Grounding and the explanatory role of generalizations

Abstract

According to Hempel’s (Aspects of scientific explanation and other essays. The Free Press, New York, 1965) influential theory of explanation, explaining why some a is G consists in showing that the truth that a is G follows from a law-like generalization to the effect that all Fs are G together with the initial condition that a is F. While Hempel’s overall account is now widely considered to be deeply flawed, the idea that some generalizations play the explanatory role that the account predicts is still often endorsed by contemporary philosophers of science. This idea, however, conflicts with widely shared views in metaphysics according to which the generalization that all Fs are G is partially explained by the fact that a is G. I discuss two solutions to this conflict that have been proposed recently, argue that they are unsatisfactory, and offer an alternative.

1 Introduction

In the last two decades, the notion of grounding has been widely discussed in metaphysics and neighbouring fields. Grounding is generally conceived as a non-causal form of priority among facts that is introduced by certain uses of the phrase ‘in virtue of’ or by the connective ‘because’. This relatively sparse characterization is often augmented with the claim that grounding is an explanatory form of determination. Grounds, thus a common assumption, explain what they ground. This assumption is frequently used to motivate various claims about grounding, for instance that it is asymmetric and non-monotonic.¹ Relying on certain criteria for what explains what has, moreover, been suggested as a heuristic to find out what grounds what.² Finally, even to acknowledge the notion in the first place is sometimes motivated by the claim that doing so is the best way to account for certain non-causal explanations.³ It thus seems safe to say that the idea that the notion of grounding is an explanatory notion is the default view in the current debate.⁴ In this paper, I will discuss a conflict that seems to arise between this view and widely shared views on explanation in the philosophy of science.

In rough outline, the conflict is as follows: It is virtually uncontested in the debate on grounding that (1) universal quantifications are partially grounded in their respective instances, that (2) material conditionals are grounded in their consequents (in case the latter hold), and that (3) grounding is transitive.⁵ These three claims entail that the fact that all Fs are G is partially grounded in the fact that a is G, provided the latter obtains. Given that grounds explain what they ground, part of the explanation of why all Fs are G is thus that a is G.⁶ This seems to stand in conflict with the view that an explanation of why a is G is at least in some cases that all Fs are G and that a is F. The latter view is part and parcel of many accounts of scientific explanation, most notably of Hempel’s influential deductive-nomological account, but by no means confined to it. Also adherents of the unificatory theory of explanation and many others take it for granted or defend it (more on this below). Even independently of any particular philosophical theory, it seems hard to deny that we sometimes answer questions as to why some particular F is G by citing the fact that all Fs are G.⁷

In order to put the conflict a bit more perspicuously, let us slightly extend the extension of the term ‘instance’. In particular, let us call any proposition semantically expressed by sentences of the form ${\ulcorner}a$ is $G\urcorner$ an instance of the proposition semantically expressed by sentences of the form $\ulcorner$All Fs are $G\urcorner$. The conflict can then be put as follows:

(Gen-By-In) :

Every true universal generalization is partially explained by its instances.

(In-By-Gen) :

Some true universal generalizations partially explain their instances.

Both claims clearly stand in some intuitive tension: how can a universal generalization be explained by, and at the same time explain, its instances?⁸ The claims entail an actual contradiction if we additionally accept the following principles:

(Unity):

The term ‘explains’ introduces the same notion in both claims above.

(Asymmetry):

For every x, y: if x partially explains y, then it is not the case that y partially explains x.

In view of this problem, Loewer (2012) and others have recently argued that Unity should be rejected. Contrary to Unity, they claim, there are different notions of explanation involved in both principles about the explanatory role of generalizations.⁹ I will call this type of solution also the pluralist solution in what follows. A first task of this paper is to argue that this solution is unsatisfactory (Sect. 3). Given that the pluralist solution is unsatisfactory, we have to reject one of the two principles on the explanatory role of generalizations. In a recent paper, Marshall (2015) has argued that we should retain Unity and In-By-Gen, but reject Gen-By-In on account of certain parsimony considerations. A second task of this paper is show that Marshall’s respective argument is inconclusive (Sect. 4). Rejecting both Marshall’s and the pluralist solution forces us to reject In-By-Gen. Given the strong standing of the latter view, this requires some account to accommodate the intuitions that have driven philosophers to accept it. In the final part of the paper (Sect. 5), I will show that at least some intuitions underlying the view can be accommodated without having to accept In-By-Gen. Roughly, the idea I will defend is that when a universally quantified statement of the form $\ulcorner$All Fs are $G\urcorner$ is used to successfully communicate an explanation, this is never the case because it semantically expresses the proposition that all Fs are G. Rather, it is explanatory because of other propositions it conveys in the pertinent context.

