Vague credence.

*(English)*Zbl 1417.03085Summary: It is natural to think of precise probabilities as being special cases of imprecise probabilities, the special case being when one’s lower and upper probabilities are equal. I argue, however, that it is better to think of the two models as representing two different aspects of our credences, which are often (if not always) vague to some degree. I show that by combining the two models into one model, and understanding that model as a model of vague credence, a natural interpretation arises that suggests a hypothesis concerning how we can improve the accuracy of aggregate credences. I present empirical results in support of this hypothesis. I also discuss how this modeling interpretation of imprecise probabilities bears upon a philosophical objection that has been raised against them, the so-called inductive learning problem.

##### MSC:

03A05 | Philosophical and critical aspects of logic and foundations |

03B52 | Fuzzy logic; logic of vagueness |

03B42 | Logics of knowledge and belief (including belief change) |

##### Keywords:

vagueness; imprecise; indeterminate; credence; subjective probability; degree of belief; aggregation
Full Text:
DOI

##### References:

[1] | Armstrong, S; Collopy, F, Error measures for generalizing about forecasting methods: empirical comparisons, International Journal of Forecasting, 8, 69-80, (1992) |

[2] | Berinsky, AJ; Huber, GA; Lenz, GS, Evaluating online labor markets for experimental research: amazon.com’s mechanical turk, Political Analysis, 20, 351-368, (2012) |

[3] | Bradley, S. (2014). Imprecise probabilities. Stanford Encyclopedia of Philosophy. Stanford: Stanford University. |

[4] | Bradley, S; Steele, K, Uncertainty, learning, and the “problem” of dilation, Erkenntnis, 79, 1287-1303, (2012) · Zbl 1329.03035 |

[5] | Chandler, J, Subjective probabilities need not be sharp, Erkenntnis, 79, 1273-1286, (2014) · Zbl 1329.03036 |

[6] | Christensen, D. (2004). Putting logic in its place. Oxford: Oxford University Press. |

[7] | Dallmann, JM, A normatively adequate credal reductivism, Synthese, 191, 2301-2313, (2014) · Zbl 1318.03008 |

[8] | Dardashti, R., Glynn, L., Thébault, K., & Frisch, M. (2014). Unsharp humean chances in statistical physics: A reply to beisbart. In M. C. Galavotti et al. (Eds.), New directions in the philosophy of science (pp. 531-542). Dordrecht: Springer. |

[9] | Elga, A, Subjective probabilities should be sharp, Philosophers’ Imprint, 10, 1-11, (2010) |

[10] | Eriksson, L; Hájek, A, What are degrees of belief?, Studia Logica, 86, 183-213, (2007) · Zbl 1124.03303 |

[11] | Gärdenfors, P; Sahlin, N-E, Unreliable probabilities, risk taking, and decision making, Synthese, 53, 361-386, (1982) · Zbl 0516.62011 |

[12] | Gudder, S, What is fuzzy probability theory?, Foundations of Physics, 30, 1663-1678, (2000) |

[13] | Hájek, A, Objecting vaguely to pascal’s wager, Philosophical Studies, 98, 1-14, (2000) |

[14] | Hájek, A, What conditional probability could not be, Synthese, 137, 273-323, (2003) · Zbl 1047.03003 |

[15] | Hájek, A; Smithson, M, Rationality and indeterminate probabilities, Synthese, 187, 33-48, (2012) · Zbl 1275.91023 |

[16] | Jeffrey, R, Bayesianism with a human face, Testing Scientific Theories, Minnesota Studies in the Philosophy of Science, 10, 133-156, (1983) |

[17] | Jeffrey, R, Indefinite probability judgment: A reply to Levi, Philosophy of Science, 54, 586-591, (1987) |

[18] | Joyce, JM, How probabilities reflect evidence, Philosophical Perspectives, 19, 153-178, (2005) |

[19] | Joyce, JM, A defense of imprecise credences in inference and decision making, Philosophical Perspectives, 24, 281-323, (2010) |

