Abstract
Priority Theory is an increasingly popular view in metaphysics. By seeing metaphysical questions as primarily concerned with what explains what, instead of merely what exists, it promises not only an interesting approach to traditional metaphysical issues but also the resolution of some outstanding disputes. In a recent paper, Louis deRosset argues that Priority Theory isn’t up to the task: Priority Theory is committed to there being explanations that violate a formal constraint on any adequate explanation. This paper critically examines deRosset’s challenge to Priority Theory. We argue that deRosset’s challenge ultimately fails: his proposed constraint on explanation is neither well-motivated nor a general constraint. Nonetheless, lurking behind his criticism is a deep problem for prominent ways of developing Priority Theory, a problem which we develop.
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Notes
Much of the recent enthusiasm for grounding takes the form of advocating a Priority Theoretic reorientation of ontological theorizing. [Schaffer (2009) is an early example. See Bliss and Trogdon (2014) for further citations.] So does enthusiasm for the distinct, but related, concept of fundamentality. Indeed, deRosset probably intends the label ‘Priority Theory’ as just a covering term for these different reorientations currently in vogue. Note that Priority Theory is more than just the claim that there are non-causal explanations, a claim probably no one ever questioned. It is rather the claim that certain non-causal explanations or explanatory relations are primarily what ontologists should be studying. (This is consistent with ontology, and metaphysics more generally, also being concerned with other questions, like existence questions. After all, existence facts are presumably quite relevant to explanatory facts.)
We’ll simply treat facts as true propositions, for reasons of simplicity. Nothing will turn on this. We will also be loose with use-mention throughout, as our intentions should make our meaning clear.
These theses are consistent with the fundamental objects being microscopic things or the whole universe. (The latter position is sometimes referred to as Priority Monism.) Other positions might be consistent with these theses but sit poorly with their motivation. For instance, composition as identity is perhaps best seen as a way to thread the needle between the broad and narrow camps by saying that the existence of (e.g.) baseballs just is the existence of certain microscopic particles organized a certain way. [For more on composition as identity, see Wallace (2011a, b).] We think endorsing Priority Theory would alleviate some (though maybe not all) of the motivation for such a position.
We switch from a generic facts, like those used in deRosset’s quote, to an atomic fact because deRosset scrutinizes atomic facts the most.
By ‘situation’ deRosset has in mind some actual situation and not a merely possible one. [Some critics, e.g. Skiles (2015), have not been sensitive to this.] But we’ll follow him in this here.
To reduce clutter, we have rewritten finite sequences of terms \(a_1,\dots ,a_n\) in vector notation, \({\mathop {a}\limits ^\rightharpoonup }\), in quotations and throughout. In what follows, lower case Latin letters late in the alphabet, x, y, ...will be reserved for variables, and any other lower case Latin letters for constants. A vector \({\mathop {x}\limits ^\rightharpoonup }\) (resp. \({\mathop {a}\limits ^\rightharpoonup }\)) will be assumed to be a sequence of variables (resp. constants) only—there won’t be need for “mixed” vectors.
Often logicians use notation like \((\exists x)\varphi (c/x)\) to indicate that x is being substituted for c in the formula \(\varphi (c)\). We follow deRosset in leaving this implicit.
This is harmless in many applications. For instance, if c doesn’t actually occur in \(\varphi (c)\) then the \(\exists x\) in \(\exists x.\varphi (x)\) will just be a vacuous quantifier. Although deRosset doesn’t make this explicit, the reason for this second condition is that it allows for a syntactically clean generalization over all explanatory proposals à la (Prop). Were we to instead require that r must in occur in the object language sentence denoted by \(\varphi (r,{\mathop {a}\limits ^\rightharpoonup })\) then the kind of explanations Priority Theorists seek to produce would not fit the form of (Prop)—as, by definition, a nonfundamental raindrop r may not occur in the explanans of some explanation of its being F. And thus (Prop) would not a general form of explanation.
Tarski (1983) defined satisfaction as one step removed from the truth: a term or sequence of terms satisfies a formula just if subbing that term (sequence) into the formula results in something true.
We represent deRosset as maintaining that (Det) is a general constraint, but his text is a little ambiguous. His discussion of (Det) doesn’t explicitly include a relativization to a certain kind of explanation. But the prefatory comments to (Det) suggest that he is interested in non-causal explanations, that is, explanations where the explanans is not a cause of the explanandum. We discuss this matter in more detail in Sect. 3.2, especially fn. 26.
