¹ Pasternak, ‘In Everything I seek to grasp…’ (1956, trans. Avril Pyman).

² Rosen, Gideon, ‘The Limits of Contingency’, Identity and Modality, (ed.) MacBride, Fraser (Oxford: Oxford University Press, 2006), 13–39Google Scholar. See also Schaffer, Jonathan, ‘Monism: The Priority of the Whole’, Philosophical Review 119 (2010), 31–76CrossRefGoogle Scholar, at 56.

³ Kment, Boris, Modality and Explanatory Reasoning (Oxford: Oxford University Press, 2014)Google Scholar, at 5 and 167–73.

⁴ Wilsch, Tobias, ‘The Nomological Account of Ground’, Philosophical Studies 172 (2015), 3293–312Google Scholar, at 3294.

⁵ Glazier, Martin, ‘Laws and the Completeness of the Fundamental’, Reality Making (ed.) Jago, Mark (Oxford: Oxford University Press, 2016): 11–37Google Scholar.

⁶ Rosen, Gideon, ‘Ground By Law’, Philosophical Issues 27: Metaphysics (2017), 279–301CrossRefGoogle Scholar.

⁷ Quine, W. V., ‘Things and their Place in Theories’, Theories and Things, 1–23 (Cambridge, MA: Harvard University Press, 1981)Google Scholar, at 98.

⁸ Lewis, David, ‘New Work for a Theory of Universals’, Australasian Journal of Philosophy 61 (1983), 343–77Google Scholar.

⁹ Armstrong, David, A World of States of Affairs (Cambridge: Cambridge University Press, 1997)CrossRefGoogle Scholar, at 50; Rosen, Gideon, ‘Metaphysical Dependence: Grounding and Reduction’, Modality: Metaphysics, Logic, and E pistemology (eds) Hale, Bob and Hoffmann, Aviv, 109–36 (Oxford: Oxford University Press, 2010)Google Scholar, at 126.

¹⁰ Fine, Kit, ‘The Logic of Essence’, Journal of Philosophical Logic 24 (1995), 241–73Google Scholar, at 271.

¹¹ Schaffer, Jonathan, ‘On What Grounds What’, Metametaphysics (eds) Chalmers, David, Manley, David, and Wasserman, Ryan, 347–83 (Oxford: Oxford University Press, 2009)Google Scholar; Schaffer, Jonathan, ‘Grounding in the Image of Causation’, Philosophical Studies 173 (2016), 49–100Google Scholar; Correia, Fabrice and Schnieder, Benjamin, ‘Grounding: An Opinionated Introduction’, Metaphysical Grounding: Understanding the Structure of Reality (eds) Correia, Fabrice and Schnieder, Benjamin, 1–36 (Cambridge: Cambridge University Press, 2012)Google Scholar.

¹² For instance, Bennett's pluralistic notion of building would work equally well (Bennett, Karen, Making Things Up (Oxford: Oxford University Press, 2017)CrossRefGoogle Scholar). Though Bennett herself (at 61–2) is non-committal with respect to explanation.

¹³ Hofweber, Thomas, Ontology and the Ambitions of Metaphysics (Oxford: Oxford University Press, 2016), at 328–40Google Scholar.

¹⁴ Wilson, Jessica, ‘No Work for a Theory of Grounding’, Inquiry 57 (2014), 535–79CrossRefGoogle Scholar; c.f. Schaffer, Jonathan, ‘Ground Rules: Lessons from Wilson’, Scientific Composition and Metaphysical Ground (eds) Aizawa, Ken and Gillett, Carl, 143–70 (London: Palgrave-MacMillan, 2016)Google Scholar; Wilson, Jessica, ‘The Priority and Unity Arguments for Grounding’, Scientific Composition and Metaphysical Ground (eds) Aizawa, Ken and Gillett, Carl, 171–204. (London: Palgrave-MacMillan, 2016)CrossRefGoogle Scholar.

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¹⁶ Salmon, Wesley, ‘Why Ask, “Why?”?: An Inquiry Concerning Scientific Explanation. Proceedings and Addresses of the American Philosophical Association 51 (1978), 683–705Google Scholar.

