Explanation, justification, and egalitarianism

This paper argues that the philosophy of explanation can help inform core debates in value theory. Specifically, it argues that there is a consistent parallelism between the properties of explanation and the properties of justification such that one can reasonably infer that any property of explanation has a counterpart property of justification. Thus, by appealing to facts about the nature of explanation, one can derive various conclusions about the justifications offered by normative theorists. The paper illustrates this point by considering a debate within political philosophy over whether inequality requires justification in a way that equality does not. Egalitarians typically presume an affirmative answer to this question. However, libertarian critics note that this justificatory asymmetry cannot be simply assumed without argument. This paper argues that, by appealing to the explanation-justification parallelism, one can resolve this debate in favor of the egalitarians, as there are two properties of explanation, the justificatory analogs of which vindicate the egalitarian presumption.

some set of conditions under which inequality is justified, with the stringency of those conditions varying from theory to theory.
For example, John Rawls famously argues that inequality is not justified if it is the result of previous inequalities in the distribution of goods (e.g., inherited wealth or private schooling), as he considers one's prior social circumstances to be "arbitrary from a moral point of view," and, thus, incapable of justifying any resulting inequality. 2 However, his Difference Principle maintains that inequality is justified if (and only if) it is necessary for improving the situation of the worst-off relative to their situation absent the inequality.
More stringently, luck egalitarians contend that only culpable choices can justify inequality. 3 Given this commitment, a core aspect of the luck egalitarian project involves showing that other properties of persons do not justify inequality. For example, G. A. Cohen considers and rejects a number of possible justifications for inequality, including people's "status based rights assignments" (i.e., social class membership), their prior social circumstances, and their traits and abilities. 4 Similarly, Michael Otsuka considers the person who ends up worse off than others due to luck and asks, "why me?". Given the apparent absence of a satisfactory answer to this question, Otsuka deems her predicament unjust-a result suggesting the more general conclusion that bad luck fails to justify inequality. 5 And Kasper Lippert-Rasmussen argues that one basis for endorsing luck egalitarianism is the rejection of legal status, social background, and innate abilities as acceptable justifications for inequality. 6 These diverse arguments all have a shared premise: inequality demands justification and is unjust if such justification cannot be given. Indeed, this premise is made explicit in Otsuka's argument, where the worse-off actually voice this demand. And both Cohen and Lippert-Rasmussen conclude that inequality is unjust when various luck-based properties of the worse-off fail to justify the inequality. Additionally, these egalitarians assume that equality does not require such justification. While Otsuka posits the person who asks, "why me?" in response to being left worse-off, he introduces no similar complaint by a person who finds herself equally situated vis-à-vis others. Similarly, luck egalitarians who find luck to be an inadequate justification for inequality typically have no such complaint about luck-based equality, as they seemingly do not take equality to demand justification. 7 Thus, egalitarians assume that inequality demands justification (Thesis 1) in a way that equality does not (Thesis 2). Call the conjunction of these two theses the egalitarian presumption. 8 2 Rawls (1971, pp. 72-3). 3 Some prominent statements of the luck egalitarian position include those made by Arneson et al. (2011, p. 243), Cohen (2006, p. 440;2009a, p. 7;2009b, pp. 17-8;2011, p. 13), Lippert-Rasmussen (2015, p. 1), Temkin (1993, p. 13) and Vallentyne (2008, p. 58). 4 Cohen (2009b). 5 Otsuka (2004, pp. 151-2). 6 Lippert-Rasmussen (2015, pp. 7-9). 7 As Susan Hurley notes, it is this kind of presumption that separates an egalitarian theory from a luckist one that holds that a distributive arrangement-either equal or unequal-is just iff holdings reflect choice rather than luck. See, Hurley (2001, 58-9). 8 Isaiah Berlin provides an explicit statement of the egalitarian presumption when he writes: To state the egalitarian presumption a bit more precisely, Thesis 1's contention that inequality demands justification can be understood as follows: for any given inequality U, U is just if and only if (a) there is some true justifying conditional of the form "If p then U is just" and (b) p is true. This presumption is what allows the egalitarians discussed above to conclude that existing inequalities are unjust when they do not satisfy conditions (a) and (b). For example, luck egalitarians like Cohen believe talent-based inequalities can be proven unjust by merely showing that differences in talent are not sufficient for rendering that inequality just-i.e., it is not the case that (a) obtains with respect to the inequality in question.
Thesis 2 can then be understood as the denial that equality needs its own justifying conditional with a true antecedent in order to be just. If an equality obtains between two people, that inequality can be just even if there is no true conditional asserting that if some condition obtains then that equality is just-or there is such a conditional but its antecedent does not obtain. For example, suppose one posits that if two people have equally contributed to the economy, then it is just that they have equal holdings. Thesis 2 maintains that, even if it turns out that this supposition is false-as are all other posited conditionals that have the justness of the equality as their consequent-the equality in question might still be just. Further, it maintains that even if this conditional is true, the equality could be just even if the two people have not contributed equally to society.
Note that this second thesis is compatible with holding that any given equality has a true justifying conditional with a true antecedent. Suppose that one believes that a given equality is just because (a) if two people have contributed equally, then it is just that they have equal holdings and (b) the two people have contributed equally; this view would be entirely consistent with Thesis 2's claim that the equality in question would still be just if either (a) or (b) did not obtain. 9 Additionally, the thesis is compatible with the view that there are true conditionals that provide the sufficient conditions of an equality being unjust. For example, someone might affirm Thesis 2 while also thinking that if equality undermines desert or Pareto efficiency, then it is unjust. Thus, one might both embrace the thesis while being a pluralist who holds that a conflict between equality and some other consideration renders that equality unjust.
Finally, it is worth noting that one might be an egalitarian while rejecting one or both of the egalitarian presumption's theses. For example, an egalitarian might believe that, contrary to Thesis 1, inequality does not demand justification (i.e., it might be just even if it lacks a justifying conditional with a true antecedent); however, she might hold that Footnote 8 continued The assumption is that equality needs no reasons, only inequality does so; that uniformity… need not be specially accounted for, whereas differences… need explanation and, as a rule, justification. If I have a cake and there are ten persons among whom I wish to divide it, then if I give exactly one tenth to each, this will not… call for justification; whereas if I depart from this principle of equal division I am expected to produce a special reason.
Berlin takes this presumption to be fundamental and, thus, not apt for philosophical defense; by contrast, this paper seeks to provide such a defense. See Berlin (1955-56, pp. 305, 326). 9 The example posits a desertist justifying conditional, but one might substitute any alternative justifying conditional while still affirming Thesis 2. Egalitarians might be more attracted to the following justifying conditional: an equality between persons is just if inequality between them would be unjust. Further, they might hold that injustice of the counterfactual inequality follows from the conjunction of Thesis 1 and the fact that such inequality would lack a justifying conditional with a true antecedent. Justifications of this kind will be discussed at length in Sect. 9. most inequalities meet some sufficient condition of injustice. 10 Alternatively, egalitarians might reject Thesis 2 and hold that equality demands justification while also maintaining that the antecedent of the justifying conditional can be readily realized. 11 Such positions would still be recognizably egalitarian in that they condemn existing inequalities as unjust and affirm the justice of egalitarian alternatives. However, they will be comparatively less egalitarian than positions that incorporate the egalitarian presumption, as the latter will declare inequality unjust and/or equality just in more possible worlds than positions that reject the presumption.

