.
Koehler, J.L. (1996) The base rate fallacy reconsidered: Descriptive, normative, and methodological challenges,Brain and Behavioral Sciences, 19, pp. 1–53.
Krosnick, J.A., Li, F. & Lehman, D.R. (1990) Conversational conventions, order of information acquisition, and the effect of base rates and individuating information on social judgments,Journal of personality and social psychology, 59, pp. 1140–1152.
Kyburg, H.E. (1970)Probability and Inductive logic (Toronto, Macmillan).
Langer, E. (1994) The illusion of calculated decisions, in R.C. Schank, & E. Langer (Eds.),Beliefs, Reasoning, and decision making (Hillsdale New Jersey, Lawrence Erlbaum).
Lindley, D.V. (1965)Introduction to probability and statistics. Part 1, Probability (Cambridge, C.U.P.).
Luchins, A.S. & Luchins, E.H. (1965)Logical foundations of mathematics for behavioral scientists (New York, Holt Rinehart & Winston).
Mandelbrot, B. (1982)The fractal geometry of nature (New York, Freeman).
Macdonald, R.R. (1976) The effect of sequential dependencies on some signal detection parameters,Quarterly Journal of Experimental Psychology, 28, pp. 643–652.
Macdonald, R.R. (1984) Combined significance test for differences between conditions and ordinal predictions,Applied Statistics, 33, pp. 245–248.
Macdonald, R.R. (1986) Credible conceptions and implausible probabilities,British Journal of Mathematical and Statistical Psychology, 39, pp. 15–27.
Macdonald, R.R. (1997) On statistical testing in psychology,British Journal of Psychology, 88, pp. 333–347.
Macdonald, R.R. (1997) Base rates and randomness,Behavioral and Brain Sciences, 20, p. 778.
Macdonald, R.R. (1998) Conditional and unconditional tests of association in 2X2 tables,British Journal of Mathematical and Statistical Psychology, 51, pp. 191–204.
Macdonald, R.R. & Gilhooly, K.J. (1990) More about Linda, or Conjunctions in context,European Journal of Cognitive Psychology, 2, pp. 57–70.
Macdonald, R.R. & Smith, P.T. (1983) Testing for differences between means with ordered hypotheses,British Journal of Mathematical and Statistical Psychology, 36, pp. 22–35.
Miller, G.A. (1956) The magical number seven, plus or minus two: some limits on our capacity for processing information,Psychological Review, 67, pp. 81–97.
Mises, von R. (1957)Probability, statistics and truth (London, Allen and Unwin).
Nickerson, R.S. (1998) Confirmation bias: A ubiquitous phenomenon in many guises,Review of General Psychology, 2, pp. 175–220.
Peterson, I. (1998)The jungles of randomness (New York, Penguin).
Popper, K. (1981)The logic of scientific discovery, 10th ed. (London, Routledge).
Popper, K. & Miller, D. W. (1987) Why probabilistic support is not inductive.Philosophical Transactions of the Royal Society, series A, 321, pp. 569–591.
Politzer, G. & Noveck, I.A. (1991) Are conjunction rule violations the result of conversational rule violations,Journal of psycholinguistic research, 20, pp. 83–103.
Portnoy, S. (1994) A Lewis Carroll pillow problem: probability of an obtuse triangle,Statistical Science, 9, pp. 279–284.
Putnam, H. (1988)Representation and reality (Cambridge Massachusetts, M.I.T. Press).
Putnam, H. (1990)The meaning of the concept of probability in application to finite sequences (Garland Publishing, New York). (Phd Thesis, University of California, Los Angeles, 1951)
Putnam, H. (1994)Words and life (Harvard, Harvard University Press).
Ramsey, F.P. (1926) Truth and Probability. in H.E. Kyburg & H.E. Smokler (1963)Studies in subjective probability (New York, Wiley).
Reichenbach, H. (1970)The theory of probability, 2nd ed. (Berkelery, University of California Press).
Ross, L., Green, D. & House, P. (1977) The “false consensus effect”: an egocentric bias in social perception and attribution processes,Journal of Experimental Social Psychology, 13, pp.279–301.
Rucker, R. (1997)Infinity and the mind (London, Penguin).
Savage, L.J. (1954)Foundations of statistics (New York, Wiley)
Schum, D.A. (1994)Evidential foundations of probabilistic reasoning (New York, Wiley).
Shafer, G. (1982) Belief functions and parametric models,Journal of the Royal Statistical Society, Series B, 44, pp. 322–352.
Shafer, G. (1986) Savage revisited,Statistical Science, 1, pp.463–501.
Synge, J.L. (1968) Letter to the editor,Mathematical Gazette, 52, p 165.
Tversky, A. (1974) Assessing uncertainty,Journal of the Royal Statistical Society, Series B, 36, pp.148–160.
Tversky, A. & Kahneman, D. (1983) Extensional versus intuitive reasoning: The conjunction fallacy in probability judgement,Psychological review, 90, 293–313.
Venn, J. (1888)The logic of chance, 3rd ed. Reprinted (1962) (New York, Chelsea Publishing Co).
Yates, F. (1984) Tests of significance of 2X2 contingency tables,Journal of the Royal Statistical Society, Series A, 147, pp.426–463.
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Macdonald, R.R. The limits of probability modelling: A serendipitous tale of goldfish, transfinite numbers, and pieces of string. Mind & Society 1, 17–38 (2000). https://doi.org/10.1007/BF02512312
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DOI: https://doi.org/10.1007/BF02512312