Reply to Gideon Rosen’s ‘The Modal Status of Generative Principles’

1 The Problem

Rosen’s problem is:

In other words: for which p does there exist an F for which □_F p?² We might call this ‘the problem of essentialist existence’. It is not the problem of essential existence, of which things essentially exist, but the problem of which essences exist, i.e. of which candidate essences are the essence of something.

One might be more or less ambitious in attempting to this question. At the one extreme, one might aim to provide a criterion, i.e. a substantive set of necessary and sufficient conditions, for being an essence p. At the other extreme, one might adopt a more skeptical position and take there to be nothing of general interest to be said in response to the question. And, under an intermediate view, one might aim to provide a constraint on what essences there are, where this constraint can take the form either of a sufficient condition or of a necessary condition for a proposition to be the essence of some property.

I think it is hard to say in advance of further inquiry what one might reasonably aim for. Consider the somewhat analogous case of set theory. We know, in the light of Russell’s paradox, that there cannot be a set corresponding to any condition φ(x) whatever, i.e. a set y for which ∀x(x ∈ y ↔ φ(x)). So the question arises as to which conditions there will be a corresponding set. There are a number of different ways in which one might attempt to answer this question. One could follow the example of Quine’s New Foundations (Quine [1937]) and use a syntactic criterion, such as stratification, to delimit the required conditions. One could follow the lead of axiomatic ZF and provide some specific sufficient conditions (such as Separation) along with some necessary conditions (such as Foundation). One could adopt a more general principle, such as ‘limitation of size’, as an axiom or as a guide to which existence axioms might be adopted. And so on. But I think it would be agreed that, in advance of inquiry, there is no saying which would turn out to be the more fruitful approach; and similarly, I suggest, in the present case.

However, in the present case (and also in the case of set theory), certain explanatory challenges seem to cry out for some sort of constraint on which propositions can legitimately be regarded as essences. To take an example from early on in Rosen’s paper,

But if not, then surely there should be some explanation as to why not. It cannot, as Rosen puts it, be ‘a mere “accident”’. So the question arises as to what extent one might meet any reasonable explanatory challenge of this sort, even if one has not set one’s sights on providing a full set of necessary and sufficient conditions.

One might also be more or less ambitious over the resources that are to be used in specifying the conditions of legitimacy. One might attempt to provide a purely syntactic formulation of these conditions (as with New Foundations). Or one might provide a more semantically oriented formulation of those conditions, one which also takes into account the meanings of the terms by which the proposition is specified. And within this second option, one might be more or less austere in the further resources to which one then appeals. As examples, we might mention Rosen’s appeal to the quasi-formal notions of dependence and ground in §§3–4 and his subsequent appeal to the more informal notion of being a genuinely further fact in §10.

We should also note that there are a number of ways in which the problem may be generalized. Rosen considers the question of definition for a single defining proposition p. But given a system of defining propositions (or propositional functions) p₁, p₂, …, we might also wish to know when there are properties (or relations) F₁, F₂, … for which □_F1 p₁, □_F2 p₂, ….³ This more general form of the problem gives rise to additional difficulties. One might think, for example, that for any property F there should be a property G such that G is by its very nature the negation of F (□_G ∀x(Gx ≡ ¬Fx)) and yet balk, on grounds of circularity, at there being two properties F₁ and F₂, of which each is by its very nature the negation of the other (□_F2 ∀x(F₂x ≡ ¬F₁x) and □_F1 ∀x(F₁x ≡ ¬F₂x)).⁴ Or, one might wish to avoid an infinite regress under which F₁ is defined in terms of F₂, F₂ is defined in terms of F₃, and so on ad infinitum. It strikes me as plausible that a system of definitions □_F1 p₁, □_F2 p₂, …. will be satisfiable by some F₁, F₂, … as long as it does not lead to a cycle or infinite regress of dependency relations and as long as each individual definition, working from the “bottom up”, is satisfiable. If this is so, then the case does not call for separate treatment.

