Skip to main content Accessibility help

The uncertain reasoner: Bayes, logic, and rationality

  • Mike Oaksford (a1) and Nick Chater (a2)


Human cognition requires coping with a complex and uncertain world. This suggests that dealing with uncertainty may be the central challenge for human reasoning. In Bayesian Rationality we argue that probability theory, the calculus of uncertainty, is the right framework in which to understand everyday reasoning. We also argue that probability theory explains behavior, even on experimental tasks that have been designed to probe people's logical reasoning abilities. Most commentators agree on the centrality of uncertainty; some suggest that there is a residual role for logic in understanding reasoning; and others put forward alternative formalisms for uncertain reasoning, or raise specific technical, methodological, or empirical challenges. In responding to these points, we aim to clarify the scope and limits of probability and logic in cognitive science; explore the meaning of the “rational” explanation of cognition; and re-evaluate the empirical case for Bayesian rationality.



Hide All
Adams, E. W. (1975) The logic of conditionals: An application of probability to deductive logic. Reidel.
Adams, E. W. (1998) A primer of probability logic. CLSI Publications.
Ali, N., Schlottmann, A., Shaw, A., Chater, N. & Oaksford, M. (in press) Causal discounting and conditional reasoning in children. In: Cognition and conditionals: Probability and logic in human thought, ed. Oaksford, M. & Chater, N.. Oxford University Press.
Anderson, A. & Belnap, N. D. (1975) Entailment: The logic of relevance and necessity, vol. 1. Princeton University Press.
Anderson, J. R. (1983) The architecture of cognition. Harvard University Press.
Anderson, J. R. (1990) The adaptive character of thought. Erlbaum.
Anderson, J. R. (1991a) Is human cognition adaptive? Behavioral and Brain Sciences 14:471–84; discussion 485–517.
Ariely, D., Loewenstein, G. & Prelec, D. (2003) Coherent arbitrariness: Stable demand curves without stable preferences. Quarterly Journal of Economics 118:73105.
Bennett, J. (2003) A philosophical guide to conditionals. Oxford University Press.
Blakemore, C., Adler, K. & Pointon, M. (1990) Vision: Coding and efficiency. Cambridge University Press.
Braine, M. D. S. & O'Brien, D. P. (1991) A theory of if: A lexical entry, reasoning program, and pragmatic principles. Psychological Review 98:182203.
Carnap, R. (1950) The logical foundations of probability. University of Chicago Press.
Carroll, S. (2005) Endless forms most beautiful. W. W. Norton.
Chater, N. (1996) Reconciling simplicity and likelihood principles in perceptual organization. Psychological Review 103:566–81.
Chater, N. & Oaksford, M. (1999b) The probability heuristics model of syllogistic reasoning. Cognitive Psychology 38:191258.
Chater, N. & Oaksford, M. (2006) Mental mechanisms: Speculations on human causal learning and reasoning. In: Information sampling and adaptive cognition, ed. Fiedler, K. & Juslin, P., pp. 210–38. Cambridge University Press.
Chater, N. & Oaksford, M. eds. (2008a) The probabilistic mind: Prospects for Bayesian cognitive science. Oxford University Press.
Chater, N. & Oaksford, M. (2008b) The probabilistic mind: Where next? In: The probabilistic mind: Prospects for Bayesian cognitive science, ed. Chater, N. & Oaksford, M., pp. 501–14. Oxford University Press.
Chater, N., Oaksford, M., Heit, E. & Hahn, U. (in press) Inductive logic and empirical psychology. In: The handbook of philosophical logic, vol. 10, ed. Hartmann, S. & Woods, J.. Springer.
Chater, N., Oaksford, M., Nakisa, R. & Redington, M. (2003) Fast, frugal and rational: How rational norms explain behavior. Organizational Behavior and Human Decision Processes 90:6386.
Chater, N., Tenenbaum, J. B. & Yuille, A. eds. (2006) Probabilistic models of cognition: Where next? Trends in Cognitive Sciences 10:335–44. [Special Issue.]
Chater, N. & Vitányi, P. (2002) Simplicity: A unifying principle in cognitive science? Trends in Cognitive Sciences 7:1922.
Copeland, D. E. (2006) Theories of categorical reasoning and extended syllogisms. Thinking and Reasoning 12:379412.
Copeland, D. E. & Radvansky, G. A. (2004) Working memory and syllogistic reasoning. Quarterly Journal of Experimental Psychology 57A:1437–57.
Cosmides, L. (1989) The logic of social exchange: Has natural selection shaped how humans reason? Studies with the Wason selection task. Cognition 31:187276.
Dawes, R. M. (1979) The robust beauty of improper linear models in decision making. American Psychologist 34:571–82.
Dempster, A. P. (1968) A generalization of Bayesian inference. Journal of the Royal Statistical Society, Series B 30:205–47.
De Neys, W., Vartanian, O. & Goel, V. (2008) Smarter than we think: When our brains detect that we are biased. Psychological Science 19:483–89.
Domingos, P. & Pazzani, M. (1997) On the optimality of the simple Bayesian classifier under zero-one loss. Machine Learning 29:103–30.
Dowty, D. R., Wall, R. E. & Peters, S. (1981) Introduction to Montague semantics. Springer.
Doya, K., Ishii, S., Rao, R. P. N. & Pouget, A., eds. (2007) The Bayesian brain: Probabilistic approaches to neural coding. MIT Press.
Earman, J. (1992) Bayes or bust? MIT Press.
Edgington, D. (1995) On conditionals. Mind 104:235329.
Evans, J. St. B. T. (1989) Bias in human reasoning: Causes and consequences. Erlbaum.
Evans, J. St. B. T. (2007) Hypothetical thinking: Dual processes in reasoning and judgement. Psychology Press.
Evans, J. St. B. T. & Frankish, K., eds. (in press) In two minds: Dual processes and beyond. Oxford University Press.
Evans, J. St. B. T. & Over, D. E. (1996a) Rationality and reasoning. Psychology Press.
Evans, J. St. B. T. & Over, D. E. (2004) If. Oxford University Press.
Field, H. (1978) A note on Jeffrey conditionalization. Philosophy of Science 45:361–67.
Fodor, J. A. (1968) Psychological explanation. Random House.
Friston, K. (2005) A theory of cortical responses. Philosophical Transactions of the Royal Society B 360:815–36.
Gärdenfors, P. (1986) Belief revisions and the Ramsey test for conditionals. Philosophical Review 95:8193.
George, C. (1997) Reasoning from uncertain premises. Thinking and Reasoning 3:161–90.
Gigerenzer, G. & Goldstein, D. (1996) Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review 103:650–69.
Gigerenzer, G., Todd, P. & the ABC Research Group. (1999) Simple heuristics that make us smart. Oxford University Press.
Glymour, C. (2001) The mind's arrow. MIT Press.
Goel, V. (2007) The anatomy of deduction. Trends in Cognitive Science 11:435–41.
Gold, J. I. & Shadlen, M. N. (2000) Representation of a perceptual decision in developing oculomotor commands. Nature 404:390–94.
Green, D. W. & Over, D. E. (1997) Causal inference, contingency tables and the selection task. Current Psychology of Cognition 16:459–87.
Green, D. W. & Over, D. E. (2000) Decision theoretical effects in testing a causal conditional. Current Psychology of Cognition 19:5168.
Gregory, R. L. (1970) The intelligent eye. Weidenfeld & Nicolson.
Grice, H. P. (1975) Logic and conversation. In: The logic of grammar, ed. Davidson, D. & Harman, G., pp. 6475. Dickenson.
Griffiths, T. L., Steyvers, M. & Tenenbaum, J. B. (2007) Topics in semantic representation. Psychological Review 114:211–44.
Griffiths, T. L. & Tenenbaum, J. B. (2005) Structure and strength in causal induction. Cognitive Psychology 51:354–84.