Brennan's challenge
While not all egalitarians endorse the egalitarian presumption, many influential egalitarian arguments tacitly assume its truth (e.g., those listed in the previous section). As a result, these arguments are vulnerable to a foundational objection advanced by those who deny the presumption. For example, Jason Brennan explicitly rejects Cohen's argument for luck egalitarianism by rejecting the egalitarian presumption: We cannot just assume, without argument, that equality is a baseline from which departures must be justified. But without a sound argument for this assumption, Cohen's strategy of refuting justifications for inequality makes no difference, even if it succeeds. A non-egalitarian can just say, 'Sure, these arguments for inequality fail, but as far as I'm concerned, they do not need to succeed, because we have no reason to presume equality is a baseline from which departures must be justified.' 12 Brennan's objection targets Thesis 1 by denying that inequality demands justification. However, he might equally have denied Thesis 2 by insisting that equality demands justification-i.e., is prima facie unjust in the same way that egalitarians presume inequality to be. If conceded, these denials would then pose a serious threat to the egalitarian project, as they would undermine the broad array of egalitarian theories that rest on the egalitarian presumption. 10 Although this position is internally consistent, there do not appear to be any philosophers who actually endorse it. 11 Larry Temkin is an egalitarian who rejects Thesis 2, as he takes undeserved equality to be morally objectionable. See Temkin (1993, pp. 138-40). Such a view sets him apart from egalitarians like Shlomi Segall (2016) who explicitly endorses Thesis 2. This paper can be understood as an intervention in this intraegalitarian debate, as its argument in defense of Thesis 2 provides a novel reason to side with Segall over Temkin. Similarly, it takes sides in the debate between Hurley and luck egalitarians discussed in footnote 7. Contra luck egalitarians like Cohen, Hurley posits a luckist position where equality does demand justification (specifically, equality must be justified by showing that it is not a product of luck). If this paper succeeds in showing that Thesis 2 is correct, then that would amount to a rejection of Hurley's luckist position. 12 Brennan (2015, p. 115).

The argument
This paper attempts to defeat Brennan's challenge by providing a defense of (both theses of) the egalitarian presumption. To briefly lay out its argumentative strategy, begin by noting that what is at issue is a metaphysical question regarding the properties of justifications understood as semantic objects (i.e., sets of propositions), where such properties include the relations in which justifications stand vis-à-vis other objects. Specifically, what has to be determined is what states of affairs are the appropriate objects of justification. Such a general theory of justification can then be applied to the particular question of whether inequality requires (or is an appropriate object of) justification in a way that equality does (is) not.
One way of developing such a theory is to simply reflect upon justification and rely upon intuitive judgments about its character to determine its properties. However, such an approach will be at high risk of question begging when it comes to rendering a judgment about justification and its relation to (in)equality. To avoid this problem, the paper will develop a theory of justification without assuming any controversial claims about the relation between justification and equality. Specifically, it will begin by considering a different sort of semantic object that, as it turns out, has much in common with justifications, namely explanations. The paper will posit (in Sect. 5) a number of properties of explanation, and, for each one, suggest that there is an analogous property characteristic of justification. It will then make the inductive inference (discussed in Sects. 6 and 8) that this pattern extends to cover two additional properties of explanation (presented in Sects. 7 and 9), where the justificatory analogs of those properties serve to vindicate the egalitarian presumption.

Varieties of explanation
In drawing an analogy between explanation and justification, one difficulty that must be confronted at the outset is that there is some reason for doubting that explanation is a unitary phenomenon. The basis for this doubt is that there are many uses of the term "explain" that appear to have little in common: the shape of a flag explains the shape of its shadow; an informative placard explains the significance of a piece of art; a philosophical argument might explain how free will and determinism are compatible; your cousin explains to you how to bake a cake; the arrangement of the atoms that make up a cube explain its shape; and so on.
Carl Hempel has argued that some of these uses of "explain" have as little in common with others as the use of "proof" in "mathematical proof" has with its use in "'86 proof Scotch." 13 By contrast, Robert J. Matthews suggests that all acts properly called an "explanation" share the property of being an effort to produce understanding. 14 However, even if Matthews is correct, one might worry that such a general account of explanation makes it impossible to identify any other shared properties of explanation 13 Hempel (1965, p. 413). 14 Matthews (1993, p. 357). that might help illuminate justification and thereby settle the debate over the egalitarian presumption.
Alternatively, one might attempt to draw the analogy between justification and some subset of explanations whose members share a larger set of properties than the single property posited by Matthews. However, one difficulty with this approach is that there are many subcategories of explanation including scientific explanation, 15 grounding explanation, 16 philosophical explanation, 17 analogical explanation, 18 rule-based explanation, 19 mathematical explanation, 20 probabilistic explanation, 21 functional explanation, 22 consequence explanation, 23 historical explanation, 24 psychological explanation, 25 and moral explanations, 26 among others. Thus, one might worry, first, that it will be difficult to identify which of these varieties of explanation is the best analog for justification vis-à-vis rules of use.
Second, one might worry that, even if one identifies such an analog, the existence of other forms of explanation that are slightly-less-perfect analogs might make the identified correspondence between the properties of justification and those of the posited analog seem to be merely a coincidence that does not allow for inductive generalization of the kind the argument requires. Indeed, the inductive claim that, because five of the analog's properties correspond to those of justification, its sixth will as well loses its plausibility if there is some inferior analog which has four properties that correspond, but a fifth that does not.
In light of these worries, the paper will attempt to strike a balance where it considers explanation in its most general form that still allows for the ascription of properties beyond the minimal "intended to produce understanding" account posited by Roberts above. Thus, the paper will attempt to posit properties of all explanations that are explanations of why certain facts obtain (in contrast to explanations of how to do things, what things are, when things will happen, etc.). 27 Or, to put this point slightly differently, what is of interest here are explanations of particular facts, where such explanations can be introduced with either the conjunction "because" or the prepositional phrase "because of." The claim is that such explanations share a number of properties including ones relevant to the debate over the egalitarian presumption. Thus, 15 See: Woodward (2017) the effort in the subsequent section to identify some of these properties will draw upon examples from across the literature on explanation.
Additionally, the hope is that, by focusing on explanation as a more general phenomenon, the paper will be able to sidestep much of the controversy that surrounds more specific subspecies of explanation. For example, while David Lewis argues that all scientific explanation is causal, 28 there are many dissenters who posit various forms non-causal scientific explanation. 29 However, because this debate is specific to scientific explanation, the paper can avoid taking a position on the matter, as it is concerned with all varieties of explanation, including explanations that are uncontroversially non-causal (e.g., mathematical explanations of mathematical facts). Granted, some of the proposed properties might still be contested via the denial that all forms of explanation have the particular property in question. However, as long as no shared property of explanation can be found that lacks an analogous property of justification, then the analogy between explanation and justification will hold and the argument will go through.

