In our usual talk about identity we say that two things are identical if they have all of their properties in common. Generally this is a perfectly reasonable way of talking about identity, at least when we restrict ourselves to considering things of the same level of abstraction. However when we try to identify an event or object at one level of abstraction with events or objects at a different level there can be problems.
For example consider the abstraction “my desk” and the arrangement of atoms before me. I want to say that my desk is identical with the atoms in front of me, but unfortunately life is not so simple. For example consider the color that I perceive my desk as having. This is certainly not a property of any of the atoms that make up the desk, and thus the desk and the atoms that make it up do not have all their properties in common (and are not identical). But wait! Can’t we argue that the collection of atoms in front of me have that property, and thus that the collection as a whole has all the same properties as my desk? Well if we take that route a different problem presents itself to us. Now consider the collection of atoms that I am currently considering to be identical to my desk and take away one of its atoms (at random). The remaining collection will still have all the properties that I consider to define my desk, but clearly my desk cannot be both identical to the entire collection and the entire collection minus one atom, since that would imply that both collections are identical, and clearly they aren’t since there are more atoms in one than the other. To fix this problem then we might be tempted to argue that my desk is not defined by the properties that I perceive as being essential to my desk, but all of its properties, and thus the two collections would not be considered to be identical to my desk, since one of them differs infinitesimally from the exact weight that defines my desk. But now I am faced with an equally difficult question: which of the two collections is my desk? Because the idea of my desk doesn’t pin down the weight that precisely I am left with the conclusion that there are an extremely large number of possible collections of atoms in front of me, each of which is equally qualified to be described as my desk. We may have solved the problem of identity then, but at the expense of discarding the idea that there is really a single thing in front of me that is my desk (or that I know what my desk is), which is equally unacceptable.
Perhaps then we should give up the idea that my desk is “identical” with the collection of atoms in front of me in the strict sense of the word. Let us then use a new word, “dependentical”, to describe the relation between a description at one level of abstraction and the more basic elements that it depends on. Dependentical is obviously not the same relation as identity. One interesting feature of being dependentical is that two things can be dependentical without being a priori dependentical (identical objects are always a priori identical). It follows from this that we won’t always know if A is dependentical with B without performing some investigation. For example an ancient Greek* would probably have denied that my desk was dependentical with a collection of atoms (since he or she “knows” that matter is continuous). How then are we to know if one thing is dependentical with another? First we have to identify which is the higher or lower abstraction (desk is higher while collection of atoms is lower). Secondly we have to know when the higher abstraction applies (I know when something is my desk because it looks and feels like my desk, which means it falls within a certain range of mass, size, location, ect). We then produce alterations in the lower of the two abstractions and see if altering the lower component can force the higher abstraction to be inapplicable (for example remove the atoms that make up the legs of the desk and suddenly it is not my desk). Finally we see if we can force the higher abstraction to be inapplicable without changes in the lower level (we see if we can make my desk not my desk without altering any of the atoms that I thought that it was dependentical with). If we can’t produce such a change, and changes in the lower level can invalidate the higher level description then we have good (although not perfectly conclusive) evidence that the two things are dependentical with each other.
Now why did I bother to go through this song and dance about dependentical? Well, I claim that the mind is dependentical with the brain (not identical, as materialists traditionally put it). This solves two objections to materialism at the same time. First it nullifies the objection that the mental and the physical must be different things since they have radically different properties, since, as the example of the desk shows, dependentical objects do not necessarily have the same properties. It also invalidates objections based on conceptual analysis, which claims that because the concepts of brain processes and consciousness can be separated that they must not be the same thing. Once again our example with the desk shows that two things which really are dependentical may not be a priori dependentical, and it may even be perfectly reasonable to deny that they are dependentical in the absence of evidence. This of course also ties in with my earlier paper.
* Obviously not Democritus