2 Preliminaries

This paper is concerned with the notion of explanation. While I won’t assume any substantial theory of explanation, there are a couple of broad assumptions about explanations I will make that should be laid out before the main discussion.

As is well known, the term ‘explanation’ has various uses. There is, first, a process-product ambiguity. ‘Explanation’ and its cognates can designate some activity (perhaps a certain kind of speech-act) or certain products that are the outcome of this activity.¹⁰ I will in what follows concentrate on the product-sense of ‘explanation’, though I will occasionally use process-talk for the sake of conciseness. I take it that explanations (in this sense) consist of fine-grained, structured propositions, one of which is the explanandum and the others the explanans.¹¹ In line with what I take to be a majority view in the debate I am concerned with in this paper, I will further assume that whether some propositions constitute an explanans for a given explanandum is an entirely subject-independent matter. That is to say, the fact that some propositions explain some other proposition is objective in the same sense in which it is objective that Goldbach’s Conjecture is true or false.¹² This notion of explanation is arguably different from the (or at least a) colloquial notion of explanation according to which whether some bit of information counts as an explanation is dependent on whether the information is appropriately related to the epistemic background of a given subject. However, it is the notion that seems to be in place in the debate I am concerned with, and it can be understood perhaps as some kind of Carnapian explication of one central aspect of the ordinary notion. By and large, we may identify explanations in this objective sense with what is expressed by true ‘because’-claims.

Note that adopting an objective notion of explanation does not entail that epistemological and pragmatic considerations have no place in the study of explanation—far from it. In particular, such considerations are of the foremost importance in deciding whether explanations are good or bad, or whether they are relevant or irrelevant in the course of some inquiry.¹³

Let us next fix some ideas about grounding. For the purposes of this paper, I take grounding to be a relation that obtains exclusively among facts. I will further assume that the relation is transitive and asymmetric.¹⁴ Finally, I will make a number of standard assumptions about the logic of ground. As these assumptions provide the main motivation for Gen-By-In, let me elaborate a bit on them.

A core intuition of virtually every account of the logic of ground is that truth-functionally complex facts are grounded in their constituents.¹⁵ Disjunctive facts are, for instance, grounded in their obtaining disjuncts: Socrates is wise or foolish because he is wise. Conjunctive facts, on the other hand, are jointly grounded in their conjuncts: Socrates is an Athenian philosopher partially because he is from Athens and partially because he is a philosopher. Now, given the tight, truth-conditional relationship between existential and universal generalizations on the one hand, and disjunctions and conjunctions on the other, the aforementioned principles for the grounds of disjunctive and conjunctive facts suggest related principles for the grounds of existential and universal facts. Since there is a tight connection between the truth-conditions of existential generalizations and the disjunctions of their instances, it seems plausible to take facts corresponding to existential generalizations to be fully grounded in their obtaining instances.¹⁶ Analogously, since there is a tight relation between the truth-conditions of universal generalizations and the conjunctions of all their instances, it seems plausible to take facts corresponding to universal generalizations to be partially grounded in each of their instances.¹⁷ The latter principle then, together with the claim that truths about grounds explain truths about facts grounded by those grounds, supports Gen-By-In, which gives rise to the conflict described earlier. So let us consider proposals towards a solution.

3 The pluralist solution

In view of the relative unfamiliarity of explanations based on grounding, a first reaction to the conflict outlined in Sect. 1 might be to suspect that there is an ambiguity hidden somewhere. Are we really talking about the same notion of explanation in all principles under consideration? According to Loewer (2012) this is indeed not the case. He maintains, in particular, that the conflict between In-By-Gen and Gen-By-In can be solved by keeping two different types of explanation apart: metaphysical and scientific explanations.¹⁸ In metaphysical explanations, he propounds, “a type of fact [...] is shown to be grounded or constituted by some other kind of fact”, whereas a scientific explanation “typically, shows why [a particular] event occurred in terms of prior events and laws”.¹⁹ (NB: Loewer equates laws with certain generalizations.) Loewer’s idea is that a generalization may scientifically explain its instances, while the instances will, at the same time, metaphysically explain the respective generalization. Thereby, he effectively denies Unity. In-By-Gen and Gen-By-In have to give way to two restricted principles:

(In-By-Gen $_S$):

Some true universal generalizations scientifically explain their instances.