Hahn, U. & Oaksford, M. (2007) The rationality of informal argumentation: A Bayesian approach to reasoning fallacies. Psychological Review 114:704–32.
Harman, G. (1986) Change in view: Principles of reasoning. MIT Press.
Hawthorn, J. (2008) Inductive logic. In: Stanford Encyclopedia of Philosophy. Available at:
Hilton, D. J., Kemmelmeier, M. & Bonnefon, J-F. (2005) Putting Ifs to work: Goal-based relevance in conditional directives. Journal of Experimental Psychology: General 134:388405.
Hochberg, J. & McAlister, E. (1953) A quantitative approach to figure “goodness.” Journal of Experimental Psychology 46:361–64.
Houdé, O., Zago, L., Mellet, E., Moutier, S., Pineau, A., Mazoyer, B. & Tzourio-Mazoyer, N. (2000) Shifting from the perceptual brain to the logical brain: The neural impact of cognitive inhibition training. Journal of Cognitive Neuroscience 12:721–28.
Hurley, S. & Nudds, M., eds. (2006) Rational animals? Oxford University Press.
Jacob, F. (1977) Evolution and tinkering. Science 196:1161–66.
Jacobs, R. A., Jordan, M. I., Nowlan, S. & Hinton, G. E. (1991) Adaptive mixtures of local experts. Neural Computation 3:112.
Jeffrey, R. C. (1967) Formal logic: Its scope and limits, 2nd edition.McGraw-Hill.
Jeffrey, R. C. (1983) The logic of decision, 2nd edition.University of Chicago Press.
Johnson-Laird, P. N. (1983) Mental models. Cambridge University Press.
Johnson-Laird, P. N. (1992) Syllogs (computer program). Available at:
Kemp, C. & Tenenbaum, J. B. (2008) The discovery of structural form. Proceedings of the National Academy of Sciences USA 105:10687–92.
Klauer, K. C. (1999) On the normative justification for information gain in Wason's selection task. Psychological Review 106:215–22.
Knill, D. & Richards, W., eds. (1996) Perception as Bayesian inference. Cambridge University Press.
Körding, K. P. & Wolpert, D. (2004) Bayesian integration in sensorimotor learning. Nature 427:244–47.
Kowalski, R. (1979) Algorithm = Logic + Control. Communications of the Association for Computing Machinery 22:424–36.
Krauth, J. (1982) Formulation and experimental verification of models in propositional reasoning. Quarterly Journal of Experimental Psychology 34:285–98.
Laplace, P. S. (1951) A philosophical essay on probabilities, trans. Truscott, F. W. & Emory, F. L.. Dover. (Original work published 1814).
Lauritzen, S. & Spiegelhalter, D. (1988) Local computations with probabilities on graphical structures and their application to expert systems. Journal of the Royal Statistical Society B 50:157224.
Leeuwenberg, E. & Boselie, F. (1988) Against the likelihood principle in visual form perception. Psychological Review 95:485–91.
Liu, I.-M. (2003) Conditional reasoning and conditionalization. Journal of Experimental Psychology: Learning, Memory, and Cognition 29:694709.
Liu, I. M., Lo, K. C. & Wu, J. T. (1996) A probabilistic interpretation of “If-Then”. The Quarterly Journal of Experimental Psychology 49A:828–44.
Ma, W. J., Beck, J., Latham, P. & Pouget, A. (2006) Bayesian inference with probabilistic population codes. Nature Neuroscience 9:1432–38.
MacKay, D. J. C. (2003) Information theory, inference, and learning algorithms. Cambridge University Press.
Manning, C. & Schütze, H. (1999) Foundations of statistical natural language processing. MIT Press.
Marr, D. (1982) Vision: A computational investigation into the human representation and processing of visual information. Freeman.
Martignon, L. & Blackmond-Laskey, K. (1999) Bayesian benchmarks for fast and frugal heuristics. In: Simple heuristics that make us smart, ed. Gigerenzer, G., Todd, P. M. & the ABC Research Group, pp. 169–88. Oxford University Press.