Another generalization, considered by Rosen in §1, concerns the essence of individuals, as opposed to properties. Rosen hopes to evade issues arising from the case of individuals by following a suggestion mooted in Fine [2015]) and reducing this case to the case of properties; we ask ‘not for the essence of Socrates, the man [but] ask instead for an account of what it is to be Socrates’ (§1). He treats the property of being Socrates as a ‘simple, non-relational property’ rather than as ‘an identity property – λx. x = s’. But the obvious objection to this proposal is that there is nothing for the property of being Socrates to be other than the property of being identical to Socrates. If only wishing a relational property to be non-relational could make it so!

Whatever the merits of his proposal,⁵ I wish to consider an approach that runs somewhat in the opposite direction. For any non-qualitative property can be regarded as the result of “plugging in” certain individuals into a purely qualitative property. Moreover, a purely qualitative property will only depend upon other purely qualitative properties, not on non-qualitative properties or on individuals. But this suggests that the general question of the essence of properties and individuals can be separated into the question of the essence of purely qualitative properties and the question of the essence of individuals. I believe that these two questions raise somewhat different issues (indeed, Rosen’s discussion from §7 onwards suggests as much); and so in what follows I shall consider them separately, dealing first (in §§2–3) with the essence of purely qualitative properties and then with the essence of individuals (in §4).

We might also wish to consider the case in which we ask, not whether there is an F for which □_F p, but whether there are F₁, F₂, … for which □_{F1, F2, …} p (corresponding to simultaneous definition) or whether, given properties G₁, G₂, …, there is an F for which □_{F, G1, G2, …} p (corresponding to parametrized definition) – either on their own or in combination with the other generalizations. Once we consider the problem in this general form and ask for which propositions are there items in whose nature it lies that the proposition be true then, under an essentialist conception of essence, we are simply asking which propositions are necessary. I take it that this is not the issue; and so there has to be some restriction, however hard it may be to articulate, on what the relevant instances of the problem should be taken to be.⁶

Finally, I should mention that there is a somewhat milder way of taking the question. For it might be thought that we are asking for too much when we ask whether a given proposition is the essence of some property, that this is to stray too far into the substantive question of what the essentialist truths actually are. Rather, what we should be asking is whether the given proposition is a plausible candidate for being the essence of some property – whether there is some principled barrier, so to speak, to its being the essence of some property. I think that Rosen may sometimes have this milder construal of the question in mind. In any case, we are always at liberty to ask the more stringent question and yet construe a partial answer to the more stringent question as a full, or fuller, answer to the less stringent form of the question.

2 Explicit Definition

Rosen begins his discussion of the problem of essentialist existence with what is the clearest and cleanest case of definition, explicit definition, under which one gives ‘non-circular necessary and sufficient conditions for being F’ (§3). In this case, he tells us (§3), a definition will take the following form:

where φ does not contain F or anything that depends on F for its definition.

Since we are taking F to be a purely qualitative property, it should also be required that φ(x) not contain any individual constants or any free individual variables other than x and that any predicate variables or constants in φ(x) should be interpreted as denoting purely qualitative properties.

Note here a subtle shift in how our original problem has (or appears to have) been framed. Before we were talking in the material mode about which propositions p constitute an essence of some property. But we are now talking in the formal mode about which formal conditions φ(x) might be used to specify the essence of some property. Both are legitimate questions – of interest in their own right, yet clearly related. However, the formulation of the proposed conditions on essentialist existence is much more problematic when stated in the material rather than in the formal mode. When working in the formal mode, we can simply appeal to the predicates that appear in the formula φ(x) but, when working in the material mode, we need to appeal to the notion of a property occurring or being contained in a given proposition. Some account of the identity of propositions is thereby presupposed. Despite the great interest of the material mode formulation, I shall, in what follows, go along with what I take to be closer to Rosen’s intent in focusing on the formulation of the problem in the formal mode.

This means that we can regard an answer to the problem of essentialist existence as being part of a formal theory or logic of essence. I am not going to be altogether precise about the form such a theory or logic should take; and I hope that most of what I say will hold regardless of what the form is taken to be. But there is one particular issue that needs to be discussed; and this is whether we take ourselves to be working with a constitutive or consequentialist notion of essence. Under a consequentialist, as opposed to a constitutive, notion, essentialist claims are closed under some appropriate form of logical consequence – if □_F φ₁, □_F φ₂, … and ψ is, in an appropriate way, a logical consequence of φ₁, φ₂, …, then also □_F ψ (and similarly when the essences of different properties are involved). I shall not be too careful in specifying the forms of logical consequence to which I appeal below, but we will sometimes need to make use of “restricted” consequences (in which the essence of an item is restricted to the items upon which it depends) and of “mediated” consequences (in which the essence of an item inherits the essences of the items upon which it depends).⁷

