Tuesday, October 11, 2011

An infinity argument for incommensurability

Consider these plausible claims:

  1. If worlds w1 and w2 contain the exact same individuals, and each individual in w2 is better off than she is in w1, then w2 is a more valuable world.
  2. A world can contain an infinite number of individuals.
  3. If worlds w1 and w2 differ only in respect of which particular individuals exist in them, and perhaps some further value-insignificant respects, if an identity of indiscernibles principle requires it, then w1 and w2 are equal in value.
  4. If worlds w1 and w2 differ only in respect of w1 lacking one individual that exists in w2 and that has a good life in w2, and perhaps in some value-insignificant way, then w2 is more valuable than w1.
  5. If a and b are equally valuable, then c more (less) valuable than b if and only if c is more (less) valuable than a.
  6. Being more valuable than is transitive.
  7. Nothing is more valuable than itself.
Now, making liberal use of (2), imagine this sequence of worlds:
  • w1: God plus an infinite sequence of spatiotemporally disconnected individuals x1, x2, ... who are almost exactly alike, differing only in that xi enjoys on balance flourishing of value i (plus any other insignificant differences needed to avoid violation of the identity of indiscernibles)
  • w2: just like w1, except that the individuals are all different: y1, y2, ...
  • w3: just like w2, except that yi now enjoys on balance flourishing of value i+1, for all i
  • w4: just like w3, except that in the place of yi we have xi+1, for all i
Now, notice that w4 is basically w1 minus the individual x1, with perhaps some value-insignificant differenecs. By (4), it follows that w1 is more valuable than w4. By (3), it follows that w2 has the same value as w1. By (1), it follows that w3 is more valuable than w2. By (3), it follows that w4 has the same value as w3. By (5) and (6) it follows that w4 is more valuable than w4. Which contradicts (7).

I think the controversial assumptions are (2) and (3). It's really hard to deny (1), (4), (5), (6) or (7). So, either there can't be an actual infinity or (3) is false. Now, the falsity of (3) would imply a really radical form of incommensurability: situations that are exactly alike except for the particular identities of the individuals involved (and whatever identity of indiscernibles further requires) can differ in value.

I want to hold on to (2). Plainly a world can have an infinite future containing an infinite number of individuals. So I reject (3), and thus accept the radical incommensurability claim above.

And I think the incommensurability claim is independently plausible.


Leonhard said...

Maybe this will be a dumb question, but isn't there a tension between premise 1 and 3? You say that w1 and w2 contain the exact same individuals, but in premise 3 you state that w1 and w2 differs from each other because of the individuals they contain.

Premise1: A and B are identical.
Premise2: A and B differ from eachother.

Alexander R Pruss said...

I mean each of the premises to hold individually for any worlds. So don't assume that the w1 and w2 in premise 1 are the same w1 and w2 in premise 3.

Leonhard said...

Okay, I think I get that now. What about premise 6? Being what? What is being uhm more than transitive? I'm having tremendous difficulty in making sense of that sentence.

Leonhard said...
This comment has been removed by the author.
Alexander R Pruss said...

The relation of being-more-valuable-than is transitive. Does that help?

Leonhard said...

Yup now the sentence makes sense to me. :)