McKenzie, C. R. M., Ferreira, V. S., Mikkelsen, L. A., McDermott, K. J. & Skrable, R. P. (2001) Do conditional statements target rare events? Organizational Behavior and Human Decision Processes 85:291309.
McKenzie, C. R. M. & Mikkelsen, L. A. (2000) The psychological side of Hempel's paradox of confirmation. Psychonomic Bulletin and Review 7:360–66.
McKenzie, C. R. M. & Mikkelsen, L. A. (2007) A Bayesian view of covariation assessment. Cognitive Psychology 54:3361.
Milne, P. (1995) A Bayesian defence of Popperian science? Analysis 55:213–15.
Milne, P. (1996) log[P(h|eb)/P(h|b)] is the one true measure of confirmation. Philosophy of Science 63:2126.
Moore, G. E. (1903) Principia ethica. Cambridge University Press.
Navarro, D. J., Griffiths, T. L., Steyvers, M. & Lee, M. D. (2006) Modeling individual differences using Dirichlet processes. Journal of Mathematical Psychology 50:101–22.
Nelson, J. D. (2005) Finding useful questions: On Bayesian diagnosticity, probability, impact, and information gain. Psychological Review 112(4):979–99.
Oaksford, M. & Chater, N. (1991) Against logicist cognitive science. Mind and Language 6:138.
Oaksford, M. & Chater, N. (1994) A rational analysis of the selection task as optimal data selection. Psychological Review 101:608–31.
Oaksford, M. & Chater, N. eds. (1998b) Rational models of cognition. Oxford University Press.
Oaksford, M. & Chater, N. (2002) Common sense reasoning, logic and human rationality. In: Common sense, reasoning and rationality, ed. Elio, R., pp. 174214. Oxford University Press.
Oaksford, M. & Chater, N. (2007) Bayesian rationality: The probabilistic approach to human reasoning. Oxford University Press.
Oaksford, M. & Chater, N. (2008) Probability logic and the Modus Ponens–Modus Tollens asymmetry in conditional inference. In: The probabilistic mind: Prospects for Bayesian cognitive science, ed. Chater, N. & Oaksford, M., pp. 97120. Oxford University Press.
Oaksford, M. & Chater, N. (in press) Conditionals and constraint satisfaction: Reconciling mental models and probabilistic approaches? In: Cognition and conditionals: Probability and logic in human thought, ed. Oaksford, M. & Chater, N.. Oxford University Press.
Oaksford, M., Chater, N. & Grainger, B. (1999) Probabilistic effects in data selection. Thinking and Reasoning 5:193244.
Oaksford, M., Chater, N. & Larkin, J. (2000) Probabilities and polarity biases in conditional inference. Journal of Experimental Psychology: Learning, Memory and Cognition 26:883–89.
Oaksford, M. & Hahn, U. (2007) Induction, deduction and argument strength in human reasoning and argumentation. In: Inductive reasoning, ed. Feeney, A. & Heit, E.. pp. 269301. Cambridge University Press.
Oaksford, M. & Moussakowski, M. (2004) Negations and natural sampling in data selection: Ecological vs. heuristic explanations of matching bias. Memory and Cognition 32:570–81.
Oaksford, M. & Wakefield, M. (2003) Data selection and natural sampling: Probabilities do matter. Memory and Cognition 31:143–54.
Oberauer, K. (2006) Reasoning with conditionals: A test of formal models of four theories. Cognitive Psychology 53:238–83.
Oberauer, K., Weidenfeld, A. & Hörnig, R. (2004) Logical reasoning and probabilities: A comprehensive test of Oaksford and Chater (2001) Psychonomic Bulletin and Review 11:521–27.
Over, D. E., Hadjichristidis, C., Evans, J. St. B. T., Handley, S. J. & Sloman, S. A. (2007) The probability of causal conditionals. Cognitive Psychology 54:6297.