It seems clear to me that Rosen wishes to work with some version of a consequentialist notion of essence.⁸ In any case, use of the constitutive notion gives rise to the difficulty in saying how exactly a constitutive essence is to be formulated (can, for example, the essence of the explicitly defined property F be given by □_F ∀x(φ(x) ↔ Fx) rather than by □_F ∀x(Fx ↔ φ(x)?); and, as I hope will become clear, the use of a consequentialist notion allows for the formulation of conditions of essentialist existence which would not otherwise be readily available.

Under a consequentialist notion of essence one might plausibly maintain, when ∧ is the empty plurality, that □∧ ψ is true just in case ψ is a logical truth, i.e. true in virtue of the nature of the logic constants. Logic, in this sense, comes for free. Of course, this still opens what we should take the logical truths to be.

As it stands, (ED) above simply tells us what it is to have an explicit definition of a property. In itself, it tells us nothing about what it is for a formula ψ(F) to be an actual essence (i.e. one for which ∃F□_F ψ(F)). But I suspect that Rosen’s line of thought, here and in the rest of his paper, is that we should be able to read off a solution to the problem of essentialist existence from the different ways in which a property might be defined. There is, so to speak, a definitional backdrop against which the legitimate essences of properties may be ascertained. I think that this approach may well be on the right track in the present case. However, I do not believe that the path from the one to the other is as straightforward as one might think; and so it may be worth spelling out in some detail what the connection between the two might reasonably be taken to be.

In the present section of the paper, Rosen takes himself to be operating under what he calls a ‘strict definitional conception’ (§3), under which ‘the essence of F is constrained to consist in a real [and explicit] definition of F’. But as he later points out, this conception is ‘too strict’. Indeed, it is not even clear whether it can be consistently maintained. For if there is a property F₁ it will have a definition in terms of some other property F₂ (and perhaps others as well), F₂ will have a definition in terms of some other property F₃, …; and so we will be saddled with an infinite regress. So all that we can reasonably hope for, on the basis of (ED), is an account of essentialist existence that covers the case of properties which can be explicitly defined.

To this end, let us first consider the question of stating sufficient conditions for essentialist existence. We are presumably meant to think, on the basis of (ED), not merely that any explicit definition of a property will be of a certain form but that anything of that form will actually provide an explicit definition of a property. Let us use F > G to indicate that F depends upon G. Suppose that G₁, G₂, …, G_n are the predicate variables (or schematic predicate letters) occurring in the formal condition φ(x) and that F is a distinct predicate variable. We are thereby led, on the basis of our expectations about (ED), to the following existence principle:

Such an F will satisfy the formal requirements from (ED) for being an explicit definition of F, since none of the properties denoted by the predicates in φ(x) can be identical to or dependent upon F, given that dependence is transitive and asymmetric.⁹ Given (ED1), not only will there be such an F, but it will be unique in the sense that, should □_F ∀x(Fx ↔ φ(x)) and □_F’ ∀x(F’x ↔ φ(x)) both hold, then □_{F, F}’ ∀x(Fx ↔ Fʹx) will hold – it will lie in the nature of the two properties F and Fʹ to be co-extensive.

If φ(x) is a first-order formula (without quantifiers over properties or the like), then I believe that (ED1) may be allowed to stand.¹⁰ There is, however, a possible, and somewhat annoying, exception. For what if φ(x) is itself an atomic formula of the form Gx? (ED1) will then tell us that ∃F (□_F ∀x(Fx ↔ Gx) ∧ F > G). But it might be thought that if it lies in the nature of F to be co-extensive with G, then F and G must be the same (or, at least, indistinguishable). I myself am happy with the thought that F and G are not the same, with the difference between them simply consisting in the fact that F is defined in terms of G while G is not. I am therefore willing to concede that there are at least two distinguishable properties F for which □_F ∀x(Fx ↔ Gx), where one is defined in terms of G and hence dependent upon G and where the other is simply G itself (given that □_G ∀x(Gx ↔ Gx)) and therefore not dependent upon G.

