Abstract
Explanation is asymmetric: if A explains B, then B does not explain A. Traditionally, the asymmetry of explanation was thought to favor causal accounts of explanation over their rivals, such as those that take explanations to be inferences. In this paper, we develop a new inferential approach to explanation that outperforms causal approaches in accounting for the asymmetry of explanation.
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Notes
Throughout this paper we will use capital Roman letters for sentences of a formal language and capital Greek letters for sets of these sentences.
The graphical separation of \( \Sigma \) from the premises of a defeasible inference via “ \( \mid \,\)” is intended to remind the reader that the conclusion of the inference follows from the information contained in the premises and not from the contents of the background set. The latter figures solely in the determination of the inference’s defeat-status.
More precisely, an inference is defeated whenever its premises or background set contain a sentence that is logically equivalent to a member of the defeater set. This characterization of defeat allows for defeaters that are false to be considered in the evaluation of inferences—what we call expedient defeaters below. Although we refrain from providing it here, all of our informal references to defeat in the present text conform to the formal definition given in Millson et al. (2018). Two features of this formal definition are worth pointing out. First, a disjunction defeats an inference if both the disjuncts (or their logical equivalents) belong to the defeater set, and, second, a conjunction defeats the inference if at least one of the conjuncts (or their logical equivalents) belongs to the defeater set.
In the formalism above, C can be bracketed with respect to B if and only if
and there exists a sentence \(D \in \Theta \) and \( D\in \Sigma \).Inference defeaters admit of a further subdivision between what Pollock (2015) calls rebutting defeaters (which provide reasons for believing the negation of the conclusion of a given inference) and undercutting defeaters (which challenge the support provided by the premises of a given inference). The distinction between rebutting and undercutting defeaters will not be relevant here.
One way to do so would be to explicitly add the relevant modal operators into the content of the explanandum, but another way to do so is to keep the explanandum fixed and add modal information to both the defeater set \(\Theta \) and background set \(\Sigma \).
To be clear: we are not claiming that explanations must be the products of this procedure. Indeed, we make no claims about the “production” of explanations whatsoever. Rather, this three-step process is simply a useful heuristic for the reader to identify the relevant inferential properties that distinguish explanations from other nontrivial inferences.
The third step in the sturdiness test is represented formally as follows. If
is the only competitor to 
, then Step 3 consists in determining whether 
is defeated.Strictly speaking, some clever counterexamples might appear to spoil this result. We address them in Sect. 4.1.
Indeed, while we will not argue for it here, Woodward’s account of interventions can be seen as exhibiting the kind of inferential sturdiness we describe. Section 5 provides some clues as to how this argument would proceed.
We also assume that \(\Sigma \) is typical in what it brackets, so as to block recherché counterexamples to Design.
Compare: suppose that we are debating whether LeBron James would beat Michael Jordan in a one-on-one match if each were at their primes at the same time. Neither basketball player shot over 35% from three-point range, so it would make no sense in this hypothetical basketball game to suppose that either player had greater accuracy from downtown. In the scenario we are considering, construction errors are like these high shooting percentages. As we’ll see in Sect. 5, what’s good for basketball also carries over to causal explanation.
In this section, we leave the background set \(\Sigma \) implicit.
In addition to the differences between the intermediate value theorem and the meteorological causes, Antipode is shown to be sturdy if and only if, for its remaining competitors, either Antipode is undefeated on the supposition that their premises are false, or these competitors are bracketed, as discussed in Sect. 3.3.
Recall from Sect. 3.2 that such inferences are defeasible because of modeling considerations.
To simplify the presentation, we use the ambiguous phrase “ball A’s velocity.” Unless otherwise specified, “ball A’s velocity” denotes \(V_{1A}\), “ball B’s velocity” denotes \(V_{2B}\), and “ball C’s velocity” denotes \(V_{0C}\).
In effect, this will underwrite a backtracking counterfactual: “Had \(V_{2B} \ne 0.8 \text { m/s}\), then it would have had to have been the case that \(V_{1A} \ne 1\,\text {m/s}\).
Since the arguments showing that Ball C is sturdy (or not) can be extrapolated to Ball A, we focus on showing that the former is a candidate for sturdiness. Analogous considerations apply to the latter.
We have not used Woodward’s ultimate formulation of actual causation, (AC*), as it introduces complexities that are unnecessary for the purposes at hand.
Since the well-known syphilis-paresis example, there is a longstanding debate as to whether such an assumption is safe. However, the leading causal theorists, e.g. Strevens (2008) and Woodward (2003), both require every explanation to have some inferential structure, even if they do not require every explanation of be an inference. We hope to link these ideas to DIME in future work.
Indeed, in Millson et al. (2018), we develop a more precise account of explanation using a broadly inferentialist semantics. This approach to modal and explanatory vocabulary gives those of a Humean bent a compelling story about how one can use modal vocabulary without having to represent or be ontologically committed to metaphysically controversial modal entities (see Brandom 2008, 2015). This idea can be traced back to Sellars (1957). For similar approaches to the semantics of modal vocabulary see Thomasson (2007) and Stovall (2015).
and there exists a sentence 
is the only competitor to 
, then Step 3 consists in determining whether 
is defeated.References
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Khalifa, K., Millson, J. & Risjord, M. Inference, explanation, and asymmetry. Synthese 198 (Suppl 4), 929–953 (2021). https://doi.org/10.1007/s11229-018-1791-y
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DOI: https://doi.org/10.1007/s11229-018-1791-y