Explanation and justification
With this groundwork in place, it is now possible to introduce the shared properties of explanation and argue that, for each such property, there is an analogous, counterpart property of justification. This correspondence will then serve as the basis for an inductive inference presented in Sect. 6. None of the posited properties is essential to the argument (though the "autonomous facts" subsection will be referenced later), so if one posited property is judged to be excessively controversial, it can be discarded without abandoning the conclusion about egalitarianism defended below. Thus, one must merely grant that enough of the posited properties (a) obtain and (b) have justificatory analogs to warrant an inductive inference from additional posited properties of explanation to analogous properties of justification.
Consider, then, the following proposed properties of explanations and justifications:

Acts and objects
Above, explanations have been described as "semantic objects"; however, as Sylvain Bromberger has noted, the term "explanation" is actually ambiguous, referring sometimes to an act of explaining (what he calls the "performance sense" of the word), or, alternatively the propositional content expressed in these speech acts (the "text sense"), with the latter being the semantic objects mentioned above. 30 Taking the latter as the primary objects of inquiry (though both kinds of explanation will be discussed below), one can then say that explanations have the property of being expressed by a particular 28 Lewis (1986). 29 See, e.g., Lipton (1993), . 30 Bromberger (1992, pp. 18-9).
variety of act, where both such acts and associated semantic objects share a referring term. 31 Additionally, it is often said that facts or things (of the non-semantic variety) explain other facts or things. 32 For example, just as one might say that the theory of gravity explains why objects adhere to the Earth's surface, one might also say that the fact of the Earth's massiveness explains the observed phenomenon. This relationship between linguistic practice, explanations qua semantic objects, and the objects referred to by those explanations is another distinctive property of explanation. 33 Analogously, the term "justification" is ambiguous, referring alternately to a particular kind of act as well as the semantic content expressed by that act. Thus, justifications can also be said to have the property of being expressed by counterpart acts, where both acts and semantic objects share a referring term. Additionally, people often speak as though facts or objects justify other facts or things (e.g., "we are doing things this way because you agreed to it!"). Given this, explanation and justification are analogous in that their referring terms are used in analogous ways.

Accomplishment terms
A second property of explanations is that explanatory acts are things that can be completed (as opposed to merely halted). Indeed, as Bromberger has noted "to explain" is an accomplishment term with distinctive semantic properties indicating the completable nature of the act to which it refers. 34 Specifically, such terms can naturally be used in sentences of the form "How long did it take for P to φ?"-with the relevant contrast being activity terms, which are used in questions of the form "How long did P φ?". 35 Similarly, justificatory acts are things that can be completed, making the associated verb "to justify" an accomplishment term. Indeed, just as one might ask how long it took someone to explain some state of affairs, one might similarly ask how long it took someone to justify some state of affairs-with the question "How long did P justify X?" seeming malformed in just the same way as does the question "How long did P explain X?". 31 There may be certain accounts of word individuation that deny that identical words can have distinct referents (e.g., perhaps on the view put forward by Kaplan (1990Kaplan ( , 2011. Thus, the linguistic relation between explanatory acts and explanatory products would be that tokens of their respective referring terms are uttered/inscribed identically (as opposed to the referring words, themselves, being identical]. 32 van Fraassen (1980, 100-101). 33 Going forward, the paper will generally stick to semantic language, but will occasionally talk of some fact or thing explaining another. In such cases, this should be understood as convenient shorthand for some set of propositions (the explanans) standing in an explanatory relation to some fact (or facts) where the description of that fact (or facts) is the explanandum. 34 Bromberger (1992, pp. 20-1). 35 Bromberger (1992, pp. 21-2).

Nomological deduction
In their classic work on scientific explanation, Carl Hempel and Paul Oppenheim argue that the relationship between explanans and explanandum is one of logical deduction where the explanans is a set of premises that, together, logically entail the explanandum (which refers to the fact to be explained). 36 Additionally, they argue that, for a set of premises to count as the explanation of the conclusion, there must be at least one of the premises that is both a law of nature and essential to the argument such that, if it were removed, the argument would be invalid-with this requirement making their theory a deductive-nomological theory of explanation, as opposed to a merely deductive one. For example, the explanation of why a sample of gas expands is that (1) if something is a gas, then it will expand if its temperature is increased and its pressure remains constant; (2) the temperature of the gas is increased; and (3) the pressure of the gas remains constant. Here, because premise 1 is an essential premise that is also a law of nature-and premises 1-3 jointly entail the conclusion that the gas expands-the set of premises 1-3 is the explanation of why the gas expands. 37 There is much debate over what, exactly counts as a law of nature, with little agreement in the literature over how the notion should be analyzed. 38 However, very roughly, Hempel and Oppenheim's motivating thought seems to be that laws of nature are universal generalizations that are sufficiently syntactically simple-where syntactic simplicity is inversely proportionate to the number of exceptions to the generalization. 39 Insofar as one is willing to work with this rough-but-intuitive notion, an analogous property of justifications seems to obtain, as justifications also take the form of deductions featuring an essential, law-like premise. For example, a pescatarian attempting to justify the permissibility of eating fish might argue that (1) if a species of animal does not feel pain, then it is permissible to eat members of that species and (2) fish do not feel pain. 40 Here, premises 1 and 2, together, function as a justification for the deduced conclusion that eating fish is permissible, where premise 1 is also a law-like universal generalization that is essential to the validity of the inference (again, using the rough, intuitive sense of what counts as law-like). Generalizing from this example, a plausible thought is that justifications necessarily take the form of arguments rest- 36 Hempel and Oppenheim (1948). 37 Hempel (1965) also suggests that there can be inductive-statistical explanations in addition to deductivenomological deductions. Very roughly, his idea is that some fact F is successfully explained by the conjunction of (a) some other fact E and (b) the fact that E makes it highly likely that F obtains. However,  presents persuasive reasons for thinking that explanations that are probabilistic in this way can and should be recast as deductive-nomological explanations. Thus, it can be maintained that all explanations (of the kind specified in Sect. 4) have the shared property of taking a deductive-nomological form. However, if Railton is wrong, then this particular property will no longer support the argument, as it will prove to be more parochial than the standard posited in Sect. 4 allows. 38 See: Woodward (2017). 39 One obvious problem with this suggestion is that syntactic simplicity exists relative to a particular ontology; however, this difficulty will be bracketed for these purposes. 40 Note that, while an explanandum is a proposition describing some fact, a justificandum is a proposition describing some state of affairs but where that proposition is within the scope of some affirmative normative operator such as an "it is just that" operator or an "it should be the case that" operator.
ing on some sort of normative principle-where such principles are the justificatory analog for laws of nature. Thus, both explanations and justifications have the property of being nomological deductions of that which they explain/justify. 41

Ellipticity and partialness
Given a conception of explanation as deductive argument, two additional properties of explanation noted by Carl Hempel can be posited. First, the explanatory speech acts in which explanations qua semantic objects are expressed rarely present the entire argument that composes the explanation; rather, they are elliptically formulated such that various premises are left unstated. 42 Putting this more strongly, it might be suggested that such formulations are not just put forward as a matter of empirical fact, but are also acceptable in some normative sense left deliberately undeveloped here.
Analogously, acts of justification are also frequently elliptically formulated, with many premises left unstated. For example, a pescatarian justifying why she is willing to eat fish might simply say, "because fish do not feel pain," without presenting the entire deduction presented above. And, insofar as one is willing to grant the stronger normative claim posited in the final sentence of the previous paragraph, it seems the same could be said of justifications that are elliptically formulated.
Second, Hempel notes that many explanations for some state of affairs are partial in that they do not actually entail the explanandum, but, rather, that the referent of the explanandum will fall into some particular category. 43 For example, he cites Freud's explanation that an erroneous calendar entry occurred because people tend to make mistakes that reflect their subconscious desires. In this case, the argument fails to deduce the specific error Freud made, as it is but one of the many things that reflect his subconscious desires. 44 Similarly, justifications can be partial in this way. For example, socialists who opt to form a non-capitalist commune might justify this choice by citing various purported harms of capitalism; however, the underlying argument might entail only that they ought to live in some non-capitalist fashion, without demonstrating that they ought to live in the particular fashion that they have chosen. Thus, an additional property of both explanation and justification is that both sorts of objects can and do take partial and elliptical forms in their expression. 41 Note that the claim is that being a nomological deduction is a necessary condition of being an explanation/justification rather than a sufficient one. As many have noted, there are numerous examples of nomological deductions where the premises of the argument are not an explanation of the conclusion. See David-Hillel Ruben (1993, p. 7), and Bromberger (1992, pp. 83-4). 42 Hempel (1993, p. 25). A similar idea is defended by Peter , who contrasts what he calls an ideal explanatory text-which includes all information relevant to explaining the fact in question-with the actual explanations that people provide being fragments of this text. 43 Hempel (1993, p. 25). 44 Hempel (1993, p. 26).