This view is much more amenable to formal development than the view under which the properties are taken to be the same; and it is one I shall adopt in what follows. The criterion of identity in terms of essential co-extension can of course no longer be accepted, but one might put in its place a criterion according to which two properties are the same when it lies in their nature to be co-extensive and when, in addition, they depend upon the same things.¹¹

The case in which φ(x) is not first-order is much more problematic. For it is not then even clear that there will an F for which ∀x(Fx ↔ φ(x)) – let alone an F for which □_F ∀x(Fx ↔ φ(x)). One problem arises from the possibility of paradox within an untyped language. Another, which also arises within a typed language, concerns the legitimacy of impredicative definitions (in which the definition suffers from what might be regarded as an indirect form of circularity, with the second-order quantifiers in φ(x) ranging over the very property F that is in question).

The case of impredicative definition is of special interest in the present context. There is, of course, a familiar issue over whether explicit impredicative definitions should even be allowed in an extensional language. Is it legitimate, for example, to define a natural number as a number that belongs to every inductive “property”, given that the property of being a natural number is one of those properties? But it strikes me that a very plausible hybrid position is one in which an impredicative definition may be legitimate in the extensional case, when only the extension of the predicate is in question, but not in the essentialist case, when the very nature of the property that it denotes is also in question. Thus, even though an impredicative definition may be allowed, it should, for this reason, always be possible in principle to back it up with a definition which is not impredicative in form and which may be regarded as getting at what the property “really” is. This illustrates how extensional, or even intensional, demands on definition may fall far short of the essentialist demands that one might wish to place.

However, the issue of impredicativity is orthogonal to our present concerns; and so I propose that we ignore the issue and focus on the case in which the defining formula φ(x) is first-order (which is the only kind of case Rosen considers in his paper). We may still use second order quantifiers elsewhere in our symbolism, but if we wished to stay within the confines of a first-order symbolism (as in Fine [1995a]), then we might replace (ED1) with a rule of definition permitting one to introduce a new predicate F subject to the condition □_F ∀x(Fx ↔ φ(x)) for any suitable formula φ(x).

Let us now turn to the question of providing necessary conditions for essentialist existence. One might think this problem is trivial, since has Rosen not told us that ‘the essence of F is constrained to consist in a real definition of F’? It is not altogether clear to me what Rosen means by saying that the essence is so constrained, but if this means that a statement of the form □_F ψ(F) can only be true when ψ(F) is of the form ∀x(Fx ↔ φ(x)), then it would appear to be far too stringent a constraint. It would mean, for example, that, given □_F ∀x(Fx ↔ φ(x)), we could not infer □_F ∀x(Fx → φ(x)), or □_F ∀x(φ(x) → Fx), or □_F ∀x(Fx ↔ Fx), or even □_F ∀x(φ(x) ↔ Fx). But under very weakly consequentialist notions of essence, we should be able to make inferences of this sort.

Rosen’s intent, I take it, is that the essence of the property should in some sense be ‘exhausted’ by an explicit definition; and this, at first blush, can be taken to mean that if F has a real definition of the form ∀x(Fx ↔ φ(x)) then □_F A, for any given sentence A, will only be the case if A follows from ∀x(Fx ↔ φ(x)).¹² But what is meant by ‘follows from’? Suppose G₁, G₂, …, G_n are the (non-logical) terms occurring in φ(x). Then if □_F B and □_{G1, G2, …, Gn} (B → A), it plausibly follows under a consequentialist notion of essence that □_F A. This suggests that we may take A to follow from ∀x(Fx ↔ φ(x)) in the present context to mean □_{G1, G2, …, Gn} ∀F(∀x(Fx ↔ φ(x)) → A). Thus, given □_F ∀x(Fx ↔ φ(x)) and given that G₁, G₂, …, G_n are the (non-logical) predicates occurring in φ(x), a necessary condition for □_F A to hold is given by the following principle:¹³

This is a substantive principle. It means, in effect, that whenever a property has an explicit definition then that explicit definition exhausts the nature of the property, that there is nothing more to what it is to be that property. The principle is very plausible in the present case; and the general idea of an essence being complete or exhaustive in this way is clearly of considerable interest (and yet has not, as far as I know, been subject to formal study).

We also have a kind of converse to (ED2):

For given that F does not occur in A, ∃F□_F∀x(Fx ↔ φ(x)) by (ED1). Given □_{G1, G2, …, Gn} ∀F(∀x(Fx ↔ φ(x)) → A), □_F ∀F(∀x(Fx ↔ φ(x)) → A) under a principle of “mediated” essence. So by logical closure (and granted also, given that F is purely qualitative, that it essentially exists), it follows that □_F A.

The above account has some general features, which may have application to other forms of definition. The first is that we have provided what one might call a canonical form of explicit definition. This is a statement of the form ∀x(Fx ↔ φ(x)). On the basis of this canonical form, we can then provide a sufficient condition, (ED1), for the existence of a property conforming to the definition and a necessary condition, (ED2), for something to be essential to a property that has been defined in this way.

This suggests that one might be able to provide a canonical form for other kinds of definition and use it, in a like manner, to provide sufficient and necessary conditions.¹⁴ If the canonical form of the alternative form of definition is ψ(F) (where G₁, G₂, … are the other predicate variables in ψ(F)) then what corresponds to (ED1) and (ED2) are:

But there are reasons for thinking that this strategy for extending the account of explicit definition will be hard to bring off. For once we consider more general forms of definition, it is hard to see why the existence principles like (GD1) should hold unless the defining condition ψ(F) is very weak. But if the condition ψ(F) is very weak then there is no reason in general to think that ψ(F) will exhaust the essence of F and hence that something like (GD2) might be true. We want, on the one hand, for the essence to exist and we want, on the other hand, for it to be exhaustive. There may, of course, be special cases in which both desiderata can be satisfied, as in the case of explicit definition. But it is hard to see what general format might cover all the kinds of case that could arise.

Rosen claims that ‘if the essence of Z is constrained to include only necessary and sufficient conditions for being Z, it will never lie in the nature of Z that Jane is a banker’ (§3).

This claim, as it stands, is not quite accurate. For suppose that there is a W for which it lies in the nature of W that Jane is a banker. Define Z to be the negation of W (any other explicit definition of Z in terms of W would do). Then we may plausibly (through a principle of mediated essence) take it to lie in the nature of Z that Jane is a banker. However, it does strike me that Rosen’s instincts are right and that it is plausible, and even capable of demonstration, that if A is not a logical truth, i.e. is a sentence for which ¬□∧ A, then □_F A can only be true if □_G A is true for some G upon which F depends and yet which is is not itself explicitly definable. Thus, the source of a bad essence can always be traced to a property that is not explicitly definable.

3 Implicit Definition

We turn to the case in which a property has an essential nature even though it lacks any explicit definition. To use a putative example from Rosen (§3), ‘it should lie in the nature of knowledge that if S knows that p then p is true even if knowledge is indefinable’.

Rosen wishes to deal with certain difficulties that arise in this case by supposing that the definitions of these properties should provide necessary or sufficient grounds for the application of the property. I am very sympathetic to the idea that grounds will often play such a role in determining a property’s nature. However, as Rosen himself points out, not all of the essential properties seem to derive in this way from the grounds for their application (§6). We should also note that we cannot arbitrarily assign grounds for the application of a property and expect the property to exist (this would give rise to analogues of the tonk problem in logic) and nor can we expect an arbitrary assignment of grounds for the application of a property to exhaust its nature. Definitions of this sort, important as they maybe, cannot therefore be expected to play the same role as explicit definitions in determining when we have an essence.

I should like to adopt a somewhat different tack, though it loosely relates to Rosen’s own solution in being “bottom-up”. It is a natural thought that if we “introduce” a property then it should leave things as they were. This corresponds to the familiar idea in the logical literature that definitions should be non-creative, or conservative – that one should not, through the introduction of new vocabulary, be able to prove anything in the old vocabulary that one was not already able to prove.

But what does this natural thought come to in the present context? Let the subvening properties of a given supervening property be those upon which it depends; and let a subvening essence of a given property be one that only involves its subvening properties. Then what we would like to say is that any subvening essence of a property should be an essence of its subvening properties, that the supervening property should not, by its very nature, tell us anything more than what was already implicit in the nature of the subvening properties.

