Thursday, July 31, 2014

A problem for easy ontology arguments

Consider this "easy ontology" argument:

  1. There are no unicorns.
  2. So, there are zero unicorns.
  3. So, there is a zero.
This seems fine. Now consider the parallel:
  1. Every leprechaun is a fairy.
  2. So, the set of leprechauns is a subset of the set of fairies.
  3. So, there is a set of leprechauns.
  4. If there is a set of leprechauns, it's empty. (There aren't any leprechauns!)
  5. So, there is an empty set.
That seems fine as well. So far so good. But now:
  1. Every non-self-membered set (set a that isn't one of its own members) is a set.
  2. So, the set of non-self-membered sets is a subset of the set of all sets.
  3. So, there is a set of non-self-membered sets (the Russell set).
But of course (11) yields a contradiction (just ask if the Russell set is a member of itself).

What to do? One move is to make the easy ontology arguments defeasible. This isn't in the spirit of the game. The other is to add to the premises of the easy ontology argument a coherence premise: that there is a coherent theory of zero, of the empty set and of the Russell and universal sets. The coherence premise will be false in the Russell case but will be true in the other cases. But the point is one that should make us take easy ontology less easily. (I wouldn't be surprised if this was in the easy ontology literature, with which I have little familiarity.)


Heath White said...

I don't know anything about easy ontologies, but I do know that "Every X is a Y" carries no ontological commitment to Xs or Ys, whereas "The X" does carry ontological commitment. So you can't infer from statements of the former type to statements of the latter type.

Alexander R Pruss said...

In Armstrongian state of affairs ontologies, from "Every X is a Y" you can infer that the state of affairs of every X being a Y exists. More generally, in a truthmaker maximalist ontology, from every truth you can infer an existential truth.

But these aren't easy inferences. In easy ontology, the idea is that you get to get something for cheap. Thus, from "There are three chairs in the room", you get "The number of chairs in the room is three", from which you get that there is a number.

Alexander R Pruss said...

Nothing new in this post, looks like. See refs to "bad company" stuff here: