Monday, June 18, 2007

Hypergunk and Modality

Something is hypergunk iff it is atomless (every proper part of it has proper parts) and for every set S containing only parts of it there is another set S* containing only parts of it which has a subset whose members are equinumerous with the members of S, but S* itself is not equinumerous with S.

Put more simply but less carefully, for any set of parts of hypergunk there is another set of parts of hypergunk such that the second set has a strictly greater cardinality than the first.

The possibility of hypergunk makes all sorts of trouble for views like Lewis's, and others as well. (Want to hold that worlds are maximal sets of propositions? There are no such sets if hypergunk is possible.) Or so argues Daniel Nolan in his "Classes, Worlds and Hypergunk".

If hypergunk is possible, then there are possibly more than set-many individuals. This is because there is no set that has as members all the parts of hypergunk. For suppose there were such a set and its cardinality were c (for reductio). Then by the definition of hypergunk, there is another set with a cardinality c* such that c* > c. But since it's not the case that c > c, our assumption that there is a set that has as members all the parts of hypergunk is false. So there is no such set.

Some immediate implications for Lewis's view:
  • There are worlds that do not correspond to any set of individuals
  • There is no set of all worlds
  • There are no necessary truths
  • There are no truths about some worlds
  • There are no truths about some individuals
The paper also discusses several other interesting cardinality issues related to modality. It's highly recommended. Feel free to post questions, comments, or objections in the comments section of this post.

(Former Priestly seminarians may also be interested in these papers by Nolan.)


Adam said...

So “Gunk” is stuff that is not made up of mereological atoms: it is not ultimately made up of parts which do not themselves have further parts.

“Hypergunk” is gunk such that, for any set S of its parts, there is another set S’ of its parts which enjoys strictly greater cardinality than S. Or, for any sized set of parts of hypergunk, there are enough parts of hypergunk to make up a larger set.

The possibility of hypergunk poses cardinality problems and paradoxes for both AR and GR conceptions of the nature of possible worlds: for AR views according to which possible worlds are sets of propositions; maximal sets of compossible facts, and for GR views according to which worlds are sets of concrete individuals.

Near then end of the paper, Nolan suggests that denying the possibility of hypergunk would be a cost to any theory of possible worlds. This seems to be because the possibility of hypergunk is consistent with our commonsense notion of infinite divisibility, it is formally consistent, and violates no obvious analytic truths.
However, if we are comfortable with paying these costs, then Nolan suggests that one way a denier of the possibility of hypergunk could go would be to argue that metaphysical or logical possibility is not tied to the limitations of set theory.

I’d like to ask about a possible approach to the problem that I take to be similar to the general strategy outlined by Nolan. It involves the thesis of “compositional pluralism” (CP).

According to the compositional pluralist, there is more than one fundamental part-whole relation. For example, the part-whole relation defined on material objects is, according to CP, different than the fundamental part-whole relation defined on regions of spacetime. While these appear to be the standard examples, it seems CP could plausibly extend the notion of different fundamental part-whole relations to cover cases such as:

(1) 12:30 pm is part of the interval ranging from 12:00 pm to 1:00 pm.
(2) The third inning was the most boring part of the ball game.
(3) {a} is a proper part of {a.b}.

(These examples come from McDaniel’s ‘Modal Realism with Overlap” paper.)

The compositional monist (CM), by contrast, will deny this. According to CM, there is exactly one fundamental part-whole relation, and that this part-whole relation is invoked in any parthood attribution (such as 1-3).

(Note, however, that CP is not committed to the denial of a ‘generic’ sense of the part-whole relation. This generic, or univocal, sense of ‘part’ is at play in all instances of part-hood attributions. But CP does not take this generic sense to be the fundamental meaning of ‘part,’ since, by definition, CP takes there to be more than one fundamental part-whole relation.)

Now take the axioms of classical mereology:

Transitivity: For any x, y and z, if x is a part of y, and y is a part of z, then x is a part of z.

Unrestricted Composition: For any xs, there is a fusion of the xs.

Uniqueness: For any xs and any y and z, if y is a fusion of the xs, and z is a fusion of the xs, then y = z.

CM will want to say that the axioms of classical mereology adequately characterize the (one and only) fundamental parthood relation. It is open to CP, however, to deny that the axioms of classical mereology adequately characterize (or characterize in the same way) each and all of the various fundamental part-whole relations that obtain.

If this is right, then it seems like this is one avenue that might be explored by someone who wants to defend a particular version of AR or GR from the problems generated by the possibility of hypergunk. For example, if it could be shown that the part-whole relation that obtains between (pieces? parts?) of hypergunk was a fundamentally different part-whole relation than the one that obtains between a physical object and its proper parts, and if the axioms of classical mereology could be shown to apply to the second, but not the first, then maybe this is a way for AR or GR to avoid the problem.

This is obviously a very loose sketch of a possible response. But I’d be interested to know where, and why, it would fail to get off the ground.

Dan said...

I see no obvious reason why parts of hypergunk would fail to satisfy the axioms of classic mereology. As long as it's a genuine part-whole relation I think it would continue to make problems for AR and GR, regardless of what other part-whole relations are around.

Chris Tillman said...

As it happens, hypergunk violates the axioms of classical extentional mereology just because CEM has an axiom that basically says "there are atoms". Otherwise I agree with Dan on this. See here for more fun with mereology:

Chris Tillman said...

(Oops. Varzi's formulation in the link is neutral on atomism.)