To give formal expression to this thought, let us suppose that ψ(G₁, G₂, …, G_n) is a formula whose (free) predicate variables are G₁, G₂, …, G_n and, given a property F, let its dependency base F↓ be the plurality of properties upon which F depends (or upon which F immediately depends).¹⁵ We may then adopt the following principle:

Conservativity (for Properties) (□_F ψ(G₁, G₂, …, G_n) ∧ F > G₁ ∧ F > G₂ ∧ … ∧ F > G_n)) → □_F↓ψ(G₁, G₂, …, G_n).

We should note that F↓ will include the properties G₁, G₂, …, G_n “occurring” in ψ(G₁, G₂, …, G_n) but may include other properties as well.

This takes care of Rosen’ problem case of a property Q for which □_Q ∀x(Red x → Qx) and □_Q ∀x(Qx → Round x). For if there were such a property then □_Q ∀x(Red x → Round x) and so, by Conservativity, □_{Red, Round} ∀x(Red x → Round x) – which is not so and which, in any case, effects a reduction of the problem to the essential nature of Red and Round. We thereby obtain an austere solution to the problem, avoiding all appeal to ground and the difficulties to which such an appeal gives rise.¹⁶

This principle yields a necessary condition for there to a property F for which □_F ψ(F, G₁, G₂, …, G_n). For suppose □_F ψ(F, G₁, G₂, …, G_n), where F depends upon each of G₁, G₂, …, G_n. Then □_F ∃Fψ(F); and so □_F↓ ∃Fψ(F) by the Conservativity Condition. We thereby obtain the following necessary condition:

I should now like to suggest that a kind of converse to (ID2) also holds. Again, suppose G₁, G₂, …, G_n are the predicate variables occurring in ψ(F) and that F is a predicate variable distinct from each of G₁, G₂, …, G_n. Then we will want:

If it lies in the nature of certain properties for there to be a property that plays a certain role then there is a property which depends upon those other properties and whose nature it is to have that role.

(ID1) is a kind of essentialist witnessing condition (or exportation principle); and we might see it as providing a vindication of implicit definition, much as (ED1) provides a vindication of explicit definition. Indeed, (ED1) can be seen to follow from (ID1). For take ψ(F) to be the formula ∀x(Fx ↔ φ(x)), where G₁, G₂, …, G_n are the predicate variables occurring in φ(x). Now presumably □_{G1, G2, …, Gn} ∃F∀x(Fx ↔ φ(x)), since ∃F∀x(Fx ↔ φ(x)) is a logical truth concerning G₁, G₂, …, G_n; and so:

by (ID1).

(ID1) is a very strong principle and of considerable interest in its own right. Suppose it lies in the nature of pain to be physically realized, i.e. it lies in the nature of pain that there should be a physical property F which is such that any instance of pain is realized by an instance of this property. It then follows from this principle that there is a property whose nature it is to be a physical realization of pain, i.e. it lies in the nature of this property to be a physical property which is such that any instance of pain is realized by an instance of this property. But it is far from clear that this does follow.

However, ψ(F) in this case involves a second order property, the property of being a physical property, and it involves the notion of realizing, which is not extensional. Suppose we insist that ψ(F) should be an extensional first-order formula so that its satisfaction only depends upon the extensions of its predicates. Then it is far more plausible that the principle should hold. Indeed, one might argue that the corresponding modal claim:

□∃Fψ(F) → ∃F□ψ(F)

should hold. For the left hand side will deliver an appropriate extension for F in each possible world and so one might take the F on the right to have that extension in each possible world. F is, so to speak, a choice function. This then gives us some reason to think that the corresponding essentialist principle, (ID1), might be true. But note that the possibility of applying the principle will depend upon the G₁, G₂, …, G_n in the antecedent □_{G1, G2, …, Gn} ∃Fψ(F) being “cognizant” of there being an F for which ψ(F) even if they are not cognizant of any particular F for which this is so.

4 Individual Definition

I come now to what Rosen regards as the “the main event”, ‘If it can lie in the nature of singleton Socrates to be a set that exists automatically whenever Socrates exists’ then how can it not ‘lie in the nature of Socrates guardian angel to be an angel that follows Socrates from world to world?’ (§10). Like Rosen, I find this problem very troubling and difficult; and I will follow his lead in making a few tentative suggestions in the direction of answer.

