According to the present account of essentiality, as stated in §7.1, a property is essential to a given entity just in case, for some systematically optimal ontology O, the property is O-essential to that entity. This proposal bears an obvious disanalogy to Lewis’s account of lawhood: according to him, a contingent generalization counts as a law only if it is a theorem or axiom in every deductive system that is maximally elegant (in the sense suggested on p. 112) among those that are true and formulated in a suitably austere vocabulary. To mark
this difference between the two approaches, one might say that Lewis’s account of lawhood universalizes, whereas the present account of essentiality particularizes. I submit that, at least initially, a particularizing approach seems more natural, both in an account of essentiality and in an account of lawhood. For why should a true generalization not count as a law if it is a theorem or axiom in only some maximally elegant system? And why should a property not count as an essence if it is an O-essence only for some systematically optimal ontology O?
As far as the Lewisian account of lawhood is concerned, a possible answer might be that, under a particularizing version of that account, there might be two maximally elegant systems S1 and S2 such that (i) there exists a generalization L1 that is a theorem in S1 but not in S2, (ii) there exists another generalization L2 that is a theorem in S2 but not in S1, and (iii) there is no maximally elegant system S3 that contains the conjunction of L1 and L2 as an axiom or theorem. Then, though both L1 and L2 would count as laws, their conjunction wouldn’t; and this might be an unwelcome consequence. To be sure, this defect (if it is a defect) could be easily repaired by saying that a law is any sentence that appears in the deductive closure of any union of maximally elegant systems, but the added complication would render this option somewhat unattractive.
Another possible answer might be that Lewis’s criterion can be naturally adapted to deal with cases in which there is no maximally elegant system (even if we only count those that are true and formulated in an austere vocabulary). Such a case would obtain if, for every true system that is formulated in an austere vocabulary, there is another one that is also true and formulated in an austere vocabulary, but more elegant than the former. This poses a problem, because it would certainly be unfortunate if, in such a case, one’s account of lawhood had the consequence that there are no laws. Analogously for the present account of essentiality: if, for every maximally elegant ontology (among those that are true and whose atomic predicates all denote basic attributes), there is another one that is more elegant, then our account should certainly not automatically have the consequence that nothing has any essential properties.
Lewis can deal with his version of this problem by a slight modification of his criterion of lawhood. In particular, he can say that “a law must appear as a theorem in all sufficiently
good systems”.17 This looks prima facie preferable to the (particularizing) alternative account on which a law must appear as a theorem in some “sufficiently good system”. For this latter account would imply that a generalization G would count as a law if it appears as a theorem in some sufficiently elegant system S, even if it does not appear as a theorem in any system that is more elegant than S. In such a case, it would seem natural to say that G should not count as a law after all, and Lewis’s formulation accommodates this intuition.
Arguably, however, his formulation can still be improved. For suppose that there is a system S that does not contain a certain generalization G, and which barely manages to be sufficiently elegant, whereas every system that is more elegant than S does contain G. Then, on Lewis’s proposal, G will not count as a law, but this is counter-intuitive, for G would have counted as a law if the required level of elegance had been set only slightly higher. We might try to remedy this defect by saying that a generalization counts as a law if there exists some level of elegance λ such that G appears as a theorem in every system whose elegance exceeds λ. But now suppose that, no matter how high we set λ, there is always some system whose elegance exceeds λ but which does not contain G.18 Should this prevent G from counting as a law? Presumably not if, for every level λ, there is also a system that does contain G and whose elegance likewise exceeds λ. However, if there is a level λ such that (i) there exists at least one system whose elegance exceeds λ and (ii) no such system contains G, then this may plausibly count as a good reason to say that G is not a law. Here we seem to have hit on a passable necessary condition on lawhood. Can this condition also be regarded as sufficient? If so, we can say:
17Op. cit., p. 73n. Lewis prefaces this proposal by expressing doubt that “our standards of simplicity would
permit an infinite ascent of better and better systems”. However, the threat of such an infinite ascent need not exclusively arise from the possibility that we might have some perverse standards of simplicity; it may also come from the universe itself. For it is at least conceivable that (i) no deductive system is large enough to provide a full description of the entire universe, (ii) for every given system that describes a certain part or aspect of the universe, there is another part or aspect that can be described even more elegantly, and (iii) by describing this other part or aspect as well, a system can achieve greater elegance than the former. This admittedly looks like a remote possibility, but arguably it ought still to count as a deficiency if an account of lawhood (or of essence) should be unable to cope with it.
(L?) A generalization G is a law just in case, for every level of elegance λ: if there exists at least one system whose elegance exceeds λ, then there will also be at least one such system that contains G.
This may at first blush look reasonable enough, but we should still be skeptical. For it seems at least conceivable that, whenever we have a level λ and a system S whose elegance exceeds λ, we could stick any true generalization G into S and go on to improve the resulting system – e.g., by making it more informative – so that the end-product will again be a system whose elegance exceeds λ, but which also contains G. If so, then any generalization whatsoever will under (L?) count as a law.
To prevent this kind of mischief, it is no use theorizing at the level of whole systems; instead, one will have to look at the individual statements. In the light of these considerations, our initial definition (SO0) (p. 114) appears relatively hopeless, given that it, like Lewis’s account of lawhood, relies on the maximization of elegance. I shall therefore now pursue the alternative route of developing a number of conditions that have to be satisfied by the individual statements of an ontology in order for the latter to count as systematically optimal. From the way in which the concept of systematic optimality will be defined in the follow- ing, it will become clear that the account of essentiality that is based on it has to be of the particularizing sort. For the definition will have the consequence that there exists a systemati- cally optimal ontology O0 that contains no individuative specifications whatsoever. Since this ontology contains no individuative specifications, every entity has the same O0-individuation graph, which consists of only a single node and no edges. Correspondingly, for any entity x, the fully concretized O0-essence of x will be simply the property of being identical with x. So the O0-essential entities of x will be limited to the properties whose instantiation by x is necessitated by x’s being self-identical. If, by contrast, the present account of essentiality were of the universalizing sort, so that a property P would be regarded as essential to x just in case it is O-essential to x for every systematically optimal ontology O, then x would on this account have no other essential properties than those that are O0-essential to it; and this would be a very unwelcome consequence. The upshot of this is that we should stick with the
particularizing account stated in §7.1.