Abstract
It is truism that accuracy is valued. Some deem accuracy to be among the most fundamental values, perhaps the preeminent value, of inquiry. Because of this, accuracy has been the focus of two different, important programs in epistemology. The truthlikeness program (instigated by Popper) pursued the notion of propositional accuracy—an ordering of propositions by closeness to the objective truth of some matter. The epistemic utility program (instigated by Hempel) pursued the notion of credal state accuracy—an ordering of credal states by closeness to the ideal credal state. It may be tempting to think that each question that informs an inquiry must be determinate of necessity. That is to say, there must be one correct, precise answer to it. That precise, correct answer is not only the maximally accurate answer to the question, but it also singles out the ideal credal state that is the epistemic target of the inquiry. An enormous amount of the effort has been focused on accuracy with respect to necessarily determinate questions. But often the focus of an inquiry is a question that is not necessarily determinate. Those who deem accuracy to be cognitively valuable, and fundamentally so, need an account of accuracy for indeterminate questions. Ideally such an account would cover both propositional accuracy and credal state accuracy. This is a tall order, and it is not clear that the order can be filled.
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Notes
Tichý (1978).
For the intension account see Tichý (1978). It is more fine-grained than the partition account but it is still too coarse-grained, ignoring as it does the hyperintensional dimension. The question Is the sum of seven and five twelve is clearly different from the question is Fermat's last theorem true despite inducing the same (trivial) intension.
The abbreviation [X = x] is fairly standard though a bit misleading. X is known as a "random variable" in probability theory (although it is not a variable and need not be random). Let \({X}_{w}\) be the result of applying the function \(X\) to arbitrary state. w [X = x] is really short for the λ-abstraction of \({[X}_{w}=x]\) w.r.t. variable \(w\): i.e. \(\lambda w[X_{w} = x\)].
More needs to be said about partial questions: the case where \(X\) is a partial function.
This usage is not universal and I take it is as stipulative here. Causal indeterminism is a source of indeterminacy of questions, and the main source to consider. Vagueness is another source of indeterminacy of questions, but one I won't explore. See Wilson (2016) for a defense of metaphysical indeterminacy, and Calosi and Wilson (2019) for a discussion of quantum indeterminacy.
I assume we have settled on a precisifications of the concepts involved. For \(\mathbf{E}\), choose a concept of living such that at each moment it is determinate whether or not a person is alive or not at that moment.
It is narrower because on the intensional account the answers to questions can be of any logical type, not just propositions.
Beginning with Kuipers (1987).
Kuipers (1992).
See Oddie (2013) for some intuitively plausible constraints of this sort.
When one is trying to determine the cost of a move one generally takes the lowest quote!
EMD has been applied to a wide range of phenomena, perhaps most interesting in this context being the measurement of similarity of images. See Wang and Guibas (2012).
We can extend the account to continuous magnitudes but there will be the usual problems with denumerably infinite sets, over which one cannot have even credal distributions without violating countable additivity.
These could be worlds, or kinds, or more generally precise answers to a question. In the likeness program we start with kinds, or attributive constituents—the simplest examples of which are Carnap's Q-predicates. Suppose we have distances between such kinds. Constituents are collections of attributive constituents. Given the distributive normal form theorem, every first-order sentence is equivalent to a disjunction (collection) of constituents. Hence each expressible proposition can be identified with the class of consistent constituents in its normal form. From here distances between expressible propositions can be derived from distances between constituents. See Niiniluoto (1987).
Oddie (1986, chapter 4). In the terminology of that work, I am going straight to absolutely fair linkages here. Niiniluoto also demonstrated that an intermediate fairness proposal also skewed the distances.
See Pettigrew (2016).
There is a more general result for credal states corresponding to imprecise incorrect answers: that a decrease in strength is invariably an increase in accuracy. This is the value of the lack of content for falsehoods. This avoids an objection to Popper's theory (the incomparability of falsehoods) but it lacks the sting of a corresponding objection to the symmetric difference measure (the value content for falsehoods).
For various ways of tinkering with the Brier score by weighting certain kinds of propositions see Oddie (2019) and Schoenfield (2020). Dunn (2019) suggests weighting atomic propositions (something already suggested and criticized in Oddie 2019). Schoenfield (2020) contains an extended and nuanced discussion of weighting convex propositions. As noted in Oddie (1987, 2019) there is in fact a close connection between convexity and atomicity so the two remedies are connected. For the relation of convexity to the notion of a natural proposition or genuine property see Gärdenfors (2000), Oddie (1987) and Oddie (2005, chapter 6).
