Abstract
Second-order uncertainty, also known as model uncertainty and Knightian uncertainty, arises when decision-makers can (partly) model the parameters of their decision problems. It is widely believed that subjective probability, and more generally Bayesian theory, are ill-suited to represent a number of interesting second-order uncertainty features, especially “ignorance” and “ambiguity”. This failure is sometimes taken as an argument for the rejection of the whole Bayesian approach, triggering a Bayes versus anti-Bayes debate which is in many ways analogous to what the classical versus non-classical debate used to be in logic. This paper attempts to unfold this analogy and suggests that the development of non-standard logics offers very useful lessons on the contextualisation of justified norms of rationality. By putting those lessons to work I will flesh out an epistemological framework suitable for extending the expressive power of standard Bayesian norms of rationality to second-order uncertainty in a way which is both formally and foundationally conservative.
To appear in Hansson, Sven Ove, (ed.) David Makinson on classical methods for non-classical problems. Trends in logic. Springer.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For a recent appraisal of the distinction see Gelman (2011).
- 2.
The so-called Ellsberg paradox, of which the coin tossing problem is the simplest example, appears as far back as in Keynes (1921) and Knight (1921). This is often acknowledged in the literature by referring to probabilistically unquantifiable belief as “Knightian uncertainty”. The synonym “ambiguity” is due to Ellsberg (1961). Much of the recent revival of the interest in “Knightian decision theory” owes to Bewley (2002).
- 3.
Roughly speaking, what is objected is that a rational agent must assign probability 1 to any tautology of the form \(\theta \vee \lnot \theta \), even when the agent knows nothing about \(\theta \).
- 4.
See e.g. Heyting (1956).
- 5.
As the content of this section is purely heuristic, I will not burden the reading with otherwise unnecessary definitions. On the general questions of providing rigorous characterisations of logical systems and context of reasoning, Gabbay (1995) is a slightly dated yet still very valuable reference.
- 6.
The restriction to single sentences is clearly immaterial here.
- 7.
- 8.
No doubt some meta-mathematical properties are lost in this extension! Supraclassical consequence relations, for instance, need not be closed under substitution (Makinson 2005).
- 9.
Daston (1988) emphasises how the virtually unanimous agreement on the “expectations of reasonable men”—one of the trademark of the Enlightment—played an important role in the construction of the theory of probability. The objective Bayesian approach of Paris (1994) takes for granted that our intuitions about rationality are largely intersubjective, an idea which is further developed in Hosni and Paris (2005).
- 10.
A set of sentences is consistent if it is not inconsistent.
- 11.
- 12.
- 13.
The first representation theorem to this effect goes all the way back to 1933 when Gödel introduced what we now call the Gödel translation and then proved that the algebra of open elements of every modal algebra for S4 is a Heyting algebra and, conversely, every Heyting algebra is isomorphic to the algebra of open elements of a suitable algebra for S4 (see, e.g. Blackburn et al. 2007especially Sect. 7.9).
- 14.
Arieli and Zamansky (2009) illustrate the applicability of non-deterministic matrices in modelling non-deterministic structures, where uncertainty is taken to be an intrinsic feature of the world, in addition to being a subjective epistemic state of the reasoning agents. Adams (2005) gives a feel for the taboo of abandoning compositionality in connection with Lewis’s Triviality results.
- 15.
This process is mostly, but not necessarily, application-driven. As Banach is often reported to have said, “Good mathematicians see analogies between theorems; great mathematicians see analogies between analogies” (as quoted by Jaynes who quotes from Ulam).
- 16.
This is precisely what the title of Gilboa et al. (2011) provocatively recommends.
- 17.
A familiar instantiation of this Norm is Lewis’s Principal Principle.
- 18.
- 19.
See Chap. 3 of Williamson (2010) for a detailed analysis of such justifications.
- 20.
Alternative denominations might have included “Dominance”, “Pareto” or even of course “Admissibility”. As they all have rather specific connotations in distinct areas of the uncertain reasoning literature, from statistics to decision and game theory to social choice theory, it appears that “Choice Norm” sits more comfortably at the desired level of generality whilst avoiding potential confusion.
- 21.
- 22.
In this respect, the concept of rationality seems to be analogous with that of democracy: there is usually more disagreement on what democracy should be than on what counts as a violation of a democratic society.
- 23.
I am adapting the terminology from Bossert and Suzumura (2010) who use it in connection to their notion of Suzumura consistency, arguably the weakest requirement in the formal theory of rational choice.
- 24.
