A Theory of DuplicationApril 8, 2008
Normally when people define a duplicate they’ll say something like x is a duplicate of y iff x and y have exactly the same intrinsic properties. This might be a fruitful way to think about intrinsic properties, but it seems to me that a theory of duplication can be theoretically separated from our theory of intrinsic properties. For example, as I understand it, duplicates are always spatial entities, yet you might still want to make room for talk about abstract objects having intrinsic properties (unless you’re a structuralist, I guess.)
Also, some people have argued that shape is not an intrinsic property, yet clearly no two duplicates can have different shapes.
What I want to do here is outline a theory of duplication from a spatial perspective. There are quite a lot of potential principles we might adopt, but I’m only going to discuss a small subset of them. Let’s start off by adding to our language of mereology, another primitive, (to be read: x is a duplicate of y.) Here are some axioms/interesting principles worth thinking about:
Explanation below the fold.
The first three principles I take to be analytic, and say that the ~ is an equivalence relation.
Then we move on to some more substantive principles. Let be a formula with one free variable.
This schema says that any duplicate of a is a . e.g. any duplicate of a mereorlogical atom, is an atom, or if x has a duplicate as a proper part, then any duplicate of x does too. Let A abbreviate ‘x is a mereological atom.’
D5 is more controversial – it says that all atoms are duplicates. Whether this principle holds will depend on whether we think that objects are duplicates in virtue of their spatial representations, and whether there can be extended simples. If you answer yes to both these questions, then presumably you will reject D4, since two extended simples might not be duplicates. Of course, when thinking of duplicates spatially, this principle doesn’t require that all atoms have the same intrinsic properties! If we want to consider duplicate theoretic versions of gunk we could add the following principle (which is stronger than saying every thing is gunky – it says everything is homogenous.)
Let abbreviate ‘the fusion of the ‘s’, and – mereological complement. Then we can have:
This principle says that if x and y are duplicates, then so are their ‘shadows’ – their complements with the whole universe. Note that this also puts restrictions on the structure of the universal fusion. It must be, in some sense, unbounded in all directions, since if it was bounded the complements of two duplicates might be too near an edge to be a duplicate (this opens interesting possibilities for defining the notion of a bounded region of space in duplication theory.) Another way thinking of the structure of the universal fusion is that it is its only duplicate – any duplicate of an unbounded region must be unbounded, and there is only one region unbounded in all directions with no holes. (I’m sure this principle can be made more general, but I haven’t really thought about it.)
Now to make the theory interesting, we’re going to add some topological notions – in particular we’re going to add meaning ‘x is connected to y’, to include cases where x and y are kissing as well as cases when they’re overlapping. (Note: one might have thought this notion could already be defined in mereological terms. It can’t. Consider the interval [0, 3) interpreting parthood as subset. Here [0, 1) and [1, 2) are kissing. Consider the following automophism which preserves parthood: [0,1) is fixed and [1, 2) and [2, 3) are switched – i.e f is identity on [0, 1), x + 1 on [1, 2) and x – 1 on [2, 3). Under this image [0, 1) and [1, 2) are no longer kissing, so it follows by elementary model theory that connection can’t be defined in mereological terms.)
There are, I think, lots of potential axioms we could consider here, but I’m only going to consider two.
Here ‘int’ is the interior operation. Along with D7 we can show the closure operation has this property too, but note however that the converse does not hold. An open and a closed unit sphere have duplicate interiors, but aren’t themselves duplicates. Note, also, that D9 falls straight from D4 if we allow | to appear in . Lastly
This says if x and y are connected to each other everywhere then they must be duplicates. This will include colocated objects (objects occupying exactly the same location.) As well as, e.g. two laminas touching everywhere (not possible in Euclidean space.) There are probably more intuitive principles about, but I haven’t really thought much about the topological duplicate theory yet.