Anonymized draft as of 18 July 2020. Comments welcome. Please contact us before citing. Essence, Modality, and Identity Fabrice Correia & Alexander Skiles Introduction In an article forthcoming in MIND, Jessica Leech (2020) raises a challenge for essentialists about metaphysical modality, who claim that facts about the essences of things account for all facts about what holds with metaphysical necessity. Suppose that Plato is necessarily human. Then according to the standard version of the view (cf. Fine 1994, 1995; Hale 2002, 2013), there is some fact about the essences of things-perhaps it is the fact that being human is part of what it is to be Plato- which accounts for the fact that he must be human. But if so, Leech writes, it may be reasonably asked why one should agree that whatever is essential is also necessary: 'Why, just because [...] Plato is essentially human, should it therefore be the case that Plato is necessarily human?' (p. 2) The answer cannot be that part of what accounts for Plato's being essentially human is that he is necessarily human, which would be patently circular. But how else could the essentialist answer? Worries about an alleged 'gap' between essence and necessity within the essentialist framework recur repeatedly in recent literature (e.g. Casullo 2020; Leech 2018; Mackie 2020; Noonan 2018; Romero 2019). Leech's distinctive and important contribution is to argue that essentialists cannot avail themselves of a promising new account of essence-in terms of generalized identity-in order to bridge it. In 'Grounding, Essence, and Identity', Correia and Skiles (2019) have argued that facts about what is essential to what are a special class of facts about what is identical to what, the relevant notion of identity being a higher-order extension of the familiar, 'objectual' notion, and expressed by statements such as those that take the form 'To be F just is to be G' (e.g. Correia 2010, 2016; Dorr 2016; Linnebo 2014; Rayo 2013). Roughly for now, on Correia and Skiles's view, the fact that Plato is essentially human is the fact that there is no difference between a thing's being Plato versus its being both Plato and human. The notion of essence has long been thought to be 'important for our understanding of the metaphysics of identity' (Fine 1994, p. 8); Correia and Skiles take this idea literally, yet in reverse, extend it to essence facts of various other types, and present various arguments in favor of their approach. 2 Suppose that identity-and therefore generalized identity, if it purports to be a bona fide notion of identity at all-not only has all the usual non-modal features (reflexivity, symmetry, transitivity, obeys something akin to Leibniz's Law, and so on), but also holds with necessity. If that is so, it seems that what might be called the identity-based essentialist can easily say why essence implies necessity. As Leech puts it, 'statements of essence are a special case of statements of identity. Statements of identity are necessary. Hence, statements of essence are necessary too' (p. 13). Yet according to Leech, identity-based essentialism 'just moves the bump in the carpet', since one may now reasonably ask why one should agree that generalized identity implies necessity (p. 14). Why, just because there is no difference between a thing's being Plato versus its being both Plato and human, should it therefore follow that Plato is necessarily human? Leech argues that in order to bridge this new gap one can either account for necessity in terms of essence, or account for essence in terms of generalized identity in the way Correia and Skiles envisage-but one cannot do both, 'on pain of circularity' (p. 16). We disagree. In fact, there are a number of safe and non-circular routes from generalized identity to necessity for the identity-based essentialist to choose from. Leech considers only one, which centers around a generalization of the classic proof of the necessity of objectual identity due to Barcan (1947) and Kripke (1971). After laying out the necessary background (§1), we present a more direct proof that shows why one should agree that generalized identity implies necessity, and argue that neither our proof nor the generalized Barcan-Kripke proof tangles the essentialist up in any circularity (§2). As we said before, Leech is not alone in worrying about a purported 'gap' between essence and necessity for the essentialist to bridge, although what that gap is, exactly, gets described by different authors in rather different ways. So after dealing with the specific one worrying Leech, we propose several full-blown accounts of necessity in terms of identity, and use them to bridge an alleged essence-to-necessity gap of another sort (§3). Thus we conclude that identity-based essentialism deserves serious consideration in discussions of the nature and source of metaphysical modality. 1. Background To begin with, we need to sketch some elements of Correia and Skiles's identity-based account of essence relevant to the discussion to come. For now, it suffices to focus on just two types of essence 3 statements: those that concern what it is to be some particular thing (i.e. objectual essence), and those that concern what it is to be some particular way (i.e. generic essence). These in turn may either concern what it is to be such-and-such at least in part, or in full. The difference consists in whether the statement leaves open whether there is 'more' to the essence of the such-and-such. Statements of partial essence (both objectual and generic, respectively) include: (1) It is essential to Plato to be human. (2) It is essential to being human to be an animal. (1) and (2) are most charitably read as statements of partial essence, for usually those who commit to (1) or (2) wish to at least remain neutral