Abstract
The thesis that probabilities of conditionals are conditional probabilities has putatively been refuted many times by so-called ‘triviality results’, although it has also enjoyed a number of resurrections. In this paper I assault it yet again with a new such result. I begin by motivating the thesis and discussing some of the philosophical ramifications of its fluctuating fortunes. I will canvas various reasons, old and new, why the thesis seems plausible, and why we should care about its fate. I will look at some objections to Lewis’s famous triviality results, and thus some reasons for the pursuit of further triviality results. I will generalize Lewis’s results in ways that meet the objections. I will conclude with some reflections on the demise of the thesis—or otherwise.
Similar content being viewed by others
Notes
Page references are to Harper et al. (1981) for all articles that are cited as appearing there.
I owe this point to Ned Hall.
I will speak interchangeably of ‘propositions’ and ‘sentences’, and I will use both set-theoretic operations and sentential connectives, in a way that I hope is perspicuous.
At the end of the paper I will briefly discuss ‘no truth value’ accounts of indicative conditionals, which seemingly evade the triviality results, suggesting that even these accounts may not evade them altogether after all.
One wonders what the conditional becomes when it appears in the scope of two or more probability functions—when, for example, P assigns probabilities to various hypotheses about another function P*’s assignments! (Shades here of the contextualist and relativist literature on reported speech and eavesdropping cases.).
I don't actually need the full strength of the assumption of boldness in the proof. All I need is that, for any non-trivial probability function P, and propositions A and B as described in the proof, the rule can revise P so as to give probability 1 to (A & B) − (A□ → □B), or so as to give probability one to (A & B). That is certainly weaker than boldness, but I don't know a neat way of characterizing it.
As I hope is obvious, P − Q is shorthand for P & ¬Q.
References
Adams E (1975) The logic of conditionals. Reidel, Dordrecht
Bacchus F, Kyburg HE, Thalos M (1990) Against conditionalization. Synthese 85:475–506
Bennett J (2003) Conditionals. Oxford University Press, Oxford
Cartwright N (1999) The dappled world: a study of the boundaries of science. Cambridge University Press, Cambridge
Collins J (1991) Belief revision. Ph.D. dissertation, Princeton University
de Finetti B (1972) Probability, induction and statistics. Wiley, New York
Edgington D (1995) On conditionals. Mind 104(414):235–329
Eells E, Skyrms B (eds) (1994) Probability and conditionals. Cambridge University Press, Cambridge
Gärdenfors P (1982) Imaging and conditionalization. Journal of Philosophy 79:747–760
Gibbard A, Harper W (1978) Counterfactuals and two kinds of expected utility. In: Hooker CA, Leach JJ, McClennen EF (eds) Foundations and applications of decision theory, vol 1. Reidel, Dordrecht
Hájek A (1989) Probabilities of conditionals—revisited. J Philos Logic 18:423–428
Hájek A (1994) Triviality on the cheap? In: E Eells, B Skyrms (eds)
Hájek A (forthcoming) The fall of Adams’ thesis? J Lang Logic Inf
Hájek A, Hall N (1994) The hypothesis of the conditional construal of conditional probability. In: E Eells, B Skyrms (eds)
Hall N (1994) Back in the (CCCP). In: E Eells, B Skyrms (eds)
Harper WL, Skyrms B (1988) Introduction to causation, chance, and credence. Kluwer, Dordrecht
Harper WL, Stalnaker R, Pearce G (eds) (1981) Ifs. Reidel, Dordrecht
Hild M (1998) The coherence argument against conditionalization. Synthese 115:229–258
Jaeger M (1995) “Minimum cross entropy reasoning” International joint conference on artificial intelligence. In: Proceedings of the 14th international joint conference on artificial intelligence, Vol 2, 1847–1852
Jaynes ET (2003) Probability theory: the logic of science. Cambridge University Press, Cambridge
Jeffrey R (1966) The logic of decision. University of Chicago Press, Chicago, 2nd edn, 1983
Joyce JM (1999) The foundations of causal decision theory. Cambridge University Press, Cambridge
Levi I (1980) The enterprise of knowledge. MIT Press, Cambridge, MA
Lewis D (1973) Counterfactuals. Blackwell and Harvard University Press
Lewis D (1974) Radical interpretation. Synthese 27: 331–344; reprinted in Philosophical Papers, Vol 1. Oxford University Press, New York and Oxford, 1983
Lewis D (1976) Probabilities of conditionals and conditional probabilities. Philos Rev 85: 297–315; reprinted in Harper et al.
Lewis D (1981) Causal decision theory. Aust J Philos 59: 5–30; reprinted in Lewis (1986a)
Lewis D (1986a) Probabilities of conditionals and conditional probabilities II. Philos Rev 95:581–589
Lewis D (1986b) Philosophical papers, vol II. Oxford University Press, Oxford
McGee Vann (1989) Conditional probabilities and compounds of conditionals. Philos Rev 98(4):485–541
Milne P (2003) The simplest Lewis-style triviality proof yet? Analysis 63.4(October): 300–303
Ramsey FP (1931/1990) Truth and probability. In: RB Braithwaite (ed) Foundations of mathematics and other essays. Routledge & P. Kegan, London; reprinted in DH Mellor (ed) (1990) Philosophical Papers. Cambridge University Press, Cambridge
Ramsey FP (1965) The foundations of mathematics (and other logical essays). Routledge and Kegan Paul, London
Rehder W (1982) Conditions for probabilities of conditionals to be conditional probabilities. Synthese 53:439–443
Savage LJ (1954) The foundations of statistics. Wiley, New York
Skyrms B (1980) Causal necessity. Yale University Press, New Haven
Skyrms B (1984) Pragmatics and empiricism. Yale University Press, New Haven
Sobel JH (1994) Taking chances: essays on rational choice. Cambridge University Press, New York
Stalnaker R (1968) A theory of conditionals. Studies in Logical Theory. American Philosophical Quarterly Monograph Series, No. 2. Blackwell, Oxford
Stalnaker R (1970) Probability and conditionals. Philos Sci 37: 64–80; reprinted in Harper et al.
Stalnaker R (1976) Letter to van Fraassen. In: Harper WL, Hooker CA (eds) Foundations of probability theory, statistical inference and statistical theories of science, vol I. Reidel, Dordrecht, pp 302–306
Titelbaum MG (2008) The relevance of self-locating beliefs. Philos Rev 117:555–606
van Fraassen B (1976) Probabilities of conditionals. In: Harper WL, Hooker CA (eds) Foundations of probability theory, statistical inference and statistical theories of science, vol I. Reidel, Dordrecht, pp 261–301
van Fraassen B (1984) Belief and the Will. J Philos 81:235–256
van Fraassen B (1989) Laws and symmetry. Clarendon Press, Oxford
Acknowledgments
I am indebted to Ned Hall, Richard Jeffrey, David Lewis, Bas van Fraassen, and Lyle Zynda for very helpful comments on a short, early precursor to this paper. I am grateful to participants in a reading group in Leuven for discussion of a resuscitated version of it—especially Jake Chandler, Igor Douven, and David Etlin. I have recently substantially revised and expanded it. For further valuable comments on this transmogrified version, I thank especially Rachael Briggs, John Cusbert, Daniel Greco, Aidan Lyon, Daniel Nolan, Wolfgang Schwarz, and Dan Singer. Thanks also to Brett Calcott and Ralph Miles for their help.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hájek, A. Triviality Pursuit. Topoi 30, 3–15 (2011). https://doi.org/10.1007/s11245-010-9083-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11245-010-9083-2