Too Many Thoughts
A while ago, I was discussing Adam Rieger's alleged paradox in Frege's ontology (here, here, and here). I'm still confident that the Russellian version of the paradox can be blocked. But on second thought, the cardinality version of the paradox appears to be much more difficult. Here it is again.
1) For any things there is at least one concept under which all and only those things fall.
2) For each of these concepts, there exists the thought that Ben Lomond falls under it.
3) All these thoughts are different.
4) All thoughts are objects.
From (1)-(3) it follows that there are more thoughts than objects (2^k if k is the number of objects), contradicting (4).
I previously argued that Frege has no reason to assume (3). My main reason for this was that "a thought can be split up in many ways" (Frege, Ãber Begriff und Gegenstand, p.199): For instance, the thought expressed by
Ben Lomond killed Ben Lomond
is both the thought that Ben Lomond falls under 'x killed x', and that he falls under 'x killed Ben Lomond', even though these two concepts are not co-extensive. But for the cardinality objection to go through, it suffices that in any such case there are two concepts F and G co-extensive with 'x killed x' and 'x killed Ben Lomond' respectively, such that "Ben Lomond is F" and "Ben Lomond is G" do not express the same thought. And this, unfortunately, seems quite probable. In the above case for example we can choose 'x killed x' itself as F and 'x is a round square' as G: Since nothing killed Ben Lomond and nothing is a round square, those latter concepts are co-extensive. But it is far from obvious that "Ben Lomond killed Ben Lomond" and "Ben Lomond is a red square" express the very same thought.
I wonder if it would help to identify all necessarily equivalent thoughts, as Frege sometimes does. If even this doesn't work then the only way out is probably to deny (1).