about whether there is more to what it is to be Plato or a human than this. In contrast, statements of full essence include: (3) To be a set with Plato as its sole member is what it is to be {Plato}. (4) To be filled with H2O molecules is what it is to be filled with water. (3) and (4), unlike (1) and (2), might be seriously put forward as saying enough to imply not only necessary but also sufficient conditions for being {Plato} or being filled with water. Likewise, for now let us focus on simple generalized identities that take the form 'For a thing, x, to be F is for x to be G', where 'F' and 'G' are simple or complex monadic predicate expressions, which we abbreviate as 'Fx ox Gx'. Since generalized identity is a type of identity, the ox operator is both reflexive and governed by a generalization of Leibniz's Law: Reflexivity Fx ox Fx (LL) If Fx ox Gx and F, then F[Gx//Fx] Three comments. First, in (LL), F[Gx//Fx] is the result of replacing one or more occurrences of Fx by Gx in sentence F, with the condition that no variable that is free in Fx ox Gx is bound in Φ or F[Gx//Fx] (cf. Correia and Skiles 2019, p. 645). In what follows we are only concerned with statements of this type with no free variables, so this condition can be safely ignored henceforth. Second, although we take these principles to be true, let us leave open for now their logical status- 4 i.e. whether these are also validities in the correct logic of generalized identity. Third, just like Leibniz's Law for objectual identity, (LL) must be restricted somehow, to include substitutions in linguistic contexts that only concern how the world is, and to exclude substitutions in contexts that are sensitive to how the world is represented. However this distinction is to be made more precise, like Dorr (2016, pp. 43-4) we take it as clear enough that belief operators, standard quotation devices, and other contexts known to cause problems belong in excluded batch, while the included batch should contain not only extensional contexts (e.g. those created by 'and', 'not', and 'all'), but also those created by notions denoting the metaphysical modalities, as well as either objectual or generalized identity. (From this last assumption follows the symmetry and transitivity of ox, as one would hope and expect: Correia and Skiles 2019, p. 645.) So, then, how do Correia and Skiles account for essence statements like (1) – (4)? In three steps.1 First, they take full generic essence to just be generalized identity: FULL-GENERIC-ESSENCE Being F is what it is in full to be G iff: Gx ox Fx.2 Thus, (4) is understood as the statement 'x is filled with water ox x is filled with H2O molecules'. Second, they take partial generic essence to be accounted for in terms of generalized identity in the following way: PARTIAL-GENERIC-ESSENCE Being F is part of what it is to be G iff: there is some H such that Gx ox (Fx Ù Hx).3 1 Harold Noonan has argued that 'the concept of essence so understood [i.e. in the sense at issue in our debate with Leech] has not been adequately explained' and 'any attempt to explain it, at least along the lines most familiar in the literature, must be flagrantly circular or appeal to de re modal notions' (Noonan 2018, p. 1). However, what follows is an account of what essence is, not how it is conceptualized (Correia and Skiles 2019, pp. 649-50), which may well require the capacity to think in modal terms. In any case, in what follows we explain what generalized identity is, and then what essence is, in what we believe are clearly non-modal terms (see pp. 16-7 below for further discussion). 2 Here and elsewhere, we use 'iff:' for sake of readability when denoting certain instances of generalized identity, following Correia and Skiles (2019, p. 649). 3 Note that 'partial essence' should be understood broadly, as it includes full essence as well: given the transitivity of ox and the plausible principle that all the instances of 'Fx ox (Fx Ù Fx)' are true, if being F is what it is in full to be G, it follows that being F is (at least) part of what it is to be G. 5 Thus (2) is understood as the statement 'For some H, x is human ox (x is an animal Ù Hx)'.4 Granted that (2) is true, Correia and Skiles accordingly call being an animal a conjunctive part of being human. Finally, they take objectual essence to be a special case of generic essence: thus (1) is understood as a statement of the form 'Being human is part of what it is to be Plato', while (3) is understood as a statement of the form 'Being a set with Plato as its sole member is what it is in full to be {Plato}'.5 We have focused on the generic case because the examples are philosophically familiar, and it subsumes cases involving the essential properties of individuals that Leech and others usually focus on (talk of 'properties' here need not be read as ontologically committing: cf. Correia and Skiles 2019, p. 643). But statements of generalized identity do not reduce to statements of the form 'To be F is to be G'. The latter statements are particular instances of the following general type: • For some things x, y, ... to be such that j is for them to be such that y (in symbols: j ox, y, ... y). Statements of generalized identity also include statements of type • For it to be the case that j is for it to be the case that y (in symbols: j o y), and statements whose regimentation requires the use of higher-order resources, like for instance statements that can be formalized by means of formulas of type 'j oa y' where 'a' is a sentential variable (illustration: for Tim to know that a isa for him to have a justified true belief that a) and statements can be formalized by means of formulas of type 'j oq y' where 'q' is a predicate variable (illustration: for Socrates to essentially q isq for him to necessarily q). To make our