This is one way of interpreting the axiom of the Independence of irrelevant alternatives, according to which a dominated alternative should not be chosen from any superset of the original set of feasible alternatives Sen (1970).
- 25.
The consensus on granting logic a normative status is far from being unanimous, and indeed the question might turn out to be ill-posed. A discussion of this point would take us too far, but the interested reader might wish to consult, among others (Gabbay and Woods 2003; van Benthem 2008; Wheeler 2008).
- 26.
As a well-known story goes, Savage was initially tricked into Allais’s “paradox”, but once Allais pointed that out, Savage acknowledged his mistake and corrected his answer accordingly. A similar reaction to the descriptive failures of Bayesian theory is condensed in the one-page paper (de Finetti 1979).
- 27.
Gabbay and Woods (2005) go some way towards developing the idea of a hierarchy of agents based on a similar relation.
- 28.
This need not pertain only to applied mathematics. In a number of widely known mathematical expositions George Polyá insists that abstraction is intrinsic to mathematical reasoning for the solution to hard mathematical problems is often best achieved by solving simpler problems from which the general idea can be extrapolated.
- 29.
This assumption is clearly related to Carnap’s “Principle of Total Evidence” and Keynes’ “Bernoulli’s maxim”. The fact that experimental subject systematically violate this principle motivates the introduction of the “editing phase” in Prospect Theory.
- 30.
Here \(Bel()\) denotes the expert’s belief function, \(\theta \) is a sentence and \(K\) is a finite set of expressions of the form \(Bel(\theta _i)= \beta _i\), where all the \(\beta _i\in [0,1]\).
- 31.
Which in turn is a variant of a problem known to both Keynes and Knight before it was revamped by Ellsberg.
- 32.
Since this is one of the most intensely studied aspect of Bayesian epistemology, I will take many details for granted and focus on the specific aspects which are directly relevant to the present discussion. Readers who are not familiar with the argument are urged to consult the original (de Finetti 1931, 1974). Paris (2001) offers a very general proof whilst Williamson (2010) provides ample background.
- 33.
The term “consistency” is also frequently used in English translations.
- 34.
In economics and political science a contract is said to be complete if it contemplates all possible contingencies. Despite being blatantly unrealistic, it is a widely used assumption in those areas (see, e.g. Tirole 1999).
- 35.
The question as to how suitable the betting problem is as an elicitation device is raised by de Finetti in his later work on proper scoring rules and especially Brier’s (see, especially de Finetti 1962, 1969, 1972). I will postpone the analysis of admissibility as “miniminum expected loss under Brier’s score” to further research.
- 36.
Binmore et al. (2012) report on a recent experiment which casts substantial doubts on the alleged universality of the “ambiguity aversion” phenomenon in Ellsberg-type problems.
- 37.
The term appears to have been coined by Willem Buiter of Citigroup, in his 6th February 2012 report.
- 38.
- 39.
Inadmissibility is referred to as the bad bet criterion in the terminology of (Fedel et al. 2011).
- 40.
See (de Finetti (1975), Appendix 19.3).
- 41.
Suffice it to mention that much of the popularity enjoyed by non-monotonic logics during the 1980s was more or less directly linked to the idea that logic would better serve the (computational) needs of artificial intellingence than probability.
References
Adams, E. W. (2005). What is at stake in the controversy over conditionals. In G. Kern-Isberner, W. Roedder, & F. Kulman (Eds.), WCII 2002 (pp. 1–11)., Lecture Notes in Artificial Intelligence. New York: Springer.
Arieli, O., & Zamansky, A. (2009). Distance-based non-deterministic semantics for reasoning with uncertainty. Logic Journal of IGPL, 17(4), 325–350.
Barwise, J., & Keisler, H. J. (Eds.). (1977). Handbook of mathematical logic. Amsterdam: North-Holland.
Bewley, T. F. (2002). Knightian decision theory, part I (1986). Decisions in Economics and Finance, 25, 79–110.
Binmore, K., Stewart, L., & Voorhoeve, A. (2012). An experiment on the Ellsberg paradox. Working paper.
Blackburn, P., van Benthem, J., & Wolter, F. (2007). Handbook of modal logic. Amsterdam: Elsevier.
Bossert, W., & Suzumura, K. (2010). Consistency, choice, and rationality. Cambridge: Harvard University Press.
Bossert, W., & Suzumura, K. (2012). Revealed preference and choice under uncertainty. SERIEs, 3(1), 247–258.