(Gen-By-In $_M$):

Every true universal generalization is metaphysically explained by its instances.

Rejecting Unity removes the contradiction, but also gives rise to a pressing question.²⁰ How exactly are metaphysical and scientific explanations related? Arguably, both types of explanation are not completely unrelated kinds of things. This is evidenced by the fact that there are plausible examples of what one might call combined explanations: explanations that are part metaphysical and part scientific in Loewer’s sense. Consider an example for illustration. We may explain why Jane parked in the no-parking zone in terms of her being under a lot of stress. This is arguably a scientific explanation in Loewer’s sense. (In any case, it seems to be a perfectly fine causal explanation.) We may, further, explain why Jane broke the law in terms of her parking in the no-parking zone. This might be a case of a metaphysical explanation in Loewer’s sense. (In any case, it is not a causal one.) That Jane was under a lot of stress and parked in the no-parking zone then seems to be a perfectly fine explanation of why she broke the law that is part metaphysical and part scientific in Loewer’s sense (in any case, part causal and part non-causal). Also the less informative explanation that Jane broke the law because she was under a lot of stress seems to be a perfectly fine combined explanation.²¹

Notably, Loewer himself seems to accept that there are such kinds of explanations.²² How are we to account for them given the pluralist solution to our conflict? Two initially plausible ideas of how to do so give rise to problems for Loewer’s account. According to the first idea, combined explanations are a sub-species of scientific explanations. According to the second idea, both types are sub-species of a more general notion of explanation.

The first idea has been discussed by Lange (2013) in response to Loewer. Lange claims that the best way to make sense of Loewer’s remarks about combined explanations is to take him as claiming that such explanations belong to one of the two types Loewer distinguishes. In particular, if a metaphysical explanation supports a scientific one, the result is again a scientific explanation. The best way for Loewer to account for the phenomenon of combined explanations, Lange argues, is to accept the following principle:²³

(Cross Transitivity):

If x metaphysically explains y and y scientifically explains z, then x scientifically explains z.

(While Lange is not explicit about this, we can plausibly read the ‘explains’ in the above principle as partially explains.) According to this principle, our example from above would count as a scientific explanation of why Jane was under a lot of stress that involves a metaphysical element. Lange then points out that on this way of accounting for combined explanations, Loewer’s solution of the conflict fails. For, if Cross Transitivity holds, In-By-Gen $_S$ and Gen-By-In $_M$ turn out inconsistent on the plausible assumption that nothing scientifically explains itself. For take some generalization that scientifically explains its instances. By Gen-By-In $_M$, any given instance will partially metaphysically explain the generalization, whence, by Cross Transitivity, the instance will also partially scientifically explain itself.

Pluralists have tried to debunk Lange’s argument by arguing that Cross Transitivity is inherently objectionable and hence not a plausible principle to account for combined explanations to begin with. In particular, Hicks and van Elswyk (2015) as well as Miller (2015) have tried to defend the pluralist solution to our conflict by presenting a number of purported counterexamples to Cross Transitivity. As Marshall (2015) has pointed out, however, there are reasons to doubt the viability of these counterexamples.²⁴ While I cannot conclusively decide the dispute at this juncture, note that even if the pluralists’ objections to Cross Transitivity would go through, they would still owe us an account of how combined explanations are to be accounted for.

Independently of that, postulating principles such as Cross Transitivity does not seem to be a particularly attractive way for pluralists to account for combined explanations in the first place. For, it is unclear whether pluralists can rest content with distinguishing merely between scientific and metaphysical explanations. They should at least not deny the existence of, for instance, purely mathematical explanations, normative ones, and others outright. But since it seems that such kinds of explanations often combine with scientific and/or metaphysical ones, pluralists would owe us an account of how these notions of explanation interact with each other. Postulating further principles along the lines of Cross Transitivity that govern the interaction between different kinds of explanations seems to result in a potentially open-ended, overly complicated account that, while perhaps not incoherent, seems rather unappealing—certainly as long as alternative accounts to accommodate combined explanations are available.²⁵ Moreover, we may ask why metaphysical explanations should always be subordinated to scientific ones as Cross Transitivity predicts. Can’t a scientific explanation also sometimes support a metaphysical one, and, if yes, under what conditions?

None of these worries constitutes a decisive refutation of using principles such as Cross Transitivity to accommodate combined explanations on a pluralist picture. I take it, however, that they provide reason enough to consider alternative accounts.