points, it will not be necessary to specify a precise higher-order language for expressing generalized 4 In this particular case, a candidate witness for the existential statement readily comes to mind: 'x is human ox (x is an animal Ù x is rational'. Many other cases are different in this respect. However, given plausible general principles about generalized identity, it can be shown that 'For some H, Gx ox (Fx Ù Hx)' is logically equivalent to 'Gx ox (Fx Ù Gx)'. The principles are that ox is symmetric, and that all the instances of 'Fx Ù (Gx Ù Hx) ox (Fx Ù Gx) Ù Hx' and 'Fx ox (Fx Ù Fx)' are true. 5 It is natural to understand the expression 'to be n', where 'n' is a nominal expression, as synonymous with 'to be identical to n', and in what follows we will indeed understand expressions of this type in exactly this way. However, we do not want to insist that these expressions should be understood that way. We trust that nothing of substance in the discussion to come would change if other (reasonable) construals of these expressions were taken for granted. 6 identities (but see Dorr 2016, pp. 46-9 for two such proposals). Like the generic operator, we take the 'is' of all the other generalized identities to be reflexive, symmetric, transitive, and governed by some adequate version of Leibniz's Law, and to allow one to formulate essentialist statements along the lines of suitable generalizations of FULL/PARTIAL-GENERIC-ESSENCE. 2. Bridging the epistemic essence-to-necessity gap Let us return to the essence-to-necessity gap worrying Leech. It concerns what she calls 'the Necessity Principle', which she borrows from Mackie (2020, pp. 248-9). (NP) If being (an) F is an essential property of x, then being (an) F is a necessary property of x. Leech does not say what she takes for a property to be 'necessary'. But for Mackie, F is a necessary property of x iff x could not have existed without being (an) F (2020, p. 249). We shall interpret (NP) in the same fashion. Moreover, with '®' the material conditional, and with '£' and ' ' denoting metaphysical necessity and possibility, respectively, and interdefinable in the standard way, we will formalize this as the condition requiring that ~ ($y(y = x) Ù ~Fx), or equivalently £($y(y = x) ® Fx), as it is a condition that essentialists would generally be on board with. With (NP) in hand, Leech characterizes the worry for the essentialist in explicitly epistemic terms: because (NP) 'is something calling for justification', 'the challenge to defend (NP) is precisely a challenge to give reasons to agree' that it is true (pp. 7-8). Notice, the challenge is not to give reasons to agree that essentialism itself is true. One can agree to (NP) yet reject the view, as Leech notes (p. 3). Nor is the challenge to show how all knowledge of necessity could be derived from a purely essentialist basis, or even to defend the claim that (NP) is a conceptual truth about essence.6 It is not at all clear to us whether either project can succeed; but in any case, they are side issues. Essentialists (as we understand them) aim to account for what necessity is, not how one could come to know what necessities there are-let alone how one could do so from a purely essentialist basis, and let alone what one could know by merely consulting how essence is conceptualized. The 6 Like Fine (2020, p. 462), we read Mackie (2020) as worrying that the denial of (NP) is a conceptual possibility, since she focuses on illustrating conceptually coherent views of essence on which (NP) is false. Leech (2020) and Noonan (2018) employ a similar argumentative strategy, so it is reasonable to read them in a similar fashion. 7 challenge is simply to justify (NP), preferably using the most uncontroversial resources available, including (but not limited to) uncontroversial modal principles that even non-essentialists would accept. The challenge so understood, then given the account of essence sketched in §1, our first task is to show how the identity-based essentialist can justify the following principle, which is how the identity-based account of essence would unpack (NP): (NP*) If there is some H such that (y = x) oy (Fy Ù Hy), then £($y(y = x) ® Fx). We do this in two steps. First, we show how anyone-including, but not limited to, the identitybased essentialist-can justify belief in the (antecedently prima facie plausible) principle that generalized identities imply corresponding necessitated universal generalizations, one instance of which is the following: (NP+) If Iy oy Jy, then £"y(Iy ® Jy). And second, we show how once this is done, anyone-including, but not limited to, the identitybased essentialist-can justify belief in (NP*).7 For the first step, consider the following argument:8 (5) If Iy oy Jy and £"y(Iy ® Iy), then £"y(Iy ® Jy) instance of (LL) for oy (6) £"y(Iy ® Iy) premise (7) If Iy oy Jy, then £"y(Iy ® Jy) from 5 and 6 7 We have been granting to Leech that (NP) is in need of justification of some suitably substantive, non-circular sort. For instance, we are granting that the essentialist must do more than simply insist that implying necessities is simply what essences do. But that may well be challenged (see e.g. Wallner and Vaidya 2020). 8 Correia and Skiles (2019, p. 646) already gestured at using (LL) to derive similar principles, to wit 'If Fx ox Gx, then £"x£(Fx « Gx)' and 'If j o y, then £(j « y)', where '«' is the material biconditional. Note that the consequents of these principles are modalized biconditionals whereas the consequent of (NP+) is a modalized conditional. 