Carnielli, W., Coniglio, M. E., & Marcos, J. (2007). Logics of formal inconsistency. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 14, pp. 1–93). New York: Springer.
Colyvan, M. (June 2008). Is probability the only coherent approach to uncertainty? Risk Analysis, 28(3), 645–652.
Daston, L. (1988). Classical probability in the enlightenment. Princeton: Princeton University Press.
de Finetti, B. (1931). Sul significato soggettivo della probabilità. Fundamenta Mathematicae, 17, 289–329.
de Finetti, B. (1962). Does it make sense to speak of good probability appraisers? In I. J. Good (Ed.), The scientist speculates (pp. 357–364). Londan: Heinemann.
de Finetti, B. (1969). Un matematico e L’economia. Milano: Franco Angeli.
de Finetti, B. (1972). Probability, induction and statistics. New York: Wiley.
de Finetti, B. (1973). Bayesianism: Its unifying role for both the foundations and the applications of statistics. Bulletin of the International Statistical Institute, 39, 349–368.
de Finetti, B. (1974). Theory of probability (Vol. 1). New York: Wiley.
de Finetti, B. (1975). Theory of probability (Vol. 2). New York: Wiley.
de Finetti, B. (1979). A short confirmation of my standpoint. In M. Allais & O. Hagen (Eds.), Expected utility hypotheses and the allais paradox (p. 161). Dordrecht: D. Reidel.
de Finetti, B. (2008). Philosophical lectures on probability. New York: Springer.
Ellsberg, D. (1961). Risk, ambiguity and the savage axioms. The Quarterly Journal of Economics, 75(4), 643–669.
Fedel, M., Hosni, H., & Montagna, F. (2011). A logical characterization of coherence for imprecise probabilities. International Journal of Approximate Reasoning, 52, 1147–1170.
Gabbay, D. M. (Ed.) (1995). What is a logical system?. Oxford: Oxford University Press.
Gabbay, D. M., & Hunter, A. (1991). Making inconsistency respectable 1 : A logical framework for inconsistency in reasoning. In Ph. Jorrand & J. Keleman (Eds.), Fundamentals of artificial intelligence research, (pp. 19–32). Lecture notes in Computer Science, Berlin: Springer.
Gabbay, D. M., & Woods, J. (Eds.). (2007). Handbook of the history of logic: The many valued and non monotonic turn in logic, (Vol. 8). Amsterdam: North-Holland.
Gabbay, D. M., & Woods, J. (2003). Normative models of rational agency: The theoretical disutility of certain approaches. Logic Journal of IGPL, 11(6), 597–613.
Gabbay, D. M., & Woods, J. (2005). The practical turn in logic. In D. M. Gabbay & F. Guenthner (Eds.), Handbook of philosophical logic (2nd ed., Vol. 13, pp. 15–122). Netherland: Springer.
Gelman, A. (2011). Induction and deduction in Bayesian data analysis. Rationality, Markets and Morals, 2, 67–78.
Gigerenzer, G., Todd, P. M., & ABC Research Group. (1999). Simple heuristics that make us smart. Oxford: Oxford University Press.
Gigerenzer, G., Todd, P. M., & ABC Research Group. (2012). Ecological rationality: Intelligence in the world. Oxford: Oxford University Press.
Gigerenzer, G. (2008). Rationality for mortals: How people cope with uncertainty. Oxford: Oxford University Press.
Gilboa, I., Postlewaite, A., & Schmeidler, D. (2011). Rationality of belief or: Why Savage’s axioms are neither necessary nor sufficient for rationality. Synthese, 187, 11–31.
Gilboa, I. (2009). Theory of decision under uncertainty. New York: Cambridge University Press.
Good, I. J. (1983). Good thinking: The foundations of probability and its applications. Minneapolis: The University of Minnesota Press.
Haenni, R., Romeijn, J. W., & Wheeler, G. (2011). Probabilistic logics and probabilistic networks. Dordrecht: Springer.
Hansson, S. O. (2009). Measuring uncertainty. Studia Logica, 93(1), 21–40.
Hasbrouck, J. (2007). Empirical market microstructure: The institutions, economics and econometrics of securities trading. Oxford: Oxford University Press.
Heyting, A. (1956). Intuitionism: An introduction. Amsterdam: North-Holland.
Hosni, H., & Paris, J. B. (2005). Rationality as conformity. Synthese, 144(2), 249–285.
Howson, C. (2009). Can logic be combined with probability? Probably. Journal of Applied Logic, 7(2), 177–187.
Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge: Cambridge University Press.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263–291.