In fact (4) and (5) are logically equivalent. Proof sketch: (4) and
- (4\('\)):
-
\((\forall x,{\mathop {y}\limits ^\rightharpoonup })(\varphi (x,{\mathop {y}\limits ^\rightharpoonup }) \rightarrow Fx)\)
are equivalent by the usual quantifier exchange rules and facts about the sentential connectives. But (4\('\)) is the prenex normal form of (5), so they’re logically equivalent.
We note that r’s not occurring in \(\varphi (r,{\mathop {a}\limits ^\rightharpoonup })\) is crucial: if it did occur the prenex normal form would be different. As a second note: the crucial step is (4)’s implying (5)—that they are equivalent is an added bonus. And as it happens logics that lack prenex normal form theorems, like intuitionistic logic and most quantified relevance logics would still validate that (4) implies (5). Thanks to David Charles McCarty for discussion here.
The sentence in Ontologese might be picked out using English sentences and a special metaphysical operator, see (2012, p. 398).
Even stranger, this proposal would require that the explanations Priority Theorists give—specifically, the explanans—must frequently straddle two languages. We can’t think of any other kind of explanation that is essentially cross-linguistic like this.
Actually, we think her project may be even more unappealing that we are letting on. She actually reformulates the positions in a way that is not logically equivalent to deRosset’s. In fact, she reformulates the view under discussion so that it is essentially metalinguistic. E.g., instead of explaining why a raindrop is translucent, metaphysicians are to explain why the English sentence ‘this raindrop is translucent’ is true. Formulating the issue in this way lets her focus on a position that denies the existence of nonfundamental objects [compare her statement of (Sparsity) (2012, p. 392) with deRosset’s (2010, p. 76)]. But some of the most influential Priority Theorists—like Schaffer or Fine—do not deny the existence of nonfundamental objects and are interested in explaining facts about those objects, not the truth of sentences in a natural language (cf. Fine 2001, p. 16). In a footnote (p. 392, fn 3) she recognizes that the position she is interested in actually inconsistent with the position of authors like Schaffer. But she does not draw the conclusion that such authors are unlikely to embrace a project inconsistent with their own.
These were found by an afternoon of googling. There are a number of wrinkles to the semantics of this expression we can’t explore, such as what role context-sensitivity plays in filling out the proposed “likeness.” Our discussion will not turn on such wrinkles.
For uniformity’s sake in what follows we’ll stick with the inevaluable/anomalous reading of sentences like (7)–(9); the same points can be made on the false reading.
Unless of course \(r^*\) is among \({\mathop {a}\limits ^\rightharpoonup }\) or \({\mathop {c}\limits ^\rightharpoonup }\) or occurs elsewhere in \(\varphi \), but it can easily be chosen so that this isn’t the case.
deRosset might urge an alternative understanding of (10). Instead of taking ‘the explanans’ to refer to a closed sentence, it might instead refer to the function that is used to generate the closed sentence. The idea would be that r and \(r^*\) are then alike in virtue of being arguments that, in conjunction with other arguments, output a true sentence when fed into the function ‘the explanans’ refers to. As a reading of (10), we find this forced. Since for reasons explained above Priority Theory would need \(\varphi \) to be a function that didn’t take the argument actually mentioned in (10), to wit, r and \(r^*\). Ignoring that point, this understanding of (10) would still generate bizarre results. For instance, we could use this proposal to make the nonsensical sentence ‘Seasons 1 and 2 of True Detective are alike as far as being low in cholesterol goes’ sensible, even true, by appealing to the true sentence: Seasons 1 and 2 of True Detective are alike in so far as, with some other objects, they partially satisfy ‘x is low in cholesterol.’
Since \(r^*\) is not F, it follows that \(r^*\) is a confounding case for the proposed explanation. Thus, we do not deny that given the way deRosset defines ‘confounding case’ Priority Theory explanations have confounding cases. The issue is whether deRosset has shown that having a confounding case, in this sense, is ever a problem for a proposed explanation and not simply a quirky formal fact, a mere artifact of a tendentious explication.