So let us turn to a second and by my lights more plausible way to accommodate combined explanations which has been suggested by Kim and Ruben. According to Kim and Ruben, whether some propositions constitute the explanans for some given explanandum is generally not a brute fact, but rather based on certain other facts. In particular, propositions constitute explanations in virtue of facts that involve certain relations between entities the respective explanans and explanandum are about (in a broad sense of ‘being about’).²⁶ For instance, that Jane was under a lot of stress explains why she parked in the no parking zone in virtue of the fact that the former proposition is about a cause of an event the latter is about. That she parked in the no-parking zone, in turn, explains why she broke the law in virtue of the former proposition being about a fact that stands in some non-causal form of determination to a fact the latter is about. The combined explanation, finally, obtains in virtue of both facts. In general, we can view metaphysical explanations in Loewer’s sense as explanations that rest on facts about grounding (and perhaps other forms of non-causal determination), whereas scientific ones rest on facts about causation and other forms of nomic dependence. Combined explanations are then precisely those explanations that rest on various different kinds of facts about causation, grounding, and other forms of dependence or determination.²⁷

The main advantage of this view is that it allows us to account for different types of explanation without having to postulate different, unrelated notions of explanation. There is one notion of explanation, but whether something constitutes an explanation of something else may be due to different kinds of facts in different cases. Of course, if an explanation solely (or mainly) rests on facts about causation, we may well call it ‘scientific’, and if it solely (or mainly) rests on facts about grounding, we may call it ‘metaphysical’. But this alone does not necessitate postulating different notions of explanation any more than the fact that there are vertebrate and invertebrate animals necessitates postulating different notions of animal.

If the pluralist takes this route to accommodate combined explanations, however, they are forced to reject Asymmetry. For on the suggested view, scientific and metaphysical explanations both count as explanations simpliciter, and thus Asymmetry still allows to derive a contradiction from In-By-Gen $_M$ and Gen-By-In $_S$.

This seems to be a highly undesirable result: that explanation is asymmetric is one of most recalcitrant intuitions in the debate on explanation. Indeed, this intuition has driven much of the critique of various proposed theories of explanation, most notably of Hempel’s. It has often been argued that no theory of explanation that does not respect the asymmetry of explanation can be adequate.²⁸ If we interpret Loewer’s remarks on scientific and metaphysical explanations in the way outlined above, there is a clear sense in which this condition is not met. In particular, not only is it not met in certain extraordinary cases. Rather, it fails across the board for all explanations involving generalizations.²⁹ Pluralists can, of course, simply deny that explanations are generally asymmetric. Perhaps Asymmetry has to be restricted to metaphysical and scientific explanations, respectively—provided we can draw sufficiently clean distinction between them.³⁰ However, pending direct arguments against Asymmetry, it seems better to offer a solution of the puzzle that allows to retain this principle.³¹

In conclusion, while the Loewerian pluralist solution to our conflict removes the contradiction, it needs to be augmented with a plausible story of how to account for combined explanations. I have argued that two stories that have been advanced in response to this problem remain wanting. Either pluralists are faced with the potentially open-ended task of accounting for the interaction between different notions of explanation, or else they have to give up the asymmetry of explanation. Neither result might conclusively refute the account, but both strongly motivate searching for a better solution.

4 Marshall against Gen-By-In

If we retain Asymmetry and Unity, either Gen-By-In or In-By-Gen have to go. In a recent paper, D. Marshall has argued for the former on broadly Humean assumptions. His case has a positive and a negative part. The positive part is a principle that he offers as an alternative to Gen-By-In and that is supposed to preserve some of the latter’s spirit. The negative part consists of an argument against Gen-By-In. In what follows, I will raise some doubts about the motivation of the principle Marshall offers as an alternative to Gen-By-In, and then have a critical look at his argument against the principle.

Since Marshall denies that Gen-By-In is true, he has to deny that instances of generalizations help explain those generalizations. Yet, he maintains, instances of generalizations do explain why those generalizations are laws, if they are. Marshall here presupposes Lewis’ Best Systems Account of laws. On this account, laws are those universal quantifications that belong to every axiomatization of the totality of all facts that “strikes as good a balance as truth will allow between simplicity and strength” (Lewis 1994, 478). Marshall now argues as follows:

Given the [Best Systems Account], whether $\underline{\hbox{All F}s\hbox{ are } G\hbox{s}}$ is a law is determined by what the best systems are, which in turn is determined by $\underline{a\hbox{ is }G}$, together with a multitude of other particular matters of fact and a totality fact. However, [...] the fact that All Fs are Gs is a law is distinct from the fact All Fs are Gs. Hence, it does not follow from it being the case that a is G partly explains the fact that All Fs are Gs is a law that a is G partly explains All Fs are Gs. (Marshall 2015, 3158)

Accordingly, Gen-By-In should be replaced by the following principle:

(Law-By-In):

If some generalization G is a law, then every instance of G explains why G is a law.

