A NOTE ON COVERING CONCEPTS, COMPARATIVES AND RELATIVE IDENTITY
What I want to do in this note is connect the semantics of one type construction with that of another. In particular I want to argue that there are certain formalities shared by two conceptions: relative identity and the logic of comparative concepts. First, I will outline a set of ideas associated with relative identity and then do the same for comaparative concepts and then link the two. I will then apply these results to the Bradley/James dispute over the nature of identity.
We sometimes employ identity concepts in a relativized way - something explored in detail by Peter Geach, Nick Griffin and others.
If we want to say that John is the same man as Bob, this is not the same thing as saying that Bob and John are identical. But if they are identical then it stands to reason that they are the same man. Yet from this it does not follow that if there is no respect in which two things differ then they are the same. This is entailed by the converse thesis, the identity of indiscernibles. I don't want to take up, at this point, the merits and demerits of the thesis of relative identity, that is, that it is sometimes the case, contra Leibniz (as Wiggins demonstrates), that a may be the same f as b, for some f, but not the same g as b, where a is g. More formally:
((Ef)(a=b)^f) & ((Eg)(g(a) & ~(a=b)^g))
Suppose we want to say: "a is the same f as b but not the same g, although a or (inclusive) b is g." Here is how we do it - and here I have in my sights the first half or so of David Wiggin's _Sameness and Substance_. Harvard 1980.
(((a=b)^f & ~(a=b)^g) & (g(a) v g(b))
There are other possibilities, all examined by Wiggins in his criticism of relative identity. And there are different characterizations of relative identity. Now let's say something about the logic of comparative concepts. This second step in our procedure breaks down into two: first, I hope to uncover a distinction between grammatical positives, which is a form of comparative and ordinary noncomparative adjectives. Then I will show that there are, just as in the formulation of relative identity, "covering concepts" essential to the formulation of certain logical descriptions of comparative concepts. This will involve noting two things. First, the relation between grammatical positives and comparatives taken in the comparative degree and second certain inferential rules that appear to obtain in the semantical treatment of the grammar of comparatives taken in the positive degree. For a understandin! g of the grammar of comparatives a review of the subject in any Latin grammar, for example, is recommended. First, a look at the difference between grammatical positives and non-comparative adjectives is in order.
The distinction I am going to make was the outcome of considering what I took to be certain problems in Russell and Reichenbach's treatment of comparative notions. Russell had discussed the matter in his Principles of Mathematics and Reichenbach included a discussion of comparatives in his book on Symbolic Logic. In discussion with Ivan Sag (circa 1977) it came out that we had independently arrived at a similar conception. I don't know if Sag has published on this;and I have moved in different directions, owing largely to my rejection of compositional approaches to semantics. Take the adjectives 'square' and 'bright'.
One difference is that to be bright a certain standard is met which if it were not we would use 'dim' instead, say. In other words the thing described must meet a certain standard in order to be called bright; there must be *more* of something to be distinguished from 'not bright' or 'dim', presumably reflected light. But in the case of 'square' such is not the case. Notice that there is such a construction as 'brighter' (comparative adjective taken in the comparative degree), but in order to make a comparative out of 'square' we need something like 'more square', but the 'more' here does involve 'more of the same' as 'more' in 'more bright' involves 'more of the same', i. e. light. 'Green' can go either way; that is, it can be regarded as a comparative or as a simple adjective like 'square'. But can 'bright' ever be used as a "simple adjective"? My claim is: "Yes, when it is used as a "covering concept" in describing the semantics of the comparative in much the same way as we describe the semantics of relative identity using "covering concepts" such as "f" and "g" as in the above. Let's take a closer look at covering concepts in the case of comparative relations.
If one thing is brighter than another then the brightness of the one is greater than the brightness of the other.
If we follow, more or less, Reichenbach we can elaborate this way (A):
bright(a)=x & bright(b)=y & x>y.
That is: a is bright to such a degree (x) and y is bright to such a degree (y) and x is greater than y. What does not follow is (B)
Is this always the case? Suppose we distinguish the 'bright' of (A) and that of (B) and regard the later as noncomparative on the assumption thatC: If 'G' is a comparative in the positive degree then it must possess 'G' in its noncomparative sense, but the converse does not hold.
In this case we have room for a distinction between covering concepts that did not at first appear to exist. We have 'G', call it 'G*', which is comparative (in the positive degree) and then we have 'G' which is a simple adjective, that is, non comparative. How is this distinction useful? I won't go into all the details on this ocassion; however, the distinction affords us a way of linking comparatives and relative identity (keep in mind that '>' and '<' are related by negation to '=', and '>' and '<' are themselves comparatives.)
One can argue that if ~g(b) then'~(a=b)^g'; so Abe may not be the same boy as Gabe as Gabe is not a boy (let's momentarily set aside Wiggins on 'phase sortals'). This circumstance is much like that where it can be said that '~(a>b>^g' where 'g(a)' and '~g(b)' - compare: 'Abe is bright & is brighter than Gabe and Gabe is not bright'. However, if 'bright' does indeed, as we have said, possess two forms (comparative and noncomparative) it may in fact be possible that a should be brighter than b, although neither a nor b is bright. They would each be 'bright' to some degree, but not 'bright*'. In this case neither would be bright in the comparative sense of 'bright'. We do need this noncomparative sense if we are to say of two non-bright things that one is brighter than the other. We might eventually want to distinguish 'brighter' from 'brighter*' as we have 'bright' and 'bright*'. Are there covering c! oncepts which are such that in order that '(a>b)^g' that 'g(a)' must be true? And if so does the ambiguity we have pointed towards persist? If I am right there are some concepts that are never used as adjectives in the comparative degree (that is, as grammatical positives). If 'a is fuzzier than b' then a must be fuzzy; but if 'a is brighter than b' that does not entail that 'a is bright'. Here to be fuzzy to any degree requires being fuzzy, whereas a dim star may be bright to some degree - even when we say of two dim stars that one is brighter than the other. There is a superlative 'fuzziest' but no distinction appears, obvious at least, between comparative and non-comparative 'fuzzy', or at least one we would make in the same way. Keep in mind that there is a distinction between comparative and noncomparative relations as well as simple adjectives on the view I take here. I will defer consideration of Bradley/James until the next posting, but I will make this observation: resemblance can be viewed comaparatively or non comparatively. Bradley, I will argue was using 'resembles' in a noncomparative sense; James in a comparative sense. Bradley was also closer to viewing resemblance as requiring covering concepts; james was not. Therein lies the basis of resolving their conflict. I may be transit over the next couple of days; not sure. But there may be a delay in approving the postings, sorry.