Perhaps one reason why this point has gone unremarked in discussion of deRosset’s paper is that at no point are any actual explanations given of the kind Priority Theory must traffic in. A general schema (Prop) is specified, with pains to taken to ensure the Priority Theorist’s explanation fits it. But it is never instantiated. Some toy examples are discussed, but none fitting the form required by Priority Theory.
If either term actually appeared in the closed formula, deRosset’s argument for the first premise would fail, for reasons pointed out in fn. 12.
For critical discussion of this simple model, see Anchinstein (1983), Scrivens (1959), Railton (1978), Ruben (1990), Salmon (1989), and van Fraassen (1980). Skiles (2015, p. 742) also critizes a stronger modal form of (Det) that deRosset mentions (2010, p. 91). Of course, Hempel himself came to think that the D–N account was inadequate and proposed a second one that departed from it in some, but not all, ways.
Of course, they don’t have all the same problems. For instance, proponents of the D–N model have to explain the difference between laws and accidentally true generalizations. Conversely, D–N proponents are not shackled to the extra constraint that a specified term mentioned in the explanandum cannot be mentioned in the explanans.
Once we recognize this, we can see why it is tempting to think of (Det) as a general constraint on explanations. deRosset’s examples implicitly contain, in the explanans, universal generalizations, which he then extrapolates from. For instance, he gives the example: “Diamond is hard because each carbon atom in its crystalline structure is bonded to each of its neighbors” (2010, p. 78). If you had never had a course in chemistry, this explanation would leave you mystified. For you would not have filled in the other assumed part of this explanation: “All substances which have a crystalline structure in which each carbon atom is bonded to its neighbor are hard.” So the explanans implicitly appeals to a universal generalization. But once we see that a fuller articulation of this explanation would include this universally quantified statement, we can see why it might be a poor example to extrapolate from.
An anonymous reviewer worries that our counterexamples to (Det) might be targeting a strawman. As we indicated earlier in fn. 11, deRosset might not intend (Det) as a general constraint. And, as the anonymous reviewer reminds us, deRosset gives toy-examples of the kinds of explanations (Det) should apply to and those examples use the locution ‘in virtue of’. So one might worry that because some of our examples are causal explanations, they do not apply. We have a variety of worries about this response. First, even if our criticism of (Det) miss their mark, we think the problem we identify below in Sect. 4 is more pressing. Second, we don’t think there is a great gap between causal explanations and “in virtue of” explanations. For instance, to our ears, there’s nothing wrong with saying ‘Why Adam turned down the offer of cake? In virtue of the fact that he is on a diet.’ Indeed, appealing to the locution ‘in virtue of’ feels very natural in Scriven’s syphilis/paresis case because only patients with syphilis develop paresis, even though this is still a causal explanation. [For more principled reasons to worry about separating out causal explanations from other kinds, see Bennett (2017).] Third, even if we restrict ourselves to non-causal explanations, there might still be counterexamples. Consider an example due to Lipton (2004, pp. 31–32). Throw some sticks into the air with spin. If you take a “freeze shot” of them, you’ll find that many of them are closer to the horizontal axis then the vertical axis. The explanation of this is, crudely put, that there are more ways for them to be near the horizontal axis than the vertical axis. But this does not imply that any freeze shot of them will fail to capture all of the sticks closer to the vertical axis than the horizontal axis. Without developing them too much, there might be other counterexamples as well. For instance, there might be counterexamples from the normative domain (perhaps we can explain the value of this belief by pointing out it is true without maintaining that all true beliefs are valuable) or perhaps even emergent phenomenon. [For emergent phenomenon in the sciences, see (e.g.) Mitchell (2009), in philosophy O’Connor and Wong (2005).]
Priority Theory is concerned with a certain kind of metaphysical explanation. In this section, any explanation mentioned is to be understood as a “Priority Theoretic” explanation, whatever that happens to be.
An admissible extension of a consequence relation is a consequence relation that preserves all theorems of the original. We interpret \(\models \) in this way primarily for reasons of familiarity and expedience. Strictly, the present argument could be given with any (CC)-like condition stated in terms of consequence relation that it is truth-preserving, in that true premise sets should only have as \(\models \) consequences true conclusions, and “normative,” in that it should model certain logically impeccable forms of reasoning. (The impeccable forms of reasoning at issue will always be deductive or nonampliative. Thanks to an anonymous referee.) The argument in Sect. 4.5 is framed in terms of these two more abstract conditions.
