A First Solution

Before this additional requirement can be formulated, we have to introduce a few auxiliary concepts. The first of these is the concept of permutation, as applied to individuative spec- ifications. In §7.2.2, I have said that a permutation of an attribute is, roughly, the result of permuting the attribute’s argument-places.33 _{Now, if σ and σ}0_{are individuative specifications,} let us say that σ0 _{is a permutation of σ just in case there are attributes A and A}0 _{such that} the predicate derived from σ denotes A, the predicate derived from σ0 _{denotes A}0_{, and A}0 _{is a} permutation of A. For example, σ∈ _{and σ}3 _{are in this sense permutations of each other.}

Second, we will need the concept of differentiation. If x and y are two entities and O some individuational ontology, then O will be said to differentiate the pair (x, y) just in case x and y have distinct abstract O-essences. For example, O3 differentiates the pair (Socrates, {Socrates}), but O∈ doesn’t: whereas Socrates and his singleton have distinct ab- stract O3-essences, they both have – at least if cardinality-related difficulties are set aside – one and the same abstract O∈-essence, because they both have one and the same O∈-individuation graph, as we have seen above. (If the mentioned cardinality-related difficulties are not set aside, one will have to say that Socrates and his singleton do not have any O∈-essences at all, because they will then not have any O∈-individuation graphs. So in this case, too, O∈ will fail to differentiate the pair (Socrates, {Socrates}).)

Let now σ and σ0 _{again be two individuative specifications, and let O and O}0 _{be the}

individuational ontologies whose only statements are, respectively, σ and σ0_{. I will then say} that σ trumps σ0 _{just in case}

(i) σ is a permutation of σ0_,

(ii) O differentiates every pair of entities that is differentiated by O0_{, and} (iii) O0 _{does not differentiate every pair of entities that is differentiated by O.}

By extension, I will say that a given individuative specification is trumped just in case there exists some specification that trumps it. With this concept in hand, the requirement that I here wish to add to the definition of ‘systematically optimal’ can be formulated as follows: If an ontology O is to be systematically optimal, then none of its individuative specifications should be trumped.

As we will see shortly, the definition of ‘trumps’ that has just been proposed still requires some revision; but before we come to that, more has to be said about the motivation for condition (i). To see the need for this condition, consider once more Fine’s asymmetry, and in particular its first half, i.e., the thesis that it is essential to {Socrates} to have Socrates as a member. If the present account of essentiality is to accommodate this thesis, it has to turn out that, for some systematically optimal ontology O, it is O-essential to {Socrates} to have Socrates as a member. And a very straightforward way in which this could happen would be for O to contain the individuative specification σ3_{. (We will see this in detail in} §8.1 below.) But now, if the above definition of ‘trumps’ did not include that first condition, then it would be possible to conceive of scenarios in which σ3 _{would have to be regarded as} trumped, while the intuitive appeal of the thesis that it is essential to {Socrates} to have Socrates as a member would seem to persist even if we took the respective scenario to be actual. Given that σ3 _{would have to be regarded as trumped, it would on the present account} of essentiality no longer be clear that, in such a scenario, the first half of Fine’s asymmetry still comes out true.

To make this point more vivid, it might help to imagine a concrete (albeit unrealistic) scenario. Suppose that two demons, Magos and Moros, set out to contemplate the richness

of both the physical and the set-theoretic universe. For every pair of entities x and y, Magos contemplates x to be a member of y if and only if x is in fact a member of y. Moros, on the other hand, is more interested in his fellow demon, and contemplates Magos to have the empty set as a member. By contrast, nothing – not even Moros himself – contemplates Moros to have anything as a member. And finally, let us suppose that Magos is the only entity by which any set is contemplated to have any members. Consider now the ontology O∗ _that contains only the following individuative specification:

(∗) Properties of the form ‘λx Cmplhx, α, βi’ are individuating,

where ‘Cmpl’ denotes the ternary relation whose instantiation by entities x, y, z (in this order) is the state of affairs that x is contemplated by y to have z as a member. By what has just been said, Moros and Magos have distinct abstract O∗_{-essences: e.g., it is O}∗_{-essential to} Moros, but not Magos, that nothing contemplates him to have anything as a member.

In addition, any two things (and in particular, any two sets) that have distinct abstract O3_{-essences also have distinct abstract O}∗_-essences.34 _{But of course, Moros and Magos, both} being memberless entities, do not have distinct abstract O3_{-essences. So here is a pair of} entities that is differentiated by O∗ _{but not by O}3_{. Consequently, if the above definition of} ‘trumps’ did not include condition (i), then σ3 _{would have to be regarded as trumped by} (∗). Yet we would still want to say that it is in this scenario essential to {Socrates} to have Socrates as a member. The present account of essentiality can accommodate this intuition only because the definition of ‘trumps’ contains that first condition.35

Let us briefly consider how the requirement that I have just now proposed to include in the definition of ‘systematically optimal’ – viz., the requirement that systematically optimal

34_{Recall from §4.5 that O}3 _{is the individuational ontology that has σ}3 _{as its only statement.}

35_{The scenario just described might be thought to raise problems for the present account even with condition}

(i) included. For intuitively, it seems obvious that nothing is essentially such that some entity y contemplates it to have some entity z as a member. In order for our account to accommodate this intuition, it has to turn out that no ontology containing the specification (∗) is systematically optimal. But it may certainly be that (∗) pays its dues and is untrumped, and for all we can tell it might also be parsimonious. In order to avoid the unwelcome consequence that it is essential to a set to be contemplated by Magos to have such-and-such members, we thus have to add yet another requirement to our definition of ‘systematically optimal’. I will return to this problem in §7.7.

ontologies should not contain any trumped specifications – helps with accommodating the second half of Fine’s asymmetry. I will defer the details till §8.2, but the basic idea is quite simple. Suppose that O∈ _{is the individuational ontology that has σ}∈ _{as its only statement.} It then turns out that O3 _{differentiates every pair of entities that is differentiated by O}∈_, whereas O∈ _{does not differentiate every pair of entities that is differentiated by O}3_{. By the} requirement in question, it follows that σ∈ _{is trumped by σ}3_{, and can hence not be part of a} systematically optimal ontology. Similar considerations hold, mutatis mutandis, for relevant variants of σ∈_{, so that no systematically optimal ontology O will be such that it is O-essential} to Socrates to be a member of {Socrates}. All this will be considered in greater detail in §8.2.