A Problem Posed by Non-Well-Founded Sets

On the one hand, let O3 be the simplest true individuational ontology that contains no individuative specifications except for σ3_{, according to which properties of the form ‘λx (α ∈} x)’ are individuating.29 _{In order to be true, O}

3 will have to contain at least one ontological restriction. In particular, it will have to contain some version of an Axiom of Extensionality

28_{The expression ‘semantically equivalent’ should here be understood in the sense that two individuative}

specifications σ and σ0 _{are semantically equivalent if and only if the two individuational ontologies that}

(respectively) have σ and σ0 _{as their only statements will entail exactly the same states of affairs.} 29_{Note that O}

3is not identical with O3(as described on p. 75): the latter contains σ3as its only statement,

that is compatible with the existence of multiple memberless entities (viz., the empty set and the urelements). If there are no non-well-founded sets, it will also have to contain some version of a Foundation Axiom. And if the number of urelements does not exceed every set- sized cardinality, it will have to contain some restriction to the effect that there are no more than so-and-so many memberless entities.30

On the other hand, let O∈ be the simplest true ontology that contains the following as its only individuative specification:

Properties of the form ‘λx (x ∈ α)’ are individuating.

Following a convention introduced in §4.4, we can refer to this specification as ‘σ∈_{’. While} the aforementioned ontology O3 has to include some ontological restrictions to keep it from falsehood, the need for such restrictions is even greater in the case of O∈. For, to describe the effect that σ∈ _{has on the existence claims of this latter ontology, one could say that it stands} the set-theoretic universe on its head. Where O3 claims that there are memberless entities, O∈ would claim, if it were not for its ontological restrictions, that there are entities that fail to be members of anything. Further, where O3 claims that there are singletons, O∈ would claim that some things are members of precisely one entity each; and so on. However, at least on standard assumptions about the abundance of sets, there are no such things as O∈ would claim there to be. For it is standardly assumed that, for any entity x and set a, there is a set a ∪ {x}, which has x as a member; and if so, everything must be a member of as many sets as there are sets. So O∈ has to contain ontological restrictions that prevent it from making the sorts of existence claim just mentioned.

In this connection, it is instructive to consider what an O∈-individuation graph of (say) Socrates would look like. Socrates, like every other entity, is a starting-point of infinitely long ascending ∈-chains: for instance, he is a member of his singleton, which is a member of its singleton, and so on ad infinitum. In fact, he is a member of as many such ∈-chains as

30_{To be sure, this ontology does not provide the resources to distinguish, among the many memberless}

entities, the one entity that is the empty set. However, I do not think that such a set is really needed, for the role that it plays in classical set theory can equally well be played by any memberless entity.

there are sets. For this reason, if there existed an O∈-individuation graph of Socrates, that graph would have to contain more nodes and edges than any set-sized cardinality. But by definition, every graph consists of a set of nodes and a set of edges. Consequently, there can be no O∈-individuation graph of Socrates. Since these considerations can be applied to any entity whatsoever, it follows that no entity has an O∈-individuation graph. But if no entity has an O∈-individuation graph, then no O∈-graph has an O∈-decoration. If O∈ is to be true, it will thus have to contain an ontological restriction that prevents it from making any existence claims at all.

We might suppose, however, that the cardinality-related difficulties described in the previ- ous paragraph can be circumvented somehow, e.g., by relaxing our conception of graph (and relatedly, our conceptions of formula, state of affairs, and attribute). In that case, Socrates will have an O∈-individuation graph, after all: Figure 7.1 gives a rough impression of what that graph would look like. Since not only Socrates, but any entity whatsoever is a starting- point of as many ascending ∈-chains as there are sets, this graph will be the O∈-individuation graph not only of Socrates, but of every other entity as well, except for those non-well-founded set that, directly or indirectly, have themselves as members. (The reason for this is that the O∈-individuation graph of such a set would have to contain at least one edge that leads back to the graph’s point.)

To see what all this has to do with Fine’s asymmetry, suppose that O∈ is systematically optimal, and that Socrates has an O∈-individuation graph like the one partially depicted in Figure 7.1. Let us also assume that there exists an O∈-decoration of this graph that assigns Socrates to the graph’s point (depicted at the bottom of Figure 7.1), and that Socrates has a fully concretized O∈-essence. The mentioned decoration will then assign Socrates’ singleton to the node that, in Figure 7.1, is depicted directly above the point. Accordingly, Socrates’ instantiation of his fully concretized O∈-essence will necessitate the state of affairs that he is a member of his singleton. Hence, if O∈ were systematically optimal, then it would on the present account be essential to Socrates to be a member of {Socrates}, and this contradicts the second half of Fine’s asymmetry. If this outcome is to be avoided, O∈ must therefore fail

. .. .