While substituting Gen-By-In for Law-By-In indeed removes our contradiction, it raises the question of whether the latter can be independently motivated. Why should we believe that the law-hood of a generalization is explained by its instances, without those instances explaining the generalization itself?

In the passage quoted above, Marshall argues for Law-By-In by claiming that (1) for any generalization G, whether G is a law is determined by what the best systems are, and (2) what the best systems are is partially determined by G’s instances. While claim (1) is plausibly entailed by the Best Systems Account that Marshall presupposes, claim (2) seems to require independent motivation. Marshall does not provide any such motivation, but here is an attempt. Whether a given axiomatization satisfies the criteria for being among the best systems in Lewis’ sense depends on two factors: first, a factor that we may dub the ‘economy factor’: the axiomatization has to exhibit an optimal combination of simplicity and deductive strength. Second, it depends on a factor that we may dub the ‘adequacy factor’: the generalizations belonging to the axiomatization have to be true. This second factor is crucial as there are numerous simple but false axiomatizations that are strong enough to cover the totality of all particular matters of fact, most notably contradictory ones.

Now, the economy factor of an axiomatization is arguably independent of the instances of the generalizations belonging to it. The same totality of particular matters of fact that is covered by one given set of generalizations can be axiomatized in many ways that differ vastly in deductive strength and simplicity. However, the truth of the generalizations belonging to such an axiomatization is not independent of their respective instances. Whether the generalizations hold is directly determined by whether the instances hold. In other words, because the instances of the generalizations belonging to a given axiomatization determine whether it satisfies the adequacy factor, they determine whether it counts as one of the best systems.

If this is the motivation for (2), however, it is difficult to see how it should favor Law-By-In over Gen-By-In. For if we accept the view that the law-hood of a generalization is explanatorily determined by the generalization’s instances simply because whether a generalization holds is determined by its instances, it is difficult to see why we should not accept that generalizations are explanatorily determined by their instances, which is precisely what Gen-By-In says.³² I take it, thus, that Marshall’s positive proposal to substitute Law-By-In for Gen-By-In is not sufficiently well motivated.³³

Marshall also has a direct argument against Gen-By-In to offer that is independent of his proposal to substitute Law-By-In for Gen-By-In. The argument tries to establish a surplus in theoretical virtues of In-By-Gen over Gen-By-In. Marshall admits that Gen-By-In leads to a certain form of unification with respect to the explanation of generalizations. After all, according to the principle, there is a single pattern for the explanation of any true generalization.³⁴ Now, consider a famous characterization of unification due to Kitcher:

Science advances our understanding of nature by showing us how to derive descriptions of many phenomena, using the same patterns of derivation again and again, and, in demonstrating this, it teaches us how to reduce the number of types of facts we have to accept as ultimate (or brute). (Kitcher 1989, 432; orig. in italics)

Since Gen-By-In radically reduces the argument-patterns necessary to explain generalizations, it scores high on one dimension of unification, at least given Kitcher’s account. However, Marshall objects, this

comes at a terrible cost in terms of explanatory parsimony, and hence in terms of overall simplicity. If laws can partly explain their instances then a vast number of particular matters of fact can plausibly be explained in terms of a much smaller number of particular matters of fact, together with a small number of laws. If [Gen-By-In] is true, on the other hand, and laws cannot partly explain their instances, then a vast number of particular matters of fact will instead have to be foundational, where a fact is foundational iff it is not explained by any other fact. (Marshall 2015, 3162–3163)

It is not entirely straightforward to see how the claim that a vast number of particular matters of fact is foundational should follow from Gen-By-In. Call a generalization that meets the criteria of the Best Systems Account a Humean law. The following argument would then establish the intended conclusion:

(P.1)	If Gen-By-In holds, Humean laws do not explain their instances.
(P.2)	If Humean laws do not explain their instances, a vast number of particular matters of fact will be explanatorily fundamental.35
(C)	If Gen-By-In holds, a vast number of particular matters of fact will be explanatorily fundamental.

