7.4 Paying One’s Dues
7.4.2 Paying One’s Dues, Relativized and Simpliciter
Suppose that O is an individuational ontology and σ some individuative specification (not necessarily contained in O), and let O+ be the simplest true ontology that results from O by adding to it σ as well as whatever ontological restrictions are needed in order to render the resulting ontology true. I will then say that σ pays its dues relative to O just in case the contribution that σ makes to the complexity of O+ is ‘outweighed’ by its contribution to the discriminatory power of O+.
This definition naturally raises the question of how the notion of outweighing should be spelled out. Suppose we write ‘Dσ,O’ to denote σ’s contribution to O’s discriminatory power, and ‘Cσ,O’ to denote σ’s contribution to O’s complexity. On the most straightforward (i.e., linear) approach, there will then be two constants c0 and c1, and Dσ,O will be said to outweigh Cσ,O just in case Dσ,O exceeds the ratio Cσ,O−c
0
c1 . This proposal may not be adequate in cases where Dσ,O is infinite, since we might not want to allow that an infinite Dσ,O will outweigh every finite increase of the respective ontology’s complexity. But even if we suppose that the linear approach is adequate, we have to face the question of how the parameters c0 and c1 are to be determined. There does unfortunately not seem to be any clear, non-arbitrary way of choosing a value for them. Consequently, the parameters could be set to whatever values are deemed appropriate, which introduces a certain subjective element into our definition of
‘systematically optimal’, and thus also into the present account of essentiality. Although the presence of this subjective element is not exactly welcome, I think that it can be tolerated, and also that it should be tolerated, at least by adherents of Lewis’s account of lawhood. For if one can accept a subjective element in one’s account of lawhood, it seems that one should also be able to live with such an element in one’s account of essence and essentiality.24
Finally, we have to specify what it means for a given specification σ to pay its dues simpliciter. This additional step is necessary because we cannot simply say that, in order for an ontology to be systematically optimal, all its individuative specifications have to pay their dues relative to that particular ontology. For in that case, there would be a danger of violating the constraint (C) from §5.4, according to which
[f]or every set S of optimal ontologies, there exists an optimal ontology O that contains every individuative specification that is contained in any given member of S. (p. 96)
If we now said that an ontology O is systematically optimal only if all its individuative specifications pay their dues relative to O, this would give rise to the following difficulty: There might be pairs of individuative specifications σ1 and σ2 such that there exists a systematically optimal ontology that contains σ1, and also a systematically optimal ontology that contains
24In his (1994), Lewis worries about the subjective element in his account of lawhood:
The worst problem about the best-system analysis is that when we ask where the standards for simplicity and strength and balance come from, the answer may seem that they come from us. (p. 479)
His way of alleviating this worry is to take resort to the “reasonable hope” that “nature is kind” and “the best system [is] robustly best – so far ahead of its rivals that it [comes] out first under any standards of simplicity and strength and balance” (ibid.). But not everyone is equally worried. For instance, Sider (2011), in commenting on Lewis’s account, seems quite comfortable with the idea that the question of “how much complexity can be tolerated to gain a given amount of strength” admits of different answers, which “correspond to different notions of law” (p. 22):
So let a “middling” assignment of cost to complexity be one that counts the generalizations of physics as laws, but only barely. This corresponds to a sense of ‘law’ in which there are laws of physics, but in which certain special-science generalizations do not count as laws. [Footnote:] I have in mind special-science generalizations that are physically contingent—perhaps because they depend on certain physically contingent “initial conditions”. [End of footnote.] If complexity is instead made cheaper, then those special-science generalizations no longer count as laws; all that remain are laws of metaphysics and logic. (ibid.)
It seems to me that one could reasonably come to accept a similar sort of scheme in a theory of essences. Another option would be to allow that essentiality ‘comes in degrees’. In the following, however, I shall not pursue these ideas any further.
σ2, but no systematically optimal ontology that contains both σ1 and σ2. For it might turn out that, if σ1 is added to an ontology that already contains σ2, then σ1’s contribution to the resulting ontology’s discriminatory power will be simply zero. If, then, on the account currently under consideration, there exists no systematically optimal ontology that contains both σ1 and σ2, (C) would clearly be violated.
To avoid this outcome, I propose the following definition for the concept of a specification’s paying its dues:
(PD) An individuative specification σ pays its dues if and only if there exists some systemat- ically optimal ontology such that σ pays its dues relative to that ontology.
Given that this definition makes use of the concept of systematic optimality, and given that the latter will in turn be defined on the basis of the concept just now introduced (viz., that of a specification’s paying its dues), one might think that (PD) is circular. In fact, however, it should be understood as being merely recursive.