We say that all things equal (GoC) ought to hold for a consequence relation because it is merely sufficient for modeling the desired reasoning, if rather naturally so, but it may not be necessary. One way around the challenge we are posing in this section is to show how to license this reasoning without (GoC). But exploring other ways of modeling this reasoning would take us too far afield—we’re primarily interested in posing the challenge.
We also note that (GoC) is not the only metatheorem that can be slotted into the crucial step in the derivation of the contradiction. The derivation could also be run, with some extra effort, with The Craig Interpolation Theorem in place of (GoC) (cf. Fefferman 2008). Alternatively, if \(\models \) were a relevant consequence relation then it will have as a metatheorem a variable-sharing property: that at least one sentence in one of the premises (\({\varGamma }\)) occur in the conclusion (\(\varphi \)) (Mares 2014). With a bit of extra effort this metatheorem could be slotted into the derivation as well. Again, our interest is in tracing the quickest logical path to the challenge, and to exhibit some philosophical commitments that would lead to it.
This provides another reason for thinking (Det) and (CC) differ. (Det) implies that what entails the explanandum will also contain the explanans. (CC) does not imply that.
To be sure, more could be said on each constraint in an attempt to foist it upon the Priority Theorist. But we find that style of argumentation more dialectically fraught. Besides, focusing on just one constraint would miss the bigger picture of the difficulty of developing Priority Theory.
Of course, this metaphor is not new; Kim (1979, pp. 40–41) used it to make a similar point.
It is not for nothing that Bolzano appears to be have thought of his notion of grounding as an explanatory consequence relation (Roski and Rumberg 2016, p. 473; Roski 2017, Chapter 2). And then there is Wittgenstein from the Tractatus:
“If a god creates a world in which certain propositions are true, then by that very act he also creates a world in which all the propositions that follow from them come true.” (Wittgenstein 1922, §5.123.)
Wilsch (2016) seems to defend a version of this view. Similarly, Skow (2016, pp. 110–111) suggests laws of nature play a role in explaining why (e.g.) the microscopic structure of a vase explains its fragility, without those laws being part of the reason, i.e. the explanans, for the vase’s being fragile.
This point is consistent with their being laws about rains drops, coffee mugs, and downtowns in general. (Though we might have worries about there being such laws.) For what its worth, in Fisher (unpublished), we discuss this position at a greater length. We argue that this position actually licenses a regress—one more general than that identified by Bennett (2011) and discussed in Sider (forthcoming) and deRosset (2013).
We should add that a lot of the language in that article is tentative and exploratory, so we don’t want to insist too much here. Plus there is much in that paper, including the rich discussion in §1.6, that we can’t discuss here, just for reasons of space. Still, nothing in the later sections directly addresses the challenge we are posing here.
cf. (2016, 23ff.)
In the inset quote, the universal generalization Glazier actually cites is ‘For all x ,if x is crimson, then that x is crimson makes it the case that x is red.’ (p. 23) But since \(\psi \) makes it the case that \(\varphi \) implies the material conditional, \(\psi \rightarrow \varphi \), this more complicated generalization logically implies the simpler universal generalization given above.
Here we mean a formal consequence relation in the sense first set out by Tarski (2002).
Schaffer (2012) argues against transitivity.
Correia (2015) also compares grounding to consequence relations.
The definition of \(\Rrightarrow \) above is just an intuitive way of defining what is the reflexive closure of the monontonic closure of the transitive closure of the relation, >. (The transitive closure of a relation is the smallest extension which is transitive. Likewise for the other closures.) It is easy to show that the property of being truth-preserving is itself preserved under each of these closure operations.
See Fine (2012)
Justification: \(\Rrightarrow \) is reflexive, so \(\varphi \Rrightarrow \varphi \); it is also monotonic, so \(\varphi , \psi \Rrightarrow \varphi \).
The justification here is essentially the same as the last one (conjunction elimination): it suffices for the consequence relation \(\Rrightarrow \) to model an impeccable form of reasoning that \(\models \) does if \(\models \) models that reasoning with \({\Gamma } \models \varphi \) and there exists a set of sentences \({\Gamma }^*\) straightforwardly logically equivalent to \({\Gamma }\) such that \({\Gamma }^*\Rrightarrow \varphi \). To model a form of reasoning one may “pivot” on a logically equivalent set of premises. To bring it back to the God metaphor: presumably, God in making something the case thereby also makes everything logically equivalent to it the case.
