Finally
If nothing goes terribly wrong, I will finish the Frege paper tomorrow. Though I'm not sure if it's really the same Frege paper I mentioned previously. Initially I just wanted to put together all the comments on Fregean thoughts and Rieger's paradox that I had already posted to this weblog. That looked like a cheap way to get a termpaper. For some reason however the paper has now evolved into a discussion about the prospects and dangers of developing a semantics that can be applied to its own metalanguage.
This self-application is where Frege runs into the problem of the concept 'horse': Frege says that the semantic value of the predicate 'horse' can never be denoted by a singular term. But if this claim is applied to the metalanguage itself, it means that the semantic value of the predicate 'horse' can't be denoted by "the semantic value of the predicate 'horse'". So it looks like Frege's claim can't be consistently expressed if it is to be true in our own language. In fact, all his statements about functions and objects have to be reformulated. But this can be done. Luckily, afterwards we don't need the notions "function" and "object" any more, so we don't have to worry why Frege says that no function could ever be denoted by a singular term.
Not only can this problem of applying a Fregean semantics to itself be avoided (by simply following Frege's theses), it turns out that the same is true of a lot of other potential problems, like Russell's paradox, Bradley's regress and Cantor's theorem. Hence the conclusion of the paper is that, unlike the set-theoretical semantics for predicate logic common nowadays, Frege's semantics can after all be consistently applied to its own metalanguage -- even though it's not the only way to achieve this.