¹⁷ Lewis, David, ‘Causal Explanation’, Philosophical Papers Volume II, 214–40 (Oxford: Oxford University Press, 1986)Google Scholar.

¹⁸ I say ‘in the vicinity’ because all of the authors I mention qualify their views in various ways. For instance, Lewis (1986) restricts his account to explanations of the occurrences of particular events. He also allows causal explanations that do not actually cite causes but merely provide some information about the causal history of the particular event at issue. If none of these authors would oppose 1 in the end, all the better.

¹⁹ See especially Lange's, Marc Because without Cause: Non-Causal Explanations in Science and Mathematics (Oxford: Oxford University Press, 2017)Google Scholar.

²⁰ Op. cit. note 16.

²¹ Ruben, David-Hillel, Explaining Explanation (New York: Routledge, 1990) at 210Google Scholar; Kim, Jaegwon, ‘Explanatory Knowledge and Metaphysical Dependence’, Philosophical Issues 5 (1994), 51–69Google Scholar; Woodward, James, Making Things Happen (Oxford: Oxford University Press, 2003), at 31Google Scholar; Strevens, Michael, Depth: An Account of Scientific Explanation (Cambridge, MA: Harvard University Press, 2008), at 179–80Google Scholar; Skow, Bradford, Reasons Why (Oxford: Oxford University Press, 2016) at 3Google Scholar; Schaffer, Jonathan, ‘Grounding in the Image of Causation’, Philosophical Studies 173 (2016), 49–100Google Scholar, at 83.

²² Op. cit. note 3, at 5.

²³ I am thus using ‘law of nature’ in Hitchcock & Woodward's non-controversial ‘inclusive sense’ (Christopher Hitchcock and James Woodward, ‘Explanatory Generalizations, Part I: A Counterfactual Account’ , Noûs 37 (2003), 1– 24, at 3). It is controversial that causal explanation requires fundamental and/or exceptionless generalizations. For instance – to take a case from Woodward (‘Explanation and Invariance in the Special Sciences’, British Journal for the Philosophy of Science 51: 197–254) – one might causally explain crop production by an equation treating crop production as a linear function from amount of rainfall, where the equation involved is neither fundamental nor exceptionless, and only holds for ‘intermediate’ rainfall values (beyond which all the crops die from drought or from flood). It may be argued that there need to be fundamental and exceptionless background laws behind this rainfall-crop equation, but in the main text I remain neutral on the matter.

²⁴ Putnam, Hilary, ‘Philosophy and Our Mental Life’, Mind, Language, and Reality: Philosophical Papers, Volume 2, 291–303 (Cambridge: Cambridge University Press, 1975) at 295–7Google Scholar.

²⁵ Friedman, Michael, ‘Explanation and Scientific Understanding’, The Journal of Philosophy 71 (1974), 5–19Google Scholar; Kitcher, Philip, ‘Explanatory Unification’, Philosophy of Science 48 (1981), 507–31Google Scholar.

²⁶ Op. cit. note 21.

²⁷ Skyrms, Brian, Causal Necessity: A Pragmatic Investigation of the Necessity of Laws (New Haven, CT: Yale University Press, 1980)Google Scholar, at 11.

²⁸ Nozick, Robert, Philosophical Explanations (Cambridge, MA: Harvard University Press, 1981)Google Scholar, at 12.

²⁹ Op. cit. note 21, at 54.

³⁰ Op. cit. note 21, at 3.

³¹ Baumberger, Christoph, Beisbart, Claus, and Brun, George, ‘What is Understanding? An Overview of Recent Debates in Epistemology and Philosophy of Science’, Explaining Understanding, (eds) Grimm, Stephen, Baumberger, Christoph, and Ammon, Sabine, 1–34 (New York: Taylor & Francis, 2017)Google Scholar, at 13.

³² Grimm, Stephen, ‘The Goal of Explanation’, Studies in History and Philosophy of Science 41 (2010), 337–44Google Scholar, at 340–1.

³³ Indeed Grimm (2010, op. cit. note 32, at 341, fn. 23; also Grimm, Stephen, ‘Is Understanding a Species of Knowledge?’ British Journal for the Philosophy of Science 57 (2006), 515–35Google Scholar, at 532–3) explicitly connects explanatory understanding with Woodward's (op. cit. note 21) idea of answering ‘what if things had been different’ questions. In this sense the manipulation and understanding roles are not independent, but interconnected aspects of the link between explanations and counterfactuals.

³⁴ Op. cit. note 13, at 331–40.

³⁵ I say ‘some’ because Hofweber says that others are backed by conceptual connections. I am focused on Hofweber's account of metaphysical explanation in the cases not backed by conceptual connections, which (as I read him) includes the Euthyphro example as well as the grounding of the moral in the natural.

³⁶ Wilsch (op. cit. n.4: 2; see also Rosen op. cit. n.9: 130, Audi op. cit. n.3: 697–8) puts this point about patterns nicely, with respect to grounding claims: ‘[C]onsider the grounding-truth expressed by the following sentence: “That the tomato t is red grounds that t is colored.” It follows from this grounding-truth that any truth expressed by a sentence of the form “o is red” grounds the corresponding truth expressed by “o is colored.” … [G]rounding-truths exhibit general patterns.’

³⁷ Hempel, , Carl, and Oppenheim, Paul, ‘Studies in the Logic of Explanation’, Philosophy of Science 15 (1948), 135–75Google Scholar.

³⁸ Op. cit. note 23.

³⁹ Op. cit. note 11; see also Alastair Wilson, ‘Metaphysical Causation’, Noûs (forthcoming).

⁴⁰ Scriven, Michael, ‘Truisms as the Grounds for Historical Explanation’, Theories of History (ed.) Gardner, Patrick, 443–75 (Glencoe, IL: The Free Press, 1959)Google Scholar.

⁴¹ Op. cit. note 21.

⁴² Scriven (op. cit. note 40) holds that laws merely play an evidential role, in justifying causal explanatory claims connecting causes to effect. Skow (op. cit. note 21, at 81–91) holds that laws do not help to give a reason why the effect occurred, but merely supply a ‘second-order reason’ as to why the causes caused the effect. I am sympathetic to Skow's claim that laws supply a second-order reason. I just think they also provide a first-order reason why, since I think they are needed for understanding why. But in any event, I need not have a dispute with Skow here, just so long as he would also accept that first-order reasons require the existence of law-involving second-order reasons. For that would suffice to make the existence of laws of nature into a requirement for the possibility of causal explanation, which is all I am after.

⁴³ Kim (1994) op. cit. note 21, at 68; see also Ruben (1990) op. cit. note 21; Woodward (2003) op. cit. note 21; Alexander Reutlinger, ‘Is There a Monist Theory of Causal and non-Causal Explanations? The Counterfactual Theory of Scientific Explanation’, Philosophy of Science 83 (2016), 733–45.

⁴⁴ Shaheen, Jonathan, ‘The Causal Metaphor Account of Metaphysical Explanation’, Philosophical Studies 174 (2017), 553–78Google Scholar; also ‘Ambiguity and Explanation’, Inquiry (forthcoming) contests the univocity of ‘because’, arguing that it is polysemous between a baseline causal meaning and a metaphorically extended meaning that covers the metaphysical case. I disagree with Shaheen's judgments about key sentences but lack the space to discuss the matter. That said, I need have no argument with Shaheen here, since he (2017, at 567–8) sees the metaphysical sense of ‘because’ as a structural metaphor, extending the baseline causal notion by holding fixed its structural aspects. Insofar as the involvement of laws is a core structural aspect of causal explanation, Shaheen's view equally predicts that metaphysical explanation requires structurally analogous laws of metaphysics.

⁴⁵ I lack the space to discuss mathematical explanation, but want to flag it as a potential difficulty for the dependence-based view. The general picture I would like to defend sees the conclusion of a theorem as a dependent result, with the premises (and ultimately the axioms) serving as sources, and inference rules serving as linking principles (/laws of logic). But – unlike in the causal and metaphysical case – there seems to be no fixed order as to what should count as axiom or theorem, especially in cases of mutual inter-derivability (such as the cluster of propositions inter-derivable with Euclid's parallel postulate, in the context of the rest of Euclidean geometry). There is also the more subtle problem of saying why certain proofs (e.g. brute force proofs) lack explanatory power.

