## Non-well-founded mereology

February 19, 2008

I was wondering recently if there has been any work on non-well-founded mereology. A preliminary google search didn’t bring up much except the SEP article on mereology, which informed me there has so far been no systematic study of well founded mereology in the literature so far.

I thought a little about what it might look like, but the prospects don’t look particularly bright to me. By a non-well-founded mereology, I mean one in which a particularly weird kind of gunk is allowed – gunk with finitely many parts. Although this is sufficient for the mereology to be non-well-founded, it is not necessary: let us simply say that a mereology is non-well-founded iff $\exists x \exists y(y \sqsubset x \wedge x \sqsubset y)$. (The SEP article quoted “Borges Aleph” as an example of this: “I saw the earth in the Aleph and in the earth the Aleph once more and the earth in the Aleph… (Borges 1949: 151)”)

The first thing to note is that anti-symmetry fails (this follows straight from the definition.) So already there is a worry that we are not talking about a parthood relation here – if there are any analytic constraints on parthood, it must surely include being a partial order.

But even granting this, many other natural principles are inconsistent with the existence of non-well-founded gunk (as usual we help ourselves to reflexivity and transitivity.) For example, take weak supplementation.

• $\forall x \forall y(x \sqsubset y \rightarrow \exists z(z \sqsubseteq y \wedge z \bot x))$

Suppose $(a \sqsubset b \wedge b \sqsubset a)$. By weak supplementation there should be a part of b, z say, disjoint from a. But since $z \sqsubseteq b \sqsubset a$, $z \sqsubseteq a$ by transitivity, which contradicts the assumption that $z$ was disjoint from $a$. (Ironically, I think the so called “strong supplementation” axiom is consistent.)

Similarly uniqueness of fusions fail radically in the presence of non-well-founded gunk. A reminder: u is a fusion of the F’s iff

$\forall t(t \circ u \leftrightarrow \exists z(z \circ t \wedge Fz))$

For example $u$ is a fusion of $a$ iff $\forall t(t \circ u \leftrightarrow t \circ a)$. Clearly, if $(a \sqsubset b \wedge b \sqsubset a)$ then both $a$ and $b$ are fusions of $a$. Even worse, whenever $a \sqsubseteq c \sqsubseteq b$, then $c$ is a fusion of $a$. Similarly, whenever $c$ is a fusion of $a$ and $b$, $c$ is a fusion of $a$.

Perhaps we could make do with “sums” instead of fusions. Say that $u$ is a sum of the $F$‘s iff

• $(\forall x(Fx \rightarrow x \sqsubseteq u) \wedge \forall v(\forall x(Fx \rightarrow x \sqsubseteq v) \rightarrow u \sqsubseteq v))$

In standard mereology, sums play exactly the same role as fusions do, but they come apart in many non-standard mereologies. However, even here we have problems. For NWF gunk, sums fail to exist: take the $a$ and $b$ from above and take $F = \{a, b\}$. Both $a$ and $b$ are upperbounds for $F$ but neither are least, so there is no sum.

It seems, then, that there isn’t anything much mereological about the resulting system. After all, where is mereology without fusions or remainders? Maybe there are other ways of setting up the axioms (e.g. I think that strong supplementation is still consistent.) Can anyone think of an alternative to play the rule of fusions? Maybe something that coincides with fusions in a standard mereology?

1. By the way, it makes an important difference what I’m taking as primitive here. I’m defining all the other symbols, $\bot, \circ, \sqsubset$ in terms of $\sqsubseteq$. $\sqsubset$ I’m defining as $x \sqsubseteq y \wedge x \not= y$ unlike the SEP article which defines it as $x \sqsubseteq y \wedge y \not\sqsubseteq x$ (otherwise a would not be a proper part of b in the above discussion.)

2. Hi Andrew. Interesting post. I think though that there are other ways of setting things up. As you note in your comments, you’re defining proper part thus: $PPxy$ iff $Pxy \wedge x \neq y$. If that’s so, your operative definition of non-well-foundedness — $PPxy \wedge PPyx$ — allows for objects which are proper parts of each other, but does not allow for objects to be proper parts of themselves.

In all non-well-founded set theories we allow that $a \in a$ for some a, and is a natural consequence of denying the axiom of foundation. In mereological terms, there’s a natural reason to want to allow for this: consider some gunky object o for which PPoo all the way down. Here too we have finite gunk. It requires a rejection of weak supplementation, but that was gone already.

3. […] Andrew, over at Possibly Philosophy, has an interesting post up on non-well-founded mereology. Casati and Varzi, in Parts and Places, suggest that a genuine […]

4. Hi Aaron. Thanks for your comment. That’s very interesting.

Let’s take the case where $(a \sqsubset b \wedge b \sqsubset a)$. One might be inclined to say that this is fine since $a \sqsubseteq a$, which is exacltly the relevant sense in which a is a part of itself. However it seems to me there are two different reasons a is a (reflexive) part of itself. 1) a is a part of itself in virtue of the fact that everything is a part of itself, 2) a is a part of itself invirtue of being a part of b, which is a part of a.

So maybe we want to keep a notion of proper part which is transitive. It seems quite plausible, actually, that the proper subject of non-well-founded mereology is the proper parthood relation, so it might be worth taking it as a primitive.

5. […] Weblog devoted to the philosophy of language, metaphysics and philosophical logic « Non-well-founded mereology More non-well-founded gunk! February 21, 2008 Aaron has an nice response to my last post […]

6. […] I have been exploring the unexplored: non-well-founded (NWF) mereology. For those catching up, see here, here, and here. Andrew suggests, I think rightly, that we should treat proper parthood as […]