Indefinite Extensibility and Contractibility of Plurals and Ensamblism
Let's introduce some definitions:
our primitive notion will be xx are amongst yy and we will assume that there are no empty plurals.
xx are ancestorally amongst yy =df either (i) xx are amongst yy or (ii) xx are amongst somethings that are ancestorally amongst yy.
xx are contractible =df for any yy amongst those xx, there is a zz such that zz are amongst yy.
xx are indefinitely contractible =df for any yy ancestorally amongst xx, there are some zz such that zz are ancestorally amongst yy.
xx are extensible =df there are some yy such that xx are among yy
xx are indefinitely extensible =df there are some yy such that xx are ancestorally amongst yy and for any zz such that xx are ancestorally amongst zz, there are some vv such that zz are ancestorally amongst vv.
Some restricted plurals are indefinitely contractible. For example, take a quantifier that is restricted to countably infinite pluralities of integers. If we restrict our attention to only plurals that satisfy the variables of that quantifier, then those plurals will be indefinitely contractible.
Some restricted plurals are indefinitely extensible. For example, take a quantifier that is restricted to finite pluralities of integers. If we restrict our attention to only plurals that satisfy the variables of that quantifier, then those plurals will be indefinitely extensible.
Two Questions: Will our fundamental theory talk about pluralities that are indefinitely extensible? Will our fundamental theory talk about pluralities that are indefinitely contractable?
A while ago I posted about Priority Ensamblism. Priority Ensamblism says that there are some things such that they are basic. Priority Ensamblism comes in two varieties. Distributive Priority Ensamblism and Non-distributive Priority Ensamblism. You can read about these views and an argument for the latter view in my previous post here. My thoughts right now is the following: If our fundamental theory contains plurals that are indefinitely contractible, then that is a reason to favor non-distributive priority ensamblism. However, if our fundamental theory contains plurals that are indefinitely extensible, then that is a reason to favor distributive priority ensamblism. I need to think this through a bit more and see what kind of premises might be lurking behind these rough thoughts.
posted by Joshua at 7:39 AM
5 Comments:
Hi Joshua,
Just some points for clarification. First, the definition of 'ancestorally amongst' is circular; see condition (ii). Second, and relatedly, it's not clear to me whether you take xx to be amongst yy even if xx is the same plurality as yy. (Given the claims you hope to establish, I take it that you mean to rule this type of case out. And if so, the way to correct the definition of 'ancestorally amongst' is obvious. But I thought I'd ask before I thought about this further.)
Hi Joshua,
One immediate problem with the two conditionals both being true is that (I think) there can be pluralities that are both indefinitely contractible AND indefinitely extensible.
For instance, suppose we quantify over only those pluralities that constitute a finite open interval of the real numbers. Wouldn't those pluralities count? Or take some plurality structurally isomorphic to the material objects in a hunky world (Bohn, "Must There be a Top Level?").
Suppose that's right. If both conditionals were true in such a scenario, distributive AND non-distributive priority ensamblism would be true. But presumably, that's impossible.
Hey Alex,
Thanks for the comments. One quick note, I intended the definition of ancestorally amongst to be a recursive definition (I take it this is the standard way of defining ancestoral relations). So, tenet (i) is the base case and tenet (ii) is the recursive case.
You are right that I should have used the relation 'properly amongst' as my primitive.
I also think it is consistent to say that some pluralities are indefinitely extensible and indefinitely contractible. The case of finite open intervals is a prime example. If there is a genuine possibility of such pluralities, then there is trouble for both distributive and non-distributive ensamblism.
I think there is an analogous puzzle for the priority pluralist and the priority monist. Suppose it is possible for everything to be gunky junk (objects, each proper part of which has further proper parts and each of which is a proper part of something further). Then there is a problem for both Priority pluralism and Priority Monism. I think that if this is a genuine possibility, then there is a reason to give up on Schaffer's non-overlap constraint on the basic entities.
I was just thinking that if there are pluralities that are indefinitely extensible, then unrestricted composition is consistent with the denial of a universal object.
Let's suppose that there are some things that are indefinitely extensible. Let's introduce a series of plural quantifiers each one of which ranges over an extension of the initial plurals. Then it should be true that:
(Craziness)
There are some-1 xx such that for any-1 yy, yy are properly amongst xx but there are some-2 zz such that xx are properly amongst zz. Moreover, this will be true (so to speak) all the way up.
Now, the best we can do in asserting unrestricted composition with the strange constraints that are in place is to quantify over quantifiers:
(UC)
For any quantifier n, for any-n xx there is a y such that xx compose y.
But, this is consistent with junk. Junk would occur in a world where (Craziness) holds.
This seems like a rather surprising result.
My previous comment assumed some sort of type theory. But, I don't think I need that. I am still working out the details and I hope to post more in a bit.
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