Non-well-founded mereology
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 case could be made for non-well-founded mereology.
Finally, concerning the anti-symmetry postulate, one may observe that this rules our non-well-founded mereological structures. [. . . ] in this case there is a legitimate concern that one of the principles that we are assuming to be constitutive of ‘part’ is in fact too restrictive. (36)
I’ve been thinking for a while about writing up the theory, and some of its more interesting consequences. Non-well-founded sets are fairly well understood, thanks to Peter Aczel (1988), and so the corresponding mereology shouldn’t be too difficult to work up. Andrew’s post gives a nice discussion, but I think something is missing.
Andrew’s tact is to relax the requirement that the parthood relation be anti-symmetric. He allows for two objects to be parts of each other without being identical. That is two objects can be proper parts of each other. Accordingly,
- A mereology is non-well-founded iff
.
- Proper parthood:
.
But I think these definitions are quite right. Here’s why: while this allows objects to be proper parts of each other, it doesn’t allow for an object to be a proper part of itself.
It is standard in non-well-founded set theory to allow sets to be members of themselves, as this is a natural consequence of denying the axiom of foundation. In mereological terms, there’s a natural reason to want to allow for non-well-foundedness is to allow for finite gunk. This is allowed by the definitions above. But consider some gunky object o for which PPoo all the way down. Here too we have finite gunk.
One interesting consequence of all this is that Andrew’s above definitions require a rejection of weak supplementation:
- Weak supplementation:
(Here O is overlap). If we are to allow for objects which a proper parts of themselves, we must likewise reject weak supplementation as well. Let PPxx. Then there is no part of x which does not overlap x.
I’m not entirely sure how to fix the definitions above to allow for what I want. I’m pretty sure a paraconsistent logic could definitely do it. But there’s probably a consistent way to do it as well. (Aside: I should note, that in a paraconsistent non-well-founded mereology, we could still have weak supplementation.)
Hi Aaron.
This is all very interesting. When I get time I’m going to think over this stuff more carefully.
Some obvious alternative definitions of proper parthood might be:
1)
2)
3)
4)
1) just makes
inconsistent, so presumably this isn’t what we want. 2) essentially forces weak supplementation to hold, so again, is inconsistent with 
. 3) and 4) don’t seem to allow $PPaa$ either. Methodologically I’m thinking it might be better to start off with proper parthood, and see what kind of mereology we can get based on that. This definitely requires some more thought.
[...] 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 (iff:) is [...]
More non-well-founded gunk! « Possibly Philosophy said this on February 22, 2008 at 1:51 am |
[...] 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 primitive in a [...]