Even More Thoughts about Thoughts
In my last posting, I argued that to escape the cardinality problem for thoughts Frege perhaps has to give up
1) For any things there is at least one concept under which all and only those things fall.
Now (1) is clearly false if, as I think, all there is are objects -- that is, if it makes sense to quantify over absolutely everything. But if not, as Frege thinks, denying (1) is not an option. A concept is a function from things to truth values. Given that functions are not themselves things, how could there fail to be such functions?
I now think that a more promising strategy is to reject
2) For each of these concepts, there exists the thought that Ben Lomond falls under it.
Let's grant (1). If there are k things there are 2^k concepts. But concepts are not grasped directly, and do not themselves figure in thoughts. Rather, they are represented by modes of presentation, or senses. I don't know any standard name for modes of presentation of concepts, so I'll just call them "properties". We can now distinguish two components in (2):
2a) For each concept there is at least one unique mode of presentation, a property.
2b) For each of these properties, there exists the thought that Ben Lomond has it.
I think one might try rejecting (2a). I can't find any compelling reason to do so other than the paradox. But, more importantly, I also can't find a compelling reason to accept (2a). Here are two uncompelling reasons.
1. Aren't properties themselves just functions from possibilities to concepts? How could some of these functions fail to exist? They couldn't. The answer, I think, is to reject the idea that properties can be represented by functions from possible worlds to sets of things (or functions from things to truth values), where "thing" is used as a blanket term for everything that is not a function. Rather, the relevant things are assumed to be inhabitants of the relevant world: A function that assigns to our world a set of unicorns living at various other worlds is not eligible as predicate intension. Many sets of things in the widest sense are outside the range of possible predicate intensions. That is, many concepts -- functions from things to truth values -- are not eligible as value of a property. (This answer presupposes that possibilia exist, and that some of them are in the domain of concepts. The first presupposition is shared by the objection it answers: If there are no possibilia, then how could properties be functions from them to anything else? The second is justified by Frege's insistence that every function must be defined for any object whatsoever.)
2. Note that properties are not functions from things to thoughts, but functions from modes of presentation of things to thoughts. Provided that for each thing there is a unique mode of presentation (which might also be rejected, since it looks much like the principle of the identity of indiscernibles), couldn't we express different properties corresponding to all concepts by long disjunctions of expressions of modes or presentation? No. Given the proviso, there are infinitely many things: Every mode of presentation of a thing is itself a thing. But we cannot express infinite disjunctions. At any rate, expressions themselves are objects, so there must be more concepts than expressions.
One could also try rejecting (2b). However, I don't see how this could be motivated: Assume there is a property F but no thought that Ben Lomond -- under some mode of presentation -- is F. F is a function from modes of presentation to thoughts, and it is defined for all modes of presentation -- in fact, for everything at all. So it is also defined for the relevant mode of presentation of Ben Lomond. That is, there is a value of F for that argument. This value is the thought that Ben Lomond -- under that mode of presentation -- is F. So this thought exists, contradicting the assumption.
Somewhat surprisingly, Michael Dummett seems to reject (2b) on p.302 of "More about Thought" (Notre Dame Journal of Formal Logic 30, 1989: 1-19; page reference to the reprint in his "Frege and other Philosophers"), where he argues that every thought can be uniquely decomposed into its constituents. So for example, the thought expressed by "Ben Lomond killed Ben Lomond" is either definitely composed of a mode of presentation of Ben Lomond and the property of killing oneself or of that mode and the property of killing Ben Lomond. I'm not sure exactly why Dummett defends this strange view, and what exactly it amounts to. I still think a rejection of (2b) would be weird and unFregean. Maybe Dummett thinks that only one of the two possibilities exists, because for instance there is no such property as killing Ben Lomond. This would not be a rejection of (2b), but probably of (1). Or maybe he thinks that the sentence "Ben Lomond killed Ben Lomond" in fact expresses two thoughts, one containing the property of killing oneself and the other the property of killing Ben Lomond.