8 As we stressed in the previous section, (LL) must come with restrictions. However, whichever the correct restrictions may be, they certainly do not rule (5) out (remember that here, '£' denotes metaphysical necessity; see our remarks on page 4). And presumably one can justifiably believe as much, regardless of one's stance on (identity-based) essentialism. Moreover, what (6) says is hardly deniable: necessarily, anything that is G is, after all, G. It is, of course, a controversial matter how it is that one might be justified in believing (6). Nonetheless, presumably one can justifiably believe (6) independently of knowing which (if any) account of necessity is correct. For the second step, suppose that (y = x) oy (Fy Ù Hy) for some given H. (NP+), which has just been established, allows one to infer that £"y((y = x) ® (Fy Ù Hy)). Consider then the following general modal principles: (8) If £"y(j ® (y Ù c)), then £"y(j ® y). (9) If £"y((y = x) ® Gy), then £($y(y = x) ® Gx). Given that £"y((y = x) ® (Fy Ù Hy)), (8) and (9) allow one to infer that £($y(y = x) ® Fx). Thus, (NP*) follows from (NP+) and principles (8) and (9). Both principles are validated in any normal propositional modal logic augmented with very weak postulates for the quantifiers and Leibniz's Law for objectual identity (a full set of postulates for classical quantification is not needed, a suitable set of postulates for free logic is indeed enough). One can accordingly surely be justified in believing these principles regardless of one's stance on (identity-based) essentialism. We have just showed how one can be justified in believing (NP+) regardless of one's stance on (identity-based) essentialism. We can therefore conclude that one can be justified in believing (NP*) regardless of one's stance on (identity-based) essentialism. As a corollary, we have shown that Leech's epistemic essence-to-necessity gap can be crossed by the identity-based essentialist. Our proof using (LL) above shows that generalized identities imply corresponding necessitated universal generalizations. But as we said in the introduction, Leech focuses on a different proof, one meant to show that generalized identities are themselves necessary. Like the original, the generalized Barcan-Kripke proof derives the necessity of generalized identity from the necessity 9 of generalized self-identity, utilizing a generalized version of Leibniz's Law (p. 15). The proof goes like this (Leech presents a proof that if j o y, then £(j o y), but we stick to generic identities in order to be closer to the issue at stake): (10) If Iy oy Jy and £(Iy oy Iy), then £(Iy oy Jy) instance of (LL) for oy (11) £(Iy oy Iy) premise (12) If Iy oy Jy, then £(Iy oy Jy) from 10 and 11 Our problem with this proof is not that it is unsound-we happily endorse it. Rather, the problem is that its conclusion, (12), appears to us more remote from (NP*) than the conclusion of our proof, (NP+). For suppose that (i) (y = x) oy (Fy Ù Hy) for some given H, and that the goal is to establish that (ii) £($y(y = x) ® Fx). The conclusion of our proof from (LL) allowed us to move from (i) to a modal statement not involving the o operator, and we were then able to reach the goal by appealing to two general modal principles also not involving the o operator. If we instead use the conclusion of Leech's proof from (LL), we would move from (i) to a modal statement that does contain the o operator. From this point, we would presumably then first have to 'get rid of' the o operator in some way, and then use general modal principles in order to reach the goal. This is an additional gap that needs to be crossed, and it is not obvious what will bridge it. Be that as it may, Leech argues that it would be circular for the identity-based essentialist to appeal to the generalized Barcan-Kripke proof in order to bridge the epistemic essence-to-necessity gap, and one may be tempted to lodge a similar complaint about our own. Recall that in Leech's proof, one must appeal to the necessity of generalized self-identity, to which Leech responds: 'How can the essentialist account for this necessity? That is, why agree that [£(Iy oy Iy)]? If they are to remain true to their essentialism, the crucial necessity must have its source in essence' (p. 15). The identity-based essentialist might reply that it is part of the essence of generalized identity that Iy oy Iy. But it only follows from this that £(Iy oy Iy) if one has already established what the proof was meant to show: that essentialist truths imply corresponding necessities (p. 16). Likewise, in our proof we assume that £"y(Iy ® Iy). And likewise, the identity-based essentialist may try to account for this fact by claiming that it is part of the essence of quantification and the material conditional that "y(Iy ® Iy). And likewise, Leech might object that it only follows from this that 10 £"y(Iy ® Iy) if we have already established what our proof was meant to show: that essentialist truths imply corresponding necessities. There is no genuine circularity here, however. This is especially clear in the case of our proof (our point extends to the generalized Barcan-Kripke proof). If essentialism is true, then every necessity, including the fact that £"y(Iy ® Iy), will have its 'source' in essence that needs to be 'accounted' for (in Leech's words). However, as we stressed before, the challenge at issue for the essentialist is to justify (NP), to give reasons to 'agree' that it is true, while remaining true to essentialism (again in Leech's words). We did so, in part, by appealing to the fact that £"y(Iy ® Iy). Hence, our only remaining burden is to argue that one is justified in believing that £"y(Iy ® Iy). Yet we take it that only the most extreme skeptic about metaphysical necessity would deny this. And more directly to the point, it is obvious that one can be-and many are-justified in believing that £"y(Iy ® Iy) on some basis or other, and regardless of whether one is justified in believing essentialism or not. And so the burden is shouldered. Similar points can be made about the other premises in our proof of (NP). Since this is the lynchpin of Leech's case against the identity-based essentialist, it is worth putting the point in a slightly different way. Suppose that one wants to provide an account of consciousness in terms of having such-and-such brain state. Moreover, suppose one (also) believes that if one is conscious, then one is in such-and-such brain state. Even so, one might try to justify the belief in the relevant consciousness-to-brain implication in a neutral way, taking no stance on what account of consciousness might be correct. For instance, one might reason as follows: (13) I am conscious. (14) If something is conscious, then it is in such-and-such brain state. (15) I am in such-and-such brain state. There are any number of many ways to justify (15) partly on the basis of (13) that even those who reject that (15) accounts for (13) might still find compelling. For instance, perhaps one is justified in believing (13) on the basis of introspection; perhaps one is justified in believing (14) on the basis of repeated and representative confirmation; and perhaps one is justified in believing that (15) logically follows from (13) and (14) on whatever basis one typically takes deduction to be 11 truth-preserving. Even so, it is obviously not a good objection to this way of justifying (15) to note that one believes that (15) accounts for (13), and moreover that the success of this account requires that (15) implies (13). For one did not rely on these facts about (13) to justify (15). Similarly goes with how we have justified (NP). It may well be that an essentialist account of £"y(Iy ® Iy) succeeds only if (NP) is true. But our justification for (NP) is not circular, since we did not rely on the success of this account to justify (NP). Rather, we justified (NP) using principles that even the anti-essentialist should find compelling. 3. Bridging the explanatory essence-to-necessity gap Thus far, we have focused on Leech's worry that the identity-based essentialist cannot give reasons to believe that essences imply necessities. However, several authors have worried about another sort of alleged essence-to-necessity gap. Even if one is justified in believing principles like (NP) are true, one may reasonably ask for a more detailed metaphysical story of how it is that essences account for necessities-the exact 'mechanism' by which this feat occurs, so to speak-a story in which (NP) follows as a consequence. Since this is a demand to explain how it is that (NP) turns out true for the identity-based essentialist, call this the explanatory gap in order to contrast it with the epistemic gap worrying Leech.9 So how, then, should such a story be told? Before we begin, two preliminaries. First, let necessity and possibility be features of propositions, and accordingly let essentialism about metaphysical modality be an account of what it is for a proposition to be necessary (i.e. necessarily true) or possible (i.e. possibly true). With this move to proposition-talk, we will take (objectual and generalized) identities to be propositions rather than statements. Adopting such talk is by no means required, but it is very convenient. (By our lights, the correct "official" idiom should be a higherorder language, but such languages are notoriously impossible to translate satisfactorily in English.) 9 As we did when discussing the epistemic gap, we will simply grant that the demand in question is a legitimate one (although see fn. 6). Casullo (2020, pp. 91-2) is explicitly worried about the explanatory gap, while others authors have been less clear about which of the two they have in mind (or both). For instance, Romero (2019) at first says that his worry consists in the fact that essentialism 'just tells us that there is an explanation of modality by the essences; it doesn't tell us how that explanation is supposed to go' (p. 125, his emphasis), which suggests the explanatory gap; but later, the challenge is described in epistemic terms: 'I am asking what reasons there are to believe that essence does explain necessity' (p. 126, our emphasis). 12 Second, a crucial notion for what follows is that of the extensional correlate of an identity. We adopt the convention that every objectual identity counts as its own extensional correlate. The extensional correlate of a generalized identity is, roughly, the proposition you get from the generalized identity by replacing the identity operator by a material biconditional and, if the original identity is not factual, by taking the universal closure of the result. Thus, to illustrate, the extensional correlate of < j o y > is < j « y > (following standard notation, we throughout use the angled bracket to achieve reference to propositions), that of < j ox y > is < "x(j « y) >, and that of < j ox,y y > is < "x"y(j « y) >. Preliminaries in tow, we can now begin to bridge the explanatory gap. Since it is identity-based essentialism that is at issue, we first need to say how it is that identities account for necessities. We believe that every true identity, be it objectual or generalized, is necessary.10 We also believe that every extensional correlate of a true identity is necessary. We wish to consider here three different accounts of necessity in terms of identity which deliver these results, either immediately or with the help of extra assumptions that have some plausibility: The strong account A proposition is necessary iff: it is a logical consequence of the true identities. The weak account A proposition is necessary iff: it is a logical consequence of the extensional correlates of the true identities. The intermediate account A proposition is necessary iff: it is a logical consequence of the true identities and their extensional correlates. 10 We believe that every true objectual identity statement involving rigid nominal expressions is necessarily true. Our claim that every true objectual identity proposition is necessarily true must be understood as similarly restricted. How exactly the restriction should be formulated depends on which theory of propositions is countenanced, and we will not propose any such theory in what follows. It is not clear to us that a similar restriction should be imposed in the case of generalized identity. For instance, it strikes us as intuitively correct to say that all statements of type 'To be F is to be G', and all corresponding propositions, are necessarily true if true. Should we be wrong on this point, then our claim that all true generalized identities are necessary should be restricted in the appropriate way. (The issue is linked to, but does not boil down to, the recently widely discussed question of whether the rigid / non-rigid distinction can be meaningfully applied to predicates. See Nimtz 2019, §1, for a recent survey.) 13 The intermediate account immediately delivers both (i) that every true identity is necessary and (ii) that every extensional correlate of a true identity is necessary. For the other two accounts, the delivery is only partly immediate. The strong account immediately delivers (i), and it delivers (ii) once certain principles are taken on board. Given that objectual identities count as their own extensional correlates, we only need to focus on generalized identities. For generalized identities of type < j o y >, the following principle does the job: (a) For all identities < j o y >, < j « y > is a logical consequence of < j o y >. The argument is straightforward. If we assume more generally that the extensional correlate of any generalized identity is a logical consequence of this identity, then we get that every extensional correlate of a true generalized identity is necessary. The situation with the weak account is symmetric: it immediately delivers (ii) and it delivers (i) once appropriate principles are taken on board. This case is less straightforward than the previous one. Here as before, we only need to focus on generalized identities since objectual identities are their own extensional correlates. For generalized identities of type < j o y >, one may invoke the principle that if to be the case that j is to be the case that y, then 'for something to be j is for it to be y', i.e. more rigorously (formally, we should require that the variable a is not free in j or y): (b) For all identities < j o y >, if < j o y > is true, then so is < (a o j) oa (a o y) >, and the principle that o is as a matter of logic reflexive, i.e.: (c) < "a(a o a) > is valid, i.e. a logical consequence of any set of propositions. One can then reason as follows. Suppose that < j o y > is true. By (b), < (a o j) oa (a o y) > is also true. By universal instantiation, its extensional correlate, namely < "a(a o j « a o y) >, entails < j o j « j o y >. By (c) and again universal instantiation, < j o j > is valid, and therefore < j o j « j o y > entails < j o y >. Given that < j o y >, as has just been shown, is a logical 14 consequence of the extensional correlate of a true generalized identity, < j o y > is necessary according to the weak account. For arbitrary generalized identities, simply replace in (b) and (c) the unindexed identity operator by an identity operator with arbitrary index. Interestingly, the previous arguments can actually be used to show that the three accounts are extensionally equivalent. The previous argument establishes that given (b) and (c), every true generalized identity of type < j o y > is a consequence of the extensional correlate of some true generalized identity. Given the suggested generalization of (b) and (c), what can be established is that every true generalized identity is a consequence of the extensional correlate of some true generalized identity. Since every objectual identity is its own extensional correlate, we then have the following general principle: (G1) Every true identity is a consequence of the extensional correlate of some true identity. The suggested generalization of (a), remember, says that the extensional correlate of any generalized identity is a logical consequence of this identity. Since, again, every objectual identity is its own extensional correlate, we then get the following further general principle: (G2) Every extensional correlate of an identity is a logical consequence of this identity. Given (G1), the intermediate account is extensionally equivalent to the weak account. Given (G2), it is extensionally equivalent to the strong account. Given both (G1) and (G2), thus, the three accounts are extensionally equivalent. What could be said in favor of (a), (b) and (c)? A version of Leibniz's Law for o can be straightforwardly invoked in the case of (a): given that < j « j > is valid, the relevant version of the Law licenses the claim that < j « y > is a logical consequence of < j o y >. This version of the Law must come with some restrictions, for sure, but here it is applied to an extensional context (remember that '«' stands for material equivalence), and such contexts certainly do not create 15 problems (see our remarks on page 4 above). For (c), one may invoke a comparison with the case of objectual identity: it is a logical fact that objectual identity is reflexive, and likewise generalized identity is reflexive as a matter of logic. For (b), both a version of Leibniz's Law and the reflexivity of generalized identity may be invoked. Given that generalized identity is reflexive, < (a o j) oa (a o j) > is true (note that here we do not need to invoke validity, since truth is enough). The relevant version of Leibniz's Law then licenses the claim that < (a o j) oa (a o y) > is true if < j o y > is. Restrictions