⁴⁶ In some cases this involves counter-metaphysical conditionals. For instance, supposing that it is metaphysically necessary that the empty set exists with, one still wants to be able to say that if the empty set did not exist, then its singleton would not exist. See Schaffer 2016 (op. cit. note 11, at 71–3) and Wilson (Alastair Wilson, ‘Grounding Entails Counterpossible Non-Triviality’ Philosophy and Phenomenological Research (forthcoming)) for further discussion.

⁴⁷ Notation: S is the signature characterizing the system under study, via a triple of exogenous variables, endogenous variables, and a range function mapping variables to allowed values. L is the linkage adding the ‘dynamics’ of the system, via a set of structural equations evaluating each endogenous variable as a function of other variables. A is the assignment setting the actual values of the exogenous variables. See Schaffer 2016 (op. cit. note 11) for further clarification of the formalism employed, and discussion of the applicability of such models to a wide range of metaphysical cases. (See also Koslicki, Kathrin, ‘Where Grounding and Causation Part Ways: Comments on Schaffer’, Philosophical Studies 173 (2016), 101–12Google Scholar; and Jessica Wilson 2016 (op. cit. note 14) for criticisms.)

⁴⁸ See Schaffer 2016 (op. cit. note 11, at 76–82) for application of the structural equation framework to a wide range of paradigm cases of grounding.

⁴⁹ Op. cit. note 3, at 167–73.

⁵⁰ Op. cit. note 4, at 3294.

⁵¹ Op. cit. note 5.

⁵² Op. cit. note 6.

⁵³ Fine, Kit, ‘The Logic of Essence’, Journal of Philosophical Logic 24 (1995), 241–73Google Scholar.

⁵⁴ Op. cit. note 9.

⁵⁵ Dasgupta, Shamik, ‘The Possibility of Physicalism’, The Journal of Philosophy 111 (2014), 557–92Google Scholar.

⁵⁶ Op. cit. note 3.

⁵⁷ Op. cit. note 5.

⁵⁸ Fine, Kit, ‘Guide to Ground’, Metaphysical Grounding: Understanding the Structure of Reality (eds) Correia, Fabric and Schnieder, Benjamin, 37–80 (Cambridge: Cambridge University Press, 2012)Google Scholar.

⁵⁹ Op. cit. note 3, at 167.

⁶⁰ It may be thought that pinning the law on {Socrates} rather than Socrates helps give direction to the explanation. But really the explanatory asymmetry in hierarchical set theory stems from the recursive structure of the set-formation machinery, which outputs set S at stage n+1 when fed the members of S at stage n. It is the structure of the machinery that does the work, not any claims about essences.

⁶¹ It is controversial whether the causal asymmetry is sourced from the dynamical laws themselves, or from the external temporal order, etc. But surely the order of causal explanation between the throwing of the rock and the shattering of the window does not await an assignment of essences. It is not as if the physicist is incapable of giving a causal explanation of the window shattering, until she first settles the matter of essences.

⁶² My own view (Schaffer 2009, op. cit. note 11) is that grounding relates entities – not just facts – so a fundamental entity is just an ungrounded entity. In the main text I treat a fundamental entity as one whose existence fact is fundamental, so that the discussion can be open to those (such as Rosen 2010, op. cit. note 9) who restrict grounding to facts.

⁶³ Carroll, John, The Laws of Nature (Cambridge: Cambridge University Press, 1994)Google Scholar; Maudlin, Tim, ‘A Modest Proposal Concerning Laws, Counterfactuals, and Explanations’, The Metaphysics within Physics, 5–49 (Oxford: Oxford University Press, 2007)CrossRefGoogle Scholar.

⁶⁴ Thanks to Paul Audi, Karen Bennett, Selim Berker, Shamik Dasgupta, Janelle Derstine, Christopher Frugé, Martin Glazier, Antonella Mallozzi, Noël Saenz, Erica Shumener, Alex Skiles, Ted Sider, Isaac Wilhelm, Tobias Wilsch, Jessica Wilson, and Richard Woodward, as well as to the Rutgers Metaphysics Group, and audiences at Sofia XXI: Grounding in Huatulco, and at Workshop: Metaphysical Laws in Hamburg.