Pearl, J. (1988) Probabilistic reasoning in intelligent systems. Morgan Kaufmann.
Pearl, J. (2000) Causality: Models, reasoning and inference. Cambridge University Press.
Rao, R. P. N., Olshausen, B. A. & Lewicki, M. S., eds. (2002) Probabilistic models of the brain: Perception and neural function. MIT Press.
Restall, G. (1996) Information flow and relevant logics. In: Logic, language and computation, ed. Seligman, J. & Westerståhl, D., pp. 463–78 CSLI Publications.
Rips, L. J. (1994) The psychology of proof. MIT Press.
Rissanen, J. J. (1989) Stochastic complexity and statistical inquiry. World Scientific.
Shafer, G. (1976) A mathematical theory of evidence. Princeton University Press.
Shannon, C. E. & Weaver, W. (1949) The mathematical theory of communication. University of Illinois Press.
Sher, S. & McKenzie, C. R. M. (2006) Information leakage from logically equivalent frames. Cognition 101:467–94.
Sloman, S. A. (1996) The empirical case for two systems of reasoning. Psychological Bulletin 119:322.
Sloman, S. A. (2005) Causal models. Oxford University Press.
Sobel, D. M., Tenenbaum, J. B. & Gopnik, A. (2004) Children's causal inferences from indirect evidence: Backwards blocking and Bayesian reasoning in preschoolers. Cognitive Science 28:303–33.
Sobel, J. H. (2004) Probable modus ponens and modus tollens and updating on uncertain evidence. Unpublished manuscript, Department of Philosophy, University of Toronto, Scarborough. Available at:
Sober, E. (2002) Intelligent design and probability reasoning. International Journal for Philosophy of Religion 52:6580.
Stanovich, K. E. (2008) Individual differences in reasoning and the algorithmic/intentional level distinction in cognitive science. In: Reasoning: Studies of human inference and its foundations, ed. Rips, L. & Adler, J., pp. 414–36. Cambridge University Press.
Stanovich, K. E. & West, R. F. (2000) Individual differences in reasoning: Implications for the rationality debate? Behavioral and Brain Sciences 23:645–65.
Stevenson, R. J. & Over, D. E. (1995) Deduction from uncertain premises. The Quarterly Journal of Experimental Psychology 48A:613–43.
Stewart, N., Chater, N. & Brown, G. D. A. (2006) Decision by sampling. Cognitive Psychology 53:126.
Tenenbaum, J. B. (1999) A Bayesian framework for concept learning. Doctoral dissertation, Brain and Cognitive Sciences Department, MIT.
Tenenbaum, J. B. & Griffths, T. L. (2001) Structure learning in human causal induction. In: Advances in neural information processing systems, vol. 13, ed. Keen, T. K., Dietterich, T. G. & Tresp, V., pp. 5965. MIT Press.
Tenenbaum, J. B., Kemp, C. & Shafto, P. (2007) Theory based Bayesian models of inductive reasoning. In: Inducive reasoning, ed. Feeney, A. & Heit, E., pp. 167204. Oxford University Press.
Thaler, R. H. (2005) Advances in behavioral finance, Vol. II. Princeton University Press.
Toussaint, M., Harmeling, S. & Storkey, A. (2006) Probabilistic inference for solving (PO)MDPs. Technical Report EDI-INF-RR-0934, University of Edinburgh.
Vitányi, P. M. B. & Li, M. (2000) Minimum description length induction, Bayesianism, and Kolmogorov complexity. IEEE Transactions on Information Theory IT-46446–64.
von Helmholtz, H. (1910/1925) Physiological optics. Volume III. The theory of the perception of vision. Dover. (Translated from 3rd German edition, 1910).
Walton, D. N. (1989) Informal logic. Cambridge University Press.
Williamson, J. & Gabbay, D., eds. (2003) Special issue on combining logic and probability. Journal of Applied Logic 1:135308.
Zadeh, L. A. (1975) Fuzzy logic and approximate reasoning. Synthese 30:407–28.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed