Inconsistent Thoughts

Another angle on Composition as Identity (CID). I now think that my earlier approach is misguided. Lewis’ argument was pretty straightforward. CID is true, and so the axiom of unrestricted composition just commits us to things (fusions or composite objects) which are identical to things (their parts) we already believed in. This is innocent because ‘identicals’ are an ontological free lunch.

The problem here is that we cannot really assess the argument unless we can assess CID and that is something we lack the conceptual resources to do. The alleged identity of the parts to the whole they compose would have to be many-one identity and we just don’t have a theory of identity on which this makes sense. Even plural logic can’t help here. Standard plural logics like that developed by Yi (“The Logic and Meaning of Plurals I and II” 2005, 2006 respectively) include a plural identity predicate, but the truth-conditions are just an extension of the standard one-one identity predicate. In other words, theories of plural identity make it diverge from classical identity only insofar as there are many-many instances, but these have to ‘many’ of the same cardinality.

Without a theory of identity on which many can be identical to one, we just can’t make sufficient sense of CID to even assess whether or not it is true. Van Inwagen makes this sort of point repeatedly (“Composition as Identity” 1994) although he takes it as evidence that Lewis is simply wrong since what he is claiming is ungrammatical and perhaps incoherent. Sider makes some headway by brutely flanking an identity sign with referring expressions of different cardinalities (“Parthood” 2007). He interprets the identity sign in a classical way, its logic governed by Leibniz’ Law, the result being some weird behavior of plural apparatus like the predicate ‘is-one-of’.

I take a different approach altogether.  The advocate of CID has a framework available.  The parts and the whole are said to be different ways of counting “the same external phenomenon” by Frege and are said to be different divisions of “the same portion of Reality” by Lewis.  To talk about the alleged phenomenon or portion which can be counted or divided in various ways I am going to use Baxter’s terminology of aspects of reality, or just ‘aspects’ for brevity (the terminology is from “The Discernibility of Identicals” 1999).  The identity of parts and wholes, then, has to do with a two-fold story about reference on which the ontological ground floor are aspects, but reference to them is always constrained to a count.  Given some aspects, they are self-identical regardless of how they are counted or divided up, so the parts and the whole are identical because ‘as parts’ and ‘as whole’ are just different counts of the same aspects of reality.

The informal sketch makes plausible how many-one identity could hold, but raises a lot of questions.  Does this identity respect Leibniz’ Law (in some appropriate form)?  To answer that, we first need to answer a separate question, namely:  what is the logical form of many-one identity?  Working out the logical form will require some revisions to classical quantification theory.  Most of the action takes place at the level of referring expressions.  I’ll try to post some thoughts on that soon.