 ## A Theory of Duplication

April 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, $\sqsubseteq$ another primitive, $\sim$ (to be read: x is a duplicate of y.) Here are some axioms/interesting principles worth thinking about:

D1. $\forall x x \sim x$
D2. $\forall xy(x \sim y \rightarrow y \sim x)$
D3. $\forall xyz((x \sim y \wedge y \sim z) \rightarrow x \sim z)$
D4. $\forall xy(x \sim y \rightarrow (\phi(y) \leftrightarrow \phi(x)))$
D5. $\forall xy((A(x) \wedge A(y)) \rightarrow x \sim y)$
D6. $\forall x \exists y(y \sqsubset x \wedge x \sim y)$
D7. $\forall xy(x \sim y \rightarrow (\sigma z(z=z) - x) \sim (\sigma z(z=z) - y))$
D8. $\forall x(\forall y y \sqsubseteq x \rightarrow \forall y(y \sim x \rightarrow x = y))$
D9. $\forall xy(x \sim y \rightarrow int(x) \sim int(y))$
D10. $\forall xy(\forall z((z \sqsubseteq y \rightarrow z \mid x) \wedge (z \sqsubseteq x \rightarrow z \mid y)) \rightarrow x \sim y)$

Explanation below the fold.

The first three principles I take to be analytic, and say that the ~ is an equivalence relation.

D1. $\forall x x \sim x$
D2. $\forall xy(x \sim y \rightarrow y \sim x)$
D3. $\forall xyz((x \sim y \wedge y \sim z) \rightarrow x \sim z)$

Then we move on to some more substantive principles. Let $\phi$ be a formula with one free variable.

D4. $\forall xy(\phi(x) \wedge x \sim y \rightarrow \phi(y))$

This schema says that any duplicate of a $\phi$ is a $\phi$. 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. $\forall xy((A(x) \wedge A(y)) \rightarrow x \sim y)$

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.)

D6. $\forall x \exists y(y \sqsubset x \wedge x \sim y)$

Let $\sigma x \phi$ abbreviate ‘the fusion of the $\phi$‘s’, and – mereological complement. Then we can have:

D7. $\forall xy(x \sim y \rightarrow (\sigma z(z=z) - x) \sim (\sigma z(z=z) - y))$

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.)

D8. $\forall x(\forall y y \sqsubseteq x \rightarrow \forall y(y \sim x \rightarrow x = y))$

Now to make the theory interesting, we’re going to add some topological notions – in particular we’re going to add $x \mid y$ 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.

D9. $\forall xy(x \sim y \rightarrow int(x) \sim int(y))$

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 $\phi$. Lastly

D10. $\forall xy(\forall z((z \sqsubseteq y \rightarrow z \mid x) \wedge (z \sqsubseteq x \rightarrow z \mid y)) \rightarrow x \sim y)$

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.

1. Hmm, interesting idea, axiomatizing duplication. I have just two thoughts on the axioms proposed here.

First, unless you have some hefty restrictions on $\phi$ I don’t know about, D5 is just Leibniz’ Law for $\sim$, so it will get you that $x\sim y$ iff $x=y$.

Second, I’m suspicious of D7. Let a and b be two duplicate atoms. Suppose a is surrounded by a ring of simples, and b is not. Then the shadow of a doesn’t seem to be a duplicate (even in a weak spatial sense) of the shadow of b. After all, the shadow of a has a point-sized “hole” inside a ring of simples, and the shadow of b doesn’t.

Or is D7 supposed to be only plausible given D6? But D6 will give us a very homogeneous world indeed; I’m wondering what it’s interest is supposed to be. If I understand it right, then (given classical mereology) it will be much stronger than the gunky counterpart of each atom being a duplicate of any other, for it will entail that for any non-empty regions R1 and R2, if something is in R1, then it has a duplicate in R2. (The fusion F of R1 and R2 has the contents of R1 and the contents of R2 as parts; by D6, it must be a duplicate of both; but since $\sim$ is an equivalence relation, R1’s contents and R2’s contents must be duplicates of each other.)

2. (Sorry, I guess I didn’t quite get the latex code right…)

Edit Andrew: I tried to fix it. You have to type ‘latex’ after the first \$.

3. I’m puzzled by D4. The minor point is that appears to be different above and below the fold. But either way, it seems to say that, without restriction on \phi, if x duplicates y, then x is \phi iff y is. Unless Lewis et al were completely wrong in defining intrinsicality with respect to duplicates, this can’t be right: just let \phi be extrinsic. And if it needs a restriction to intrinsic \phi, which might at least make D4 true, we don’t really have an axiomatisation.

I share Jason’s worries about D7, especially in a Lewisian framework. Recall that the idea was an intrinsic property was one a lonely duplicate could have. Since you aren’t lonely, your lonely duplicates can’t be worldmates of you. So I guess the quantifiers in D7 must range over all possibilia. But then D7 looks almost always false, since (assuming one reject purely haecceitistic differences between worlds) cutting an object out of one world will be different from cutting its duplicate from another. The only exceptions will be when the duplicates are worldmates in symmetrical worlds. Or am I missing something?

And D6 doesn’t seem to be homogenity: D6 simply says everything has a part which duplicates it. I may be mistaken about this, but consider a doll’s house which has a doll’s house in it which duplicates it, which then (of course) has a doll’s house in it that duplicates it. Doesn’t this satisfy D6 while being pretty far from homogeneous in the intuitive sense? (It is gunky.)

4. Jason + Antony – thank you both for your comments.

Jason: regarding D4, it’s a schema only involving formulae (without paramaters) definable in this language (i.e parthood and duplication.) It basically says any two duplicates are indistinguishable with respect to mereological and duplication theoretic properties. This isn’t sufficient to get you identity I don’t think.

Antony: sorry about the different formulations, I wasn’t checking what I’d already written as I was typing this (they are of course equivalent though.) There is a restriction on $\phi$, namely that it has only one free variable (which means it’s a monadic, non-relational, property definable from $\sqsubseteq$ and $\sim$.) Given that I can’t can’t easily think of a way to define an extrinsic property (there is a worry, if you don’t have pairwise fusions in your mereology, that being a proper part of something might do that.) If D4 had paramaters you could have, for example, properties like overlapping with Antony, which some but not all duplicates of your overlappers have. But D4 doesn’t allow parameters.

Secondly regarding D6. Your description of the dolls house was incomplete, so it’s not determinate if it satisfies D6. If, e.g. the complement of the second largest dolls house from the largest dolls house is simple then no. If this complement contains a duplicate of its self, and so equally for all of the dolls house parts, then it would satisfy D6 – but then I would be happy calling it homogenous (it probably wouldn’t look much like a dolls house anymore though.)

I’m beginning to feel uneasy with D7 now though. I guess I was getting carried away with the idea of being able to get boundedness just from duplication. But regarding Antony’s point – if my quantifiers range over all possibilia (in a Lewisian framework) then the universal fusion, $\sigma x(x=x)$, will be the fusion of all worlds, and so the shadow of an object inside a world will never be itself inside a world (assuming there are 2 or more worlds.) BTW, I’m not so moved by all of these worries about intrinsicness – part of what I was trying to do here was give the notion of duplication a firmer footing independent of its characterisation in terms of intrinsic properties.

5. Oh, I forgot Jasons last point – yes in a homogenous world everything will have a duplicate inside everything else. It’s supposed to be like an extreme kind of gunk – I don’t know what it’s interest is supposed to be (why do people find gunk interesting? It makes for good science fiction anyway…)

6. OK, great — I was wondering if you had something like that in mind for D4. I’m not sure what exactly you mean by “paramaters” here, though, but if you allow identity and either (i) names or (ii) impredicative formulas, you can get it just with $\phi$ being $= \alpha$ (with $\alpha$ bound out front if it’s a variable). Somehow those formulae have to be ruled out, too.

On gunk: I take it that a world is gunky iff it has at least one object with proper parts each of which have proper parts. But for all this says, our world might be one like this. (Maybe everything is built up out of (spatially extended) quarks, which are themselves infinitely divisible, for instance…) But I was thinking that the thesis of homogeneity is pretty clearly inconsistent with what we see, unless you have some sort of “plenum”-style view of the contents of space(-time).

7. By no parameters, I mean $\phi$ doesn’t have free variables taking values from assignments. The language doesn’t have names either, if you added names you would have to restrict D4 further. D4 is supposed to say that duplicates have the same mereological (and duplication theoretic) properties. If we add names then we get properties like ‘overlaps with John’, which is clearly not a mereological, or duplication theoretic property. I don’t understand you second example (ii) either. Is being identical to something supposed to be an extrinsic property? (Lonely objects have this.)

I think the gunk thing may be terminological. I say a world is gunky iff everything in it has a proper part, and is homogenous iff everything in it has a duplicate proper part. Alternatively I could have used your definition and said a world is gunky iff it contains an object such that all of its parts have proper parts, and is homogenous iff it contains an object such that all of its parts have duplicate proper parts. So if the actual world contained Antony’s homogenous dolls house it would be a homogenous world on the second but not the first definition. I take it that the existence of a homogenous dolls house is consistent with what we have so far seen, insofar as gunk is.

8. I’ve actually been working on a paper for the last week that deals with something similar. (Well, not really that similar, but related.) I was working on mereologies w/o antisymmetry, for the purposes of getting around Varzi’s recent argument for extensionality principles in PQ. Anyway, if you reject antisymmetry, then parthood is a preorder. You can then define an equivalence relation thus: $x \sim y \Leftrightarrow x \sqsubseteq y \wedge y \sqsubseteq x$

We then take the set of all equivalence classes of our domain D given by $\sim$ as our new domain. Now parthood is a partial order on $D/\sim$. This gives us all the relevant mereological structure without having the philosophical oddness of extensionality principles. In particular, the oddness arises precisely because of principles like your D4. (But of course, your axioms are for a completely different purpose for which you *would* want D4).

Anyway, if you want to have a look, email me, since we’ve talked about mereology without extensionality before.

9. Very nice style and superb articles , nothing else we need : D.