Keynes, J. M. (1921). Treatease on probability. New York: Harper & Row.
Knight, F. H. (1921). Risk, uncertainty and profit. New York: Beard Books Inc.
Levi, I. (1986). The paradoxes of Allais and Ellsberg. Economics and Philosophy, 2, 23–53.
Makinson, D. (1965). The paradox of the preface. Analysis, 25(6), 205.
Makinson, D. (1994). General patterns in nonmonotonic reasoning. In J. Robinson (Ed.), Handbook of logic in artificial intelligence and logic programming (Vol. 3, pp. 35–110). Oxford: Oxford University Press.
Makinson, D. (2005). Bridges from classical to nonmonotonic logic. London: College Publications.
Makinson, D. (2012). Logical questions behind the lottery and preface paradoxes: Lossy rules for uncertain inference. Synthese, 186, 511–529.
Miranda, E. (2008). A survey of the theory of coherent lower previsions. International Journal of Approximate Reasoning, 48(2), 628–658.
Moss, L. S. (2005). Applied logic: A manifesto. In D. M. Gabbay, S. Gonchargov, & M. Zakharyaschev (Eds.), Mathematical problems from applied logic I (pp. 317–343). Netherland: Springer.
Paris, J.B. (2001). A Note on the Dutch Book Method. In T. Seidenfeld, G. De Cooman & T. Fine (Eds.), ISIPTA ’01, Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications, Ithaca, NY, USA. Shaler.
Paris, J. B., & Vencovská, A. (1990). A note on the inevitability of maximum entropy. International Journal of Approximate Reasoning, 4(3), 183–223.
Paris, J. B. (1994). The uncertain reasoner’s companion: A mathematical perspective. Cambridge: Cambridge University Press.
Paris, J. B. (1998). Common sense and maximum entropy. Synthese, 117(1), 75–93.
Paris, J. B. (2004). Deriving information from inconsistent knowledge bases: A completeness theorem. Logic Journal of IGPL, 12, 345–353.
Rubinstein, A. (2006). Lecture notes in microeconomic theory. Princeton: Princeton University Press.
Rukhin, A. L. (1995). Admissibility: Survey of a concept in progress. International Statistical Review, 63(1), 95–115.
Savage, L. J. (1972). The foundations of statistics (2nd ed.). New York: Dover.
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3), 571–587.
Sen, A. K. (1970). Collective choice and, social welfare. San Francisco: Holden Day.
Shafer, G. (1986). Savage revisited. Statistical Science, 1(4), 463–485.
Tirole, J. (1999). Incomplete contracts: Where do we stand? Econometrica, 67(4), 741–781.
van Benthem, J. (2008). Logic and reasoning: Do the facts matter? Studia Logica, 88, 67–84.
van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic epistemic logic. New York: Springer.
Wakker, P. P. (2010). Prospect theory for risk and ambiguity. Cambridge: Cambridge University Press.
Walley, P. (1991). Statistical reasoning with imprecise probabilities. New York: Wiley.
Wheeler, G. (2008). Applied logic without psychologism. Studia Logica, 88(1), 137–156.
Williamson, J. (2010). In defence of objective bayesianism. Oxford: Oxford University Press.
Acknowledgments
I have presented the main ideas of this paper at Kent’s Centre for Reasoning and at the LSE Choice Group in London. I would like to thank Jon Williamson and Richard Bradley for inviting me to speak at those seminars and both audiences for their very valuable feedback. I am very grateful to Gregory Wheeler for his comments on an earlier draft and for many stimulating discussions on the topics covered in this paper. Thanks also to two referees, whose thorough reviews helped me to improve the chapter in many ways. Finally, readers familiar with David Makinson’s work will certainly have spotted a number of terms and expressions which are easily associated with, and sometimes directly coming from, David’s papers and books. I realised this only when the first draft of this chapter was completed. As I started fetching all the originals to give credit where credit was due, it occurred to me that leaving those paraphrases uncredited would probably be the most direct way to express how influential David’s way of doing logic is to my way of thinking about logic. So I stopped.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Hosni, H. (2014). Towards a Bayesian Theory of Second-Order Uncertainty: Lessons from Non-Standard Logics. In: Hansson, S. (eds) David Makinson on Classical Methods for Non-Classical Problems. Outstanding Contributions to Logic, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7759-0_11
Download citation
DOI: https://doi.org/10.1007/978-94-007-7759-0_11
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-7758-3
Online ISBN: 978-94-007-7759-0
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)