Evidently, this argument only poses a problem for a Humean conception of laws. For, P.2 can easily be resisted on ‘anti-Humean’ conceptions of laws according to which laws are not identical to universal generalizations, but rather sui-generis facts or facts about relations between universals.³⁶ On such a conception, Gen-By-In is perfectly compatible with there being only a few explanatorily fundamental facts that explain a vast number of particular matters of fact which in turn explain the universally quantified facts they are instances of (including Humean laws). Anti-Humeans need thus not be moved by the argument.

What the argument might show, however, is that Gen-By-In scores badly in terms of quantitative parsimony when it is combined with a broadly Humean picture; a picture, namely, according to which the explanatorily most fundamental level consists of “a vast mosaic of local matters of particular fact” (Lewis 1986b, ix) and does, in particular, not include sui-generis law-facts or suchlike. On such a conception, P.2 seems plausible. However, even Humean adherents of Gen-By-In need not be worried by the argument’s conclusion. For, it is questionable whether a loss in mere quantitative parsimony is indeed as “terrible” a cost as Marshall claims it to be (2015, 3163). Indeed, while Humean adherents of Gen-By-In score badly in terms of quantitative parsimony, their account fares well in terms of qualitative parsimony. There is only a very limited variety of kinds of explanatorily fundamental facts: kinds of particular facts that constitute the Humean mosaic.³⁷ Moreover, as we have seen, Marshall concedes the adherent of Gen-By-In an advantage in another theoretical value, namely unification. Hence, Humean adherents of Gen-By-In may argue that the gain in unification and qualitative parsimony afforded by their account outweighs the loss in quantitative parsimony. Of course, weighing theoretical values is notoriously difficult and hardly ever conclusive, but the previous consideration should at least establish that there are viable replies to Marshall’s argument even for Humean adherents of Gen-By-In.

Of course, Humeans who take elements of the mosaic of particular matters of fact to be fundamental and accept Gen-By-In have to deny that the generalizations that figure in the best systems (i.e. the laws on their account) explain their instances. But even that might not be such a terrible result after all. For, at least some epistemic values that are often associated with laws can be retained also on such an account. A number of authors have recently suggested that the notions of understanding and explanation are not as tightly connected as often assumed. In particular, they argue that theories can provide understanding by unifying a range of facts even if they do not explain why those facts obtain.³⁸ When adopting such a view, Humean adherents of Gen-By-In could argue that generalizations belonging to the best systems contribute to an understanding of regularities in nature by exhibiting unifying structures among them, even though they strictly speaking don’t play any explanatory role with respect to them. They thereby could retain the idea that generalizations belonging to the best systems are of specific interest to scientists to the extent that scientists pursue understanding. Needless to say, much more work would be necessary to develop this suggestion into a satisfactory account. But, again, the suggestion points to a way in which Gen-By-In may be found plausible even for Humeans.

In conclusion, Marshall’s attack on Gen-By-In seems unsatisfactory. Neither does he convincingly establish the positive substitute for the claim he offers, nor does his argument against the claim succeed—even Humeans can resist it.

5 Against In-By-Gen

Thus far, we have discussed two solutions to the conflict between Gen-By-In and In-By-Gen and dismissed both as unsatisfactory. The remaining option on the table is to reject In-By-Gen. This may initially seem to be a strategy not worth pursuing given the wide acceptance of the latter. As I will argue in this section, however, there are reasons to reject In-By-Gen that are independent of its conflict with Gen-By-In. In addition to that, I will sketch a way to accommodate at least some of the intuitions that have driven philosophers towards accepting In-By-Gen.

5.1 A direct argument against In-By-Gen

While In-By-Gen has occasionally been rejected, I am only aware of one direct argument against the view.³⁹ This argument is suggested by Dretske (1977) as well as Hoeltje et al. (2013, §1.d). Dretske summarizes his main point succinctly as follows:

Subsuming an instance under a universal generalization has exactly as much explanatory power as deriving Q from $P\wedge Q$. None. (Dretske 1977, 262)⁴⁰

We can make his argument a bit more explicit. It starts from the following assumption:

• A.1 There is no true instance of the following scheme:

  That a is F and b is F explains why a is F.