on the applications of this version of the Law are not a problem here either (see again our remarks on page 4 above). Similar arguments can be given to support the generalized versions of (a), (b), and (c). One might reply here that Leibniz's Law for generalized identity, even suitably restricted, is not a logical law, and that it is not a logical fact that generalized identity is reflexive, on the grounds that generalized identity is not a logical concept. This reply steps into muddy waters: drawing a line between the logical concepts and the non-logical concepts is notoriously tricky. Be that as it may, a possible rejoinder is to broaden the notion of logical consequence at work in the accounts so that logical validity includes all conceptual truths in its scope, and to claim that Leibniz's Law for generalized identity, suitably restricted, and the reflexivity of generalized identity both count as conceptual truths-which sounds very plausible. For what follows, we do not want to decide on the issues just discussed. We simply make the following recommendations: • If you are happy with generalized (a) but not with generalized (b) + (c), go for the strong account. • If you are happy with generalized (b) + (c) but not with generalized (a), go for the weak account. • If you are happy with neither, go for the intermediate account. • If you are happy with both, pick your choice between the strong account and the weak account (given either (a) or (b) + (c), the intermediate account involves a redundancy in the basis from which the necessities flow, and for this reason should be discarded). 16 We have now said how it is that identities account for necessities. So how does all this help the identity-based essentialist say how it is that essences account for necessities, and thus bridge the explanatory essence-to-necessity gap? Assuming Correia and Skiles's identity-based account of essence, all three accounts are essentialist: they account for metaphysical necessity in terms of essence. The reason is simply that on Correia and Skiles's account of essence, every generalized identity is an essentialist proposition, one that ascribes a full essence (recall FULL-GENERICESSENCE from §1). It is important to appreciate this, since this distinguishes the way the accounts under focus proceed from another way one might proceed: by first accounting for necessity in terms of essence in the standard way that Fine and Hale do that we discussed at the onset, and then accounting for essence in terms of identity. In contrast, on any of the three accounts of necessity introduced above coupled with Correia and Skiles's account of essence, one directly accounts for necessity in terms of identity. Unlike the accounts proposed by Fine and Hale, the three accounts of necessity do not give an essentialist treatment of logical necessity (nor of conceptual necessity if one adopts the broad notion of logical consequence mentioned before). Rather, they simply take the notion of logical consequence as a primitive. This is not itself a problem: our accounts are simply less ambitious than Fine's and Hale's. And as Leech notes, there may well be reason to give metaphysical and logical necessity separate treatments anyway (p. 18). Leech does worry that if metaphysical and logical necessity have difference sources, then one will have trouble explaining why it is that every logical necessity is a metaphysical necessity (p. 18). But there is no such trouble for our three accounts. Every logical necessity is (trivially) a logical consequence of the true generalized identities and/or their extensional correlates, and thus is metaphysically necessary on all three.11 It is also worth noting that our strong account is very different from the account of modality offered in Rayo 2013, first appearances notwithstanding, and in a way that is crucial to how we have bridged the explanatory gap. Rayo endorses the view that '[a] first-order sentence (or set of firstorder sentences) describes a metaphysically possible scenario if and only if it is logically consistent with the set of true 'just is'-statements' (p. 49). Putting the linguistic tone and the restriction to 11 Fine (2002, pp. 264-6) raises objections which, in effect, affect any view that takes necessities of a particular kind to be the logical consequences of a particular class of truths. Although we lack the space to discuss Fine's objections here, see Leech (2015, pp. 161-5) for replies that we find convincing, which can be adapted to the present context. 17 first-order aside, the view looks very much like the strong account. However, when Rayo gives the details, a very different picture emerges. He characterizes necessity by giving a worlds semantics for a first-order modal language, and what the true 'just-is' statements do is provide constraints on the construction of the model: Rayo selects a range of 'basic' true 'just-is' statements, and the constraint imposed e.g. by the true statement 'φ ≡x ψ' is to the effect that '£"x(φ « ψ)' comes out as true (he also has special "conditional" 'just-is'-statements that impose constraints represented by other kinds of modal statements). It is not clear to us that what Rayo proposes is an account of how it is that identities imply necessities, much less one that would be useful for bridging the explanatory essence-to-necessity gap. At any rate, if this is an account, and if it could be put to this use, a Rayo-style approach is clearly different from one that utilizes the strong account (or the other two accounts, for that matter).12 Let us finally address how the identity-based essentialist might explain how it is that (NP) in particular turns out true, and thus bridge the explanatory essence-to-necessity gap. She can simply do so by adopting either of the three accounts of necessity in