Saving Frege from another contradiction
In the October issue of Analysis, Adam Rieger presents the following paradox in Frege's ontology.
For any object b and first order concept F, there is the thought [Fb] that b falls under F. Let Con and Obj be functions that yield the (referents of the) constituents of such thoughts: Con([Fb])=F, and Obj([Fb])=b. We stipulate moreover that 'b' shall denote the mountain Ben Lomond, and define O ("ordinary") as follows:
O(x) iff Obj(x)=b
(Con(x))x
In words: x is a thought that Ben Lomond falls under some concept C and x does not itself fall under C.
Now we get a contradiction if we ask whether O([Ob]): Obj([Ob]) = b, so O([Ob]) iff [Ob] does not fall under Con([Ob]), iff [Ob] does not fall under O, iff O([Ob]).
Rieger concludes that Frege should not have admitted thoughts as objects. But obviously the paradox does not depend on any details of Frege's ontology. For example, it can easily be construed in modal semantics with sets of possible worlds replacing Frege's thoughts:
For any object b and property (or, if you prefer, set) F, there is a proposition [Fb] containing all and only those worlds, where b is (in) F. Let Con and Obj be functions that yield the (referents of the) constituents of such propositions: Con([Fb])=F, and Obj([Fb])=b. And so on. The argument goes through as before.
I think one can see now what is wrong with Rieger's paradox. To make it even more plain, here is another version:
For any object b and property F, there is a truth value [Fb] such that [Fb]=true iff b falls under F. Let Con and Obj be functions that yield the (referents of the) constituents of such truth values: Con([Fb])=F, and Obj([Fb])=b. Again, the argument goes through as before.
The crucial assumption in Rieger's paradox is that there is a unique way back from the semantic value of 'Fb' to the (referents of the) values of 'F' and 'b'. If there isn't, there are no such functions as Con and Obj. Let's see how this blocks the paradox in Frege's case.
We have to redefine the predicate O to get rid of Con and Obj:
O(x) iff 
F([Fb]=xFx)
Now is O([Ob])? That is, is there an F such that [Fb]=[Ob], and F([Ob])? We still get the contradiction if we assume that the only F with [Fb]=[Ob] is O itself. (In fact, this is how the contradiction should have been stated in the first place, since Frege rules out functions whose arguments are not objects, like Con. See Dummett, 'Frege: Philosophy of Language', pp.40-44.)
But if we drop that assumption, there may well be another concept P, different from O, such that [Pb]=[Ob] and P([Ob]). It then follows that O([Ob]) and P([Ob]), but this is not a contradiction. It merely proves that O and P are indeed different.
(Rieger derives another contradiction from the crucial assumption: The cardinality of concepts is larger than the cardinality of objects. But if for every concept C there is a thought [Cb], and all these thoughts are distinct, then there must be at least as many thoughts, and hence objects, as concepts.)
Does Frege make the crucial assumption? I don't think so, but I'm not aware of any passage where he clearly states the identity conditions of thoughts (or senses in general). However, it seems to me that far more vulnerable to the paradox than Frege are accounts on which the semantic value of a sentence is a structure literally containing the semantic value of its (the sentence's) contituents.
Rieger did not discover this problem for Frege. Russell did in 1902, and indeed, told Frege about it a few months after telling him about Russell's paradox. Indeed, there was a literature about this problem long before Rieger's paper. I published a paper in 2001 in _History and Philosophy of Logic_ that discusses the Frege/Russell correspondence over this problem. ("Russell's Paradox from Appendix B of the Principles of Mathematics: Was Frege's Reply Adequate?") In my 2002 book _Frege and the Logic of Sense and Reference_, written primarily in 1999, I axiomatized what I took to be Frege's theory of the identity conditions of thoughts, and proved that it yields contradictions similar to, but not identical, to Rieger's problem.
You're right that Rieger's version paradox demands that from [Ob] = [Pb] we must be able to derive that O = P. And this is not justified -- which is pointed out by Denyer in the followup paper. But there are better ways of generating the problem.
You're wrong when you say that Frege does not accept functions whose arguments are not objects. Frege explicitly regards quantifiers as functions with functions as arguments. That's the whole idea of the distinction between different "levels" of functions that he talks about in both "Function and Concept" and in Grundgesetze. Dummett does not attribute that view to Frege either. The view that Dummett attributes to Frege is that functions never have VALUES that are not object. There's a huge difference there.
So, if rather than making use of Ben Lomond thoughts, we form thoughts from concepts by putting the concepts in argument position to a higher type function, the paradox can go through. For each concept F, consider the thought that [everything is F]. I believe this yields distinct thoughts for each concept, and possibly more than thought for each concept, since the same concept can be presented by more than one incomplete sense.
And I am in good company. Alonzo Church said so too, and hence his "Logic of Sense and Denotation" (1951) was proven inconsistent by the resulting paradox by John Myhill in 1959. (Roughly consider the concept of being a universal thought that does not fall under the concept it generalizes, and ask about whether some thought that claims that it holds universally falls under it.)
You're right that it requires making certain assumptions about the identity conditions of thoughts. However, Frege says far more about such things than you let on. Indeed, at least a half dozen times in his writings, Frege explicitly says that the thought expressed by a complete sentence is composed of the senses of the parts. You can find this, e.g., in Grundgesetze, in Notes for Ludwig Darmstaedter, in "Compound Thoughts", etc.
Indeed, Frege writes, "if in a sentence or part of a sentence one constituent is replaced by another with a different referent, the different sentence or part that results does not have to have a different referent from the original; ON THE OTHER HAND, IT ALWAYS HAS A DIFFERENT SENSE." (Posthumous Writings, p. 255) That's precisely the sort of principle that generates this sort of paradox.
If we don't make such assumptions, then there's no reason to suppose that "Hesperus is planet" and "Phosphorus is a planet" are distinct thoughts because the names differ in sense, and Frege cannot solve the belief puzzle.
I talk about these issues at length in my book, and give many more citations. But that's enough plugging my own work.