Theory acceptance
Bas Van Fraassen has noted that the fact that a theory explains some phenomenon is generally taken to be a fact that counts in favor of accepting the theory. 45 However, this entails that a theory can explain a fact even prior to it being accepted as correct (or, perhaps, merely adequate). 46 Similarly, the fact that some normative theory justifies permitting or restricting a behavior that is pre-theoretically taken to warrant permitting or restricting is generally taken to count in favor of accepting the theory. Thus, both explanations and justifications have the property of (a) counting in favor of theory acceptance and (b) standing in an explanatory/justificatory relation to some proposition even prior to their acceptance. 47

Why-questions
One popular view is that explanations function as answers to why-questions. If one asks why an event occurred or a thing exists in the way that it does, the appropriate response would be an explanation. Indeed, Hempel and Oppenheim's paper opens with the observation that "To explain the phenomena in the world of our experience" is "to answer the question of 'why?' rather than only the question 'what? '" (1948).
The exact relation between explanations and why-questions is disputed. Bromberger argues that the answer to the why-question is simply part of the deductivenomological argument that is an explanation, with there sometimes being multiple correct answers that can be selected from within that argument. 48 For example, the answer to the question of why a floating piece of ice does not raise the water level might simply be that ice has the same density as water-even though this is but one premise among many in a Hempelian deduction. 49 Alternatively, Bas Van Fraassen takes the answer to a why-question to be, itself, an explanation as opposed to a mere part of an explanation, (where this identification of answers and explanations is part of his broader pragmatic, question-based approach to theorizing about explanation and its role in science). 50 Fortunately, this disagreement need not be resolved here, as it is possible to posit a substantive property of explanation vis-à-vis its relation to why-questions while remaining agnostic with respect to the exact nature of that relation. Specifically, it can be maintained that an additional property of explanation is that, for any given 45 van Fraassen (1980, p. 98). 46 van Fraassen (1980, p. 98). 47 Robert Nozick (1981, p. 14) suggests that the difference between an explanation and a proof is that the premises that constitute the explanans need not be accepted, while all the premises of a proof must already be accepted as true. See, Nozick. 48 Bromberger (1992, p. 92). The exact details of which premises of an explanation count as answers to a why-question will not be discussed here, as the problems with Bromberger's account make such detailed examination unfruitful (see, for example, Paul Teller (1974, pp. 374-5)). 49 Bromberger (1992, p. 93). explanation, a certain subset (note: not a proper subset) of its premises is the answer to a corresponding why-question. 51 Here, again, there is a corresponding property of justification that is analogous to this property of explanation. Specifically, note that justifications are also answers to whyquestions. The pescatarian provides a justification in response to questions like, "Why don't you eat steak?" (from a curious stranger); "Why can't we eat meat?" (from her child); or "Why shouldn't I eat meat?" (from a challenger). Further, when she replies, "Because it's wrong to consume creatures that suffer in the process," that response might be thought to be either a complete justification or a part of a justification; however, on either account it would still be true that, for any given justification, some subset of its premises is the answer to a corresponding why-question.

Asymmetries
Another property of explanation is that there is an asymmetry in the direction of explanation such that, while explanans E explains some fact F, the description of F cannot explain any of the facts whose descriptions compose E. In a critical discussion of the issue, Bas Van Fraassen provides a number of examples of this asymmetry: propositions describing the movement of galaxies explain the red shift in the light received from those galaxies, but a description of the red shift cannot explain their movement; a description of changes in the weather can explain the behavior of a barometer, but a description of that behavior cannot explain the weather; and a description of the size and shape of a flagpole can explain the shape of its shadow, but the description of the shadow cannot explain the flagpole's shape and size. 52 51 Some recent commentators have raised doubts about how closely connected answers to why-questions are to explanations. For example, Bradford Skow (2016) argues that there are three reasons why a theory of explanation cannot also serve as a theory of why-questions. First, he argues that explanations include answers to other sorts of interrogative-based questions besides why-questions (2016, pp. 7-8). However, this divergence does not cause problems for the thesis defended here, as such explanations have been ruled out via stipulation for the reasons discussed in Sect. 4. Second, Skow argues that, while answers to why-questions are strictly semantic objects, explanation has the dual character mentioned above where both speech acts and semantic objects are labeled explanation. A theory of explanation, Skow argues, must account for both sorts of things meaning that it must be distinct from a theory of answers to why-questions (pp. 8-9). Fortunately, this point similarly causes no problems for the proposed analogy, as the explanationrelated objects relevant to that analogy are strictly the semantic objects called explanations, not all objects and events referred to by the word "explanation." Finally, Skow argues that there are distinct varieties of answers to why-questions including both partial and complete answers to such questions; however, because a theory of explanation does not specify which kind of answer explanations amount to, it cannot serve as an adequate theory of answers to why-questions (pp. 14-15). Here, again, the proposed analogy sidesteps this worry, as the proposal here can be understood as maintaining that the explanations of interest are merely some variety of an answer to a why-question; one can, thus, remain agnostic about whether explanations are partial or complete answers to such questions. Given that the claim here is merely about the relation between explanations qua semantic objects and why-questions-as opposed to a claim about the relation between all varieties of explanation and all varieties of answers to why-questions-the proposed analogy avoids Skow's worries about more ambitious efforts to identify theories of explanation and theories of answers to why-questions. 52 van Fraassen (1980, pp. 104, 112). Van Fraassen actually denies that explanation is asymmetrical in this way, and presents a case wherein he believes a change in context transposes explanans and explanandum (i.e., P explains Q in one context, but Q explains P in another) (pp. 130-4). However, for a persuasive rejection of van Fraassen's proposal, see Kitcher and Salmon (1993).
Similarly, there is a directional asymmetry between justifications and their objects. For example, the fact that a person skips family game night to get dinner with a friend might be justified by the proposition that she promised the friend that she would. However, the choice to make this promise cannot be justified by appealing to the proposition that she skips family game night to get dinner. Thus, the analogy between explanation and justification persists.