.

. _...

Figure 7.1: Extremely simplified sketch of a hypothetical O∈-individuation graph of an ure- lement or well-founded set. The chevron indicates the graph’s point. For simplicity, nodes to which an O∈-decoration would have to assign non-well-founded sets, or sets with more than three members (among many others), are not depicted.

to be systematically optimal. And on the supposition that our definition of ‘systematically optimal’ is by now complete, this means that at least one of the following must be the case: O∈ is not a true individuational ontology, or its single individuative specification σ∈ fails to pay its dues, or σ∈ _{is not parsimonious.}

Now, by hypothesis, O∈ is a true individuational ontology. Further, there does not seem to be any compelling reason to think that σ∈ _{is not parsimonious. For the predicate derived} from σ∈_{, i.e., ‘λx, α (x ∈ α)’, simply denotes the relation of set-membership, and it certainly} seems possible that this relation might be basic.31 _{And if it is basic, then, by what was said} in the previous section (in particular, on p. 144), σ∈ _{will not fail to be parsimonious. Yet} even if set-membership were not basic, we could not very well rely on this fact in order to save the present account’s ability to accommodate the second half of Fine’s asymmetry. For the intuitive appeal of Fine’s asymmetry does certainly not seem to stand or fall with the question of whether set-membership is basic. So now the only remaining hope for the account in its present form is that σ∈ _{might turn out not to pay its dues.}

The question of whether σ∈ _{pays its dues depends on how many non-well-founded sets have,} either directly or indirectly, themselves as members. If there are none, then there will also be

31_{However, see Lewis (1991) for an analysis of set-membership in terms of parthood and singleton-}

no systematically optimal ontology with respect to which σ∈ _{pays its dues. For as we have} seen above, if there are no non-well-founded sets that (directly or indirectly) have themselves as members, then everything will have the same abstract O∈-essence, which we may call ‘E∈’. So, for every ontology O, and for every O-essence E, there will be exactly one instantiated property that is a conjunction of E and some O∈-essence, namely, λx (Ehxi ∧ E∈hxi). For any ontology that contains σ∈_{, the contribution that σ}∈ _{makes to that ontology’s discriminatory} power will therefore be exactly zero. And so, by any reasonable standard of what it takes for a specification to pay its dues relative to a given ontology, there will be no ontology (let alone a systematically optimal one) relative to which σ∈ _{pays its dues.}

Suppose, however, that there are some non-well-founded sets that (directly or indirectly) have themselves as members. If we accept that there are some, we will presumably also have to accept that there are infinitely many, barring arbitrary restrictions. So we would have to accept that there are infinitely many instantiated abstract O∈-essences. This in turn means that σ∈ _{will make a rather large contribution to O}

∈’s discriminatory power; and so the specification will very plausibly count as paying its dues.32 _{But if σ}∈ _{is parsimonious and pays} its dues, then O∈, being a true individuational ontology, will according to our definition, as so far developed, be systematically optimal; and, on the present account of essentiality, this will then mean that Socrates is essentially a member of {Socrates}.

One might want to avoid this outcome by insisting that there are no non-well-founded sets, and a fortiori none that (directly or indirectly) have themselves as members. But this would clearly not do. For the intuition that Socrates is not essentially a member of his singleton does not in the least seem to depend on the question of whether there are non-well-founded sets of any particular sort. Similarly, we should not try to avert the mentioned outcome by appealing to the fact that, due to the mentioned limitations of cardinality, Socrates does not have any O∈-individuation graphs or O∈-essences in the first place, because those limitations

32_{Recall that any true individuational ontology that contains no individuative specifications at all is thereby}

systematically optimal. So, if σ∈_{pays its dues relative to such an ontology (which it plausibly does if it makes}

an infinitely large contribution to O∈’s discriminatory power), then it pays its dues relative to a systematically

are intuitively no less irrelevant to the issue of essentiality. This becomes clear if one switches the example to one that involves singleton-membership instead of set-memberhip: In this case, the analogous individuation graph of Socrates would need to contain only countably many nodes and edges, but nothing would change as far as the relevant intuitions of essentiality are concerned. The upshot is that the definition of ‘systematically optimal’, as it has so far been developed, is incomplete, and that at least one further requirement is needed.