terms of essence just discussed. We have already established that on any of the three accounts, every true generalized identity is necessary and has a necessary extensional correlate: unconditionally in the case of the intermediate account, conditionally upon (a) in the case of the strong account, and conditionally upon (b) and (c) in the case of the weak account. Since on Correia and Skiles's account of essence in terms of identity, generalized identities are essentialist propositions, once this account is combined with any of the proposed accounts of necessity, some essence-to-necessity gaps are already bridged. In fact, many more gaps, and in particular the one highlighted by the Leech-Mackie 'Necessity Principle' (NP), are bridged on any of these combinations of views. The key point here is that on any of the proposed accounts of necessity, the following 'closure' principle holds: (Closure) Any proposition that logically follows from true identities, or extensional correlates of true identities, or a mix of both, is necessary. 12 Dorr (2016, p. 69) discusses yet another account of necessity in terms of generalized identity, according to which 'It is metaphysically necessary that φ' is understood as 'φ ≡ T', where 'T' is an arbitrary tautology. Needless to say, this idea presupposes a very deflationary conception of generalized identity. Dorr discusses the idea in the context of what he calls Booleanism, which is a very deflationary conception of the notion. 18 Recall that on the account of essence in terms of identity under focus, (NP) translates into (NP*) If there is some H such that (y = x) oy (Fy Ù Hy), then £($y(y = x) ® Fx). (NP*) clearly follows from (Closure). For suppose that (y = x) oy (Fy Ù Hy) for some given H. Then < (y = x) oy (Fy Ù Hy) > is true. Its extensional correlate is < "y((y = x) « (Fy Ù Hy) >. A logical consequence of this proposition is < $y(y = x) ® Fx) > (very weak quantificational principles need to be involved here, these are indeed the same that need to be used to establish (9) above). By (Closure), it follows that < $y(y = x) ® Fx) > is necessary. Thus it follows that £($y(y = x) ® Fx), as desired.13 We said before when providing reasons to believe that essences give rise to necessities (which was the goal of §2), that doing so neither entails nor requires providing an account of how it is that they manage to do so (which was the goal of the present section). The same holds in reverse. We have not, after all, considered what reasons there are to believe the accounts of essence, modality, and identity that we have been relying upon. Perhaps those are in the offing. In that case, and if those reasons are not themselves based on prior justification for believing (NP), then our bridge over the explanatory gap also provides a route over the epistemic gap. But perhaps they are in the offing, but the order of justification is the opposite: one starts with reasons to believe (NP), such as those we provided before, and then one's reasons to believe these accounts derive in part from their joint capacity to explain (NP). In that case, the route is blocked. Nevertheless, although they are 13 One might also consider deriving a stronger version of (NP), which reads 'necessary property' not in the weak way as the condition that £($y(y = x) ® Fx), but rather as the condition that £Fx. We get this result with the weak and intermediate accounts, and on the strong account if (a) is true, if we take propositions of type < $y(y = x) > to be valid. These are valid by the lights of classical logic, which for some may be reason enough. But one must be careful here. If all such propositions are valid, then all three accounts also count them as necessary-that is, all three counts are committed to a form of 'necessitism' (cf. Williamson 2013). Whether that is a result to be accepted or avoided is, of course, a topic of considerable recent debate, a debate in which we do not wish to enter here. The reader may have noticed that we have not treated first-order quantification and higher-order quantification in the same way in this paper. When arguing that the weak account entails that all generalized identities are necessary, we made use of principle that generalized identity holds as a matter of logic, plus the view that universal instantiation for higher-order quantification is valid. Given the proposed accounts of necessity, together these yield the higherorder correlate of the form of (first-order) necessitism mentioned above. In contrast, in our previous remark on the stronger version of (NP) we express our desire to stay neutral regarding first-order contingentism. Now it may be that, as Williamson (2013) argues, higher-order necessitism does not go well with first-order contingentism, in which case our stance is problematic. If this is the case, then there is an easy fix that would allow us to be neutral regarding both first-order and higher-order necessitism: ignore the weak account. 19 independent of each other, the bridges we have constructed over these two gaps are compatible with each other, regardless whether one wishes to travel over one or both, and in what order. Conclusion A number of authors, including Leech (2020), have worried about how essentialists are supposed to 'deliver a modal rabbit out of a non-modal hat' (Mackie 2020, p. 252). We have shown that once one is equipped with the notion of generalized identity, and is careful to distinguish epistemic versions of the worry from more metaphysically-oriented ones, no magic is required.14 References Barcan, Ruth 1947, 'The Identity of Individuals in a Strict Functional Calculus of Second Order', Journal of Symbolic Logic, 12, pp. 12-15. Casullo, Albert 2020, 'Essence and Explanation', Metaphysics, 2, pp. 88-96. 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