Consider the cumulative hierarchy of sets. This is given by a collection of domains V_α, one for each ordinal α, defined in the usual way. Let us now generalize the conception of a cumulative hierarchy to generated entities in general.¹⁷ At the base level, we have the domain of entities U₀ that do not immediately depend upon any other individuals. At the next level, we add the individuals that immediately depend upon individuals in U₀ (and upon no other individuals).¹⁸ And so on up the hierarchy in the usual way. In the special case of set theory, immediate dependency will be set-theoretic membership and U₀ will be the domain of urelements, those individuals that are not themselves sets. But we, of course, are supposing that there may be many other forms of dependency. So U₀ will be restricted to those urelements that are not dependent upon any of the other urelements and the subsequent domains, U_α, will grow to include many other individuals besides sets.

We thereby obtain a cumulative ontology of generated entities. But there is also an associated cumulative ideology of generative relations. These are the properties and relations – $R_{\alpha }^1$, $R_{\alpha }^2$, … – by which the behavior of the generated entities is fixed. Each relation (or property) $R_{\alpha }^k$ is defined over the domain U_α; and just as the ontology is cumulative, with U₀ ⊆ U₁ ⊆ …, so the ideology is cumulative, with $R_0^k$ ⊆ $R_1^k$ ⊆ …. Indeed, the relations will be conservative in the sense that the restriction of $R_{\alpha }^k$ to U_β, for β < α, will be $R_{\beta }^k$; no $R_{\alpha }^k$ will introduce new relationships over pre-existing entities. There may well be some additional constraints on the cumulative ideology. So, for example, if <e₁, e₂, … > is in the extension of R then one might want to insist that exactly one of e₁, e₂, … ∈ U_α (this will be the entity generated from the other entities).

For each entity e in U_α – U₀ for some α > 0, there will be a least ordinal α for which e ∈ U_α. This is the stage at which e is “introduced” into the ontology. There will then be a family of relationships <e₁, e₂, … > that involve e and belong to the extension of some generative relation R. This family of relationships can then be taken to constitute a definition of the entity e. So, for example, a definition of a set will be given by its being a set and having certain entities as its members.

What I would now like to suggest is that we think of these relationships as constituting a definitional backdrop, much as in the case of explicit definition, through which the existence of the essences of the entities may then be determined. However, the case is by no means as straightforward. For we cannot arbitrarily generate entities. We cannot, for example, generate an entity which is the sole member of Socrates. The principles by which the existence of an entity is determined would therefore seem to depend upon the character of the relations by which the entity is generated, though possibly there are more or less systematic ways in which this can happen.¹⁹ However, we can perhaps reasonably claim that the proposed definitions exhaust the nature of the entities in the sense previously explained. Thus, even though we will no longer have a sufficient condition for essentialist existence, along the lines of (ED1), we will still have a necessary condition, along the lines of (ED2) or (GD2).

But just as not all properties have an explicit definition, not all individuals will be generated, even when they depend upon other individuals. It may well be thought to be essential to an electron, for example, to be negatively charged or essential to a material body in a gunky universe to be the fusion of other material bodies. But in this case, we can perhaps uphold a version of the Conservativity Condition for individuals. Indeed, it is plausible that Conservativity should hold in a more general form. Suppose that ψ(X₁, X₂, …, X_n) is a formula whose free variables (of any type) are X₁, X₂, …, X_n and now, given an item X (of any type), let us use X↓ for the plurality of items upon which X depends (or upon which X immediately depends). We may then adopt the following principle:

General Conservativity (□_X ψ(X₁, X₂, …, X_n) ∧ X > X₁ ∧ X > X₂ ∧ … ∧ X > X_n → □_X↓ψ(X₁, X₂, …, X_n).