Understanding
An additional property of explanations is that, under the right conditions, their expression brings about a state of understanding for the audience of the explanatory act. This claim is perhaps most explicit in Robert C. Matthews' pragmatic account of explanation wherein explanations qua semantic objects are defined in terms of their disposition to bring about understanding when conveyed to an audience. 53 A similar account is provided by Bromberger, who introduces the dual notions of ppredicaments-cognitive states that obtain when people have a question to which they can think of no answer that is not ruled out by their prior beliefs-and b-predicaments, which obtain when people have a why-question but lack the capacity to come up with the correct answer to that question. 54 To lack understanding, he suggests, is, roughly, to be in one of these predicaments. 55 Acts of explaining attempt to resolve b-and p-predicaments via the provision of relevant information. 56 To do this, the explainer might have to furnish the audience with novel concepts with which the latter was not previously familiar and/or correct the audience's prior false beliefs that conflicted with the correct answer. 57 Similarly, in his discussion of philosophical explanations, Robert Nozick suggests that explanation is a subset of the kinds of acts that increase understanding. 58 Specifically, he suggests that philosophical explanations provide such understanding by demonstrating that, contrary to what the audience previously believed, their prior commitments do not imply the negation of some proposition-with this demonstration being an explanation of the possibility of that proposition. 59 Additionally, an audience can be brought to understand how a proposition could be true by demonstrating how it can be deduced from plausible premises. 60 Here, again, an analogous relation can be shown to obtain between justification and understanding. Note, first, that justifications bring about a state of understanding, with this achievement revealed by the common locutions that accompany the delivery of such justifications. For example, a person who demands justification and receives a 53 Matthews (1993, p. 354). 54 Bromberger (1992, p. 81). 55 Bromberger (1992, pp. 29, 37fn13). 56 Bromberger (1992, p. 41). 57 Bromberger (1992, pp. 42-3). 58 Nozick (1981, p. 12). 59 Nozick (1981, pp. 9-10). 60 Nozick (1981, p. 11). satisfactory one will often say, "I understand," or "That makes sense," to indicate that no further information need be conveyed.
Further, the means by which justifications bring about this understanding seem identical to those described immediately above. First, note that those who demand justifications via a why-question appear equally to be in a p-or b-predicament in that they are unable to conceptualize the correct answer to the question and/or posit an answer that does not conflict with prior beliefs. Justifications, then, resolve these predicaments via either the provision of novel concepts (e.g., normative notions) or correction of false beliefs (either empirical or normative) that generate a contradiction with the proposition to be justified. Or, alternatively, one might follow Nozick by showing how already-held beliefs do not, in fact, contradict the justificandum. Thus, not only do successful justifications have the property of bringing about states of understanding, they bring about understanding in the same way that explanations bring about understanding.

Autonomous facts
Another posited property of explanation is that there are certain limits to what can be explained. Or, to put this point another way, there is a certain set of facts such that, for any given explanation, no member of the set is the object of that explanation. 61 Specifically, these facts don't seem apt for explanation, with any demand for an explanation revealing some sort of conceptual confusion on the part of the asker. Shamik Dasgupta, for example, proposes the analogy of a person who demands a mathematical proof of a definition within a particular mathematical system. 62 As Dasgupta notes, to demand a proof of a mathematical definition betrays a failure to grasp the nature of mathematical definitions and proofs, as a satisfactory understanding of those notions would make it sufficiently clear that such definitions are simply not apt for proof. 63 His suggestion, then, is that there are also certain facts such that a demand for explanation of those facts would betray a failure to fully understand the relevant concepts involved (e.g., the fact that water is made up of H 2 O). 64 Analogously, there are certain actions or states of affairs that seem similarly inapt for justification, with any request for a corresponding justificans betraying some sort of conceptual confusion. For example, a person who demands why one ought to do that which is morally obligatory betrays a failure to grasp the relevant normative notions in the same way that a person who asks why water is made up of H 2 O betrays a failure to grasp the relevant metaphysical notions.

Irreflexivity
As Nozick notes, explanation is irreflexive in the sense that the explanans cannot be identical to the explanandum, with "explanations of the form 'p because p' [being] inadequate and unsatisfactory." 65 Similarly, the property of irreflexivity is shared by justification, where any justificans that is identical to its justificandum is unsatisfactory.

Explanatory distance
A closely related (potentially identical) property of explanation is that one proposition can only explain another if there is adequate explanatory distance between them. 66 In a helpful explication of the notion, Elanor Taylor presents two paradigmatic cases of purported explanations that are unacceptable due to a lack of explanatory distance. 67 First, there is the case of opium's "dormitive virtue" being posited as an explanation of why the drug tends to make people fall asleep. 68 And, second, there is the case where a person attempts to explain some medical symptom R by citing syndrome S, where that syndrome is defined strictly in terms of observed symptoms, P, Q, and R. 69 While Taylor proposes a necessary and sufficient condition of acceptable explanatory distance obtaining, one can remain agnostic on this point and merely hold that a property of all explanations is that they are appropriately distant from their explananda-with distance then construed in accordance with one's preferred account.
Similarly, all justifications appear to have the property of being sufficiently distant from that which they justify. To mirror the syndrome case, for example, note that one cannot justify why some particular property rights schemeR should obtain by positing that society type S should obtain, where S is defined in terms of imposed rights schemes P, Q, and R. Thus, the property of explanatory distance also has a justificatory analog.

Contrast cases
The final property of explanation to be discussed here is that explanations take a contrastive form. 70 There are a few different accounts of this contrastivity, but the core idea is, again, best conveyed by example: in explaining why geese migrate south 65 Nozick (1981, p. 118). Nozick does raise the possibility that explanation might be reflexive under very specialized conditions; however, this complication will be ignored for these purposes. Hempel and Oppenheim also design their influential account of explanation to specifically rule out the possibility of reflexive explanations (1948, p. 160). 66 It may be the case that all instances of inadequate explanatory distance are really just cases of reflexive explanation; however, there is no need for the paper to take a position on this point. 67 Taylor (forthcoming). 68 Taylor (forthcoming, pp. 5-6). As Taylor notes, this explanation might, in fact, be distant enough to be genuine if one accepts an ontology that includes powers (p. 5). 69 Taylor (forthcoming, pp. 2-3). 70 There is some disagreement as to whether all explanations are contrastive (see, e.g., Lipton (1993, p. 221)). The paper will avoid picking sides in this debate, suggesting, instead, that, even if there are some non-contrastive explanations, their justificatory analogs will also be non-contrastive (for reasons analogous to why those particular explanations are non-contrastive). as winter sets in, one might explain either (a) why geese migrate south as winter sets in (as opposed to robins); (b) why geese migrate south as winter sets in (as opposed to north); or (c) why geese migrate south as winter sets in (as opposed to summer). The general thought, then, is that there can be different explanations of the same fact, with each explanation corresponding to a distinct contrast case. 71 Van Fraassen puts this point in question-centric terms, suggesting that any whyquestion is identical with a triple consisting of (1) a proposition to be explained, (2) a contrast class of rival propositions (which includes the explanandum), and (3) a relation of explanatory relevance which helps to settle what counts as an explanation-where 2 and 3 are both determined by context. 72 Because he takes explanations to be answers to why-questions, it follows that the same explanandum can have multiple explanantia, with distinct explanantia corresponding to distinct contrast classes.
Alternatively, Peter Lipton puts things in terms of facts and foils, where the same fact might have many different foils and many distinct explanations, with each explanation corresponding to a different foil. 73 Specifically, he suggests that a fact is explained relative to a foil by citing some event that caused the fact where this event lacks "a corresponding event in the history of" the foil not obtaining. 74 For example, if both Kate and Frank submit an essay to a contest, Kate's act of submitting cannot explain why she wins, as there is the same corresponding event in the history of Frank not winning the contest (the foil being Frank winning the contest); by contrast, the fact that Kate wrote the best essay explains why she won, as there is no corresponding event in the history of Frank not winning. 75 Unfortunately, Lipton states his account in causal terms that make it inapplicable to non-causal varieties of explanation. However, it can, perhaps, be made more general by restating the account in terms of dependence: a fact is explained relative to a foil by citing some sub-fact upon which the fact depends, where there is no corresponding sub-fact upon which the negation of the foil depends. For example, in Lipton's case of a group of sticks tossed spinning in the air, one would explain why more ended up near the horizontal axis-as opposed to the vertical axis-by appealing to the fact that there are more ways for a stick to be horizontal than to be vertical. 76 The thought here is that the statistical fact about the position of the sticks depends in some sense on the geometric fact about the possible orientations of a line in three-dimensional space. Additionally, this speculative account would posit that there is no corresponding geometric fact (e.g., that there are more ways for a line to be vertical than horizontal) 71 One might potentially sidestep the debate mentioned in the previous footnote by claiming that all explanations are contrastive, but only in the sense that if some fact has at least one contrast case, then there will be distinct explanations for each such contrast case. This statement sustains the claim that all explanation is contrastive (at least in some sense) while avoiding the counterexamples that lead Lipton to deny that all explanation is contrastive (1993, p. 221). 72 van Fraassen (1980, pp. 141-3). Van Fraassen only explicitly says that the relevance relation is determined by context, but it seems the same must be true of the contrast class (p. 142). 73 Lipton (1993, p. 210). 74 Lipton (1993, p. 217). 75 Lipton (1993, p. 217). 76 Lipton (1993, p. 208). upon which the negated foil depends. Thus, the original geometric fact would explain why most sticks end up near horizontal, as opposed to near vertical.
Justification, too, is contrastive. Consider, for example, one parent who demands the other justify why their elder child, Anne, should get a car for Christmas. In justifying this (normative) fact, one might justify either (1) why Anne should get a car for Christmas (as opposed to her sibling); (2) why Anne should get a car for Christmas (as opposed to a bicycle); or why Anne should get a car for Christmas (as opposed to her birthday). Given that 1-3 will each call for a distinct justification, justification is contrastive in just the way explanation is.
Further, the apparent details of contrastive justification are analogous to the speculative generalization of Lipton's account presented immediately above. To justify why Anne-as opposed to her sibling should get the car-one might note that only Anne is old enough to legally drive. Further, note that the truth of the proposition that Anne should get the car depends on this fact, and there is no corresponding fact (e.g., that her sibling is old enough to drive) upon which the negation of the foil (i.e., that the sibling should not receive a car) depends. Thus, in addition to justification being generally contrastive, the specific contrastive property of explanation posited by Lipton seems to have a justificatory analog.
Admittedly this proposal is a bit underdeveloped, as developing a complete theory of non-causal contrastive explanation goes beyond the scope of this paper. However, the hope is the above analysis provides reason for thinking that the analogy will persist even as additional details regarding the contrastive character of explanation are filled in.

A quick inductive step
The previous section has attempted to show that, for each examined property of explanation, there is an analogous property of justification. Given the voluminous nature of the literature on explanation, it is not possible to provide an exhaustive list of all such properties that have a justificatory analog; however, the hope is to have captured a diverse array of the most notable characteristics of explanation (and justification). The suggestion, then, is that, given this strong and persistent parallelism between explanation and justification, it is reasonable to make the inductive inference that any additional properties of explanation will also have justificatory counterparts. Thus, when two additional properties of explanation are introduced below, enumerative induction warrants inferring that each such property entails the existence of a counterpart property of justification-where these properties of justification serve to vindicate the egalitarian presumption.
More will be said in defense of this move in Sect. 8. However, for now, one need merely note the intuitive implausibility of the contrary suggestion that explanation and justification share the roughly dozen properties described above, but suddenly diverge when it comes to the two additional principles presented below. Indeed, it is unclear on what grounds one would argue against the inductive conclusion that the pattern persists (at least, beyond a more general skepticism about induction that many philosophers would be reluctant to embrace).

The difference-difference property
So far the paper has argued that, for any given property of explanation, there is a corresponding property of justification. It argued that: 1. Every listed property of explanation has a corresponding property of justification. 2. If every listed property of explanation has a corresponding property of justification, then every property of explanation has a corresponding property of justification.
Thus, 3. Every property of explanation has a corresponding property of justification. The paper will now introduce two additional properties of explanation and, for each, posit a corresponding property of justification, with one of these latter properties entailing Thesis 1 of the egalitarian presumption and the other entailing Thesis 2. This section will introduce the property entailing the former thesis. Specifically, it will argue that: 4. Any difference between two things demands explanation.
This premise, when taken in conjunction with premise 3, entails that: 5. Any difference in advantage between two persons demands justification.
Given that premise 5 is a restatement of Thesis 1 of the egalitarian presumption, the section concludes that, contra Brennan's challenge, there is reason for thinking that this component of the egalitarian presumption is true.
To begin, consider two of the cases presented by Michael Della Rocca in his defense of the principle of sufficient reason. 77 Della Rocca, first, introduces the case of a balance with equal weights hung on each side, arguing that "absent any difference between the sides of the balance, one naturally concludes that the whole will be at rest," as any movement would be unacceptably inexplicable. 78 Similarly, in the case of two blocks of matter that have identical chemical and physical structures, he contends that it could not be the case that only one would dissolve when placed in water, as "given their exact categorical similarity, nothing could ground this dispositional difference between the two objects." 79 Della Rocca's contention here is that difference demands explanation. If one block dissolves in water and another does not, there must be an explanation for that difference in behavior; absent such an explanation, it simply could not be the case that the posited behavior obtains. While Della Rocca does not put things quite this way, his cases suggest that any difference in property between two things must be explained via appeal to some relevant difference in some other property of the two things. Or, to put this point a bit more precisely, given any two things x and y and some property P, there is some non-identical property O such that, if it is the case that Px but~Py, then the proposition that Px &~Py has an explanans that includes Ox and~Oy (or some conjunction of these two propositions) as essential premises (unless either or~Py is an autonomous fact of the kind discussed in Sect. 5). 80 Call this property of explanation the difference-difference property. 81 Importantly, the difference-difference property can be put as follows: for any property difference Px &~Py, this difference obtains if and only if (a) there is some true explanatory conditional of the form "If Ox and~Oy then Px &~Py" and (b) Ox and~Oy is true. This restatement follows from the explication of the differencedifference property posited just above paired with the fact that explanations are nomological deductions, as discussed in Sect. 5. It will also help to clarify how the justificatory analog of the difference-difference property vindicates Thesis 1 of the egalitarian presumption.
So, what is the analogous property of justification? Rather than explaining differences that obtain in terms of other property difference(s), one would, instead, need to justify such differences in terms of other property differences if the former are to be considered just. 82 Applying this general statement to the specific case of inequality of advantage, one would note that such inequality amounts to a property difference across people: some people are rich while others are not. Thus, for any two persons x and y and non-identical properties P and O-where P is the property of possessing some particular level of advantage-if the state of affairs Px &~Py is just, then it must be the case that (a) there is some true justifying conditional of the form "If Ox and~Oy then Px &~Py is just" and (b) Ox and~Oy obtains.
With this result, Thesis 1 of the egalitarian presumption is vindicated. This thesis posits that inequality demands justification and it has now been shown that it does: any difference in possessed advantage is just only if it has a true justifying conditional with a true antecedent (where that antecedent makes reference to some relevant property difference between the unequal parties). Of course, one cannot derive from this result what kind of property difference will factor into the justificans of the inequality 80 The parenthetical is included to prevent an infinite regress of explanation where the fact that Ox and~Oy cited in the explanans generates its own explanandum, with this property difference demanding explanation in terms of some further property difference represented by the proposition Nx &~Ny, where N is a property that is identical to neither O nor P-with, this difference, in turn, demanding explanation… etc. However, if there are autonomous facts, then the explanatory regress will terminate once one posits an explanans where the cited property difference involves a property Z such that the fact that Zx (or~Zx) is autonomous. Indeed, this is the move made by  who is sympathetic to something like this proposed property of explanation but wants a way of terminating regress.
By contrast, Della Rocca (2010), who most explicitly endorses something like the proposed property, does not include any discussion of what might terminate an explanatory regress. For him-and those who are skeptical that there might be such a thing as autonomous facts-there are three apparent ways to deal with the regress problem. First, one might simply bite the bullet and posit that there is no problem with explanatory regress. Alternatively, one might take there to be certain self-explainers that terminate the regress-i.e., the fact that Px and~Py is explained by the fact that Ox &~Oy, but P is identical to O (though this might demand rejecting that explanation has the property of irreflexivity posited above). Finally, one might allow for circularity of explanation where the fact that Px and~Py is explained by the fact that Ox &~Oy and the fact that Ox &~Oy is explained by the fact that Px and~Py (where P is not identical to O). Or, perhaps there is some other way to address the regress problem. Ultimately, it seems that some variant of the proposed property could be posited that avoids such a problem while still yielding the conclusion arrived at below. (My thanks go out to Justin Tiehen for pressing me on this point). 81 John Stuart Mill also endorses something like this principle, though he focuses only on causal differences (See: Mill 1884, pp. 256-7). 82 Given that what must ultimately be justified are acts and omissions, differences that are beyond the control of agents are excluded from the set of differences that can and must be justified. in question. Indeed, that is a task for those who posit specific theories of distributive justice. 83 Rather, what has been shown is that at least some such justification is necessary if the inequality is to be just. Thus, the egalitarians discussed in Sect. 1 are correct in their presumption that they can demonstrate the injustice of some inequality by showing that no suitable justification exists for that inequality. Indeed, contra Brennan's challenge, we do have "reason to presume equality is a baseline from which departures must be justified," as such departures represent a property difference that, as such, demands justification. 84 83 In arguing against the egalitarian principle that "like people must be treated alike," Peter Westen (1982) notes that any two persons are both alike in some respects and different in others. Thus, one must specify which property similarities are relevant in the sense that they render similar treatment obligatory. Westen believes that there is only one such specification that is plausible: two persons are relevantly alike iff they share the property that they ought to be treated alike. Thus, he takes the thesis that like people must be treated alike to be trivial, as it merely asserts that people who ought to be treated alike ought to be treated alike (1982, pp. 544-7).
Given that the difference-difference property can be seen as correlative of the principle that like people must be treated alike, one might worry that it is similarly trivial. Specifically, one might worry that the only relevant property difference that might render inequality just is one person having the property of being entitled to some quantity of advantage and another lacking that property (i.e., having the property of being entitled to a different quantity of advantage). And such a statement of the difference-difference principle might seem hopelessly trivial.
Three things can be said in response to this worry. First, even if the principle lacks content in this way, it would still entail that inequality demands justification in the way that egalitarians presume and Brennan denies. Second, Westen's claim that there is only one plausible specification of relevant similarity rests upon the questionable premise that "moral alikeness is established only when people define categories." Finally-and most importantly-when one considers specific egalitarian theories, one finds lots of nontrivial claims about what property differences justify differences in advantage (e.g., one person has chosen culpably while another has not, one person has contributed economically in a way that another has not, etc.). Thus, Westen-inspired worries about triviality do not seem to get traction in this context. 84 One might wonder if this argument can be extended to undermine Thesis 2 of the egalitarian presumption. If property difference must be explained via appeal to some further property difference, might we also think that two things sharing a property must be explained by appealing to some further property similarity? If so, that would entail that there is an analogous property of justification: the fact that two persons have an identical quantity of advantage would be a shared property that would demand justification via appeal to some other property that they share. This result would negate Thesis 2's contention that equality does not demand justification.
The reply to this worry is that the starting premise is false: property similarity need not be explained in terms of a further property similarity. The clearest example in support of this point is the case of two poisons that share the functional property of killing a person when ingested. It is entirely possible that the two poisons share this property despite there being no further shared property that explains this similarity (e.g., they might be different colors, made up of different chemicals, act on different parts of the body in different ways, etc.). Thus, while functional property difference demands explanation in terms of some further property difference (as in Della Rocca's case of the two blocks placed in water), similarity in functional property does not demand explanation in terms of some further shared property. Thus, one cannot infer that there is an analogous property of justification where equality demands justification in terms of some further shared property such as shared basic capacities or history of desert-relevant actions. Of course, this does not show that similarity does not demand explanation at all as Thesis 2 maintains; this claim will be defended in Sect. 9. I am indebted to an anonymous reviewer for pressing me on this point.

The quick inductive step, briefly revisited
Now that the difference-difference property has been posited, it is possible to provide additional support for the inductive inference made in Sect. 6. Specifically, note that having a justificatory counterpart is, itself, a property of each of the properties of explanation listed in Sect. 5. Thus, if one rejects the inductive step and insists that either the difference-difference property or the inegalitarian property to be discussed in Sect. 9 lacks a justificatory counterpart, that would amount to a difference in property between these things (i.e., the difference-difference/inegalitarian property and any of the properties discussed in Sect. 5). Given such a difference-coupled with the difference-difference property-it follows that some additional property difference would have to obtain between the properties discussed in Sect. 5 and those discussed in Sects. 7 and 9 that could explain why the former have justificatory counterparts and the latter do not. However, it is entirely unclear what this property difference could be. Thus, given the apparent absence of such a grounding property difference, it cannot be the case that the properties discussed in Sect. 5 have a property not also possessed by the properties discussed in Sects. 7 and 9-which is to say, the latter properties must also have justificatory counterparts.
This argument can be formally stated as a sub-argument for premise 2 of the argument synopsis presented in the previous section: i. When it comes to possessing the meta-property of having a counterpart property of justification, the two posited properties differ from the properties presented in Sect. 5-i.e., lack this meta-property-only if there is some further property difference between the posited properties and the Sect. 5 properties that explains this difference. (This is an invocation of the difference-difference property). ii. There is no further property difference between the posited properties and the properties of Sect. 5 that explains this difference. Thus iii. The posited properties do not differ from the properties of Sect. 5 when it comes to the property of having a counterpart party of justification.
In other words, if all the properties of explanation listed in Sect. 5 have counterpart properties of justification, then so do the properties of explanation posited in Sects. 7 and 9. This conclusion vindicates premise 2 of the argument synopsis presented in the previous section. 85 85 The difference-difference property can also be used to formulate an alternative argument in defense of the conclusion that the posited properties of explanation have justificatory counterparts. Specifically, one might argue as follows: a. A property difference between explanation and justification obtains-i.e., there is some property of explanation for which there is no counterpart property of justification-only if there is some further property difference between explanation and justification that explains this difference (this is an application of the difference-difference principle). b. When it comes to the two posited properties of explanation (namely, the difference-difference property and the property discussed in the subsequent section), there is no further property difference between explanation and justification that explains this difference. Thus, c. There is no property difference between explanation and justification when it comes to the two posited principles of explanation.
It is, of course, possible that some relevant grounding property difference might be found that would explain why the posited properties lack justificatory counterparts. However, beyond the difficulty of actually coming up with such a difference, there is a structural reason for doubting that there could be such a difference, namely, the abstraction of the property-bearers. Because properties are abstract objects, they have a fairly limited set of properties relative to non-abstract objects. Thus, there are few things to which a critic of the inductive step could point to explain a purported property difference between the various properties of explanation.

