Consider again the somewhat fantastic scenario described in §7.6.2 (p. 152), in which two demons contemplate the richness of the set-theoretic and physical universe. In this connection, we have discussed the following individuative specification:
(1) Properties of the form ‘λx Cmplhx, α, βi’ are individuating,
where the constant ‘Cmpl’ denotes the relation that is instantiated by entities x, y, z just in case x is contemplated by y to have z as a member. Let now O1be a true ontology that contains (1) as its only individuative specification. In the mentioned scenario, (1) certainly pays its dues. Moreover, for all that has just been said, the specification may also be parsimonious, and we may in addition suppose that it is untrumped. In other words, O1 may well satisfy all the conditions that we have so far included in the definiens of ‘systematically optimal’. This further means that, if the definition of that phrase were by now complete, it would on the present account be essential to any given set to be contemplated by Magos to have such- and-such members. As unrealistic as the scenario may be, it still seems that this latter result should be avoided.
Fortunately, it is also fairly clear just why that consequence is so counter-intutive: It cannot be essential to a set to be contemplated by Magos to have such-and-such members, because the set already exists when Magos sets out to contemplate it! Moreover, in order to explain how Magos is able to contemplate such-and-such sets to have such-and-such members,
one would presumably have to appeal to some way of differentiating between sets. This might be done, e.g., by making use of their respective O3-essences. An adequate explanation as to why O1 is true will therefore not merely consist of O1 itself, but will have to go back to an ontology like O3. To a first approximation, we might thus say that the specification (1) fails to be ‘principial’ (or perhaps: explanatorily fundamental) in the following sense:
(Pr0) If σ is an individuative specification and Oσ a true ontology that contains σ as its only specification, then σ is principial just in case an adequate explanation of the fact that Oσ is true is provided by Oσ itself.
However, this is not yet satisfactory. Consider, e.g., the ontology O3: We do want to say that O3 is in some sense explanatorily fundamental; certainly, in order to explain why this ontology is true, it does not seem as if we should have to appeal to another ontology of sets. However, we do have to appeal to an at least rudimentary ontology of urelements, for O3 does after all make claims about the number of memberless entities, just as it makes claims about the number of singletons, etc. So in what sense, exactly, does O1 but not O3 fail to be explanatorily fundamental?
To answer this, I would suggest that we look once more at the discriminatory power of the ontologies in question. While O3 does not differentiate any pairs of urelements, it does differentiate pairs of sets, and so does O1. Hence we can say that the truth of O1 has to be explained by appeal to some ontology O (which in this case is O3) that is not only such that an adequate explanation of O’s own truth need not make any appeal to O1,45 but also such that at least one pair of entities that is differentiated by O1 is likewise differentiated by O. (By contrast, at least if set-membership is basic, the same will plausibly not be true of O3, as we will see in §8.1.1 below.) In this sense, I would suggest that O1 fails to be explanatorily fundamental. Accordingly, I propose to redefine the notion of principiality as follows:
45It might be thought that explanatory dependence can never be reciprocal, but this is not so clear. Consider,
e.g., a pair of true ontologies that have as their respective domains two species that have co-evolved (coyotes and jack rabbits, say). Then, in order adequately to explain the truth of either ontology, one may conceivably have to appeal to the respectively other member of the pair.
(Pr) If σ is an individuative specification and Oσ a true ontology that contains σ as its only specification, then σ is principial just in case no adequate explanation of the fact that Oσ is true relies on any ontology O that satisfies the following two conditions:
(i) No adequate explanation of the fact that O is true relies on Oσ; and
(ii) Some pair of entities that is differentiated by Oσ is also differentiated by O. This might be paraphrased by saying that a specification σ is principial just in case it allows us to differentiate a pair of entities only if that pair ‘had not been differentiated before’, which is to say, only if that pair is not differentiated by any ontology that is relied on by some adequate explanation as to why Oσ is true, but whose own truth cannot be adequately explained on the basis of Oσ. A noteworthy consequence of this definition is that, in cases where Oσ does not differentiate any pairs of entities at all, σ will invariably count as principial.46
It may also be worth noting that the references to differentiation that this definition contains cannot be replaced by talk of differentiationsc, on pain of losing any assurance that σ3 should be regarded as principial. The reason for this is that an ontology O of urelements might quite possibly, as in the case of O0, be such that every set is ultimately O-individuated only by itself. Such an ontology will thus manage to differentiatesc every single pair of sets. Hence, if it should happen that an explanation of the truth of O3 has to rely on that ontology, then σ3 will no longer count as principial under the definition in question.