Perhaps along the lines of Russell and Restall (2010)’s barriers to implication.
This is why the results of §3.2 do not help Priority Theorists. Even if it need not be the case that explanans must entail explanandum in general, which we find plausible, they must never entail in this class of cases for Priority Theory to be consistent.
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Thanks to Tim Leisz, David Charles McCarty, Nick Montgomery, Tim O’Connor, Harrison Waldo, Phil Woodward, and audiences at the Society for Exact Philosophy.
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Appendix: Further formal issues
Appendix: Further formal issues
In our argument in Sect. 4.5 we urged a defeasible proposal: all things equal, we would expect a truth-preserving consequence relation that modeled generalizing from an arbitrary case will have (GoC). We recognize that there might be other ways of modeling this kind of reasoning. However, we contend that alternative ways of modeling this reasoning will not avoid the challenge we are trying to press here. We’ll illustrate this point by considering a proposal due to an anonymous reviewer.
There is a familiar way of taking extensions of a consequence relation in mathematical logic. For example, if \(\vdash \) is a consequence relation and \({\Lambda }\) is a set of sentences, we can define the extension \(\vdash _{{\Lambda }}\) by
Since \(\vdash _{{\Lambda }}\) essentially treats the members of \({\Lambda }\) as tacit premises we’ll call this a premise extension of \(\vdash \). \(\vdash _{{\Lambda }}\) is a truth-preserving consequence relation if \(\vdash \) is, and it needn’t have (GoC) in full generality—for instance, if some constant c occurred in \({\Lambda }\) then c won’t act like an arbitrary constant that can be universally generalized over. Nevertheless, \(\vdash _{{\Lambda }}\) may still model generalizing from an arbitrary case: provided \(\vdash \) itself had (GoC), it is easy to see that for a given premise set \({\Gamma }\), the constants that can be universally generalized are just those that don’t occur anywhere in \({\Gamma }\) or \({\Lambda }\).
Hence a premise extension like \(\vdash _{{\Lambda }}\) shows how a truth-preserving consequence relation can model generalizing from arbitrary case without having (GoC) in full generality. However, we maintain that if the consequence relation that figures in (Extended Grounding Explanation) is a relation like this that will still be enough to pose our challenge.
To see how, let’s return to Priority Theory developed in terms of grounding (Sect. 4.5). Suppose the consequence relation \(\Rrightarrow \) we define as an extension the grounding relation, >, didn’t have (GoC) in full generality but that it modeled generalizing from an arbitrary case by being a premise extension of a consequence relation, \(\Rrightarrow ^{\!\circ }\), which itself has (GoC). In particular, for a fixed set \({\Lambda }\), for any \({\Delta }\) and \(\psi \),
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\({\Delta } \Rrightarrow \psi \) if and only \({\Delta } \cup {\Lambda } \Rrightarrow ^{\!\circ } \psi \).
Return now to the kind of example we have been discussing: let a be nonfundamental, \(\varphi (a)\) be a non-total fact, and let \({\Gamma }\) be the explanantia of some explanation, \(\varphi (a)\) because \({\Gamma }\), where a doesn’t occur in \({\Gamma }\). The principle (Extended Grounding Explanation) will still hold for \(\Rrightarrow \), so \({\Gamma } \Rrightarrow \varphi (a)\). And, since \(\Rrightarrow \) is a premise extension of \(\Rrightarrow ^{\!\circ }\), \({\Gamma } \cup {\Lambda } \Rrightarrow ^{\!\circ } \varphi (a)\). The question now is whether a appears in \({\Lambda }\) or not. If it does not, then, given that \(\Rrightarrow ^{\!\circ }\) has (GoC), a contradiction is derivable just as before. If, however, a does occur in \({\Lambda }\), then we are in a situation that exactly parallels the case of Metaphysical Laws we discussed in Sect. 4.4. And what we had to say there applies equally well here.
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Fisher, D., Hong, H. & Perrine, T. A challenge to the new metaphysics: deRosset, Priority, and explanation. Synthese 198, 6403–6433 (2021). https://doi.org/10.1007/s11229-019-02468-3
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DOI: https://doi.org/10.1007/s11229-019-02468-3