More non-well-founded gunk!

February 21, 2008

Aaron has an nice response to my last post on non-well-founded gunk. He notes that my definition of proper parthood ($x \sqsubset y$iff:$x \sqsubseteq y \wedge x \not= y$) is too restrictive for the kinds things normally associated with non-well-foundedness. In particular objects can’t be proper parts of themselves, diverging from the set theoretic analogue of non-well-foundedness where $a \in a$ is a paradigmatic kind of non-well-founded set.

I think maybe I should put my reasons for discussing this particular kind of non-well-foundedness in a broader context. I was thinking of cases a bit like Goliath and the lump of clay. In this case they are not identical, nonetheless they are both parts of each other. So on my definition they are proper parts of each other, and so form an instance of NWF gunk. This case poses weird problems for mereology, for example neither Goliath nor Lump have a part disjoint from the other, even though they are distinct, they have several fusions, and no sum (see my last post for reasons.)

Nonetheless I think there should be another way of setting up NWF gunk which is more closely related to the non-well-founded set theory. One that’s transitive unlike my earlier definition, and e.g. captures examples like “Borges Aleph” – here the intuition is that Borges Aleph is a proper part of itself, while, of course, remaining identical with itself.

At the end of his post Aaron poses the problem of what the definition of proper parthood should be to allow this. Here are some candidates, all of which are equivalent to my definition in standard mereology

• $(x \sqsubseteq y \wedge y \not\sqsubseteq x)$
• $(x \sqsubseteq y \wedge \exists z (z \sqsubseteq y \wedge z \not\sqsubseteq x)$
• $(x \sqsubseteq y \wedge \exists z (z \sqsubseteq y \wedge z \bot x)$
• $(y \not\sqsubseteq x \wedge \exists z(x + z = y))$
• $(y \not\sqsubseteq x \wedge \exists z (z \sqcup x = y))$

(Here + means mereological fusion, and $\sqcup$ means mereological sum. See last post for definitions.) However, it’s quick to check that non of these allow an object to be a proper part of itself, and I’m not convinced that there will be such a definition in terms of parthood.

I think the best way forward would be to take proper parthood as a primitive, a good starting point would be

• A1 $((x \sqsubset y \wedge y \sqsubset z) \rightarrow x \sqsubset z)$
• A2 $(\exists x Fx \rightarrow \exists x \forall y(x \circ y \leftrightarrow \exists z(Fz \wedge z \circ y)))$

Just take the two element digraph where every node is connected to every other node in each direction to see this is at least consistent. (It’s a bit of a mouthful, so call it “Borges Aleph” for short.)

Of course, fusions will never be unique, nor will supplementation hold, but I think you are pretty much committed to this however you set it up. We can obviously add, for each axiom of standard mereology, an axiom to restricted to well-founded-objects (Define WF(x) as $\forall y(y \sqsubset x \rightarrow y \not\sqsubset y)$. My question now is, what substantial principles can we consistently add?