Admittedly, (Pr) is still in need of clarification. In particular, more has to be said as to what is meant by an ‘adequate explanation’ of a given ontological fact. This is a difficult notion to explicate, but an intuitive grasp will here hopefully be sufficient.47 The final requirement
46It is partly for this reason that, in the present context, I prefer the term ‘principial’ over ‘explanatorily
fundamental’. In view of the consequence just mentioned, the latter term would have sounded like a gross misnomer, whereas the semantics of the rarely-used term ‘principial’ seems rather more malleable.
47If pressed to explicate the notion, I would start with the ideas sketched in §6.1 above. Of course, the
present explicandum is not quite the same, since we are here concerned not with explanations of ‘what there is’, but rather with explanations of more circumscribed ontological facts, viz., of facts that consist in the truth of a particular ontology. Thus, if F is such a to-be-explained fact, then a best explanation of F may, roughly, be taken to be a theory that (i) in some suitable sense entails F , and (ii) is a part of the simplest possible theory that is true, comprehensive, and formulated in an austere vocabulary in which all atomic predicates
to be included in our definition of ‘systematically optimal’ can then be stated as follows: If O is a systematically optimal ontology, then all of O’s individuative specifications should be principial.
It might be thought that, in the presence of this last requirement, both the parsimony and the no-trumping requirement of the previous two sections are superfluous. Thus, to begin with the no-trumping requirement, one might think that (e.g.) the fact that σ∈ is trumped by σ3 should in the first place be taken to mean that O
∈’s truth has to be explained on the basis of O3, given that the latter allows for finer distinctions between sets. And further, if O∈ does not differentiate any pairs of entities at all (which will be the case if there are no non-well-founded sets), then σ∈ will not pay its dues, though it will be principial. But if O
∈ does differentiate at least one pair of entities, then any such pair will also be differentiated by O3 (as we will see in §8.2), so that σ∈ will fail to be principial, given that – as it seems plausible to hold – an adequate explanation of O∈’s truth has to rely on O3. So, at least when it comes to excluding σ∈ from the class of systematically optimal ontologies, there does not seem to be any need for the no-trumping requirement.
A similar concern could be raised for the parsimony requirement. For example, consider the non-parsimonious specification discussed at the beginning of §7.5:
(2) Properties of the form ‘λx (P hxi ∧ Qhαi)’ are individuating, and compare it with the simpler specification
(3) Properties of the form ‘λx P hxi’ are individuating.
Suppose that O2 and O3 are true ontologies that contain (2) and (3), respectively, as their only specifications. Given that (3) results from a pruning of (2), O3 is correspondingly simpler
refer to basic attributes.
A theory may here be understood to be comprehensive (relative to F ) if and only if it entails all the facts, or at least all the purely qualitative ones, in addition to all those that involve any entity that is involved in F . (For the relevant meaning of ‘involves’, see the footnote on p. 105 above.) Since the theories in question are here not intended to reveal essences, this proposal does not lead to the sort of problem that we encountered at the end of §6.3. However, it does lead to a certain cardinality-related difficulty, for the class of facts just mentioned might conceivably be so large and complex that no single theory manages to entail all the facts in that class. I do not know how this difficulty might best be dealt with, but it is perhaps reasonable to hope that the possibility that gives rise to it does not obtain.
than O2, and it seems natural to say that some adequate explanations of the fact that O2 is true will rely on O3. Moreover, as long as there exists at least one entity that has P , one entity that lacks P , and also at least one entity that has Q, there will be a pair of entities that is differentiated by both O2 and O3. In such a case, then, the specification (2) will fail to be principial. So one might think that the parsimony requirement does not do any work that is not already done by the principiality requirement.
It would certainly be a welcome result if both the parsimony and the no-trumping re- quirement should in this way turn out to be superfluous: our definition of ‘systematically optimal’ could then be drastically simplified. However, as long as we have not yet settled on a precise explication of ‘adequate explanation’, our definition of ‘systematically optimal’ will benefit from the fact that it also includes the two other requirements, since these will help to delineate this latter concept more sharply.