Abstract
This paper deals with inference guided by explanatory considerations –specifically with the prospects for a probabilistic interpretation of it. After pointing out some differences between two sorts of explanatory reasoning – i.e.: abduction and “inference to the best explanation” – in the first section I distinguish two tasks: (a) to discern which explanation is the best one; (b) to assess whether the best explanation deserves to be legitimately believed. In Sect. 20.2 I discuss some recent definitions of explanatory power based on “reduction of uncertainty” (Schupbach and Sprenger 2011; Crupi and Tentori 2012). Even though a probabilistic framework is a promising option here, I will argue that explanatory power so defined is not a convincing characterization of what makes a particular hypothesis better, from an explanatory point of view, that an alternative option. Then, in Sect. 20.3 I will suggest a sufficient condition (rule R1*) as my answer to (a). Regarding (b) I will propose a probabilistic threshold as a minimal condition for entitlement to believe (Sect. 20.4). The rule R1* and the threshold condition are intended as a partial explication of explanatory value (and, consequently, also as a partial explication of “inference to the best explanation”).
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Notes
- 1.
All this fits well with Peter Lipton’s “two-filter model” (see below). It could be recalled here that perceptual judgments in ordinary conditions were understood by Peirce as “extreme cases of abductive inferences” where the abductive conclusion “comes to us as a flash” (CP, 5.181). Apparently no comparison is done here, although unconscious comparative processes could underlay those judgments. However, provided that these are legitimate examples of abductive reasoning, they are not typical instances.
- 2.
A good example can be found in Josephson and Josephson (1994) where it is proposed a definition of an “abduction problem” “intended to formalize the notion of best explanation” (p. 160 and ff.).
- 3.
Minnameier (2004) also discusses abduction and IBE in relation to Peirce’s cyclical view of scientific inquiry. He concludes that IBE should not be included in the abductive stage, but in the inductive one.
- 4.
For more details about differences between “Hansonian” and “Harmanian” models of explanatory reasoning, see Paavola (2006).
- 5.
Two different measures M 1 y M 2 are ordinally equivalent just in case, for all H, D, H′ and D′ it is true that M 1 (D, H) ≥ M 1 (D′,H′) iff M 2 (D, H) ≥ M 2 (D′, H′).
- 6.
This is obvious for Ep CT . Concerning Ep SS , it should be noticed that \( \frac{p\left(H/D\right)-p\left(H/\neg D\right)}{p\left(H/D\right)+p\left(H/\neg D\right)} \) equates to \( \frac{p\left(\neg D\right)p\left(D/H\right)-p(D)p\left(\neg D/H\right)}{p\left(\neg D\right)p\left(D/H\right)+p(D)p\left(\neg D/H\right)} \) (proof omitted).
- 7.
For a detailed comparison of both measures, see Crupi and Tentori (2012).
- 8.
For Bayesians all probabilities are relative to the background knowledge, so strictly speaking there are no unconditional probabilities. Formalization demands a specific term K at the right of the symbol “|” and p (B) and p (M) should properly be rephrased as p (B|K) and p (M|K). Although K is omitted here for ease of exposition, notice that E would be a relevant item included in K.
- 9.
Weisberg 2009, 129–130, appeals to a different example that highlights the differences about simplicity among the rival hypotheses. However, his point is, again, that our intuitions about comparative explanatory merit are affected by prior probabilities.
- 10.
Likelihoods are also crucial to discern the best explanation when priors are even. In those situations that explanation which enjoys the highest likelihood would be also the best one. But it is easy to see that Bayes’ Theorem entails that it would be the more probable option as well.
- 11.
J. Weisberg maintains that genuine compatibility between Bayesianism and IBE requires a perfect match between probability and truth, that is, the best explanation is always the most probable hypothesis (Weisberg 2009, 137). The following remarks are intended to show that this is a very demanding condition.
- 12.
Scientists considerably agree on the “perception” of explanatory goodness, at least on its more general features. This agreement could be a by-product of a highly institutionalized training process that reinforces some heuristics. For details on the cognitive mechanism involved here, see Kuipers (2002).
- 13.
Perhaps we could assume that in the search space are included only those H i that satisfy this condition: p(D|H i ) > p(¬D|H i ). Otherwise, they would not even be considered as putative explanations of D. I will argue in the next section that a similar condition seems reasonable concerning whether to infer the best explanation (as a true conclusion that should be believed) or not.
- 14.
See Mackonis 2013, for a recent proposal.
- 15.
My particular proposal can be found at Iranzo (2008).
- 16.
- 17.
Of course, we could eventually discover that the best potential explanation, and also the most probable alternative given the evidence, is false. A justified belief obtained by means of IBE may be false since justified belief ≠ true belief, but this is a different question to be dealt with in the next section.
- 18.
Even though we have assumed, ex-hypothesi, that H b is the best explanation, it should be noticed that this condition also guarantees that there is only one best explanation. Thus, if p (H b |D) > 0.5, then ∑ p (H i≠b |D) < 0.5; therefore, p (H b |D) > p (H i≠b |D).
- 19.
A different argument for this very same condition was proposed in my (2007).
- 20.
Incidentally, this is another sort of situations where the most probable alternative given D is not the best explanation of D (see above, Sect. 20.3).
- 21.
Peirce himself toyed with an alleged “instinctive” ability to stick at the true option. See the discussion in Paavola (2005).
- 22.
See, for instance, Glass (2012) –where simulations are employed to vindicate a measure of explanatory goodness derived from coherence measures–, and Douven (2013), where the Bayesian standard rule for updating probabilities is disadvantageously compared to an explanationist alternative that gives an extra-bonus to good explanations.
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Iranzo, V. (2016). Explanatory Reasoning: A Probabilistic Interpretation. In: Redmond, J., Pombo Martins, O., Nepomuceno Fernández, Á. (eds) Epistemology, Knowledge and the Impact of Interaction. Logic, Epistemology, and the Unity of Science, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-26506-3_20
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