Inconsistent Thoughts

I have recently been expounding on an approach to understanding the thesis of Composition as Identity (CID) that is inspired by the work of Donald Baxter. I came across this paper by Lowell Friesen that challenges Baxter’s view and I want to reply to one of his criticisms which also applies to the approach I am taking.

My approach to CID shares with Baxter the notion that individuation is relative to some standard which we could call a “carving” or a “count”. What is being carved, counted, etc. is what we could call “substrate” to follow in an influential vein. The story goes that when we talk of all the parts of a composite object what we are doing is referring to the same substrate that we refer to when we talk of the composite object, only we are referring to that substrate under a different count. The composite object is some substrate individuated as one thing and the parts are that same substrate individuated as many things. The identity of parts to whole alleged by CID is underwritten by the identity of the underlying substrate, which is why Baxter calls this phenomenon “cross-count identity”. Friesen argues as follows:

Suppose we make a cross-count identity assertion: the deck is identical to the cards. We would be saying something false. One of the terms flanking the identity predicate doesn’t denote anything; if the deck exists, the cards don’t. And there can be no identity between something that exists and something that doesn’t.

The conclusion is clearly a problem for the plausibility of cross-count identity. If the truth of CID rests on the truth of cross-count identity claims and those are uniformly false, we should give up on CID. The rationale for the premise that one of the terms in a cross-count identity claim doesn’t denote anything is as follows. Fix a count of some substrate under which that substrate is individuated to be 52 playing cards. Where is the one deck of cards? Under this count, the way that this substrate is divided up, there is no one deck of cards because that (the deck) would have to be an individuation of this same substrate, and this substrate is already ’spoken for’ under this count as it were.
 
However, there is a misunderstanding of the position here. The misunderstanding has to do with the scope of the “in a count” operator. This functions somewhat like the familiar “in a world” operator of hybrid logic, but there is a significant difference: the modal operators are sentential operators whereas the count operator is a term (or more generally, referring expression) operator. When the advocate of CID says that the deck is identical to the 52 cards, they are not saying something equivalent to the following.

1. (the deck = the cards) in a count

What they are saying has the following form.

1. (the deck, in one count) = (the cards, in another count)

The criticism depends on thinking that the first form is right. If we had to evaluate the identity within a given count, then it would always come out false, as discussed. But in one way, it should be clear that this is wrong. How would that be “cross-count” identity? The second form is what is intended when the CIDer says that the deck is identical to the cards.