Consider a concrete case: that Socrates is a philosopher and Kant is a philosopher explains why Socrates is a philosopher. That sounds wrong. The proposition that Socrates is a philosopher arguably cannot explain itself, so the first conjunct of the purported explanans is at best explanatorily irrelevant. What we want to say here is, of course, not merely that it is explanatorily irrelevant, but rather that it makes the whole explanation circular. The second conjunct on the other hand—that Kant is a philosopher—in no way helps explain why Socrates is one. Even if we consider an instance where one of the conjuncts does help explain the other, the fact that the explanandum is a conjunctive part of the purported explanans deprives the latter of its explanatory power.⁴¹

Arguably, this point holds independently of how many conjuncts are involved. In particular, if A.1 is accepted, one should also accept:

A.1$'$ :

There is no true instance of the following scheme:

That a is F and b is F and c is F, and ...ad. inf. explains why a is F.

Also an infinite conjunction cannot explain one of its conjuncts, since it will both have the explanandum and various other, explanatorily irrelevant facts as parts.

Now consider an infinite conjunction together with a proposition stating that the objects mentioned in the conjuncts are all objects there are. The same point seems to apply. That is, if A.1 and A.1 $'$ are accepted, one should also accept:

A.1$^\tau$ :

There is no true instance of the following scheme:

That a is F and b is F and c is F, and ...ad. inf. and $a,b,c,\ldots$ are all and only the objects there are explains why a is F.

Since also in this case the explanandum is part of the explanans, the attempted explanation fails due to circularity and irrelevance. But this point puts the adherent of In-By-Gen under pressure. For an infinite conjunction that involves a totality claim logically entails the corresponding universal generalization. The generalization, in turn, logically entails the conjunction of each of its instances. This raises the question why one should expect that the generalization can explain something that the infinite conjunction cannot. Of course, that two propositions stand in a close logical relationship does not in general mean that they are interchangeable in explanations. But the proponent of In-By-Gen ought to provide us with some reason for why there is an explanatorily relevant difference between them.

One may try to argue for such a difference by claiming that both propositions have different epistemic properties: perhaps universal quantifications can be grasped by finite subjects while infinite conjunctions cannot. The viability of this strategy hinges, however, on substantial commitments on the nature of propositions and the conditions under which they are graspable. While it is clear that finite human beings cannot process infinitely long linguistic items, it is not so clear whether they cannot grasp the propositions expressed by them. Adherents of In-By-Gen pursuing this strategy thus have to come up with a suitable account of the relata of explanation that establishes an epistemic difference between infinite conjunctions and universal generalizations. But apart from that, recall that the notion of explanation that we are dealing with here is an objective one (cf. Sect. 2). Whether or not a given purported explanans is graspable by finite subjects ought not make a difference with respect to the latter’s status within an explanation. That status should be determined solely by the information contained in the purported explanans. But the information an infinite conjunction together with a totality fact contains about a given conjunct, it seems, does not differ in any explanatorily relevant aspect from the information a corresponding universal generalization contains about the corresponding instance. Hence, the above argument provides us with a strong reason to reject In-By-Gen.

5.2 Accommodating intuitions: laws and other generalizations

The principle In-By-Gen has dominated the philosophical discussions on explanation for the last decades. Even if one finds the arguments presented above convincing, one should thus not rest content with simply rejecting the principle. In the remainder of the paper, I will show that there are indeed ways to accommodate intuitions that arguably underlie the principle without having to commit to it.

Note, to begin with, that no adherent of In-By-Gen would agree to the claim that generalizations explain merely in virtue of being generalizations. Rather, the idea is that if a generalization figures within an explanation, it does so by virtue of possessing a certain trait. The historically most popular idea in this respect is that some generalizations explain in virtue of being laws. When rejecting In-By-Gen, identifying laws with generalizations of course means to deny that laws play an explanatory role with respect to their instances. We have seen in Sect. 4 that there are ways to retain the idea that laws play an epistemically important role with respect to their instances even on such a view by severing the link between the notions of explanation and understanding. Moreover, the view that laws are nothing but generalizations of a certain kind is not mandatory. In particular, it has often been argued that even though laws are closely related to generalizations, they are not generalizations themselves but rather second-order facts about necessary connections between universals (thus Armstrong 1983 and Dretske 1977), or perhaps some kind of sui generis facts (thus Maudlin 2007). For present purposes, I don’t need to commit myself to any of those views. The crucial point is that it is perfectly consistent to claim that laws are explanatory while rejecting In-By-Gen.