We should note that, even when X is an individual, X↓ will in general include properties as well as individuals. We might, for example, take it to be essential to the null set to be a set and have no members. The null set will then depend upon the property of sethood and the relation of membership.²⁰

General Conservativity gives rise to an issue (which, to some extent, has already arisen for some of the other principles we have considered). Consider the null set. It lies in the nature of this entity to exist and so it lies in the nature of this entity that there is something. Now this entity presumably depends upon sethood and membership and so, by the conservativity condition, it lies in the nature of sethood and membership, that there is something. But then by a further application of the condition, it lies in the nature of the null plurality that there is something and hence it is a logical truth that there is something.²¹

This conclusion may not appear so bothersome. But the reasoning extends to the singleton of the null set, to its singleton, and so on; and so, for each finite number n, we are able to establish that it is a logical truth that there are at least n things (and we may even extend the argument into the infinite).²²

Although I have presented the issue as one concerning individuals, it also arises for properties. For presumably it is a logical truth that for any individual there is a corresponding identity property (□∧ ∀y∃F∀x(Fx ↔ x = y)). So given that it is a logical truth that there are at least n individuals, for any finite number n, it will also be a logical truth that there are at least n non-coextensive properties.

One may be willing to accept these consequences. One will then think of logic, or the null plurality, as already having access to the whole ontology of generated entities, though not to the generative relations by which they might be described. But if one is not willing to accept these consequences, then the conservativity condition, and some of our other principles, must be modified. In the case of the conservativity, we might insist that the quantifiers in ψ(X₁, X₂, …, X_n) be explicitly restricted to entities already recognized by X₁, X₂, …, X_n. And one might go even further and say that in claiming that it lies in the nature of the null set to have no members, □_∅∀x(x ∉ ∅), the inner quantifier ∀x cannot be taken to range over any individuals which depend upon ∅, since then the claim would suffer from a kind of impredicative circularity. Clearly, this latter option would severely impede our ability to state the essence of things. However, a discussion of how an alternative view of this sort might proceed would take us too far afield.

Let us see how the present considerations might bear upon Rosen’s problem of the guardian angel (stated now in terms of the essences of individuals rather than of properties). There is an entity – viz. singleton Socrates – which, by its very nature, exists and is a set containing Socrates when Socrates exists (∃y□_y (Es →(Ey ∧ S(y) ∧ s ∈ y))). But there is no entity (even if Socrates is fortunate enough to have a guardian angel) which, by its very nature, exists and is an angel who guards Socrates when Socrates exists (¬∃y□_y (Es →(Ey ∧ A(y) ∧ G(y, s)))). But how come the one essence exists and the other does not?

The two cases only differ in the properties and relations that they involve – with the pair, set and member, on the one side, and the pair, angel and guardianship, on the other. So this would seem to show that there is a material, rather than a formal difference, between the two kinds of case – much as in Goodman’s riddle of induction. But what is that difference?

If Socrates had such a guardian angel y then it would be true in virtue of the nature of y that if Socrates exists then there is an angel who guards Socrates and hence true in virtue of the nature of y that a guardian angel exists if Socrates exists (□_y (Es → ∃y(A(y) ∧ G(y, s)))). So by Conservativity, it should follow that it is true in virtue of the nature of Socrates, angelhood and guardianship (and perhaps some other things upon which y depends) that a guardian angel exists if Socrates exists.

We have therefore achieved some kind of reduction of the problem; a putative essence concerning Socrates’ guardian angel has been replaced by a putative essence concerning Socrates, angelhood and guardianship. But a problem remains. For how come it is true in virtue of the nature of Socrates, sethood and membership, say, that a set exists if Socrates exists yet not true in virtue of the nature of Socrates, angelhood, and guardianship that a guardian angel exists if Socrates exist?

I am inclined at this point to appeal to the generative character of these properties and relations.²³ The nature of sethood and membership wholly consists in their generative role whilst, under anything like the usual understanding of angelhood and guardianship, their nature does not wholly consist, or even partly consist, in their generative role. My solution is not so very different from Rosen’s. The critical difference for him turns on the fact that ‘the existence of the singleton should be nothing over and above the existence of its member’ (§11). But I would like to explain why this is so in terms of the generative character of the properties and relations by which the singleton is defined. It is because of the insubstantiality of these properties and relations that the existence of the object is not a further substantial fact that makes any real difference to the world.²⁴

This is a partial solution to the problem Rosen has posed. But clearly much more needs to be done both in further elaborating on this solution and in considering the other kinds of case that can arise. And, ideally, we should have a general account of definition that covers both real definition, as in the present case, but also the more familiar case of nominal definition that have been so extensively investigated within formal logic.²⁵

Competing Interests

The author has no competing interests to declare.