Justificatory inegalitarianism
So far, the paper has demonstrated that egalitarians are correct in their presupposition that inequality that lacks adequate justification is unjust. However, what about Thesis 2 of the egalitarian presumption? Does equality also require justification, or are egalitarians correct in assuming that equal distributions are prima facie just?
Here, again, there is a relevant property of explanation whose induced justificatory counterpart helps to answer the question in favor of the egalitarians. Specifically, note that, in addition to the propositions discussed in the "autonomous facts" subsection, there is another set of propositions that are not apt for explanation. Consider, for example, the case of an object at rest that remains at rest (or, alternatively an object in motion that stays in motion). Is there an explanation for this constancy? As Nozick notes, Newton thought that such perpetuation was not in need of explanation. 86 Similarly, Van Fraassen suggests that we "reject such questions as… why does a body free of impressed forces retain its velocity?" and, from this, concludes that not everything is a legitimate object of explanation. 87 And, Nozick quotes R. Harré, who writes: I come to the most fundamental and the most powerful of methodological principles. It is this. Enduring is in no need of explanation. We are not required to explain the fact that something remains the same; only if there is a change is explanation called for. 88 To see the intuitive appeal of these claims, consider the above case of an object that remains at rest, unmoving. Here, explanation seems not only unnecessary, but potentially inappropriate. Were a person to ask, "Why did that happen?" while pointing Footnote 85 continued While this argument resembles the argument presented immediately above in its invocation of the differencedifference property, it avoids any reference to the meta-property of having a counterpart property of justification, which some might find controversial. 86 Nozick (1981, p. 121). 87 Van Fraassen (1980, pp. 111-12). 88 R. Harré (1970, p. 248). As an anonymous reviewer notes, one might see these various authors as advancing two distinct claims. On the one hand, they might be contending that one need not explain certain facts, i.e., those facts can obtain even if they lack a true explanatory conditional with a true antecedent. On the other hand, they might be read as arguing that there cannot be an explanation of these facts. Here the claim might be either that these facts do not have true explanatory conditionals with true antecedents or, perhaps, that there are such conditionals but these do not count as genuine explanations of the facts in question. This paper will adopt the former interpretation as it is both logically weaker than the latter (i.e., it is entailed by the latter) and is sufficient for establishing the posited conclusion. at the object, her query would likely elicit only puzzlement regarding the identity of the explanandum. Even if she were to refine her question to, "Why is that object not moving right now?" she would likely be met with confusion and perhaps a response of "Why would it move?" Such confusion, itself, is seemingly best explained by positing that the object's non-movement is not in need of explanation. 89 To maintain that there is such a class of propositions that do not call for explanation is to endorse what Nozick calls (somewhat confusingly in this context) an inegalitarian theory of explanation, with natural states being those that need not be explained. 90 Unfortunately, Nozick does not provide an account of which states are natural states. However, this gap can be perhaps filled in as follows.
Suppose that one were to attempt to explain why an object at rest remains at rest when no force acts upon it. The only thing that could seemingly be cited in such an effort is the absence of any external force that would have caused-and, thereby explained-movement on the part of the object. Thus, rather than pick out some fact in the causal history of the object as one standardly would when presenting a scientific explanation, one, instead, proceeds as follows. First, one identifies all of the possible worlds where a rival proposition from the explanandum's contrast class is true-or, to use Lipton's language, a foil obtains. So, in this case, one would consider all possible worlds where the object moves. Second, one would identify the true propositions in those worlds that (a) explain the foil and (b) are false in the actual world (e.g., that force F acted on the object at time T ). The purported explanans, then, functions by citing (via shorthand) the falsity of all such propositions.
Consideration of other paradigmatic cases of natural states (e.g., the endurance of an object, or, to use Nozick's example, there being nothing rather than something) will yield similar results, with any attempt at explanation being forced to cite the absence of those factors that would have explained contrastive behavior, had it occurred. 91 The suggestion, then, is that the necessity of referencing the absence of explanatory factors is the indicator that a state is a natural one: if some fact can only be "explained" via the indirect approach described immediately above, then it is a natural state-and, 89 An anonymous reviewer suggests that perhaps the response to the refined question would be to clarify the relevant foil. Such a response would be of a kind with the natural response to someone asking why an object did move. In response to this latter question, one might seek to clarify the relevant foil by asking, "As opposed to what? Not moving? Some other object moving? Moving now as opposed to ten seconds from now?" etc. Similarly, the reviewer suggests that one might respond to the question "Why didn't that object move?" by asking, "As opposed to what? Moving? Some other object not moving? Not moving now as opposed to ten seconds from now?" However, the paradigmatic case of non-movement does not seem to allow for this response, as there is only one plausible foil, namely, the object moving. Consider, for example, the block that sits on a table unmoving. In this case, it does not seem that one can ask why the block did not move as opposed to some other object not moving, as there is no other moving object present to serve as a contrast. Nor can the foil be temporal (e.g., the block not moving 10 s earlier) as the block is, by hypothesis, permanently unmoving. Thus, it seems there is only one apparent foil, namely, the object moving, thereby precluding any reply that seeks to clarify the foil. In other words, it is clear in this case what is being asked: why is the block not moving rather than moving? And, as noted just above, the natural response to this why-question is the dismissive "Why would it move?"-which, in turn, suggests that the non-movement is not something that needs to be explained. 90 Nozick (1981, p. 121). 91 Nozick (1981, p. 122). thus, lacks a genuine explanation, with any request for explanation of that proposition being inappropriate.
Given this property, one can, given the inductive step, posit an analogous property of justification: there are certain arrangements that are "natural" in the sense that they cannot-and, thus, need not-be justified. More specifically, these arrangements will be those that, if one were to attempt to justify them, one would have to, first, identify all of the possible worlds where a rival proposition from the justificandum's contrast class is true (i.e., the fact's foil obtains). Second, one would identify the true propositions in those worlds that (a) justify the foil and (b) are false in the actual world, with the purported justification functioning by citing (via shorthand) the falsity of all such propositions.
Consider, then, the case of an arrangement where advantage levels are equal. If pressed, how might one attempt to justify such an arrangement? Seemingly, one would cite the falsity of any proposition that might justify one of the arrangement's foils. For example, a luck egalitarian might point to the fact that no one made a culpable choice of the kind that would justify inequality, while a Rawlsian would cite the fact that inequality is not "to the greatest benefit to the least advantaged." 92 Thus, in each case, one considers a rival proposition from the justificandum's contrast class-namely that some unequal arrangement is justified-and identifies those propositions that, if true, would justify the arrangement (e.g., that a culpable choice was made or that the arrangement satisfies the Difference Principle). One, then, cites the falsity of these propositions when purporting to justify the rival egalitarian arrangement. 93 Given that attempts to justify equality will take the form described immediately above, it follows that equal arrangements meet the sufficient condition of being a natural state and, thus, cannot and need not be justified. Thus, the second thesis of the egalitarian presumption has also been vindicated, thereby defeating Brennan's challenge.
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Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 92 Rawls (1971, p. 302). 93 It has been maintained here that the described "indirect" justifications are not genuine justifications. However, the argument does not rest on this point, as one might concede that equality does need to be justified, but with any such justification taking the indirect form described above. Given that there is no difficulty in providing such a justification on any given egalitarian theory, Brennan's challenge would be defanged (if not entirely defeated), as it will have shown only that egalitarians must note that no justification for inequality obtains prior to declaring an equal arrangement just.