Proximity does not generalize in any obvious way to the indeterminate case. Suppose that the indeterminate truth A is \(\{1,6\}\) and \(B\) is \(\{2,5\}\). The distance of B from A is less than that of either {2} or {5} since it is less costly to shift from B to A than to shift from either {2} or {5} to A. A more plausible candidate for Proximity might be this. Let \({Closest}_{A}(C)\) be class of subsets of B that have the same cardinality as A. Then any \(C\in\) \(\textit{Closest}_{A}(B)\) is no more inaccurate than is \(B\) itself.
There are some interesting exceptions to this but it is a good rule of thumb and the exceptions need not detain us.
I myself endorsed this in Oddie (1997).
Schoenfield (2020).
See Oddie (2013) for an extended discussion of this.
I owe special thanks to the editors and two anonymous referees who made extensive and detailed suggestions which greatly improved the paper.
References
Belnap, N. D., & Steel, T. B. (Eds.). (1976). The logic of questions and answers. Yale University Press.
Calosi, C., & Wilson, J. (2019). Quantum Metaphysical Indeterminacy. Philosophical Studies, 176(10), 2599–2627.
Dunn, J. (2019). Accuracy, verisimilitude, and scoring rules. Australasian Journal of Philosophy, 97(1), 151–166.
Gärdenfors, P. (2000). Conceptual spaces: the geometry of thought. MIT Press.
Groenendijk, J., & Stokhof, M. (1997). Questions. In J. van Benthem & A. ter Meulen (Eds.), Handbook of logic and language (pp. 1055–1124). Elsevier.
Hájek, A. l. (unpublished). A puzzle about degree of belief. Unpublished manuscript, Australian National University. http://fitelson.org/coherence/hajek_puzzle.pdf.
Kuipers, T. (1987). A structuralist approach to truthlikeness. In T. Kuipers (Ed.), What is closer-to-the-truth? (pp. 79–99). Rodopi.
Kuipers, T. (1992). Naive and refined truth approximation. Synthese, 93, 299–341.
Kuipers, T. A. F., & Wiśniewski, A. (1994). An erotetic approach to explanation by specification. Erkenntnis, 40(3), 377–402.
Miller, D. (1974). Popper’s qualitative theory of verisimilitude. The British Journal for the Philosophy of Science, 25, 166–177.
Miller, D. (1977). On distance from the truth as a true distance. Bulletin of the Section of Logic, 6(1), 15–23.
Niiniluoto, I. (1987). Truthlikeness. Reidel.
Oddie, G. (1986). Likeness to Truth, Western Ontario Series in Philosophy of Science. Reidel.
Oddie, G. (1987). Truthlikeness and the convexity of propositions. In T. Kuipers (Ed.), What is closer-to-the-truth? (pp. 197–217). Rodopi.
Oddie, G. (1997). Conditionalization, cogency, and cognitive value. British Journal for the Philosophy of Science, 48(4), 533–541.
Oddie, G. (2005). Value, reality and desire. Oxford University Press.
Oddie, G. (2013). The content, consequence and likeness approaches to verisimilitude: compatibility, trivialization, and underdetermination. Synthese, 190(9), 1647–1687.
Oddie, G. (2019). What accuracy could not be. British Journal for the Philosophy of Science, 70(2), 551–580.
Pettigrew, R. (2012). Accuracy, chance and the principal principle. Philosophical Review, 121(2), 241–275.
Pettigrew, R. (2016). Accuracy and the laws of credence. Oxford University Press.
Schoenfield, M. (2020). Accuracy and verisimilitude: The good, the bad, and the ugly. The British Journal for the Philosophy of Science (forthcoming, published online https://doi.org/10.1093/bjps/axz032).
Tichý, P. (1974). On Popper’s definitions of verisimilitude. The British Journal for the Philosophy of Science, 25, 155–160.
Tichý, P. (1976). Verisimilitude redefined. British Journal for the Philosophy of Science, 27(1), 25–42.
Tichý, P. (1978). Questions, answers, and logic. American Philosophical Quarterly, 15(4), 275–284.
Wang, F., & Guibas, L. J., et al. (2012). Supervised earth mover’s distance learning and its computer vision applications. In S. Lazebnik (Ed.), Computer Vision ECCV 2012 (pp. 442–455). Springer.
Wilson, J. M. (2016). Are there indeterminate states of affairs? Yes. In E. Barnes (Ed.), Current controversies in metaphysics.Taylor & Francis.
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This article belongs to the topical collection “Approaching Probabilistic Truths”, edited by Theo Kuipers, Ilkka Niiniluoto, and Gustavo Cevolani.
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Oddie, G. Propositional and credal accuracy in an indeterministic world. Synthese 199, 9391–9410 (2021). https://doi.org/10.1007/s11229-021-03207-3
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DOI: https://doi.org/10.1007/s11229-021-03207-3