This is possible because the concept of a systematically optimal ontology will here be defined in such a way that an individuational ontology counts as systematically optimal if and only if (i) it is true and (ii) all of its individuative specifications fulfill certain conditions. Hence, an individuational ontology that does not contain any individuative specifications at all will be systematically optimal as long as it is true. And in order to render a specificationless ontology true, all that is needed is, at most, an ontological restriction to the effect that there are no more than a certain number of entities. For if an individuational ontology contains no individuative specifications, then any existence claim that it generates will only be to the effect that, for some set-sized cardinality κ, there are at least κ-many entities. Thus, if it should happen that there is no limit to the number of entities, no ontological restriction will be needed.
By what has just been said, there exists a systematically optimal ontology, call it ‘O0’, that contains no individuative specifications. Using this ontology as a starting-point, one can construct a further systematically optimal ontology as follows. Suppose that there is
some individuative specification σ that pays its dues relative to O0. By (PD), σ will then count as paying its dues simpliciter, and thereby satisfy one of the four conditions that have to be met by the individuative specifications of any systematically optimal ontology. If σ also satisfies the three other conditions (to be formulated in the next three sections), then a true individuational ontology that contains σ as its only individuative specification will be systematically optimal. Similarly, one could now ask what individuative specifications pay their dues relative to this new ontology. In this way, one can recursively construct ever larger systematically optimal ontologies. (Of course, the process will come to an end once no further specifications can be found that satisfy the mentioned conditions.)
It might at this point be asked why (PD) has to make reference to systematically optimal ontologies in the first place. Would it not have been enough simply to speak of individuational ontologies, or perhaps of ones that are both true and individuational? The definition would then have been much more straightforward. However, it would not have been enough to put ‘individuational’ or ‘true individuational’ where (PD) has ‘systematically optimal’, because it would then have been all-too easy for an individuative specification to count as paying its dues. For example, consider:
(1) Properties of the form ‘λx whitehxi’ are individuating.
As long as there are sufficiently many white things as well as non-white things, one can easily construct an ontology relative to which (1) seems to pay its dues by practically any reasonable standard. Thus, let N be some large number that is smaller than the total number of white things and also smaller than the total number of non-white things. Further, let x1, . . . , xN be pairwise distinct white things, and let y1, . . . , yN be pairwise distinct non-white things. For each i = 1, . . . , N, let now σi be the individuative specification
Properties of the form ‘λx (x = X ∨ x = Y )’ are individuating,
where ‘X’ is replaced by a name of xi and ‘Y ’ by a name of yi. Finally, let O be the ontology that consists of (1) together with σ1, . . . , σN, and let O0 be the ontology that consists only of σ1, . . . , σN. Then it can be seen that the number of abstract O0-essences that O0
correctly claims to be instantiated is N + 1, and for every such O0-essence E, there are two relevant properties that O correctly claims to be instantiated, viz., λx (Ehxi ∧ whitehxi) and λx (Ehxi ∧ ¬whitehxi). So the contribution that (1) makes to the discriminatory power of O will be exactly N + 1.
What about the contribution that (1) makes to the complexity of O? Let us say that P1, . . . , PN are the properties that are denoted by the predicates derived from the various σi. Then, in order not be false, already O0 has to contain an ontological restriction to the effect that none of the Pi are co-instantiated. But the addition of (1) does not require any further ontological restrictions. So the contribution that (1) makes to the complexity of O will be quite modest, and, at least if N is large enough, it should seem reasonable to suppose that the increase in complexity will be outweighed by the contribution that (1) makes to O’s discriminatory capacity. If so, then (1) will have to count as paying its dues relative to O, but it is very questionable that it should also count as paying its dues simpliciter.
Admittedly, (PD) is not an easy definition to work with. In particular, it makes it difficult to show that a given specification fails to pay its dues; for to show this, one would (at least on a brute-force approach) have to consider all systematically optimal ontologies. By contrast, the definition is much more user-friendly – at least in some cases – when it comes to showing that a specification σ does pay its dues. For we have already seen that there is a systematically optimal ontology that contains no individuative specifications whatsoever, and so we can show that σ pays its dues if we can show that it pays its dues relative to such an ontology. To do the latter, we have to establish that σ’s contribution to the complexity of Oσ is outweighed by its contribution to Oσ’s discriminatory power, where Oσ is the simplest true ontology that has σ as its only individuative specification. Given that this ontology contains no other specifications, σ’s contribution to Oσ’s discriminatory power is simply the number of instantiated abstract Oσ-essences (or the number of instantiated abstract Oσ-essences minus 1, if that number is finite).
The requirement that the individuative specifications of a systematically optimal ontology should pay their dues is needed if the present account of essentiality is not to become overly
liberal. However, it is not the only, or even the most important, condition that we have to impose. In order for the account to accommodate Fine’s asymmetry, we still have to add three further requirements.