More important for present purposes is that rejecting In-By-Gen is compatible with acknowledging that universally quantified sentences are often used to successfully communicate explanations. After all, it is a commonplace that the information someone intends to convey by using a certain sentence need not coincide (or even overlap) with the sentence’s semantic content.⁴² In particular, when using a sentence of the form $\ulcorner$All Fs are $G\urcorner$ within an explanation, its contribution to communicating the explanation need not be the proposition semantically expressed by the sentence (i.e. the proposition that all Fs are G); it can also be a proposition conveyed by such a sentence, e.g. the Armstrongian law that F-hood necessitates G-hood.⁴³

That the apparent appeal to generalizations in scientific explanations need not always be taken at face value is a point familiar from the debate on ceteris paribus clauses. Even though scientists are aware that certain laws do not hold unrestrictedly, or only in cases that are never actualized, they nonetheless often state laws by unrestricted quantifications in the context of explanations.⁴⁴ But from this one of course does not have to conclude that scientists engaged in this practice try to explain truths by falsities. Rather, one usually does not take the explanatory import of what is said in scientific discourse to be exhausted by what is semantically expressed. The present suggestion is essentially a generalization of this familiar idea.

Rejecting In-By-Gen thus in no way forces us to deny that sentences which semantically express generalizations often have an explanatory import: such sentences may be used to convey laws. Note, however, that the account is by no means wedded to the view that conveying laws is the only source from which the explanatory import of such sentences may derive. This is important, since there seem to be generalizations that do not stand in any obvious connection to laws, but nonetheless apparently have explanatory power. Let me elaborate this point briefly. First, whatever counts as a genuine law, it seems initially plausible that laws can confer their explanatory power to generalizations that are themselves not laws. Even if the generalizations corresponding to Kepler’s ‘laws’ of planetary motion do not express any law (because they are too restricted in scope), it is often argued that they possess explanatory power vis-a-vis their instances because they are entailed by certain fundamental laws of physics (whether or not those laws are taken to be universal quantifications).⁴⁵ Second, it seems initially plausible that generalizations have an explanatory role in special sciences such as biology or economy. While those sciences perhaps do not deal with anything that passes the criteria for being a law in the strictest sense of the term, it seems utterly plausible that they provide explanations of phenomena they investigate by subsuming them under certain generalizations that stand in no direct connection with laws. J. Woodward has argued extensively for this.⁴⁶ Think, for instance, of the so-called laws of supply and demand in economics, e.g. the law that the average price for a product will go down if the demand for it decreases while the supply remains unchanged. This is clearly no law in the sense of the term that philosophers generally have in mind (because it arguably only holds for a very restricted range of phenomena), but it seems to do explanatory work in economic theorizing.

None of this puts the present account under pressure. The only aspect of the account that is essential to ensure consistency with Gen-By-In, Unity, and Asymmetry is the idea that the explanatory role of universally quantified sentences does not derive from the propositions semantically expressed by them, but rather from something else they convey. That something else need not be a law. The account is, for instance, perfectly compatible with mechanistic views on explanation: in particular, we may take a given universally quantified sentence to convey (in a suitable context) that there is a mechanism that, given certain initial conditions, bring about an event the explanandum is about.⁴⁷ The idea also seems compatible with Woodward’s influential counterfactual theory of explanation. Admittedly, Woodward officially sticks with the view that explanations involve generalizations.⁴⁸ Yet, he emphasizes that “it is physical dependency relations, as expressed by the relevant counterfactuals about what would happen under interventions, that are primary or fundamental in causal explanations” Woodward (2003, 202). Hence, what the explanation rests on is not the general proposition semantically expressed by a suitable universally quantified sentence, but rather a claim about a relation of dependency (that Woodward cashes out in terms of manipulability). It seems to be compatible with this view to factor out the explanatory role of the former proposition and take universally quantified claims that appear in causal explanations to contribute only the latter. Admittedly, it would require some more work to fill in the details of such an account. But that is for another day.

6 Conclusion

If one accepts standard assumptions about the grounds of logically complex facts and takes grounding-links to give rise to explanations, one is committed to take universal generalizations to be explained by their instances. I have argued that adherents of this view can escape an apparent conflict between it and widely shared views about the role of universal generalizations in scientific explanations. I have moreover claimed that this does not require to postulate different notions of explanation. Nor does it require to give up the asymmetry of explanation. Finally, I have tried to show that even if one rejects the view that the propositions semantically expressed by universal generalizations play an explanatory role, there are ways to account for the apparently widespread use of universally generalized sentences in scientific explanations, and for certain epistemic traits they are often taken to possess.⁴⁹