Abstract
In this essay we investigate how deductive and probabilistic explanations add value to the theory providing the explanation. We offer an analysis of how, e.g., an explanation in the form of an abduction differs from a mere prediction in this regard. We apply this analysis to respond to a challenge posed by Glymour (Theory and Evidence. Princeton University Press, Princeton, 1980) regarding Bayesian confirmation of new theories using old data. Last, we consider additional criteria involving subjunctive conditionals and counterfactual conditionals for distinguishing explanations from mere predictions, which help illuminate why an explanation carries different cognitive value than does a mere prediction.
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Notes
- 1.
We imagine that explanation might run as follows. Use the more general arithmetic law that the sum of the first k positive integers, 1 + 2 + … + k, equals k(k + 1)∕2 to show that the sum of the first k positive odd integers, \(\sum _{n=0}^{k-1} (2n + 1) \), equals \(2 \cdot [ \sum _{n=1}^{k-1} n ] + k = 2 [( k - 1) k/2 ] + k = k^2 \).
- 2.
Here, we do not address cases of Direct Inference: see Levi (1980, Chap. 12). In Direct Inference, the premise of the reasoning is a statement of a chance distribution over outcomes on a kind of trial, and the conclusion is a conditional credence distribution for outcomes on an instance of that kind of trial.
- 3.
See Roseveare (1982). See Jeffreys (1973, pp. 170–171) and Levenson (2015) for discussions of the historical account about the failure to defend Newton by speculating an unobserved planet, Vulcan, orbiting between Mercury and the Sun. The combination of Newton Theory and the Vulcan hypothesis provided an abductory explanation for the observed advance of Mercury’s perihelion. However, that theory also entailed failed predictions, at odds with GTR, that were refuted by evidence known by 1915.
- 4.
We use subscripts on the probability function to indicate a difference between the two theories. Where the theories assign the same probabilities, we avoid adding a subscript.
- 5.
See Duhem’s (1916/1953) well known criticism of “crucial experiments.”
- 6.
In Savage’s (1954) theory, all decision problems are of this first kind. In his decision theory, states and options are probabilistically independent.
- 7.
For some background on this controversy, see Levi (2007). For a contrary position to the one we endorse here, see Jeffrey (1965). In Jeffrey’s theory, there is no corresponding difference in the decision theoretic assessments for what here are called “options” and “states.” In Jeffrey’s theory each proposition is assigned both a probability and a utility.
- 8.
The assumption of random mating is not satisfied, for instance, in Mendel’s classic experiments with pea plants. Pea plants are self-fertilizing—pollen is not randomly scattered—which feature was essential for Mendel’s experimental design. Then, over successive generations, the two homozygous types are absorbing and the hybrid type is transient.
- 9.
Specifically, by the Law of Total Probability, expectations and conditional expectations for bounded random variables X and Y satisfy, E[X] = E[E[X|Y ]].
- 10.
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Gong, R., Kadane, J.B., Schervish, M.J., Seidenfeld, T., Stern, R.B. (2022). The Value Provided by a Scientific Explanation. In: Augustin, T., Cozman, F.G., Wheeler, G. (eds) Reflections on the Foundations of Probability and Statistics. Theory and Decision Library A:, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-031-15436-2_2
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