[20] | Kaplan, M, Decision theory as philosophy, Philosophy of Science, 50, 549-577, (1983) · Zbl 1223.91020 |

[21] | Kaplan, M. (1996). Decision theory as philosophy. Cambridge: Cambridge University Press. · Zbl 0885.62004 |

[22] | Keynes, J. M. (1921). A treatise on probability. London: Macmillan. |

[23] | Koopman, BO, The bases of probability, Bulletin of the American Mathematical Society, 46, 763-774, (1940) · Zbl 0024.05002 |

[24] | Kyburg, H. (1983). Epistemology and Inference. Minneapolis: University of Minnesota Press. |

[25] | Kyburg, H. E, Jr. (1961). Probability and the logic of rational belief. Middletown: Wesleyan University Press. |

[26] | Kyburg, H. E, Jr., & Pittarelli, M. (1996). Set-based bayesianism. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 26(3), 324-339. |

[27] | Levi, I, On indeterminate probabilities, Journal of Philosophy, 71, 391-418, (1974) |

[28] | Levi, I. (1980). The enterprise of knowledge: an essay on knowledge, credal probability, and chance. Cambridge, MA: MIT Press. |

[29] | Levi, I, Imprecision and indeterminacy in probability judgment, Philosophy of Science, 52, 390-409, (1985) |

[30] | Levi, I, Imprecise and indeterminate probabilities, Risk, Decision and Policy, 5, 111-122, (2000) |

[31] | Levi, I, Why indeterminate probability is rational, Journal of Applied Logic, 7, 364-376, (2009) · Zbl 1191.03017 |

[32] | Maher, P. (2006). Book review: David Christensen. Putting logic in its place: Formal constraints on rational belief. Notre Dame Journal of Formal Logic, 47(1), 133-149. |

[33] | Milgram, E. (2009). Hard truths. Malden: Wiley. |

[34] | Moss, S. (2014). Credal dilemmas. Noûs. doi:10.1111/nous.12073. · Zbl 1275.91023 |

[35] | Paolacci, G; Chandler, J; Ipeirotis, P, Running experiments on amazon mechanical turk, Judgment and Decision Making, 5, 411-419, (2010) |

[36] | Pedersen, P; Wheeler, G, Demystifying dilation, Erkenntnis, 79, 1305-1342, (2014) · Zbl 1329.03038 |

[37] | Rinard, S, Against radical credal imprecision, Thought: A Journal of Philosophy, 2, 157-165, (2013) |

[38] | Seidenfeld, T; Schervish, MJ; Kadane, JB, Forecasting with imprecise probabilities, International Journal of Approximate Reasoning, 53, 1248-1261, (2012) · Zbl 1284.60082 |

[39] | Seidenfeld, T; Wasserman, L, Dilation for sets of probabilities, The Annals of Statistics, 21, 1139-1154, (1993) · Zbl 0796.62005 |

[40] | Singer, DJ, Sleeping beauty should be imprecise, Synthese, 191, 3159-3172, (2014) · Zbl 1307.03006 |

[41] | Sturgeon, S, Reason and the grain of belief, Noûs, 42, 139-165, (2008) |

[42] | Van Fraassen, B. C. (1990). Figures in a probability landscape, chapter 21. In Truth or consequences (pp. 345-356). Dordrecht: Kluwer. |

[43] | Fraassen, BC, Vague expectation value loss, Philosophical Studies, 127, 483-491, (2006) |

[44] | Walley, P. (1991). Statistical reasoning with imprecise probabilities. London: Chapman Hall. · Zbl 0732.62004 |

[45] | Wheeler, G; Epistemology, Context (ed.), Character matching and the locke pocket of belief, 187-195, (2014), cham |

[46] | White, R. (2010). Evidential symmetry and mushy credence. In J. Hawthorne (Eds.), Oxford studies in epistemology. Oxford: Oxford University Press.. |

[47] | Williamson, T. (1994). Vagueness. London: Routledge. |

[48] | Zadeh, LA, Probability measures of fuzzy events, Journal of mathematical analysis and applications, 23, 421-427, (1968) · Zbl 0174.49002 |

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