Frege's Semantics and Bradley's Regress
Frege believes that predicate expressions have semantic values (Sinne and Bedeutungen) which can't be denoted by singular terms. Hence "the Bedeutung of 'is a horse'" does not denote the Bedeutung of 'is a horse'. Before the discovery of Russell's paradox, the only reason he ever gave for this view -- apart from claiming that it is a fundamental logical fact that just has to be accepted -- is that otherwise the semantic values of a sentence's constituents wouldn't "stick together". The more I think about this reason, the less convincing I find it.
Assume that every meaningful expression of our language (or an idealisation of our language) can be assigned a semantic value such that the semantic value of every sentence is determined by the semantic values of its constituents. (These semantic values might e.g. be Fregean senses, instensions, extensions, or something different altogether.)
For every meaningful expression (of our language) x, let 'V(x)' denote its semantic value (these quotes are corner quotes). If we now want to specify exactly how the value of a sentence, say "Socrates is mortal", depends on the value of its constituents, we have to say something like
Rule) V("Socrates is mortal") = V("Socrates") blah blah V("mortal").
For example,
V("Socrates is mortal") = True iff V("Socrates") is an Element of V("mortal"), otherwise V("Socrates is mortal") = False.
Since (Rule) is a sentence in our language, and so by assumption every meaningful constituent of this sentence has a semantic value, this must also be true of whatever takes the place of "blah blah". Thus it seems that the specification of a complex expression's semantic value, on the right-hand side of (Rule), has to be more complex than the original expression. This obviously leads to a regress. For instance, to determine the semantic value of "V('Socrates') blah blah V('mortal')" we have to apply a rule like
Rule 2) V("V('Socrates') blah blah V('mortal')") = V("V('Socrates')") blah V("blah blah") blah V("V('mortal')").
And so on.
The obvious move to avoid this regress is to restrict the assumption that every meaningful expression of our language can be assigned a semantic value. Maybe "blah blah", that is, "is an Element of", or "instantiates", does not have a semantic value. Then the rule that determines the semantic value of the right-hand side of (Rule) can look like this:
Rule 2') V("V('Socrates') blah blah V('mortal')") = V("V('Socrates')") blah blah V("V('mortal')")
which doesn't give rise to a regress. (At least in ordinary cases, iterations of V() can presumably be canceled: The extension of "the extension of 'mortal'" = the extension of "mortal".)
Frege's strategy is different: He tries to avoid the regress without restricting the assumption about semantic values. To this end, he stipulates that the semantic value of a predicate expression must itself be specified by a predicate expression. That is, if x is a predicate expression, then 'V(x)' also has to be a predicate expression. Hence (Rule) can be replaced by:
Rule F) V("Socrates is mortal") = V("Socrates")(V("mortal")).
Given the cancelation of iterated V()s, this blocks the regress because the semantic value for the right-hand side can likewise be specified by
Rule F2) V("V('Socrates')(V('mortal'))") = V('V("Socrates")')(V('V("mortal")')).
Now does Frege's strategy actually require that the semantic values of a predicate expression cannot be denoted by a singular term? I don't think so. All that is required is that the semantic value of a predicate expression can be picked out by a predicate expression. From this it doesn't follow that it can't also be picked out by a singular term. If it can, the semantic values of singular terms and predicate expressions ('objects' and 'functions', as Frege calls them), do not comprise distinct parts of reality.
The real problem with this alternative is that once anything like Frege's strategy is adopted, it becomes impossible to talk about semantic values of predicate expressions. If the semantic value of "mortal" is picked out by the predicate "V('mortal')", what does it even mean to say that the same value is also picked out by a term like "mortality"? "V('mortal') = mortality" is not a well-formed expression, neither is "V('V('mortal')') = V('mortality')". Neither is, of course, "the semantic value of 'mortal' = V('mortal')". And neither is "all meaningful expressions have a semantic value".
Conclusion 1: I think that nothing in Frege's logic or semantics forces him to say that functions can never be denoted by singular terms, and therefore that functions aren't objects. In fact, these claims do not play any role at all in his theories: Once we adopt his strategy to analyse complex expressions, singular terms for the semantic values of predicate expressions (that is, for functions), are never needed. So whether or not we add a theory of special entities called "functions", and whether or not we claim that such entities are the semantic value of predicate expressions is completely irrelevant. The very fact that this addition isn't used for anything is actually a good reason to reject it.
Conclusion 2: If we want to avoid Bradley's regress, there is no reasonable way to defend the principle that every meaningful expression of our language has a semantic value. (Russell's paradox is an independent argument for the same conclusion.)